title
sequencelengths 0
18
| author
sequencelengths 0
4.41k
| authoraffiliation
sequencelengths 0
6.45k
| venue
sequencelengths 0
9
| abstract
stringlengths 1
37.6k
| doi
stringlengths 10
114
⌀ | pdfurls
sequencelengths 1
3
⌀ | corpusid
int64 158
259M
| arxivid
stringlengths 9
16
| pdfsha
stringlengths 40
40
| text
stringlengths 66
715k
| github_urls
sequencelengths 0
36
|
|---|---|---|---|---|---|---|---|---|---|---|---|
[
"DOUGLAS-RACHFORD SPLITTING AND ADMM FOR NONCONVEX OPTIMIZATION: TIGHT CONVERGENCE RESULTS",
"DOUGLAS-RACHFORD SPLITTING AND ADMM FOR NONCONVEX OPTIMIZATION: TIGHT CONVERGENCE RESULTS"
] | [
"Andreas Themelis ",
"Panos Patrinos "
] | [] | [] | Although originally designed and analyzed for convex problems, the alternating direction method of multipliers (ADMM) and its close relatives, Douglas-Rachford splitting (DRS) and Peaceman-Rachford splitting (PRS), have been observed to perform remarkably well when applied to certain classes of structured nonconvex optimization problems. However, partial global convergence results in the nonconvex setting have only recently emerged. In this paper we show how the Douglas-Rachford envelope (DRE), introduced by the authors in 2014, can be employed to unify and considerably simplify the theory for devising global convergence guarantees for ADMM, DRS and PRS applied to nonconvex problems under less restrictive conditions, larger prox-stepsizes and over-relaxation parameters than previously known. In fact, our bounds are tight whenever the over-relaxation parameter ranges in (0, 2]. Moreover, a novel primal equivalence of ADMM and DRS extends to any problem the known duality of the algorithms holding in the convex case.Here, ϕ 1 , ϕ 2 : IR p → IR are proper, lower semicontinuous (lsc), extended-real-valued functions (IR := IR ∪ {∞} denotes the extended-real line). Starting from some s ∈ IR p , one iteration of DRS applied to (1.1) with stepsize γ > 0 and relaxation parameter λ > 0 amounts to | 10.1137/18m1163993 | [
"https://arxiv.org/pdf/1709.05747v4.pdf"
] | 73,681,470 | 1709.05747 | c7eae3bc6783c4cb39128dd509fa32c45dd03887 |
DOUGLAS-RACHFORD SPLITTING AND ADMM FOR NONCONVEX OPTIMIZATION: TIGHT CONVERGENCE RESULTS
Andreas Themelis
Panos Patrinos
DOUGLAS-RACHFORD SPLITTING AND ADMM FOR NONCONVEX OPTIMIZATION: TIGHT CONVERGENCE RESULTS
Although originally designed and analyzed for convex problems, the alternating direction method of multipliers (ADMM) and its close relatives, Douglas-Rachford splitting (DRS) and Peaceman-Rachford splitting (PRS), have been observed to perform remarkably well when applied to certain classes of structured nonconvex optimization problems. However, partial global convergence results in the nonconvex setting have only recently emerged. In this paper we show how the Douglas-Rachford envelope (DRE), introduced by the authors in 2014, can be employed to unify and considerably simplify the theory for devising global convergence guarantees for ADMM, DRS and PRS applied to nonconvex problems under less restrictive conditions, larger prox-stepsizes and over-relaxation parameters than previously known. In fact, our bounds are tight whenever the over-relaxation parameter ranges in (0, 2]. Moreover, a novel primal equivalence of ADMM and DRS extends to any problem the known duality of the algorithms holding in the convex case.Here, ϕ 1 , ϕ 2 : IR p → IR are proper, lower semicontinuous (lsc), extended-real-valued functions (IR := IR ∪ {∞} denotes the extended-real line). Starting from some s ∈ IR p , one iteration of DRS applied to (1.1) with stepsize γ > 0 and relaxation parameter λ > 0 amounts to
Introduction
First introduced in [8] for finding numerical solutions of heat differential equations, the Douglas-Rachford splitting (DRS) is now considered a textbook algorithm in convex optimization or, more generally, in monotone inclusion problems. As the name suggests, DRS is a splitting scheme, meaning that it works on a problem decomposition by addressing each component separately, rather than operating on the whole problem which is typically too hard to be tackled directly. In optimization, the objective to be minimized is split as the sum of two functions, resulting in the following canonical framework addressed by DRS: minimize s∈IR p ϕ(s) ≡ ϕ 1 (s) + ϕ 2 (s).
(1.1)
The case λ = 1 corresponds to the classical DRS, whereas for λ = 2 the scheme is also known as Peaceman-Rachford splitting (PRS). If s is a fixed point for the DRiteration -that is, such that s + = s -then it can be easily seen that u satisfies the first-order necessary condition for optimality in problem (1.1). When both ϕ 1 and ϕ 2 are convex functions, the condition is also sufficient and DRS iterations are known to converge for any γ > 0 and λ ∈ (0, 2), in the sense that the residual vanishes. Closely related to DRS and possibly even more popular is the alternating direction method of multipliers (ADMM), first appeared in [12,10], see also [11] for a recent historical overview. ADMM addresses linearly constrained optimization problems
minimize (x,z)∈IR m ×IR n f (x) + g(z) subject to Ax + Bz = b (1.2)
where f : IR m → IR, g : IR n → IR, A ∈ IR p×m , B ∈ IR p×n and b ∈ IR p . ADMM is an iterative scheme based on the following recursive steps
x + ∈ argmin L β ( · , z, y) y + /2 = y − β(1 − λ)(Ax + + Bz − b) z + ∈ argmin L β (x + , · , y + /2 ) y + = y + /2 + β(Ax + + Bz + − b).
(ADMM)
Here, β > 0 is a penalty parameter, λ > 0 is a possible relaxation parameter, and
L β (x, z, y) := f (x) + g(z) + y, Ax + Bz − b + β 2 Ax + Bz − b 2 (1.3)
is the β-augmented Lagrangian of (1.2) with y ∈ IR p as Lagrange equality multiplier. 1 It is well known that for convex problems ADMM is simply DRS applied to a dual formulation [9], and its convergence properties for λ = 1 and arbitrary penalty parameters β > 0 are well documented in the literature, see e.g., [7].
1.1. Equivalence gap for nonconvex problems. Recently, DRS and ADMM have been observed to perform remarkably well when applied to certain classes of structured nonconvex optimization problems and partial or case-specific convergence results have also emerged. In order to compensate the lack of convexity some additional assumptions are in order, and all results in the literature seem to agree that a sufficient condition for ensuring convergence of DRS is Lipschitz-continuous differentiability of function ϕ 1 . Due to the lack of a strong duality theory, this requirement could not be directly translated into equivalent conditions for ADMM applied to nonconvex problems, and indeed the literature only offers standalone results, possibly involving implicit constants and burdened with non-trivial assumptions.
Contributions.
Our contributions can be summarized as follows. 1) New tight convergence results for nonconvex DRS. We provide novel convergence results for DRS applied to nonconvex problems with one function being Lipschitz-differentiable (Theorem 4.3). Differently from the results in the literature, we make no a priori assumption on the existence of accumulation points and we consider all relaxation parameters λ ∈ (0, 4), as opposed to λ ∈ {1, 2}. Moreover, our results are tight for all λ ∈ (0, 2] (Theorem 4.8). Figures 1a and 1b highlight the extent of the improvement with respect to the state of the art.
2) Novel primal equivalence of DRS and ADMM. We prove the equivalence of DRS and ADMM for arbitrary problems, so extending the well known duality of the algorithms holding in the convex case. 3) New convergence results for ADMM. Thanks to the equivalence with DRS, not only do we provide new convergence results for the ADMM scheme, but we also offer an elegant unifying framework that greatly simplifies and generalizes the theory in the literature, is based on less restrictive assumptions, and provides explicit bounds for stepsizes and possible other coefficients. A comparison with the state of the art is shown in Figure 1c .1) and (1.2). In particular, we show that the DRE serves as an exact, continuous and real-valued (as opposed to extended-real-valued) merit function for the original problem, computable with quantities obtained in the iterations of DRS (or ADMM). Finally, we propose out-of-the-box implementations of DRS and ADMM where the stepsize γ and the penalty parameter β are adaptively tuned, so that no prior knowledge of quantities such as Lipschitz moduli is needed.
1.3.
Comparisons & related work. We now compare our results with a selection of recent related works which, to the best of our knowledge, represent the state of the art for generality and contributions.
1.3.1. ADMM. The primal equivalence of ADMM and DRS exploits a reformulation of problem (1.2) into the DRS-form (1.1) which, up to a sign switch, was used in [30] to show self-dual symmetry of ADMM in the convex case.
In [29] convergence of ADMM is studied for problems of the form Despite addressing a more general class of problem than (1.2), when specialized to the standard two-function formulation analyzed in this paper it relies on numerous assumptions. These include Lipschitz continuous minimizers of all ADMM subproblems (in particular, uniqueness of their solution) and uniform boundedness of the subgradient of the nonsmooth term, whereas we allow for multiple minimizers and make almost no requirement on the nonsmooth function. For instance, the requirements rule out interesting cases involving discrete variables or rank constraints. Moreover, the analysis is limited to showing convergence for 'large enough' penalty parameters, and the given bounds involve implicit constants that are not readily available.
In [18] a class of nonconvex problems with more than two functions is presented and variants of ADMM with deterministic and random updates are discussed. The paper provides a nice theory and explicit bounds for the penalty paramenter in ADMM, which agree with ours when the smooth function is convex but are more restrictive by a factor of √ 2 otherwise (cf. Fig. 1c for a more detailed comparison). The main limitation of the proposed approach, however, is that the theory only allows for functions either convex or smooth, differently from ours where the nonsmooth term can basically be anything. Once again, many interesting applications are not covered.
The work [20] studies a proximal ADMM where a possible Bregman divergence term in the second block update is considered. By discarding the Bregman term so as to recover the original ADMM scheme, the same bound on the stepsize as in [18] is found. Another proximal variant is proposed in [13], under less restrictive assumptions related to the concept of smoothness relative to a matrix that we will introduce in Definition 5.7. When matrix B has full-column rank, the proximal term can be discarded and their method reduces to the classical ADMM.
The problem addressed in [14] is fully covered by our analysis, as they consider ADMM for (1.2) where g is L-Lipschitz continuously differentiable and B is the identity matrix. Their bound β > 2L for the penalty parameter is more conservative than ours; in fact, the two coincide only in a worst case scenario.
1.3.2. Douglas-Rachford splitting. Few exceptions apart [21,19], advances in nonconvex DRS theory are problem-specific and only provide local convergence results, at best. These mainly focus on feasibility problems, where the goal is to find points in the intersection of nonempty closed sets A and B subjected to some regularity conditions. This is done by applying DRS to the minimization of the sum of ϕ 1 = δ A and ϕ 2 = δ B , where δ C is the indicator function of a set C (see §2.1). The minimization subproblems in DRS then reduce to (set-valued) projections onto either set, regardless of the stepsize parameter γ > 0. This is the case of [2], for instance, where A and B are finite unions of convex sets. Local linear convergence when A is affine, under some conditions on the (nonconvex) set B, are shown in [16,15].
Although this particular application of DRS does not comply with our requirements, as ϕ 1 fails to be Lipschitz-differentiable, however replacing δ A with ϕ 1 = 1 2 dist 2 A yields an equivalent problem which fits into our framework when A is a convex set. In terms of DRS iterations, this simply amounts to replacing Π A , the projection onto set A, with a "relaxed" version Π A,t := (1 − t)Id + t Π A for some t ∈ (0, 1). Then, it can be easily verified that for any α, β ∈ (0, +∞] one DRS-step applied to
minimize s∈IR p α 2 dist 2 A (s) + β 2 dist 2 B (s) (1.4) results in s + ∈ (1 − λ /2)s + λ /2 Π B,q Π A,p s (1.5)
for p = 2αγ 1+αγ and q = 2βγ 1+βγ . Notice that (1.5) is the λ /2-relaxation of the "method of alternating (p, q)-relaxed projections" ((p, q)-MARP) [4]. The (non-relaxed) (p, q)-MARP is recovered by setting λ = 2, that is, by applying PRS to (1.4). Local linear convergence of MARP was shown when A and B, both possibly nonconvex, satisfy some constraint qualifications, and also global convergence when some other requirements are met. When set A is convex, then α 2 dist 2 A is convex and α-Lipschitz differentiable; our theory then ensures convergence of the fixed-point residual and subsequential convergence of the iterations (1.5) for any λ ∈ (0, 2), p ∈ (0, 1) and q ∈ (0, 1], without any requirements on the (nonempty closed) set B. Here, q = 1 is obtained by replacing β 2 dist 2 B with δ B , which can be interpreted as the hard penalization obtained by letting β = ∞. Although the non-relaxed MARP is not covered due to the non-strong convexity of dist 2 A , however λ can be set arbitrarily close to 2.
The work [21] presents the first general analysis of global convergence of (nonrelaxed) DRS for fully nonconvex problems where one function is Lipschitz differentiable. In [19] PRS is also considered under the additional requirement that the smooth function is strongly convex with strong-convexity/Lipschitz moduli ratio of at least 2 /3. Both papers show that for sufficiently small stepsizes one iteration of DRS or PRS yields a sufficient decrease on an augmented Lagrangian. However, due to the lower unboundedness of the augmented Lagrangian they cannot infer the very convergence of the algorithms, namely that the fixed-point residual vanishes and that therefore with finitely many iterations a stopping criterion is satisfied. 1a) and PRS (Fig. 1b), and maximum inverse of the penalty paramenter 1 /β in ADMM (Fig. 1c); comparison between our bounds (blue plot) and [21] for DRS, [19] for PRS and [13,14,18,20] for ADMM; [29] is not considered due to the unknown range of parameters. On the x-axis the ratio between (possibly negative) strong convexity parameter σ and the Lipschitz modulus L of the gradient of the smooth function.
−1 −0.5 0 0.5 1 1 /4L 1 /2L 3 /4L 1 /L str. convexity/Lipschitz ratio σ /L Range of γ in DRS (λ = 1) Ours Li-Pong (a) −1 −0.5 0 0.5 1 1 /4L 1 /2L 3 /4L 1 /L convexity/Lipschitz ratio σ /L Range of γ in PRS (λ = 2) Ours Li-Liu-Pong (b) −1 −0.5 0 0.5 1 1 /4L 1 /2L 3 /4L
On the y-axis, the supremum of stepsize γ such that the algorithms converge. For ADMM, the analysis is made for a common framework: 2-block ADMM with no Bregman or proximal terms, A full rank and B identity; L and σ are relative to the transformed problem. Notice that, due to the proved analogy of DRS and ADMM, our theoretical bounds coincide in Fig. 1a and 1c.
Other than completing the analysis to all relaxation parameters λ ∈ (0, 4), as opposed to λ ∈ {1, 2}, we improve their results by showing convergence for a considerably larger range of stepsizes and, in the case of PRS, with no restriction on the strong convexity modulus of the smooth function. We also show that our bounds are optimal whenever λ ∈ (0, 2]. The extent of the improvement is evident in the comparisons outlined in Figure 1.
1.4.
Organization of the paper. The paper is organized as follows. Section 2 introduces some notation and offers a brief recap of the needed theory. In Section 3, after formally stating the needed assumptions for the DRS problem formulation (1.1) we introduce the DRE and analyze in detail its key properties. Based on these properties, in Section 4 we prove convergence results of DRS and show the tightness of our findings by means of suitable counterexamples. In Section 5 we deal with ADMM and show its equivalence with DRS; based on this, convergence results for ADMM are derived from the ones already proven for DRS. Section 6 concludes the paper. For the sake of readability, some proofs and auxiliary results are deferred to the Appendix. The identity matrix of suitable size is denoted as I, and for a nonzero matrix M ∈ IR p×n we let σ + (M ) denote its smallest nonzero singular value. The linear subspaces parallel and orthogonal to an affine subspace V are denoted as V and
V ⊥ .
For a set E and a sequence (x k ) k∈IN we write (x k ) k∈IN ⊂ E with the obvious meaning of x k ∈ E for all k ∈ IN. We say that (x k ) k∈IN ⊂ IR n is summable if k∈IN x k is finite, and square-summable if ( x k 2 ) k∈IN is summable.
The indicator function of a set S ⊆ IR n is denoted as δ S : IR n → IR, namely δ S (x) = 0 if x ∈ S and δ S (x) = ∞ otherwise, and Π S : IR n ⇒ IR n is the (set-valued)
projection x → argmin z∈S z − x . A function h : IR n → IR is proper if h > −∞ and h ≡ ∞, in which case its domain is defined as the set dom h := {x ∈ IR n | h(x) < ∞}. For α ∈ IR, lev ≤α h is the α-level set of h, i.e., lev ≤α h := {x ∈ IR n | h(x) ≤ α}. We say that h is level bounded if lev ≤α h is bounded for all α ∈ IR. A vector v ∈ ∂h(x) is a subgradient of h at x, where the subdifferential ∂h(x) is considered in the sense of [26, Def. 8.3] ∂h(x) = v ∈ IR n | ∃(x k , v k ) k∈IN s.t. x k → x, h(x k ) → h(x),∂h(x k ) v k → v ,
and∂h(x) is the set of regular subgradients of h at x, namelŷ
∂h(x) = v ∈ IR n | h(z) ≥ h(x) + v, z − x + o( z − x ), ∀z ∈ IR n .
The set of horizon subgradients of h at x is ∂ ∞ h(x), defined as ∂h(x) except that v k → v is meant in the "cosmic" sense, namely λ k v k → v for some λ k 0.
(a) h − σ h 2 · 2 is convex; (b) σ h u − v 2 ≤ ∇ h(u) −∇h(v), u − v ≤ L h u − v 2 for all u, v ∈ dom h; (c) σ h 2 u−v 2 ≤ h(v)−h(u)− ∇ h(u), v−u ≤ L h 2 u−v 2 for all u, v ∈ dom h.ψ dom h , where ψ := h − σ h 2 · 2 .
In particular, 2.2(a) emphasizes that the largest σ h is, the closer h to a convex function: if σ h ≥ 0 then h is convex, and if σ h > 0 then h is strongly convex, in which case σ h is its modulus of strong convexity.
For notational convenience, we denote the convexity-smoothness ratio of a smooth function h as p h := σ h/L h ∈ [−1, 1].
2.3.
Proximal mapping. The proximal mapping of h : IR n → IR with parameter γ > 0 is the (possibly set-valued) map prox γh : IR n ⇒ dom h defined as
prox γh (x) := argmin w∈IR n h(w) + 1 2γ w − x 2 . (2.3)
We say that a function h is prox-bounded if h + 1 2γ · 2 is lower bounded for some γ > 0. The supremum of all such γ is the threshold of prox-boundedness of h, denoted as γ h . If h is lsc, then prox γh is nonempty-and compact-valued over IR n for all γ ∈ (0, γ h ) [26, Thm. 1.25]. Consequently, the value function of the minimization problem defining the proximal mapping, namely the Moreau envelope with parameter γ, denoted h γ : IR n → IR and defined as
h γ (x) := inf w∈IR n h(w) + 1 2γ w − x 2 , (2.4)
is everywhere finite and, in fact, strictly continuous [26,Ex. 10.32].
Douglas-Rachford envelope
We now list the blanket assumptions for the functions in problem (1.1).
Assumption I (Requirements for the DRS formulation (1.1)). ϕ 1 , ϕ 2 : IR p → IR are proper and lsc functions such that
(i) (Smoothness) dom ϕ 1 is an affine set and ϕ 1 ∈ C 1,1 (dom ϕ 1 ); (ii) (DRS-feasibility) ϕ 2 is prox-bounded and γ < min {γ ϕ2 , 1 /[σϕ 1 ]−}; (iii) (Domain inclusion) dom ϕ 2 ⊆ dom ϕ 1 .
We will say that a stepsize γ is feasible if it complies with Assumption I(ii). The bound γ < γ ϕ2 is equivalent to imposing that γ is small enough such that the subproblem defining v in (DRS) admits a (not necessarily unique) solution. Further constraining γ < 1 /[σϕ 1 ]− ensures that also the subproblem defining u admits a solution; in fact, u is unique and depends Lipschitz continuously on s, and for all
s ∈ IR p∇ ϕ 1 (u) = 1 γ (Π dom ϕ1 (s) − u) (3.1) where u = prox γϕ1 (s) (cf. Prop. A.3)
. Although the threshold γ ϕ1 might be larger than 1 /[σϕ 1 ]−, because of these favorable properties we constrain it as such. From (3.1) we can easily verify that the v-update in (DRS) is equivalent to selecting v ∈ argmin
w∈IR p ϕ 2 (w) + ϕ 1 (u) + ∇ ϕ 1 (u), w − u + 1 2γ w − u 2 . (3.2)
This shows that v is the result of a minimization of a majorization model for the original function ϕ = ϕ 1 + ϕ 2 , where the smooth function ϕ 2 is replaced by the quadratic upper bound emphasized by the under-bracket in (3.2). Closely related to (3.2) and first introduced in [25] for convex problems, the
Douglas-Rachford envelope (DRE) is the function ϕ DR γ : dom ϕ 1 → IR defined as ϕ DR γ (s) := min w∈IR p ϕ 2 (w) + ϕ 1 (u) + ∇ ϕ 1 (u), w − u + 1 2γ w − u 2 (3.3)
where u := prox γϕ1 (s). Namely, rather than the minimizer v, ϕ DR γ (s) is the value of the minimization problem (3.2) defining the v-update in (DRS). By plugging the minimizer w = v in (3.3) we obtain the following useful intepretation of the DRE:
ϕ DR γ (s) = L1 /γ (u, v, γ −1 (u − s)) (3.4)
where u and v come from the DRS iteration and
L β (x, z, y) := ϕ 1 (x) + ϕ 2 (z) + y, x − z + β 2 x − z 2 (3.5)
is the β-augmented Lagrangian relative to the equivalent problem formulation
minimize x,z∈IR p ϕ 1 (x) + ϕ 2 (z) subject to x − z = 0. (3.6)
Therefore, computing ϕ DR γ (s) requires the same operations as performing one DRS update s → (s, u, v). Moreover, by expressing s in terms of the Lagrange multiplier
y := 1 γ (u − s), so that∇ϕ 1 (u) = − Π dom ϕ1 y (3.7) (cf. (3.1))
, with simple algebra we may rewrite iterations (DRS) in the equivalent form
y + /2 = y − 1 γ (1 − λ)(u − v) u + ∈ prox γϕ1 (v − γy + /2 ) y + = y + /2 + 1 γ (u + − v) v + ∈ prox γϕ2 (u + + γy + ).v ∈ prox γϕ2 (u − γ∇ϕ 1 (u)), (3.9)
see e.g., [6,28] for an extensive discussion on nonconvex FBS. In particular, the definition (3.3) emphasizes the close connection that the DRE has with the forwardbackward envelope (FBE) as in [28], namely
ϕ DR γ (s) = ϕ FB γ (u) for u = prox γϕ1 (s). (3.10)
A first immediate consequence, due to Proposition A.4 in the Appendix, is that the DRE is "flat" along lines orthogonal to dom ϕ 1 :
ϕ DR γ (s + ν) = ϕ DR γ (s) for all s ∈ IR p and ν ∈ dom ϕ ⊥ 1 . (3.11)
The FBE, first introduced in [24] and further studied and extended in [27,28,22], is an exact penalty function for FBS, in the same way as the DRE is for DRS, as we will see later on in this section. Strictly speaking, the FBE is defined when the smooth term ϕ 1 is full domain. However, when the domain of ϕ 1 is a proper affine subspace of IR p there is no ambiguity in regarding the FBE rather as a function ϕ FB γ : dom ϕ 1 → IR. This causes no troubles, since the composition with prox γϕ1 in (3.10) ensures that the argument of ϕ FB γ is always in its domain. For the sake of notational simplicity we will then write ϕ FB γ rather than ϕ FB γ dom ϕ1 , and accordingly we shall write ∇ϕ FB γ in place of∇ϕ DR γ whenever the latter exists.
3.2.
Properties. The equivalence (3.10) shows that the DRE ϕ DR γ and the FBE ϕ FB γ are basically the same function up to a change of variable. As a consequence, regularity properties of the DRE can be shown with minimal effort by invoking analogous properties of the FBE. For instance, since the mapping We now investigate the fundamental connections between the DRE ϕ DR γ and the original function ϕ. We show, for γ small enough and up to an (invertible) change of variable, that infima and minimizers of the two functions coincide, as well as equivalence of level boundedness of ϕ and ϕ DR γ dom ϕ1 . Due to the fact that ϕ DR γ is constant on lines orthogonal to dom ϕ 1 (cf. (3.11)) we clearly cannot expect the DRE to have bounded level sets, unless ϕ 1 has full domain. Therefore, it should not be surprising that some properties of ϕ are enjoyed by the DRE (and viceversa) up to a suitable restriction of its domain. All results are based on a key property of the DRE which we state below. The proof is similar to that of [28,Prop. 4.3], but we briefly outline it here for the sake of self-inclusiveness.
s → u = prox γϕ1 (s) ∈ dom ϕ 1 is Lipschitz continuous for γ < 1 /[σϕ 1 ]− (see Prop. A.3(ii)),
Proposition 3.2 (Sandwiching property). Let s ∈ IR p and a feasible γ be fixed, and consider u, v generated by one DRS iteration. Then,
(i) ϕ DR γ (s) ≤ ϕ(u); (ii) ϕ(v) + 1−γLϕ 1 2γ u − v 2 ≤ ϕ DR γ (s) ≤ ϕ(v) + 1−γσϕ 1 2γ u − v 2 .
Equivalently, this holds for all u ∈ dom ϕ 1 ,
s ∈ u + γ∇ϕ 1 (u) + dom ϕ ⊥ 1 and v ∈ prox γϕ2 (u − γ∇ϕ 1 (u)). Proof. Plugging w = u in (3.3) proves 3.2(i). The minimizer is instead w = v ∈ dom ϕ 2 ⊆ dom ϕ 1 , resulting in ϕ DR γ (s) = ϕ 2 (v) + ϕ 1 (u) + ∇ ϕ 1 (u), v − u + 1 2γ v − u 2 Rem. 2.2(c) ≥ ϕ 2 (v) + ϕ 1 (v) − Lϕ 1 2 v − u 2 + 1
2γ v − u 2 and one inequality of 3.2(ii) follows. The other inequality is readily proven by using the quadratic lower bound in Rem. 2.2(c) instead. Finally, the last claim follows from the expression (3.9) and Prop. A.3(i).
Theorem 3.3 (Minimization and level-boundedness equivalence).
For any feasible γ < 1 /Lϕ 1 the following hold:
(i) inf ϕ = inf ϕ DR γ ; (ii) argmin ϕ = prox γϕ1 argmin ϕ DR γ . (iii) ϕ is level bounded iff so is ϕ DR γ dom ϕ1 .
Proof. That ϕ DR γ and ϕ • prox γϕ1 have same infima and minimizers easily follows from Prop. 3.2, and clearly inf ϕ
• prox γϕ1 = inf ϕ since prox γϕ1 is invertible on dom ϕ 1 ⊇ dom ϕ for any feasible γ (cf. Prop. A.3(ii)). To show 3.3(iii)
, observe first that, due to the equivalence of infima and (3.11), if either ϕ or ϕ DR γ dom ϕ1 is level bounded then ϕ := inf ϕ = inf ϕ DR γ is finite and attained by both functions. Let α > ϕ be arbitrary.
♠ Suppose first that ϕ DR γ dom ϕ1 is level bounded and let u ∈ lev ≤α ϕ. Then, clearly u ∈ dom ϕ 1 and s := u + γ∇ϕ 1 (u) is such that prox γϕ1 (s) = u (cf. Prop. A.3(i)). Then, from Prop. 3.2 it follows that s ∈ lev ≤α ϕ DR γ . In particular,
lev ≤α ϕ ⊆ dom ϕ 1 ∩ [I + γ∇ϕ 1 ] lev ≤α ϕ DR γ ⊆ [I + γ∇ϕ 1 ] lev ≤α ϕ DR γ dom ϕ1 . Since I + γ∇ϕ 1 is Lipschitz continuous, necessarily lev ≤α ϕ is bounded.
♠ Suppose now that ϕ is level bounded, and contrary to the claim suppose that for
all k ∈ IN there exists s k ∈ lev ≤α ϕ DR γ dom ϕ1 \ B(0; k). Let u k = prox γϕ1 (s k ) so that s k = u k + γ∇ϕ 1 (u k ) (since s k ∈ dom ϕ 1 ), and v k ∈ prox γϕ2 (u k − γ∇ϕ 1 (u k )); from Prop. 3.2 it follows that v k ∈ lev ≤α ϕ, and that α − ϕ ≥ ϕ DR γ (s k ) − ϕ ≥ ϕ DR γ (s k ) − ϕ(v k ) ≥ 1−γLϕ 1 2γ u k − v k 2 . Therefore, v k ≥ u k − u 0 − u 0 − u k − v k A.3(ii) ≥ 1 1+γLϕ 1 s k − s 0 − u 0 − u k − v k ≥ k− s0 1+γLϕ 1 − u 0 − 2γ(α−ϕ ) 1−γLϕ 1 → + ∞ as k → ∞,
and therefore lev ≤α ϕ cannot be bounded.
Convergence of Douglas-Rachford splitting
Closely related to the DRE, the augmented Lagrangian (3.5) was used in [21] under the name of Douglas-Rachford merit function to analyze DRS for the special case λ = 1. It was shown that for sufficiently small γ there exists c > 0 such that the iterates generated by DRS satisfy
L1 /γ (u k+1 , v k+1 , y k+1 ) ≤ L1 /γ (u k , v k , y k ) − c u k − u k+1 2 , (4.1)
to infer that (u k ) k∈IN and (v k ) k∈IN have same accumulation points, all of which are stationary for ϕ. However, it wasn't actually proved that DRS does converge, namely the fact that the fixed-point residual vanishes and that therefore for any ε > 0 the stopping criterion u k − v k ≤ ε eventually is satisfied. This limitation comes from using L1 /γ : IR p × IR p × IR p → IR as a function of u, v and y (or s) separately in (4.1), which in fact is not guaranteed to be bounded below. Because of this, the standard practice of telescoping the inequality (4.1) does not ensure that u k+1 − u k vanishes, and consequently nor that (the proportional quantity) u k − v k does. Similar remarks apply to [19], where the Peaceman-Rachford merit function was introduced to analyze PRS, that is, DRS with λ = 2.
On the contrary, the DRE is defined as a section of L1 /γ which is tightly connected both to ϕ and to DRS iterations: namely, ϕ DR
γ (s) = L1 /γ u γ (s), v γ (s), γ −1 (u γ (s) − s) , where u γ (s)
and v γ (s) are respectively the u-and v-updates of point s through DRS with stepsize γ (cf. (3.4)). 2 As shown in the previous section, our interpretation of the DRE as a function of the sole variable s overcomes this limitation and preserves important properties the original function ϕ may have such as level boundedness.
We now generalize the decrease property (4.1) shown in [21,19] by considering arbitrary relaxation parameters λ ∈ (0, 4) (as opposed to λ ∈ {1, 2}) and providing tight ranges for the stepsize γ whenever λ ∈ (0, 2]. We are only interested in the case γ < 1 /Lϕ 1 , for otherwise the DRE may fail to be lower bounded. Morever, it will be shown in Section 4 that the bound γ < 1 /Lϕ 1 is necessary for ensuring the convergence of DRS, unless the generality of Assumption I is sacrificed.
ϕ DR γ (s) − ϕ DR γ (s + ) ≥ c u − v 2 (4.2)
where c is a strictly positive constant defined as
c := λ (1+γLϕ 1 ) 2 2−λ 2γ − max [σ ϕ1 ] − , L ϕ1 (γL ϕ1 − λ 2 ) .
If ϕ 1 is strongly convex, then (4.2) also holds for 2 ≤ λ < 4 1+ √ 1−pϕ 1 and feasible γ such that
pϕ 1 λ−δ 4σϕ 1 < γ < pϕ 1 λ+δ 4σϕ 1 , where δ := (p ϕ1 λ) 2 − 8p ϕ1 (λ − 2), in which case c := λ (1+γLϕ 1 ) 2 2−λ 2γ + σ ϕ1 ( λ 2 − γL ϕ1 )
. For the sake of readability, the proof of Theorem 4.1 is referred to Appendix B.
Remark 4.2 (Simpler bounds for DRS). By using the (more conservative) estimate σ ϕ1 = 0 when the smooth function ϕ 1 is convex, and σ ϕ1 = −L ϕ1 otherwise, the range of γ can be simplified as follows in case λ ∈ (0, 2]:
λ ∈ (0, 2): γ < 1 Lϕ 1 if ϕ 1 is convex γ < 2−λ 2Lϕ 1 otherwise λ = 2: γ < 1 Lϕ 1 if ϕ 1 is str. convex ∅ otherwise.
Theorem 4.3 (Subsequential convergence). Suppose that the cost function ϕ is lower bounded. Then, the following hold for the iterates generated by DRS with stepsize γ and relaxation λ as in Theorem 4.1:
(i) the residuals (u k − v k ) k∈IN vanish with rate min i≤k u i − v i = o( 1 / √ k);
(ii) (u k ) k∈IN and (v k ) k∈IN have same cluster points, all of which are stationary for ϕ and on which ϕ has same value, this being the limit of (ϕ DR γ (s k )) k∈IN ; Moreover, if ϕ has bounded level sets, then the sequences are bounded.
Proof. To avoid trivialities, we assume that the stopping criterion v k = u k is never met, so that the algorithm runs infinite many iterations.
c k∈IN u k − v k 2 ≤ k∈IN ϕ DR γ (s k ) − ϕ DR γ (s k+1 ) ≤ ϕ DR γ (s 0 ) − inf ϕ DR γ . Since inf ϕ DR γ = inf ϕ > −∞ and ϕ DR γ
is real valued (cf. Prop. 3.1 and Thm. 3.3), it follows that (u k −v k ) k∈IN is square summable, hence the claimed rate of convergence. Moreover, since ϕ DR γ (s k ) is decreasing it admits a limit, be it ϕ > −∞.
♠ 4.3(ii) Since (u k − v k ) k∈IN → 0, necessarily (u k ) k∈IN and (v k ) k∈IN have same cluster points. From Prop. A.3(i) it follows that for all k ∈ IN there exists ν k ∈ dom ϕ ⊥ 1 such that s k = u k + γ∇ϕ 1 (u k ) + ν k . Then, λ[v k − u k ] = s k+1 − s k = u k+1 − u k + γ ∇ ϕ 1 (u k+1 ) −∇ϕ 1 (u k ) + ν k+1 − ν k .
All the terms in square brackets belong to dom ϕ 1 whereas ν k+1 − ν k ∈ dom ϕ ⊥ 1 , and from the equation above we conclude that ν k+1 − ν k = 0. Namely, there exists
ν ∈ dom ϕ ⊥ 1 (independent of the iteration k ∈ IN) such that s k = u k + γ∇ϕ 1 (u k ) + ν for all k ∈ IN. (4.3)
Suppose now that (u k ) k∈K → u for some K ⊆ IN and u ∈ IR p . Then, (v k ) k∈K → u and (s k ) k∈K → s = u + γ∇ϕ 1 (u ) + ν, due to continuity of∇ϕ 1 on dom ϕ 1 . From Prop. A.3(i) we infer that u = prox γϕ1 (s ). Therefore,
v k ∈ prox γϕ2 (2u k −s k ) = prox γϕ2 (u k −γ∇ϕ 1 (u k )−ν) = prox γϕ2 (u k −γ∇ϕ 1 (u k ))
where the last equality follows from Prop. A.4. Since (u k − γ∇ϕ 1 (u k )) k∈K → u − γ∇ϕ 1 (u ), the outer semicontinuity of prox γϕ2 [26, Ex. 5.23(b)] implies that
u = lim K k→∞ u k ∈ lim sup K k→∞ prox γϕ2 u k − γ∇ϕ 1 (u k ) ⊆ prox γϕ2 (u − γ∇ϕ 1 (u )).
From the definition of prox γϕ2 (u − γ∇ϕ 1 (u )) it follows that u minimizes the function ψ := ϕ 2 + 1 2γ · − u + γ∇ϕ 1 (u ) 2 , and therefore 0 ∈∂ψ(u ) =∂ϕ 2 (u ) + 1 γ (u − u + γ∇ϕ 1 (u )) ⊆∂ϕ(u ) where the first equality follows from [26, Thm. 10.1] and the last inclusion from (2.1). Finally, since v k → u ,
ϕ(u ) ≤ lim inf K k→∞ ϕ(v k ) ≤ lim sup K k→∞ ϕ(v k ) ≤ lim sup K k→∞ ϕ DR γ (s k ) = ϕ DR γ (s ) ≤ ϕ(u )
where in the first inequality we used lower semicontinuity of ϕ, in the third and last the sandwiching property (Prop. 3.2), and in the equality the continuity of ϕ DR γ . This shows that (ϕ(u k )) k∈K → ϕ(u ) = ϕ DR γ (s ), and since (ϕ DR γ (s k )) k∈IN → ϕ , then necessarily ϕ(u ) = ϕ DR γ (s ) = ϕ independently of the cluster point u . Finally, suppose that ϕ has bouneded level sets. Then, it follows from Thm. 3.3(iii) that so does ϕ DR γ dom ϕ1 , and since s k ∈ lev ≤ϕ DR γ (s 0 ) ϕ DR γ ∩ dom ϕ 1 for all k ∈ IN, then the sequence (s k ) k∈IN is bounded. Due to Lipschitz continuity of prox γϕ1 (cf. Prop. A.3(ii)), also (u k ) k∈IN is bounded and since v k − u k → 0 we conclude that so is (v k ) k∈IN .
In [21] it was shown that the augmented Lagrangian decreases along the iterates generated by the non-relaxed DRS. This fact was then used to prove global convergence in case the sequence remains bounded, which was later shown to be the case in [19] when ϕ has bounded level sets. Due to the equivalence of the DRE and the augmented Lagrangian evaluated at points generated by DRS, cf. (3.4), by invoking Theorem 4.1 we can extend their result to the tight ranges we provided. Notice that boundedness of the sequence for any λ is ensured by the level boundedness of the DRE, which holds for all γ < 1 /Lϕ 1 and independently of λ when ϕ is level bounded. Remark 4.5 (Adaptive variant when L ϕ1 is unknown). When σ ϕ1 is not known, then a conservative estimate σ ϕ1 = −L ϕ1 is always feasible or, in case ϕ 1 is convex, σ ϕ1 = 0 can be fixed (cf. Remark 4.2). In most applications, however, it is the very Lipschitz constant L ϕ1 which is not known, in which case it can be adaptively retrieved. This is done by replacing L ϕ1 with an initial estimate L > 0, and by checking at each iteration if the quadratic upper bound for ϕ 1 as in Remark 2.2(c) holds with L in place of the unknown L ϕ1 . Whenever the bound is violated, it suffices to, say, double the estimate L and decrease the stepsize γ accordingly. Notice that there is no need to compute∇ϕ 1 , as it holds that
∇ ϕ 1 (u), u + − u (3.1) = 1 γ Π dom ϕ1 s − u, u + − u = 1 γ s − u, u + − u where
the last equality is due to the fact that u, u + ∈ dom ϕ 1 . The procedure is summarized in Algorithm 1. It is important to observe that, since replacing L ϕ1
Algorithm 1 DRS with adaptive stepsize
Require s 0 ∈ IR p , L > 0, γ, λ as in Rem. 4.2 with L in place of L ϕ1 Initialize u 0 , v 0 and s 1 as in DRS For k = 1, 2 . . .
1: u k = prox γϕ1 (s k ) 2: if ϕ 1 (u k ) > ϕ 1 (u k−1 ) + 1 γ s k−1 − u k−1 , u k − u k−1 + L 2 u k − u k−1 2 then L ← 2L, γ ← γ /2, and go back to step 1 3: v k ∈ prox γϕ2 (2u k − s k ) 4: s k+1 = s k + λ(v k − u k )
with any L ≥ L ϕ1 still satisfies the (upper) bound in Remark 2.2(c), it follows that L is incremented only a finite number of times. Therefore, there exists an iteration k 0 starting from which γ and L are constant; in particular, the convergence result stated in Theorem 4.3 covers this adaptive variant as well.
4.1.
Tightness of the results. When both ϕ 1 and ϕ 2 are convex and ϕ 1 + ϕ 2 attains a minimum, well known results of monotone operator theory guarantee that for any λ ∈ (0, 2) and γ > 0 the residual u k − v k generated by DRS iterations vanishes (see e.g., [3]). In fact, the whole sequence (u k ) k∈IN converges and ϕ 1 needs not even be differentiable in this case. On the contrary, when ϕ 2 is nonconvex then the bound γ < 1 /Lϕ 1 plays a crucial role, as the next example shows. 2 otherwise. Notice that dom ϕ = {±1}, and therefore ±1 are the unique stationary points of ϕ (in fact, they are also global minimizers). Moreover, ϕ 1 ∈ C 1,1 (IR) with
ϕ 1 (x) = L 2 x 2 if x ≤ t L 2 x 2 − L−σ 2 (x − t)u k − v k ∈ s k 1+γL + sgn(s k ) if s k ≤ t(1 + γL) s k 1+γσ − γ(L−σ)t 1+γσ − v k otherwise, where v k is either 1 or −1 in the second case. In particular, if u k − v k → 0, then min s k 1+γL + sgn(s k ) , s k 1+γσ − L−σ 1+γσ γt − 1 , s k 1+γσ − L−σ 1+γσ γt + 1 → 0.
Notice that the first element in the set above is always larger than 1, and therefore eventually s k will be always close to either (L−σ)γt+(1+γσ) or (L−σ)γt−(1+γσ), both of which are strictly smaller than t(1+γL) (since t > 1). Therefore, eventually s k ≤ t(1 + γL) and the residual will then be u k − v k = s k 1+γL + sgn(s k ) which is bounded away from zero, contradicting the fact that u k − v k → 0.
Notice that in Example 4.6 we actually showed that for γ ≥ 1 /Lϕ 1 DRS fails to converge for any λ > 0 regardless of the starting point s 0 . This was possible by allowing a too large stepsize γ; the next example shows the necessity of bounding λ.
Example 4.7 (Necessity of 0 < λ < 2(1 + γσ)). Fix L > 0 and σ ∈ [−L, L], and consider ϕ = ϕ 1 + ϕ 2 where ϕ 2 = δ {0} and
ϕ 1 (x) = σ 2 x 2 if x ≤ 1 σ 2 x 2 + L−σ 2 (x − 1) 2 otherwise.
For any s k one DRS iteration produces v k = 0, and in particular the DR-residual is
u k − v k = u k = s k 1+γσ if s k ≤ 1 + γσ s k +γ(L−σ) 1+γL otherwise. 1 0 ϕ1 • dom ϕ2
Suppose that λ ≥ 2(1 + γσ); then it is easy to check that starting from s 0 = 0 we have s k = 0 for all k. Moreover, if the DRS-residual converges to 0, then min |s k |, |s k + γ(L − σ)| → 0 and in particular, eventually s k ≤ 1 + γσ. The iterations will then reduce to
s k+1 = s k + λ(v k − u k ) = 1 − λ 1+γσ s k .
Since λ ≥ 2(1 + γσ) and s k = 0 for all k, we have s k+1 /s k ≥ 1, contradicting the fact that s k → 0.
Let us draw some conclusions:
• the nonsmooth function ϕ 2 is (strongly) convex in Example 4.7, therefore even for fully convex formulations the bound 0 < λ < 2(1 + γσ ϕ1 ) needs be satisfied; • if λ > 2 (which is feasible only if ϕ 1 is strongly convex, i.e., if σ ϕ1 > 0), then, regardless of whether also ϕ 2 is (strongly) convex or not, we obtain that the stepsize must be lower bounded as γ > λ−2 2σϕ 1 ; • if ϕ 1 is not strongly convex, i.e., if σ ϕ1 ≤ 0, we infer the bound λ ∈ (0, 2): this means, for instance, that even in the fully convex case, plain (nonstrong) convexity of ϕ 1 is not enough to guarantee convergence of the Peaceman-Rachford splitting; • combined with the bound γ < 1 /Lϕ 1 shown in Example 4.6, we infer that (at least when ϕ 2 is nonconvex) necessarily 0 < λ < 2(1 + p ϕ1 ) and consequently λ ∈ (0, 4).
In particular we can infer the following:
Theorem 4.8 (Tightness). Unless the generality of Assumption I is sacrificed, when λ ∈ (0, 2) or ϕ 1 is not strongly convex the bound γ < min 1
Lϕ 1 , 2−λ 2[σϕ 1 ]−
is tight for ensuring convergence of DRS. Similarly, PRS (i.e., DRS with λ = 2) is ensured to converge iff ϕ 1 is strongly convex and γ < 1 /Lϕ 1 .
Alternating Direction Method of Multipliers
In convex optimization, it is well known that ADMM and DRS are essentially the same algorithm applied to the respective dual formulations. Unfortunately, this equivalence does not extend to the nonconvex case due to the limitation of duality theory. In this section we provide a primal equivalence of the two algorithms that cope with nonconvexity and serves as universal framework. This will allow to extend the theory developed for DRS to ADMM. To this end, we invoke the notion of image function, also known as infimal post-composition or epi-composition [1,3,26].
minimize (x,z,s)∈IR m ×IR n ×IR p f (x) + g(z) subject to Ax = b − s, Bz = s.
Since the problem is independent of the order of minimization [26,Prop. 1.35] we may minimize first with respect to (x, z) to arrive to
minimize s∈IR p (Af )(b−s) inf x∈IR m {f (x) | Ax = b − s} + (Bg)(s) inf z∈IR n {g(z) | Bz = s}.
Therefore, ADMM problem formulation (1.2) can be expressed in the equivalent DRS formulation (1.1) as
minimize s∈IR p ϕ1(s) (Bg)(s) + ϕ2(s) (Af )(b − s). (5.1)
Restricting the analysis to the convex case, from [30] it can be inferred that DRS on problem (5.1) is equivalent to ADMM on problem (1.2). However, the arguments still rely on the known dual equivalence of ADMM and DRS, which cannot be exploited in the general nonconvex framework here discussed. Our result is rather based on properties of image functions which also hold for nonconvex functions. It is convenient to eliminate the DRS variable s by expressing the iteration in terms of the dual multiplier y. To this end, we consider the equivalent DRS iteration described in (3.8).
For ϕ 1 = (Bg) and ϕ 2 = (Af )(b − · ), it follows from Proposition C.1(iii) in the Appendix that prox γϕ1 (s) = B argmin z {g(z) + 1 2γ Bz − s 2 } and, by means of a simple change of variable, prox γϕ2 (s
) = b − A argmin x {f (x) + 1 2γ Ax + s 2 }. Then, (3.8) becomes y k+ 1 /2 = y − 1 γ (1 − λ)(u k − v k ) u k+1 = B argmin z g(z) + 1 2γ Bz − v k + γy k+ 1 /2 2 y k+1 = y k+ 1 /2 + 1 γ (u k+1 − v k ) v k+1 ∈ b − A argmin x f (x) + 1 2γ Ax + u k+1 − b + γy k+1 ) .
By introducing suitable variables x k ∈ IR m and z k ∈ IR n , so as to express u k = Bz k and v k = b − Ax k , this reduces to
y k+ 1 2 = y − 1 γ (1 − λ)(Ax k + Bz k − b) z k+1 ∈ argmin z g(z) + 1 2γ Ax k + Bz − b + γy k+ 1 /2 2 = argmin z L1 /γ (x k ,z,y k+ 1 /2 ) y k+1 = y k+ 1 /2 + 1 γ (Ax k + Bz k+1 − b) x k+1 ∈ argmin x f (x) + 1 2γ Ax + Bz k+1 − b + γy k+1 2 = argmin x L1 /γ (x,z k+1 ,y k+1 )
which is exactly ADMM with β = 1 /γ, up to one update order switch (that is, starting from the y + /2 -update rather than from the x + -update). To sum up, we have shown the following. where γ ↔ 1 /β and (s, u, v, y) ↔ (Bz − γy, Bz, b − Ax, y). Moreover, for all k ≥ 1 the following hold
(i) ϕ DR 1 /β (s k ) = L β (x k , z k , y k ); (ii) (Af )(Ax k ) = f (x k ); (iii) (Bg)(Bz k ) = g(z k ); (iv) if (Bg) ∈ C 1,1 (rng B), then∇(Bg)(Bz k ) = − Π rng B y k .
Proof. By definition of x k , we have that L β (x k , z k−1 , y k−1 ) ≤ L β (s, z k−1 , y k−1 ) for all s ∈ IR m . If s ∈ IR m is such that As = Ax k , then the inequality reduces to f (x k ) ≤ f (s), and 5.2(ii) follows from the definition of image function. A similar reasoning shows 5.2(iii); in turn, 5.2(i) follows from (3.4). Finally, 5.2(iv) follows from (3.7).
Convergence of ADMM.
In order to extend the theory developed for DRS to ADMM we shall impose that ϕ 1 and ϕ 2 as in (5.1) comply with Assumption I. This motivates the following blanket requirement.
Assumption II (Requirements for the ADMM formulation (1.2)). Functions f : IR m → IR and g : IR n → IR, and matrices A ∈ IR p×m , B ∈ IR p×n and b ∈ IR p are such that (Af ) is a lsc function, and (i) (Smoothness) (Bg) is C 1,1 on its (affine) domain rng B;
(ii) (ADMM-feasibility) for β large enough, the subproblems in (ADMM) admit a (not necessarily unique) solution;
(iii) (Domain inclusion) A dom f ⊆ b + rng B.
These requirements generalize those in Assumption I by allowing linear constraints more generic than x − z = 0, cf. (3.6). Notice that prox-boundedness of (Af ) is ensured by ADMM-feasibility, as it follows from Proposition C.1(iii) in the Appendix. For ϕ 1 as in (5.1) it holds that dom ϕ 1 = B dom g, and therefore unless B is surjective ϕ 1 cannot have full domain. Thankfully, the theory for DRS developed in the previous sections accounted for this event by considering functions with affine domain. Although we could still work under this more generic assumption, at this stage this generalization is no longer needed, which is why we consider full domain functions g.
Theorem 5.3 (Convergence of ADMM)
. Suppose that f + g is lower bounded on the feasible set {(x, z) ∈ IR m × IR n | Ax + Bz = b}. Let λ and γ be as in Theorem 4.1 for ϕ 1 = (Bg) and ϕ 2 = (Af )(b − · ). Then, the following hold for the iterates generated by ADMM with penalty β = 1 /γ and relaxation λ:
(i) L β (x k+1 , z k+1 , y k+1 ) ≤ L β (x k , z k , y k ) − β−L (Bg) 2 Ax k + Bz k − b 2 , and the residual (Ax k +Bz k −b) k∈IN vanishes with min i≤k Ax i +Bz i −b = o( 1 / √ k);
(ii) all cluster points (x, z, y) of ((x k , z k , y k )) k∈IN satisfy the KKT conditions
• −A y ∈ ∂f (x) • −B y ∈ ∂g(z)
• Ax + Bz = b; and attain the same cost f (x) + g(z), this being the limit of the sequence (L β (x k , z k , y k )) k∈IN ;
(iii) the sequence ((Ax k , Bz k , y k )) k∈IN is bounded provided that f (x)+g(z) is level bounded on the feasible set C := {(x, z) ∈ IR m × IR n | Ax + Bz = b}.
Proof. ♠ 5.3(i) It follows from Thm. 5.2 that s k = Bz k − 1 β y k , u k = Bz k and v k = b−Ax k are the iterates generated by DRS with stepsize γ = 1 /β applied to ϕ 1 + ϕ 2 , where ϕ 1 = (Bg) and ϕ 2 = (Af )(b − · ). Then the claim follows from Thm.s 4.1 and 4.3, by observing that u k − v k = Ax k + Bz k − b. ♠ 5.3(ii) Suppose now that for some K ⊆ IN the subsequence ((x k , z k , y k )) k∈K converges to (x, z, y); then, necessarily Ax + Bz = b. Moreover, the optimality conditions defining x k and z k in ADMM read
0 ∈ ∂f (x k ) + A y k−1 + βA (Ax k + Bz k−1 − b) 0 ∈ ∂g(z k ) + B y k− 1 /2 + βB (Ax k + Bz k − b) = ∂g(z k ) + B y k .
Continuity of∇(Bg) on its domain then yields
−B y = − B Π rng B y k∈K ← −− − − B Π rng B y k 5.2(iv) = B ∇ (Bg)(Bz k ) k∈K − −− → B ∇ (Bg)(Bz)
and from Prop. C.2 we conclude that −B y ∈∂g(z).
From Thm.s 4.3 and 5.2(i) it follows that
L β (x k , z k , y k ) = ϕ DR γ (s k ) ϕ DR γ (s) = L β (x, z, y) = f (x) + g(z) (5.2)
where s = Bz − 1 β y and the last equality follows from the fact that Ax + Bz = b. Clearly, Bz k ∈ B dom g = dom(Bg) for all k, and since g is lsc and (Bg) is continuous on its domain, we have
(Bg)(Bz) ≤ g(z) ≤ lim inf k∈K g(z k ) 5.2(iii) = lim inf k∈K (Bg)(Bz k ) = (Bg)(Bz).
Therefore, g(z k ) → g(z) and (5.2) then implies that f (x k ) → f (x). Notice that, since ϕ 1 = (Bg) is smooth, Prop. A.3(ii) implies that B(z k − z k−1 ) = u k − u k−1 vanishes, and consequently so does Ax k + Bz k−1 − b. In particular,
−A y ← −− − k∈K − A y k−1 + β(Ax k + Bz k−1 − b) ∈ ∂f (x k ),
and f -attentive outer semicontinuity of ∂f [26,Prop. 8.7] ensures −A y ∈ ∂f (x). ♠ 5.3(iii) It suffices to show that ϕ = ϕ 1 + ϕ 2 is level bounded, as boundedness of the sequence ((Ax k , Bz k , y k )) k∈IN will then follow from Thm. 4.3 in light of the DRS equivalence stated in Thm. 5
.2. Let F (x, z) := f (x) + g(z) + δ C (x, z), where C := {(x, z) ∈ IR m × IR n | Ax + Bz = b} is the feasible set. For α ∈ IR we have lev ≤α ϕ = s | inf x {f (x) | Ax = b − s} + inf z {g(x) | Bz = s} ≤ α = s | inf x,z {f (x) + g(z) | Ax = b − s, Bz = s} ≤ α = {Bz | f (x) + g(z) ≤ α, ∃x : Ax + Bz = b} = {Bz | (x, z) ∈ lev ≤α F, ∃x}.
Since Bz ≤ B z ≤ B (x, z) for any x, z, it follows that if lev ≤α F is bounded then so is lev ≤α ϕ.
As a consequence of the Tarski-Seidenberg theorem, functions ϕ 1 := (Bg) and ϕ 2 := (Af )(b − · ) are semialgebraic provided f and g are, see e.g., [5]. Therefore, sufficient conditions for global convergence of ADMM follow from the similar result for DRS stated in Theorem 4.4, through the primal equivalence of the algorithms illustrated in Theorem 5.2. We should emphasize, however, that the equivalence identifies u k = Bz k and v k = b − Ax k , and thus only convergence of (Ax k ) k∈IN and (Bz k ) k∈IN can be deduced. Theorem 5.4 (Global convergence of ADMM). Suppose that f (x) + g(z) is level bounded on the feasible set {(x, z) ∈ IR m × IR n | Ax + Bz = b}, and that f and g are semi-algebraic. Then, the sequence ((Ax k , Bz k , y k )) k∈IN generated by ADMM with β and λ as in Theorem 5.3 converges. Remark 5.5 (Simpler bounds for ADMM). In parallel with the simplifications outlined in Remark 4.2 for DRS, and by observing that (Bg) is strongly convex if so is g, simpler (more conservative) bounds for the penalty parameter β in ADMM are
λ ∈ (0, 2): β > L if g is convex β > 2L 2−λ otherwise λ = 2: β > L if g is str. convex ∅ otherwise
where L := L (Bg) . In particular, if B is full-column rank or g is convex and B is full row-rank, then L = Lg /σ+(B B) (cf. Thm. 5.8 and commentary thereafter).
As it was the case for DRS, the knowledge of L (Bg) (or σ (Bg) ) is actually not needed for ensuring convergence of ADMM iterations. In fact, the adaptive variant of DRS outlined in Remark 4.5 can be easily translated into an adaptive version of ADMM in which the penalty β is suitably increased whenever a quadratic upper bound fails to hold. Notice that, thanks to the equivalences outlined in Theorems 5.2(iii) and 5.2(iv), as it was the case for DRS this condition can be verified by using quantities which are already available. Putting all the pieces together, the adaptive variant of ADMM is outlined in Algorithm 2. For the sake of simplicity, we only consider the case λ = 1, so that the half-update y + /2 can be discarded.
Algorithm 2 (Non-relaxed) ADMM with adaptive stepsize
Require (x 0 , z 0 , y 0 ) ∈ IR m × IR n × IR p , L > 0, β > L if g is convex, β > 2L otherwise For k = 0, 1, 2 . . . 1: x k+1 ∈ argmin x L β (x, z k , y k ) 2: z k+1 ∈ argmin z L β (x k+1 , z, y k ) 3: if g(z k+1 ) > g(z k ) − y k , B(z k+1 − z k ) + L 2 B(z k+1 − z k ) 2 then L ← 2L, β ← 2β
, and go back to step 1 4: y k+1 = y k + β(Ax k+1 + Bz k+1 − b) 5.2. Sufficient conditions. We conclude the section by providing sufficient conditions on f and g ensuring that Assumption II is satisfied.
Lower semicontinuity of the image function.
Proposition 5.6 (Lsc of (Af )). Suppose that A ∈ IR p×n and the proper and lsc function f : IR m → IR satisfy Assumption II(ii). Then, (Af ) is proper. Moreover, it is also lsc provided that for allx ∈ dom f , either (i) the set X(s) := argmin x {f (x) | Ax = s} is nonempty and dist(0, X(s)) is bounded for all s ∈ A dom f close to Ax,
(ii) or lim inf d→d, t→∞ f (td) ≥ inf d∈ker A f (x + d) for alld ∈ ker A \ {0}.
Proof. See Appendix C.
The requirement in Proposition 5.6(i) is much weaker than Lipschitz continuity of the map s → X(s), which is the standing assumption in [29] for the analysis of ADMM. In fact, we do not even require uniqueness or boundedness of the sets of minimizers. Notice that Assumption II(ii) is only invoked to infer that (Af ) is proper. Moreover, whenever the inequality in Proposition 5.6(ii) holds, then it is actually an equality. Its role can be better visualized by considering f : IR 2 → IR defined as
f (x, y) = 1 if y ≤ 1, −|x| if 0 < |x|y < 1, 1 − q(|xy|)(1 + |x|) otherwise (5.3) where q(t) = 1 2 (1 − cos πt). Notice that f ∈ C 1 (IR 2 )
, and that f and A = [1 0] are ADMM-feasible, meaning that argmin w∈IR 2 f (w) + β 2 Aw − s 2 = ∅ for any β > 0 and s ∈ IR. However, (Af )(s) = − |s| if s = 0 while (Af )(0) = 1, resulting in the lack of lsc at s = 0. Along ker A = {0} × IR, by keeping x constant f attains minimum at {x} × [x −1 , ∞) for x = 0, which escapes to infinity as x → 0, and f (x, x −1 ) = −|x| → 0. However, if instead x = 0 is fixed (as opposed to x → 0), then the pathology comes from the fact that f (0, · ) ≡ 1 > 0, which contradicts the condition imposed in Proposition 5.6(ii). This example also shows that f ∈ C 1 (IR m ) is not enough a requirement for (Af ) to be lsc.
The limit inferior is somehow related to the asymptotic function, defined as f ∞ (d) := lim inf d→d, t→∞ f (td) t , see e.g., [1]. The referenced book provides other sufficient conditions based on the behavior of f ∞ on ker A \ {0}. Unlike ours, such conditions ensure also nonemptyness of the set X(s) for all s ∈ A dom f , and are in this sense less general. To see this, it suffices to modify (5.3) as follows
f (x, y) = 1 if y ≤ 1, −|x| if 0 < |x|y < 1, e −x 2 − q(|xy|)(e −x 2 + |x|) otherwise.
The sufficient condition dictated by Proposition 5.6(ii) is satisfied. In fact, the function (Af )(s) = −|s| is lsc, however argmin w {f (w) | Aw = 0} is empty.
5.2.2.
Smoothness of the image function. We now turn to the smoothness requirement of (Bg). To this end, we introduce the notion of smoothness with respect to a matrix, as follows Definition 5.7 (Smoothness relative to a matrix). We say that a function h : IR n → IR is smooth relative to a matrix C ∈ IR p×n , and we write h ∈ C 1,1 C (dom h), if h is differentiable on its domain and ∇h satisfies the following Lipschitz condition: there exist L h,C and σ h,C with |σ h,C | ≤ L h,C such that
σ h,C C(x − y) 2 ≤ ∇h(x) − ∇h(y), x − y ≤ L h,C C(x − y) 2 (5.4)
whenever ∇h(x), ∇h(y) ∈ rng C . This condition is similar to that considered in [13], where Π rng B ∇g is required to be Lipschitz. The paper analyzes convergence of a proximal ADMM; standard ADMM can be recovered when matrix B is invertible, in which case both conditions reduce to Lipschitz differentiability of g. In general, our condition applies to a smaller set of points only, as it can be verified with g(x, y) = 1 2 x 2 y 2 and B = [1 0]. In fact, Π rng g ∇g(x, y) = xy 2 0 is not Lipschitz continuous; however, ∇g(x, y) ∈ rng A iff xy = 0, in which case ∇g ≡ 0. Then, g is smooth relative to B with L g,B = 0.
To better understand how this notion of regularity comes into the picture, notice that if g is differentiable, then ∇g(x) ∈ rng B on some domain U if there exists a differentiable function q : BU → IR such that g(x) = q(Bx). Then, it is easy to verify that g is smooth relative to B if the local "reparametrization" q is smooth (on its domain). From an a posteriori perspective, if (Bg) is smooth (in the classical sense), then due to the relation B ∇(Bg)(Bz s ) = ∇g(z s ) holding for z s ∈ argmin z:Bz=s g(z) (cf. Prop. C.2), it is apparent that q serves as (Bg). Therefore, smoothness relative to B is somewhat a minimal requirement for ensuring smoothness of (Bg).
Theorem 5.8 (Smoothness of (Bg)). Suppose that g : IR n → IR and B ∈ IR p×n satisfy Assumption II(ii). Suppose that there exists β ≥ 0 such that g + β 2 B · − s 2 is level bounded for all s ∈ IR p . Then, the image function (Bg) is smooth on its (affine) domain, provided that either (i) g ∈ C 1,1 B (IR n ), in which case L (Bg) = L g,B and σ (Bg) = σ g,B , (ii) or g ∈ C 1,1 (IR n ), and Z(s) := argmin {g(z) | Bz = s} is single-valued and Lipschitz continuous with modulus M , in which case
L (Bg) = L g M 2 and σ (Bg) = σg / B 2 if σ g ≥ 0 σ g M 2 if σ g < 0,
(iii) or g ∈ C 1,1 (IR n ) is convex and B full-row rank, in which case L (Bg) = Lg σ+(B B) and σ (Bg) = σg / B 2 .
Proof. See Appendix C.
Notice that the condition in Theorem 5.8(ii) covers the case when g ∈ C 1,1 (IR n ) and B has full column rank, in which case M = 1 /σ+(B). This is somehow trivial, since necessarily (Bg)(s) = g • (B B) −1 B if B has full column rank.
Conclusive remarks
This paper provides new convergence results for nonconvex Douglas-Rachford splitting (DRS) and ADMM with an all-inclusive analysis of all possible relaxation parameters λ ∈ (0, 4). Under the only assumption of Lipschitz differentiability of one function, convergence is shown for larger prox-stepsizes and relaxation parameters than was previously known. The results are tight when λ ∈ (0, 2] or when the differentiable function is nonconvex, covering in particular classical (non-relaxed) DRS and PRS. The necessity of λ < 4 and of a lower bound for the stepsize when λ > 2 is also shown.
Our theory is based on the Douglas-Rachford envelope (DRE), a continuous, realvalued, exact penalty function for DRS, and on a new primal equivalence of DRS and ADMM that extends the well known connection of the algorithms to arbitrary (nonconvex) problems. The DRE is shown to be a better Lyapunov function for DRS than the augmented Lagrangian, due to its closer connections with the cost function and with DRS iterations.
Therefore, ψ(z) ≥ ψ(x) + ∇ ψ(x), z − x + 1 2L ψ ∇ ψ(z) −∇ψ(x) 2 for any x, z ∈ dom h = dom ψ, that is, h(z) ≥ σ 2 z 2 + h(x) − σ 2 x 2 + ∇ h(x) − σx, z − x + 1 2(L−σ) ∇ h(z) −∇h(x) − σ(z − x) 2 = h(x) + ∇ h(x), z − x + σ 2 z − x 2 + 1 2(L−σ) ∇ h(z) −∇h(x) 2 + σ 2 2(L−σ) z − x 2 − σ L−σ ∇ h(z) −∇h(x), z − x = h(x) + ∇ h(x), z − x + σL 2(L−σ) z − x 2 + 1 2(L−σ) ∇ h(z) −∇h(x) 2 − σ L−σ ∇ h(z) −∇h(x), z − x
and since σ ≤ 0, by using (A.1),
≥ h(x) + ∇ h(x), z − x + σL 2(L−σ) z − x 2 + 1 2(L−σ) ∇ h(z) −∇h(x) 2 − σ L−σ σL σ+L z − x 2 + 1 σ+L ∇ h(z) −∇h(x) 2 = h(x) + ∇ h(x), z − x + σL 2(L+σ) z − x 2 + 1 2(L+σ) ∇ h(z) −∇h(x) 2 .
Proposition A.2 (Proximal inequalities). Given a proper function h : IR n → IR and x ∈ IR n letx ∈ prox γh (x) for some γ > 0. Then, for all w ∈ IR n h(w)
+ 1 2γ w − x 2 ≥ h(x) + 1 2γ x − x 2 + 1 2γ x − w 2 + r(w,x) where either (i) r(w,x) = − 1 2γ x − w 2 , or (ii) r(w,x) = σ 2 x − w 2 in case h − σ 2 · 2
is convex for some σ ∈ IR. Moreover, if h ∈ C 1,1 (dom h) with dom h affine, and x ∈ dom h, then for any L ≥ L h and σ ≤ min {0, σ h } such that L + σ > 0 we can also take (iii) r(w,x) = σL 2(σ+L) x − w 2 + 1 2(σ+L) ∇h(x) − ∇h(w) 2 . If h is additionally strongly convex, so that σ h > 0, then for any α ∈ [0, 1] we can take (iv) r(w,
x) = (1 − α) σ h 2 x − w 2 + α 1 2L h ∇h(x) − ∇h(w) 2 .
Proof. A.2(i) is a trivial consequence of the characterization of prox γh (x). In the other cases, from the optimality conditions ofx it follows that 1 γ (x −x) ∈ ∂h(x). In particular, by combining Lem. A.1 with the standard inequality for σ-strongly convex functions (allowing here σ ≤ 0) and for convex smooth functions (see e.g.,
[23, Thm. 2.1.5]), we have h(w) ≥ h(x) + 1 γ x −x, w −x + r(w,x) for all w ∈ IR n . The claim now follows from the identity 2 x −x, w −x = x −x 2 + w −x 2 − x − w 2 .
Proposition A.3 (Proximal properties of smooth functions). Let h : IR n → IR be a function with affine domain and suppose that h ∈ C 1,1 (dom h). Then, h is prox-bounded with γ h ≥ 1 /[σ h ]− and for all γ < 1 /[σ h ]− the following hold (i) for all s ∈ IR n , having u = prox γh (s) is equivalent to having ν ∈ dom h ⊥ such that s = u + γ∇h(u) + ν;
(ii) prox γh is ( 1 1+γL h )-strongly monotone on dom h and ( 1 1+γσ h )-Lipschitz continuous everywhere. In particular, for all s, s ∈ dom h 1 1+γL h s − s ≤ u − u ≤ 1 1+γσ h s − s where u = prox γh (s) and u = prox γh (s );
(iii) h γ ∈ C 1,1 (IR n ) with σ h γ = σ h 1+γσ h and L h γ = max L h 1+γL h , [σ h ]− 1+γσ h , with ∇h γ (s) = 1 γ (s−prox γh (s)) and∇h(prox γh (s)) = 1 γ Π dom h s−prox γh (s) .
Proof. For all γ ∈ (0, 1 /[σ h ]−) the function h + 1 2γ · 2 is strongly convex and as such lower bounded; it follows that γ h ≥ 1 /[σ h ]−. ♠ A.3(i) For all s ∈ IR n we have that u = prox γh (s) is the (unique) minimizer of the strongly convex function h(w) + 1 2γ w − s 2 , and is therefore characterized by
0 ∈ ∂h(u) + 1 γ (u − s) + ∂ δ dom h (u) (2.1) =∇h(u) + 1 γ (u − s) + dom h ⊥ . ♠ A.3(ii) For all γ < 1 /[σ h ]− the function h(u) = 1 2 u 2 + γh(u) is (1 + γσ h )
strongly convex and (1+γL h )-smooth over dom h. Moreover, for u = prox γh (s) and
u = prox γh (s ), since s − s ∈ dom h A.3(i) implies that s − s =∇h(u) −∇h(u ). Then, s − s , u − u = ∇ h(u) −∇h(u ), u − u ≥ (1 + γσ h ) u − u 2 .
This shows Lipschitz continuity, and from [23, Thm.s 2.1.5 and 2.1.10] we also have
1 1+γL h s − s 2 ≤ s − s , u − u ≤ 1 1+γσ h s − s 2 , (A.2)
the first inequality of which shows strong monotonicity.
(s) = 1 γ (s − u)
. Moreover, u minimizes h+ 1 2γ · −s 2 and in particular 1 γ (s−u) ∈ ∂h(u). Therefore, ∇h(u)
(2.2) = Π dom h ∂h(u) = 1 γ (Π dom h s − u)
, where the last equality follows from the fact that u ∈ dom h. By using the first inequality in (A.2) we then obtain
∇h γ (s) − ∇h γ (s ), s − s = 1 γ s − s 2 − s − s , u − u ≤ 1 γ 1 − 1 1+γL h s − s 2 = L h 1+γL h s − s 2 .
Similarly, by using also the second inequality in (A.2) we arrive to
σ h 1+γσ h s − s 2 ≤ ∇h γ (s) − ∇h γ (s ), s − s ≤ L h
1+γL h s − s 2 and the claimed values of σ h γ and L h γ follow.
Proposition A.4. Let h : IR n → IR and γ > 0. Then, prox γh (x) = prox γh (x + ν) for all x ∈ IR n and ν ∈ V ⊥ , where V is the affine hull of dom h.
Proof. Trivial.
Appendix B. Proof of Theorem 4. 1 We first prove a preliminary result.
Lemma B.1. Suppose that ϕ 1 and ϕ 2 comply with Assumption I, and starting from any s ∈ IR p consider one DRS update (s, u, v) → (s + , u + , v + ) for some feasible γ and λ > 0. Then,
ϕ γ (s) − ϕ γ (s + ) ≥ 2−λ 2λγ u − u + 2 − γ λ y − y + 2 + r(s, s + ) (B.1)
where r(s, s + ) = (1 − α)
σϕ 1 2 u − u + 2 + α 1 2Lϕ 1 y − y + 2 if σ ϕ1 > 0 σϕ 1 L 2(L+σϕ 1 ) u − u + 2 + 1
2(L+σϕ 1 ) y − y + 2 otherwise with α ∈ [0, 1] and L ≥ L ϕ1 such that L + σ ϕ1 > 0.
Proof. Observe first that
ϕ DR γ (s) = ϕ 1 (u) + ϕ 2 (v) + 1 2γ u − v + γy 2 − γ 2 y 2 . (B.2)
By applying Prop. A.2 with w = u, x = v − γy + /2 andx = u + we obtain
ϕ 1 (u) + 1 2γ u − v + γy + /2 2 ≥ ϕ 1 (u + ) + 1 2γ u + − v + γy + /2 2 + 1 2γ u + − u 2 + r and again Prop. A.2(i) with w = v, x = u + + γy + andx = v + we obtain ϕ 2 (v) + 1 2γ v − u + − γy + 2 ≥ ϕ 2 (v + ) + 1 2γ v + − u + − γy + 2
where r = r(s, s + ). Summing and invoking the expression (B.2) of the DRE yield
ϕ DR γ (s) − ϕ DR γ (s + ) ≥ γ 2 y + 2 − γ 2 y 2 + 1 2γ v − u − γy 2 − 1 2γ u − v + γy + /2 2 − 1 2γ v − u + − γy + 2 + 1 2γ u + − v + γy + /2 2 + 1 2γ u + − u 2 + r.
We now expand all the squares with three terms to arrive to
ϕ DR γ (s) − ϕ DR γ (s + ) ≥ − v − u, y − u − v, y + /2 + v − u + , y + + u + − v, y + /2 + 1 2γ u + − u 2 + r = u − v, y − y + /2 + v − u + , y + − y + /2 + 1 2γ u + − u 2 + r = 1−λ γ u − v 2 − 1 γ v − u + 2 + 1 2γ u + − u 2 + r = − λ γ u − v 2 − 1 2γ u + − u 2 − 2 γ v − u, u − u + + r. By replacing u − v = 1 λ (s − s + ) = 1 λ (u − u + ) − γ λ (y − y + )
, after simple algebra the claimed inequality follows.
For the sake of simplicity, we split the proof of Theorem 4.1 into two cases, depending on whether the smooth function ϕ 1 is strongly convex or not. In both cases, we will rather show that ϕ DR γ (s) − ϕ DR γ (s + ) ≥c u + − u 2 , for somec > 0 and Theorem 4.1 will follow from the inequality
u + − u 2 ≥ 1 1+γLϕ 1 2 s + − s 2 = λ 1+γLϕ 1 2 u − v 2 (B.3)
due to Proposition A.3(ii), so that one can consider c = 1 (1+γLϕ 1 ) 2c . Theorem B.2 (Sufficient decrease -Non-strongly convex case). Suppose that σ ϕ1 ≤ 0. Starting from s ∈ IR p consider one DRS update (s, u, v) → (s + , u + , v + ) for some λ ∈ (0, 2) and some feasible γ < min 1 /Lϕ 1 , 2−λ 2[σϕ 1 ]− . Then, ϕ γ (s) − ϕ γ (s + ) ≥c u − u + 2 wherec is a positive constant defined as Now, let λ ∈ (0, 2) be fixed and let us consider two cases:
c = 2−λ 2λγ + Lϕ 1 2λ min pϕ 1 λ 1+pϕ 1 , λ − 2γL ϕ1 if 0 < λ < 2(1 + p ϕ1 ),
♠ Case 1: λ ≤ 2(1 + p ϕ1 ). Then, σ ϕ1 ≥ − 2−λ 2 L ϕ1 > −L ϕ1 and we can take L = L ϕ1 . Consequently, p = p ϕ1 and ξ = γL ϕ1 , and (B.4) becomes
c = 2−λ 2λγ + σϕ 1 2(1+pϕ 1 ) if γ < λ 2(Lϕ 1 +σϕ 1 ) Lϕ 1 2 − γL 2 ϕ 1 λ otherwise.
By imposingc > 0 we obtain the following conditions on γ and λ:
0 < 2−λ 2λγ + σϕ 1 2(1+pϕ 1 ) γ < λ 2(Lϕ 1 +σϕ 1 ) ∪ 0 < 2−λ 2λγ + Lϕ 1 2 − γL 2 ϕ 1 λ γ ≥ λ 2(Lϕ 1 +σϕ 1 )
and since we are considering the case σ ϕ1 ≥ − 2−λ 2 L ϕ1 , this reduces to γ < λ 2(Lϕ 1 +σϕ 1 ) ∪ λ 2(Lϕ 1 +σϕ 1 ) ≤ γ < 1 /Lϕ 1 , that is, γ < 1 /Lϕ 1 . As to the coefficientc, since and with simple algebra we arrive to Theorem B.3 (Sufficient decrease -Strongly convex case). Suppose that σ ϕ1 > 0. Starting from any s ∈ dom ϕ 1 consider one DRS update (s, u, v) → (s + , u + , v + ) for some feasible γ < 1 /Lϕ 1 and λ > 0. Then, ϕ γ (s) − ϕ γ (s + ) ≥c u − u + 2 forc = 2−λ 2λγ + σ ϕ1 Since λ ≥ 2, by imposing ξ < 1 we can make the coefficient of y − y + 2 vanish by taking α = 2ξ /λ ∈ (0, 1). Therefore, we obtainc Lϕ 1 = 2−λ 2λξ + (λ−2ξ)pϕ 1 2λ
γ < 2−λ −2σϕ 1 ∪ 2−λ −2σϕ 1 ≤ γ < 2−λ −2σϕ 1 that is, γ < 2−λ −
. By imposinḡ c > 0 we are left with the condition
pϕ 1 λ−δ 4σϕ 1 < γ < pϕ 1 λ+δ 4ϕ 1 σ and 2 ≤ λ < 4 1+ √ 1−pϕ 1 ∨ λ > 4 1− √ 1−pϕ 1
where δ = (p ϕ1 λ) 2 − 8p ϕ1 (λ − 2). However, since
pϕ 1 λ−δ 4σϕ 1 > 1 Lϕ 1 for λ ≥ 4, it
follows that the lower bound λ > is to be discarded.
Appendix C. Proofs of Section 5 We first prove two useful results.
for all s i ∈ dom(Bg) and y i ∈ ∂(Bg)(s i ), i = 1, 2. Otherwise, if 5.8(ii) holds, then σ g z 1 − z 2 2 ≤ y 1 − y 2 , s 1 − s 2 ≤ L g z 1 − z 2 2 and from the bound 1 B s 1 − s 2 ≤ z 1 − z 2 ≤ M s 1 − s 2 we obtain σ (Bg) s 1 − s 2 2 ≤ y 1 − y 2 , s 1 − s 2 ≤ L (Bg) s 1 − s 2 2 with the constants σ (Bg) and L (Bg) as in the statement. Either way, for all s i there is at most one subgradient y i ∈ ∂(Bg) dom(Bg) , and from [26,Thm. 9.18] we conclude that necessarily such is y i = ∇(Bg)(s i ). In particular, the inequalities also show that (Bg) ∈ C 1,1 (dom(Bg)) with moduli as in the statement. Since B is injective and ∂(Bg)(s) = ∅, it follows that (Bg) is differentiable. Using Fenchel duality, the claim will follow once we show that (Bg) * is strongly convex with modulus σ+(B B) /Lg, and smooth with modulus B 2 /σg if σ g > 0. Let s i = ∇(Bg) * (y i ), i = 1, 2. In particular, y i = ∇(Bg)(s i ) and B y i = ∇g(z i ) where Bz i = s i . We have
. 4 )
4A continuous and exact merit function for DRS and ADMM. Our results are based on the Douglas-Rachford Envelope (DRE), first introduced in [25] for convex problems and here generalized. The DRE extends the known properties of the Moreau envelope and its connections to the proximal point algorithm, to composite functions as in (1
i (x i ) + h(z) subject to Ax + Bz = 0.
Figure 1 .
1Maximum stepsize γ ensuring convergence of DRS (Fig.
.
Notation. The extended-real line is IR = IR ∪ {∞}. The positive and negative parts of r ∈ IR are defined respectively as [r] + := max {0, r} and [r] − := max {0, −r}, so that r = [r] + − [r] − . We adopt the convention that 1 /0 = ∞.
Connections with the forward-backward envelope. The majorizationminimization formulation(3.2) shows that v is the result of a forward-backward splitting (FBS) step at u = prox γϕ1 (s), written more compactly as
Proposition 3. 1 (
1Strict continuity). For all feasible γ the DRE ϕ DR γ is a realvalued and strictly continuous function over IR p .
Theorem 4 . 1 (
41Sufficient decrease). Consider one DRS update s → (u, v, s + ) for some relaxation λ ∈ (0, 2) and feasible stepsize γ < min
♠ 4.3(i) Let c = c(γ, λ) be as in Thm. 4.1. Telescoping the inequality (4.2) yields
Theorem 4. 4 (
4Global convergence of DRS[21, Thm. 2]). Suppose that ϕ is level bounded and that ϕ 1 and ϕ 2 are semi-algebraic. Then, the sequences (u k ) k∈IN and (v k ) k∈IN generated by DRS with γ and λ as in Theorem 4.3 converge to (the same) stationary point for ϕ.
Example 4 . 6 (
46Necessity of γ < 1 /Lϕ 1 ). Fix L > 0, σ ∈ [−L, L] and t > 1, and let ϕ = ϕ 1 + ϕ 2 , where ϕ 2 = δ {0,1} and
L
ϕ1 = L and σ ϕ1 = σ, and prox γϕ1 is well defined iff γ < 1 /[σ]−. More precisely, γϕ2 = sgn where sgn(0) = {±1}. Suppose that γ ≥ 1 /L and fix λ > 0. Then, one DRS iteration produces v k = − sgn(s k ) if s k ≤ t(1 + γL) and in particular the DRS-residual is
Definition 5 . 1 (
51Image function). Given h : IR n → IR and a linear operator C ∈ IR m×n , the image function (Ch) : IR m → [−∞, +∞] is defined as (Ch)(s) := inf w∈IR n {h(w) | Cw = s}. In formulation (1.2) the linear constraint between x and z can be decoupled by adding a slack variable s ∈ IR p and by rewriting the problem in the equivalent form
Theorem 5. 2 ((
2Primal equivalence of ADMM and DRS). For proper and lsc functions f and g and matrices A, B and b of suitable size, up to an update order shift, one step of ADMM applied to minimize x∈IR m ,z∈IR n f (x) + g(z) subject to Ax + Bz = b is equivalent to one step of DRS applied to Af )(b − s)
From Lem. B.1 it follows that for any feasible γ and any L ≥ L ϕ1 such that L + σ ϕ1 > 0 we have ϕγ (s)−ϕγ (s + ) ξ := γL and p := σϕ 1/L ∈ (−1, 0]. Since y − y + 2 = ∇ ϕ 1 (u + ) −∇ϕ 1 (u) 2 ≤ L 2 ϕ1 u + − u 2 , the claim holds provided that
♠
Case 2: λ > 2(1 + p ϕ1 ). Then, σ ϕ1 < − 2−λ 2 L ϕ1 and we can take L = −2σϕ 1 2−λ > L ϕ1 . Therefore, p = −
2σϕ 1 .
1From (B.5) it follows thatc = of the first part of Theorem 4.1 now follows from (B.3) by observing that the constant c in the statement of Theorem 4.1 satisfies 0
where δ := (p ϕ1 λ) 2 − 8p ϕ1 (λ − 2) is a strictly positive constant.Proof. Suppose λ ≥ 2 and let ξ := γL ϕ1 . From Lem. B.
♠ 5.8(iii). We know from [3, Prop.s 12.34(ii)] and [17, Thm. D.4.5.1] that (Bg) is a convex function and that ∂(Bg)(s) = y | B y = ∇g(z) where z ∈ Z(s).
s 1 − s 2 , y 1 − y 2 = Bz 1 − Bz 2 , y 1 − y 2 = z 1 − z 2 , B y 1 − B first inequality follows from [23, Thm. 2.1.5], the second is a well known property (see e.g., [13, Lem. A.2]), and the last equality is due to the fact that rng B = IR n . These inequalities prove the sought properties.
In case dom h is an affine subspace of IR n , then for all u ∈ dom h2.2. Smoothness. For a function h : IR n → IR we say that h ∈ C 1 (dom h) if the
restriction h dom h : dom h → IR is continuously differentiable, and we definẽ
∇h := ∇(h dom h ).
∂h(u) =∇h(u) + dom h ⊥
(2.1)
and in particular∇
h(u) = Π dom h ∂h(u) = ∂h(u) ∩ dom h
(2.2)
see e.g., [1, Prop. 3.2.3]. Alternatively,∇h(x) can be defined as the unique vector
in dom h such that h(y) = h(x) + ∇ h(x), y − x + o( y − x ) for all y ∈ dom h.
Definition 2.1 (Smoothness). We say that h : IR n → IR is smooth on its domain,
and we write h ∈ C 1,1 (dom h), if h ∈ C 1 (dom h) and∇h := ∇(h dom h ) is Lipschitz
continuous, in which case the Lipschitz modulus is denoted L h .
Remark 2.2. If h is smooth on its domain, then there exists σ h ∈ [−L h , L h ]
satisfying any of the following equivalent properties:
This equivalence is readily proven by applying [23, Thm. 2.1.5] to the function
it is immediate to infer that the DRE inherits the strict continuity property of the FBE[28, Prop. 4.2].
♠ A.3(iii) From[26, Ex. 10.32] it follows that h γ is a strictly continuous function on IR n with ∂h γ (s) ⊆ { (s−u) /γ}, where u := prox γh (s). By invoking[26, Thm. 9.18] we conclude that h γ is everywhere differentiable with ∇h γ
The half-update y +/2 is introduced only for simplification purposes; in fact, replacing y +/2 with y and all occurrences of Ax + in the z-and y-update with λAx + − (1 − λ)(Bz − b) results in the same update. However, it is often the case that λ is set to 1, in which case y +/2 = y.
Although vγ (s) is not necessarily unique, this definition does not depend on the specific choice.
Appendix A. Auxiliary resultsLemma A.1. For a function h ∈ C 1,1 (dom h) with affine domain, let L ≥ L h and σ ≤ min {0, σ h } be such that L + σ > 0. Then, for all x, z ∈ dom hProof. The function ψ := h − σ 2 · 2 ∈ C 1,1 (dom h) is convex with L ψ ≤ L − σ. In particular, using the same reasoning as in[23,Thm. 2.1.12]it is easy to verify thatProposition C.1. Let h : IR n → IR be a proper lsc function and C ∈ IR p×n . Suppose there existsβ ≥ 0 such that the set X β (s) := argmin x h(x) + β 2 Cx − s 2 is nonempty for all β >β and s ∈ IR p . Then, (i) the image function (Ch) is proper;Proof. ♠ C.1(i). Lets ∈ C dom h and β >β be fixed. Then,Cx −s 2 = (Ch)(s) and since (Ch)(s) < ∞ for alls ∈ C dom h = ∅ we conclude that (Ch) is proper.♠ C.1(iii). Fix β >β ands ∈ IR p , and let x β ∈ X β (s). Then, from C.1(ii) and the optimality of x β we haveIn particular, this holds for all s ∈ IR p and x such that Cx = s, hence. Similar reasonings yield the other inclusion too.Proof of Proposition 5.6 (Lsc of (Af )). Properness is shown in Prop. C.1. Suppose now that (s k ) k∈IN ⊆ lev ≤α f for some α ∈ IR and that s k →s. Then, due to the characterization of [26, Thm. 1.6] it suffices to show thats ∈ lev ≤α f . ♠ 5.6(i). The assumption ensures the existence of a bounded sequence (x k ) k∈IN such that eventually Ax k = s k and (Af )(s k ) = f (x k ). By possibly extracting, x k →x and necessarily Ax =s. Then,hences ∈ lev ≤α f .If there exists a convergent subsequence, then the claim follows with a similar reasoning as in the proof of 5.6(i). Suppose that x k → ∞, and let t k = x k and d k := x k t k . By possibly extracting, d k →d for somed with d = 1, and since A(x k −x) = s k −s → 0, necessarilyd ∈ ker A. Then,Proof of Theorem 5.8 (Smoothness of (Bg)). That (Bg) is proper follows from Prop. C.1(i). The level boundedness condition ensures that for all α ∈ IR and s ∈ B dom g the set {z | g(x) ≤ α, Bz − s < ε} is bounded for some ε > 0 (in fact, for all ε > 0). Then, we may invoke [26, Thm. 1.32] to infer that (Bg) is lsc, that the set Z(s) := argmin z {g(z) | Bz = s} is nonempty for all s ∈ dom(Bg), and that the function H(z, s) := g(z) + δ {0} (Bz − s) is uniformly level bounded in z locally uniformly in s, in the sense of[26,Def. 1.16]. ♠ 5.8(i) and5.8(ii). Observe that ∂ ∞ H(z, Bz) = ν−B y y | ν ∈ dom g ⊥ , y ∈ IR p for all z ∈ dom g. We may then apply[26,Cor. 10.14] to arrive to{y | (0, y) ∈ ∂ ∞ H(z, s)} = y | B y ∈ dom g ⊥ = (dom(Bg)) ⊥ for all s ∈ dom(Bg) = B dom g. It follows that ∂ ∞ (Bg) dom(Bg) (s) = {0}, and by virtue of[26,Thm. 9.13]from the arbitrarity of s ∈ dom(Bg) we conclude that (Bg) is strictly continuous on its domain and has nonempty subdifferential. Now, fix s i ∈ dom(Bg) = rng B and y i ∈ ∂(Bg)(s i ), i = 1, 2. It follows from Prop. C.2 and (2.1) that B y i = ∇g(z i ) + ν i for some ν i ∈ dom g ⊥ and z i ∈ Z(s i ), i = 1, 2. Moreover, from the optimality conditions z i ∈ Z(s i ) = argmin g +δ {0} (B · − s i ), we obtain ∇g(z i ) ∈ rng B and Bz i = s i , i = 1, 2. Therefore, y 1 − y 2 , s 1 − s 2 = y 1 − y 2 , Bz 1 − Bz 2 = B y 1 − B y 2 , z 1 − z 2 = ∇g(z 1 ) + ν 1 − ∇g(z 2 ) − ν 2 , z 1 − z 2 = ∇g(z 1 ) − ∇g(z 2 ), z 1 − z 2 .If 5.8(i) holds, since ∇g(z i ) ∈ rng B , i = 1, 2, smoothness of g relative to B implies σ g,B s 1 − s 2 2 = σ g,B Bx 1 − Bx 2 2 ≤ y 1 − y 2 , s 1 − s 2 ≤ L g,B Bz 1 − Bz 2 2 = L g,B s 1 − s 2
Asymptotic Cones and Functions in Optimization and Variational Inequalities. Alfred Auslender, Marc Teboulle, SpringerNew YorkAlfred Auslender and Marc Teboulle. Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer Monographs in Mathematics. Springer New York, 2002.
On the local convergence of the Douglas-Rachford algorithm. Heinz Bauschke, Dominikus Noll, Archiv der Mathematik. 1026Heinz Bauschke and Dominikus Noll. On the local convergence of the Douglas-Rachford algorithm. Archiv der Mathematik, 102(6):589-600, Jun 2014.
Convex Analysis and Monotone Operator Theory in Hilbert Spaces. H Heinz, Patrick L Bauschke, Combettes, CMS Books in Mathematics. SpringerHeinz H. Bauschke and Patrick L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer, 2011.
The method of alternating relaxed projections for two nonconvex sets. H Heinz, Hung M Bauschke, Xianfu Phan, Wang, Vietnam Journal of Mathematics. 424Heinz H. Bauschke, Hung M. Phan, and Xianfu Wang. The method of alternating relaxed projections for two nonconvex sets. Vietnam Journal of Mathematics, 42(4):421-450, Dec 2014.
Jacek Bochnak, Michel Coste, Marie-Françoise Roy, Real Algebraic Geometry. A Series of Modern Surveys in Mathematics. Berlin HeidelbergSpringerJacek Bochnak, Michel Coste, and Marie-Françoise Roy. Real Algebraic Geometry. A Series of Modern Surveys in Mathematics. Springer Berlin Heidelberg, 2013.
Proximal Alternating Linearized Minimization for nonconvex and nonsmooth problems. Jérôme Bolte, Shoham Sabach, Marc Teboulle, Mathematical Programming. 1461-2Jérôme Bolte, Shoham Sabach, and Marc Teboulle. Proximal Alternating Linearized Mini- mization for nonconvex and nonsmooth problems. Mathematical Programming, 146(1-2):459- 494, 2014.
Distributed optimization and statistical learning via the alternating direction method of multipliers. Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, Found. Trends Mach. Learn. 31Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein. Distributed op- timization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn., 3(1):1-122, January 2011.
On the numerical solution of heat conduction problems in two and three space variables. Jim Douglas, Henry H Rachford, Transactions of the American Mathematical Society. 822Jim Douglas and Henry H. Rachford. On the numerical solution of heat conduction prob- lems in two and three space variables. Transactions of the American Mathematical Society, 82(2):421-439, 1956.
Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. Daniel Gabay, Studies in Mathematics and Its Applications. Michel Fortin and Roland Glowinski15ElsevierChapter ix applications of the method of multipliers to variational inequalitiesDaniel Gabay. Chapter ix applications of the method of multipliers to variational inequali- ties. In Michel Fortin and Roland Glowinski, editors, Augmented Lagrangian Methods: Ap- plications to the Numerical Solution of Boundary-Value Problems, volume 15 of Studies in Mathematics and Its Applications, pages 299-331. Elsevier, 1983.
A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Daniel Gabay, Bertrand Mercier, Computers & Mathematics with Applications. 21Daniel Gabay and Bertrand Mercier. A dual algorithm for the solution of nonlinear variational problems via finite element approximation. Computers & Mathematics with Applications, 2(1):17-40, 1976.
On Alternating Direction Methods of Multipliers: A Historical Perspective. Roland Glowinski, SpringerNetherlands, DordrechtRoland Glowinski. On Alternating Direction Methods of Multipliers: A Historical Perspective, pages 59-82. Springer Netherlands, Dordrecht, 2014.
Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de dirichlet non linéaires. Roland Glowinski, A Marroco, ESAIM: Mathematical Modelling and Numerical Analysis -Modélisation Mathématique et Analyse Numérique. 9R2Roland Glowinski and A. Marroco. Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité d'une classe de problèmes de dirichlet non linéaires. ESAIM: Mathematical Modelling and Numerical Analysis -Modélisation Mathématique et Analyse Numérique, 9(R2):41-76, 1975.
Convergence rate bounds for a proximal ADMM with over-relaxation stepsize parameter for solving nonconvex linearly constrained problems. L N Max, Jefferson G Goncalves, Renato D C Melo, Monteiro, ArXiv e-printsMax L. N. Goncalves, Jefferson G. Melo, and Renato D. C. Monteiro. Convergence rate bounds for a proximal ADMM with over-relaxation stepsize parameter for solving nonconvex linearly constrained problems. ArXiv e-prints, February 2017.
Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints. Ke Guo, Deren Han, Ting-Ting Wu, International Journal of Computer Mathematics. 948Ke Guo, Deren Han, and Ting-Ting Wu. Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints. International Journal of Computer Mathematics, 94(8):1653-1669, 2017.
Alternating projections and Douglas-Rachford for sparse affine feasibility. Robert Hesse, Russel Luke, Patrick Neumann, IEEE Transactions on Signal Processing. 6218Robert Hesse, Russel Luke, and Patrick Neumann. Alternating projections and Douglas- Rachford for sparse affine feasibility. IEEE Transactions on Signal Processing, 62(18):4868- 4881, Sept 2014.
Nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems. Robert Hesse, Russell Luke, SIAM Journal on Optimization. 234Robert Hesse and Russell Luke. Nonconvex notions of regularity and convergence of funda- mental algorithms for feasibility problems. SIAM Journal on Optimization, 23(4):2397-2419, 2013.
Fundamentals of Convex Analysis. Grundlehren Text Editions. Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, SpringerBerlin HeidelbergJean-Baptiste Hiriart-Urruty and Claude Lemaréchal. Fundamentals of Convex Analysis. Grundlehren Text Editions. Springer Berlin Heidelberg, 2012.
Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems. Mingyi Hong, Meisam Zhi-Quan Luo, Razaviyayn, SIAM Journal on Optimization. 261Mingyi Hong, Zhi-Quan Luo, and Meisam Razaviyayn. Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems. SIAM Journal on Opti- mization, 26(1):337-364, 2016.
Peaceman-Rachford splitting for a class of nonconvex optimization problems. Guoyin Li, Tianxiang Liu, Ting Kei Pong, Computational Optimization and Applications. Guoyin Li, Tianxiang Liu, and Ting Kei Pong. Peaceman-Rachford splitting for a class of nonconvex optimization problems. Computational Optimization and Applications, May 2017.
Global convergence of splitting methods for nonconvex composite optimization. Guoyin Li, Ting Kei Pong, SIAM Journal on Optimization. 254Guoyin Li and Ting Kei Pong. Global convergence of splitting methods for nonconvex com- posite optimization. SIAM Journal on Optimization, 25(4):2434-2460, 2015.
Douglas-Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems. Guoyin Li, Ting Kei Pong, Mathematical Programming. 1591Guoyin Li and Ting Kei Pong. Douglas-Rachford splitting for nonconvex optimization with application to nonconvex feasibility problems. Mathematical Programming, 159(1):371-401, Sep 2016.
Further properties of the forward-backward envelope with applications to difference-of-convex programming. Tianxiang Liu, Ting Kei Pong, Computational Optimization and Applications. 673Tianxiang Liu and Ting Kei Pong. Further properties of the forward-backward envelope with applications to difference-of-convex programming. Computational Optimization and Applica- tions, 67(3):489-520, Jul 2017.
Introductory lectures on convex optimization: a basic course. Applied optimization. Yurii Nesterov, Springer USYurii Nesterov. Introductory lectures on convex optimization: a basic course. Applied opti- mization. Springer US, 2004.
Proximal Newton methods for convex composite optimization. Panagiotis Patrinos, Alberto Bemporad, 52nd IEEE Conference on Decision and Control. Panagiotis Patrinos and Alberto Bemporad. Proximal Newton methods for convex composite optimization. In 52nd IEEE Conference on Decision and Control, pages 2358-2363, Dec 2013.
Douglas-Rachford splitting: Complexity estimates and accelerated variants. Panagiotis Patrinos, Lorenzo Stella, Alberto Bemporad, 53rd IEEE Conference on Decision and Control. Panagiotis Patrinos, Lorenzo Stella, and Alberto Bemporad. Douglas-Rachford splitting: Complexity estimates and accelerated variants. In 53rd IEEE Conference on Decision and Control, pages 4234-4239, Dec 2014.
Variational analysis. R , Tyrrell Rockafellar, Roger J B Wets, Springer317R. Tyrrell Rockafellar and Roger J.B. Wets. Variational analysis, volume 317. Springer, 2011.
Forward-backward quasi-Newton methods for nonsmooth optimization problems. Lorenzo Stella, Andreas Themelis, Panagiotis Patrinos, Computational Optimization and Applications. 673Lorenzo Stella, Andreas Themelis, and Panagiotis Patrinos. Forward-backward quasi-Newton methods for nonsmooth optimization problems. Computational Optimization and Applica- tions, 67(3):443-487, Jul 2017.
Forward-backward envelope for the sum of two nonconvex functions: Further properties and nonmonotone line-search algorithms. ArXiv preprint. Andreas Themelis, Lorenzo Stella, Panagiotis Patrinos, Andreas Themelis, Lorenzo Stella, and Panagiotis Patrinos. Forward-backward envelope for the sum of two nonconvex functions: Further properties and nonmonotone line-search algo- rithms. ArXiv preprint, jun 2016.
Global convergence of ADMM in nonconvex nonsmooth optimization. Yu Wang, Wotao Yin, Jinshan Zeng, ArXiv e-printsYu Wang, Wotao Yin, and Jinshan Zeng. Global convergence of ADMM in nonconvex non- smooth optimization. ArXiv e-prints, November 2015.
Self Equivalence of the Alternating Direction Method of Multipliers. Ming Yan, Wotao Yin, Springer International PublishingChamMing Yan and Wotao Yin. Self Equivalence of the Alternating Direction Method of Multipli- ers, pages 165-194. Springer International Publishing, Cham, 2016.
| [] |
[
"Radiative neutrino masses from modular A 4 symmetry and supersymmetry breaking",
"Radiative neutrino masses from modular A 4 symmetry and supersymmetry breaking"
] | [
"Hajime Otsuka *[email protected]†[email protected] \nInstitute of Particle and Nuclear Studies\nKEK Theory Center\n1-1 Oho305-0801TsukubaIbarakiJapan\n\nDepartment of Physics\nKyushu University\n744 Motooka, Nishi-ku819-0395FukuokaJapan\n",
"Hiroshi Okada \nAsia Pacific Center for Theoretical Physics (APCTP)\nHeadquarters San 31, Hyoja-dong, Nam-gu790-784PohangKorea\n\nDepartment of Physics\nPohang University of Science and Technology\nPohang 37673Republic of Korea\n"
] | [
"Institute of Particle and Nuclear Studies\nKEK Theory Center\n1-1 Oho305-0801TsukubaIbarakiJapan",
"Department of Physics\nKyushu University\n744 Motooka, Nishi-ku819-0395FukuokaJapan",
"Asia Pacific Center for Theoretical Physics (APCTP)\nHeadquarters San 31, Hyoja-dong, Nam-gu790-784PohangKorea",
"Department of Physics\nPohang University of Science and Technology\nPohang 37673Republic of Korea"
] | [] | We investigate a modular A 4 invariant two-loop neutrino mass model in a supersymmetric framework, where we introduce new fields as minimum as possible, expecting contributions of superpartners to the neutrino masses. We successfully reproduce the neutrino oscillation data thanks to the superpartner contributions in case of normal hierarchy, and predict several observables such as phases and neutrino masses, concentrating on three specific regions at nearby fixed points of modulus τ = i, ω, i × ∞, where ω ≡ e 2πi/3 . These points are statistically favored in flux compactifications of the string theory. We show several results in each the points by performing global χ 2 analysis, and demonstrate benchmark points with the minimum χ 2 . | null | [
"https://arxiv.org/pdf/2202.10089v1.pdf"
] | 247,011,350 | 2202.10089 | 3d80c36885b6d9693318b29c43db9a0ae7d800b0 |
Radiative neutrino masses from modular A 4 symmetry and supersymmetry breaking
21 Feb 2022
Hajime Otsuka *[email protected]†[email protected]
Institute of Particle and Nuclear Studies
KEK Theory Center
1-1 Oho305-0801TsukubaIbarakiJapan
Department of Physics
Kyushu University
744 Motooka, Nishi-ku819-0395FukuokaJapan
Hiroshi Okada
Asia Pacific Center for Theoretical Physics (APCTP)
Headquarters San 31, Hyoja-dong, Nam-gu790-784PohangKorea
Department of Physics
Pohang University of Science and Technology
Pohang 37673Republic of Korea
Radiative neutrino masses from modular A 4 symmetry and supersymmetry breaking
21 Feb 2022(Dated: February 22, 2022)1
We investigate a modular A 4 invariant two-loop neutrino mass model in a supersymmetric framework, where we introduce new fields as minimum as possible, expecting contributions of superpartners to the neutrino masses. We successfully reproduce the neutrino oscillation data thanks to the superpartner contributions in case of normal hierarchy, and predict several observables such as phases and neutrino masses, concentrating on three specific regions at nearby fixed points of modulus τ = i, ω, i × ∞, where ω ≡ e 2πi/3 . These points are statistically favored in flux compactifications of the string theory. We show several results in each the points by performing global χ 2 analysis, and demonstrate benchmark points with the minimum χ 2 .
I. INTRODUCTION
Active neutrino and dark matter (DM) candidate may be related each other, since their features are similar in view of electric neutrality and rather difficulty of detection. In fact, the nature of neutrino is partially discovered by experiments, and DM is not directly found even though so many different kinds of experiments are on going. From a theoretical point of view, it could suggest that we propose more imaginable ideas than the other three fermion sector in the standard model (SM). One of the ideas is a radiative seesaw model [1], in which the neutrino mass matrix is constructed at loop levels. When DM runs in the loop, we might interpret the neutrinos indirectly interact with the SM Higgs and get the masses only through DM. Hence, we might understand the origin of tiny neutrino masses. In order to realize this kind of neutrino masses, we often need a symmetry such as Z 2 that could simultaneously stabilizes DM. 1 This symmetry is usually introduced by hand. But, if it arises from the other symmetry, the model would be more attractive. Also, this scenario could accommodate other phenomenologies such as lepton flavor violations (LFVs) due to not so small Yukawa couplings. It implies that the model can be within a low energy scale that might reach at current experiments, thus the testability is enhanced. The neutrino mixing patterns, phases, and LFVs do depend on the structure of Yukawa matrix, but the SM does not provide any prescriptions to fix this structure.
Several years ago, a finite modular group has been applied to the lepton sector to predict the neutrino oscillation data in ref. [2]. 2 Interestingly, this scenario does not require many flavons that are traditionally introduced to get a desired texture, and the group includes a new degree of freedom: "modular weight", that comes from the modular group.
If we apply this group to a radiative seesaw model, DM can be stable by assigning nonzero modular weight. In this way, we might obtain predictions to the lepton sector, and a specific interaction pattern of LFVs could be tested by current experiments. Triggering this paper, a lot of groups have applied to various phenomenologies such as quark, lepton, DM, and so on.
For example, the modular A 4 flavor symmetry has been discussed in refs. [2,[4][5][6], S 3 in refs. [49][50][51][52][53][54], S 4 in refs. [55][56][57][58][59][60][61][62][63][64][65][66][67], A 5 in refs. [60,68,69], double covering of A 4 in refs. [70][71][72], 1 In this framework, a leptonic DM is favored. 2 Charged-lepton and neutrino sectors have been discussed in ref. [3] by embedding subgroups of various finite modular flavor symmetries. double covering of S 4 in refs. [73,74], and double covering of A 5 in refs. [75][76][77][78]. Other types of modular symmetries have also been proposed to understand masses, mixings, and phases of the standard model (SM) in refs. [79][80][81][82][83][84][85][86][87]110]. 3 Different applications to physics such as dark matter and origin of CP are found in refs. [7,8,12,15,59,[97][98][99][100][101][102][103][104][105]. Mathematical studies such as possible correction from Kähler potential, systematic analysis of the fixed points, and moduli stabilization are discussed in refs. [106][107][108][109][110]. Recently, the authors of ref. [111] proposed a scenario to derive four-dimensional modular flavor symmetric models from higher-dimensional theory on extra-dimensional spaces with the modular symmetry.
It constrains modular weights and representations of fields and modular couplings in the four-dimensional effective field theory. Higher-dimensional operators in the SM effective field theory are also constrained in the higher-dimensional theory, in particular, the string theory [112]. Non-perturbative effects relevant to neutrino masses are studied in the context of modular symmetry anomaly [113].
In this paper, we apply a modular A 4 symmetry into a two-loop induced neutrino model in refs. [114,115], where we introduce new fields as minimum as possible, expecting contributions of superpartners to the neutrino masses. 4 We focus on the SUSY-breaking sector inducing modular symmetric soft SUSY-breaking terms. (See for the SUSY-breaking phenomenology in modular flavor models [54,118]). In fact, we successfully reproduce the neutrino oscillation data thanks to these contributions in case of normal hierarchy, and predict several observables such as phases and neutrino masses at nearby three fixed points of modulus τ = i, ω, i × ∞, where ω ≡ e 2πi/3 . These fixed points are statistically favored in the flux compactification of Type IIB string theory [45]. We show several results in each the points by performing global χ 2 analysis, and demonstrate benchmark points with the minimum χ 2 . This paper is organized as follows. In Sec. II, we review our model, giving superpotential and SUSY-breaking terms. Then, we formulate valid mass matrices for bosons and fermions that are needed to construct the neutrino mass matrix. In Sec. III, we show several predictions at nearby three fixed points via global χ 2 analysis, and demonstrate benchmark points with the minimum χ 2 . Finally, we conclude and summarize our model in Sec. IV, in which we briefly discuss the possibility of DM candidate. In Appendix A, we summarize formulas in the framework of modular A 4 symmetry.
{L e ,L µ ,L τ } {ê c ,μ c ,τ c } {N c 1 ,N c 2 }ŜĤ 1Ĥ2η1η2χ SU (2) L 2 1 1 1 2 2 2 2 1 U (1) Y − 1 2 1 0 0 1 2 − 1 2 1 2 − 1 2 0 A 4 {1, 1 , 1 } {1, 1 , 1 } {1, 1 } 3 1 1 1 1 1 −k I −4 0 −1 0 0 0 −3 −3 −3
II. MODEL
Here, we review our model in order to obtain the two-loop neutrino masses in a similar mechanism discussed in the non-SUSY framework [115]. In addition to the minimal supersymmetric SM (MSSM), we introduce matter superfields including two right-handed neutral fermions N c 1,2 and three left-handed neutral fermions S. N c 1,2 and S respectively belong to 1, 1 and 3 under the modular A 4 group. Here, the modular A 4 group is one of the finite modular subgroups of SL(2, Z), parametrized by the modulus τ . We also add two superfields including inert bosons η 1 and χ where these are true singlets under the A 4 group. Chiral superfields {Ĥ 2 ,η 2 } including two bosons {H 2 , η 2 } are just required in order to retain the holomorphic feature. Here and in what follows, we denote byφ their superpartners in matter superfieldsφ. Nonzero modular weights are imposed by −k (Le,Lµ,Lτ ) = −4, k N c 1,2 = −1, −k (η 1,2 ,χ) = −3. All the fields and their assignments are summarized in Table I. Under these symmetries, one writes renormalizable superpotential as follows:
W = y eê cL eĤ2 + y µμ cL µĤ2 + y ττ cL τĤ2
denote the K mod and K matter for the modulus and matter Kähler potentials, respectively, 7 the explicit form of soft scalar masses m 2 φ i and the A-terms in the canonical normalization are written as [116]:
m 2 φ i = m 2 3/2 − X |F X | 2 ∂ X ∂X ln ∂ φ i ∂φ i K matter , A ijk = A i + A j + A k − X F X y ijk ∂ X (y ijk ), (II.3) with A i = X F X ∂ X ln e −K mod ∂ φ i ∂φ i K matter , (II.4)
where m 3/2 is the gravitino mass and y ijk denote the Yukawa couplings of fields. Since the modulus τ does not appear in gauge kinetic functions at the tree-level in the context of Type IIB string theory, we assume that the gaugino masses are generated by F -terms of other moduli (see for the stabilization of moduli fields, e.g., refs. [16, 101-103, 108, 110]).
In this paper, we randomly search for soft terms to study the structure of neutrino masses and mixing angles without specifying the SUSY-breaking sector.
Inert boson and fermion mixings
Inert bosons χ, η 1 , and η 2 mix each other through the soft SUSY-breaking terms of A a,b and µ Bη , after the spontaneous electroweak symmetry breaking. Here, we suppose to be µ Bη , A a << A b for simplicity, then the mixing dominantly comes from χ and η 1 only.
This assumption does not affect the structure of the neutrino mass matrix. Then the mass eigenstate is defined by
χ R,I η 1 R,I = c θ R,I −s θ R,I s θ R,I c θ R,I ξ 1 R,I ξ 2 R,I , (II.5)
where c θ R,I , s θ R,I are respectively the shorthand notations of sin θ R,I and cos θ R,I , and ξ 1,2 are mass eigenstates for χ, η 1 and their mass eigenvalues are denoted by m i R,I , (i = 1, 2). Notice that the mixing angle θ simultaneously diagonalizes the mass matrix of real and imaginary part.
These superpartners of χ and η;χ andη, mix each other via b, and its mass matrix is given by
Mχη = Y (6) 1 m 1 m 2 m 2 0 , (II.6) where m 1 ≡ µ χ , m 2 ≡ v 2 b 2 √ 2 .
Then, the above matrix is diagonalized by a unitary matrix:
OMχηO T = diag(mξ 1 , mξ 2 ), (II.7) O = i 0 0 1 cθ −sθ sθ cθ , tanθ = m 1 + m 2 1 + 4m 2 2 2m 2 , (II.8) where mξ 1 ≡ 1 2 ( m 2 1 + 4m 2 2 − m 1 ), mξ 2 ≡ 1 2 ( m 2 1 + 4m 2 2 + m 1 )
, and cθ, sθ are respectively the shorthand notations of sinθ and cosθ. Similar to the boson sector, the mass eigenstate is defined by
χ η 1 = icθ sθ −isθ cθ ξ 1 ξ 2 . (II.9)
Neutral fermion mass matrix of S
The mass matrix of S is found to be
M S = M 0 1 0 0 0 0 1 0 1 0 . (II.10) Since M S M † S = M † S M S ∼ 1 3×3 , the mixing matrix V S = 1. Here, we define S = V T S ψ = ψ, where ψ is the mass eigenstate of S.
Boson mass matrix of the superpartner S:S
SinceS i consists of real and imaginary scalars, we redefine them to beS i ≡ (s R i +is I i )/ √ 2.
Then, we explicitly write the mass matrices as follows:
M 2 S R = |M 0 | 2 + m 2 S + µ 2 SB 0 0 0 4|M 0 | 2 + m 2 S 2µ 2 SB 0 2µ 2 SB 4|M 0 | 2 + m 2 S , (II.11) M 2 S I = |M 0 | 2 + m 2 S − µ 2 SB 0 0 0 4|M 0 | 2 + m 2 S −2µ 2 SB 0 −2µ 2 SB 4|M 0 | 2 + m 2 S . (II.12)
Both of the above mass matrices are diagonalized by an orthogonal matrix OS as follows:
D 2 S R = diag[|M 0 | 2 + m 2 S + µ 2 SB , 4|M 0 | 2 + m 2 S − 2µ 2 SB , 4|M 0 | 2 + m 2 S + 2µ 2 SB ] = OSM 2 S R O T S , (II.13) D 2 S I = diag[|M 0 | 2 + m 2 S − µ 2 SB , 4|M 0 | 2 + m 2 S + 2µ 2 SB , 4|M 0 | 2 + m 2 S − 2µ 2 SB ] = OSM 2 S R O T S , (II.14) OS = 1 0 0 0 1/ √ 2 −1/ √ 2 0 1/ √ 2 1/ √ 2 . (II.15)
Then, we finds R,I = O T SS R,I , wheres R,I is the mass eigenstate ofS R,I .
Neutral fermion mass matrix of N c
The mass of N c is induced at one-loop level, running ψ and χ. The valid Lagrangian in terms of mass eigenstate of S is found as follows: 8
−L = N c Y N S √ 2 ψ(c θ R ξ 1 R − s θ R ξ 2 R ) − N c Y N S √ 2 ψ(c θ I ξ 1 I − s θ I ξ 2 I ) + h.c., Y N S = α N S 0 0 β N S y 1 y 3 y 2 y 2 y 1 y 3 . (II.16)
Then, the mass matrix of N c is derived as follows: The mass matrix ofÑ c is also induced at one-loop level, running several fields. The valid Lagrangian is given by
M N c = − M 0 (4π) 2 Y N S Y T N S × 2F 0 (M 0 , m 2 1 R , m 2 1 I )(c 2 θ R − c 2 θ I ) + 2F 0 (M 0 , m 2 2 R , m 2 2 I )(s 2 θ R − s 2 θ I ) (II.17) +F I (M 0 , m 2 1 R , m 2 1 I )(c 2 θ I m 2 1 R − c 2 θ R m 2 1 I ) + F I (M 0 , m 2 2 R , m 2 2 I )(s 2 θ I m 2 2 R − s 2 θ R m 2 2 I ) , F 0 (M 0 , m 1 , m 2 ) ≡ [dx] 3 ln[xM 2 0 + ym 2 1 + zm 2 2 ], F I (M 0 , m 1 , m 2 ) ≡ [dx] 3 xM 2 0 + ym 2 1 + zm 2 2 , (II.18) where [dx] 3 ≡−L =Ñ c Y N S ψ(icθξ 1 + sθξ 2 ) +Ñ c AY N S O T S √ 2 (s R + is I ) [(c θ R ξ 1 R − s θ R ξ 2 R ) + i(c θ I ξ 1 I − s θ I ξ 2 I )] + h.c. (II.19) =Ñ c Y N S ψ(icθξ 1 + sθξ 2 ) + c θÑ c G(s R + is I )(ξ 1 R + iξ 1 I ) − s θÑ c G(s R + is I )(ξ 2 R + iξ 2 I ) + h.c., AY N S = Y (4) 1 A α N S 0 0 A β N S y 1 y 3 y 2 y 2 y 1 y 3 , G ≡ AY N S O T S . (II.20)
Then, the mass matrixÑ c is found as
(M 2 N c ) ij = 4 (YÑ ) ij (4π) 2 2(−c 2 θ mξ 1 + s 2 θ mξ 2 )F 0 (M 0 , mξ 1 , mξ 2 ) + mξ 1 mξ 2 (c 2 θ mξ 2 + s 2 θ mξ 1 )F I (M 0 , mξ 1 , mξ 2 ) − G ia G T aj (4π) 2 2(c 2 θ R − c 2 θ I )[F 0 (DS Ra , m ξ 1 R , m ξ 1 I ) + F 0 (DS Ia , m ξ 1 R , m ξ 1 I )] +2(s 2 θ R − s 2 θ I )[F 0 (DS Ra , m ξ 2 R , m ξ 2 I ) + F 0 (DS Ia , m ξ 2 R , m ξ 2 I )] +(c 2 θ I m 2 1 R − c 2 θ R m 2 1 I )[F 1 (DS Ra , m ξ 1 R , m ξ 1 I ) + F 1 (DS Ia , m ξ 1 R , m ξ 1 I )] +(s 2 θ I m 2 2 R − s 2 θ R m 2 2 I )[F 1 (DS Ra , m ξ 2 R , m ξ 2 I ) + F 1 (DS Ia , m ξ 2 R , m ξ 2 I )] , (II.21) YÑ = α 2 N S (y 2 1 + 2y 2 y 3 ) α N S β N S (y 2 2 + 2y 1 y 3 ) α N S β N S (y 2 2 + 2y 1 y 3 ) β 2 N
Mass matrix of m ν
Now the active neutrino mass matrix m ν is induced at one-loop level via the following Lagrangian in terms of mass eigenstate:
−L = 1 √ 2 (ψ c N ) a (V N ) ai (Y η ) ij ν j (s θ R ξ 1 R + c θ R ξ 2 R ) + i √ 2 (ψ c N ) a (V N ) ai (Y η ) ij ν j (s θ I ξ 1 I + c θ I ξ 2 I ) + (ñ c ) α (OÑ ) αβ (Y η ) βj ν j (−isθξ 1 + cθξ 2 ) + h.c., (II.23) Y η = 1 √ 2 a η Y (8) 1 b η Y (8) 1 e η Y (8) 1 f η Y (8) 1 c η Y (8) 1 d η Y (8) 1 , (II.24)
Then, the neutrino mass matrix is given by (II. 30) The effective mass for the neutrinoless double beta decay is depicted by m ee = |m 1 cos 2 θ 12 cos 2 θ 13 + m 2 sin 2 θ 12 cos 2 θ 13 e iα 21 + m 3 sin 2 θ 13 e i(α 31 −2δ CP ) |, (II. 31) where δ CP , (α 21 , α 31 ) are respectively Dirac and two Majorana CP phases appearing in the U P M N S matrix. m ee could be tested by KamLAND-Zen in future [122].
m ν = − 1 2(4π) 2 (Y T η ) iα (V T N ) αa D Na (V N ) aβ (Y η ) βj × s 2 θ R f (m ξ 1 R , D Na ) + c 2 θ R f (m ξ 2 R , D Na ) − s 2 θ I f (m ξ 1 I , D Na ) − c 2 θ I f (m ξ 2 I , D Na ) − µ χ Y (6) 1 (4π) 2 mξ 1 mξ 2 (Y T η ) iβ (O T N ) βα F I (DÑ α , mξ 1 , mξ 2 )(OÑ ) αβ (Y η ) β j , (II.25) f (m 1 , m 2 ) = 1 0 ln x m 2 1 m 2 2 − 1 + 1 , (II.26)
III. NUMERICAL ANALYSIS
In this section, we show numerical ∆χ 2 analysis at nearby three fixed points, employing the five reliable experimental data; ∆m 2 atm , ∆m 2 sol , sin 2 θ 13 , sin 2 θ 23 , sin 2 θ 12 in ref. [123]. Notice here that we consider CP phases δ CP , α 21 , α 31 as predictive values, and three chargedlepton masses are supposed to be fitted by the best fit values. In case of IH, we would not find any allowed region within 5σ. Thus, we focus on the case of NH only. The dimensionful input parameters are randomly selected by the range of [10 2 − 10 7 ] GeV except for µ χ with [10 −5 − 10] GeV, while the dimensionless ones [10 −10 − 10 −1 ] except for τ . We work on three fixed points in the fundamental region of τ , and all the input parameters are supposed to be real. Therefore, the origin of CP comes from τ .
A. τ = i
In case where the region is at nearby τ = i, we show four plots in Fig. 1. Each of color represents blue ≤ 1σ, 1σ < green ≤ 2σ, 2σ < yellow ≤ 3σ, 3σ < red ≤ 5σ. The top left Also, we show a benchmark point in case of nearby τ = i in Table II that provides minimum ∆χ 2 in our numerical analysis. Also, we show a benchmark point in case of nearby τ = ω in Table III Also, we show a benchmark point in case of nearby τ = i × ∞ in Table IV
τ = i × ∞ in NH.
Here, this BP is taken such that ∆χ 2 is minimum.
IV. CONCLUSION AND DISCUSSION
We have studied a modular A 4 invariant two-loop neutrino mass model in a SUSY framework, in which we have focused on regions at nearby three fixed points of τ = i, ω, i × ∞.
These points with residual symmetries are motivated by flux compactifications of the string theory.
Thanks to contributions of SUSY partnersÑ c ,χ,η 1 to the neutrino mass matrix, we have successfully obtained several predictions in NH such as phases and neutrino masses for each of fixed points through our global ∆χ 2 analysis. Moreover, the non-SUSY contributions to the neutrino masses are negligibly small at some points, since the mass of N c is minuscule in addition to rather small Yukawa couplings 10 −2 ∼ 10 −3 . Here, 10 −3 comes from the conservative upper limit to satisfy the lepton flavor violations such as µ → eγ. On the other hand, the mass ofÑ c can be larger than the mass of N c , which could reach at the order TeV, due to soft SUSY-breaking terms such as A α N S , A β N S , mS, µS B .
Before closing our paper, we will briefly mention dark matter (DM) candidates. In our model, the mass of N c (as well as its superpartner) is induced at one-loop level. 9 It implies that the mass of DM candidate is naturally smaller than the other particles. But the problem would arise from the thermal averaged cross section to satisfy the observed relic density of DM since valid interactions come from Yukawa couplings only, and their order is 10 −2 at most. In fact, order one Yukawa couplings are needed in order to explain the relic density.
Thus, we need to rely on bosonic DM candidate; i.e., χ or neutral component of η 1,2 [41].
Since χ and η 1 directly interact with N c through Yukawa couplings and η 2 mixes with χ and η 1 , one component DM is favored when the bosonic DM mass is lighter than the mass of N c . For simplicity, let us consider η ≡ η 1,2 dominant DM or χ dominant DM, neglecting the mixing among them. In case where η is the DM candidate, the main interaction comes from kinetic terms and it is known that there exist some solutions to satisfy the relic density. Here we cannot rely on the Yukawa interactions due to the similar reason as the case of fermionic DM. According to the systematic analysis by ref. [124], it tells us the dark matter mass is at around 534 GeV when the mass is larger than the mass of W/Z mass. In the lighter region, one also finds the dark matter mass is at around the half of Higgs mass; 63 GeV. In order to avoid the constraint of direct detection searches, we need mass difference between the real and imaginary part of η that is more than 100 keV. It implies that we need a little mixings among neutral bosons. In case where χ is the DM candidate, the main interaction comes from Higgs potential, and any mass range could be possible through these interactions [125]. where f (τ ) denotes holomorphic functions of τ with the modular weight k.
In a similar way, the modular transformation of a matter chiral superfield φ (I) with the modular weight −k I is given by = (y 2 1 − 2y 2 y 3 , y 2 3 − 2y 1 y 2 , y 2 2 − 2y 1 y 3 ).
I: Field contents of matter chiral superfields and their charge assignments under SU (2) L × U (1) Y × A 4 in the lepton and boson sector, where k is the number of modular weight, and the quark sector is the same as the SM.
1 0
1dxdydzδ(1 − x − y − z) and m i R,I is the mass of ξ i R,I , (i = 1, 2). Similar to S, we find N c = V T N ψ c N , where ψ c N is the mass eigenstate of N c and D N = V N M N V T N . 5. Boson mass matrix of the superpartner N c :Ñ c
N c , we findÑ c = O T Nñ c , whereñ c is the mass eigenstate ofÑ c and D 2Ñ = OÑ M 2 N c O T N .
m ν is diagonalized by a unitary matrix U ν as follows: U ν m ν U T ν ≡ diag[m 1 , m 2 , m 3 ], where i=1,2,3 m i 0.12 eV is given by the recent cosmological data [120]. Since the mixing matrix for charged-lepton is three by three unit matrix, one finds U ν = U P M N S . It is remarkable that the second term in eq. (II.25) represents the loop effects of superpartners, and these contributions play an important role of obtaining the rank-3 neutrino mass matrix. mixing angle is given in terms of the component of U P M N S as follows: sin 2 θ 13 = |(U P M N S ) 13 | 2 , sin 2 θ 23 = |(U P M N S ) 23 | 2 1 − |(U P M N S ) 13 | 2 , sin 2 θ 12 = |(U P M N S ) 12 | 2 1 − |(U P M N S ) 13 | 2 .
figure shows allowed region of τ , the top right one m ee in terms of the lightest neutrino mass m 1 , the bottom left one Majorana phases α 21 , α 31 , and the bottom right one Dirac CP phase δ CP versus sum of neutrino masses m i . These figures suggest 1meV m ee 4.5meV, m 1 0.033µeV, 57meV m i 61meV, δ CP 0 • . Allowed regions of Majorana phases tend to be localized at nearby α 21 = α 31 and α 21 = α 31 = 0.
FIG. 1 :
1By focusing on nearby τ = i, we show an allowed region of τ in the top left panel, m ee in terms of the lightest neutrino mass m 1 in the top right one , Majorana phases α 21 , α 31 in the bottom left one, and Dirac CP phase δ CP versus sum of neutrino masses m i in the bottom right θ R , s θ I , sθ] [−0.0000658, 0.000751, −0.535] [A α N S , A β N S ]/GeV [6.12 × 10 5 , 1.49 × 10 4 ] [M 0 , mS, µS B , µ χ ]/GeV [2.16 × 10 4 , 3.41 × 10 5 , 998, 0.000250] [m 1 R , m 1 I , m 2 R , m 2 I , mξ 1 , mξ 2 ]/GeV [1.54 × 10 4 , 8.75 × 10 5 , 242.552, 262.181, 2049.28, 5329.CP , α 21 , α 31 ] [0.0242851 • , 180.081 • , 180.131 • ] : Numerical benchmark point (BP) of our input parameters and observables at nearby the fixed point τ = i in NH. Here, this BP is taken such that ∆χ 2 is minimum. FIG. 2: We analyze nearby the fixed point τ = ω, where the legends and the colors are the same as the case of nearby τ = i. B. τ = ω In case where the region is at nearby τ = ω, we show four plots in Fig. 2. The legends and the colors are the same as the case of nearby τ = i. These figures suggest 2.5meV m ee 4meV, m 1 0.04µeV, 56meV m i 61meV, 10 • δ CP 100 • . Allowed regions of Majorana phases tend to be localized at nearby α 21 = α 31 /2, α 21 = α 31 0 • .
FIG. 3 :
3We analyze nearby the fixed point τ = i × ∞, where the legends and the colors are the same as the case of nearby τ = i. C. τ = i × ∞ In case where the region is at nearby τ = i × ∞, we show four plots in Fig. 3. The legends and the colors are the same as the case of nearby τ = i. These figures suggest 3.0meV m ee 4.5meV, m 1 0.035µeV, 57meV m i 61meV, δ CP 0 • . Majorana phases are localized at nearby α 21 = α 31 = [0 • , 180 • ].
and was supported by the Korean Local Governments-Gyeongsangbuk-do Province and Pohang City. Hiroshi O. is sincerely grateful for all the KIAS members.Appendix A: Formulas in modular A 4 framework In this appendix, we summarize some formulas in the framework of A 4 modular symmetry belonging to the SL(2, Z) symmetry. The modulus τ transforms as τ −→ γτ = aτ + b cτ + d , (A.1) with {a, b, c, d} ∈ Z satisfying ad − bc = 1 and Im[τ ] > 0. The transformation of modular forms f (τ ) are given by f (γτ ) = (cτ + d) k f (τ ) , γ ∈ Γ(N ) , (A.2)
φ
(I) → (cτ + d) −k I ρ (I) (γ)φ (I) , (A.3) where ρ (I) (γ) stands for an unitary matrix corresponding to A 4 transformation. Note that the superpotential is invariant when the sum of modular weight from fields and modular form is zero. It restricts a form of the superpotential as shown in Eq. (II.1). Modular forms are constructed on the basis of weight 2 modular form, Y a triplet of A 4 . Their explicit forms are written by the Dedekind eta-function η(τ ) and its derivative with respect to τ [2]: τ ) = q 1/24 Π ∞ n=1 (1 − q n ), q = e 2πiτ , ω = e 2πi/3 .Modular forms of higher weight can be obtained from tensor products of Y number in superscript denotes the modular weight. The A 4 triplet of the modular weight 4 is given by Y
TABLE
TABLE III :
IIINumerical BP of our input parameters and observables at nearby the fixed point τ = ω in NH. Here, this BP is taken such that ∆χ 2 is minimum.
TABLE IV :
IVNumerical BP of our input parameters and observables at nearby the fixed point
Here, we provide useful review references for beginners[89][90][91][92][93][94][95][96].4 The neutrino mass scenario from the soft supersymmetry (SUSY)-breaking terms are proposed by ref.[117].
The matter Kähler metric is assumed to be a diagonal form.
We neglect a contribution from gauginos to the neutrino masses[119], assuming the masses are much heavier than the masses of N c ,Ñ c , ξ 1 R,I , and , ξ 2 R,I . This assumption will be justified when gaugino masses are generated by moduli fields without τ .
To make our discussion simple, we focus on non-SUSY particles only.
. E Ma, hep-ph/0601225Phys. Rev. D. 7377301E. Ma, Phys. Rev. D 73, 077301 (2006) [hep-ph/0601225].
. F Feruglio, arXiv:1706.08749hep-phF. Feruglio, [arXiv:1706.08749 [hep-ph]].
. R De Adelhart Toorop, F Feruglio, C Hagedorn, arXiv:1112.1340Nucl. Phys. B. 858hep-phR. de Adelhart Toorop, F. Feruglio and C. Hagedorn, Nucl. Phys. B 858, 437-467 (2012) [arXiv:1112.1340 [hep-ph]].
. J C Criado, F Feruglio, arXiv:1807.01125SciPost Phys. 5542hep-phJ. C. Criado and F. Feruglio, SciPost Phys. 5, no.5, 042 (2018) [arXiv:1807.01125 [hep-ph]].
. T Kobayashi, N Omoto, Y Shimizu, K Takagi, M Tanimoto, T H Tatsuishi, arXiv:1808.03012JHEP. 11196hep-phT. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, JHEP 11, 196 (2018) [arXiv:1808.03012 [hep-ph]].
. H Okada, M Tanimoto, arXiv:1812.09677Phys. Lett. B. 791hep-phH. Okada and M. Tanimoto, Phys. Lett. B 791, 54-61 (2019) [arXiv:1812.09677 [hep-ph]].
. T Kobayashi, H Okada, Y Orikasa, arXiv:2111.05674hep-phT. Kobayashi, H. Okada and Y. Orikasa, [arXiv:2111.05674 [hep-ph]].
. T Nomura, H Okada, arXiv:1904.03937Phys. Lett. B. 797134799hep-phT. Nomura and H. Okada, Phys. Lett. B 797, 134799 (2019) [arXiv:1904.03937 [hep-ph]].
. H Okada, M Tanimoto, arXiv:1905.13421Eur. Phys. J. C. 811hep-phH. Okada and M. Tanimoto, Eur. Phys. J. C 81, no.1, 52 (2021) [arXiv:1905.13421 [hep-ph]].
. F J De Anda, S F King, E Perdomo, arXiv:1812.05620Phys. Rev. D. 101115028hep-phF. J. de Anda, S. F. King and E. Perdomo, Phys. Rev. D 101, no.1, 015028 (2020) [arXiv:1812.05620 [hep-ph]].
. P P Novichkov, S T Petcov, M Tanimoto, arXiv:1812.11289Phys. Lett. B. 793hep-phP. P. Novichkov, S. T. Petcov and M. Tanimoto, Phys. Lett. B 793, 247-258 (2019) [arXiv:1812.11289 [hep-ph]].
. T Nomura, H Okada, arXiv:1906.03927Nucl. Phys. B. 966115372hep-phT. Nomura and H. Okada, Nucl. Phys. B 966, 115372 (2021) [arXiv:1906.03927 [hep-ph]].
. H Okada, Y Orikasa, arXiv:1907.13520hep-phH. Okada and Y. Orikasa, [arXiv:1907.13520 [hep-ph]].
. G J Ding, S F King, X G Liu, arXiv:1907.11714JHEP. 0974hep-phG. J. Ding, S. F. King and X. G. Liu, JHEP 09, 074 (2019) [arXiv:1907.11714 [hep-ph]].
. T Nomura, H Okada, O Popov, arXiv:1908.07457Phys. Lett. B. 803135294hep-phT. Nomura, H. Okada and O. Popov, Phys. Lett. B 803, 135294 (2020) [arXiv:1908.07457 [hep-ph]].
. T Kobayashi, Y Shimizu, K Takagi, M Tanimoto, T H Tatsuishi, arXiv:1909.05139Phys. Rev. D. 1001139904Phys. Rev. D. hep-phT. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, Phys. Rev. D 100, no.11, 115045 (2019) [erratum: Phys. Rev. D 101, no.3, 039904 (2020)] [arXiv:1909.05139 [hep-ph]].
. T Asaka, Y Heo, T H Tatsuishi, T Yoshida, arXiv:1909.06520JHEP. 01144hep-phT. Asaka, Y. Heo, T. H. Tatsuishi and T. Yoshida, JHEP 01, 144 (2020) [arXiv:1909.06520 [hep-ph]].
. D Zhang, arXiv:1910.07869Nucl. Phys. B. 952114935hep-phD. Zhang, Nucl. Phys. B 952, 114935 (2020) [arXiv:1910.07869 [hep-ph]].
. G J Ding, S F King, X G Liu, J N Lu, arXiv:1910.03460JHEP. 1230hep-phG. J. Ding, S. F. King, X. G. Liu and J. N. Lu, JHEP 12, 030 (2019) [arXiv:1910.03460 [hep-ph]].
. T Kobayashi, T Nomura, T Shimomura, arXiv:1912.00637Phys. Rev. D. 102335019hep-phT. Kobayashi, T. Nomura and T. Shimomura, Phys. Rev. D 102, no.3, 035019 (2020) [arXiv:1912.00637 [hep-ph]].
. T Nomura, H Okada, S Patra, arXiv:1912.00379Nucl. Phys. B. 967115395hep-phT. Nomura, H. Okada and S. Patra, Nucl. Phys. B 967, 115395 (2021) [arXiv:1912.00379 [hep-ph]].
. X Wang, arXiv:1912.13284Nucl. Phys. B. 957115105hep-phX. Wang, Nucl. Phys. B 957, 115105 (2020) [arXiv:1912.13284 [hep-ph]].
. H Okada, Y Shoji, arXiv:2003.13219Nucl. Phys. B. 961115216hep-phH. Okada and Y. Shoji, Nucl. Phys. B 961, 115216 (2020) [arXiv:2003.13219 [hep-ph]].
. H Okada, M Tanimoto, arXiv:2005.00775hep-phH. Okada and M. Tanimoto, [arXiv:2005.00775 [hep-ph]].
. M K Behera, S Singirala, S Mishra, R Mohanta, arXiv:2009.01806hep-phM. K. Behera, S. Singirala, S. Mishra and R. Mohanta, [arXiv:2009.01806 [hep-ph]].
. M K Behera, S Mishra, S Singirala, R Mohanta, arXiv:2007.00545hep-phM. K. Behera, S. Mishra, S. Singirala and R. Mohanta, [arXiv:2007.00545 [hep-ph]].
. T Nomura, H Okada, arXiv:2007.04801hep-phT. Nomura and H. Okada, [arXiv:2007.04801 [hep-ph]].
. T Nomura, H Okada, arXiv:2007.15459hep-phT. Nomura and H. Okada, [arXiv:2007.15459 [hep-ph]].
. T Asaka, Y Heo, T Yoshida, arXiv:2009.12120Phys. Lett. B. 811135956hep-phT. Asaka, Y. Heo and T. Yoshida, Phys. Lett. B 811, 135956 (2020) [arXiv:2009.12120 [hep-ph]].
. H Okada, M Tanimoto, arXiv:2009.14242Phys. Rev. D. 103115005hepphH. Okada and M. Tanimoto, Phys. Rev. D 103, no.1, 015005 (2021) [arXiv:2009.14242 [hep- ph]].
. K I Nagao, H Okada, arXiv:2010.03348hep-phK. I. Nagao and H. Okada, [arXiv:2010.03348 [hep-ph]].
. H Okada, M Tanimoto, arXiv:2012.01688JHEP. 0310hep-phH. Okada and M. Tanimoto, JHEP 03, 010 (2021) [arXiv:2012.01688 [hep-ph]].
. C Y Yao, J N Lu, G J Ding, arXiv:2012.13390JHEP. 05102hep-phC. Y. Yao, J. N. Lu and G. J. Ding, JHEP 05 (2021), 102 [arXiv:2012.13390 [hep-ph]].
. P Chen, G J Ding, S F King, arXiv:2101.12724JHEP. 04239hep-phP. Chen, G. J. Ding and S. F. King, JHEP 04 (2021), 239 [arXiv:2101.12724 [hep-ph]].
. M Kashav, S Verma, arXiv:2103.07207hep-phM. Kashav and S. Verma, [arXiv:2103.07207 [hep-ph]].
. H Okada, Y Shimizu, M Tanimoto, T Yoshida, arXiv:2105.14292hep-phH. Okada, Y. Shimizu, M. Tanimoto and T. Yoshida, [arXiv:2105.14292 [hep-ph]].
. I De Medeiros Varzielas, J Lourenço, arXiv:2107.04042hep-phI. de Medeiros Varzielas and J. Lourenço, [arXiv:2107.04042 [hep-ph]].
. T Nomura, H Okada, Y Orikasa, arXiv:2106.12375hep-phT. Nomura, H. Okada and Y. Orikasa, [arXiv:2106.12375 [hep-ph]].
. P T P Hutauruk, D W Kang, J Kim, H Okada, arXiv:2012.11156hep-phP. T. P. Hutauruk, D. W. Kang, J. Kim and H. Okada, [arXiv:2012.11156 [hep-ph]].
. G J Ding, S F King, J N Lu, arXiv:2108.09655hep-phG. J. Ding, S. F. King and J. N. Lu, [arXiv:2108.09655 [hep-ph]].
. K I Nagao, H Okada, arXiv:2108.09984hep-phK. I. Nagao and H. Okada, [arXiv:2108.09984 [hep-ph]].
. Georgianna Charalampous, Stephen F King, George K Leontaris, Ye-Ling Zhou, arXiv:2109.11379hep-phGeorgianna Charalampous, Stephen F. King, George K. Leontaris, Ye-Ling Zhou [arXiv:2109.11379 [hep-ph]].
. H Okada, Y H Qi, arXiv:2109.13779hep-phH. Okada and Y. h. Qi, [arXiv:2109.13779 [hep-ph]].
. T Nomura, H Okada, Y H Qi, arXiv:2111.10944hep-phT. Nomura, H. Okada and Y. h. Qi, [arXiv:2111.10944 [hep-ph]].
. T Kobayashi, H Otsuka, M Tanimoto, K Yamamoto, arXiv:2112.00493hep-phT. Kobayashi, H. Otsuka, M. Tanimoto and K. Yamamoto, [arXiv:2112.00493 [hep-ph]].
. A Dasgupta, T Nomura, H Okada, O Popov, M Tanimoto, arXiv:2111.06898hepphA. Dasgupta, T. Nomura, H. Okada, O. Popov and M. Tanimoto, [arXiv:2111.06898 [hep- ph]].
. X G Liu, G J Ding, arXiv:2112.14761hep-phX. G. Liu and G. J. Ding, [arXiv:2112.14761 [hep-ph]].
. T Nomura, H Okada, arXiv:2201.10244hep-phT. Nomura and H. Okada, [arXiv:2201.10244 [hep-ph]].
. T Kobayashi, K Tanaka, T H Tatsuishi, arXiv:1803.10391Phys. Rev. D. 98116004hep-phT. Kobayashi, K. Tanaka and T. H. Tatsuishi, Phys. Rev. D 98, no.1, 016004 (2018) [arXiv:1803.10391 [hep-ph]].
. T Kobayashi, Y Shimizu, K Takagi, M Tanimoto, T H Tatsuishi, H Uchida, arXiv:1812.11072Phys. Lett. B. 794hep-phT. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi and H. Uchida, Phys. Lett. B 794, 114-121 (2019) [arXiv:1812.11072 [hep-ph]].
. T Kobayashi, Y Shimizu, K Takagi, M Tanimoto, T H Tatsuishi, arXiv:1906.10341PTEP. 20205hep-phT. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, PTEP 2020, no.5, 053B05 (2020) [arXiv:1906.10341 [hep-ph]].
. H Okada, Y Orikasa, arXiv:1907.04716Phys. Rev. D. 10011115037hepphH. Okada and Y. Orikasa, Phys. Rev. D 100, no.11, 115037 (2019) [arXiv:1907.04716 [hep- ph]].
. S Mishra, arXiv:2008.02095hep-phS. Mishra, [arXiv:2008.02095 [hep-ph]].
. X Du, F Wang, arXiv:2012.01397JHEP. 02221hep-phX. Du and F. Wang, JHEP 02, 221 (2021) [arXiv:2012.01397 [hep-ph]].
. J T Penedo, S T Petcov, arXiv:1806.11040Nucl. Phys. B. 939hepphJ. T. Penedo and S. T. Petcov, Nucl. Phys. B 939, 292-307 (2019) [arXiv:1806.11040 [hep- ph]].
. P P Novichkov, J T Penedo, S T Petcov, A V Titov, arXiv:1811.04933JHEP. 045hep-phP. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov, JHEP 04, 005 (2019) [arXiv:1811.04933 [hep-ph]].
. T Kobayashi, Y Shimizu, K Takagi, M Tanimoto, T H Tatsuishi, arXiv:1907.09141JHEP. 0297hep-phT. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, JHEP 02, 097 (2020) [arXiv:1907.09141 [hep-ph]].
. S F King, Y L Zhou, arXiv:1908.02770Phys. Rev. D. 101115001hepphS. F. King and Y. L. Zhou, Phys. Rev. D 101, no.1, 015001 (2020) [arXiv:1908.02770 [hep- ph]].
. H Okada, Y Orikasa, arXiv:1908.08409hep-phH. Okada and Y. Orikasa, [arXiv:1908.08409 [hep-ph]].
. J C Criado, F Feruglio, S J D King, arXiv:1908.11867JHEP. 021hep-phJ. C. Criado, F. Feruglio and S. J. D. King, JHEP 02, 001 (2020) [arXiv:1908.11867 [hep-ph]].
. X Wang, S Zhou, arXiv:1910.09473JHEP. 0517hep-phX. Wang and S. Zhou, JHEP 05, 017 (2020) [arXiv:1910.09473 [hep-ph]].
. Y Zhao, H H Zhang, arXiv:2101.02266JHEP. 032hep-phY. Zhao and H. H. Zhang, JHEP 03 (2021), 002 [arXiv:2101.02266 [hep-ph]].
. S F King, Y L Zhou, arXiv:2103.02633JHEP. 04291hep-phS. F. King and Y. L. Zhou, JHEP 04 (2021), 291 [arXiv:2103.02633 [hep-ph]].
. G J Ding, S F King, C Y Yao, arXiv:2103.16311hep-phG. J. Ding, S. F. King and C. Y. Yao, [arXiv:2103.16311 [hep-ph]].
. X Zhang, S Zhou, arXiv:2106.03433hep-phX. Zhang and S. Zhou, [arXiv:2106.03433 [hep-ph]].
Bu-Yao Qu, Xiang-Gan Liu, Ping-Tao Chen, arXiv:2106.11659Gui-Jun Ding. hep-phBu-Yao Qu, Xiang-Gan Liu, Ping-Tao Chen, Gui-Jun Ding [arXiv:2106.11659 [hep-ph]].
. T Nomura, H Okada, arXiv:2109.04157hep-phT. Nomura and H. Okada, [arXiv:2109.04157 [hep-ph]].
. P P Novichkov, J T Penedo, S T Petcov, A V Titov, arXiv:1812.02158JHEP. 04174hep-phP. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov, JHEP 04, 174 (2019) [arXiv:1812.02158 [hep-ph]].
. G J Ding, S F King, X G Liu, arXiv:1903.12588Phys. Rev. D. 10011115005hep-phG. J. Ding, S. F. King and X. G. Liu, Phys. Rev. D 100, no.11, 115005 (2019) [arXiv:1903.12588 [hep-ph]].
. X G Liu, G J Ding, arXiv:1907.01488JHEP. 08134hep-phX. G. Liu and G. J. Ding, JHEP 08, 134 (2019) [arXiv:1907.01488 [hep-ph]].
. P Chen, G J Ding, J N Lu, J W F Valle, arXiv:2003.02734Phys. Rev. D. 102995014hep-phP. Chen, G. J. Ding, J. N. Lu and J. W. F. Valle, Phys. Rev. D 102, no.9, 095014 (2020) [arXiv:2003.02734 [hep-ph]].
. C C Li, X G Liu, G J Ding, arXiv:2108.02181hep-phC. C. Li, X. G. Liu and G. J. Ding, [arXiv:2108.02181 [hep-ph]].
. P P Novichkov, J T Penedo, S T Petcov, arXiv:2006.03058Nucl. Phys. B. 963115301hep-phP. P. Novichkov, J. T. Penedo and S. T. Petcov, Nucl. Phys. B 963, 115301 (2021) [arXiv:2006.03058 [hep-ph]].
. X G Liu, C Y Yao, G J Ding, arXiv:2006.10722Phys. Rev. D. 103556013hep-phX. G. Liu, C. Y. Yao and G. J. Ding, Phys. Rev. D 103, no.5, 056013 (2021) [arXiv:2006.10722 [hep-ph]].
. X Wang, B Yu, S Zhou, arXiv:2010.10159Phys. Rev. D. 103776005hep-phX. Wang, B. Yu and S. Zhou, Phys. Rev. D 103, no.7, 076005 (2021) [arXiv:2010.10159 [hep-ph]].
. C Y Yao, X G Liu, G J Ding, arXiv:2011.03501Phys. Rev. D. 103995013hep-phC. Y. Yao, X. G. Liu and G. J. Ding, Phys. Rev. D 103, no.9, 095013 (2021) [arXiv:2011.03501 [hep-ph]].
. X Wang, S Zhou, arXiv:2102.04358hep-phX. Wang and S. Zhou, [arXiv:2102.04358 [hep-ph]].
. M K Behera, R Mohanta, arXiv:2108.01059hep-phM. K. Behera and R. Mohanta, [arXiv:2108.01059 [hep-ph]].
. I De Medeiros, S F Varzielas, Y L King, Zhou, arXiv:1906.02208Phys. Rev. D. 101555033hep-phI. de Medeiros Varzielas, S. F. King and Y. L. Zhou, Phys. Rev. D 101, no.5, 055033 (2020) [arXiv:1906.02208 [hep-ph]].
. T Kobayashi, S Tamba, arXiv:1811.11384Phys. Rev. D. 99446001hepthT. Kobayashi and S. Tamba, Phys. Rev. D 99, no.4, 046001 (2019) [arXiv:1811.11384 [hep- th]].
. S Kikuchi, T Kobayashi, H Otsuka, S Takada, H Uchida, arXiv:2007.06188JHEP. 11101hep-thS. Kikuchi, T. Kobayashi, H. Otsuka, S. Takada and H. Uchida, JHEP 11, 101 (2020) [arXiv:2007.06188 [hep-th]].
. Y Almumin, M C Chen, V Knapp-Pérez, S Ramos-Sánchez, M Ratz, S Shukla, arXiv:2102.11286JHEP. 0578hep-thY. Almumin, M. C. Chen, V. Knapp-Pérez, S. Ramos-Sánchez, M. Ratz and S. Shukla, JHEP 05 (2021), 078 [arXiv:2102.11286 [hep-th]].
. G J Ding, F Feruglio, X G Liu, arXiv:2102.06716SciPost Phys. 10133hep-phG. J. Ding, F. Feruglio and X. G. Liu, SciPost Phys. 10 (2021), 133 [arXiv:2102.06716 [hep-ph]].
. F Feruglio, V Gherardi, A Romanino, A Titov, arXiv:2101.08718JHEP. 05242hep-phF. Feruglio, V. Gherardi, A. Romanino and A. Titov, JHEP 05 (2021), 242 [arXiv:2101.08718 [hep-ph]].
. S Kikuchi, T Kobayashi, H Uchida, arXiv:2101.00826hep-thS. Kikuchi, T. Kobayashi and H. Uchida, [arXiv:2101.00826 [hep-th]].
. P P Novichkov, J T Penedo, S T Petcov, arXiv:2102.07488JHEP. 04206hep-phP. P. Novichkov, J. T. Penedo and S. T. Petcov, JHEP 04 (2021), 206 [arXiv:2102.07488 [hep-ph]].
. S Kikuchi, T Kobayashi, Y Ogawa, H Uchida, arXiv:2112.01680hep-phS. Kikuchi, T. Kobayashi, Y. Ogawa and H. Uchida, [arXiv:2112.01680 [hep-ph]].
. P P Novichkov, J T Penedo, S T Petcov, arXiv:2201.02020hep-phP. P. Novichkov, J. T. Penedo and S. T. Petcov, [arXiv:2201.02020 [hep-ph]].
. G Altarelli, F Feruglio, arXiv:1002.0211Rev. Mod. Phys. 82hepphG. Altarelli and F. Feruglio, Rev. Mod. Phys. 82, 2701-2729 (2010) [arXiv:1002.0211 [hep- ph]].
. H Ishimori, T Kobayashi, H Ohki, Y Shimizu, H Okada, M Tanimoto, arXiv:1003.3552Prog. Theor. Phys. Suppl. 183hep-thH. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, Prog. Theor. Phys. Suppl. 183, 1-163 (2010) [arXiv:1003.3552 [hep-th]].
. H Ishimori, T Kobayashi, H Ohki, H Okada, Y Shimizu, M Tanimoto, Lect. Notes Phys. 858H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto, Lect. Notes Phys. 858, 1-227 (2012)
. D Hernandez, A Y Smirnov, arXiv:1204.0445Phys. Rev. D. 8653014hep-phD. Hernandez and A. Y. Smirnov, Phys. Rev. D 86, 053014 (2012) [arXiv:1204.0445 [hep-ph]].
. S F King, C Luhn, arXiv:1301.1340Rept. Prog. Phys. 7656201hep-phS. F. King and C. Luhn, Rept. Prog. Phys. 76, 056201 (2013) [arXiv:1301.1340 [hep-ph]].
. S F King, A Merle, S Morisi, Y Shimizu, M Tanimoto, arXiv:1402.4271New J. Phys. 1645018hep-phS. F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto, New J. Phys. 16, 045018 (2014) [arXiv:1402.4271 [hep-ph]].
. S F King, arXiv:1701.04413Prog. Part. Nucl. Phys. 94hep-phS. F. King, Prog. Part. Nucl. Phys. 94, 217-256 (2017) [arXiv:1701.04413 [hep-ph]].
. S T Petcov, arXiv:1711.10806Eur. Phys. J. C. 789hep-phS. T. Petcov, Eur. Phys. J. C 78, no.9, 709 (2018) [arXiv:1711.10806 [hep-ph]].
. A Baur, H P Nilles, A Trautner, P K S Vaudrevange, arXiv:1908.00805Nucl. Phys. B. 947114737hep-thA. Baur, H. P. Nilles, A. Trautner and P. K. S. Vaudrevange, Nucl. Phys. B 947, 114737 (2019) [arXiv:1908.00805 [hep-th]].
. T Kobayashi, Y Shimizu, K Takagi, M Tanimoto, T H Tatsuishi, H Uchida, arXiv:1910.11553Phys. Rev. D. 101555046hep-phT. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi and H. Uchida, Phys. Rev. D 101, no.5, 055046 (2020) [arXiv:1910.11553 [hep-ph]].
. P P Novichkov, J T Penedo, S T Petcov, A V Titov, arXiv:1905.11970JHEP. 07165hep-phP. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov, JHEP 07, 165 (2019) [arXiv:1905.11970 [hep-ph]].
. A Baur, H P Nilles, A Trautner, P K S Vaudrevange, arXiv:1901.03251Phys. Lett. B. 795hep-thA. Baur, H. P. Nilles, A. Trautner and P. K. S. Vaudrevange, Phys. Lett. B 795, 7-14 (2019) [arXiv:1901.03251 [hep-th]].
. T Kobayashi, H Otsuka, arXiv:2001.07972Phys. Rev. D. 10110106017hep-thT. Kobayashi and H. Otsuka, Phys. Rev. D 101, no.10, 106017 (2020) [arXiv:2001.07972 [hep-th]].
. T Kobayashi, H Otsuka, arXiv:2004.04518Phys. Rev. D. 102226004hep-thT. Kobayashi and H. Otsuka, Phys. Rev. D 102, no.2, 026004 (2020) [arXiv:2004.04518 [hep-th]].
. K Ishiguro, T Kobayashi, H Otsuka, arXiv:2010.10782Nucl. Phys. B. 973115598hep-thK. Ishiguro, T. Kobayashi and H. Otsuka, Nucl. Phys. B 973, 115598 (2021) [arXiv:2010.10782 [hep-th]].
. M Tanimoto, K Yamamoto, arXiv:2106.10919hep-phM. Tanimoto and K. Yamamoto, [arXiv:2106.10919 [hep-ph]].
. K Ishiguro, T Kobayashi, H Otsuka, arXiv:2107.00487JHEP. 0120hep-thK. Ishiguro, T. Kobayashi and H. Otsuka, JHEP 01, 020 (2022) [arXiv:2107.00487 [hep-th]].
. M C Chen, S Ramos-Sánchez, M Ratz, arXiv:1909.06910Phys. Lett. B. 801135153hep-phM. C. Chen, S. Ramos-Sánchez and M. Ratz, Phys. Lett. B 801, 135153 (2020) [arXiv:1909.06910 [hep-ph]].
. I De Medeiros, M Varzielas, Y L Levy, Zhou, arXiv:2008.05329JHEP. 1185hep-phI. de Medeiros Varzielas, M. Levy and Y. L. Zhou, JHEP 11, 085 (2020) [arXiv:2008.05329 [hep-ph]].
. K Ishiguro, T Kobayashi, H Otsuka, arXiv:2011.09154JHEP. 03161hep-phK. Ishiguro, T. Kobayashi and H. Otsuka, JHEP 03, 161 (2021) [arXiv:2011.09154 [hep-ph]].
. H Abe, T Kobayashi, S Uemura, J Yamamoto, arXiv:2003.03512Phys. Rev. D. 102445005hep-thH. Abe, T. Kobayashi, S. Uemura and J. Yamamoto, Phys. Rev. D 102, no.4, 045005 (2020) [arXiv:2003.03512 [hep-th]].
. P P Novichkov, J T Penedo, S T Petcov, arXiv:2201.02020hep-phP. P. Novichkov, J. T. Penedo and S. T. Petcov, [arXiv:2201.02020 [hep-ph]].
. S Kikuchi, T Kobayashi, H Otsuka, M Tanimoto, H Uchida, K Yamamoto, arXiv:2201.04505hep-phS. Kikuchi, T. Kobayashi, H. Otsuka, M. Tanimoto, H. Uchida and K. Yamamoto, [arXiv:2201.04505 [hep-ph]].
. T Kobayashi, H Otsuka, arXiv:2108.02700Eur. Phys. J. C. 82125hep-phT. Kobayashi and H. Otsuka, Eur. Phys. J. C 82, no.1, 25 (2022) [arXiv:2108.02700 [hep-ph]].
. S Kikuchi, T Kobayashi, K Nasu, H Uchida, S Uemura, arXiv:2202.05425hep-thS. Kikuchi, T. Kobayashi, K. Nasu, H. Uchida and S. Uemura, [arXiv:2202.05425 [hep-th]].
. Y Kajiyama, H Okada, K Yagyu, arXiv:1303.3463Nucl. Phys. B. 874hep-phY. Kajiyama, H. Okada and K. Yagyu, Nucl. Phys. B 874 (2013), 198-216 [arXiv:1303.3463 [hep-ph]].
. Y Kajiyama, H Okada, T Toma, arXiv:1303.7356Phys. Rev. D. 88115029hep-phY. Kajiyama, H. Okada and T. Toma, Phys. Rev. D 88 (2013) no.1, 015029 [arXiv:1303.7356 [hep-ph]].
. V S Kaplunovsky, J Louis, arXiv:hep-th/9303040Phys. Lett. B. 306hep-thV. S. Kaplunovsky and J. Louis, Phys. Lett. B 306 (1993), 269-275 [arXiv:hep-th/9303040 [hep-th]].
. N Arkani-Hamed, L J Hall, H Murayama, D Tucker-Smith, N Weiner, arXiv:hep-ph/0006312Phys. Rev. D. 64115011hep-phN. Arkani-Hamed, L. J. Hall, H. Murayama, D. Tucker-Smith and N. Weiner, Phys. Rev. D 64 (2001), 115011 [arXiv:hep-ph/0006312 [hep-ph]].
. T Kobayashi, T Shimomura, M Tanimoto, arXiv:2102.10425Phys. Lett. B. 819136452hep-phT. Kobayashi, T. Shimomura and M. Tanimoto, Phys. Lett. B 819, 136452 (2021) [arXiv:2102.10425 [hep-ph]].
. L Megrelidze, Z Tavartkiladze, arXiv:1609.07344Nucl. Phys. B. 914hep-phL. Megrelidze and Z. Tavartkiladze, Nucl. Phys. B 914 (2017), 553-576 [arXiv:1609.07344 [hep-ph]].
. N Aghanim, Planck CollaborationarXiv:1807.06209[astro-ph.CON. Aghanim et al. [Planck Collaboration], arXiv:1807.06209 [astro-ph.CO].
. M Tanabashi, Particle Data GroupPhys. Rev. D. 9830001M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018).
. A Gando, KamLAND-Zen CollaborationarXiv:1605.02889Phys. Rev. Lett. 1178109903Phys. Rev. Lett.. hep-exA. Gando et al. [KamLAND-Zen Collaboration], Phys. Rev. Lett. 117, no. 8, 082503 (2016) Addendum: [Phys. Rev. Lett. 117, no. 10, 109903 (2016)] [arXiv:1605.02889 [hep-ex]].
. I Esteban, M C Gonzalez-Garcia, A Hernandez-Cabezudo, M Maltoni, T Schwetz, arXiv:1811.05487JHEP. 1901106hep-phI. Esteban, M. C. Gonzalez-Garcia, A. Hernandez-Cabezudo, M. Maltoni and T. Schwetz, JHEP 1901, 106 (2019) [arXiv:1811.05487 [hep-ph]].
. T Hambye, F S Ling, L Lopez Honorez, J Rocher, arXiv:0903.4010JHEP. 0790erratum: JHEP 05 (2010), 066. hep-phT. Hambye, F. S. Ling, L. Lopez Honorez and J. Rocher, JHEP 07 (2009), 090 [erratum: JHEP 05 (2010), 066] [arXiv:0903.4010 [hep-ph]].
. S Kanemura, S Matsumoto, T Nabeshima, N Okada, arXiv:1005.5651Phys. Rev. D. 8255026hep-phS. Kanemura, S. Matsumoto, T. Nabeshima and N. Okada, Phys. Rev. D 82, 055026 (2010) [arXiv:1005.5651 [hep-ph]].
| [] |
[
"Phonon renormalisation in doped bilayer graphene",
"Phonon renormalisation in doped bilayer graphene"
] | [
"A Das \nDepartment of Physics\nIndian Institute of Science\n560012BangaloreIndia\n",
"B Chakraborty \nDepartment of Physics\nIndian Institute of Science\n560012BangaloreIndia\n",
"S Piscanec \nEngineering Department\nCambridge University\nCB3 0FACambridgeUK\n",
"S Pisana \nEngineering Department\nCambridge University\nCB3 0FACambridgeUK\n",
"A K Sood \nDepartment of Physics\nIndian Institute of Science\n560012BangaloreIndia\n",
"A C Ferrari \nEngineering Department\nCambridge University\nCB3 0FACambridgeUK\n"
] | [
"Department of Physics\nIndian Institute of Science\n560012BangaloreIndia",
"Department of Physics\nIndian Institute of Science\n560012BangaloreIndia",
"Engineering Department\nCambridge University\nCB3 0FACambridgeUK",
"Engineering Department\nCambridge University\nCB3 0FACambridgeUK",
"Department of Physics\nIndian Institute of Science\n560012BangaloreIndia",
"Engineering Department\nCambridge University\nCB3 0FACambridgeUK"
] | [] | We report phonon renormalisation in bilayer graphene as a function of doping. The Raman G peak stiffens and sharpens for both electron and hole doping, as a result of the non-adiabatic Kohn anomaly at the Γ point. The bilayer has two conduction and valence subbands, with splitting dependent on the interlayer coupling. This results in a change of slope in the variation of G peak position with doping, which allows a direct measurement of the interlayer coupling strength.Graphene is the latest carbon allotrope to be discovered[1,2,3,4,5]. Near-ballistic transport at room temperature and high carrier mobilities[2,3,4,5,6,7,8], make it a potential material for nanoelectronics[9,10,11], especially for high frequency applications. It is now possible to produce areas exceeding thousands of square microns by means of micro-mechanical cleavage of graphite. An ongoing effort is being devoted to large scale deposition and growth on different substrates of choice.Unlike single layer graphene (SLG), where electrons disperse linearly as massless Dirac fermions[1, 2, 3, 4, 5], bilayer graphene (BLG) has two conduction and valence bands, separated by γ 1 , the interlayer coupling[12,13]. This was measured to be∼0.39eV by angle resolved photoelectron spectroscopy[14]. A gap between valence and conduction bands could be opened and tuned by an external electric field (∼100meV for∼10 13 cm −2 doping)[15,16], making BLG a tunable-gap semiconductor.Graphene can be identified in terms of number and orientation of layers by means of elastic and inelastic light scattering, such as Raman[17]and Rayleigh spectroscopies[18,19]. Raman spectroscopy also allows V TGLaserSpectrometerX ObjectiveSLG BLG Si Si SiO SiO 2 2 Pt A u A u FIG. 1: (color online). Experimental setup. The black dotted box on SiO2 indicates the polymer electrolyte (PEO + LiClO4). The left inset shows an SEM image of the SLG and BLG. Scale bar: 4µm. The right inset, the 2D Raman band monitoring of doping and defects[4,20,21,22,23,24,25]. Indeed, Raman spectroscopy is a fast and non-destructive characterization method for carbons[26]. They show common features in the 800-2000 cm −1 region: the G and D peaks, around 1580 and 1350 cm −1 , respectively. The G peak corresponds to the E 2g phonon at the Brillouin zone center (Γ). The D peak is due to the breathing modes of sp 2 atoms and requires a defect for its activation[27,28,29]. The most prominent feature in SLG is the second order of the D peak: the 2D peak[17]. This lies at ∼ 2700 cm −1 and involves phonons at K+∆q[17,23]. ∆q depends on the excitation energy, due to double-resonance, and the linear dispersion of the phonons around K[17, 29, 30]. 2D is a single peak in SLG, whereas it splits in four in BLG, reflecting the evolution of the band structure[17]. The 2D peak is always seen, even when no D peak is present, since no defects are required for overtone activation.In SLG, the effects of back and top gating on Gpeak position (Pos(G)) and Full Width at Half Maximum (FWHM(G)) were reported in Refs[20,21,24]. Pos(G) increases and FWHM(G) decreases for both electron and hole doping. The G peak stiffening is due to the non-adiabatic removal of the Kohn-anomaly at Γ[20,31]. FWHM(G) sharpening is due to blockage of phonon decay into electron-hole pairs due to the Pauli exclusion principle, when the electron-hole gap is higher than the phonon energy[20,32], and saturates for a Fermi shift bigger than half phonon energy[20,21,32]. A similar behavior is observed for the LO-G − peak in metallic nanotubes[33], for the same reasons. The conceptually different BLG band structure is expected to renormalize the phonon response to doping differently from SLG[13,34]. Here we prove this, by investigating the effect of doping on the BLG G and 2D peaks. The G peak of doped BLG was recently investigated[35], and reproduced that of SLG, due to the very low doping range(∼ 5 × 10 12 cm −2 ), not enough to cross the second BLG subband. Here we reach much higher values (∼ 5×10 13 cm −2 ), probing the further renormalisation resulting from crossing to the second BLG subband.We recently demonstrated a SLG top-gated by poly-arXiv:0807.1631v1 [cond-mat.mes-hall] | 10.1103/physrevb.79.155417 | [
"https://arxiv.org/pdf/0807.1631v1.pdf"
] | 118,413,730 | 0807.1631 | 43cb6139b6c422fc8f8b7108db88fe117c86dfc0 |
Phonon renormalisation in doped bilayer graphene
10 Jul 2008
A Das
Department of Physics
Indian Institute of Science
560012BangaloreIndia
B Chakraborty
Department of Physics
Indian Institute of Science
560012BangaloreIndia
S Piscanec
Engineering Department
Cambridge University
CB3 0FACambridgeUK
S Pisana
Engineering Department
Cambridge University
CB3 0FACambridgeUK
A K Sood
Department of Physics
Indian Institute of Science
560012BangaloreIndia
A C Ferrari
Engineering Department
Cambridge University
CB3 0FACambridgeUK
Phonon renormalisation in doped bilayer graphene
10 Jul 2008
We report phonon renormalisation in bilayer graphene as a function of doping. The Raman G peak stiffens and sharpens for both electron and hole doping, as a result of the non-adiabatic Kohn anomaly at the Γ point. The bilayer has two conduction and valence subbands, with splitting dependent on the interlayer coupling. This results in a change of slope in the variation of G peak position with doping, which allows a direct measurement of the interlayer coupling strength.Graphene is the latest carbon allotrope to be discovered[1,2,3,4,5]. Near-ballistic transport at room temperature and high carrier mobilities[2,3,4,5,6,7,8], make it a potential material for nanoelectronics[9,10,11], especially for high frequency applications. It is now possible to produce areas exceeding thousands of square microns by means of micro-mechanical cleavage of graphite. An ongoing effort is being devoted to large scale deposition and growth on different substrates of choice.Unlike single layer graphene (SLG), where electrons disperse linearly as massless Dirac fermions[1, 2, 3, 4, 5], bilayer graphene (BLG) has two conduction and valence bands, separated by γ 1 , the interlayer coupling[12,13]. This was measured to be∼0.39eV by angle resolved photoelectron spectroscopy[14]. A gap between valence and conduction bands could be opened and tuned by an external electric field (∼100meV for∼10 13 cm −2 doping)[15,16], making BLG a tunable-gap semiconductor.Graphene can be identified in terms of number and orientation of layers by means of elastic and inelastic light scattering, such as Raman[17]and Rayleigh spectroscopies[18,19]. Raman spectroscopy also allows V TGLaserSpectrometerX ObjectiveSLG BLG Si Si SiO SiO 2 2 Pt A u A u FIG. 1: (color online). Experimental setup. The black dotted box on SiO2 indicates the polymer electrolyte (PEO + LiClO4). The left inset shows an SEM image of the SLG and BLG. Scale bar: 4µm. The right inset, the 2D Raman band monitoring of doping and defects[4,20,21,22,23,24,25]. Indeed, Raman spectroscopy is a fast and non-destructive characterization method for carbons[26]. They show common features in the 800-2000 cm −1 region: the G and D peaks, around 1580 and 1350 cm −1 , respectively. The G peak corresponds to the E 2g phonon at the Brillouin zone center (Γ). The D peak is due to the breathing modes of sp 2 atoms and requires a defect for its activation[27,28,29]. The most prominent feature in SLG is the second order of the D peak: the 2D peak[17]. This lies at ∼ 2700 cm −1 and involves phonons at K+∆q[17,23]. ∆q depends on the excitation energy, due to double-resonance, and the linear dispersion of the phonons around K[17, 29, 30]. 2D is a single peak in SLG, whereas it splits in four in BLG, reflecting the evolution of the band structure[17]. The 2D peak is always seen, even when no D peak is present, since no defects are required for overtone activation.In SLG, the effects of back and top gating on Gpeak position (Pos(G)) and Full Width at Half Maximum (FWHM(G)) were reported in Refs[20,21,24]. Pos(G) increases and FWHM(G) decreases for both electron and hole doping. The G peak stiffening is due to the non-adiabatic removal of the Kohn-anomaly at Γ[20,31]. FWHM(G) sharpening is due to blockage of phonon decay into electron-hole pairs due to the Pauli exclusion principle, when the electron-hole gap is higher than the phonon energy[20,32], and saturates for a Fermi shift bigger than half phonon energy[20,21,32]. A similar behavior is observed for the LO-G − peak in metallic nanotubes[33], for the same reasons. The conceptually different BLG band structure is expected to renormalize the phonon response to doping differently from SLG[13,34]. Here we prove this, by investigating the effect of doping on the BLG G and 2D peaks. The G peak of doped BLG was recently investigated[35], and reproduced that of SLG, due to the very low doping range(∼ 5 × 10 12 cm −2 ), not enough to cross the second BLG subband. Here we reach much higher values (∼ 5×10 13 cm −2 ), probing the further renormalisation resulting from crossing to the second BLG subband.We recently demonstrated a SLG top-gated by poly-arXiv:0807.1631v1 [cond-mat.mes-hall]
We report phonon renormalisation in bilayer graphene as a function of doping. The Raman G peak stiffens and sharpens for both electron and hole doping, as a result of the non-adiabatic Kohn anomaly at the Γ point. The bilayer has two conduction and valence subbands, with splitting dependent on the interlayer coupling. This results in a change of slope in the variation of G peak position with doping, which allows a direct measurement of the interlayer coupling strength.
PACS numbers: 73. 63.-b, 63. 20.Kr, 81. 05.Uw, 78. 30.Na, Graphene is the latest carbon allotrope to be discovered [1,2,3,4,5]. Near-ballistic transport at room temperature and high carrier mobilities [2,3,4,5,6,7,8], make it a potential material for nanoelectronics [9,10,11], especially for high frequency applications. It is now possible to produce areas exceeding thousands of square microns by means of micro-mechanical cleavage of graphite. An ongoing effort is being devoted to large scale deposition and growth on different substrates of choice.
Unlike single layer graphene (SLG), where electrons disperse linearly as massless Dirac fermions [1,2,3,4,5], bilayer graphene (BLG) has two conduction and valence bands, separated by γ 1 , the interlayer coupling [12,13]. This was measured to be∼0.39eV by angle resolved photoelectron spectroscopy [14]. A gap between valence and conduction bands could be opened and tuned by an external electric field (∼100meV for∼10 13 cm −2 doping) [15,16], making BLG a tunable-gap semiconductor.
Graphene can be identified in terms of number and orientation of layers by means of elastic and inelastic light scattering, such as Raman [17] and Rayleigh spectroscopies [18,19]. Raman spectroscopy also allows V TG [4,20,21,22,23,24,25]. Indeed, Raman spectroscopy is a fast and non-destructive characterization method for carbons [26]. They show common features in the 800-2000 cm −1 region: the G and D peaks, around 1580 and 1350 cm −1 , respectively. The G peak corresponds to the E 2g phonon at the Brillouin zone center (Γ). The D peak is due to the breathing modes of sp 2 atoms and requires a defect for its activation [27,28,29]. The most prominent feature in SLG is the second order of the D peak: the 2D peak [17]. This lies at ∼ 2700 cm −1 and involves phonons at K+∆q [17,23]. ∆q depends on the excitation energy, due to double-resonance, and the linear dispersion of the phonons around K [17,29,30]. 2D is a single peak in SLG, whereas it splits in four in BLG, reflecting the evolution of the band structure [17]. The 2D peak is always seen, even when no D peak is present, since no defects are required for overtone activation.
Laser
Spectrometer
X
In SLG, the effects of back and top gating on Gpeak position (Pos(G)) and Full Width at Half Maximum (FWHM(G)) were reported in Refs [20,21,24]. Pos(G) increases and FWHM(G) decreases for both electron and hole doping. The G peak stiffening is due to the non-adiabatic removal of the Kohn-anomaly at Γ [20,31]. FWHM(G) sharpening is due to blockage of phonon decay into electron-hole pairs due to the Pauli exclusion principle, when the electron-hole gap is higher than the phonon energy [20,32], and saturates for a Fermi shift bigger than half phonon energy [20,21,32]. A similar behavior is observed for the LO-G − peak in metallic nanotubes [33], for the same reasons. The conceptually different BLG band structure is expected to renormalize the phonon response to doping differently from SLG [13,34]. Here we prove this, by investigating the effect of doping on the BLG G and 2D peaks. The G peak of doped BLG was recently investigated [35], and reproduced that of SLG, due to the very low doping range(∼ 5 × 10 12 cm −2 ), not enough to cross the second BLG subband. Here we reach much higher values (∼ 5×10 13 cm −2 ), probing the further renormalisation resulting from crossing to the second BLG subband.
We recently demonstrated a SLG top-gated by poly-arXiv:0807.1631v1 [cond-mat.mes-hall] 10 Jul 2008 mer electrolyte [24], able to span a large doping range, up to∼5×10 13 cm −2 [24]. This is possible because the nanometer thick Debye layer [24,36,37] gives a much higher gate capacitance compared to the usual 300nm SiO 2 back gate [5]. We apply here this approach to BLG. Fig.1 shows the scheme of our experiment. A sample is produced by micromechanical cleavage of graphite. This consists of a SLG extending to a BLG, as proven by the characteristic SLG and BLG 2D peaks in the inset of Fig.1 [17]. An Au electrode is then deposited by photolithography covering both SLG and BLG, Fig.1. Top gating is achieved by using a solid polymer electrolyte consisting of LiClO 4 and polyethelyne oxide (PEO) in the ratio 0.12:1 [24]. The gate voltage is applied by placing a platinum electrode in the polymer layer. Note that the particular shape of our sample, consisting of a BLG, with a protruding SLG, ensures the top gate to be effectively applied to both layers at the same time. This would not necessarily be the case for a monolithic BLG, where, due to screening effects, the gate would give a separate evolution of the Raman spectra of the top and bottom layers [38]. Measurements are done with a WITEC confocal (X50 objective) spectrometer with 600 lines/mm grating, 514.5 nm excitation, at<1mW to avoid heating. For a given top gate voltage, V T G , spectra are recorded after 10 mins. Figs.2(a,b) plot the spectra as a function of V T G . We use Voigt functions to fit the G peak in both SLG and BLG. The SLG 2D band is fitted to one Lorentzian. The BLG 2D band is fitted to four Lorentzians,2D 1A ,2D 1B ,2D 2A ,2D 2B [17], Fig.1. As previously discussed, two of these, 2D 1A and 2D 2A , are much stronger [17]. Thus, we focus on these.
To get a quantitative understanding, it is necessary to convert V T G into a E F shift. For electrolytic gating, the chemical potential is eV T G = E SLG F + eφ SLG = E BLG F + eφ BLG . The electrostatic potential φ = ne C T G is determined by the geometrical capacitance C T G and carrier concentration n (e is the electron charge), while E F /e by the chemical (quantum) capacitance of graphene. For
SLG, n SLG = µE 2 F , where µ = gsgv 4πγ 2 = 1 π( v F ) 2 , g s =g v =2 are spin and valley degeneracies, γ = √ 3
2 γ 0 a, with γ 0 the nearest-neighbor tight binding parameter, a the graphene lattice parameter, and v F is the Fermi velocity. Thus:
eV T G = E F + νE 2 F(1)
For BLG [13,39,40]
n BLG =µ[γ 1 E F + E 2 F ] for E F < γ 1 and n BLG = 2µE 2 F for E F > γ 1 . Thus: eV T G = (1 + νγ 1 )E F + νE 2 F , E F < γ 1 (2) = E F + 2νE 2 F , E F > γ 1 where ν = e 2 πC T G ( v F ) 2 .
We take C T G = 2.2 × 10 −6 Fcm −2 [33], and γ 1 =0.39eV constant with doping (since its variation for n up to∼10 13 cm −2 is <5%) [14,15]). Eqs.1,2 then give E F as a function of V T G . Fig.3 plots the resulting Pos(G), FWHM(G) as a function of E F . In SLG, Pos(G) does not increase up to E F ∼0.1eV (∼ ω 0 /2), where ω 0 is the frequency of the E 2g phonon in the undoped case ( ω 0 /(2π c)=Pos(G 0 ), with c the speed of light), and then increases with E F . Fig.3b,d indicate that in SLG and BLG, FWHM(G) decreases for both electron and hole doping, as expected since phonons decay into real electron-hole pairs when E F < ω 0 /2 [20]. Fig.3c plots Pos(G) of BLG.(i) Pos(G) does not increase until E F ∼0.1eV (∼ ω 0 /2).(ii) Between 0.1 and 0.4eV, the BLG slope R= dP os(G) dE F is smaller than the SLG one.(iii) A kink is observed in Fig.3b at E F ∼0.4eV.(iv)Beyond E F >0.4eV the slope is larger than in SLG.(v) The kink position does not significantly depend on γ 1 used to convert V T G in E F (e.g. a ∼ 66% change in γ 1 modifies E F by ∼ 6%).
These trends can be explained by considering the effects doping on the phonons:(i) a change of the equilib-rium lattice parameter with a consequent "static" stiffening/softening, ∆P os(G) st ;(ii) the onset of "dynamic" effects beyond the adiabatic Born-Oppenheimer approximation, that modify the phonon dispersion close to the Kohn anomalies, ∆P os(G) dyn [20,31]. Thus, the total phonon renormalization can be written as [20,31]:
Pos(G E F )−Pos(G 0 ) = ∆P os(G) = ∆P os(G) st +∆P os(G) dyn
(3) For SLG, we compute ∆P os(G) st by converting E F into the corresponding electron density n SLG , then using Eq.3 of Ref. [31]. For BLG, we assume n BLG equally distributed on the two layers, each behaving as a SLG with an electron concentration n BLG /2. Eq.3 of Ref. [31] is then used to compute ∆P os(G) st for BLG. ∆P os(G) dyn is calculated from the phonon self-energy Π [41]:
∆P os(G) dyn = Re[Π(E F ) − Π(E F = 0)].(4)
The electron-phonon coupling (EPC) contribution to FWHM(G) is given by [41,42,43]:
FWHM(G) EP C = 2Im[Π(E F )](5)
The self-energy for the E 2g mode at Γ in SLG is [20,31]:
Π(E F ) SLG = α ∞ −∞ f ( ) − f (− ) 2 + ω 0 + iδ | |d ,(6)
while for BLG it is given by [13]:
Π(E F ) BLG = α ∞ 0 γ 2 kdk s,s j,j φ + jj × [f ( sjk ) − f ( s j k )][ sjk − s j k ] ( sjk − s j k ) 2 − ( ω 0 + iδ) 2 (7)
where α = AucEP C(Γ) 2 πM ω0( v F ) 2 , A uc = 5.24Å 2 is the graphene unit-cell area, M is the carbon atom mass, f ( ) = 1/[exp( −E F k B T ) + 1] is the Fermi-Dirac distribution, δ is a broadening factor accounting for charge inhomogeneity, EPC(Γ) is the electron phonon coupling [46]. s = ±1 and s =±1 label the conduction (+1) and valence (-1) bands, while j = 1, 2 and j =1,2 label the two parabolic subbands. sjk is computed from Eq.2.8 of Ref. [13], and φ + jj is given by Eq. 3.1 of Ref. [13]. By using Eqs.6,7 in Eqs.4,5, we get ∆P os(G) dyn ,FWHM(G) EP C for SLG and BLG.
To compare Eqs.3,5 with the experimental data, we use α = 4.4 × 10 −3 (obtained from the DFT values of EPC(Γ) and v F [20,30]), the experimental ω 0 for SLG and BLG, and T=300K. δ is fitted from the experimental FWHM(G) to FWHM(G)=FWHM(G) EP C +FWHM(G) 0 , with FWHM(G) 0 a constant accounting for non-EPC effects (e.g. resolution and anharmonicity). For SLG (BLG) we get δ = 0.13eV (0.03eV) and FWHM(G) 0 =4.3cm −1 (5.1cm −1 ). These δ values are then used to compute Pos(G). Note that the relation between n and E F FIG. 4: (color online). Phonon renormalization for BLG: (I) EF < ω0, (II) ω0 < EF < γ1, (III) EF > γ1. Blue and red arrows correspond respectively to positive and negative contributions to Π. Solid and dashed arrows correspond to interband and intraband processes respectively. implies that charge inhomogeneity causes different E F broadening in SLG and BLG (e.g. δn∼10 12 cm −2 would give 0.13eV and 0.03eV in SLG and BLG,respectively).
The solid lines in Fig. 3 are the theoretical Pos(G) and FWHM(G) trends. The experimental and theoretical FWHM(G) are in excellent agreement, as expected since the latter was fitted to the former. The theoretical Pos(G) captures the main experimental features. In particular, the flat dependence for |E f | < 0.1 eV in both SLG and BLG, and the kink at ∼ 0.4 eV in BLG. This kink is the most striking difference between SLG and BLG. It is the signature of the second subband filling in BLG. Indeed, a shift of E F , by acting on f ( ) in Eq.7, modifies the type and number of transitions contributing to Π. The only transitions giving a positive contribution to Π are those for which | s,j,k − s ,j ,k | < ω 0 , i.e. a subset of those between (s = −1; j = 1) and (s = 1; j = 1) (interband transitions, solid blue lines in Fig. 4). Interband transitions with | s,j,k − s ,j ,k | > ω 0 (solid red lines in Fig. 4) and all intraband (between (s = ±1; j = 1) and (s = ±1, j = 2), dashed red lines in Fig. 4) contribute to Π as negative terms. It is convenient to distinguish three different cases: (I), |E F | < ω 0 , (II) ω 0 < |E F | < γ 1 , and (III) |E F | > γ 1 . For simplicity let us assume E F > 0 (the same applies for E F < 0).In case (I), positive contributions from interband transitions are suppressed, and new negative intraband transitions are created. This results in strong phonon softening at low temperatures [35]. At T=300K, these effects are blurred by the fractionary occupation of the electronic states, resulting in an almost doping independent phonon energy (see Fig.3b). In case (II), a shift of E F suppresses negative interband contributions and creates new negative intraband transitions. By counting their number and relative weight (given by Φ jj /( s,j,k − s ,j ,k )), one can show that interband transitions outweight intraband ones, resulting in phonon hardening. Case (III) is similar to (II), with the difference that the second subband filling suppresses negative intraband transitions at k∼K, further enhancing the phonon hardening. It is also possible to demonstrate that, for T and δ → 0, the slope of ∆P os(G) dyn just above E F = γ 1 is double than that just below. Thus, the kink in Fig.3 is a direct measurement of the interlayer coupling strength from Raman spectroscopy.
In SLG the intensity ratio of 2D and G, I(2D)/I(G), has a strong dependence on doping [24]. Fig.5a plots I(2D)/I(G) as a function of doping. For BLG we take the highest amongst 2D 1A and 2D 2A . The SLG dependence reproduces our previous results [24]. However, we find an almost constant ratio in BLG. Fig.5b plots the doping dependence of Pos(2D) in SLG, and Pos(2D 1A ), Pos(2D 2A ) in BLG. To a first approximation, this is governed by lattice relaxation, which explains the overall stiffening for hole doping and softening for electron doping [24]. A quantitative understanding is yet to emerge, and beyond DFT many body effects need be considered.
To conclude, we have simultaneously measured the behavior of optical phonons in single and bilayer graphene as a function of doping. In the latter, the G peak renormalizes as the Fermi energy moves from the 1st to the 2nd subband, allowing a direct measurement of γ 1 ∼0.4eV.
We thank D. Basko for useful and stimulating discussions. AKS acknowledges funding from the Department of Science and Technology, India, SP from Pembroke College and the Maudslay society, ACF from The Royal Society and The Leverhulme Trust.
FIG. 2 :
2(color online). Raman spectra of (a) SLG; (b) BLG at several VT G. Red lines fits to the experimental data.
FIG. 3 :
3(color online). Pos(G) for (a) SLG; (c) BLG as a function of Fermi energy. FWHM(G) of (b) SLG;(d) BLG as a function of Fermi energy. Solid lines:theoretical predictions.
FIG. 5 :
5(a) Ratio of 2D and G peaks intensities for SLG (solid circles) and BLG (open circles) as a function of n. (b) Position of 2D for SLG (solid circles) and 2D main components for BLG (open circles) as a function of n.
Objective FIG. 1: (color online). Experimental setup. The black dotted box on SiO2 indicates the polymer electrolyte (PEO + LiClO4). The left inset shows an SEM image of the SLG and BLG. Scale bar: 4µm. The right inset, the 2D Raman band monitoring of doping and defectsSLG
BLG
Si
Si
SiO
SiO
2
2
Pt
A u
A u
* Electronic address: [email protected] † Electronic address: [email protected]. * Electronic address: [email protected] † Electronic address: [email protected]
. K S Novoselov, Science. 306666K.S. Novoselov et al. Science 306, 666 (2004)
. K S Novoselov, Nature. 438197K.S. Novoselov, et al. Nature 438,197 (2005).
. Y Zhang, Nature. 438201Y. Zhang et al. Nature, 438, 201 (2005).
. J C Charlier, Topics Appl. Phys. 111673J. C. Charlier et al. Topics Appl. Phys. 111, 673 (2008)
. A K Geim, K S Novoselov, Nat. Mater. 6183A. K. Geim, K. S. Novoselov; Nat. Mater., 6, 183 (2007).
. S V Morozov, Phys. Rev. Lett. 10016602S. V. Morozov et al. Phys. Rev. Lett., 100 016602 (2008).
. X Du, arXiv:0802.2933cond matX. Du et al. cond mat. arXiv:0802.2933
. K I Bolotin, arXiv:0802.2389arXiv:0805.1830K. I. Bolotin et al arXiv:0802.2389; arXiv:0805.1830
. M C Lemme, IEEE El. Dev. Lett. 28282M. C. Lemme et al. IEEE El. Dev. Lett. 28, 282 (2007).
. M Y Han, Phys. Rev. Lett. 98206805M. Y. Han et al. Phys. Rev. Lett. 98, 206805 (2007).
. Z Chen, Physica E. 40228Z. Chen et al. Physica E, 40, 228 (2007).
. E Mccann, V I Falko, Phys. Rev. Lett. 9686805E.McCann,V.I.Falko, Phys. Rev. Lett.,96 086805 (2006)
. T Ando, J. Phys. Soc. Jpn. 76104711T. Ando, J. Phys. Soc. Jpn. 76, 104711 (2007).
. T Ohta, Science. 313951T. Ohta et al. Science 313, 951 (2007).
. E Mccann, Phys. Rev. B. 74R161403E. McCann, Phys. Rev. B 74, 161403 (R) (2006).
. E V Castro, Phys. Rev. Lett. 99216802E. V. Castro et al. Phys. Rev. Lett. 99, 216802 (2007).
. A C Ferrari, Phys. Rev. Lett. 97187401A. C. Ferrari et al. Phys. Rev. Lett. 97, 187401 (2006).
. C Casiraghi, Nano. Lett. 72711C. Casiraghi et al. Nano. Lett. 7, 2711 (2007).
. P Blake, Appl. Phys. Lett. 9163124P. Blake et al. Appl. Phys. Lett. 91, 063124 (2007).
. S Pisana, Nature Mat. 6198S. Pisana et al., Nature Mat. 6,198 (2007).
. J Yan, Phys.Rev.Lett. 98166802J. Yan et al. Phys.Rev.Lett. 98, 166802 (2007).
. C Casiraghi, Appl. Phys. Lett. 91233108C. Casiraghi et al. Appl. Phys. Lett. 91, 233108 (2007)
Solid State Comm. A C Ferrari, 14347A. C. Ferrari, Solid State Comm. 143, 47 (2007).
. A Das, Nature Nanotech. 3210A. Das et al., Nature Nanotech. 3, 210 (2008).
. A Das, cond-mat/0710.4160A. Das et al., cond-mat/0710.4160 (2007).
Raman spectroscopy in carbons: from nanotubes to diamond, Theme Issue. Phil. Trans. Roy. Soc. A. A. C. Ferrari, J. Robertson362A. C. Ferrari, J. Robertson (eds), Raman spectroscopy in carbons: from nanotubes to diamond, Theme Issue, Phil. Trans. Roy. Soc. A 362, 2267-2565 (2004).
. F Tuinstra, J L Koenig, J. Chem. Phys. 531126F.Tuinstra,J.L. Koenig, J. Chem. Phys.53, 1126 (1970).
. A C Ferrari, J Robertson, ibid. 64Phys. Rev. B. 6175414A.C. Ferrari, J. Robertson Phys. Rev. B 61, 14095 (2000); ibid. 64, 075414 (2001).
. C Thomsen, S Reich, Phys. Rev.Lett. 855214C. Thomsen, S. Reich, Phys. Rev.Lett. 85, 5214 (2000).
. S Piscanec, Phys. Rev. Lett. 93185503S. Piscanec et al. Phys. Rev. Lett. 93, 185503 (2004).
. M Lazzeri, F Mauri, Phys. Rev. Lett. 97266407M. Lazzeri, F. Mauri, Phys. Rev. Lett. 97, 266407 (2006).
. M Lazzeri, Phys. Rev. B. 73155426M. Lazzeri et al. Phys. Rev. B 73, 155426 (2006).
. A Das, Phys Rev Lett. 99136803A. Das et al. Phys Rev Lett. 99, 136803 (2007).
. T Ando, J. Phys. Soc. Jpn. 75124701T. Ando, J. Phys. Soc. Jpn. 75, 124701 (2006).
. J Yan, cond-mat/0712J. Yan et al. cond-mat/0712.3879v1 (2007).
. K T Nguyen, Phys. Rev. Lett. 98145504K. T. Nguyen, Phys. Rev. Lett. 98, 145504 (2007).
. C Lu, Nano Lett. 4623C. Lu et al. Nano Lett. 4, 623 (2004).
. A Das, A C Ferrari, A K Sood, A. Das, A. C. Ferrari, A. K. Sood, unpublished (2008)
Note that in Fig.3 of Ref.[13] both SLG density of states and electron concentration are multiplied by a factor 2. Note that in Fig.3 of Ref.[13] both SLG density of states and electron concentration are multiplied by a factor 2.
. A H Castro Neto, cond-mat/0709.1163v1A. H. Castro Neto et al. cond-mat/0709.1163v1 (2007).
. W E Pickett, P B Allen, Phys. Rev. 163127W.E.Pickett,P.B.Allen, Phys. Rev. B16, 3127 (1977).
. P B Allen, Phys. Rev. B. 62577P. B. Allen, Phys. Rev. B 6, 2577 (1972).
Note that the phonon self-energy imaginary part corresponds to G half width at half maximum, HWHM(G), as for Eq.8 in Ref. 42Note that the phonon self-energy imaginary part corre- sponds to G half width at half maximum, HWHM(G), as for Eq.8 in Ref.[42].
This is sometimes neglected in literature. For example, ∆Γ in Eq.1 of Ref.[21] represents HWHM(G), and not FWHM(G). Thus, the factor 2 to compute FWHM(G) in Eq.5. Ref.[21] then compares this with FWHM(G) calculated in Eq.3 of Ref.[32], finding D 2 /4 = D 2Thus, the factor 2 to compute FWHM(G) in Eq.5. This is sometimes neglected in lit- erature. For example, ∆Γ in Eq.1 of Ref.[21] represents HWHM(G), and not FWHM(G). Ref.[21] then compares this with FWHM(G) calculated in Eq.3 of Ref.[32], find- ing D 2 /4 = D 2
However, the correct relation should. Γ F , Γ F . However, the correct relation should
Because of this, the coupling constant of Ref.[21] is λ = 2α instead of λ = α . Similarly, "broadening" in Figs.4,6 of Ref. Γ F , 4 of Ref.[34] is HWHM(G), not FWHM(G). Also, Fig.6 in Ref.[44] mistakenly compares the experimental FWHM of the G − peak of metallic SWNTs with the theroetical HWHMΓ F . Because of this, the coupling con- stant of Ref.[21] is λ = 2α instead of λ = α . Similarly, "broadening" in Figs.4,6 of Ref.[13] and Fig.4 of Ref.[34] is HWHM(G), not FWHM(G). Also, Fig.6 in Ref.[44] mistakenly compares the experimental FWHM of the G − peak of metallic SWNTs with the theroetical HWHM.
. K Ishikawa, T Ando, J.Phys.Soc.Jap. 7584713K.Ishikawa,T.Ando, J.Phys.Soc.Jap.75, 084713 (2006).
Note that the prefactor of Eq.7 of Ref. 20] should be ω 0 α 4cNote that the prefactor of Eq.7 of Ref.[20] should be ω 0 α 4c
EPC(Γ) is equivalent to G 2 Γ F as defined in Ref. EPC(Γ) is equivalent to G 2 Γ F as defined in Ref.[32]
| [] |
[
"WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES EQUATION FOR TWO FLUIDS WITH SURFACE TENSION",
"WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES EQUATION FOR TWO FLUIDS WITH SURFACE TENSION"
] | [
"Julian Fischer ",
"Sebastian Hensel "
] | [] | [] | In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension -like, for example, the evolution of oil bubbles in water. Our main result is a weak-strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier-Stokes equation: As long as a strong solution exists, any varifold solution must coincide with it. In particular, in the absence of physical singularities the concept of varifold solutions -whose global in time existence has been shown by Abels [2] for general initial data -does not introduce a mechanism for non-uniqueness. The key ingredient of our approach is the construction of a relative entropy functional capable of controlling the interface error. If the viscosities of the two fluids do not coincide, even for classical (strong) solutions the gradient of the velocity field becomes discontinuous at the interface, introducing the need for a careful additional adaption of the relative entropy. | 10.1007/s00205-019-01486-2 | [
"https://arxiv.org/pdf/1901.05433v1.pdf"
] | 119,142,503 | 1901.05433 | f701971c7a09c1eabe62636381f5916e63b5856b |
WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES EQUATION FOR TWO FLUIDS WITH SURFACE TENSION
Julian Fischer
Sebastian Hensel
WEAK-STRONG UNIQUENESS FOR THE NAVIER-STOKES EQUATION FOR TWO FLUIDS WITH SURFACE TENSION
In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension -like, for example, the evolution of oil bubbles in water. Our main result is a weak-strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier-Stokes equation: As long as a strong solution exists, any varifold solution must coincide with it. In particular, in the absence of physical singularities the concept of varifold solutions -whose global in time existence has been shown by Abels [2] for general initial data -does not introduce a mechanism for non-uniqueness. The key ingredient of our approach is the construction of a relative entropy functional capable of controlling the interface error. If the viscosities of the two fluids do not coincide, even for classical (strong) solutions the gradient of the velocity field becomes discontinuous at the interface, introducing the need for a careful additional adaption of the relative entropy.
Introduction
In evolution equations for interfaces, topological changes and geometric singularities often occur naturally, one basic example being the pinchoff of liquid droplets (see Figure 1). As a consequence, strong solution concepts for such PDEs are naturally limited to short-time existence results or particular initial configurations like perturbations of a steady state. At the same time, the transition from strong to weak solution concepts for PDEs is prone to incurring unphysical non-uniqueness of solutions: For example, Brakke's concept of varifold solutions for mean curvature flow admits sudden vanishing of the evolving surface at any time [19]; for the Euler equation, even for vanishing initial data there exist nonvanishing solutions with compact support [81], and the notion of mild solutions to the Navier-Stokes equation allows any smooth flow to transition into any other smooth flow [23]. In the context of fluid mechanics, the concept of relative entropies has proven successful in ruling out the aforementioned examples of non-uniqueness: Energy-dissipating weak solutions e. g. to the incompressible Navier-Stokes equation are subject to a weak-strong uniqueness principle [68,73,85], which states that as long as a strong solution exists, any weak solution satisfying the precise form of the energy dissipation inequality must coincide with it. However, in the context of evolution equations for interfaces, to the best of our knowledge the concept of relative entropies has not been applied successfully so far to obtain weak-strong uniqueness results.
In the present work, we are concerned with the most basic model for the evolution of two fluids separated by a sharp interface (like, for instance, the evolution of oil For this free boundary problem for the flow of two immiscible incompressible fluids with surface tension, Abels [2] has established the global existence of varifold solutions for quite general initial data.
The main result of the present work is a weak-strong uniqueness result for this free boundary problem for the Navier-Stokes equation for two fluids with surface tension: In Theorem 1 below we prove that as long as a strong solution to this evolution problem exists, any varifold solution in the sense of Abels [4] must coincide with it.
1.1. Free boundary problems for the Navier-Stokes equation. The free boundary problem for the Navier-Stokes equation has been studied in mathematical fluid mechanics for several decades. Physically, it describes the evolution of a viscous incompressible fluid surrounded by or bordering on vacuum. The (local-intime) existence of strong solutions for the free boundary problem for the Navier-Stokes equation has been proven by Solonnikov [87,88,89] in the presence of surface tension and by Shibata and Shimizu [86] in the absence of surface tension; see also Beale [17,18], Abels [1], and Coutand and Shkoller [34] for related or further results. While the existence theory for global weak solutions for the Navier-Stokes equation in a fixed domain like R d , d ≤ 3, has been developed starting with the seminal work of Leray [68] in 1934, the question of the global existence of any kind of solution to the free boundary problem for the Navier-Stokes equation has remained an open problem. An important challenge for a global existence theory of weak solutions to the free boundary problem for the Navier-Stokes equation is the possible formation of "splash singularities", which are smooth solutions to the Lagrangian formulation of the equations which develop self-interpenetration. Such solutions have been constructed by Castro, Cordoba, Fefferman, Gancedo, and Gomez-Serrano [27], see also [26,36,47] for splash singularities in related models in fluid mechanics.
In the present work we consider a closely related problem, namely the flow of two incompressible and immiscible fluids with surface tension at the fluid-fluid interface, like for example the flow of oil bubbles immersed in water or vice versa. For this free boundary problem for the Navier-Stokes equation for two fluids -described by the system of PDEs (1) below -, a global existence theory for generalized solutions is in fact available: In a rather recent work, Abels [2] has constructed varifold solutions which exist globally in time. In an earlier work, Plotnikov [72] had treated the case of non-Newtonian (shear-thickening) fluids. The local-in-time existence of strong solutions has been established by Denisova [43]; for an interface close to the half-space, an existence and instant analyticity result has been derived by Prüss and Simonett [75,76]. Existence results for the two-phase Stokes and Navier-Stokes equation in the absence of surface tension have been established by Giga and Takahashi [57] and Nouri and Poupaud [71]. Note that in contrast to the case of a single fluid in vacuum, for the flow of two incompressible immiscible inviscid fluids splash singularities cannot occur as shown by Fefferman, Ionescu, and Lie [50] and Coutand and Shkoller [35]; one would expect a similar result to hold for viscous fluids. However, solutions may be subject to the Rayleigh-Taylor instability as proven by Prüss and Simonett [74].
In terms of a PDE formulation, the flow of two immiscible incompressible fluids with surface tension may be described by the indicator function χ = χ(x, t) of the volume occupied by the first fluid, the local fluid velocity v = v(x, t), and the local pressure p = p(x, t). The fluid-fluid interface moves just according to the fluid velocity, the evolution of the velocity of each fluid and the pressure are determined by the Navier-Stokes equation, and the fluid-fluid interface exerts a surface tension force on the fluids proportional to the mean curvature of the interface. Together with the natural no-slip boundary condition and the appropriate boundary conditions for the stress tensor on the fluid-fluid interface, one may assimilate the Navier-Stokes equations for the two fluids into a single one, resulting in the system of equations
∂ t χ + (v · ∇)χ = 0, (1a) ρ(χ)∂ t v + ρ(χ)(v · ∇)v = −∇p + ∇ · (µ(χ)(∇v + ∇v T )) + σH|∇χ|, (1b) ∇ · v = 0,(1c)
where H denotes the mean curvature vector of the interface ∂{χ = 0} and |∇χ| denotes the surface measure H d−1 | ∂{χ=0} . Here, µ(0) and µ(1) are the shear viscosities of the two fluids and ρ(0) and ρ(1) are the densities of the two fluids. The constant σ is the surface tension coefficient. The total energy of the system is given by the sum of kinetic and surface tension energies
E[χ, v] :=ˆR d 1 2 ρ(χ)|v| 2 dx + σˆR d 1 d|∇χ|.
It is at least formally subject to the energy dissipation inequality
E[χ, v](T ) +ˆT 0ˆR d µ(χ) 2 ∇v + ∇v T 2 dx ≤ E[χ, v](0).
Note that the concept of varifold solutions requires a slight adjustment of the definition of the energy: The surface area´R d 1 d|∇χ| is replaced by the corresponding quantity of the varifold, namely its mass. A widespread numerical approximation method for the free boundary problem (1a)-(1c) capable of capturing geometric singularities and topological changes in the fluid phases are phase-field models of Navier-Stokes-Cahn-Hilliard type or Navier-Stokes-Allen-Cahn type, see for example the review [12], [6,69,59,61,70] for modeling aspects, [3,5] for the existence analysis of the corresponding PDE systems, and [7,8,9] for results on the sharp-interface limit.
Weak solution concepts in fluid mechanics and (non-)uniqueness.
In the case of the free boundary problem for the Navier-Stokes equation -both for a single fluid and for a fluid-fluid interface -, a concept of weak solutions is expected to play an even more central role in the mathematical theory than in the case of the standard Navier-Stokes equation: In three spatial dimensions d = 3, even for smooth initial interfaces topological changes may occur naturally in finite time, for example by asymptotically self-similar pinchoff of bubbles [46] (see Figure 1). In contrast, for the incompressible Navier-Stokes equation without free boundary the global existence of strong solutions for any sufficiently regular initial data remains a possibility.
However, in general weakening the solution concept for a PDE may lead to artificial (unphysical) non-uniqueness, even in the absence of physically expected singularities. A particularly striking instance of this phenomenon is the recent example of non-uniqueness of mild (distributional) solutions to the Navier-Stokes equation by Buckmaster and Vicol [25] and Buckmaster, Colombo, and Vicol [23]: In the framework of mild solutions to the Navier-Stokes equation, any smooth flow may transition into any other smooth flow [23]. The result of [23,25] are based on convex integration techniques for the Euler equation, which have been developed starting with the works of De Lellis and Székelyhidi [40,41] (see also [22,24,39,63]).
In contrast to the case of distributional or mild solutions, for the stronger notion of weak solutions to the Navier-Stokes equation with energy dissipation in the sense of Leray [68] a weak-strong uniqueness theorem is available: As long as a strong solution to the Navier-Stokes equation exists, any weak solution with energy dissipation must coincide with it. Recall that for a weak solution to the Navier-Stokes equation v, besides the Ladyzhenskaya-Prodi-Serrin regularity criterion v ∈ L p ([0, T ]; L q (R 3 ; R 3 )) with 2 p + 3 q ≤ 1 and p ≥ 2 [73,84], both a lower bound on the pressure [82] and a geometric assumption on the vorticity [33] are known to imply smoothness of v. Interestingly, weak-strong uniqueness of energydissipating solutions fails if the Laplacian in the Navier-Stokes equation is replaced by a fractional Laplacian −(−∆) α with power α < 1 3 , see Colombo, De Lellis, and De Rosa [32] and De Rosa [78].
Another way of interpreting a weak-strong uniqueness result is that nonuniqueness of weak solutions may only arise as a consequence of physical singularities: Only when the unique strong solution develops a singularity, the continuation of solutions beyond the singularity -by means of the weak solution concept -may be nonunique. The main theorem of our present work provides a corresponding result for the flow of two incompressible immiscible fluids with surface tension: Varifold solutions to the free boundary problem for the Navier-Stokes equation for two fluids are unique until the strong solution for the free boundary problem develops a singularity.
1.3. (Non)-Uniquenesss in interface evolution problems. Weak solution concepts for the evolution of interfaces are often subject to nonuniqueness, even in the absence of topology changes. For example, Brakke's concept of varifold solutions for the evolution of surfaces by mean curvature [19] suffers from a particularly drastic failure of uniqueness: The interface is allowed to suddenly vanish at an arbitrary time. In the context of viscosity solutions to the level-set formulation of two-phase mean curvature flow, the formation of geometric singularities may still cause fattening of level-sets [16] and thereby nonuniqueness of the mean-curvature evolution, even for smooth initial surfaces [13].
To the best of our knowledge, the only known uniqueness result for weak or varifold solutions for an evolution problem for interfaces is a consequence of the relation between Brakke solutions and viscosity solutions for two-phase mean-curvature flow, see Ilmanen [62]: As long as a smooth solution to the level-set formulation exists, the support of any Brakke solution must be contained in the corresponding levelset of the viscosity solution. As a consequence, as long as a smooth evolution of the interface by mean curvature exists, the "maximal" unit-density Brakke solution corresponds to the smoothly evolving interface. The proof of this inclusion relies on the properties of the distance function to a surface undergoing evolution by mean curvature respectively the comparison principle for mean curvature flow. Both of these properties do not generalize to other interface evolution equations.
Besides Ilmanen's varifold comparison principle, the only uniqueness results in the context of weak solutions to evolution problems for lower-dimensional objects that we are aware of are a weak-strong uniqueness principle for the highercodimension mean curvature flow by Ambrosio and Soner [11] and a weak-strong uniqueness principle for binormal curvature motion of curves in R 3 by Jerrard and Smets [65]. The interface contribution in our relative entropy (11) may be regarded as the analogue for surfaces of the relative entropy for curves introduced in [65]. 1.4. The concept of relative entropies. The concept of relative entropies in continuum mechanics has been introduced by Dafermos [37,38] and DiPerna [44] in the study of the uniqueness properties of systems of conservation laws. Proving weak-strong uniqueness results for conservation laws or even the incompressible Navier-Stokes equation typically faces the problem that an error between a weak solution u and a strong solution v must be measured by a quantity E[u|v] which is nonlinear even as a function of u alone, like a norm ||u − v||. To evaluate the time evolution d dt E[u|v] of such a quantity, one would need to test the evolution equation for u by the nonlinear function D u E[u|v], which is often not possible due to the limited regularity of u. The concept of relative entropies overcomes this issue if the physical system possesses a strictly convex entropy ( [u|v] for the relative entropy and thereby a weak-strong uniqueness result.
Since Dafermos [37], the concept of relative entropies has found many applications in the analysis of continuum mechanics, providing weak-strong uniqueness results for the compressible Navier-Stokes equation [51,55], the Navier-Stokes-Fourier system [52], fluid-structure interaction problems [28], renormalized solutions for dissipative reaction-diffusion systems [29,54], as well as weak-strong uniqueness results for measure-valued solutions for the Euler equation [21], compressible fluid models [60], wave equations in nonlinear elastodynamics [42], and models for liquid crystals [48], to name just a few.
The concept of relative entropies has also been employed in the justification of singular limits of PDEs, see for example the work of Yau [90] on the hydrodynamic limit of the Ginzburg-Landau lattice model, the works of Bardos, Golse, and Levermore [15], Saint-Raymond [79,80], and Golse and Saint-Raymond [58] on the derivation of the Euler equation and the incompressible Navier-Stokes equation from the Boltzmann equation, the work of Brenier [20] on the Euler limit of the Vlasov-Poissson equation, and the works of Serfaty [83] and Duerinckx [45] on mean-field limits of interacting particles. In the context of numerical analysis, it may also be used to derive a posteriori estimates for model simplification errors [53,56].
Jerrard and Smets [65] have used a relative entropy ansatz to establish a weakstrong uniqueness principle for the evolution of curves in R 3 by binormal curvature flow. Their relative entropy may be regarded as the analogue for curves of the interfacial energy contribution to our relative entropy (i. e. the terms σ´R d 1−ξ(·, T )· ∇χu(·,T ) |∇χu(·,T )| d|∇χ u (·, T )| + σ´R d 1 − θ T d|V T | S d−1 in (11) below). It has subsequently been used by Jerrard and Seis [64] to prove that the evolution of solutions to the Euler equation with near-vortex-filament initial data is governed by binormal curvature flow, as long as a strong solution to the latter (without self-intersections) exists and as long as the vorticity remains concentrated along some curve.
One of the key challenges in the derivation of our result is the development of a notion of relative entropy which provides strong enough control of the interfacial error. The key idea to control the error between an interface ∂{χ u (·, t) = 1} and a smoothly evolving interface I v (t) = ∂{χ v (·, t) = 1} by a relative entropy is to introduce a vector field ξ which is an extension of the unit normal of I v (t), multiplied with a cutoff. The interfacial contribution σ´R d 1 − ξ(·, T ) · ∇χu(·,T ) |∇χu(·,T )| d|∇χ u (·, T )| to the relative entropy then controls the interface error in a sufficiently strong way, see Section 3 for details.
However, in the case of different viscosities µ + = µ − of the two fluids, the velocity gradient of the strong solution ∇v at the interface will be discontinuous. This necessitates an additional adaption of our relative entropy: If one were to directly compare the velocity fields u and v of two solutions by the relative entropy, the difference of the viscous stresses µ(χ u )D sym u − µ(χ v )D sym v could not be estimated appropriately to derive a Gronwall-type estimate. We rather have to compare the velocity field u to an adapted velocity field v + w, where w is constructed in a way that the adapted velocity gradient ∇v + ∇w approximately accounts for the shifted location of the interface.
The approximate adaption of the interface of the strong solution to the higherorder approximation for the interface is distantly reminiscent of an ansatz by Leger and Vasseur [67] and Kang, Vasseur, and Wang [66], who establish L 2 contractions up to a shift for solutions to conservation laws close to a shock profile. However, it differs both in purpose and in the actual construction from [66,67]: The interfacial shift in [66,67] essentially serves the purpose of compensating the difference in the propagation speed of the shocks of the two solutions, while we need the higherorder approximation of the interface to compensate for the discontinuity in the velocity gradient at the interface. While the interfacial shift in [66] is given as the solution to an appropriately defined time-dependent PDE, in our case we obtain the interfacial shift by applying at any fixed time a suitable regularization operator to the interface of the weak solution near the interface of the strong solution.
1.5. Derivation of the model. Let us briefly comment on the derivation of the system of equations (1). We consider the flow of two viscous, immiscible, and incompressible fluids. Each fluid occupies a domain Ω + t resp. Ω − t , t ≥ 0, and the interface separating both phases will be denoted by I(t). In particular, R d = Ω + t ∪Ω − t ∪I(t) for every t ≥ 0. Within each of these domains Ω ± t , the evolution of the fluid velocity is modeled by means of the incompressible Navier-Stokes equations for a Newtonian fluid
∂ t (ρ ± v ± t ) + ∇ · (ρ ± v ± t ⊗ v ± t ) = −∇p ± t + µ ± ∆v ± t , (2a) ∇ · v ± t = 0,(2b)
where v + t : Ω + t → R d and v − t : Ω − t → R d denote the velocity fields of the two fluids, p + t : Ω + t → R and p − t : Ω − t → R the pressure, ρ + , ρ − > 0 the densities of the two fluids, and µ + and µ − the shear viscosities. On the interface of the two fluids I(t) a no-slip boundary condition v + t = v − t is imposed. As the two velocities v + t and v − t are defined on complementary domains and coincide on the interface, this enables us to assimilate the two velocity fields into a single velocity
field v : R d × [0, T ) → R d , v t := v + t χ Ω + t + v − (1 − χ Ω + t )
. Note that the velocity field v inherits the incompressibility (1c) from the incompressibility of v + and v − (2b). We also assimilate the pressures p + t and p − t into a single pressure p, which however may be discontinuous across the interface.
Additionally, we assume that the evolution of the interface I(t) occurs only as a result of the transport of the two fluids along the flow. Denoting by n the outward unit normal vector field of the interface I(t) and by V n the associated normal speed of the interface, this gives rise to the equation
V n = n · v on I(t) for all t ≥ 0.(3)
This condition may equivalently be rewritten as the transport equation for the indicator function χ of the first fluid phase ∂ t χ + (v · ∇)χ = 0, see for example Remark 8 below for the (standard) arguments.
In order to assimilate the equations (2a) for the velocities v ± of the two fluids into the single equation (1b), a condition on the jump of the normal component of the stress tensor T = µ ± (∇v + ∇v T ) − ∇p Id at the interface I(t) is required. In the case of positive surface tension constant σ > 0 at the interface, the balance of forces at the interface reads
[[T n ]] = σH,(4)
where the right-hand side σH accounts for the surface tension force. Here, H denotes the mean curvature vector of the interface and [[f ]] denotes the jump in normal direction of a quantity f . In combination with (2a) and the no-slip boundary condition v + = v − on I(t), this yields the equation for the momentum balance (1b).
Main results
The main result of the present work is the derivation of a weak-strong uniqueness principle for varifold solutions to the free boundary problem for the Navier-Stokes equation for two immiscible incompressible fluids with surface tension: As long as a strong solution to the free boundary problem (1a)-(1c) exists, any varifold solution must coincide with it. In particular, the concept of varifold solutions developed by Abels [2] (see Definition 2 below for a precise definition) does not introduce an additional mechanism for non-uniqueness, at least as long as a classical solution exists. At the same time, the concept of varifold solutions of Abels allows for the construction of globally existing solutions [2], while any concept of strong solutions is limited to the absence of geometric singularities and therefore -at least in three spatial dimensions d = 3 -to short-time existence results.
Furthermore, we prove a quantitative stability result (5) for varifold solutions with respect to changes in the data: As long as a classical solution exists, any varifold solution with slightly perturbed initial data remains close to it.
Theorem 1 (Weak-strong uniqueness principle). Let d ∈ {2, 3}. Let (χ u , u, V ) be a varifold solution to the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) in the sense of Definition 2 on some time interval [0, T vari ). Let (χ v , v) be a strong solution to (1a)-(1c) in the sense of Definition 6 on some time interval [0, T strong ) with T strong ≤ T vari . Let the relative entropy E[χ u , u, V |χ v , v](t) be defined as in Proposition 9.
Then there exist constants C, c > 0 such that the stability estimate
E χ u , u, V χ v , v (T ) ≤ C(E[χ u , u, V |χ v , v](0)) e −CT(5)
holds for almost every T ∈ [0, T strong ), provided that the initial relative entropy satisfies E[χ u , u, V |χ v , v](0) ≤ c. The constants c > 0 and C > 0 depend only on the data and the strong solution.
In particular, if the initial data of the varifold solution and the strong solution coincide, the varifold solution must be equal to the strong solution in the sense that
χ u (·, t) = χ v (·, t) and u(·, t) = v(·, t)
hold almost everywhere for almost every t ∈ [0, T strong ), and the varifold is given for almost every t ∈ [0, T strong ) by
dV t = δ ∇χv |∇χv | d|∇χ v |.
The following notion of varifold solutions for the free boundary problem associated with the flow of two immiscible incompressible viscous fluids with surface tension has been introduced by Abels [2]. For the notion of an oriented varifold, see the section on notation just prior to Section 3.
Definition 2 (Varifold solution for the two-phase Navier-Stokes equation). Let a surface tension constant σ > 0, the densities and shear viscosities of the two fluids ρ ± , µ ± > 0, a finite time T vari > 0, a solenoidal initial velocity profile v 0 ∈ L 2 (R d ; R d ), and an indicator function of the volume occupied initially by the first fluid χ 0 ∈ BV(R d ) be given.
A triple (χ, v, V ) consisting of a velocity field v, an indicator function χ of the volume occupied by the first fluid, and an oriented varifold
V with v ∈ L 2 ([0, T vari ]; H 1 (R d ; R d )) ∩ L ∞ ([0, T vari ]; L 2 (R d ; R d )), χ ∈ L ∞ ([0, T vari ]; BV(R d ; {0, 1})), V ∈ L ∞ w ([0, T vari ]; M(R d ×S d−1 )),
is called a varifold solution to the free boundary problem for the Navier-Stokes equation for two fluids with initial data (χ 0 , v 0 ) if the following conditions are satisfied:
i) The velocity field v has vanishing divergence ∇ · v = 0 and the equation for the momentum balancê
R d ρ(χ(·, T ))v(·, T ) · η(·, T ) dx −ˆR d ρ(χ 0 )v 0 · η(·, 0) dx =ˆT 0ˆR d ρ(χ)v · ∂ t η dx dt +ˆT 0ˆR d ρ(χ)v ⊗ v : ∇η dx dt (6a) −ˆT 0ˆR d µ(χ)(∇v + ∇v T ) : ∇η dx dt − σˆT 0ˆR d ×S d−1 (Id − s ⊗ s) : ∇η dV t (x, s) dt is satisfied for almost every T ∈ [0, T vari ) and every smooth vector field η ∈ C ∞ cpt (R d × [0, T vari ); R d ) with ∇ · η = 0.
For the sake of brevity, we have used the abbreviations ρ(χ)
:= ρ + χ + ρ − (1 − χ) and µ(χ) := µ + χ + µ − (1 − χ).
ii) The indicator function χ of the volume occupied by the first fluid satisfies the weak formulation of the transport equation
R d χ(·, T )ϕ(·, T ) dx −ˆR d χ 0 ϕ(·, 0) dx =ˆT 0ˆR d χ (∂ t ϕ + (v · ∇)ϕ) dx dt (6b) for almost every T ∈ [0, T vari ) and all ϕ ∈ C ∞ cpt (R d × [0, T vari )). iii) The energy dissipation inequalitŷ R d 1 2 ρ(χ(·, T ))|v(·, T )| 2 dx + σ|V T |(R d × S d−1 ) +ˆT 0ˆR d µ(χ) 2 ∇v + ∇v T 2 dx dt (6c) ≤ˆR d 1 2 ρ(χ 0 (·))|v 0 (·)| 2 dx + σ|∇χ 0 (·)|(R d )
is satisfied for almost every T ∈ [0, T vari ), and the energy
E[χ, v, V ](t) :=ˆR d 1 2 ρ(χ(·, t))|v(·, t)| 2 dx + σ|V t |(R d × S d−1 ) (6d)
is a nonincreasing function of time. iv) The phase boundary ∂{χ(·, t) = 0} and the varifold V satisfy the compatibility conditionˆR
d ×S d−1 ψ(x)s dV t (x, s) =ˆR d ψ(x) d∇χ(x) (6e)
for almost every T ∈ [0, T vari ) and every smooth function ψ ∈ C ∞ cpt (R d ).
Let us continue with a few comments on the relation between the varifold V t and the interface described by the indicator function χ(·, t).
Remark 3. Let V t ∈ M(R d ×S d−1 )
denote the non-negative measure representing (at time t) the varifold associated to a varifold solution (χ, v, V ) to the free boundary problem for the incompressible Navier-Stokes equation for two fluids. The compatibility condition (6e) entails that |∇χ u (t)| is absolutely continuous with respect to |V t | S d−1 . Hence, we may define the Radon-Nikodym derivative
θ t := d|∇χ u (t)| d|V t | S d−1 , (7) which is a |V t | S d−1 -measurable function with |θ t (x)| ≤ 1 for |V t | S d−1 -almost every x ∈ R d . In particular, we havê R d f (x) d|∇χ(·, t)|(x) =ˆR d θ t (x)f (x) d|V t | S d−1 (x) (8)
for every f ∈ L 1 (R d , |∇χ(·, t)|) and almost every t ∈ [0, T vari ).
The compatibility condition between the varifold V t and the interface described by the indicator function χ(·, t) has the following consequence.
Remark 4. Consider a varifold solution (χ, v, V ) to the free boundary problem for the incompressible Navier-Stokes equation for two fluids. Let E t be the measurable set {x ∈ R d : χ(x, t) = 1}. Note that for almost every t ∈ [0, T vari ) this set is then a Caccioppoli set in R d . Let n(·, t) = ∇χ |∇χ| denote the measure theoretic unit normal vector field on the reduced boundary ∂ * E t . By means of the compatibility condition (6e) and the definition (7) we obtain
d´S d−1 s dV t (·, s) d|V t | S d−1 (·) = θ t (x)n(x, t) for x ∈ ∂ * E t , 0 else,(9)
for almost every t ∈ [0, T vari ) and |V t | S d−1 -almost every x ∈ R d .
In order to define a notion of strong solutions to the free boundary problem for the flow of two immiscible fluids, let us first define a notion of smoothly evolving domains.
Definition 5 (Smoothly evolving domains and surfaces).
Let Ω + 0 be a bounded domain of class C 3 and consider a family (Ω + t ) t∈[0,Tstrong) of open sets in R d . Let
I(t) = ∂Ω + t and Ω − t = R d \ (Ω + t ∪ I(t))
for every t ∈ [0, T strong ]. We say that Ω + t , Ω − t are smoothly evolving domains and that I(t) are smoothly evolving surfaces if we have
Ω + t = Ψ t (Ω + 0 ), Ω − t = Ψ t (Ω − 0 ), and I(t) = Ψ t (I(0)) for a map Ψ : R d × [0, T strong ) → R d , (t, x) → Ψ(t, x) = Ψ t (x)
, subject to the following conditions:
i) We have Ψ 0 = Id. ii) For any fixed t ∈ [0, T strong ), the map Ψ t : R d → R d is a C 3 -diffeomorphism. Moreover, we assume Ψ L ∞ t W 3,∞ x < ∞. iii) We have ∂ t Ψ ∈ C 0 ([0, T strong ); C 2 (R d ; R d )) and ∂ t Ψ L ∞ t W 2,∞ x < ∞.
Moreover, we assume that there exists r c ∈ (0, 1 2 ] with the following property: For all t ∈ [0, T strong ) and all x ∈ I(t) there exists a function g : B 1 (0) ⊂ R d−1 → R with ∇g(0) = 0 such that after a rotation and a translation, I(t) ∩ B 2rc (x) is given by the graph {(x, g(x)) : x ∈ R d−1 }. Furthermore, for any of these functions g the pointwise bounds |∇ m g| ≤ r −(m−1) c hold for all 1 ≤ m ≤ 3.
We have everything in place to give the definition of a strong solution to the free boundary problem for the Navier-Stokes equation for two fluids.
Definition 6 (Strong solution for the two-phase Navier-Stokes equation). Let a surface tension constant σ > 0, the densities and shear viscosities of the two fluids ρ ± , µ ± > 0, a finite time T strong > 0, a solenoidal initial velocity profile v 0 ∈ L 2 (R d ; R d ), and a domain Ω + 0 occupied initially by the first fluid be given. Let the initial interface between the fluids ∂Ω + 0 be a compact C 3 -manifold. A pair (χ, v) consisting of a velocity field v and an indicator function χ of the volume occupied by the first fluid with
v ∈ W 1,∞ ([0, T strong ]; H 1 (R d ; R d )) ∩ W 1,∞ ([0, T strong ]; W 1,∞ (R d ; R d )), ∇v ∈ L 1 ([0, T strong ]; BV(R d ; R d×d )), χ ∈ L ∞ ([0, T strong ]; BV(R d ; {0, 1})),
is called a strong solution to the free boundary problem for the Navier-Stokes equation for two fluids with initial data (χ 0 , v 0 ) if the volume occupied by the first fluid
Ω + t := {x ∈ R d : χ(x, t) = 1}
is a smoothly evolving domain and the interface I v (t) := ∂Ω + t is a smoothly evolving surface in the sense of Definition 5 and if additionally the following conditions are satisfied:
i) The velocity field v has vanishing divergence ∇ · v = 0 and the equation for the momentum balancê
R d ρ(χ(·, T ))v(·, T ) · η(·, T ) dx −ˆR d ρ(χ 0 )v 0 · η(·, 0) dx =ˆT 0ˆR d ρ(χ)v · ∂ t η dx dt +ˆT 0ˆR d ρ(χ)v ⊗ v : ∇η dx dt (10a) −ˆT 0ˆR d µ(χ)(∇v + ∇v T ) : ∇η dx dt + σˆT 0ˆIv(t)
H · η dS dt is satisfied for almost every T ∈ [0, T strong ) and every smooth vector field
η ∈ C ∞ cpt (R d × [0, T strong ); R d ) with ∇ · η = 0.
Here, H denotes the mean curvature vector of the interface I v (t). For the sake of brevity, we have used the abbreviations ρ(χ) := ρ + χ + ρ − (1 − χ) and µ(χ)
:= µ + χ + µ − (1 − χ).
ii) The indicator function χ of the volume occupied by the first fluid satisfies the weak formulation of the transport equation
R d χ(·, T )ϕ(·, T ) dx −ˆR d χ 0 ϕ(·, 0) dx =ˆT 0ˆR d χ (∂ t ϕ + (v · ∇)ϕ) dx dt (10b) for almost every T ∈ [0, T strong ) and all ϕ ∈ C ∞ cpt (R d × [0, T strong )). iii) In the domain t∈[0,Tstrong) (Ω + t ∪ Ω − t ) ×sup x∈Ω + t ∪Ω − t |∇ 3 v(x, t)| < ∞.
Before we state the main ingredient for the proof of Theorem 1, we proceed with two remarks on the notion of strong solutions. The first concerns the consistency with the notion of varifold solutions due to Abels [2]. Remark 7. Every strong solution (χ, v) to the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) in the sense of Definition 6 canonically defines a varifold solution in the sense of Definition 2. Indeed, we can define the varifold V by means of dV t = δ ∇χ |∇χ| d|∇χ|. Due to the regularity requirements on the family of smoothly evolving surfaces I(t), see Definition 5, it then followŝ Lemma 3.4]. Moreover, it follows from the regularity requirements of a strong solution that the velocity field v also satisfies the energy dissipation inequality (6c). This proves the claim.
T 0ˆI(t) H · ϕ dS dt = −ˆT 0ˆR d (Id − n ⊗ n) : ∇ϕ d|∇χ(·, t)| dt = −ˆT 0ˆR d ×S d−1 (Id − s ⊗ s) : ∇ϕ dV t (x, s) dt, for almost every T ∈ [0, T strong ) and all ϕ ∈ C ∞ cpt (R d × [0, T vari ); R d ), see for instance [4,
The second remark concerns the validity of (3) in a strong sense for a strong solution, i.e., that the evolution of the interface I(t) occurs only as a result of the transport of the two fluids along the flow.
Remark 8. Let (χ, v) be a strong solution to the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) in the sense of Definition 6 on some time interval [0, T strong ). Let V n (x, t) denote the normal speed of the interface at x ∈ I v (t), i.e., the normal component of ∂ t Ψ(x, t) where Ψ : R d × [0, T strong ) → R d is the family of diffeomorphisms from the definition of a family of smoothly evolving domains (Definition 5). Furthermore, let ϕ ∈ C ∞ cpt (R d × (0, T strong )). Due to the regularity requirements on a family of smoothly evolving domains, see Definition 5, we obtain (see for instance [4,Theorem 2.6
]) Tstrong 0ˆR d χ∂ t ϕ dx dt = −ˆT strong 0ˆIv(t) V n ϕ dS dt.
On the other side, subtracting from the former identity the equation (10b) satisfied by the indicator function χ and making use of the incompressibility of the velocity field v we deduceˆT strong 0ˆIv(t)
(V n − n · v)ϕ dS dt = 0.
Since ϕ ∈ C ∞ cpt (R d × (0, T strong )) was arbitrary we recover the identity
V n = n · v on t∈(0,Tstrong) {t} × I v (t),
that is to say, the kinematic condition (3) of the interface being transported with the flow is satisfied in its strong formulation.
Our weak-strong uniqueness result in Theorem 1 relies on the following relative entropy inequality. The regime of equal shear viscosities µ + = µ − corresponds to the choice of w = 0 in the statement below. Note also that in this case the viscous stress term R visc disappears due to µ(χ u ) − µ(χ v ) = 0. Proposition 9 (Relative entropy inequality). Let d ≤ 3. Let (χ u , u, V ) be a varifold solution to the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) in the sense of Definition 2 on some time interval [0, T vari ). Let (χ v , v) be a strong solution to (1a)-(1c) in the sense of Definition 6 on some time interval [0, T strong ) with T strong ≤ T vari and let
w ∈ L 2 ([0, T strong ); H 1 (R d ; R d )) ∩ H 1 ([0, T strong ); L 4/3 (R d ; R d ) + L 2 (R d ; R d ))
be a solenoidal vector field with bounded spatial derivative ∇w L ∞ < ∞. Suppose furthermore that for almost every t ≥ 0, for every x ∈ R d either x is a Lebesgue point of ∇w(·, t) or there exists a half-space H x such that x is a Lebesgue point for both ∇w(·, t)| Hx and ∇w(·, t)| R d \Hx .
For a point (x, t) such that dist(x, I v (t)) < r c , denote by P Iv(t) x the projection of x onto the interface I v (t) of the strong solution. Introduce the extension ξ of the unit normal n v of the interface of the strong solution defined by
ξ(x, t) := n v (P Iv(t) x)(1 − dist(x, I v (t)) 2 )η(dist(x, I v (t)))
for some cutoff η with η(s) = 1 for s ≤ 1 2 r c and η ≡ 0 for s ≥ r c . LetV n (x, t) := (n(P Iv(t) x, t) · v(P Iv(t) x, t))n(P Iv(t) x, t) be an extension of the normal velocity of the interface of the strong solution I v (t) to an r c -neighborhood of I v (t). Let θ be the density θ t = d|∇χu(·,t)| d|Vt| S d−1 as defined in (7) and let β : R → R be a truncation of the identity with β(r) = r for |r| ≤ 1 2 , |β | ≤ 1, |β | ≤ C, and β (r) = 0 for |r| ≥ 1. Then the relative entropy
E χ u , u, V χ v , v (T ) := σˆR d 1 − ξ(·, T ) · ∇χ u (·, T ) |∇χ u (·, T )| d|∇χ u (·, T )| (11) +ˆR d 1 2 ρ χ u (·, T ) u − v − w 2 (·, T ) dx +ˆR d χ u (·, T ) − χ v (·, T ) β dist ± (·, I v (T )) r c dx + σˆR d 1 − θ T d|V T | S d−1
is subject to the relative entropy inequality
E χ u , u, V χ v , v (T ) +ˆT 0ˆR d 2µ(χ u ) D sym (u − v − w) 2 dx dt ≤ E χ u , u, V χ v , v (0) + R surT en + R dt + R visc + R adv + R weightV ol + A visc + A dt + A adv + A surT en + A weightV ol
for almost every T ∈ (0, T strong ), where we made use of the abbreviations
R surT en := − σˆT 0ˆR d ×S d−1 (s − ξ) · (s − ξ) · ∇ v dV t (x, s) dt + σˆT 0ˆR d (1 − θ t ) ξ · (ξ · ∇)v d|V t | S d−1 dt + σˆT 0ˆR d (χ u − χ v ) (u − v − w) · ∇ (∇ · ξ) dx dt − σˆT 0ˆR d ξ · ∇χ u |∇χ u | n v (P Iv(t) x) · n v (P Iv(t) x) · ∇ v − ξ · (ξ · ∇)v d|∇χ u | dt + σˆT 0ˆR d ∇χ u |∇χ u | · (Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x))(∇V n −∇v) T · ξ d|∇χ u | dt + σˆT 0ˆR d ∇χ u |∇χ u | · (V n − v) · ∇ ξ d|∇χ u | dt and R dt := −ˆT 0ˆR d ρ(χ u ) − ρ(χ v ) (u − v − w) · ∂ t v dx dt, R visc := −ˆT 0ˆR d 2 µ(χ u ) − µ(χ v ) D sym v : D sym (u − v) dx dt, R adv := −ˆT 0ˆR d ρ(χ u ) − ρ(χ v ) (u − v − w) · (v · ∇)v dx dt −ˆT 0ˆR d ρ(χ u )(u − v − w) · (u − v − w) · ∇ v dx dt,
as well as
R weightV ol := T 0ˆR d (χ u −χ v ) V n −(v · n v (P Iv(t) x))n v (P Iv(t) x) · ∇ β dist ± (·, I v ) r c dx dt +ˆT 0ˆR d (χ u −χ v ) (u−v−w) · ∇ β dist ± (·, I v ) r c dx dt.
Moreover, we have abbreviated
A visc :=ˆT 0ˆR d 2 µ(χ u ) − µ(χ v ) D sym v : D sym w dx dt −ˆT 0ˆR d 2µ(χ u )D sym w : D sym (u − v − w) dx dt, and A dt := −ˆT 0ˆR d ρ(χ u )(u − v − w) · ∂ t w dx dt −ˆT 0ˆR d ρ(χ u )(u − v − w) · (v · ∇)w dx dt, A adv := −ˆT 0ˆR d ρ(χ u )(u − v − w) · (w · ∇)(v + w) dx dt −ˆT 0ˆR d ρ(χ u )(u − v − w) · (u − v − w) · ∇ w dx dt, A weightV ol :=ˆT 0ˆR d (χ u −χ v )(w · ∇)β dist ± (·, I v ) r c dx dt,
as well as
A surT en := − σˆT 0ˆR d ×S d−1 (s−ξ) · (s−ξ) · ∇ w dV t (x, s) dt + σˆT 0ˆR d (1 − θ t ) ξ · (ξ · ∇)w d|V t | S d−1 (x) dt + σˆT 0ˆR d (χ u − χ v )(w · ∇)(∇ · ξ) dx dt + σˆT 0ˆR d (χ u − χ v )∇w : ∇ξ T dx dt − σˆT 0ˆR d ξ · (n u − ξ) · ∇ w d|∇χ u | dt.
Notation. We use a∧b (respectively a∨b) as a shorthand notation for the minimum (respectively maximum) of two numbers a, b ∈ R.
Let Ω ⊂ R d be open. For a function u : Ω × [0, T ] → R, we denote by ∇u its distributional derivative with respect to space and by ∂ t u its derivative with respect to time. For p ∈ [1, ∞] and an integer k ∈ N 0 , we denote by L p (Ω) and W k,p (Ω) the usual Lebesgue and Sobolev spaces. In the special case p = 2 we use as usual H k (Ω) := W k,2 (Ω) to denote the Sobolev space. For integration of a function f with respect to the d-dimensional Lebesgue measure respectively the d − 1-dimensional surface measure, we use the usual notation´Ω f dx respectivelý I f dS. For measures other than the natural measure (the Lebesgue measure in case of domains Ω and the surface measure in case of surfaces I), we denote the corresponding Lebesgue spaces by L p (Ω, µ). The space of all compactly supported and infinitely differentiable functions on Ω is denoted by C ∞ cpt (Ω). The closure of C ∞ cpt (Ω) with respect to the Sobolev norm · W k,p (Ω) is W k,p 0 (Ω), and its dual will be denoted by W −1,p (Ω) where p ∈ [0, ∞] is the conjugated Hölder exponent of p, i.e. 1/p + 1/p = 1. For vector-valued fields, say with range in R d , we use the notation L p (Ω; R d ), and so on. For a Banach space X, a finite time T > 0 and a number p ∈ [1, ∞] we denote by L p ([0, T ]; X) the usual Bochner-Lebesgue space. If X itself is a Sobolev space W k,q , we denote the norm of L p ([0, T ]; X) as · L p t W k,q x . When writing L ∞ w ([0, T ]; X ) we refer to the space of bounded and weak- * measurable maps f : [0, T ] → X , where X is the dual space of X. By L p (Ω) + L q (Ω) we denote the normed space of all functions u : Ω → R which may be written as the sum of two functions v ∈ L p (Ω) and w ∈ L q (Ω). The space C k ([0, T ]; X) contains all k-times continuously differentiable and X-valued functions on [0, T ].
In order to give a suitable weak description of the evolution of the sharp interface, we have to recall the concepts of Caccioppoli sets as well as varifolds. To this end, let Ω ⊂ R d be open. We denote by BV(Ω) the space of functions with bounded variation in Ω. A measurable subset E ⊂ Ω is called a set of finite perimeter in Ω (or a Caccioppoli subset of Ω) if its characteristic function χ E is of bounded variation in Ω. We will write ∂ * E when referring to the reduced boundary of a Caccioppoli subset E of Ω; whereas n denotes the associated measure theoretic (inward pointing) unit normal vector field of ∂ * E. For detailed definitions of all these concepts from geometric measure theory, we refer to [49,30]. In case Ω has a C 2 boundary, we denote by H(x) the mean curvature vector at x ∈ ∂Ω. Recall that for a convex function g : R d → R the recession function g rec : R d → R is defined as g rec (x) := lim τ →∞ τ −1 g(τ x).
An oriented varifold is simply a non-negative measure V ∈ M(Ω×S d−1 ), where Ω ⊂ R d is open and S d−1 denotes the (d−1)-dimensional sphere. For a varifold V , we denote by |V | S d−1 ∈ M(Ω) its local mass density given by |V | S d−1 (A) := V (A × S d−1 ) for any Borel set A ⊂ Ω. For a locally compact separable metric space X we write M(X) to refer to the space of (signed) finite Radon-measures on X. If A ⊂ X is a measurable set and µ ∈ M(X), we let µ A be the restriction of µ on A. The k-dimensional Hausdorff measure on R d will be denoted by H k , whereas we write L d (A) for the d-dimensional Lebesgue measure of a Lebesgue measurable set A ⊂ R d .
Finally, let us fix some tensor notation. First of all, we use (∇v) ij = ∂ j v i as well as ∇ ·v = i ∂ i v i for a Sobolev vector field v : R d → R d . The symmetric gradient is denoted by D sym v := 1 2 (∇v + ∇v T ). For time-dependent fields v : R d × [0, T ) → R n we denote by ∂ t v the partial derivative with respect to time. Tensor products of vectors u, v ∈ R d will be given by
(u ⊗ v) ij = u i v j . For tensors A = (A ij ) and B = (B ij ) we write A : B = ij A ij B ij .
3. Outline of the strategy 3.1. The relative entropy. The basic idea of the present work is to measure the "distance" between a varifold solution to the two-phase Navier-Stokes equation (χ u , u, V ) and a strong solution to the two-phase Navier-Stokes equation (χ v , v) by means of the relative entropy functional
E χ u , u, V χ v , v (t) :=σˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | +ˆR d ρ(χ u ) 2 |u − v − w| 2 dx + σˆR d 1 − θ t d|V t | S d−1 (12) +ˆR d |χ u − χ v | β dist ± (x, I v (t)) r c dx where ξ : R d × [0, T strong ) → R d
is a suitable extension of the unit normal vector field of the interface of the strong solution and where w is a vector field that will be constructed below and that vanishes in case of equal viscosities µ + = µ − . More precisely, we choose ξ as
ξ(x, t) := n v (P Iv(t) x)(1 − dist(x, I v (t)) 2 )η(dist(x, I v (t)))
for some cutoff η with η(s) = 1 for s ≤ 1 2 r c and η ≡ 0 for s ≥ r c , where P Iv(t) x denotes for each t ≥ 0 the projection of x onto the interface I v (t) of the strong solution and where the unit normal vector field n v of the interface of the strong solution is oriented to point towards {χ v (·, t) = 1}.
Rewriting the relative entropy functional in the form
E χ u , u, V χ v , v (t) = E[χ u , u, V ](t) +ˆR d χ u ∇ · ξ dx −ˆR d ρ(χ u )u · (v + w) dx +ˆR d 1 2 ρ(χ u )|v + w| 2 dx +ˆR d (χ u − χ v )β dist ± (x, I v (t)) r c dx
with the energy (6d), we see that we may estimate the time evolution of the relative entropy E χ u , u, V χ v , v (t) by exploiting the energy dissipation property (6c) of the varifold solution and by testing the weak formulation of the two-phase Navier-Stokes equation (6a) and (6b) against the (sufficiently regular) test functions v + w respectively 1 2 |v + w| 2 , ∇ · ξ, and β( dist ± (x,Iv(t)) rc ). As usual in the derivation of weak-strong uniqueness results by the relative entropy method of Dafermos [37] and Di Perna [44], in the case of equal viscosities µ + = µ − the goal is the derivation of an estimate of the form
E χ u , u, V χ v , v (T ) + cˆT 0ˆR d |∇u − ∇v| 2 dx dt (13) ≤ C(v, I v , data)ˆT 0 E χ u , u, V χ v , v (t) dt
which implies uniqueness and stability by means of the Gronwall lemma and by the coercivity properties of the relative entropy functional discussed in the next section.
In the case of different viscosities µ + = µ − , we will derive a slightly weaker (but still sufficient) result of roughly speaking the form
E χ u , u, V χ v , v (T ) + cˆT 0ˆR d |∇u − ∇v − ∇w| 2 dx dt (14) ≤ C(v, I v , data)ˆT 0 E χ u , u, V χ v , v (t) log E χ u , u, V χ v , v (t) dt,
along with estimates on w which include in particular the bound
R d |w(·, T )| 2 dx ≤ C(v, I v , data)E χ u , u, V χ v , v (T ).
3.2. The error control provided by the relative entropy functional. The relative entropy functional (12) provides control of the following quantities (up to bounded prefactors): Velocity error control. The relative entropy E[χ u , u, V |χ v , v](t) controls the square of the velocity error in the L 2 norm R d |u(·, t) − v(·, t)| 2 dx at any given time t. In the case of equal viscosities, this is immediate from (12) by w ≡ 0, while in the case of different viscosities this follows by the estimaté
R d |w| 2 dx ≤ C ∇v L ∞´R d 1 − ξ · ∇χu
|∇χu| d|∇χ u | which is a consequence of the construction of w and the choice of ξ, see below.
Interface error control. The relative entropy provides a tilt-excess type control of the error in the interface normal
R d 1 − ξ · n u d|∇χ u |.
In particular, it controls the squared error in the interface normal
R d |n u − ξ| 2 d|∇χ u |.
The term also controls the total length respectively area (for d = 2 respectively d = 3) of the part of the interface I u which is not locally a graph over I v , see Figure 2. For example, in the region around the left purple half-ray the interface of the weak solution is not a graph over the interface of the weak solution. Furthermore, the term controls the length respectively area (for d = 2 respectively d = 3) of the part of the interface with distance to I v (t) greater than the cutoff length r c , as there we have ξ ≡ 0.
h + (x) {χ u = 1} {χ u = 0} {χ v = 1} {χ v = 0}
Denote the local height of the one-sided interface error by h + : I v (t) → R + 0 as measured along orthogonal rays originating on I v (t) (with some cutoff applied away from the interface I v (t) of the strong solution); denote by h − the corresponding height of the interface error as measured in the other direction. For example, in Figure 2 the quantity h + (x) for some base point x ∈ I v (t) would correspond to the accumulated length of the solid segments in each of the purple rays, the dotted segments not being counted. Note that the rays are orthogonal on I v (t). Then the tilt-excess type term in the relative entropy also controls the gradient of the one-sided interface error heightŝ
Iv(t) min{|∇h ± | 2 , |∇h ± |} dS.
Note that wherever I u (t) is locally a graph over I v (t) and is not too far away from I v (t), it must be the graph of the function h + − h − . Here, the graph of a function g over the curved interface I v (t) is defined by the set of points obtained by shifting the points of I v (t) by the corresponding multiple of the surface normal, i. e. {x + g(x)n v (x) : x ∈ I v (t)}. Varifold multiplicity error control. For varifold solutions, the relative entropy controls the multiplicity error of the varifold
R d 1 − θ t (x) d|V t | S d−1 (note that 1
θt(x) corresponds to the multiplicity of the varifold), which in turn by the compatibility condition (6e) and the definition of θ t (see (7)) controls the squared error in the normal of the varifold
h e(t) (x) h + (x) {χ u = 1} {χ u = 0} {χ v = 1} {χ v = 0}R dˆSd−1 |s − n u | 2 dV t (s, x).
Weighted volume error control. Furthermore, the error in the volume occupied by the two fluids weighted with the distance to the interface of the strong solution
R d |χ u − χ v | min{dist(x, I v ), 1} dx
is controlled. Note that this term is the only term in the relative entropy which is not obtained by the usual relative entropy ansatz
E[x|y] = E[x] − DE[y](x − y) − E[y]
. We have added this lower-order term -as compared to the term´R d 1 − ξ · ∇χu |∇χu| d|∇χ u | which provides tilt-excess-type control -to the relative entropy in order to remove the lack of coercivity of the term´R d 1 − ξ · ∇χu |∇χu| d|∇χ u | in the limit of vanishing interface length of the varifold solution. Control of velocity gradient error by dissipation. By means of Korn's inequality, the dissipation term controls the L 2 -error in the gradient T 0ˆR d |∇u − ∇v − ∇w| 2 dx dt.
3.3. The case of equal viscosities. For equal viscosities µ + = µ − , one may choose w ≡ 0. As a consequence, the right-hand side in the relative entropy inequality -see Proposition 9 above -may be post-processed to yield the Gronwall-type estimate (13). The details are provided in Section 5.
3.4.
Additional challenges in the case of different viscosities. In the case of different viscosities µ 1 = µ 2 of the two fluids, even for strong solutions the normal derivative of the tangential velocity features a discontinuity at the interface: By the no-slip boundary condition, the velocity is continuous across the interface [v] = 0 and the same is true for its tangential derivatives [(t·∇)v] = 0. As a consequence of this, the discontinuity of µ(χ v ) across the interface and the equilibrium condition for the stresses at the interface [[µ(χ)t · (n · ∇)v + µ(χ)n · (t · ∇)v]] = 0 entail for generic data a discontinuity of the normal derivative of the tangential velocity t · (n · ∇)v across the interface.
As a consequence, it becomes impossible to establish a Gronwall estimate for the standard relative entropy (12) with w ≡ 0. To see this, consider in the twodimensional case d = 2 two strong solutions u and v with coinciding initial velocities u(·, 0) = v(·, 0) = u 0 (·), but slightly different initial interfaces χ v (·, 0) = χ {|x|≤1} and χ u (·, 0) = χ {|x|≤1−ε} for some ε > 0. The initial relative entropy is then of the order ∼ ε 2 . Suppose that (in polar coordinates) the initial velocity u 0 has a profile near the interface like
u 0 (x, y) = µ − (r − 1)e φ for r = x 2 + y 2 < 1, µ + (r − 1)e φ for r > 1.
Note that this velocity profile features a kink at the interface. As one verifies readily, as far as the viscosity term is concerned this corresponds to a near-equilibrium profile for the solution (χ v , v) (in the sense that the viscosity term is bounded). However, in the solution (χ u , u) the interface is shifted by ε and the profile is no longer an equilibrium profile. By the scaling of the viscosity term, the timescale within which the profile u 0 equilibrates in the annulus of width ε towards a nearaffine profile is of the order of ε 2 . After this timescale, the velocity u will have changed by about ε in a layer of width ∼ ε around the interface; at the same time, due to the mostly parallel transport at the interface the solution will not have changed much otherwise. As a consequence, the term´1 2 ρ(χ u )|u − v| 2 dx will be of the order of at least cε 3 after a time T ∼ ε 2 , while the other terms in the relative entropy are essentially the same. Thus, the relative entropy has grown by a factor of 1 + cε within a timescale ε 2 , which prevents any Gronwall-type estimate. At the level of the relative entropy inequality (see Proposition 9), the derivation of the Gronwall inequality is prevented by the viscosity terms, which read for w ≡ 0
−ˆµ (χ u ) 2 ∇u + ∇u T − (∇v + ∇v T ) 2 dx +ˆ(µ(χ v ) − µ(χ u ))∇v : ∇u + ∇u T − (∇v + ∇v T ) dx.
The latter term prevents the derivation of a dissipation estimate: While it is formally quadratic in the difference of the two solutions (χ u , u) and (χ v , v), due to the (expected) jump of the velocity gradients ∇v and ∇u at the respective interfaces it is in fact only linear in the interface error.
The key idea for our weak-strong uniqueness result in the case of different viscosities is to construct a vector field w which is small in the L 2 norm but whose gradient compensates for most of the problematic term (µ(χ v ) − µ(χ u ))(∇v + ∇v T ). To be precise, it is only the normal derivative of the tangential component of v which may be discontinuous at the interface; the tangential derivatives are continuous by the no-slip boundary condition, while the normal derivative of the normal component is continuous by the condition ∇ · v = 0.
Let us explain our construction of the vector field w at the simple two-dimensional example of a planar interface of the strong solution I v = {(x, 0) : x ∈ R}. In this setting, we would like to set for y > 0
w + (x, y, t) :=c(µ + , µ − )ˆy ∧h + (x) 0 (e x · ∂ y v)(x,ỹ)e x dỹ
(where e x just denotes the first vector of the standard basis). Note that due to the bounded integrand, this vector field w + (x, y) is bounded by Ch + (x), i. e. it is bounded by the interface error. As we shall see in the proof, the time derivative of w + is also bounded in terms of other error terms. The tangential spatial derivative of this vector field ∂ x w + (x, y, t) is given (up to a constant factor) bý
y∧h + (x) 0 (e x · ∂ x ∂ y v)(x,ỹ)e x dỹ + χ y≥h + (x) (e x · ∂ y v)(x, h + (x))∂ x h + (x)e x which is also a term controlled by Ch + (x) + C|∂ x h + (x)|.
The normal derivative, on the other hand, is given by
∂ y w + (x, y, t) = c(µ + , µ − )χ {0≤y≤h + (x)} (e x · ∂ y v)(x, y)e x
which (upon choosing c(µ + , µ − )) would precisely compensate the discontinuity of ∂ y (e x · v) in the region in which the interface of the weak solution is a graph of a function over I v . Note that our relative entropy functional provides a higher-order control of the size of the region in which the interface of the weak solution is not a graph over the interface of the strong solution.
However, with this choice of vector field w + (x, y, t), two problems occur: First, the vector field is not solenoidal. For this reason, we introduce an additional Helmholtz projection. Second -and constituting a more severe problem -, the vector field would not necessarily be (spatially) Lipschitz continuous (as the derivative contains a term with ∂ x h + (x) which is not necessarily bounded), which due to the surface tension terms would be required for the derivation of a Gronwall-type estimate. For this reason, we first regularize the height function h + by mollification on a scale of the order of the error. See Proposition 25 and Proposition 26 for details of our construction of the regularized height function. The actual construction of our compensation function w is performed in Proposition 27. We then derive an estimate in the spirit of (14) in Proposition 33.
Time evolution of geometric quantities and further coercivity
properties of the relative entropy functional 4.1. Time evolution of the signed distance function. In order to describe the time evolution of various constructions, we need to recall some well-known properties of the signed distance function. Let us start by introducing notation. For a family (Ω + t ) t∈[0,Tstrong) of smoothly evolving domains with smoothly evolving interfaces I(t) in the sense of Definition 5, the associated signed distance function is given by
dist ± (x, I(t)) := dist(x, I(t)), x ∈ Ω + t , −dist(x, I(t)), x / ∈ Ω + t .(15)
From Definition 5 of a family of smoothly evolving domains it follows that the family of maps Φ t :
I(t) × (−r c , r c ) → R d given by Φ t (x, y) := x + yn(x, t) are C 2 -diffeomorphisms onto their image {x ∈ R d : dist(x, I(t)) < r c } subject to the bounds |∇Φ t | ≤ C, |∇Φ −1 t | ≤ C.(16)
The signed distance function (resp. its time derivative) to the interface of the strong solution is then of class C 0 t C 3 x (resp. C 0 t C 2 x ) in the space-time tubular neighborhood t∈[0,Tstrong) im(Φ t ) × {t} due to the regularity assumptions in Definition 5. We also have the bounds
|∇ k+1 dist ± (x, I(t))| ≤ Cr −k c , , k = 1, 2,(17)
and in particular for the mean curvature vector
|H| ≤ Cr −1 c .(18)
Moreover, the projection P I(t) x of a point x onto the nearest point on the manifold I(t) is well-defined and of class C 0 t C 2 x in the same tubular neighborhood. After having introduced the necessary notation we study the time evolution of the signed distance function to the interface of the strong solution. Because of the kinematic condition that the interface is transported with the flow, we obtain the following statement.
Lemma 10. Let χ v ∈ L ∞ ([0, T strong ); BV(R d ; {0, 1})) be an indicator function such that Ω + t := {x ∈ R d : χ v (x, t) = 1}
is a family of smoothly evolving domains and I v (t) := ∂Ω + t is a family of smoothly evolving surfaces in the sense of Defi-
nition 5. Let v ∈ L 2 loc ([0, T strong ]; H 1 loc (R d ; R d )) be a continuous solenoidal vector field such that χ v solves the equation ∂ t χ v = −∇ · (χ v v).
The time evolution of the signed distance function to the interface I v (t) is then given by
(19) ∂ t dist ± (x, I v (t)) = − V n (x, t) · ∇ dist ± (x, I v (t))
for any t ∈ [0, T strong ] and any x ∈ R d with dist(x, I v (t)) ≤ r c , whereV n is the extended normal velocity of the interface given bȳ
V n (x, t) = v(P Iv(t) x, t) · n v (P Iv(t) x, t) n v (P Iv(t) x, t).(20)
Moreover, the following formulas hold true
∇ dist ± (x, I v (t)) = n v (P Iv(t) x, t),(21)∇ dist ± (x, I v (t)) · ∂ t ∇ dist ± (x, I v (t)) = 0,(22)∇ dist ± (x, I v (t)) · ∂ j ∇ dist ± (x, I v (t)) = 0, j = 1, . . . , d,(23)∂ t dist ± (x, I v (t)) = ∂ t dist ± (y, I v (t)) y=P Iv (t) x ,(24)
for all (x, t) such that dist(x, I v (t)) ≤ r c . The gradient of the projection onto the nearest point on the interface I v (t) is given by
∇P Iv(t) x = Id − n v (P Iv(t) x) ⊗ n v (P Iv(t) x) − dist ± (x, I v (t))∇ 2 dist ± (x, I v (t)).(25)
In particular, we have the bound
|∇P Iv(t) x| ≤ C (26) for all (x, t) such that dist(x, I v (t)) ≤ r c .
Proof. Recall that ∇ dist ± (x, I v (t)) for a point x ∈ I v (t) on the interface equals the inward pointing normal vector n v (x, t) of the interface I v (t). This also extends away from the interface in the sense that
∇ dist ± (y, I v (t)) y=P Iv (t) x = n v (P Iv(t) x, t) = ∇ dist ± (y, I v (t)) y=x(27)
for all (x, t) such that dist(x, I v (t)) < r c , i. e. (21) holds. Hence, we also have the
formula P Iv(t) x = x − dist ± (x, I v (t))∇ dist ± (x, I v (t))
. Differentiating this representation of the projection onto the interface and using the fact that n v is a unit vector we obtain using also (28)
∇ dist ± (y, I v (t)) y=P Iv (t) x · ∂ t P Iv(t) x = −∂ t dist ± (x, I v (t)) − dist ± (x, I v (t))∇ dist ± (P Iv(t) x, I v (t))∂ t dist ± (x, I v (t)) = −∂ t dist ± (x, I v (t)) − dist ± (x, I v (t))∂ t 1 2 |∇ dist ± (x, I v (t))| = −∂ t dist ± (x, I v (t)).
Hence, we obtain in addition to (27) the formula
∂ t dist ± (x, I v (t)) = ∂ t dist ± (y, I v (t)) y=P Iv (t) x .
On the other side, on the interface the time derivative of the signed distance function equals up to a sign the normal speed. In our case, the latter is given by the normal component of the given velocity field v evaluated on the interface, see Remark 8. This concludes the proof of (19). Moreover, (22) as well as (23) follow immediately from differentiating |∇ dist ± (x, I v (t))| 2 = 1. Finally, (25) and (26) follow immediately from (17) and
P Iv(t) x = x − dist ± (x, I v (t))n v (P Iv(t) x).
In the above considerations, we have made use of the following result: Consider the auxiliary function g(
x, t) = dist ± (P Iv(t) x, I v (t)) for (x, t) such that dist(x, I v (t)) < r c . Since this function vanishes on the space-time tubular neighborhood of the in- terface t∈(0,Tstrong) {x ∈ R d : dist(x, I v (t)) < r c } × {t} we compute 0 = d dt g(x, t) = ∂ t dist ± (y, I v (t)) y=P Iv (t) x + ∇ dist ± (y, I v (t)) y=P Iv (t) x · ∂ t P Iv(t) x.(28)
Remark 11. Consider the situation of Lemma 10. We proved that
∂ t dist ± (x, I v (t)) = −v(P Iv(t) x, t) · n v (P Iv(t) x, t).
The right hand side of this identity is of class L ∞ t W 2,∞
x , as the normal component n v (P Iv(t) ) · ∇v of the velocity gradient ∇v of a strong solution is continuous across the interface I v (t). To see this, one first observes that the tangential derivatives ((Id −n v (P Iv(t) ⊗ n v (P Iv(t) )∇)v are naturally continuous across the interface; one then uses the incompressibility constraint ∇ · v = 0 to deduce that n v (P Iv(t) · (n v (P Iv(t) · ∇)v is also continuous across the interface.
4.2.
Properties of the vector field ξ. The vector field ξ -as defined in Proposition 9 -is an extension of the unit normal vector field n v associated to the family of smoothly evolving domains occupying the first fluid of the strong solution. We now provide a more detailed account of its definition. The construction in fact consists of two steps. First, we extend the normal vector field n v to a (space-time) tubular neighborhood of the evolving interfaces I v (t) by projecting onto the interface. Second, we multiply this construction with a cutoff which decreases quadratically in the distance to the interface of the strong solution (see (35)).
Definition 12. Let χ v ∈ L ∞ ([0, T strong ); BV(R d ; {0, 1})) be an indicator function such that Ω + t := {x ∈ R d : χ v (x, t) = 1}
is a family of smoothly evolving domains and I v (t) := ∂Ω + t is a family of smoothly evolving surfaces in the sense of Definition 5. Let η be a smooth cutoff function with η(s) = 1 for s ≤ 1 2 and η ≡ 0 for s ≥ 1. Define another smooth cutoff function ζ : R → [0, ∞) as follows:
ζ(r) = (1 − r 2 )η(r), r ∈ [−1, 1],(29)
and ζ ≡ 0 for |r| > 1. Then, we define a vector field ξ :
R d × [0, T strong ) → R d by ξ(x, t) := ζ dist ± (x,Iv(t)) rc n v (P Iv(t) x, t) for (x, t) with dist(x, I v (t)) < r c , 0 else.(30)
The definition of ξ has the following consequences.
Remark 13. Observe that the vector field ξ is indeed well-defined in the spacetime domain R d × [0, T strong ) due to the action of the cut-off function ζ; it also satisfies |ξ| ≤ 1 or, more precisely, the sharper inequality |ξ| ≤ (1−dist(x, I v (t)) 2 ) + . Furthermore, the extension ξ inherits its regularity from the regularity of the signed distance function to the interface I v (t). More precisely, it follows that the vector field ξ (resp. its time derivative) is of class
L ∞ t W 2,∞ x (resp. W 1,∞ t W 1,∞ x ) globally in R d × [0, T strong ), and the restrictions to the domains {χ v = 0} and {χ v = 1} are of class L ∞ t C 2
x . This turns out to be sufficient for our purposes.
The time derivative of our vector field ξ is given as follows.
Lemma 14. Let χ v ∈ L ∞ ([0, T strong ); BV(R d ; {0, 1})) be an indicator function such that Ω + t := {x ∈ R d : χ v (x, t) = 1}
is a family of smoothly evolving domains and I v (t) := ∂Ω + t is a family of smoothly evolving surfaces in the sense of Defi-
nition 5. Let v ∈ L 2 loc ([0, T strong ]; H 1 loc (R d ; R d )) be a continuous solenoidal vector field such that χ v solves the equation ∂ t χ v = −∇ · (χ v v).
LetV n be the extended normal velocity of the interface (20). Then the time evolution of the vector field ξ from Definition 12 is given by
∂ t ξ = −(V n · ∇)ξ − Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇V n ) T ξ (31)
in the space-time domain dist(x, I v (t)) < r c . Here, we made use of the abbreviation
n v (P Iv(t) x) = n v (P Iv(t) x, t).
Proof. We start by deriving a formula for the time evolution of the normal vector field n v (P Iv(t) x, t) in the space-time tubular neighborhood dist(x, I v (t)) < r c . By (21), we may use the formula for the time evolution of the signed distance function from Lemma 10. More precisely, due to the regularity of the signed distance function to the interface of the strong solution and the regularity of the vector fieldV (Remark 11), we can interchange the differentiation in time and space to obtain
∂ t ∇ dist ± (x, I v (t)) = ∇∂ t dist ± (x, I v (t)) (19) = −∇ (V n · ∇) dist ± (x, I v (t)) = −(V n · ∇)n v (P Iv(t) x) − (∇V n ) T · n v (P Iv(t) x).
Next, we show that the normal-normal component of ∇V n vanishes. Observe that by Remark 11 and (21) it holds
V n (x, t) = −∂ t dist ± (x, I v (t))∇ dist ± (x, I v (t)).
Hence, by (21)- (24) and this formula we obtain
(∇V n ) T (x, t) : n v (P Iv(t) x) ⊗ n v (P Iv(t) x) = ∇V n (x, t)∇ dist ± (x, I v (t)) · ∇ dist ± (x, I v (t)) = −∇ dist ± (x, I v (t)) · ∂ t ∇ dist ± (x, I v (t)) +V n (x, t) ⊗ ∇ dist ± (x, I v (t)) : ∇ 2 dist ± (x, I v (t))
= 0 as desired. In summary, we have proved so far that
∂ t n v (P Iv(t) x) = −(V n · ∇)n v (P Iv(t) x) (32) − Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇V n ) T · n v (P Iv(t) x),
which holds in the space-time domain dist(x, I v (t)) < r c . However, applying the chain rule to the cut-off function r → ζ(r) from (29) together with the evolution equation (19) for the signed distance to the interface shows that the cut-off away from the interface is also subject to a transport equation:
∂ t ζ dist ± (x, I v (t)) r c = −(V n (x, t) · ∇)ζ dist ± (x, I v (t)) r c .
By the definition of the vector field ξ, see (30), and the product rule, this concludes the proof.
4.3.
Properties of the weighted volume term. We next discuss the weighted volume contribution´R d |χ u − χ v | dist(x, I v (t)) dx to the relative entropy in more detail.
Remark 15. Let β be a truncation of the identity as in Proposition 9. Let χ v ∈ L ∞ ([0, T strong ); BV(R d ; {0, 1})) be an indicator function such that Ω + t := {x ∈ R d : χ v (x, t) = 1} is a family of smoothly evolving domains, and I v (t) := ∂Ω + t is a family of smoothly evolving surfaces, in the sense of Definition 5. The map
R d × [0, T strong ) (x, t) → β dist ± (x, I v (t))/r c
inherits the regularity of the signed distance function to the interface I v (t). More precisely, this map (resp. its time derivative) is of class
C 0 t C 3 x (resp. C 0 t C 2 x ). Lemma 16. Let χ v ∈ L ∞ ([0, T strong ); BV(R d ; {0, 1})) be an indicator function such that Ω + t := {x ∈ R d : χ v (x, t) = 1}
is a family of smoothly evolving domains and I v (t) := ∂Ω + t is a family of smoothly evolving surfaces in the sense of Defi-
nition 5. Let v ∈ L 2 loc ([0, T strong ]; H 1 loc (R d ; R d )) be a continuous solenoidal vector field such that χ v solves the equation ∂ t χ v = −∇ · (χ v v).
LetV n be the extended normal velocity of the interface (20). Then the time evolution of the weight function β composed with the signed distance function to the interface I v (t) is given by the transport equation
(33) ∂ t β dist ± (·, I v ) r c = − V n · ∇ β dist ± (·, I v ) r c for space-time points (x, t) such that dist(x, I v (t)) < r c .
Proof. This is immediate from the chain rule and the time evolution of the signed distance function to the interface of the strong solution, see Lemma 10.
4.4.
Further coercivity properties of the relative entropy. We collect some further coercivity properties of the relative entropy functional E χ u , u, V χ v , v as defined in (11). These will be of frequent use in the post-processing of the terms occurring on the right hand side of the relative entropy inequality from Proposition 9. We start for reference purposes with trivial consequences of our choices of the vector field ξ and the weight function β.
Lemma 17. Consider the situation of Proposition 9. In particular, let β be the truncation of the identity from Proposition 9. By definition, it holds
min dist(x, I v (t)) r c , 1 ≤ β dist ± (x, I v (t)) r c .(34)
Let ξ be the vector field from Definition 12 with cutoff multiplier ζ as given in (29). By the choice of the cutoff ζ, it holds
| dist ± (x, I v (t))| 2 r 2 c ≤ 1 − ζ dist ± (x, I v (t)) r c .(35)
We will also make frequent use of the fact that for any unit vector b ∈ R d we have
1 − ζ dist ± (x, I v (t)) r c ≤ 1 − b · ξ and |b − ξ| 2 ≤ 2(1 − b · ξ).(36)
We also want to emphasize that the relative entropy functional controls the squared error in the normal of the varifold.
Lemma 18. Consider the situation of Proposition 9. We then havê
R d ×S d−1 1 2 |s − ξ| 2 | dV t (x, s) ≤ E χ u , u, V χ v , v (t) (37)
for almost every t ∈ [0, T strong ).
Proof. Observe first that by means of the compatibility condition (6e) we havê
R d ×S d−1 (1 − s · ξ ) dV t (x, s) =ˆR d ×S d−1 1 dV t (x, s) −ˆR d n u · ξ d|∇χ u (·, t)| =ˆR d 1 d|V t | S d−1 −ˆR d n u · ξ d|∇χ u (·, t)|,
which holds for almost every t ∈ [0, T strong ). In addition, due to (8) one obtainŝ
R d 1 − θ t d|V t | S d−1 =ˆR d 1 d|V t | S d−1 −ˆR d 1 d|∇χ u (·, t)|
for almost every t ∈ [0, T strong ). This in turn entails the following identitŷ
R d 1 − n u · ξ d|∇χ u | +ˆR d 1 − θ t d|V t | S d−1 =ˆR d ×S d−1 (1 − s · ξ ) dV t (x, s),
which holds true for almost every t ∈ [0, T strong ). However, the functional on the right hand side controls the squared error in the normal of the varifold: |s − ξ| 2 ≤ 2(1 − s · ξ). This proves the claim.
We will also refer multiple times to the following bound. In the regime of equal shear viscosities µ + = µ − we may apply this result with the choice w = 0. In the general case, we have to include the compensation function w for the velocity gradient discontinuity at the interface.
Lemma 19. Let (χ u , u, V ) be a varifold solution to (1a)-(1c) in the sense of Defini- tion 2 on a time interval [0, T vari ) with initial data (χ 0 u , u 0 ). Let (χ v , v) be a strong solution to (1a)-(1c) in the sense of Definition 6 on a time interval [0, T strong ) with T strong ≤ T vari and initial data (χ 0 v , v 0 ). Let w ∈ L 2 ([0, T strong ); H 1 (R d ; R d )) be an arbitrary vector field, and let F ∈ L ∞ (R d × [0, T strong ); R d ) be a bounded vector field. Then ˆT 0ˆR d (χ u −χ v )(u − v − w) · F dx dt ≤ δˆT 0ˆR d |∇(u − v − w)| 2 dx dt + C 1 + F 2 L ∞ δˆT 0ˆR d ρ(χ u )|u−v−w| 2 dx dt + C F L ∞ δˆT 0ˆR d |χ u −χ v | β dist ± (·, I v ) r c dx dt
for almost every T ∈ [0, T strong ) and all 0 < δ ≤ 1. The absolute constant C > 0 only depends on the densities ρ ± .
Proof. We first argue how to control the part away from the interface of the strong solution, i.e., outside of {(x, t) : dist(x, I v (t)) ≥ r c }. A straightforward estimate using Hölder's and Young's inequality yields ˆT 0ˆ{dist(x,Iv(t))≥rc}
(χ u −χ v )(u−v−w) · F dx dt ≤ F L ∞ 2ˆT 0ˆ{dist(x,Iv(t))≥rc} |χ u − χ v | dx dt + F L ∞ 2ˆT 0ˆ{dist(x,Iv(t))≥rc} |u − v − w| 2 dx dt.
Note that by the properties of the truncation of the identity β, see Proposition 9, it follows that |β(dist ± (x,
I v (t))/r c )| ≡ 1 on {(x, t) : dist(x, I v (t)) ≥ r c }. Hence, we obtain (38) ˆT 0ˆ{dist(x,Iv(t))≥rc} (χ u −χ v )(u−v−w) · F dx dt ≤ F L ∞ 2ˆT 0ˆR d |χ u − χ v | · β dist ± (·, I v ) r c dx dt + F L ∞ 2(ρ + ∧ ρ − )ˆT 0ˆR d ρ(χ u )|u − v − w| 2 dx dt,
which is indeed a bound of required order. We proceed with the bound for the contribution in the vicinity of the interface of the strong solution. To this end, recall that we are equipped with a family of maps Φ t : (16). We then move on with a change of variables, the one-dimensional Gagliardo-Nirenberg-Sobolev interpolation inequality
I v (t) × (−r c , r c ) → R d given by Φ t (x, y) := x + yn v (x, t), which are C 2 -diffeomorphisms onto their image {x ∈ R d : dist(x, I v (t)) < r c }. Recall the estimatesg L ∞ (−rc,rc) ≤ C g 1 2 L 2 (−rc,rc) ∇g 1 2 L 2 (−rc,rc) + C g L 2 (−rc,rc)
as well as Hölder's and Young's inequality to obtain the bound ˆT
0ˆ{dist(x,Iv(t))<rc} (χ u − χ v )(u − v − w) · F dx dt ≤ C F L ∞ˆT 0ˆIv(t)ˆr c −rc |(χ u −χ v )|(Φ t (x, y)) |(u−v−w)|(Φ t (x, y)) dy dS(x) dt ≤ C F L ∞ˆT 0ˆIv(t) sup y∈[−rc,rc] |u − v − w|(x + yn v (x, t)) × ˆr c −rc |(χ u −χ v )|(x + yn v (x, t)) dy dS(x) dt ≤ C F L ∞ + F 2 L ∞ δˆT 0ˆR d |u − v − w| 2 dx dt + δˆT 0ˆR d |∇(u − v − w)| 2 dx dt + C F L ∞ˆT 0ˆIv(t) ˆr c −rc |(χ u −χ v )|(x + yn v (x, t)) dy 2 dS(x) dt.
It thus suffices to derive an estimate for the L 2 -norm of the local interface error height in normal direction
h(x) =ˆr c −rc |(χ u −χ v )|(x + yn v (x, t)) dy.
The proof of Proposition 25 below, where we establish next to the required L 2bound also several other properties of the local interface error height, shows that (see (58)
)ˆI v (t) |h(x)| 2 dS ≤ CˆR d |χ u −χ v | min dist(x, I v (t)) r c , 1 dx.(39)
This then concludes the proof.
We conclude this section with an L 2 tan L ∞ nor -bound for H 1 -functions on the tubular neighborhood around the evolving interfaces as well as a bound for the derivatives of the normal velocity of the interface of a strong solution in terms of the associated velocity field v, both of which will be used several times in the post-processing of the terms on the right hand side of the relative entropy inequality of Proposition 9.
Lemma 20. Consider the situation of Proposition 9. We have the estimatê
Iv(t) sup y∈[−rc,rc] |g(x + yn v (x, t))| 2 dS ≤ C( g L 2 ∇g L 2 + g 2 L 2 ) (40) valid for any g ∈ H 1 (R d ).
Proof. Let f ∈ H 1 (−r c , r c ). The one-dimensional Gagliardo-Nirenberg-Sobolev inerpolation inequality then implies
f L ∞ (−rc,rc) ≤ C f 1 2 L 2 (−rc,rc) f 1 2 L 2 (−rc,rc) + C f L 2 (−rc,rc) .
WEAK-STRONG UNIQUENESS FOR TWO-PHASE FLOW WITH SHARP INTERFACE 29
From this we obtain together with Hölder's inequalitŷ
Iv(t) sup y∈[−rc,rc] |g(x + yn v (x, t))| 2 dS ≤ CˆI v (t)ˆr c −rc |g(x+yn v (x, t))| 2 dy dS + C ˆI v (t)ˆr c −rc |g(x+yn v (x, t))| 2 dy dS 1 2 × ˆI v (t)ˆr c −rc |∇g(x+yn v (x, t))| 2 dy dS 1 2 .
This implies (40)
by making use of the C 2 -diffeomorphisms Φ t : I v (t) × (−r c , r c ) → R d given by Φ t (x, y) = x + yn v (x, t)
and the associated change of variables, using also the bound (16).
Lemma 21.
Consider the situation of Proposition 9 and define the vector field
V n (x, t) := v(x, t) · n v (P Iv(t) x, t) n v (P Iv(t) x, t), for (x, t) ∈ R d × [0, T strong ) such that dist(x, I v (t)) < r c . Then ∇V n L ∞ (O) ≤ Cr −1 c v L ∞ + C ∇v L ∞ ,(41)∇ 2 V n L ∞ (O) ≤ Cr −2 c v L ∞ + Cr −1 c ∇v L ∞ + C ∇ 2 v L ∞ t L ∞ x (R d \Iv(t)) ,(42)
where O = t∈(0,Tstrong) {x ∈ R d : dist(x, I v (t)) < r c } × {t} denotes the space-time tubular neighborhood of width r c of the evolving interface of the strong solution.
In particular, we have forV n (x, t) := V n (P Iv(t) x, t) the estimate
|V n (x, t) − V n (x, t)| ≤ Cr −1 c ||v|| W 1,∞ dist(x, I v (t)).(43)
Proof. The estimates (41) and (42) are a direct consequence of the regularity requirements on the velocity field v of a strong solution, see Definition 6, the pointwise bounds (17) and the representation of the normal vector field on the interface in terms of the signed distance function (21).
Weak-strong uniqueness of varifold solutions to two-fluid
Navier-Stokes flow: The case of equal viscosities
In this section we provide a proof of the weak-strong uniqueness principle to the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) in the case of equal shear viscosities µ + = µ − . Note that in this case the problematic viscous stress term R visc in the relative entropy inequality (see Proposition 9) vanishes because of µ(χ u ) − µ(χ v ) = 0. In this setting, it is possible to choose w ≡ 0 which directly implies A visc = 0, A adv = 0, A dt = 0, A weightV ol = 0, and A surT en = 0. What remains to be done is a post-processing of the terms R surT en , R adv , R dt , and R weightV ol which remain on the right-hand side of the relative entropy inequality. 5.1. Estimate for the surface tension terms. We start by post-processing the terms related to surface tension R surT en .
Lemma 22. Consider the situation of Proposition 9. The terms related to surface tension R surT en are estimated by
R surT en ≤ δˆT 0ˆR d |∇(u − v − w)| 2 dx dt + C(δ)r −4 c 1 + v 2 L ∞ t W 2,∞ x ˆT 0 E[χ u , u, V |χ v , v](t) dt (44)
for any δ > 0.
Proof. We start by using (36) and (30)
to estimate − σˆT 0ˆR d ξ · ∇χ u |∇χ u | n v (P Iv(t) x) · n v (P Iv(t) x) · ∇ v − ξ · (ξ · ∇)v d|∇χ u | dt = σˆT 0ˆR d 1 − ξ · ∇χ u |∇χ u | n v (P Iv(t) x) · n v (P Iv(t) x) · ∇ v d|∇χ u | dt + σˆT 0ˆR d ξ · (ξ · ∇)v − n v (P Iv(t) x) · n v (P Iv(t) x) · ∇ v d|∇χ u | dt ≤ C ∇v L ∞ˆT 0ˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | dt + C ∇v L ∞ˆT 0ˆR d 1 − ζ dist ± (x, I v (t)) r c d|∇χ u | dt ≤ C v L ∞ t W 1,∞ xˆT 0 E[χ u , u, V |χ v , v](t) dt.(45)
Recall from (37) that the squared error in the varifold normal is controlled by the relative entropy functional. Together with the bound from Lemma 19, (17) as well as (45) we get an estimate for the first four terms of R surT en R surT en
(46) ≤ C(δ)r −4 c (1 + v L ∞ t W 1,∞ x )ˆT 0 E[χ u , u, V |χ v , v](t) dt + δ 2ˆT 0ˆR d |∇(u − v − w)| 2 dx dt + σˆT 0ˆR d ∇χ u |∇χ u | · (Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x))(∇V n −∇v) T ξ d|∇χ u | dt + σˆT 0ˆR d ∇χ u |∇χ u | · (V n − v) · ∇ ξ d|∇χ u | dt
for almost every T ∈ [0, T strong ) and all δ ∈ (0, 1]. To estimate the remaining two terms we decomposeV n − v as
V n − v = (V n − V n ) + (V n − v),(47)
where the vector field V n is given by
V n (x, t) := v(x, t) · n v (P Iv(t) x, t) n v (P Iv(t) x, t)(48)
in the space-time domain {dist(x, I v (t)) < r c } (i. e. in contrast to V vecn , for V n the velocity v is evaluated not at the projection of x onto the interface, but at x itself). Note that it will not matter as to how V n and similar quantities are defined outside of the area {dist(x, I v (t)) < r c }, as the terms will always be multiplied by suitable cutoffs which vanish outside of {dist(x, I v (t)) < r c }. In the next two steps, we compute and bound the contributions from the two different parts in the decomposition (47) of the errorV n − v.
First step: Controlling the error V n − v. By definition of the vector field V n in (48),
we may write V n − v = − Id − n v (P Iv(t) x) ⊗ n v (P Iv(t) x) v. It is then not clear why the term σˆT 0ˆR d ∇χ u |∇χ u | · (Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x))(∇V n −∇v) T ξ d|∇χ u | dt + σˆT 0ˆR d ∇χ u |∇χ u | · (V n − v) · ∇ ξ d|∇χ u | dt
should be controlled by our relative entropy functional. However, the integrands enjoy a crucial cancellation
(Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x))(∇V n −∇v) T ξ + (V n − v) · ∇ ξ = 0 (49) in the space-time domain {(x, t) ∈ R d × [0, T strong ) : dist(x, I v (t)) < r c }.
To verify this cancellation, we first recall from (21) that ∇ dist ± (x, I v (t)) = n v (P Iv(t) x, t). We then start by rewriting
(V n − v) · ∇ ξ = −∇ξ (Id − ∇ dist ± (·, I v ) ⊗ ∇ dist ± (·, I v ))v.
Note that when the derivative hits the cutoff multiplier in the definition of ξ (see (30)), the resulting term on the right hand side of the last identity vanishes. Hence, we obtain together with (23)
(V n − v) · ∇ ξ = −ζ r −1 c dist ± (·, I v ) ∇ 2 dist ± (·, I v ) (Id − ∇ dist ± (·, I v ) ⊗ ∇ dist ± (·, I v ))v = −ζ r −1 c dist ± (·, I v ) ∇ 2 dist ± (·, I v ) v.
On the other side, another application of (23) yields
(∇V n −∇v) T ξ = −(∇v) T Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) ξ + ζ r −1 c dist ± (·, I v ) ∇ 2 dist ± (·, I v ) v = ζ r −1 c dist ± (·, I v ) ∇ 2 dist ± (·, I v ) v.
Therefore, the desired cancellation (49) indeed holds true since by (23) the righthand side of the last computation remains unchanged after projecting via Id − n v ⊗ n v .
Second step: Controlling the errorV n − V n . It remains to control the contributions from the following two quantities:
I :=ˆT 0ˆR d n u · Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇V n −∇V n ) T ξ d|∇χ u | dt, II :=ˆT 0ˆR d n u · (V n − V n ) · ∇ ξ d|∇χ u | dt.
Note first that we can write
I =ˆT 0ˆR d (n u − ξ) · Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇V n −∇V n ) T ξ d|∇χ u | dt.
Moreover, recall from (25) the formula for the gradient of the projection onto the nearest point on the interface I v (t). The definition of V n (see (48)) andV n (x) = V vecn (P Iv(t) x), the product rule, (21), (17), and (23) imply using the definition of ξ and the property |ξ| ≤ 1
Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇V n −∇V n ) T ξ ≤ Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇(v(P Iv(t) x)) − ∇v(x)) T + Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇(n v (P Iv(t) x)) T (v(P Iv(t) x) − v(x)) + ||v|| L ∞ (∇(n v (P Iv(t) x))) T ξ ≤ Cr −1 c ||v|| W 2,∞ (R d \Iv(t)) dist(x, I v (t))
where in the last step we have used also (25). Together Young's inequality and the coercivity properties of the relative entropy (35) and (36) we then immediately get the estimate
I ≤ CˆT 0ˆR d |n u − ξ| 2 d|∇χ u | dt + Cr −4 c v 2 L ∞ t W 2,∞ x (R d \Iv(t))ˆT 0ˆR d | dist(x, I v (t))| 2 d|∇χ u | dt ≤ C(1 + r −4 c v 2 L ∞ t W 2,∞ x (R d \Iv(t)) )ˆT 0 E[χ u , u, V |χ v , v](t) dt.(50)
To estimate the second term II, we start by adding zero and then use again (17) as well as (35) and (36)
V n (x, t) = V n (P Iv(t) x, t), (41),II =ˆT 0ˆR d (n u − ξ) · (V n − V n ) · ∇ ξ d|∇χ u | dt +ˆT 0ˆR d ξ · (V n − V n ) · ∇ ξ d|∇χ u | dt ≤ C(1 + r −2 c v 2 L ∞ t W 1,∞ x )ˆT 0 E[χ u , u, V |χ v , v](t) dt +ˆT 0ˆR d ξ · (V n − V n ) · ∇ ξ d|∇χ u | dt.
Using (23), we continue by computinĝ
T 0ˆR d ξ · (V n − V n ) · ∇ ξ d|∇χ u | dt = r −1 cˆT 0ˆR d ζ dist ± (x, I v (t)) r c ξ ⊗ (V n − V n ) : n v (P Iv(t) ) ⊗ n v (P Iv(t) ) d|∇χ u | dt
Hence, it follows from ζ (0) = 0 and |ζ | ≤ C as well as (43) that
II ≤ C(1 + r −2 c v 2 L ∞ t W 1,∞ x )ˆT 0 E[χ u , u, V |χ v , v](t) dt + Cr −3 c v L ∞ t W 1,∞ xˆT 0ˆR d | dist(x, I v (t))| 2 d|∇χ u | dt ≤ C(1 + r −3 c v 2 L ∞ t W 1,∞ x )ˆT 0 E[χ u , u, V |χ v , v](t) dt.(51)
Third step: Summary. Inserting (49), (50), and (51) into (46) entails the bound
R surT en ≤ C(δ) r 4 c (1+ v L ∞ t W 2,∞ x (R d \Iv(t)) ∨ v 2 L ∞ t W 2,∞ x (R d \Iv(t)) )ˆT 0 E[χ u , u, V |χ v , v](t) dt + δˆT 0ˆR d |∇(u − v − w)| 2 dx dt.
This yields the desired estimate.
5.2.
Estimate for the remaining terms R adv , R dt , and R weightV ol . To bound the advection-related terms
R adv = −ˆT 0ˆR d ρ(χ u ) − ρ(χ v ) (u − v − w) · (v · ∇)v dx dt −ˆT 0ˆR d ρ(χ u )(u − v − w) · (u − v − w) · ∇ v dx dt
from the relative entropy inequality, the time-derivative related terms R dt , and the terms resulting from the weighted volume control term in the relative entropy
R weightV ol :=ˆT 0ˆR d (χ u −χ v ) V n −V n · ∇ β dist ± (·, I v ) r c dx dt +ˆT 0ˆR d (χ u −χ v ) (u−v−w) · ∇ β dist ± (·, I v ) r c dx dt (with V n (x, t) := (n v (P Iv(t) x, t) ⊗ n v (P Iv(t) x, t))v(x, t))
, we use mostly straightforward estimates.
Lemma 23. Consider the situation of Proposition 9. The terms R adv , R dt , and R weightV ol are subject to the bounds
R adv ≤ C(δ)(1 + v 4 L ∞ t W 1,∞ x )ˆT 0 E[χ u , u, V |χ v , v](t) dt (52) + δˆT 0ˆR d |∇(u − v − w)| 2 dx dt, R dt ≤ δˆT 0ˆR d |∇(u − v − w)| 2 dx dt + C(δ) ∂ t v L ∞ x,tˆT 0 E[χ u , u, V |χ v , v](t) dt,(53)
and
R weightV ol ≤ δˆT 0ˆR d |∇(u − v − w)| 2 dx dt (54) + C(δ)r −2 c (1 + v L ∞ t W 1,∞ x )ˆT 0 E[χ u , u, V |χ v , v](t) dt
for any δ > 0.
Proof. To derive (52), we use a direct estimate for the second term in R adv as well as Lemma 19 for the first term. The bound (53) is derived similarly. Finally, we show estimate (54). Note that by definition we haveV n (x, t) = V n (P Iv(t) x, t). Hence, we obtain using the bound (43) as well as (34) and |β | ≤ C
R weightV ol ≤ C v L ∞ t W 1,∞ xˆT 0ˆR d |χ u −χ v | β dist ± (x, I v (t)) r c dx dt + Cr −1 cˆT 0ˆ{dist(x,Iv(t))≤rc} |χ u − χ v ||u − v − w| dx dt.
An application of Lemma 19 yields (54).
5.3.
The weak-strong uniqueness principle in the case of equal viscosities. We conclude our discussion of the case of equal shear viscosities µ + = µ − for the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) with the proof of the weak-strong uniqueness principle.
Proposition 24. Let d ≤ 3. Let (χ u , u, V ) be a varifold solution to the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) in the sense of Definition 2 on some time interval [0, T vari ) with initial data (χ 0 u , u 0 ). Let (χ v , v) be a strong solution to (1a)-(1c) in the sense of Definition 6 on some time interval [0, T strong ) with T strong ≤ T vari and initial data (χ 0 v , v 0 ). We assume that the shear viscosities of the two fluids coincide, i.e., µ + = µ − .
Then, there exists a constant C > 0 which only depends on the data of the strong solution such that the stability estimate
E[χ u , u, V |χ v , v](T ) ≤ E[χ u , u, V |χ v , v](0)e CT
holds. In particular, if the initial data of the varifold solution and the strong solution coincide, the varifold solution must be equal to the strong solution in the sense
χ u (·, t) = χ v (·, t) and u(·, t) = v(·, t)
almost everywhere for almost every t ∈ [0, T strong ). Furthermore, in this case the varifold is given by
dV t = δ ∇χv |∇χv | d|∇χ v |
for almost every t ∈ [0, T strong ).
Proof. Applying the relative entropy inequality from Proposition 9 with w = 0, using the fact that the problematic term R visc vanishes in the case of equal shear viscosities µ + = µ − , as well as using the bounds from (44), (52), (53) and (54), we observe that we established the following bound
E[χ u , u, V |χ v , v](T ) + cˆT 0ˆR d |∇u − ∇v| 2 dx dt (55) ≤ E[χ u , u, V |χ v , v](0) + δˆT 0ˆR d |∇u − ∇v| 2 dx dt + C(δ) r 4 c (1+ ∂ t v L ∞ x,t + v L ∞ t W 2,∞ x (R d \Iv(t)) ∨ v 2 L ∞ t W 2,∞ x (R d \Iv(t)) ) ׈T 0 E[χ u , u, V |χ v , v](t) dt
for almost every T ∈ [0, T strong ). An absorption argument along with a subsequent application of Gronwall's lemma then immediately yields the asserted stability estimate.
Consider the case of coinciding initial conditions, i.e., E[χ u , u, V |χ v , v](0) = 0. In this case, we deduce from the stability estimate that the relative entropy vanishes for almost every t ∈ [0, T strong ). From this it immediately follows that u(·, t) = v(·, t) as well as χ u (·, t) = χ v (·, t) almost everywhere for almost every t ∈ [0, T strong ).
The asserted representation of the varifold V of the varifold solution follows from the following considerations. First, we deduce |∇χ u (·, t)| = |V t | S d−1 for almost every t ∈ [0, T strong ) as a consequence of the fact that the density of the varifold satisfies θ t = d|∇χu(·,t)| d|Vt| S d−1 ≡ 1 almost everywhere for almost every t ∈ [0, T strong ). The remaining fact that the measure on S d−1 is given by δ nu(x,t) for |V t | S d−1 -almost every x ∈ R d for almost every t ∈ [0, T strong ) then follows from the control of the squared error in the normal of the varifold by the relative entropy functional, see (37). This concludes the proof.
6. Weak-strong uniqueness of varifold solutions to two-fluid Navier-Stokes flow: The case of different viscosities
We turn to the derivation of the weak-strong uniqueness principle in the case of different shear viscosities of the two fluids. In this regime, we cannot anymore ignore the viscous stress term (µ(χ v ) − µ(χ u ))(∇v + ∇v T ). The key idea is to construct a solenoidal vector field w which is small in the L 2 -norm but whose gradient compensates for most of this problematic term, and then use the relative entropy inequality from Proposition 9 with this function. The precise definition as well as a list of all the relevant properties of this vector field are the content of Proposition 27.
A main ingredient for the construction of w are the local interface error heights as measured in orthogonal direction from the interface of the strong solution (see Figure 2). For this reason, we first prove the relevant properties of the local heights of the interface error in Proposition 25. However, in order to control certain surfacetension terms in the relative entropy inequality, we actually need the vector field w to have bounded spatial derivatives. To this aim, we perform an additional regularization of the height functions. This will be carried out in detail in Proposition 26 by a (time-dependent) mollification. After all these preparations, in Section 6.4-6.8 we then perform the post-processing of the additional terms A visc , A dt , A adv , and A surT en in the relative entropy inequality from Proposition 9. Based on these bounds, in Section 6.9 we finally provide the proof of the stability estimate and the weak-strong uniqueness principle for varifold solutions to the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) from Theorem 1. For the family (Ω + t ) t∈[0,Tstrong) of smoothly evolving domains of the strong solution, the associated signed distance function is given by
dist ± (x, I v (t)) = dist(x, I v (t)), x ∈ Ω + t , −dist(x, I v (t)), x / ∈ Ω + t .
From Definition 5 of a family of smoothly evolving domains it follows that the family of maps Φ t :
I v (t) × (−r c , r c ) → R d given by Φ t (x, y) := x + yn v (x, t) are C 2 -diffeomorphisms onto their image {x ∈ R d : dist(x, I v (t)) < r c }.
Here, n v (·, t) denotes the normal vector field of the interface I v (t) pointing inwards {x ∈ R d : χ v (x, t) = 1}. The signed distance function (resp. its time derivative) to the interface I v (t) of the strong solution is then of class C 0 t C 3 x (resp. C 0 t C 2 x ) in the space-time tubular neighborhood t∈[0,Tstrong) im(Φ t ) × {t} due to the regularity assumptions in Definition 5. Moreover, the projection P Iv(t) x of a point x onto the nearest point on the manifold I v (t) is well-defined and of class C 0 t C 2 x in the same tubular neighborhood. Observe that the inverse of Φ t is given by is given by
Φ −1 t (x) = (P Iv(t) x, dist ± (x, I v (t))) for all x ∈ R d such that dist(x, I v (t)) < r c .
In Lemma 10, we computed the time evolution of the signed distance function to the interface I v (t) of a strong solution. Recall also the various relations for the projected inner unit normal vector field n v (P Iv(t) x, t) from Lemma 10, which will be of frequent use in subsequent computations. Finally, we remind the reader of the definition of the vector field ξ from Definition 12, which is a global extension of the inner unit normal vector field of the interface I v (t).
Proposition 25. Let χ v ∈ L ∞ ([0, T strong ); BV(R d ; {0, 1})) be an indicator func- tion such that Ω + t := {x ∈ R d : χ v (x, t) = 1}
is a family of smoothly evolving domains and I v (t) := ∂Ω + t is a family of smoothly evolving surfaces in the sense of Definition 5. Let ξ be the extension of the unit normal vector field n v from Definition 12.
Let θ : [0, ∞) → [0, 1] be a smooth cutoff with θ ≡ 0 outside of [0, 1 2 ] and θ ≡ 1 in [0, 1 4 ]. For an indicator function χ u ∈ L ∞ ([0, T strong ]; BV(R d ; {0, 1})) and t ≥ 0, we define the local height of the one-sided interface error h + (·, t) :
I v (t) → R + 0 as h + (x, t) :=ˆ∞ 0 (1 − χ u )(x + yn v (x, t), t) θ y r c dy.(56)
Similarly, we introduce the local height of the interface error in the other direction
h − (x, t) :=ˆ∞ 0 χ u (x − yn v (x, t), t)θ y r c dy.
Then h + and h − have the following properties: a) (L 2 -bound) We have the estimates |h ± (x, t)| ≤ rc 2 and
Iv(t) |h ± (x, t)| 2 dS(x) ≤ CˆR d |χ u −χ v | min dist(x, I v (t)) r c , 1 dx. (57a) b) (H 1 -bound) Moreover, the estimate holdŝ Iv(t) min{|∇ tan h ± (x, t)| 2 , |∇ tan h ± (x, t)|} dS + |D s h ± |(I v (t)) (57b) ≤ CˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | + C r 2 cˆR d |χ u −χ v | min dist(x, I v (t)) r c , 1 dx.
c) (Approximation property) The functions h + and h − provide an approximation of the set {χ u = 1} in terms of a subgraph over the set I v (t) by setting
χ v,h + ,h − := χ v − χ 0≤dist ± (x,Iv(t))≤h + (P Iv (t) x,t) + χ −h − (P Iv (t) x,t)≤dist ± (x,Iv(t))≤0 ,
up to an error of
R d χ u − χ v,h + ,h − dx ≤ CˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | + CˆR d |χ u − χ v | min dist(x, I v (t)) r c , 1 dx. (57c) d) (Time evolution) Let v be a solenoidal vector field v ∈ L 2 ([0, T strong ]; H 1 (R d ; R d )) ∩ L ∞ ([0, T strong ]; W 1,∞ (R d ; R d ))
such that in the domain t∈[0,Tstrong) (Ω + t ∪Ω − t )×{t} the second spatial derivatives of the vector field v exist and satisfy sup t∈[0,Tstrong) sup x∈Ω +
t ∪Ω − t |∇ 2 v(x, t)| < ∞. As- sume that χ v solves the equation ∂ t χ v = −∇·(χ v v).
If χ u solves the equation ∂ t χ u = −∇ · (χ u u) for another solenoidal vector field u ∈ L 2 ([0, T strong ]; H 1 (R d ; R d )), we have the following estimate on the time derivative of the local interface error heights
h ± : d dtˆI v (t) η(x)h ± (x, t) dS(x) −ˆI v (t) h ± (x, t)(Id−n v ⊗ n v )v(x, t) · ∇η(x) dS(x) (57d) ≤ C r 2 c η W 1,4 (Iv(t)) ˆI v (t) |h ± | 4 dS 1/4 × ˆI v (t) sup y∈[−rc,rc] |u − v| 2 (x + yn v (x, t), t) dS(x) 1/2 + C 1 + v W 2,∞ (R d \Iv(t)) r 3 c η L 2 (Iv(t)) × ˆR d |χ u (x, t) − χ v (x, t)| min dist(x, I v (t)) r c , 1 dx 1 2 + C(1 + v W 1,∞ ) r 2 c max p∈{2,4} η W 1,p (Iv(t))ˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | + C η L 2 (Iv(t)) ˆI v (t) |u − v| 2 dS 1/2 for any test function η ∈ C ∞ cpt (R d ) with n v · ∇η = 0 on the interface I v (t)
, and whereh ± is defined as h ± but now with respect to the modified cut-offθ(·) = θ · 2 . Proof.
Step 1: Proof of the estimate on the L 2 -norm. The trivial estimate |h ± (x, t)| ≤ rc 2 follows directly from the definition of h ± . To establish the L 2estimate, let + (x) :=´r c 0 (1 − χ u )(x + yn v (x, t), t) dy. A straighforward estimate then gives
| + (x)| 2 = 2ˆ + (x) 0 y dy ≤ Cˆr c 0 |χ u (Φ t (x, y), t) − χ v (Φ t (x, y), t)| y r c dy.(58)
Note that the term on the left hand side dominates |h + | 2 since we dropped the cutoff function. Hence, the desired estimate on the L 2 -norm of h + follows at once by a change of variables and recalling the fact that dist(Φ t (x, y), I v (t)) = y. The corresponding bound for h − then follows along the same lines.
Step 2: Proof of the estimate on the spatial derivative (57b). The definition (56) is equivalent to
h + (Φ t (x, 0), t) =ˆ∞ 0 (1 − χ u )(Φ t (x, y)) θ y r c dy.
We compute for any smooth vector field η ∈ C ∞ cpt (R d ; R d ) (recall that Φ t (x, 0) = x and dist(Φ t (x, y), I v (t)) = y for any x ∈ I v (t) and any y with |y| ≤ r c )
Iv(t) η(x) · d(D tan x h + (·, t))(x) = −ˆI v (t) h + (x, t)∇ tan · η(x) dS(x) −ˆI v (t) h + (x, t)η(x) · H(x, t) dS(x) = −ˆr c 0ˆIv(t) (1 − χ u )(Φ t (x, y), t)θ y r c ∇ tan · η(x) dS(x) dy −ˆr c 0ˆIv(t) (1 − χ u )(Φ t (x, y), t)θ y r c η(x) · H(Φ t (x, 0), t) dS(x) dy = −ˆR d (1 − χ u )(x, t)θ dist(x, I v (t)) r c | det ∇Φ −1 t (x)| × (Id −n v (P Iv(t) x) ⊗ n v (P Iv(t) x)) : ∇η(P Iv(t) x) dx −ˆR d (1 − χ u )(x, t)θ dist(x, I v (t)) r c η(P Iv(t) x) · H(P Iv(t) x)| det ∇Φ −1 t (x)| dx = −ˆR d θ dist(x, I v (t)) r c | det ∇Φ −1 t (x)|η(P Iv(t) x)(Id −n v (P Iv(t) x) ⊗ n v (P Iv(t) x)) · d∇χ u +ˆR d (1 − χ u )(x, t)θ dist(x, I v (t)) r c η(P Iv(t) x) · ∇ · (Id −n v (P Iv(t) x) ⊗ n v (P Iv(t) x))| det ∇Φ −1 t | − H(P Iv(t) x)| det ∇Φ −1 t | dx,
where in the last step we have used ∇ dist ± (x, I v (t)) = n v (P Iv(t) x). This yields by another change of variables in the second integral, the fact that χ v (Φ t (x, y), t) = 1 for any y > 0, (17), (18), | det ∇Φ −1 t | ≤ C as well as by abbreviating n u = ∇χu |∇χu| U ∩Iv(t)
1 d|D tan x h + (·, t)| ≤ Cˆ{ x+ynv(x,t): x∈U ∩Iv(t),y∈(−rc,rc)} n v (P Iv(t) x) − n u d|∇χ u (·, t)| + C r cˆU ∩Iv(t)ˆr c 0 |χ u (Φ t (x, y), t) − χ v (Φ t (x, y), t)| dy dS(x)
for any Borel set U ⊂ R d . Recall that the indicator function χ u (·, t) of the varifold solution is of bounded variation in I := {x ∈ R d : dist ± (x, I v (t)) ∈ (−r c , r c )}. In particular, E + := {x ∈ R d : χ u > 0} ∩ I is a set of finite perimeter in I. Applying Theorem 35 in local coordinates the sections E + x = {y ∈ (−r c , r c ) : χ u (x + yn v (x, t)) > 0} are guaranteed to be one-dimensional Caccioppoli sets in (−r c , r c ) for H d−1 -almost every x ∈ I v (t). Note that whenever |n v · n u | ≤ 1 2 then 1 − n v · n u ≥ 1 2 , and therefore using also the co-area formula for rectifiable sets (see [10, (2.72)
]) U ∩Iv(t) 1 d|D tan x h + (·, t)|(59)≤ C r cˆU ∩Iv(t)ˆr c 0 |χ u (Φ t (x, y), t) − χ v (Φ t (x, y), t)| dy dS(x) + CˆU ∩Iv(t)ˆ∂ * E + x ∩{nv(x)·nu(x+ynv(x,t))≥ 1 2 }∩(−rc,rc) |n v (x) − n u | |n v (x) · n u | dH 0 (y) dS(x) + Cˆ{
x+ynv(x,t): x∈U ∩Iv(t),y∈(−rc,rc),nv(x)·nu(x+ynv(x,t))≤ 1 2 }
We now distinguish between different cases depending on x ∈ I v (t) up to H d−1measure zero. We start with the set of points
x ∈ A 1 ⊂ I v (t) such that rc 0 |χ u (Φ t (x, y), t) − χ v (Φ t (x, y), t)| dy (60) +ˆ∂ * E + x ∩{nv(x)·nu(x+ynv(x,t))≥ 1 2 }∩(−rc,rc) |n v (x) − n u | |n v (x) · n u | dH 0 (y) + sup y∈{ỹ∈(−rc,rc)∩∂ * E + x : nv(x)·nu(x+ỹnv(x,t))≤ 1 2 } 1 − n v (P Iv(t) x) · n u (x+yn v (x, t)) ≤ 1 4 .
By splitting the measure D tan x h + into a part which is absolutely continuous with respect to the surface measure on I v (t), for which we denote the density by ∇ tan h + , as well as a singular part D s h + , we obtain from (59) (note that the third integral in (59) does not contribute to this estimate by the definition of the set
A 1 ⊂ I v (t)) U ∩Iv(t)∩A1 |∇ tan h + |(x) dS(x) ≤ˆU ∩Iv(t)∩A1 C r cˆr c 0 |χ u (Φ t (x, y), t) − χ v (Φ t (x, y), t)| dy dS(x) +ˆU ∩Iv(t)∩A1 Cˆ∂ * E + x ∩{nv(x)·nu(x+ynv(x,t))≥ 1 2 }∩(−rc,rc) |n v (x) − n u | |n v (x) · n u | dH 0 (y) dS(x)
for every Borel set U ⊂ R d . Since U was arbitrary, we deduce that |∇ tan h + | is bounded on A 1 by the two integrands on the right hand side of the last inequality. Hence, we obtain
A1 |∇ tan h + | 2 (x) dS(x) + |D s h + |(A 1 ) ≤ Cr −2 cˆI v (t) ˆr c 0 |χ u (Φ t (x, y), t) − χ v (Φ t (x, y), t)| dy 2 dS(x) + CˆI v (t)∩A1 ˆ∂ * E + x ∩{nv(x)·nu(x+ynv(x,t))≥ 1 2 }∩(−rc,rc) |n v − n u | dH 0 (y) 2 dS(x).
The first term on the right hand side can be estimated as in the proof of the L 2bound for h ± . To bound the second term, we make the following observation. First, we may represent the one-dimensional Caccioppoli sets E + x as a finite union of disjoint intervals (see [10,Proposition 3.52]). It then follows from property iv) in Theorem 35 that ∂ * E +
x ∩ (−r c , r c ) can only contain at most one point. Indeed, otherwise we would find at least one point y ∈ ∂ * E +
x ∩ (−r c , r c ) such that n v (x) · n u (x+yn v (x, t)) < 0 which is a contradiction to the definition of A 1 . By another application of the co-area formula for rectifiable sets (see [10, (2.72)]) we therefore
getˆA 1 |∇ tan h + | 2 (x) dS(x) + |D s h + |(A 1 ) ≤ C r 2 cˆR d |χ u − χ v | min dist(x, I v (t)) r c , 1 dx (61) + Cˆ{ dist(x,Iv(t))<rc} 1 − n v (P Iv(t) x) · ∇χ u |∇χ u | d|∇χ u |(x).
We now turn to the second case, namely the set of points A 2 := I v (t) \ A 1 . We begin with a preliminary computation. When splitting E + x into a finite family of disjoint open intervals as before, it again follows from property iv) in Theorem 35 that every second point y ∈ ∂ * E +
x ∩ (−r c , r c ) has to have the property that n v (x) · n u (x+yn v (x, t)) < 0, i.e., |n v (x) − n u | ≤ 2 ≤ 2(1 − n v (x) · n u ). In particular, by another application of the co-area formula for rectifiable sets (see [10, (2.72)]) we obtain the bound
A2ˆ∂ * E + x ∩{nv(x)·nu(x+ynv(x,t))≥ 1 2 }∩(−rc,rc) |n v (x) − n u | |n v (x) · n u | dH 0 (y) dS(x) ≤ 8ˆ{ dist(x,Iv(t))<rc} 1 − n v (P Iv(t) x) · ∇χ u |∇χ u | d|∇χ u |(x).(62)
Now, we proceed as follows. By definition of A 2 , either one of the three summands in (60) has to be ≥ 1 12 . We distinguish between two cases. If the third one is not, then this actually means that the set {ỹ ∈ (−r c , r c ) ∩ ∂ * E + x : n v (x) · n u (x+ỹn v (x, t)) ≤ 1 2 } is empty, i.e., the third summand has to vanish. Hence, either one of the first two summands in (60) has to be ≥ 1 8 . If the first one is not, we use that´r c 0 |χ u (Φ t (x, y), t) − χ v (Φ t (x, y), t)| dy ≤ r c and bound this by the second term and then (62). If the second one is not, then
+ (x) :=ˆr c 0 |χ u (Φ t (x, y), t) − χ v (Φ t (x, y), t)| dy ≤ r c ≤ C r cˆ + (x) 0 y dy ≤ Cˆr c 0 |χ u (Φ t (x, y), t) − χ v (Φ t (x, y), t)| y r c dy.(63)
Now, we move on with the remaining case, i.e., that the third summand in (60) does not vanish. In other words,
{ỹ ∈ (−r c , r c ) ∩ ∂ * E + x : n v (x) · n u (x+ỹn v (x, t)) ≤ 1 2 } is non-empty. We then estimatê rc 0 |χ u (Φ t (x, y), t) − χ v (Φ t (x, y), t)| dy ≤ r c ≤ 2r cˆ∂ * E + x ∩(−rc,rc) 1 − n v (x) · n u (x+ỹn v (x, t)) dH 0 (y).(64)
Taking finally U = A 2 in (59), the conclusions of the above case study together with the three estimates (62), (63) and (64) followed by another application of the co-area formula for rectifiable sets (see [10, (2.72)]) to further estimate the latter, then imply that
A2 |∇ tan h + |(x) dS(x) + |D s h + |(A 2 ) ≤ C r cˆR d |χ u − χ v | min dist(x, I v (t)) r c , 1 dx (65) + Cˆ{ dist(x,Iv(t))<rc} 1 − n v (P Iv(t) x) · ∇χ u |∇χ u | d|∇χ u |(x).
The two estimates (61) and (65) thus entail the desired upper bound (57b) for the (tangential) gradient of h ± with ξ replaced by n v (P Iv(t) x). However, one may replace n v (P Iv(t) x) by ξ because of (36).
Step 3: Proof of the approximation property for the interface (57c). In order to establish (57c), we rewrite using the coordinate transform Φ t (recall that dist ± (Φ t (x, y), I v (t)) = y and that |h ± | ≤ r c )
R d |χ u − χ v,h + ,h − | dx =ˆI v (t)ˆr c 0 det ∇Φ t (x, y)|χ u (Φ t (x, y)) − 1 + χ {y≤h + (x)} | dy dS(x) (66) +ˆI v (t)ˆ0 −rc det ∇Φ t (x, y)|χ u (Φ t (x, y)) − χ {y≥−h − (x)} | dy dS(x) +ˆ{ dist(x,Iv(t))≥rc} |χ u − χ v | dx.
In order to derive a bound for the first term on the right-hand side of (66), we distinguish between different cases depending on x ∈ I v (t) up to H d−1 -measure zero. We first distinguish between h + (x) ≥ rc 4 and h + (x) < rc 4 . In the former case, a straightforward estimate yields (recall (16)) ˆr
c 0 det ∇Φ t (x, y)|χ u (Φ t (x, y)) − 1 + χ {y≤h + (x)} | dy ≤ Cr c ≤ C r cˆh + (x) 0 y dy ≤ Cˆr c 0 |χ u (Φ t (x, y)) − χ v (Φ t (x, y))| y r c dy,(67)
which is indeed of required order after a change of variables. We now consider the other case, i.e., h + (x) < rc 4 . Recall that the indicator function χ u (·, t) of the varifold solution is of bounded variation in
I + := {x ∈ R d : dist ± (x, I v (t)) ∈ (0, r c )}. In particular, E + := {x ∈ R d : 1 − χ u > 0} ∩ I + is a set of finite perimeter in I + . Recall also that E + = I + ∩ {x ∈ R d : (χ v − χ u ) + > 0} since χ v ≡ 1 in I + .
Applying Theorem 35 in local coordinates, the sections t)) > 0} are guaranteed to be one-dimensional Caccioppoli sets in (0, r c ) for H d−1 -almost every x ∈ I v (t). Hence, we may represent the one-dimensional section E + x for such x ∈ I v (t) as a finite union of disjoint intervals (see [10,Proposition 3.52])
E + x = {y ∈ (0, r c ) : 1 − χ u (x + yn v (x,E + x ∩ (0, r c ) = K(x) m=1
(a m , b m ).
If K(x) = 0 then h + (x) = 0, and the inner integral in the first term on the right hand side of (66) vanishes for this x. If K(x) = 1 and a 1 = 0, then by definition of h + (x) we have (a 1 , b 1 ) = (0, h + (x)) (recall that we now consider the case h + (x) ≤ rc 4 ). Thus, again the inner integral in the first term on the right hand side of (66) vanishes for this x. Hence, it remains to discuss the case that there is at least one non-empty interval in the decomposition of E + x , say (a, b), such that a ∈ (0, r c ). From property iv) in Theorem 35 it then follows that
n v (x, t) · −∇χ E + |∇χ E + | (x + an v (x, t)) ≤ 0.
Hence, we may bound
ˆr c 0 det ∇Φ t (x, y)|χ u (Φ t (x, y)) − 1 + χ {y≤h + (x)} | dy ≤ Cr c ≤ Cˆ( 0,rc)∩(∂ * E + )x 1 − n v (x, t) · −∇χ E + |∇χ E + | (x + yn v (x, t)) dH 0 (y)
Gathering the bounds from the different cases together with the estimate in (67), we therefore obtain by the co-area formula for rectifiable sets (see [10, (2.72)]) together with the change of variables Φ t (x, y)
ˆI v (t)ˆr c 0 det ∇Φ t (x, y)|χ u (Φ t (x, y)) − 1 + χ {y≤h + (x)} | dy dS(x) ≤ CˆI v (t)ˆ(0,rc)∩(∂ * E + )x 1 − n v (x, t) · −∇χ E + |∇χ E + | (x + yn v (x, t)) dH 0 (y) dS(x) + CˆR dˆr c −rc |χ u (Φ t (x, y)) − χ v (Φ t (x, y))| y r c dy dx ≤ Cˆ{ dist(x,Iv(t))<rc} 1 − n v (P Iv(t) x) · ∇χ u |∇χ u | d|∇χ u |(x) + CˆR d |χ u − χ v | min dist(x, I v (t)) r c , 1 dx,
which is by (36) as well as (35) indeed a bound of desired order. Moreover, performing analogous estimates for the second term on the right-hand side of (66) and estimating the third term on the right-hand side of (66) trivially, we then get
R d |χ u − χ v,h + ,h − | dx ≤ CˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | + CˆR d |χ u − χ v | min dist(x, I v (t)) r c , 1 dx
which is precisely the desired estimate (57c).
Step 4: Proof of estimate on the time derivative (57d). To bound the time derivative, we compute using the weak formulation of the continuity equation ∂ t χ u = −∇ · (χ u u) and abbreviating I + (t) := {x ∈ R d : dist ± (x, I v (t)) ∈ [0, r c )} (recall that the boundary ∂I + (t) = I v (t) moves with normal speed n
v · v) d dtˆI v (t) η(x)h + (x, t) dS(x) = d dtˆI v (t)ˆ∞ 0 η(x)(1 − χ u )(x + yn v (x, t), t) θ y r c dy dS(x) = d dtˆI+ (t) η(P Iv(t) x)| det ∇Φ −1 t |(x)(1 − χ u )(x, t) θ dist(x, I v (t)) r c dx =ˆI + (t) (1 − χ u )(x, t)u · ∇ η(P Iv(t) x)| det ∇Φ −1 t |(x) θ dist(x, I v (t)) r c dx +ˆI v (t) (n v · u)(x, t)(1−χ u )(x, t)η(P Iv(t) x)| det ∇Φ −1 t |(x)θ dist(x, I v (t)) r c dS(x) +ˆI + (t) (1 − χ u )(x, t) d dt η(P Iv(t) x)| det ∇Φ −1 t |(x)θ dist(x, I v (t)) r c dx −ˆI v (t) (n v · v)(x, t)(1−χ u )(x, t)η(P Iv(t) x)| det ∇Φ −1 t |(x)θ dist(x, I v (t)) r c dS(x).
Recall from (25) the formula for the gradient of the projection onto the nearest point on the interface I v (t). Recalling also the definitions of the extended (48) respectively (20), we also have −ˆI
normal velocity V n (x, t) := v(x, t) · n v (P Iv(t) x, t) n v (P Iv(t) x, t) and its projection V n (x, t) := V n (P Iv(t) x, t) from+ (1 − χ u (x, t))| det ∇Φ −1 t |(x)θ dist(x, I v (t)) r c (∇η)(P Iv(t) x) · (v(P Iv(t) x, t) −V n (x, t)) · ∇ P Iv(t) x dx = −ˆI v (t)ˆr c 0 (1 − χ u (Φ t (x, y), t))θ y r c ∇η(x) · (v(x, t) − V n (x, t)) · ∇ P Iv(t) (Φ t (x, y)) dy dS(x) = −ˆI v (t) h + (x, t)(Id −n v (x) ⊗ n v (x))v(x, t) · ∇η(x) dS(x) +ˆI + (t) (1 − χ u (x, t))| det ∇Φ −1 t |(x)θ dist(x, I v (t)) r c dist(x, I v (t))(∇η)(P Iv(t) x) · (v(P Iv(t) x, t) −V n (x, t)) · ∇ n v (P Iv(t) x) dx.
Adding this formula to the above formula for d
dt´I v (t) η(x)h + (x, t) dS(x), intro- ducing the abbreviation f := | det ∇Φ −1 t |(x) θ( dist(x,Iv(t))
rc ), and using the fact that
χ v = 1 in I + (t), we obtain d dtˆI v (t) η(x)h + (x, t) dx −ˆI v (t) h + (x, t)(Id −n v ⊗ n v )v(x, t) · ∇η(x) dS(x) =ˆI + (t) (χ u (x, t) − χ v (x, t))f (x) dist(x, I v (t))(∇η)(P Iv(t) x) · (v(P Iv(t) x, t) −V n (x, t)) · ∇ n v (P Iv(t) x) dx −ˆI + (t) (χ u (x, t) − χ v (x, t))η(P Iv(t) x)(u − v) · ∇f dx (68) −ˆI + (t) (χ u (x, t) − χ v (x, t))f (x)(∇η)(P Iv(t) x) · ((u − v) · ∇)P Iv(t) x dx −ˆI + (t) (χ u (x, t) − χ v (x, t))f (x)(∇η)(P Iv(t) x) · (v(x, t) − (v(P Iv(t) x, t) −V n (x, t))) · ∇ P Iv(t) x dx −ˆI + (t) (χ u (x, t) − χ v (x, t))f (x)(∇η)(P Iv(t) x) · d dt P Iv(t) x dx −ˆI + (t) (χ u (x, t) − χ v (x, t)) η(P Iv(t) x) d dt f + v · ∇f dx +ˆI v (t) n v · (u − v)(1 − χ u )η dS. Note that f (x) = | det ∇Φ −1 t |(x) θ( dist(x,Iv(t))
rc ) = 1 for any t and any x ∈ I v (t). (17), the corresponding estimate (41) for the gradient of V n as well as the formula (25) for the gradient of P Iv(t) . Because of (21) and the equation (32) for the time evolution of the normal vector, we thus get the bounds
Thus, we have d dt f + v · ∇f = 0 on I v (t). Furthermore, we have |∇V n | ≤ C r 2 c v W 1,∞ and |∇ 2V n | ≤ C r 3 c v W 2,∞ (R d \Iv(t)) because ofV n (x) = V n (P Iv(t) x),| d dt ∇ dist ± (·, I v (t))| ≤ C r 2 c v W 1,∞ and |∇ d dt ∇ dist ± (·, I v (t))| ≤ C r 3 c v W 2,∞ (R d \Iv(t))
. Taking all of these bounds together,
we obtain |f | ≤ C rc , |∇f | ≤ C r 2 c and |∇ 2 f | + |∇ d dt f | ≤ C r 3 c (1 + v W 2,∞ (R d \Iv(t))
). As a consequence, we get
d dt f + v · ∇f ≤ C r 3 c (1 + v W 2,∞ (R d \Iv(t)) ) dist(·, I v (t)).(69)
Moreover, we may compute
d dt P Iv(t) x = −n v (P Iv(t) x) d dt dist ± (x, I v (t)) − dist ± (x, I v (t)) d dt (n v (P Iv(t) x)).(70)
Since n v · ∇η = 0 holds on the interface I v (t) by assumption, we obtain from (70) −ˆI
+ (t) (χ u (x, t) − χ v (x, t))f (x)(∇η)(P Iv(t) x) · d dt P Iv(t) x dx =ˆI + (t) (χ u (x, t) − χ v (x, t)) dist ± (x, I v (t))f (x)(∇η)(P Iv(t) x) · d dt (n v (P Iv(t) x)) dx.
In what follows, we will by slight abuse of notation use ∇ tan g(x) as a shorthand for (Id − n v (P Iv(t) x) ⊗ n v (P Iv(t) x))∇g(x) for scalar fields as well as (∇ tan · g)(x) instead of (Id − n v (P Iv(t) x) ⊗ n v (P Iv(t) x)) : ∇g(x) for vector fields. Let us also abbreviate P tan x := (Id − n v (P Iv(t) x) ⊗ n v (P Iv(t) x)). Note that by assumption (∇η)(P Iv(t) x) = (∇ tan η)(P Iv(t) x). Moreover, it follows from (22), (23) and (21) that n v (P Iv(t) x) · d dt (n v (P Iv(t) x)) = 0. Hence, we may rewrite with an integration by parts (recall the notation P tan (x) = (Id −n v ⊗ n v )(P Iv(t) x, t))
I + (t) (χ u (x, t) − χ v (x, t)) dist ± (x, I v (t))f (x)(∇ tan η)(P Iv(t) x) · d dt (n v (P Iv(t) x)) dx(71)
= −ˆI
+ (t) (χ u − χ v )(x, t) dist ± (x, I v (t))η(P Iv(t) x) × d dt (n v (P Iv(t) x)) ⊗ ∇ : f (x)P tan (x) dx −ˆI + (t) (χ u − χ v )(x, t) dist ± (x, I v (t))f (x)η(P Iv(t) x)∇ tan · d dt (n v (P Iv(t) x)) dx −ˆR d dist ± (x, I v (t))f (x)η(P Iv(t) x) ∇χ u |∇χ u | − n v (P Iv(t) x) · d dt (n v (P Iv(t) x)) d|∇χ u |.
Using from (23) and (21) that the spatial partial derivatives of the extended normal vector field are orthogonal to the gradient of the signed distance function, the same argument also shows that
I + (t) (χ u (x, t) − χ v (x, t))f (x) dist(x, I v (t))(∇ tan η)(P Iv(t) x) (72) · (v(P Iv(t) x, t) −V n (x, t)) · ∇ n v (P Iv(t) x) dx = −ˆI + (t) (χ u (x, t) − χ v (x, t)) dist(x, I v (t))η(P Iv(t) x) × ((v(P Iv(t) x, t) −V n (x, t)) · ∇)n v (P Iv(t) x) ⊗ ∇ : f (x)P tan (x) dx −ˆI + (t) (χ u (x, t) − χ v (x, t)) dist(x, I v (t))f (x)η(P Iv(t) x) × ∇ tan · (v(P Iv(t) x, t) −V n (x, t)) · ∇ n v (P Iv(t) x) dx −ˆR d dist ± (x, I v (t))f (x)η(P Iv(t) x) ∇χ u |∇χ u | − n v (P Iv(t) x) · (v(P Iv(t) x, t) −V n (x, t)) · ∇ n v (P Iv(t) x) d|∇χ u |.
It follows from (25) as well as (23) and (21) that (n v (P Iv(t) x)·∇)P Iv(t) x = 0. Hence, we obtain
I + (t) (χ u (x, t) − χ v (x, t))f (x)(∇η)(P Iv(t) x)(73)· (v(x, t) − (v(P Iv(t) x, t) −V n (x, t))) · ∇ P Iv(t) (x) dx =ˆI + (t) (χ u − χ v )(x, t)f (x)(∇η)(P Iv(t) x) · (v(x, t) − v(P Iv(t) x, t)) · ∇ P Iv(t) x dx.
Since the domain of integration is I + (t), we may write
v(x, t) − v(P Iv(t) x, t) = dist ± (x, I v (t))ˆ( 0,1] ∇v P Iv(t) x + λ dist ± (x, I v (t))n v (P Iv(t) x) dλ · n v (P Iv(t) x).
From this and the fact n v (P Iv(t) ) · ∇P Iv(t) (x) = 0, we deduce by another integration by parts that (where |F | ≤ r −1
c v W 2,∞ (R d \Iv(t)) ) I + (t) (χ u (x, t) − χ v (x, t))f (x)(∇ tan η)(P Iv(t) x) · ((v(x, t) − v(P Iv(t) x, t)) · ∇)P Iv(t) x dx(74)
= −ˆI
+ (t) (χ u (x, t) − χ v (x, t))η(P Iv(t) x) × ((v(x, t) − v(P Iv(t) x, t)) · ∇)P Iv(t) x ⊗ ∇ : f (x)P tan x dx −ˆI + (t) (χ u (x, t) − χ v (x, t))f (x)η(P Iv(t) x)((v(x, t) − v(P Iv(t) x, t)) · ∇(∇ tan · P Iv(t) x) dx −ˆI + (t) (χ u (x, t) − χ v (x, t)) dist(x, I v (t))f (x)η(P Iv(t) x)F (x, t) : ∇P Iv(t) x dx −ˆR d f (x)η(P Iv(t) x) ∇χ u |∇χ u | −n v (P Iv(t) x) · (v(x, t)−v(P Iv(t) x, t)) · ∇ P Iv(t) x d|∇χ u |.
Hence, plugging in (73), (72) and (74), (71) into (68) and using the estimates
|∇V n | ≤ C r 2 c v W 1,∞ , | d dt n v (P Iv(t) x)| ≤ C r 2 c v W 1,∞ , |∇ d dt n v (P Iv(t) x)| ≤ C r 3 c v W 2,∞ (R d \Iv(t)) , and |∇f | ≤ C r 2 c , we obtain d dtˆI v (t) η(x)h + (x, t) dx −ˆI v (t) h + (x, t)(Id−n v ⊗ n v )v(x, t) · ∇η(x) dS(x) ≤ C r 2 cˆ{dist(x,Iv(t))≤rc} |χ u (x, t) − χ v (x, t)||u(x, t) − v(x, t)||η(P Iv(t) x)| dx + C r cˆ{dist(x,I v (t))≤rc} |χ u (x, t) − χ v (x, t)||u(x, t) − v(x, t)||∇η(P Iv(t) x)| dx + C(1+ v W 1,∞ ) r cˆ{dist(x,I v (t))≤rc} ∇χ u |∇χ u | − n v (P Iv(t) x) | dist ± (x, I v (t))| r c |η(P Iv(t) x)| d|∇χ u |(x) + C(1+ v W 2,∞ (R d \Iv(t)) ) r 3 cˆ{dist(x,Iv(t))≤rc} |χ u (x, t)−χ v (x, t)| | dist ± (x, I v (t))| r c |η(P Iv(t) x)| dx + CˆI v (t) |u − v||η| dS.
This yields by the change of variables Φ t (x, y) and a straightforward estimate
d dtˆI v (t) η(x)h + (x, t) dx −ˆI v (t) h + (x, t)(Id−n v ⊗ n v )v(x, t) · ∇η(x) dS(x) ≤ C r 2 c η W 1,4 (Iv(t)) ˆI v (t) ˆrc 2 0 |χ u − χ v |(x + yn v (x, t), t) dy 4 dS 1/4 × ˆI v (t) sup y∈[−rc,rc] |u − v| 2 (x + yn v (x, t), t) dS(x) 1/2 + C(1+ v W 2,∞ (R d \Iv(t)) ) r 3 c η L 2 (Iv(t)) × ˆR d |χ u (x, t) − χ v (x, t)| min dist(x, I v (t)) r c , 1 dx 1 2 + C(1+ v W 1,∞ ) r c η L ∞ (Iv(t)) ˆ{ dist(x,Iv(t))≤rc} ∇χ u |∇χ u | − n v (P Iv(t) x) 2 d|∇χ u | 1 2 × ˆ{ dist(x,Iv(t))≤rc} | dist ± (x, I v (t))| 2 r 2 c d|∇χ u | 1 2 + C ˆI v (t) |u − v| 2 dS 1/2 η L 2 (Iv(t)) .
Using finally the Sobolev embedding to bound the L ∞ -norm of η on the interface (which is either one-or two-dimensional; note that the constant in the Sobolev embedding may be bounded by Cr −1 c for our geometry), we infer from this estimate the desired bound (57d), using also (36) and (35). This concludes the proof.
6.2.
A regularization of the local height of the interface error. In order to modify our relative entropy to compensate for the velocity gradient discontinuity at the interface, we need regularized versions of the local heights of the interface error h + and h − which in particular have Lipschitz regularity. To this aim, we fix some function e(t) > 0 and basically apply a mollifier on scale e(t) to the local interface error heights h + and h − at each time. These regularized versions h + e(t) and h − e(t) of the local interface error heights then have the following properties:
Proposition 26. Let χ v ∈ L ∞ ([0, T strong ); BV(R d ; {0, 1})) be an indicator func- tion such that Ω + t := {x ∈ R d : χ v (x, t) = 1}h ± e(t) (x, t) :=´I v (t) θ |x−x| e(t) h ± (x, t) dS(x) Iv(t) θ |x−x| e(t)
dS(x)
.
(75)
Then h + e(t) and h − e(t) have the following properties: a) (H 1 -bound) If the interface error terms from the relative entropy are bounded byˆR
d 1 − ξ(·, t) · ∇χ u (·, t) |∇χ u (·, t)| d|∇χ u (·, t)| +ˆR d χ u (·, t) − χ v (·, t) β dist ± (·, I v (t)) r c dx ≤ e(t) 2 ,
we have the Lipschitz estimate |∇h ± e(t) (·, t)| ≤ Cr −2 c , the global bound |∇ 2 h ± e(t) (·, t)| ≤ Ce(t) −1 r −4 c , and the bound
Iv(t) |∇h ± e(t) | 2 + |h ± e(t) | 2 dS ≤ C r 2 cˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | (76a) + C r 4 cˆR d |χ u − χ v | min dist(x, I v (t)) r c , 1 dx.
b) (Improved approximation property) The functions h + e(t) and h − e(t) provide an approximation for the interface of the weak solution
χ v,h + e(t) ,h − e(t) :=χ v − χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x,t) (76b) + χ −h − e(t) (P Iv (t) x,t)≤dist ± (x,Iv(t))≤0 , up to an error of R d χ u − χ v,h + e(t) ,h − e(t) dx ≤ CˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | + CˆR d |χ u − χ v | min dist(x, I v (t)) r c , 1 dx (76c) + Ce(t) ˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | 1/2 H d−1 (I v (t)) 1/2 + C e(t) r c ˆR d |χ u − χ v | min dist(x, I v (t)) r c , 1 dx 1/2 H d−1 (I v (t)) 1/2 . c) (Time evolution) Let v be a solenoidal vector field v ∈ L 2 ([0, T strong ]; H 1 (R d ; R d )) ∩ L ∞ ([0, T strong ]; W 1,∞ (R d ; R d ))
such that in the domain t∈[0,Tstrong) (Ω + t ∪Ω − t )×{t} the second spatial derivatives of the vector field v exist and satisfy sup t∈[0,Tstrong) sup x∈Ω +
t ∪Ω − t |∇ 2 v(x, t)| < ∞. As- sume that χ v solves the equation ∂ t χ v = −∇·(χ v v).
If χ u solves the equation ∂ t χ u = −∇ · (χ u u) for another solenoidal vector field u ∈ L 2 ([0, T strong ]; H 1 (R d ; R d )), we have the following estimate on the time derivative of h ± e(t) :
d dtˆI v (t) η(x)h ± e(t) (x, t) dx −ˆI v (t) h ± e(t) (x, t)(Id−n v ⊗ n v )v(x, t) · ∇η(x) dS(x) (76d) ≤ C e(t)r 2 c η L 4 (Iv(t)) ˆI v (t) |h ± | 4 dS 1/4 × ˆI v (t) sup y∈[−rc,rc] |u − v| 2 (x + yn v (x, t), t) dS(x) 1/2 + C (1 + v W 1,∞ ) e(t)r c max p∈{2,4} η L p (Iv(t))ˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | + Cr −4 c v W 1,∞ (1 + e (t)) ˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | 1/2 ||η|| L 2 (Iv(t)) + C 1 + v W 2,∞ (R d \Iv(t)) r c + v W 1,∞ r 6 c (1 + e (t)) η L 2 (Iv(t)) × ˆR d |χ u (x, t) − χ v (x, t)| min dist(x, I v (t)) r c , 1 dx 1 2 + C η L 2 (Iv(t)) ˆI v (t) |u − v| 2 dS 1 2
for any smooth test function η ∈ C ∞ cpt (R d ) with n v · ∇η = 0 on the interface I v (t), and whereh ± is defined as h ± but now with respect to the modified cut-off function θ(·) = θ · 2 . Proof. Proof of a). In order to estimate the spatial derivative ∇h ± e(t) , we compute using the fact that ∇ x θ |x−x|
e(t) = −∇xθ |x−x| e(t)
(note that all of the subsequent gradients are to be understood in the tangential sense on the manifold I v (t))
∇h ± e(t) (x, t) = −´I v (t) ∇xθ |x−x| e(t) h ± (x, t) dS(x) Iv(t) θ |x−x| e(t) dS(x) +´I v (t) θ |x−x| e(t) h ± (x, t) dS(x)´I v (t) ∇xθ |x−x| e(t) dS(x) ´I v (t) θ |x−x| e(t) dS(x) 2 =´I v (t) θ |x−x| e(t) ∇h ± (x, t) dS(x) Iv(t) θ |x−x| e(t) dS(x) +´I v (t) θ |x−x| e(t) dD s h ± (x) Iv(t) θ |x−x| e(t) dS(x) +´I v (t) θ |x−x| e(t) h ± (x, t)H(x, t) dS(x) Iv(t) θ |x−x| e(t) dS(x) −´I v (t) θ |x−x| e(t) h ± (x, t) dS(x)´I v (t) θ |x−x| e(t) H(x, t) dS(x) ´I v (t) θ |x−x| e(t) dS(x) 2 .
Introduce the convex function G(p) := |p| 2 for |p| ≤ 1, 2|p| − 1 for |p| ≥ 1.
Using the estimate (18), the obvious bounds G(p +p) ≤ CG(p) + CG(p) and G(λp) ≤ C(λ + λ 2 )G(p) for any p,p, and λ > 0, and Jensen's inequality, we obtain (as the recession function of G is given by 2|p|)
G(|∇h ± e(t) (x, t)|) ≤ C´I v (t) θ |x−x| e(t) G(|∇h ± (x, t)|) + G(r −1 c |h ± (x, t)|) dS(x) Iv(t) θ |x−x| e(t) dS(x)(78)+ C´I v (t) θ |x−x| e(t) d|D s h ± |(x, t) Iv(t) θ |x−x| e(t)
dS(x)
. Consider x ∈ I v (t). By the assumption from Definition 5, there is a C 3 -function g : B 1 (0) ⊂ R d−1 → R with ∇g L ∞ ≤ 1, g(0) = 0, and ∇g(0) = 0, and such that I v (t) ∩ B 2rc (x) is after rotation and translation given as the graph {(x, g(x)) : x ∈ R d−1 }. Using the fact that θ ≡ 0 on R \ [0, 1 2 ] and e(t) < r c ≤ 1, i.e., the map I v (t) x → θ( |x−x| e(t) ) is supported in a coordinate patch given by the graph of g, we then may bound
Iv(t) θ |x − x| e(t) dS(x) ≤ˆI v (t)∩B e(t) 2 (x) 1 dS(x) ≤ Cˆ{x ∈R d−1 : |x|< e(t) 2 } 1 dx ≤ Ce(t) d−1 .
We also obtain a lower bound using that θ ≡ 1 on [0, 1 4 ] and again e(t) < r c ≤ 1
Iv(t) θ |x − x| e(t) dS(x) ≥ˆI v (t)∩B e(t) 4 (x) 1 dS(x) ≥ cˆ{x ∈R d−1 : |x|<ce(t)} 1 dx ≥ ce(t) d−1 .
In summary, we infer that
ce(t) d−1 ≤ˆI v (t) θ |x − x| e(t) dS(x) ≤ Ce(t) d−1 .(79)
Making use of (79), the assumptions´R d 1 − ξ · n u d|∇χ u | ≤ e(t) 2 ≤ r 2 c < 1 and d ≤ 3, the upper bounds |θ| ≤ 1 and G(λp) ≤ C(λ + λ 2 )G(p), as well as the already established L 2 -resp. H 1 -bound for the local interface error heights h ± from (57a) resp. (57b) we deduce
G(|∇h ± e(t) (x, t)|) ≤ Cr −2 c ,
which is precisely the first assertion in a). Similarly, one derives the other desired estimate G(e(t)|∇ 2 h ± e(t) (x, t)|) ≤ Cr −4 c . Integrating (78) over I v (t) and employing the global upper bound |∇h ± e(t) (·, t)| ≤ Cr −2 c , which in turn entails G(|∇h ± e(t) (·, t)|) ≥ cr 2 c |∇h ± e(t) (·, t)| 2 , we get
Iv(t) |∇h ± e(t) (x, t)| 2 dS(x) ≤ Cr −2 cˆI v (t)´I v (t) θ |x−x| e(t) G(|∇h ± (x, t)|) + G(r −1 c |h ± (x, t)|) dS(x) Iv(t) θ |x−x| e(t) dS(x) dS(x) (80) + Cr −2 cˆI v (t)´I v (t) θ |x−x| e(t) d|D s h ± |(x, t) Iv(t) θ |x−x| e(t) dS(x) dS(x).
Applying Fubini's theorem and using the bounds (79), G(λp) ≤ C(λ + λ 2 )G(p), as well as (57a) and (57b) we deduce the estimate on´I v (t) |∇h ± e(t) | 2 dS stated in a). The estimate on´I v (t) |h ± e(t) | 2 dS follows by an analogous argument, first squaring (75) and applying Jensen's inequality, then integrating over I v (t), and finally using (79), Fubini as well as (57a) and (57b).
Proof of b). We start with a change of variables to estimate (recall (16))
R d |χ v,h + e(t) ,h − e(t) − χ v,h + ,h − | dx ≤ CˆI v (t)ˆr c 0 |χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x,t) − χ 0≤dist ± (x,Iv(t))≤h + (P Iv (t) x,t) | dy dS + CˆI v (t)ˆr c 0 |χ −h − e(t) (P Iv (t) x,t)≤dist ± (x,Iv(t))≤0 − χ −h − (P Iv (t) x,t)≤dist ± (x,Iv(t))≤0 | dy dS ≤ CˆI v (t) |h + e(t) (x, t) − h + (x, t)| + |h − e(t) (x, t) − h − (x, t)| dS(x).
By adding zero and using (57c) we therefore obtain
R d |χ u − χ v,h + e(t) ,h − e(t) | dx ≤ˆR d |χ u − χ v,h + ,h − | dx +ˆR d |χ v,h + e(t) ,h − e(t) − χ v,h + ,h − | dx ≤ CˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | + CˆR d |χ u − χ v | min dist(x, I v (t)) r c , 1 dx + CˆI v (t) |h + e(t) (x, t) − h + (x, t)| + |h − e(t) (x, t) − h − (x, t)| dS(x).
Observe that one can decompose
h ± (x, t) = h ± e(t) (x, t) + ∞ k=0 h ± 2 −k−1 e(t) (x, t) − h ± 2 −k e(t) (x, t) .
A straightforward estimate in local coordinates then yieldŝ
Iv(t) h ± 2 −k e(t) − h ± 2 −k−1 e(t) dS ≤ C2 −k e(t)ˆI v (t) 1 d|D tan h ± | ≤ C2 −k e(t)ˆI v (t) 1 d|D s h ± | + C2 −k e(t)ˆI v (t) |∇h + |χ {|∇h + |≥1} dS + C2 −k e(t) ˆI v (t) |∇h + | 2 χ {|∇h + |≤1} dS 1/2 H d−1 (I v (t)) 1/2 .
Using (57b) and summing with respect to k ∈ N, we get the desired estimate (76c). Proof of c). Note that
Iv(t) η(x)h ± e(t) (x, t) dS =ˆI v (t) h ± (x, t)ˆI v (t) θ |x−x| e(t) η(x) Iv(t) θ |x−x| e(t) dS(x) dS(x) dS(x).
Abbreviating
η e (x, t) :=ˆI v (t) θ |x−x| e(t) η(x) Iv(t) θ |x−x| e(t) dS(x) dS(x), we compute |∇ tañ x η e (x, t)| = ˆI v (t) ∇ tañ x θ |x−x| e(t) η(x) Iv(t) θ |x−x| e(t) dS(x) dS(x) ≤ˆI v (t) θ |x−x| e(t) ´I v (t) θ |x−x| e(t) dS(x) η(x) e(t) dS(x).
As in the argument for (79), one checks that´I v (t) |θ |( |x−x| e(t) ) dS(x) ≤ Ce(t) d−1 . Using the lower bound from (79), the proof for the standard L p -inequality for convolutions carries over and we obtain η e L p (Iv(t)) ≤ C η L p (Iv(t)) as well aŝ
Iv(t) |∇η e (x, t)| p dS(x) ≤ C e(t) pˆI v (t) |η(x, t)| p dS(x)
for any p ≥ 1. As a consequence of (57d) and these considerations, we deduce
d dtˆI v (t) η(x)h ± e(t) (x, t) dx −ˆI v (t) h ± (x, t) d dt η e (x, t) dS(x) −ˆI v (t) h ± (x, t)(Id−n v ⊗ n v )v(x, t) · ∇xη e (x, t) dS(x) ≤ C e(t)r 2 c η L 4 (Iv(t)) ˆI v (t) |h ± | 4 dS 1/4 (81) × ˆI v (t) sup y∈[−rc,rc] |u − v| 2 (x + yn v (x, t), t) dS(x) 1/2 + C 1 + v W 2,∞ (R d \Iv(t)) r c η L 2 (Iv(t)) × ˆR d |χ u (x, t) − χ v (x, t)| min dist(x, I v (t)) r c , 1 dx 1 2 + C (1 + v W 1,∞ ) r c e(t) max p∈{2,4} η L p (Iv(t))ˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | + C η L 2 (Iv(t)) ˆI v (t) |u − v| 2 dS 1/2 . Using the estimate |v(x, t) − v(x, t)| ≤ C|x −x| ∇v L ∞ , we infer ˆI v (t) h ± (x, t)v(x, t) · ∇xˆI v (t) θ |x−x| e(t) η(x) Iv(t) θ |x−x| e(t) dS(x) dS(x) dS(x) (82) +ˆI v (t) η(x)(v(x, t) · ∇)h ± e(t) (x, t) dS(x) = ˆI v (t)ˆIv(t) η(x)h ± (x, t)v(x, t) · ∇x θ |x−x| e(t) ´I v (t) θ |x−x| e(t) dS(x) dS(x) dS(x) +ˆI v (t)ˆIv(t) η(x)h ± (x, t)v(x, t) · ∇ x θ |x−x| e(t) ´I v (t) θ |x−x| e(t) dS(x) dS(x) dS(x) ≤ˆI v (t)ˆIv(t) h ± (x, t) ∇v L ∞ |θ | |x−x| e(t) |x − x||η(x)| e(t)´I v (t) θ |x−x| e(t) dS(x) dS(x) dS(x) +ˆI v (t)ˆIv(t) h ± (x, t) v L ∞ θ |x−x| e(t) |η(x)| ∇ x´I v (t) θ |x−x| e(t) dS(x) ´I v (t) θ |x−x| e(t) dS(x) 2 dS(x) dS(x) ≤ Cr −1 c v W 1,∞ ˆI v (t) |h ± (x, t)| 2 dS(x) 1/2 ˆI v (t) |η(x)| 2 dS(x) 1/2
where in the last step we have used the simple equality
∇ tan xˆI v (t) θ |x − x| e(t) dS(x) = −ˆI v (t) ∇ tan x θ |x − x| e(t) dS(x) (83) =ˆI v (t) θ |x − x| e(t) H(x) dS(x)
and the bounds (18) and (79). Recall from the transport theorem for moving hypersurfaces (see [77]) that we have for any (70)), we then compute for
f ∈ C 1 (R d × [0, T strong )) d dtˆI v (t) f (x, t) dS(x) =ˆI v (t) ∂ t f (x, t) dS(x) +ˆI v (t) V n · ∇f (x, t) dS(x) (84) +ˆI v (t) f (x, t) H · V n dS(x) with the normal velocity V n (x, t) = (v(x, t) · n v (P Iv(t) x, t))n v (P Iv(t) x, t). Making use of (84) and d dt P Iv(t)x = −V n (x, t) forx ∈ I v (t) (seeeveryx ∈ I v (t) d dtˆI v (t) θ |x − x| e(t) dS(x) = d dtˆI v (t) θ |P Iv(t)x − P Iv(t) x| e(t) dS(x) = − e (t) e(t) 2ˆI v (t) θ |x − x| e(t) |x − x| dS(x) + 1 e(t)ˆI v (t) θ |x − x| e(t) (x − x) · (V n (x, t) − V n (x, t)) e(t)|x − x| dS(x) +ˆI v (t) θ |x − x| e(t) V n (x) · H(x) dS(x).
This together with another application of (84) and the fact that n v · ∇η = 0 on the interface
I v (t) implies forx ∈ I v (t) d dt η e (x, t) = d dtˆI v (t) θ |P Iv (t)x −P Iv (t) x| e(t) η(x) Iv(t) θ |P Iv (t)x −P Iv (t) x| e(t) dS(x) dS(x) (85) =ˆI v (t) θ |x−x| e(t) η(x) Iv(t) θ |x−x| e(t) dS(x) V n (x) · H(x) dS(x) −ˆI v (t) θ |x−x| e(t) η(x) ´I v (t) θ |x−x| e(t) V n (x) · H(x) dS(x) ´I v (t) θ |x−x| e(t) dS(x) 2 dS(x) +ˆI v (t) η(x)θ |x−x| e(t) (x−x)·(Vn(x)−Vn(x)) e(t)|x−x| Iv(t) θ |x−x| e(t) dS(x) dS(x) −ˆI v (t) θ |x−x| e(t) η(x)´I v (t) θ |x−x| e(t) (x−x)·(Vn(x,t)−Vn(x,t)) e(t)|x−x| dS(x) ´I v (t) θ |x−x| e(t) dS(x) 2 dS(x) − e (t) e(t)ˆI v (t) F e,θ (x, x)η(x) Iv(t) θ |x−x| e(t) dS(x) dS(x)
where F e,θ (t) :
I v (t) × I v (t) → R is the kernel F e,θ (t)(x, x) := θ |x − x| e(t) |P Iv(t)x − P Iv(t) x| e(t) (86) − θ |x−x| e(t) ´I v (t) θ |x−x| e(t) |P Iv (t)x −P Iv (t) x| e(t) dS(x) Iv(t) θ |x−x| e(t)
dS(x)
.
Observe that we haveˆI
v (t) F e,θ (t)(x, x) dS(x) = 0.(87)
By the choice of the cutoff θ, we see that for every given x ∈ I v (t) the kernel F e,θ (t) is supported in B e(t)/2 (x) ∩ I v (t). Moreover, the exact same argumentation which led to the upper bound in (79) (we only used the support and upper bound for θ as well as e(t) ≤ r c ) shows that the kernel F e,θ satisfies the upper bound
Iv(t) |F e,θ (x, x)| p dS(x) ≤ C(p)e(t) d−1 (88)
for any 1 ≤ p < ∞. We next intend to rewrite the function F e,θ (x, x) for fixed x as the divergence of a vector field. By the property (87), we may consider Neumann problem for the (tangential) Laplacian with right hand side F e,θ (·, x) in some neighborhood (of scale e(t)) of the point x. To do this we first rescale the setup, i.e., we consider the kernel F 1 (x, x) := F e,θ (e(t)x, e(t)x) forx, x ∈ e(t) −1 I v (t). By scaling and the fact that F e,θ is supported on scale e(t)/2, it follows that F 1 (·, x) has zero average on e(t) −1 I v (t) ∩ B 1 (x) for every point x ∈ e(t) −1 I v (t) and that
e(t) −1 Iv(t) |F 1 (x, x)| p dS(x) ≤ C(p).(89)
We fix x ∈ e(t) −1 I v (t) and solve on e(t) −1 I v (t) ∩ B 1 (x) the weak formulation of the equation −∆ tañ xF 1 (·, x) = F 1 (·, x) with vanishing Neumann boundary condition. More precisely, we requireF 1 (·, x) to have vanishing average on e(t) −1 I v (t) ∩ B 1 (x) (note that in the weak formulation the curvature term does not appear because it gets contracted with the tangential derivative of the test function). By elliptic regularity and (89), it follows
||∇ tanF 1 (x, x)|| L ∞ ≤ C.(90)
We now rescale back to I v (t) and defineF e,θ (x, x) := e(t) 2F 1 (e(t) −1x , e(t) −1 x) for x ∈ I v (t) andx ∈ I v (t) ∩ B e(t) (x). For fixed x ∈ I v (t),F e,θ (·, x) has vanishing average on I v (t) ∩ B e(t) (x) and solves −∆ tañ xF e,θ (·, x) = F e,θ (·, x) on I v (t) ∩ B e(t) (x) with vanishing Neumann boundary condition. We finally introduce F e,θ (x, x) := ∇ tañ xF e,θ (x, x) for x ∈ I v (t) andx ∈ I v (t) ∩ B e(t) (x). It then follows from scaling, (90) as well as e(t) < r c that ∇x · F e,θ (x, x) = F e,θ and
||e −1 (t)F e,θ (x, x)|| L ∞ ≤ C.(91)
We now have everything in place to proceed with estimating the term ˆI
v (t) h ± (x, t) d dt η e (x, t) dS(x) .
To this end, we will make use of (85) and estimate term by term. Because of (18), (79), η e L p (Iv(t)) ≤ C η L p (Iv(t)) , the estimatê
Iv(t) |θ | |x − x| e(t) dS(x) ≤ Ce(t) d−1 ,
the Lipschitz property |V n (x) − V n (x)| ≤ ||∇v|| L ∞ |x −x|, and the fact that θ(s) = 0 for s ≥ 1, the first four terms on the right-hand side of (85) are straightforward to estimate and result in the bound
Cr −1 c v W 1,∞ h ± (·, t) L 2 (Iv(t)) η L 2 (Iv(t)) .(92)
To estimate the fifth term, we first apply Fubini's theorem and then perform an integration by parts (recall that we imposed vanishing Neumann boundary conditions) which entails because of the above considerations
1 e(t)ˆI v (t) h ± (x, t)ˆI v (t) F e,θ (x, x) Iv(t) θ |x−x| e(t) dS(x) η(x) dS(x) dS(x) =ˆI v (t) ˆI v (t)∩B 3 4 e(t) (x) h ± (x, t) e(t) −1 F e,θ (x, x) Iv(t) θ |x−x| e(t) dS(x) dS(x) η(x) dS(x) = −ˆI v (t) ˆI v (t)∩B 3 4 e(t) ∇xh ± (x, t) · e(t) −1 F e,θ (x, x) Iv(t) θ |x−x| e(t) dS(x) dS(x) η(x) dS(x) −ˆI v (t) h ± (x, t)H(x, t) · ˆI v (t) e(t) −1 F e,θ (x, x) Iv(t) θ |x−x| e(t) dS(x) η(x) dS(x) dS(x).
Using (91) as well as the lower bound from (79) we see that the second term can be estimated by a term of the form (92). For the first term, note that by the properties of F e,θ we may interpret the integral in brackets as the mollification of ∇h ± on scale e(t). Applying the argument which led to (80) (for this, we only need the upper bound (91) for F e,θ , a lower bound as in (79) is only required for θ) we observe that one can bound this term similar to ∇h ± e(t) (·, t) L 2 (Iv(t)) . We therefore obtain the bound ˆI
v (t) h ± (x, t) d dt η e (x, t) dS(x) ≤ Cr −4 c v W 1,∞ (1 + e (t)) ˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | 1/2 ||η|| L 2 (Iv(t)) + Cr −6 c v W 1,∞ (1 + e (t)) ˆR d |χ u − χ v | min dist(x, I v (t)) r c , 1 dx 1/2 ||η|| L 2 (Iv(t)) .
Hence, combining (81) with these estimates for the fourth term from (85) as well as (92) and (82), we obtain the desired estimate on the time derivative. This concludes the proof.
6.3. Construction of the compensation function w for the velocity gradient discontinuity. We turn to the construction of a compensating vector field, which shall be small in the L 2 -norm but whose associated viscous stress µ(χ u )D sym w shall compensate for (most of) the problematic viscous term (µ(χ u ) − µ(χ v ))D sym v appearing on the right hand side of the relative entropy inequality from Proposition 9 in the case of different shear viscosities. Before we state the main result of this section, we introduce some further notation. Let h + e(t) be defined as in Proposition 26. We then denote by P h + e(t) the downward projection onto the graph of h + e(t) , i.e.,
P h + e(t) (x, t) := P Iv(t) x + h + e(t) (P Iv(t) x, t)n v (P Iv(t) x, t).
for all (x, t) such that dist(x, I v (t)) < r c . Note that this map does not define an orthogonal projection. Analogously, one introduces the projection Then there exists a solenoidal vector field w ∈ L 2 ([0, T strong ]; H 1 (R d )) such that w is subject to the estimateŝ
R d |w| 2 dx ≤ C(r −4 c R 2 v 2 W 2,∞ (R d \Iv(t)) + 1) (93) ׈I v (t) |h + e(t) | 2 +|∇h + e(t) | 2 + |h − e(t) | 2 +|∇h − e(t) | 2 dS, where R > 0 is such that I v (t) + B rc ⊂ B R (0), and {dist ± (x,Iv(t))≥0} ∇w − χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) W ⊗ n v (P Iv(t) x, t) 2 dx (94) +ˆ{ dist ± (x,Iv(t))≤0} ∇w − χ −h − e(t) (P Iv (t) x)≤dist ± (x,Iv(t))≤0 W ⊗ n v (P Iv(t) x, t) 2 dx +ˆR d χ dist ± (x,Iv(t)) / ∈[−h − e(t) (P Iv (t) x),h + e(t) (P Iv (t) x)] |∇w| 2 dx ≤ Cr −4 c v 2 W 2,∞ (R d \Iv(t))ˆI v (t) |h + e(t) | 2 + |∇h + e(t) | 2 + |h − e(t) | 2 + |∇h − e(t) | 2 dS,
where the vector field W is given by
W (x, t) := 2(µ + − µ − ) µ + (1−χ v ) + µ − χ v Id −n v ⊗ n v (P Iv(t) x) D sym v · n v (P Iv(t) x) ,(95)
with the symmetric gradient defined by D sym v := 1 2 (∇v + ∇v T )), as well as the estimateŝ
Iv(t) sup y∈(−rc,rc) |w(x + yn v (x, t))| 2 dS(x) (96) ≤ Cr −4 c v 2 W 2,∞ (R d \Iv(t))ˆI v (t) |h + e(t) | 2 + |∇h + e(t) | 2 + |h − e(t) | 2 + |∇h − e(t) | 2 dS, ∇w L ∞ ≤ Cr −4 c | log e(t)| v W 2,∞ (R d \Iv(t)) + Cr −3 c ∇ 3 v L ∞ (R d \Iv(t)) (97) + Cr −9 c 1+H d−1 (I v (t)) v W 2,∞ (R d \Iv(t)) , ˆI v (t) sup y∈[−rc,rc] |(∇w) T (x + yn v (x, t))n v (x, t)| 2 dS(x) 1 2 (98) ≤ Cr −9 c (1 + H d−1 (I v (t))) v W 2,∞ (R d \Iv(t)) e(t) + Cr −2 c v W 3,∞ (R d \Iv(t)) e(t) + Cr −1 c v W 2,∞ (R d \Iv(t)) | log e(t)| 1 2 e(t)
and
∂ t w(·, t) = − v(·, t) · ∇ w(·, t) + g +ĝ,(99)
where the vector fields g andĝ are subject to the bounds
ĝ L 4 3 (R d ) (100) ≤ C v W 1,∞ v W 2,∞ (R d \Iv(t)) e(t)r 3 c ˆI v (t) |h ± | 4 dS 1 4 × ˆI v (t) |h + e(t) | 2 + |∇h + e(t) | 2 + |h − e(t) | 2 + |∇h − e(t) | 2 dS 1 2 + C v W 1,∞ e(t)r 2 c ˆI v (t) |h ± | 4 dS 1 4 ( u−v−w 1 2 L 2 ∇(u−v−w) 1 2 L 2 + u−v−w L 2 ) + C v W 1,∞ (1 + v W 1,∞ ) e(t)ˆRd 1 − ξ · ∇χ u |∇χ u | d|∇χ u |, and g L 2 (R d ) (101) ≤ C 1+ v W 1,∞ r 2 c ( ∂ t ∇v L ∞ (R d \Iv(t)) +(R 2 +1) v W 2,∞ (R d \Iv(t)) ) × ˆI v (t) |h + e(t) | 2 + |∇h + e(t) | 2 + |h − e(t) | 2 + |∇h − e(t) | 2 dS 1 2 + C v W 1,∞ (1 + v W 1,∞ ) e(t)r cˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | + Cr −2 c (1 + e (t)) v 2 W 1,∞ ˆI v (t) |h ± | 2 dS 1 2 + C v W 1,∞ (1+ v W 2,∞ (R d \Iv(t)) ) r c ˆR d |χ u −χ v | min dist(x, I v (t)) r c , 1 dx 1 2 + C v W 1,∞ ( u−v−w 1 2 L 2 ∇(u−v−w) 1 2 L 2 + u−v−w L 2 ),
whereh ± is defined as h ± but now with respect to the modified cut-off function θ(·) = θ · 2 , see Proposition 25. Furthermore, w may be taken to have the regularity
∇w(·, t) ∈ W 1,∞ (R d \ (I v (t) ∪ I h + e (t) ∪ I h + e (t))) for almost every t, where I h ± e (t) denotes the C 3 -manifold {x ± h ± e(t) (x)n v (x) : x ∈ I v (t)}.
Proof.
Step 1: Definition of w. Let η be a cutoff supported at each t ∈ [0, T strong ) in the set I v (t) + B rc/2 with η ≡ 1 in I v (t) + B rc/4 and |∇η| ≤ Cr −1 c , Define the vector field W as given in (95) and set (making use of the notation a ∧ b = min{a, b} and a ∨ b = max{a, b})
|∇ 2 η| ≤ Cr −2 c as well as |∂ t η| ≤ Cr −1 c v L ∞ and |∂ t ∇η| ≤ Cr −2 c v W 1,∞ .w + (x, t) := ηˆ( dist ± (x,Iv(t))∨0)∧h + e(t) (P Iv (t) x) 0 W (P Iv(t) x + yn v (P Iv(t) x, t)) dy (102)
as well as
w − (x, t) := ηˆ0 (dist ± (x,Iv(t))∧0)∨−h − e(t) (P Iv (t) x) W (P Iv(t) x + yn v (P Iv(t) x, t)) dy.(103)
For this choice, we have
∇w + (x, t) (104) = χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) W (x) ⊗ n v (P Iv(t) x) + η χ dist ± (x,Iv(t))>h + e(t) (P Iv (t) x) W (P h + e(t) x) ⊗ ∇h + e(t) (P Iv(t) x)∇P Iv(t) (x) + ηˆ( dist ± (x,Iv(t))∨0)∧h + e(t) (P Iv (t) x) 0 ∇W (P Iv(t) x+yn v (P Iv(t) x))(∇P Iv(t) x+y∇n v (P Iv(t) x)) dy + ∇ηˆ( dist ± (x,Iv(t))∨0)∧h + e(t) (P Iv (t) x) 0 W (P Iv(t) x+yn v (P Iv(t) x)) dy
(note that this directly implies the last claim about the regularity of w, namely ∇w(·, t) ∈ W 1,∞ (R d \ (I v (t) ∪ I h + e (t) ∪ I h + e (t))) for almost every t) as well as
∂ t w + (x, t) (105) = χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) W (x)∂ t dist ± (x, I v (t)) + η χ dist ± (x,Iv(t))>h + e(t) (P Iv (t) x) W (P h + e(t) x) ∂ t h + e(t) (P Iv(t) x) + ∂ t P Iv(t) x · ∇h + e(t) (P Iv(t) x) + ηˆ( dist ± (x,Iv(t))∨0)∧h + e(t) (P Iv (t) x) 0 ∂ t W (P Iv(t) x+yn v (P Iv(t) x)) dy + ηˆ( dist ± (x,Iv(t))∨0)∧h + e(t) (P Iv (t) x) 0 ∇W (P Iv(t) x+yn v (P Iv(t) x))(∂ t P Iv(t) x+y∂ t n v (P Iv(t) x)) dy + ∂ t ηˆ( dist ± (x,Iv(t))∨0)∧h + e(t) (P Iv (t) x) 0 W (P Iv(t) x + yn v (P Iv(t) x)) dy.
Moreover, note that (104) entails by the definition of the vector field W ∇ · w + (x, t)
(106) = η χ dist ± (x,Iv(t))>h + e(t) (P Iv (t) x) W (P h + e(t) x) · ∇h + e(t) (P Iv(t) x)∇P Iv(t) (x) + ηˆ( dist ± (x,Iv(t))∨0)∧h + e(t) (P Iv (t) x) 0 tr ∇W (P Iv(t) x+yn v (P Iv(t) x))(∇P Iv(t) x+y∇n v (P Iv(t) x)) dy + ∇η ·ˆ( dist ± (x,Iv(t))∨0)∧h + e(t) (P Iv (t) x) 0 W (P Iv(t) x+yn v (P Iv(t) x)) dy.
Analogous formulas and properties can be derived for w − . The function w + + w − would then satisfy our conditions, with the exception of the solenoidality ∇ · w = 0. For this reason, we introduce the (usual) kernel
θ(x) := 1 H d−1 (S d−1 ) x |x| d and set w(x, t) := w + (x, t) − (θ * ∇ · w + )(x, t) + w − (x, t) − (θ * ∇ · w − )(x, t).(107)
It is immediate that ∇ · w = 0.
Step 2: Estimates on w and ∇w. From (104), |∇η| ≤ Cr −1 c as well as the bounds (17) and (26) we deduce the pointwise bound
∇w + − χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) W (x) ⊗ n v (P Iv(t) x) ≤ Cχ supp η r −1 c ∇v L ∞ |∇h + e(t) (P Iv(t) x)| (108) + Cχ supp η r −2 c ∇v L ∞ + r −1 c ∇ 2 v L ∞ (R d \Iv(t)) |h + e(t) (P Iv(t) x)| + Cr −1 c χ supp η ∇v L ∞ |h + e(t) (P Iv(t) x)|
and therefore by integration and a change of variables Φ t
R d ∇w + − χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) W (x) ⊗ n v (P Iv(t) x) 2 dx (109) ≤ C(r −4 c ∇v 2 L ∞ + r −2 c ∇ 2 v 2 L ∞ (R d \Iv(t)) )ˆR d χ supp η (|h + e(t) | 2 + |∇h + e(t) | 2 )(P Iv(t) x) dx ≤ Cr −4 c v 2 W 2,∞ (R d \Iv(t))ˆI v (t)
|h + e(t) | 2 + |∇h + e(t) | 2 dS.
Observe that this also implies by (95)
R d |∇ · w + | 2 dx ≤ Cr −4 c v 2 W 2,∞ (R d \Iv(t))ˆI v (t) |h + e(t) | 2 + |∇h + e(t) | 2 dS.(110)
From this, Theorem 34, and the fact that ∇θ is a singular integral kernel subject to the assumptions of Theorem 34, we deducê
R d ∇(θ * (∇ · w + )) 2 dx ≤ Cr −4 c v 2 W 2,∞ (R d \Iv(t))ˆI v (t) |h + e(t) | 2 + |∇h + e(t) | 2 dS.(111)
Combining the estimates (109) and (111) with the corresponding inequalities for w − and θ * ∇ · w − , we deduce our estimate (94).
The trivial estimate |w + (x, t)| ≤ χ supp η (x, t) ∇v L ∞ h + e(t) (P Iv(t) x) gives by the change of variables Φ tˆR d |w + | 2 dx ≤ Cr cˆI v (t) |h + e(t) | 2 dS.(112)
Now, let R > 1 be big enough such that I v (t) + B rc ⊂ B R (0) for all t ∈ [0, T strong ). We then estimate with an integration by parts and Theorem 34 applied to the singular integral operator ∇θ
R d \B 3R (0) θ * (∇ · w + ) 2 dx =ˆR d \B 3R (0) ˆB R (0) θ(x −x)(∇ · w + (x)) dx 2 dx ≤ˆR d ˆB R (0) ∇θ(x −x)w + (x) dx 2 dx ≤ CˆB R (0) |w + | 2 dx.(113)
By Young's inequality for convolutions, (110), (112) and (113) we then obtain
R d θ * (∇ · w + ) 2 dx =ˆB 3R (0) θ * (∇ · w + ) 2 dx +ˆR d \B 3R (0) θ * (∇ · w + ) 2 dx ≤ C ˆB 3R (0) 1 |x| d−1 dx 2ˆR d |∇ · w + | 2 dx + CˆR d |w + | 2 dx (114) ≤ C(r −4 c R 2 v 2 W 2,∞ (R d \Iv(t)) + 1)ˆI v (t) |h + e(t) | 2 + |∇h + e(t) | 2 dS.
Together with the respective estimates for w − and θ * (∇ · w − ), this implies (93). The estimate (96) follows directly from (102) and the estimates (111) and (114) on the H 1 -norm of θ * (∇ · w + ) as well as the definition of w − and the analogous estimates for θ * (∇ · w − ).
Step 3: L ∞ -estimates for ∇w. Regarding the estimate (97) on ∇w L ∞ we have by (108) and the estimates |∇h + e(t) | ≤ Cr −2 c and |h + e(t) | ≤ r c ≤ 1 from Proposition 26
∇w + L ∞ ≤ Cr −4 c v W 2,∞ (R d \Iv(t)) .(115)
To estimate |∇(θ * (∇ · w + ))|, we first compute starting with (106)
∇(∇ · w + )(x, t) (116) = η χ dist ± (x,Iv)>h + e(t) (P Iv (t) x) W (P h + e(t) x) · ∇ 2 h + e(t) (P Iv(t) x)∇P Iv(t) (x)∇P Iv(t) (x) + W (P h + e(t) x) · ∇h + e(t) (P Iv(t) x)∇P Iv(t) (x) ∇χ dist ± (x,Iv)>h + e(t) (P Iv (t) x) + F (x, t), where F (x, t) is subject to a bound of the form |F (x, t)| ≤ Cr −5 c v W 3,∞ (R d \Iv(t))
) and supported in I v (t) + B rc . Next, we decompose the kernel θ as θ = ∞ k=−∞ θ k with smooth functions θ k with supp θ k ⊂ B 2 k+1 \ B 2 k−1 . More precisely, we first choose a smooth function ϕ : R + → [0, 1] such that ϕ(s) = 0 whenever s / ∈ [−1/2, 2] and such that k∈Z ϕ(2 k s) = 1 for all s > 0. Such a function indeed exists, see for instance [14]. We then let θ k (x) := ϕ(2 k |x|)θ(x). Note that
θ k L 1 (R d ) ≤ C2 k , ∇θ k L 1 (R d ) ≤ C as well as |∇θ k | ≤ C(2 k ) −d . We estimate |∇(θ * (∇ · w + ))| ≤ 0 k= log e 2 (t) |∇(θ k * (∇ · w + ))| + ∞ k=1 |∇(θ k * (∇ · w + ))| (117) + log e 2 (t) −1 k=−∞ |θ k * ∇(∇ · w + )|.
Using Young's inequality for convolutions as well as the estimate ∇θ k L 1 (R d ) ≤ C we obtain 0 k= log e 2 (t)
|∇(θ k * (∇ · w + ))| ≤ 2C| log e(t)| ∇ · w + L ∞ .(118)
Moreover, it follows from |∇θ k | ≤ C(2 k ) −d , the precise formula for ∇ · w + in (106), (17), (26), a change of variables and Hölder's inequality that
∞ k=1 |∇(θ k * (∇ · w + ))| (119) ≤ Cr −2 c v W 2,∞ (R d \Iv(t)) ∞ k=1 (2 k ) −dˆI v (t)+B rc /2 |∇h + e(t) (P Iv(t) x)| + |h + e(t) (P Iv(t) x)| dx ≤ Cr −2 c v W 2,∞ (R d \Iv(t)) H d−1 (I v (t)) ˆI v (t) |∇h + e(t) | 2 + |h + e(t) | 2 dS 1 2 .
Using (116), the estimate |∇ 2 h ± e(t) (·, t)| ≤ Cr −4 c e(t) −1 from Proposition 26, (17), (26) and again Young's inequality for convolutions (recall that θ k L 1 (R d ) ≤ C2 k ), we get log e 2 (t) −1
k=−∞ |θ k * ∇(∇ · w + )|(x, t) ≤ I + II + III(120)
where the three terms on the right hand side are given by
I := log e 2 (t) −1 k=−∞ 2 k Cr −5 c v W 3,∞ (R d \Iv(t)) ≤ Cr −5 c v W 3,∞ (R d \Iv(t)) e 2 (t)(121)
and
II := Cr −5 c v W 1,∞ e(t) −1 log e 2 (t) −1 k=−∞ 2 k ≤ Cr −5 c v W 1,∞ e(t)(122)
as well as
III := log e 2 (t) −1 k=−∞ ˆR d θ k (x−x) ⊗ W (P h + e(t) x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x) (123) d∇χ dist ± (x,Iv(t))>h + e(t) (P Iv (t) x) (x)
. To estimate the latter term, we proceed as follows. First of all, note that by the definition of h + e(t) in (75) as well as the trivial bound |h + | ≤ r c it holds |h + e(t) | ≤ r c . Then for allx ∈ I v (t) + {|x| > r c + 2 log e 2 (t) } and all k ≤ log e 2 (t) − 1 we observe that χ {dist ± (x,Iv(t))>h + e(t) (P Iv (t) x)} (x) = 1 for all x ∈ R d such that |x −x| ≤ 2 k+1 . In particular, for suchx the third term on the right hand side of (120) vanishes since the corresponding second term in the formula for ∇(∇ · w + ) (see (116)) does not appear anymore.
Hence, letx ∈ I v (t) + {|x| ≤ r c + 2 log e 2 (t) } and denote by F the tangent plane to the manifold {dist ± (x, I v (t)) = h + e(t) (P Iv(t) x)} at the nearest point tox. We then have for any ψ ∈ C ∞ cpt (R d )
R d ψ(x) d∇χ {dist ± (x,Iv(t))>h + e(t) (P Iv (t) x)} (x) −ˆR d ψ(x) d∇χ {dist ± (x,F )>0} (x) =ˆ{ dist ± (x,Iv(t))>h + e(t) (P Iv (t) x)} ∇ψ(x) dx −ˆ{ dist ± (x,F )>0}
∇ψ(x) dx and as a consequencê
R d θ k (x −x) ⊗ W (P h + e(t) x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x) d∇χ {dist ± (x,Iv(t))>h + e(t) (P Iv (t) x)} (x) =ˆF θ k (x −x) ⊗ W (P h + e(t) x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x) n F dS(x) +ˆR d (χ {dist ± (x,Iv(t))>h + e(t) (P Iv (t) x)} − χ {dist ± (x,F )>0} ) ∇ θ k (x −x) ⊗ W (P h + e(t) x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x) dx.
Recall that we defined θ k (x) := ϕ(2 k |x|)θ(x) where ϕ : R + → [0, 1] is a smooth function such that ϕ(s) = 0 whenever s / ∈ [−1/2, 2] and such that k∈Z ϕ(2 k s) = 1 for all s > 0. Hence,
|n F ·θ k (x−x)| ≤ C |n F ·(x−·x)| |x−x| d ≤ C dist(x,F )
|x−x| d for all x ∈ F . It also follows from the definition of θ that´F (Id −n F ⊗ n F )θ k (x −x) dS(x) = 0. Hence we may solve (Id −n F ⊗ n F )θ k (· −x) = ∆ tan xθk (·,x) on B 2 k+2 (x) ∩ F with vanishing Neumann boundary conditions. In particular, forθ k (x,x) := ∇ tan xθk (x,x) we obtain (Id −n F ⊗ n F )θ k (x −x) = ∇ tan x · ∇ xθk (x,x). It follows from elliptic regularity that θ(·,x) is C ∞ . Moreover, since we could have rescaled θ k first to unit scale, then solved the associated problem on that scale, and finally rescaled the solution back to the dyadic scale k we see that |θ k (x,x)| ≤ C(2 k ) 2−d . We then have by an integration by parts
ˆF (Id −n F ⊗ n F )θ k (x −x) ⊗ ψ dS(x) ≤ˆF ∩B 2 k+1 (x) |θ k (x,x)||∇ tan ψ| dS(x) ≤ C(2 k ) 2−dˆF ∩B 2 k+1 (x) |∇ tan ψ| dS(x) for any ψ ∈ C 1 cpt (R d ; R d ). Furthermore, it holdŝ B 2 k (x) |χ {dist ± (x,Iv(t))>h + e(t) (P Iv (t) x)} − χ {dist ± (x,F )>0} | dx ≤ C ∇ 2 h + e(t) L ∞ (2 k ) d+1 .
Using these considerations in the previous formula, we obtain
ˆR d θ k (x −x) ⊗ W (P h + e(t) x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x) d∇χ {dist ± (x,Iv(t))>h + e(t) (P Iv (t) x)} (x) (124) ≤ˆF ∩B 2 k+1 (x)\B 2 k−1 (x) dist(x, F ) |x − x| d |W (P h + e(t) x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x)| dS(x) +ˆF ∩B 2 k+1 (x) C(2 k ) 2−d |∇(W (P h + e(t)
x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x))| dS(x)
+ C ∇ 2 h + e(t) L ∞ (2 k ) d+1 ∇ θ k (x −x) ⊗ W (P h + e(t) x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x) L ∞ .
Making use of the fact that the integral vanishes for dist(x, F ) ≥ 2 k+1 and the bounds (17) and (26) we obtain
F ∩B 2 k+1 (x)\B 2 k−1 (x) dist(x, F ) |x − x| d |W (P h + e(t) x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x)| dS(x) (125) ≤ χ {dist(x,F )<2 k } Cr −3 c v W 1,∞ dist(x, F ) 2 kˆF ∩B 2 k+1 (x)\B 2 k−1 (x) |∇h + e(t) (P Iv(t) x)| |x − x| d−1 dS(x).
Using also |∇h + e(t) | ≤ Cr −2 c and |∇ 2 h + e(t) | ≤ Cr −4 c e(t) −1 from Proposition 26, we getˆF
∩B 2 k+1 (x) C(2 k ) 2−d |∇(W (P h + e(t) x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x))| dS(x) (126) ≤ C2 k e(t) −1 r −5 c v W 1,∞ + r −4 c v W 2,∞ (R d \Iv(t))
and
C ∇ 2 h + e(t) L ∞ (2 k ) d+1 ∇ θ k (x −x) ⊗ W (P h + e(t) x) · (∇P Iv(t) ) T (x)∇h + e(t) (P Iv(t) x) L ∞ (127) ≤ Cr −4 c e(t) −1 2 k r −3 c v W 1,∞ + Cr −4 c e(t) −1 (2 k ) 2 e(t) −1 r −5 c v W 1,∞ + r −4 c v W 2,∞ (R d \Iv(t)) .
Using (124), (125), (126) and (127) to estimate the term in (123), we get
III ≤ C v W 1,∞ r 3 c log e 2 (t) −1 k=−∞ χ {dist(x,F )<2 k } dist(x, F ) 2 kˆF ∩B 2 k+1 (x)\B 2 k−1 (x) |∇h + e(t) (P Iv(t) x)| |x − x| d−1 dS(x)(128)+ Cr −9 c v W 2,∞ (R d \Iv(t)) e(t).
In turn, combining this with (121) and (122) and gathering also (118), (119), (115) as well as the corresponding bounds for ∇w − and ∇(θ * ∇ · w − ), we then finally deduce (97).
Step 4: L 2 L ∞ -estimate for ∇w. By making use of the precise formula (104) for ∇w + and the definition of the vector field W in (95), we immediately get
Iv(t) sup y∈[−rc,rc] |(∇w + ) T (x + yn v (x, t)) · n v (x, t)| 2 dS(x) (129) ≤ Cr −2 c v W 2,∞ (R d \Iv(t))ˆI v (t) |h + e(t) | 2 + |∇h + e(t) | 2 dS.
To estimate the contribution from |∇(θ * (∇ · w + ))| we use the same dyadic decomposition as in (117). We start with the terms in the range k = log e 2 (t) , . . . , 0. Let x ∈ I v (t) and y ∈ (−r c , r c ) be fixed. We abbreviatex := x+yn v (x, t). Denote by F x the tangent plane of the interface I v (t) at the point x. Let Φ Fx : F x ×R → R d be the diffeomorphism given by Φ Fx (x,ŷ) :=x+ŷn Fx (x). We start estimating using the change of variables Φ Fx , the bound |∇θ k (x)| ≤ Cχ 2 k−1 ≤|x|≤2 k+1 |x| −d , as well as the fact thatx + yn Fx (x) =x + yn v (x, t) is exactly the point on the ray originating fromx ∈ F x in normal direction which is closest tox
| ∇(θ k * (∇ · w + )) T (x + yn v (x, t))| ≤ˆ( B 2 k+1 (x)\B 2 k−1 (x))∩(Iv(t)+B rc /2 ) |∇θ k (x−x)||(∇ · w + )(x)| dx ≤ CˆF x ∩(B 2 k+1 (x)\B 2 k−1 (x)) sup y∈[−rc,rc] |(∇ · w + )(x+ŷn Fx (x))| |x −x| d−1 dS(x).
Note that the right hand side is independent of y. Hence, we may estimate with Minkowski's inequality
ˆI v (t) sup y∈[−rc,rc] 0 k= log e 2 (t) −1 ∇(θ k * (∇ · w + ))(x + yn v (x, t)) 2 dS(x) 1 2 ≤ C| log e(t)| ˆI v (t) ˆF x sup y∈[−rc,rc] |(∇ · w + )(x+ŷn Fx (x))| |x −x| d−1 dS(x) 2 dS(x) 1 2
The inner integral is to be understood in the Cauchy principal value sense. To proceed we use the L 2 -theory for singular operators of convolution type, the precise formula (106) for ∇ · w + as well as (17) and (26)
|(∇ · w + )(x+ŷn Fx (x))| |x −x| d−1 dS(x) 2 dS(x) 1 2 ≤ C ˆI v (t) sup y∈[−rc,rc] |(∇ · w + )(x+yn v (x, t))| 2 dS(x) 1 2 ≤ Cr −1 c v 1 2 W 2,∞ (R d \Iv(t)) ˆI v (t) |h + e(t) | 2 + |∇h + e(t) | 2 dS 1 2 .
An application of (76a) and the assumption E[χ u , u, V |χ v , v](t) ≤ e 2 (t) finally yields
ˆI v (t) sup y∈[−rc,rc] 0 k= log e 2 (t) −1 ∇(θ k * (∇ · w + ))(x + yn v (x, t)) 2 dS(x) 1/2 (130) ≤ Cr −5 c v W 2,∞ (R d \Iv(t)) | log e(t)|e(t).
We move on with the contributions in the range k = 1, . . . , ∞. Note that by (119) we may directly infer from (76a) and the assumption
E[χ u , u, V |χ v , v](t) ≤ e 2 (t) Iv(t) sup y∈[−rc,rc] ∞ k=1 ∇(θ k * (∇ · w + )) T (x + yn v (x, t)) · n v (x, t) 2 dS(x) (131) ≤ Cr −8 c v 2 W 2,∞ (R d \Iv(t)) H d−1 (I v (t)) 2 e 2 (t)
. Moreover, the contributions estimated in (121) and (122) result in a bound of the form (recall that e(t) < r c )
Cr −4 c v 2 W 3,∞ (R d \Iv(t)) e 2 (t) + Cr −8 c v 2 W 1,∞ e 2 (t).(132)
Note that when summing the respective bounds from (126) and (127) over the relevant range k = −∞, . . . , log e 2 (t) − 1, we actually gain a factor e(t), i.e., the contributions estimated in (126) and (127) then directly yield a bound of the form
Cr −18 c v 2 W 2,∞ (R d \Iv(t)) e 2 (t).(133)
Finally, the contribution from (125) may be estimated as follows. Let x ∈ I v (t), y ∈ [−r c , r c ] and denote by Fx the tangent plane to the manifold {dist ± (x, I v (t)) = h + e(t) (P Iv(t) x)} at the nearest point tox = x + yn v (x, t). In light of (125), we start estimating for k ≤ log e 2 (t) − 1 by using Jensen's inequality, the bound |∇h + e(t) | ≤ Cr −2 c from Proposition 26, as well as the fact that |x −x| ≥ |x −x| for allx ∈ I v (t) (since x = P Iv(t)x is the closest point tox on the interface I v (t)) ˆFx
∩B 2 k+1 (x)\B 2 k−1 (x) |∇h + e(t) (P Iv(t)x )| |x −x| d−1 dS(x) 2 ≤ˆFx ∩B 2 k+1 (x)\B 2 k−1 (x) |∇h + e(t) (P Iv(t)x )| 2 |x −x| d−1 dS(x) ≤ Cr −2(d−1) cˆI v (t)∩B Cr −2 c 2 k+1 (x)
|∇h + e(t) (x)| 2 |x −x| d−1 dS(x).
Since this bound does not depend anymore on y ∈ [−r c , r c ], we may estimate the contributions from (125) using Minkowski's inequality as well as once more the L 2 -theory for singular operators of convolution type to reduce everything to the H 1 -bound (76a) for the local interface error heights. All in all, the contributions from (125) are therefore bounded by
Cr −14 c v 2 W 1,∞ e 2 (t).(134)
The asserted bound (98) then finally follows from collecting the estimates (129), (130), (131), (132), (133) and (134) together with the analogous bounds for ∇w − and ∇(θ * ∇ · w − ).
Step 5: Estimate on the time derivative ∂ t w. To estimate ∂ t w + , we first deduce using (105), |∂ t η| ≤ Cr −1 (32)), (19) and finally (70) that (19) as well as (70) we may compute
c v L ∞ , | d dt n v (P Iv(t) x)| ≤ C r 2 c v W 1,∞ (which follows from∂ t w + (x, t) = χ 0≤dist ± (x,Iv)≤h + e(t) (P Iv (t) x) W (x)∂ t dist ± (x, I v (t)) + η χ dist ± (x,Iv)>h + e(t) (P Iv (t) x) W (P h + e(t) x) ∂ t h + e(t) (P Iv(t) x) + ∂ t P Iv(t) x · ∇h + e(t) (P Iv(t) x) +g + for some vector fieldg + subject to g + (·, t) L 2 ≤ Cr −2 c (1 + v W 1,∞ )( v W 1,∞ + ∂ t ∇v L ∞ + v W 2,∞ (R d \Iv(t)) )(´I v (t) |h + e(t) (·, t)| 2 dS) 1/2 . Using (104),(v(x) · ∇)w + (x, t) + χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) W (x)∂ t dist ± (x, I v (t)) + η χ dist ± (x,Iv(t))>h + e(t) (P Iv (t) x) W (P h + e(t) x)∂ t P Iv(t) x · ∇h + e(t) (P Iv(t) x) = η χ dist ± (x,Iv(t))>h + e(t) (P Iv (t) x) W (P h + e(t) x)(Id−n v ⊗ n v )v(P Iv(t) x) · ∇h + e(t) (P Iv(t) x) + χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) W (x) (v(x) − v(P Iv(t) x) · n v (P Iv(t) ) + η χ dist ± (x,Iv(t))>h + e(t) (P Iv (t) x) W (P h + e(t) x) ∇P Iv(t) (x)v(x) − v(P Iv(t) x) · ∇h + e(t) (P Iv(t) x) +g + 1 , for some g + 1 L 2 ≤ Cr −2 c v W 2,∞ (R d \Iv(t)) (´I v (t) |h + e(t) (·, t)| 2 + |∇h + e(t) (·, t)| 2 dS) 1 2 .
This computation in turn implies
∂ t w + (x, t) (135) = −(v(x) · ∇)w + (x, t) + η χ dist ± (x,Iv(t))>h + e(t) (P Iv (t) x) W (P h + e(t) x) ∂ t h + e(t) (P Iv(t) x) + (Id−n v ⊗ n v )v(P Iv(t) x) · ∇h + e(t) (P Iv(t) x) + g + for some g + with g + L 2 ≤ Cr −2 c (1+ v W 1,∞ )( ∂ t ∇v L ∞ + v W 2,∞ (R d \Iv(t)) ) ˆI v (t)
|h + e(t) | 2 +|∇h + e(t) | 2 dS 1 2
.
We now aim to make use of (76d) to further estimate the second term in the right hand side of (135). To establish the corresponding L 2 -resp. L 4 3 -contributions, we first need to perform an integration by parts in order to use (76d). The resulting curvature term as well as all other terms which do not appear in the third term of (135) can be directly bounded by a term whose associated L 2 -norm is controlled by Cr −1
c v W 1,∞ v W 2,∞ (R d \Iv(t)) (´I v (t) |h + e(t) (·, t)| 2 +|∇h + e(t) (·, t)| 2 dS) 1 2 . Hence, using (76d) in (135) implies ∂ t w + (x, t) = −(v · ∇)w + (x, t) +ḡ + +ĝ + (136)
with the corresponding L 2 -bound
ḡ + L 2 (R d ) (137) ≤ C 1+ v W 1,∞ r 2 c ( ∂ t ∇v L ∞ + v W 2,∞ (R d \Iv(t) ) ˆI v (t) |h + e(t) | 2 +|∇h + e(t) | 2 dS 1 2 + C v W 1,∞ (1 + v W 1,∞ ) e(t)r cˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | + Cr −2 c v 2 W 1,∞ (1 + e (t)) h ± (·, t) L 2 (Iv(t)) + ∇h ± e(t) (·, t) L 2 (Iv(t)) + C v W 1,∞ (1+ v W 2,∞ (R d \Iv(t)) ) r c ˆR d |χ u −χ v | min dist(x, I v (t)) r c , 1 dx 1 2 + C v W 1,∞ ˆI v (t) |u − v| 2 dS 1 2 and L 4 3 -estimate ĝ + L 4 3 (R d ) (138) ≤ C v W 1,∞ e(t)r 2 c ˆI v (t) |h ± | 4 dS 1 4 ˆI v (t) sup y∈[−rc,rc] |u−v| 2 (x+yn v (x, t), t) dS(x) 1 2 + C v W 1,∞ (1 + v W 1,∞ ) e(t)ˆRd 1 − ξ · ∇χ u |∇χ u | d|∇χ u |.
In both bounds, we add and subtract the compensation function w and therefore obtain together with (96) and (40)
Iv(t) |u − v| 2 dS ≤ˆI v (t) sup y∈[−rc,rc] |u−v| 2 (x+yn v (x, t), t) dS(x) ≤ˆI v (t) sup y∈[−rc,rc] |u − v − w| 2 (x + yn v (x, t), t) dS(x) +ˆI v (t) sup y∈[−rc,rc] |w(x + yn v (x, t), t)| 2 dS(x) ≤ Cr −4 c v 2 W 2,∞ (R d \Iv(t))ˆI v (t)
|h ± e(t) | 2 + |∇h ± e(t) | 2 dS (139)
+ C( u−v−w L 2 ∇(u−v−w) L 2 + u−v−w 2 L 2 ).
Analogous estimates may be derived for w − . We therefore proceed with the terms related to θ * ∇ · w ± . First of all, note that the singular integral operator (θ * ∇·) satisfies (see Theorem 34) θ * ∇ ·ĝ
L 4 3 (R d ) ≤ C ĝ L 4 3 (R d ) , θ * ∇ ·ḡ L 2 (R d ) ≤ C ḡ L 2 (R d ) . (140) Furthermore, to estimate θ * ∇ · ((v · ∇)w + ) − (v · ∇)(θ * ∇ · w + ) L 2 (R d ) we first replace v with its normal velocity V n (x) := (v(x) · n v (P Iv(t) x))n v (P Iv(t) x)
. We want to exploit the fact that the vector field V n has bounded derivatives up to second order, see (41) and (42). Moreover, the kernel ∇ 2 θ(x −x) ⊗ (x − x) gives rise to a singular integral operator of convolution type, as does ∇θ. To see this, we need to check whether its average over S d−1 vanishes. We write
x ⊗ ∇ 2 θ(x) = ∇F (x) − δ ij e i ⊗ ∇θ ⊗ e j , where F (x) = x ⊗ ∇θ(x). Now, since ∇θ is homogeneous of degree −d, F itself is homogeneous of degree −(d − 1). Hence, we computé B1\Br ∇F dx =´S d−1 n ⊗ F dS −´r S d−1 n ⊗ F dS = 0
for every 0 < r < 1. Passing to the limit r → 1 shows that ∇F , and therefore also ∇ 2 θ(x) ⊗ x, have vanishing average on S d−1 . We may now compute (where the integrals are well defined in the Cauchy principal value sense due to the above considerations) for almost every
x ∈ R d R d ∇θ(x −x) · (V n (x, t) · ∇x)w + (x, t) − (V n (x, t) · ∇ x )∇θ(x −x) · w + (x, t) dx =ˆR d ∇θ(x −x)((V n (x, t) − V n (x, t)) · ∇x)w + (x, t) dx =ˆR d ∇ 2 θ(x −x) : (V n (x, t) − V n (x, t) − (x − x) · ∇V n (x, t)) ⊗ w + (x, t) dx −ˆR d ∇θ(x −x) · (∇ · V n )(x, t)w + (x, t) dx +ˆR d ∇ 2 θ(x −x) : ((x − x) · ∇)V n (x, t) ⊗ w + (x, t) dx. Note that we have |V n (x, t) − V n (x, t) − (x − x) · ∇V n (x, t)| ≤ ∇ 2 V n L ∞ |x − x| 2 and |V n (x, t) − V n (x, t) − (x − x) · ∇V n (x, t)| ≤ ||∇V n || L ∞ |x − x|.
We then estimate using Young's inequality for convolutions and
|∇ 2 θ(x)| ≤ |x| −d−1 R d \B 3R (0) ˆB R (0) ∇ 2 θ(x −x) : (V n (x) − V n (x) − (x − x) · ∇V n (x)) ⊗ w + (x) dx 2 dx (141) ≤ C ∇V n 2 L ∞ˆR d \B 3R (0) ˆB R (0) 1 |x −x| d |w + (x)| dx 2 dx ≤ C ∇V n 2 L ∞ || | · | −d || 2 L 2 (R d \B R ) ˆB R (0) |w + | dx 2 ≤ CR −d R dˆB R (0) |w + | 2 dx.
As a consequence, we obtain from (141), Young's inequality for convolutions, (112) as well as (42)
R d ˆR d ∇ 2 θ(x −x) : (V n (x) − V n (x) − (x − x) · ∇V n (x)) ⊗ w + (x) dx 2 dx (142) ≤ C ∇ 2 V n 2 L ∞ˆB 3R (0) ˆR d |w + (x)| |x −x| d−1 dx 2 dx + C ∇V n 2 L ∞ˆB R (0) |w + | 2 dx ≤ Cr −4 c v 2 W 2,∞ (R d \Iv(t)) (1+R 2 )ˆI v (t)
|h + e(t) | 2 dS.
Applying Theorem 34 to the singular integral operators ∇θ resp. ∇ 2 θ ⊗ x as well as making use of (41), (112) and (142) we then obtain the estimatê
R d |θ * ∇ · ((V n · ∇)w + ) − (V n · ∇)(θ * ∇ · w + )| 2 dx (143) ≤ Cr −4 c v 2 W 2,∞ (R d \Iv(t)) (1+R 2 )ˆI v (t) |h + e(t) | 2 dS + C ∇V n 2 L ∞ˆR d |w + | 2 dx ≤ Cr −4 c v 2 W 2,∞ (R d \Iv(t)) (1+R 2 )ˆI v (t) |h + e(t) | 2 dS.
It remains to estimate θ * ∇ · ((V tan · ∇)w + ) − (V tan · ∇)(θ * ∇ · w + ) L 2 (R d ) with V tan (x) = (Id − n v (P Iv(t) x) ⊗ n v (P Iv(t) x))v(x) denoting the tangential velocity of v. To this end, note that we may rewritê
R d ∇θ(x −x) · (V tan (x, t) · ∇x)w + (x, t) − (∇ · w + (x, t))(V tan (x, t) · ∇ x )θ(x −x) dx =ˆR d ∇θ(x−x) ∇w + (x)−χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t)x ) W (x) ⊗ n v (P Iv(t)x ) V tan (x, t) dx −ˆR d (∇ · w + (x, t))(V tan (x, t) · ∇ x )θ(x −x) dx.
Using Theorem 34, (109) as well as (110) we then obtain
θ * ∇ · ((V tan · ∇)w + ) − (V tan · ∇)(θ * ∇ · w + ) 2 L 2 (R d ) ≤ Cr −4 c v 2 L ∞ v 2 W 2,∞ (R d \Iv(t))ˆI v (t) |h + e(t) | 2 + |∇h + e(t) | 2 dS.(144)
Putting all the estimates (137), (138), (139), (140), (143) and (144) together, we get ∂ t w(x, t) + (v · ∇)w(x, t) = g +ĝ with the asserted bounds. This concludes the proof.
6.4. Estimate for the additional surface tension terms. Having established all the relevant properties of the compensating vector field w in Proposition 27, we can move on with the post-processing of the additional terms in the relative entropy inequality from Proposition 9. To this end, we start with the additional surface tension terms given by
A surT en = −σˆT 0ˆR d ×S d−1 (s−ξ) · (s−ξ) · ∇ w dV t (x, s) dt (145) + σˆT 0ˆR d (1 − θ t ) ξ · (ξ · ∇)w d|V t | S d−1 (x) dt + σˆT 0ˆR d (χ u − χ v )(w · ∇)(∇ · ξ) dx dt + σˆT 0ˆR d (χ u − χ v )∇w : ∇ξ T dx dt − σˆT 0ˆR d ξ · (n u − ξ) · ∇ w d|∇χ u | dt =: I + II + III + IV + V.
A precise estimate for these terms is the content of the following result.
Lemma 28. Let the assumptions and notation of Proposition 27 be in place. In particular, we assume that there exists a C 1 -function e : [0, T strong ) → [0, r c ) such that the relative entropy is bounded by E[χ u , u, V, |χ v , v](t) ≤ e 2 (t). Then the additional surface tension terms A surT en are bounded by a Gronwall-type term
A surT en ≤ C r 10 c (1 + v 2 L ∞ t W 2,∞ x (R d \Iv(t)) + v L ∞ t W 3,∞ x (R d \Iv(t)) ) (146)ˆT 0 (1 + | log e(t)|)E[χ u , u, V |χ v , v](t) dt + C r 10 c (1 + v 2 L ∞ t W 2,∞ x (R d \Iv(t)) + v L ∞ t W 3,∞ x (R d \Iv(t)) ) T 0 (1 + | log e(t)|)e(t)E[χ u , u, V |χ v , v] 1 2 (t) dt.
Proof. We estimate term by term in (145). A straightforward estimate for the first two terms using also the coercivity property (37) yields
I + II ≤ CˆT 0 ∇w(t) L ∞ xˆR d ×S d−1 |s − ξ| 2 dV t (x, s) dt (147) + CˆT 0 ∇w(t) L ∞ xˆR d (1 − θ t ) d|V t | S d−1 (x) dt ≤ CˆT 0 ∇w(t) L ∞ x E[χ u , u, V |χ v , v](t) dt.
Making use of (17), a change of variables Φ t , Hölder's and Young's inequality, (96), (39), (76a) as well as the coercivity property (34) the term III may be bounded by rc] |w(x+yn v (x, t))| 2 dS dt
III ≤ C r 2 cˆT 0ˆIv(t) sup y∈[−rc,rc] |w(x+yn v (x, t))|ˆr c −rc |χ u −χ v |(x+yn v (x, t)) dy dS dt (148) ≤ C r 2 cˆT 0ˆIv(t) sup y∈[−rc,+ C r 2 cˆT 0ˆIv(t) ˆr c −rc |χ u −χ v |(x+yn v (x, t)) dy 2 dS dt ≤ C r 6 c v 2 L ∞ t W 2,∞ x (R d \Iv(t))ˆT 0ˆIv(t) |h ± e(t) | 2 + |∇h ± e(t) | 2 dS dt + C r 2 cˆT 0ˆR d |χ u −χ v | min dist(x, I v (t)) r c , 1 dx dt ≤ C r 10 c v 2 L ∞ t W 2,∞ x (R d \Iv(t))ˆT 0ˆR d 1 − ξ · ∇χ u |∇χ u | d|∇χ u | dt + C r 10 c (1 + v 2 L ∞ t W 2,∞ x (R d \Iv(t)) )ˆT 0ˆR d |χ u −χ v | min dist(x, I v (t)) r c , 1 dx dt ≤ C r 10 c (1 + v 2 L ∞ t W 2,∞ x (R d \Iv(t)) )ˆT 0 E[χ u , u, V |χ v , v](t) dt.
For the term IV , we first add zero, then perform an integration by parts which is followed by an application of Hölder's inequality to obtain
IV ≤ CˆT 0 ˆR d |χ u − χ v,h + e(t) ,h − e(t) | dx 1 2 ˆR d |(∇w) T : ∇ξ| 2 dx 1 2 dt (149) + CˆT 0 ˆR d (χ v − χ v,h + e(t) ,h − e(t) )(w · ∇)(∇ · ξ) dx dt + CˆT 0 ˆR d ((w · ∇)ξ) · d∇(χ v − χ v,h + e(t) ,h − e(t) ) dt =: (IV ) a + (IV ) b + (IV ) c .
By definition of ξ, see (30), recall that
∇ξ = ζ dist ± (x,Iv(t)) rc r c n v (P Iv(t) x) ⊗ n v (P Iv(t) x) + ζ dist ± (x, I v (t)) r c ∇ 2 dist ± (x, I v (t)).
Recalling also (94), (95) and (111) as well as making use of (76c), (17), (26), (76a) and finally the coercivity property (34) the term (IV ) a from (149) is estimated by as well as (76a)
(IV ) a ≤ C r cˆT 0 E[χ u , u, V |χ v , v](t) + e(t)E[χ u , u, V |χ v , v] 1 2 (t) dt(150)+ C r 4 c v 2 L ∞ t W 2,∞ x (R d \Iv(t))ˆT 0ˆIv(t) |h ± e(t) | 2 + |∇h ± e(t) | 2 dS dt ≤ C r 8 c (1+ v 2 L ∞ t W 2,∞ x (R d \Iv(t)) )ˆT 0 E[χ u , u, V |χ v , v](t)+e(t)E[χ u , u, V |χ v , v](IV ) b ≤ C r 2 cˆT 0ˆIv(t)
|h ± e(t) | 2 dS dt (151)
+ C r 2 cˆT 0ˆIv(t) sup y∈[−rc,rc] |w(x+yn v (x, t))| 2 dS dt ≤ C r 10 c v 2 L ∞ t W 2,∞ x (R d \Iv(t))ˆT 0 E[χ u , u, V |χ v , v](t) dt.
To estimate the term (IV ) c from (149), we again make use of the definition of
χ v,h + e(t) ,h − e(t)
, (17), Hölder's and Young's inequality, (96) as well as (76a) which yields the following bound
(IV ) c ≤ C r cˆT 0ˆIv(t) |∇h ± e(t) | sup y∈[−rc,rc] |w(x+yn v (x, t))| dS dt (152) ≤ C r 9 c v L ∞ t W 2,∞ x (R d \Iv(t))ˆT 0 E[χ u , u, V |χ v , v](t) dt.
Hence, taking together the bounds from (150), (151) and (152) we obtain
IV ≤ C r 10 c (1+ v 2 L ∞ t W 2,∞ x (R d \Iv(t)) )ˆT 0 E[χ u , u, V |χ v , v](t) dt (153) + C r 10 c (1+ v 2 L ∞ t W 2,∞ x (R d \Iv(t)) )ˆT 0 e(t)E 1 2 [χ u , u, V |χ v , v](t) dt.
In order to estimate the term V , we argue as follows. In a first step, we split R d into the region I v (t) + B rc near to and the region R d \ (I v (t) + B rc ) away from the interface of the strong solution. Recall then that the indicator function χ u (·, t) of the varifold solution is of bounded variation in I v (t) + B rc . In particular,
E + := {x ∈ R d : χ u > 0} ∩ (I v (t) + B rc ) is a set of finite perimeter in I v (t) + B rc .
Applying Theorem 35 in local coordinates, the sections
E + x = {y ∈ (−r c , r c ) : χ u (x + yn v (x, t)) > 0}
are guaranteed to be one-dimensional Caccioppoli sets in (−r c , r c ), and such that all of the four properties listed in
(∂ * E + x ) ≤ 2 or H 0 (∂ * E + x ) > 2.
In other words, we distinguish between those sections which consist of at most one interval and those which consist of at least two intervals. It also turns out to be useful to further keep track of whether n v · n u ≤ 1 2 or n v · n u ≥ 1 2 holds.
We then obtain by Young's and Hölder's inequality as well as the fact that due to Definition 12 the vector field ξ is supported in
I v (t) + B rc V ≤ˆT 0 ˆ{ x+ynv(x,t)∈∂ * E + : x∈Iv(t), |y|<rc, H 0 (∂ * E + x )≤2, nv(x)·nu(x+ynv(x,t))≥ 1 2 } |(∇w) T ξ| 2 dH d−1 1/2 (154) × ˆR d |n u − ξ| 2 d|∇χ u | 1/2 dt + CˆT 0 ∇w(t) L ∞ x ˆ{ x+ynv(x,t)∈∂ * E + : x∈Iv(t), |y|<rc, H 0 (∂ * E + x )>2, nv(x)·nu(x+ynv(x,t))≥ 1 2 } 1 dH d−1 dt + CˆT 0 ∇w(t) L ∞ x ˆ{ x+ynv(x,t)∈∂ * E + : x∈Iv(t), |y|<rc, nv(x)·nu(x+ynv(x,t))≤ 1 2 } 1 dH d−1 dt + CˆT 0 ∇w(t) L ∞ x ˆR d \(Iv(t)+Br c ) 1 d|∇χ u | dt ≤ CˆT 0 ∇w(t) L ∞ x E[χ u , u, V |χ v , v](t) dt + CˆT 0 ˆ{ x+ynv(x,t)∈∂ * E + : x∈Iv(t), |y|<rc, H 0 (∂ * E + x )≤2, nv(x)·nu(x+ynv(x,t))≥ 1 2 } |(∇w) T ξ| 2 dH d−1 1 2 × ˆR d |n u − ξ| 2 d|∇χ u | 1/2 dt + CˆT 0 ∇w(t) L ∞ x ˆ{ x+ynv(x,t)∈∂ * E + : x∈Iv(t), |y|<rc, H 0 (∂ * E + x )>2, nv(x)·nu(x+ynv(x,t))≥ 1 2 } 1 dH d−1 dt =: CˆT 0 ∇w(t) L ∞ x E[χ u , u, V |χ v , v](t) dt + V a + V b .
To estimate V a from (154), we use the co-area formula for rectifiable sets (see [10, (2.72)]), (98), Hölder's inequality and the coercivity property (36) which together yield (we abbreviate in the first line F (x, y, t) :
= (∇w) T (x+yn v (x, t))n v (x, t)) V a ≤ CˆT 0 ˆ{ x∈Iv(t) : H 0 (∂ * E + x )≤2}ˆ{y∈∂ * E + x : nv(x)·nu(x+ynv(x,t))≥ 1 2 } |F (x, y, t)| 2 dH 0 (y) dS(x) 1 2 (155) × ˆR d |n u − ξ| 2 d|∇χ u | 1/2 dt ≤ CˆT 0 ˆI v (t) sup y∈[−rc,rc] |(∇w) T (x+yn v (x, t)) · n v (x, t)| 2 dS(x) 1 2 × ˆR d |n u − ξ| 2 d|∇χ u | 1/2 dt ≤ C r 9 c v L ∞ t W 3,∞ x (R d \Iv(t))ˆT 0 (1 + | log e(t)|)e(t)E[χ u , u, V |χ v , v] 1 2 (t) dt.
It remains to bound the term V b from (154). To this end, we make use of the fact that it follows from property iv) in Theorem 35 that every second point y ∈ ∂ * E + x ∩ (−r c , r c ) has to have the property that n v (x) · n u (x+yn v (x, t)) < 0, i.e., 1 ≤ 1 − n v (x) · n u (x+yn v (x, t)). We may therefore estimate with the help of the co-area formula for rectifiable sets (see [10, (2.72)]) and the bound (97)
V b ≤ CˆT 0 ∇w(t) L ∞ xˆ{ x∈Iv(t) : H 0 (∂ * E + x )>2}ˆ{y∈∂ * E + x : nv(x)·nu(x+ynv(x,t))≥ 1 2 } 1 dH 0 (y) dS(x) dt (156) ≤ CˆT 0 ∇w(t) L ∞ xˆI v (t)ˆ∂ * E + x 1 − n v (x, t) · n u (x+yn v (x, t)) dH 0 (y) dS(x) dt ≤ C r 9 c | log e(t)| v L ∞ t W 3,∞ x (R d \Iv(t))ˆT 0 E[χ u , u, V |χ v , v](t) dt.
All in all, we obtain from the assumption E[χ u , u, V |χ v , v](t) ≤ e 2 (t) as well as (154), (155), (156) and (97)
V ≤ C r 9 c v L ∞ t W 3,∞ x (R d \Iv(t))ˆT 0 (1 + | log e(t)|)e(t)E[χ u , u, V |χ v , v] 1 2 (t) dt.(157)
Hence, we deduce from the bounds (147), (148), (153), (157) as well as (97) the asserted estimate for the additional surface tension terms. 6.5. Estimate for the viscosity terms. In contrast to the case of equal shear viscosities µ + = µ − , we have to deal with the problematic viscous stress term given by (µ(χ v ) − µ(χ u ))(∇v + ∇v T ). We now show that the choice of w indeed compensates for (most of) this term in the sense that the viscosity terms from Proposition 9
R visc + A visc = −ˆT 0ˆR d 2 µ(χ u ) − µ(χ v ) D sym v : D sym (u − v) dx dt (158) +ˆT 0ˆR d 2 µ(χ u ) − µ(χ v ) D sym v : D sym w dx dt −ˆT 0ˆR d 2µ(χ u )D sym w : D sym (u − v − w) dx dt
may be bounded by a Gronwall-type term.
Lemma 29. Let the assumptions and notation of Proposition 27 be in place. In particular, we assume that there exists a C 1 -function e : [0, T strong ) → [0, r c ) such that the relative entropy is bounded by
E[χ u , u, V, |χ v , v](t) ≤ e 2 (t).
Then, for any δ > 0 there exists a constant C > 0 such that the viscosity terms R visc + A visc may be estimated by
R visc + A visc ≤ C r 8 c v 2 L ∞ t W 2,∞ x (R d \Iv(t))ˆT 0 E[χ u , u, V |χ v , v](t) dt (159) + C r c v 2 L ∞ t W 1,∞ xˆT 0 e(t)E[χ u , u, V |χ v , v] 1 2 (t) dt + δˆT 0ˆR d |D sym (u − v − w)| 2 dx dt.
Proof. We argue pointwise for the time variable and start by adding zero
R visc + A visc (160) = −2ˆR d (µ(χ u ) − µ(χ v ))D sym v : D sym (u−v−w) dx − 2ˆR d µ(χ u )D sym w : D sym (u − v − w) dx = −2ˆR d µ(χ u ) − µ(χ v ) − (µ − − µ + )χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) − (µ + − µ − )χ −h − e(t) (P Iv (t) x)≤dist ± (x,Iv(t))≤0 D sym v : D sym (u−v−w) dx − 2ˆR d χ dist ± (x,Iv(t)) / ∈[−h − e(t) (P Iv (t) x),h + e(t) (P Iv (t) x)] µ(χ u )D sym w : D sym (u−v−w) dx − 2ˆR d χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) (µ(χ u ) − µ − )D sym w : D sym (u−v−w) dx − 2ˆR d χ −h − e(t) (P Iv (t) x)≤dist ± (x,Iv(t))≤0 (µ(χ u ) − µ + )D sym w : D sym (u−v−w) dx − 2ˆR d χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) ((µ − −µ + )D sym v + µ − D sym w) : ∇(u−v−w) dx − 2ˆR d χ −h − e(t) (P Iv (t) x)≤dist ± (x,Iv(t))≤0 ((µ + −µ − )D sym v + µ + D sym w) : ∇(u−v−w) dx =: I + II + III + IV + V + V I.
We start by estimating the first four terms. Note that µ(
χ u ) − µ − = (µ + − µ − )χ u . Recalling the definition of χ v,h + e(t) ,h − e(t)
from (76b) we see that
χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) χ u = χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) (χ u − χ v,h + e(t) ,h − e(t)
).
Hence, we may rewrite
III = −2ˆR d χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) (µ + − µ − )(χ u − χ v,h + e(t) ,h − e(t) ) × (W ⊗ n v (P Iv(t) x)) : D sym (u−v−w) dx − 2ˆR d χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) (µ + − µ − ) × (∇w − W ⊗ n v (P Iv(t) x)) : D sym (u−v−w) dx.
Carrying out an analogous computation for IV , using again the definition of the smoothed approximation
χ v,h + e(t) ,h − e(t)
for χ u from (76b) and using (94) as well as (95), we then get the bound
I + II + III + IV ≤ C v W 1,∞ ˆR d |χ u − χ v,h + e(t) ,h − e(t) | dx 1/2 ˆR d |D sym (u−v−w)| 2 dx 1/2 + C r 2 c v W 2,∞ (R d \Iv(t)) ˆI v (t)
|h ± e(t) | 2 +|∇h ± e(t) | 2 dS
1/2 ˆR d |D sym (u−v−w)| 2 dx 1/2 .
Plugging in the estimates (76a) and (76c), we obtain by Young's inequality
I + II + III + IV ≤ Cδ −1 r 8 c v 2 W 2,∞ (R d \Iv(t)) E[χ u , u, V |χ v , v](t) (161) + Cδ −1 r c v 2 W 1,∞ e(t)E[χ u , u, V |χ v , v] 1 2 (t) + Cδ −1 v 2 W 1,∞ E[χ u , u, V |χ v , v](t) + δ D sym (u − v − w) L 2
for every δ ∈ (0, 1). To estimate the last two terms V and V I in (160), we may rewrite making use of the definition (95) of the vector field W and abbreviating
n v = n v (P Iv(t) x), dist ± = dist ± (x, I v (t)) as well as h + e(t) = h + e(t) (P Iv(t) x) −ˆR d χ 0≤dist ± ≤h + e(t) ((µ − −µ + )D sym v + µ − D sym w) : ∇(u−v−w) dx = −ˆR d χ 0≤dist ± ≤h + e(t) ((µ − −µ + )(Id −n v ⊗ n v )(D sym v · n v ) ⊗ n v + µ − D sym w) : ∇(u−v−w) dx −ˆR d χ 0≤dist ± ≤h + e(t) (µ − −µ + )D sym v (Id −n v ⊗ n v ) : ∇(u−v−w) dx −ˆR d χ 0≤dist ± ≤h + e(t) (µ − −µ + )(n v · D sym v · n v )(n v ⊗ n v ) : ∇(u−v−w) dx = −ˆR d χ 0≤dist ± ≤h + e(t) ((µ − −µ + )(Id −n v ⊗ n v )(D sym v · n v ) ⊗ n v + µ − D sym w) : ∇(u−v−w) dx −ˆR d χ 0≤dist ± ≤h + e(t) (µ − −µ + )D sym v (Id −n v ⊗ n v ) : ∇(u−v−w) dx +ˆR d χ 0≤dist ± ≤h + e(t) (µ − −µ + )(n v · D sym v · n v )(Id −n v ⊗ n v ) : ∇(u−v−w) dx, = 1 2ˆRd χ 0≤dist ± ≤h + e(t) ((W ⊗ n v − ∇w) + (W ⊗ n v − ∇w) T ) : ∇(u−v−w) dx + (µ − − µ + )ˆR d χ 0≤dist ± ≤h + e(t) ((Id−n v ⊗ n v )(D sym v · n v ) ⊗ n v ) : ∇(u−v−w) dx −ˆR d χ 0≤dist ± ≤h + e(t) (µ − −µ + )D sym v (Id −n v ⊗ n v ) : ∇(u−v−w) dx +ˆR d χ 0≤dist ± ≤h + e(t) (µ − −µ + )(n v · D sym v · n v )(Id −n v ⊗ n v ) : ∇(u−v−w) dx,
where in the penultimate step we have used the fact that ∇ · (u − v − w) = 0, and in the last step we added zero. This yields after an integration by parts
−ˆR d χ 0≤dist ± ≤h + e(t) ((µ − −µ + )D sym v + µ − D sym w) : ∇(u−v−w) dx = 1 2ˆRd χ 0≤dist ± ≤h + e(t) ((W ⊗ n v − ∇w) + (W ⊗ n v − ∇w) T ) : ∇(u−v−w) dx − (µ − −µ + )ˆR d χ 0≤dist ± ≤h + e(t) ∇ · (n v ⊗ (Id−n v ⊗ n v )(D sym v · n v )) · (u−v−w) dx + (µ − −µ + )ˆR d (n v · (u−v−w))(Id−n v ⊗ n v )(D sym v · n v ) · d∇χ 0≤dist ± ≤h + e(t) + (µ − −µ + )ˆR d χ 0≤dist ± ≤h + e(t) ∇ · (D sym v−(n v · D sym v · n v ) Id)(Id −n v ⊗ n v ) · (u−v−w) dx + (µ − −µ + )ˆR d (u−v−w) · (D sym v−(n v · D sym v · n v ) Id)(Id −n v ⊗ n v ) d∇χ 0≤dist ± ≤h + e(t)
.
As a consequence of (94), (76a), (17) and the global Lipschitz estimate |∇h ± e (·, t)| ≤ Cr −2 c from Proposition 26, we obtain ˆR
d χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) ((µ − − µ + )D sym v + µ − D sym w) : ∇(u − v − w) dx ≤ C r 7/2 c v W 2,∞ (R d \Iv(t)) E χ u , u, V χ v , v 1/2 ∇(u − v − v) L 2 + C r c v W 2,∞ (R d \Iv(t))ˆR d χ 0≤dist ± (x,Iv(t))≤h + e(t) (P Iv (t) x) |u − v − w| dx + C r 2 c v W 1,∞ˆI v (t) sup y∈(−rc,rc) |u − v − w|(x + yn v (x, t))|∇h + e(t) (x)| dS(x).
By a change of variables Φ t , (16), (40), (76a) and an application of Young's and Korn's inequality, the latter two terms may be further estimated by
C r 2 c v W 2,∞ (R d \Iv(t)) ˆI v (t) sup y∈(−rc,rc) |u − v − w| 2 (x + yn v (x, t)) dS 1 2 × ˆI v (t)
|h + e(t) | 2 + |∇h + e(t) | 2 dS
1 2 ≤ C r 3 c v W 2,∞ (R d \Iv(t)) E[χ u , u, V |χ v , v] 1 2 (t) u − v − w L 2 + C r 2 c v W 2,∞ (R d \Iv(t)) E[χ u , u, V |χ v , v] 1 2 (t) ∇(u − v − w) L 2 ≤ Cδ −1 r 4 c v 2 W 2,∞ (R d \Iv(t)) E[χ u , u, V |χ v , v](t) + δ D sym (u − v − w) L 2
for every δ ∈ (0, 1]. In total, we obtain the bound
V ≤ Cδ −1 r 4 c v 2 W 2,∞ (R d \Iv(t)) E[χ u , u, V |χ v , v](t) + δ D sym (u − v − w) L 2(162)
where δ ∈ (0, 1) is again arbitrary. Analogously, one can derive a bound of the same form for the last term V I in (160). Together with the bounds from (161) as well as (162) this concludes the proof. 6.6. Estimate for terms with the time derivative of the compensation function. We proceed with the estimate for the terms from the relative entropy inequality of Proposition 9
A dt := −ˆT 0ˆR d ρ(χ u )(u − v − w) · ∂ t w dx dt (163) −ˆT 0ˆR d ρ(χ u )(u − v − w) · (v · ∇)w dx dt,
which are related to the time derivative of the compensation function w.
Lemma 30. Let the assumptions and notation of Proposition 27 be in place. In particular, we assume that there exists a C 1 -function e : [0, T strong ) → [0, r c ) such that the relative entropy is bounded by
E[χ u , u, V, |χ v , v](t) ≤ e 2 (t).
Then, for any δ > 0 there exists a constant C > 0 such that A dt may be estimated by
A dt ≤ C r 22 c v 2 L ∞ t W 1,∞ x (1+ v L ∞ t W 2,∞ x (R d \Iv(t)) ) 2ˆT 0 (1+| log e(t)|)E[χ u , u, V |χ v , v](t) dt (164) + C r 11 c v L ∞ t W 1,∞ x (1+ v L ∞ t W 2,∞ x (R d \Iv(t)) )ˆT 0 (1+| log e(t)|)E[χ u , u, V |χ v , v](t) dt + C r 8 c (1+ v L ∞ t W 1,∞ x )( ∂ t ∇v L ∞ x,t +(R 2 +1) v L ∞ t W 2,∞ x (R d \Iv(t)) ) ׈T 0 E[χ u , u, V |χ v , v](t) dt + C r 2 c v 2 L ∞ t W 1,∞ xˆT 0 (1 + e (t))E[χ u , u, V |χ v , v](t) dt + δˆT 0ˆR d |D sym (u − v − w)| 2 dx dt.
Proof. To estimate the terms involving the time derivative of w we make use of the decomposition of ∂ t w + (v · ∇)w from (99):
−ˆT 0ˆR d ρ(χ u )(u−v−w) · ∂ t w dx dt −ˆT 0ˆR d ρ(χ u )(u−v−w) · (v · ∇)w dx dt ≤ˆT 0 g L 2 u − v − w L 2 dt +ˆT 0 ĝ L 4 3 u − v − w L 4 dt.
Employing the bounds (57a), (57b) and the assumption
E[χ u , u, V |χ v , v](t) ≤ e(t)
Making use of (76a), the bound for the vector fieldĝ from (100), the Gagliardo-
Nirenberg-Sobolev embedding u−v−w L 4 ≤ C ∇(u−v−w) 1−α L 2 u−v−w α L 2 ,ĝ L 4 3 u − v − w L 4 (166) ≤ C v W 1,∞ v W 2,∞ (R d \Iv(t)) r 11 c 1 + log 1 e(t) 1 4 × ( ∇(u−v−w) L 2 + u−v−w L 2 )E[χ u , u, V |χ v , v] 1 2 (t) + C v W 1,∞ r 8 c 1 + log 1 e(t) 1 4 ( ∇(u−v−w) L 2 + u−v−w L 2 ) u−v−w L 2 + C v W 1,∞ r 8 c 1 + log 1 e(t) 1 4 ∇(u−v−w) 3 2 −α L 2 u−v−w 1 2 +α L 2 + C v W 1,∞ (1+ v W 1,∞ )E[χ u , u, V |χ v , v] 1 2 (t)( ∇(u−v−w) L 2 + u−v−w L 2 ).
Now, by an application of Young's and Korn's inequality for all the terms on the right hand side of (166) which include an L 2 -norm of the gradient of u − v − w (in the case d = 3 we use a
u − v − w L 4 ≤ C δ 5 3 r 22 c v 2 W 1,∞ (1+ v W 2,∞ (R d \Iv(t)) ) 2 (1+| log e(t)|)E[χ u , u, V |χ v , v](t)(167)+ C r 11 c v W 1,∞ (1+ v W 2,∞ (R d \Iv(t)) )(1+| log e(t)|)E[χ u , u, V |χ v , v](t) + δ D sym (u−v−w) 2 L 2 ,
where δ ∈ (0, 1) is arbitrary. This gives the desired bound for the L 4 3 -contribution of ∂ t w + (v · ∇)w. Concerning the L 2 -contribution, we estimate using (57a), (76a), the bound for g L 2 from (101) as well as the assumption E[χ u , u, V |χ v , v](t) ≤ e(t) 2
g L 2 u − v − w L 2 (168) ≤ C 1+ v W 1,∞ r 8 c ( ∂ t ∇v L ∞ (R d \Iv(t)) +(R 2 +1) v W 2,∞ (R d \Iv(t)) )E[χ u , u, V |χ v , v] 1 2 (t) u−v−w L 2 + C v W 1,∞ (1 + v W 1,∞ )E[χ u , u, V |χ v , v] 1 2 (t) u−v−w L 2 + C r 2 c (1 + e (t)) v 2 W 1,∞ E[χ u , u, V |χ v , v] 1 2 (t) u−v−w L 2 + C v W 1,∞ (1+ v W 2,∞ (R d \Iv(t)) ) r c E[χ u , u, V |χ v , v] 1 2 (t) u−v−w L 2 + C v W 1,∞ ( ∇(u−v−w) L 2 + u−v−w L 2 ) u−v−w L 2 .
Hence, by another application of Young's and Korn's inequality, we may bound
g L 2 u − v − w L 2 (169) ≤ C r 8 c (1+ v W 1,∞ )( ∂ t ∇v L ∞ (R d \Iv(t)) +(R 2 +1) v W 2,∞ (R d \Iv(t)) )E[χ u , u, V |χ v , v](t) + C r 2 c v 2 W 1,∞ (1 + e (t))E[χ u , u, V |χ v , v](t) + Cδ −1 v 2 W 1,∞ E[χ u , u, V |χ v , v](t) + δ D sym (u−v−w) 2 L 2
where δ ∈ (0, 1] is again arbitrary. All in all, (167) and (169) therefore imply the desired bound. 6.7. Estimate for the additional advection terms. We move on with the additional advection terms from the relative entropy inequality of Proposition 9
A adv = −ˆT 0ˆR d ρ(χ u )(u − v − w) · (w · ∇)(v + w) dx dt (170) −ˆT 0ˆR d ρ(χ u )(u − v − w) · (u − v − w) · ∇ w dx dt.
A precise estimate is the content of the following result.
Lemma 31. Let the assumptions and notation of Proposition 27 be in place. In particular, we assume that there exists a C 1 -function e : [0, T strong ) → [0, r c ) such that the relative entropy is bounded by E[χ u , u, V, |χ v , v](t) ≤ e 2 (t). Then the additional advection terms A adv may be bounded by a Gronwall-type term
A adv ≤ C r 14 c (1+R) v 2 L ∞ t W 3,∞ x (R d \Iv(t))ˆT 0 (1+| log e(t)|)E[χ u , u, V |χ v , v](t) dt.(171)
Proof. A straightforward estimate yields
A adv ≤ C( v L ∞ t W 1,∞ x + ∇w L ∞ x,t ) u−v−w L 2 x,t ˆT 0ˆR d |w| 2 dx dt 1 2 + C ∇w L ∞ x,t u−v−w 2 L 2 x,t .
Making use of (93), (97) as well as (76a) immediately shows that the desired bound holds true.
6.8. Estimate for the additional weighted volume term. It finally remains to state the estimate for the additional weighted volume term from the relative entropy inequality of Proposition 9
A weightV ol :=ˆT
0ˆR d (χ u −χ v )(w · ∇)β dist ± (·, I v ) r c dx dt.(172)
Lemma 32. Let the assumptions and notation of Proposition 27 be in place. In particular, we assume that there exists a C 1 -function e : [0, T strong ) → [0, r c ) such that the relative entropy is bounded by E[χ u , u, V, |χ v , v](t) ≤ e 2 (t). Then the additional weighted volume term A weightV ol may be bounded by a Gronwall term
A weightV ol ≤ C r 10 c (1 + v 2 L ∞ t W 2,∞ x (R d \Iv(t)) )ˆT 0 E[χ u , u, V |χ v , v](t) dt.(173)
Proof. We may use the exact same argument as in the derivation of the estimate for the term III from the additional surface tension terms A surT en , see (148).
6.9. The weak-strong uniqueness principle with different viscosities. Before we proceed with the proof of Theorem 1, let us summarize the estimates from the previous sections in the form of a post-processed relative entropy inequality. The proof is a direct consequence of the relative entropy inequality from Proposition 9 and the bounds (44) Let ξ be the extension of the inner unit normal vector field n v of the interface I v (t) from Definition 12. Let w be the vector field contructed in Proposition 27. Let β be the truncation of the identity from Proposition 9, and let θ be the density θ t = d|∇χu(·,t)| d|Vt| S d−1 . Let e : [0, T strong ) → (0, r c ] be a C 1 -function and assume that the relative entropy
E χ u , u, V χ v , v (T ) := σˆR d 1 − ξ(·, T ) · ∇χ u (·, T ) |∇χ u (·, T )| d|∇χ u (·, T )| +ˆR d 1 2 ρ χ u (·, T ) u − v − w 2 (·, T ) dx +ˆR d χ u (·, T ) − χ v (·, T ) β dist ± (·, I v (T )) r c dx + σˆR d 1 − θ T d|V T | S d−1 is bounded by E[χ u , u, V |χ v , v](t) ≤ e(t) 2 .
Then the relative entropy is subject to the estimate
E[χ u , u, V |χ v , v](T ) + cˆT 0ˆR d |∇(u − v − w)| 2 dx dt (174) ≤ E[χ u , u, V |χ v , v](0) + CˆT 0 (1 + | log e(t)|) E[χ u , u, V |χ v , v](t) dt + CˆT 0 (1 + | log e(t)|) e(t) E[χ u , u, V |χ v , v](t) dt + CˆT 0 d dt e(t) E[χ u , u, V |χ v , v](t) dt
for almost every T ∈ [0, T strong ). Here, C > 0 is a constant which is structurally of the form C = Cr −22
c with a constantC =C(r c , v L ∞ t W 3,∞ x , ∂ t v L ∞ t W 1,∞ x ),
depending on the various norms of the velocity field of the strong solution, the regularity parameter r c of the interface of the strong solution, and the physical parameters ρ ± , µ ± , and σ.
We have everything in place to to prove the main result of this work.
Proof of Theorem 1. The proof of Theorem 1 is based on the post-processed relative entropy inequality of Proposition 33. It amounts to nothing but a more technical version of the upper bound
E(t) ≤ e e −Ct log E(0)
valid for all solutions of the differential inequality d dt E(t) ≤ CE(t)| log E(t)|. However, it is made more technical by the more complex right-hand side (33) in the relative entropy inequality (which involves the anticipated upper bound e(t) 2 ) and the smallness assumption on the relative entropy E[χ u , u, V |χ v , v](t) needed for the validity of the relative entropy inequality.
We start the proof with the precise choice of the function e(t) as well as the necessary smallness assumptions on the initial relative entropy. We then want to exploit the post-processed form of the relative entropy inequality from Proposition 33 to compare E[χ u , u, V |χ v , v](t) with e(t).
Let C > 0 be the constant from Proposition 33 and choose δ > 0 such that δ < 1 6(C+1) . Let ε > 0 (to be chosen in a moment, but finally we will let ε → 0) and consider the strictly increasing function
e(t) := e 1 2 e − t δ log(E[χu,u,V |χv,v](0)+ε) .(175)
Note that e 2 (0) = E[χ u , u, V |χ v , v](0) + ε which strictly dominates the relative entropy at the initial time. To ensure the smallness of this function, let us choose c > 0 small enough such that whenever we have E[χ u , u, V |χ v , v](0) < c and ε < c, it holds that
e(t) < 1 3C ∧ r c (176)
for all t ∈ [0, T strong ). This is indeed possible since the condition in (176) is equivalent to 1 2 log(E[χ u , u, V |χ v , v](0) + ε) < e T strong δ log( 1 3C ∧ r c ). For technical reasons to be seen later, we will also require c > 0 be small enough such that
e − T strong δ 1 6δ log(E[χ u , u, V |χ v , v](0) + ε) > C(177)
whenever E[χ u , u, V |χ v , v](0) < c and ε < c. We proceed with some further computations. We start with d dt
e(t) = 1 2δ | log(E[χ u , u, V |χ v , v](0) + ε)|e(t)e − t δ = 1 δ | log e(t)|e(t).(178)
This in particular entails
e 2 (T ) − e 2 (τ ) =ˆT τ d dt e 2 (t) dt = 1 δ | log(E[χ u , u, V |χ v , v](0) + ε)|ˆT τ e 2 (t)e − t δ dt.(179)
After these preliminary considerations, let us consider the relative entropy inequality from Proposition 9. Arguing similarly to the derivation of the relative entropy inequality in Proposition 9 but using the energy dissipation inequality in its weaker form E[χ u , u, V |χ v , v](T ) ≤ E[χ u , u, V |χ v , v](τ ) for a. e. τ ∈ [0, T ], we may deduce (upon modifying the solution on a subset of [0, T strong ) of vanishing measure)
lim sup T ↓τ E[χ u , u, V |χ v , v](T ) ≤ E[χ u , u, V |χ v , v](τ )(180)
for all τ ∈ [0, T strong ). Now, consider the set T ⊂ [0, T strong ) which contains all τ ∈ [0, T strong ) such that lim sup T ↓τ E[χ u , u, V |χ v , v](T ) > e 2 (τ ). Note that 0 ∈ T . We define
T * := inf T .
Since E[χ u , u, V |χ v , v](0) < e 2 (0) and e 2 is strictly increasing, we deduce by the same argument which established (180) that T * > 0. Hence, we can apply Proposition 33 at least for times T < T * (with τ = 0). However, by the same argument as before the relative entropy inequality from Proposition 9 shows that
E[χ u , u, V |χ v , v](T * ) ≤ E[χ u , u, V |χ v , v](T ) + C(T * − T ) for all T < T * , whereas E[χ u , u, V |χ v , v]
(T ) may be bounded by means of the post-processed relative entropy inequality. Hence, we obtain using also (175) and (178)
E[χ u , u, V |χ v , v](T * ) ≤ E[χ u , u, V |χ v , v](0) (181) + CˆT * 0 e 2 (t) dt + C 1 2δ log(E[χ u , u, V |χ v , v](0) + ε) ˆT * 0 e 3 (t)e − t δ dt + C 1 2 log(E[χ u , u, V |χ v , v](0) + ε) ˆT * 0 e 2 (t)e − t δ dt.
We compare this to the equation (179) for e 2 (t) (with τ = 0 and T = T * ). Recall that e 2 (0) strictly dominates the relative entropy at the initial time. Because of (177), the second term on the right hand side of (181) is dominated by one third of the right hand side of (179). Because of (176) and the choice δ < 1 6(C+1) the same is true for the other two terms on the right hand side of (181). In particular, we obtain using also (180) lim sup
T ↓T * E[χ u , u, V |χ v , v](T ) − e 2 (T * ) ≤ E[χ u , u, V |χ v , v](T * ) − e 2 (T * ) < 0,
which contradicts the defining property of T * . This concludes the proof since the asserted stability estimate as well as the weak-strong uniqueness principle is now a consequence of letting ε → 0.
Derivation of the relative entropy inequality
Proof of Proposition 9. We start with the following observation. Since the phasedependent density ρ(χ v ) depends linearly on the indicator function χ v of the volume occupied by the first fluid, it consequently satisfieŝ
R d ρ(χ v (·, T ))ϕ(·, T ) dx −ˆR d ρ(χ 0 v )ϕ(·, 0) dx =ˆT 0ˆR d ρ(χ v )(∂ t ϕ + (v · ∇)ϕ) dx dt (182)
for almost every T ∈ [0, T strong ) and all ϕ ∈ C ∞ cpt (R d × [0, T strong )). By approximation, the equation holds for all ϕ ∈ W 1,∞ (R d × [0, T strong )). Testing this equation
with v · η, where η ∈ C ∞ cpt (R d × [0, T strong ); R d )
is a smooth vector field, we then obtain
(183)ˆR d ρ(χ v (·, T ))v(·, T ) · η(·, T ) dx −ˆR d ρ(χ 0 v )v 0 · η(·, 0) dx =ˆT 0ˆR d ρ(χ v )(v · ∂ t η + η · ∂ t v) dx dt +ˆT 0ˆR d ρ(χ v )(η · (v · ∇)v + v · (v · ∇)η) dx dt
for almost every T ∈ [0, T strong ). Note that the velocity field v of a strong solution has the required regularity to justify the preceding step. Next, we subtract from (183) the equation for the momentum balance (10a) of the strong solution evaluated with a test function η ∈ C ∞ cpt (R d × [0, T strong ); R d ) such that ∇ · η = 0. This shows that the velocity field v of the strong solution satisfies
(184) 0 =ˆT 0ˆR d ρ(χ v )η · (v · ∇)v dx dt +ˆT 0ˆR d µ(χ v )(∇v + ∇v T ) : ∇η dx dt +ˆT 0ˆR d ρ(χ v )η · ∂ t v dx dt − σˆT 0ˆIv(t)
H · η dS dt which holds for almost every T ∈ [0, T strong ) and all η ∈ C ∞ cpt (R d × [0, T strong ); R d ) such that ∇ · η = 0. The aim is now to test the latter equation with the field u − v − w. To this end, we fix a radial mollifier φ : R d → [0, ∞) such that φ is smooth, supported in the unit ball and´R d φ dx = 1. For n ∈ N we define φ n (·) := n d φ(n ·) as well as u n := φ n * u and analogously v n and w n . We then test (184) with the test function u n − v n − w n and let n → ∞. Since the traces of u n , v n and w n on I v (t) converge pointwise almost everywhere to the respective traces of u, v and w, we indeed may pass to the limit in the surface tension term of (184). Hence, we obtain the identity
−ˆT 0ˆR d µ(χ v )(∇v + ∇v T ) : ∇(u − v − w) dx dt =ˆT 0ˆR d ρ(χ v )(u − v − w) · (v · ∇)v dx dt (185) +ˆT 0ˆR d ρ(χ v )(u − v − w) · ∂ t v dx dt − σˆT 0ˆIv(t) H · (u − v − w) dS dt,
which holds true for almost every T ∈ [0, T strong ).
In the next step, we test the analogue of (182) for the phase-dependent density ρ(χ u ) of the varifold solution with the test function 1 2 |v + w| 2 and obtain
(186)ˆR d 1 2 ρ(χ u (·, T ))|v + w| 2 (·, T ) dx −ˆR d 1 2 ρ(χ 0 u )|v 0 + w(·, 0)| 2 dx =ˆT 0ˆR d ρ(χ u )(v + w) · ∂ t (v + w) dx dt +ˆT 0ˆR d ρ(χ u )(v + w) · (u · ∇)(v + w) dx dt
for almost every T ∈ [0, T strong ). Recall also from the definition of a varifold solution that we are equipped with the energy dissipation inequalitŷ
R d 1 2 ρ(χ u (·, T ))|u(·, T )| 2 dx + σ|V T |(R d × S d−1 ) +ˆT 0ˆR d µ(χ u ) 2 ∇u + ∇u T 2 dx dt (187) ≤ˆR d 1 2 ρ(χ 0 u )|u 0 | 2 dx + σ|∇χ 0 u |(R d ),
which holds for almost every T ∈ [0, T strong ). Finally, we want to test the equation for the momentum balance (6a) of the varifold solution with the test function v + w. Since the normal derivative of the tangential velocity of a strong solution may feature a discontinuity at the interface, we have to proceed by an approximation argument, i.e., we use the mollified version v n + w n as a test function. Note that v n resp. w n are elements of L ∞ ([0, T strong ); C 0 (R d )). Hence, we may indeed use v n + w n as a test function in the surface tension term of the equation for the momentum balance (6a) of the varifold solution. However, it is not clear a priori why one may pass to the limit n → ∞ in this term.
To argue that this is actually possible, we choose a precise representative for ∇v resp. ∇w on the interface I v (t). This is indeed necessary also for the velocity field of the strong solution since the normal derivative of the tangential component of v may feature a jump discontinuity at the interface. However, by the regularity assumptions on v, see Definition 6 of a strong solution, and the assumptions on the compensating vector field w, for almost every t ∈ [0, T strong ) every point x ∈ R d is either a Lebesgue point of ∇v (respectively ∇w) or there exist two half spaces H 1 and H 2 passing through x such that x is a Lebesgue point for both ∇v| H1 and ∇v| H2 (respectively ∇w| H1 and ∇w| H2 ). In particular, by the L ∞ bounds on ∇v and ∇w the limit of the mollifications ∇v n respectively ∇w n exist at every point x ∈ R d and we may define ∇v respectively ∇w at every point x ∈ R d as this limit.
Recall then that we have chosen the mollifiers φ n to be radially symmetric. Hence, the approximating sequences ∇v n resp. ∇w n converge pointwise everywhere to the precise representation as chosen before. Since both limits are bounded, we may pass to the limit n → ∞ in every term appearing from testing the equation for the momentum balance (6a) of the varifold solution with the test function v n + w n .
Moreover, collecting all advection terms on the right hand side of (185), (186), and (188) as well as adding zero gives the contribution
RHS adv = −ˆT 0ˆR d ρ(χ u )(u − v − w) · (v · ∇)w dx dt −ˆT 0ˆR d ρ(χ u ) − ρ(χ v ) (u − v − w) · (v · ∇)v dx dt −ˆT 0ˆR d ρ(χ u )(u − v − w) · (u − v) · ∇ (v + w) dx dt = −ˆT 0ˆR d ρ(χ u )(u − v − w) · (v · ∇)w dx dt (194) −ˆT 0ˆR d ρ(χ u ) − ρ(χ v ) (u − v − w) · (v · ∇)v dx dt −ˆT 0ˆR d ρ(χ u )(u − v − w) · (u − v − w) · ∇ v dx dt −ˆT 0ˆR d ρ(χ u )(u − v − w) · (w · ∇)(v + w) dx dt −ˆT 0ˆR d ρ(χ u )(u − v − w) · (u − v − w) · ∇ w dx dt.
Next, we may rewrite those terms on the right hand side of (185), (186), and (188) which contain a time derivative as follows
RHS dt = −ˆT 0ˆR d ρ(χ u ) − ρ(χ v ) (u − v − w) · ∂ t v dx dt (195) −ˆT 0ˆR d ρ(χ u )(u − v − w) · ∂ t w dx dt.
Furthermore, the terms related to surface tension on the right hand side of (185) and (188) are given by RHS surT en = σˆT H · w dS dt.
We proceed by rewriting the surface tension terms. For the sake of brevity, let us abbreviate from now on n u = ∇χu |∇χu| . Using the incompressibility of v and adding zero, we start by rewriting
σˆT 0ˆR d ×S d−1 (Id−s ⊗ s) : ∇v dV t (x, s) dt = σˆT 0ˆR d n u · (n u · ∇)v d|∇χ u | dt − σˆT 0ˆR d ×S d−1 s · (s · ∇)v dV t (x, s) dt − σˆT 0ˆR d n u · (n u · ∇)v d|∇χ u | dt.
Next, by means of the compatibility condition (6e) we can write σˆT 0ˆIv(t) n u · (n u · ∇)v dS dt − σˆT
0ˆR d ×S d−1 s · (s · ∇)v dV t (x, s) dt = −σˆT 0ˆR d ×S d−1 (s−ξ) · (s−ξ) · ∇ v dV t (x, s) dt − σˆT 0ˆR d ×S d−1 ξ · (s−ξ) · ∇ v dV t (x, s) dt + σˆT 0ˆR d n u · (n u − ξ) · ∇ v d|∇χ u | dt.
Moreover, the compatibility condition (6e) also ensures that −σˆT
0ˆR d ×S d−1 ξ · (s · ∇)v dV t (x, s) dt = −σˆT 0ˆR d ξ · (n u · ∇)v d|∇χ u | dt,
whereas it follows from (8) σˆT
0ˆR d ×S d−1 ξ · (ξ · ∇)v dV t (x, s) dt = σˆT 0ˆR d (1 − θ t ) ξ · (ξ · ∇)v d|V t | S d−1 (x) dt + σˆT 0ˆR d ξ · (ξ · ∇)v d|∇χ u | dt.
Using that the divergence of ξ equals the divergence of n v (P Iv(t) x) on the interface of the strong solution (i. e. H = −(∇ · ξ)n v ; see Definition 12, i.e., the cutoff function does not contribute to the divergence on the interface), that the latter quantity equals the scalar mean curvature (recall that n v = ∇χv |∇χv| points inward) as well as once more the incompressibility of the velocity fields v resp. u we may also rewrite −σˆT
0ˆIv(t) H · (u − v) dS dt = −σˆT 0ˆR d χ v (u − v) · ∇ (∇ · ξ) dx dt.
The preceding five identities together then imply that
σˆT 0ˆR d ×S d−1 (Id−s ⊗ s) : ∇v dV t (x, s) dt − σˆT 0ˆIv(t) H · (u−v) dS dt = −σˆT 0ˆR d n u · (n u · ∇)v d|∇χ u | dt (197) − σˆT 0ˆR d χ v (u − v) · ∇ (∇ · ξ) dx dt − σˆT 0ˆR d ×S d−1 (s−ξ) · (s−ξ) · ∇ v dV t (x, s) dt + σˆT 0ˆR d (1 − θ t ) ξ · (ξ · ∇)v d|V t | S d−1 (x) dt + σˆT 0ˆR d (n u − ξ) · (n u − ξ) · ∇ v d|∇χ u | dt.
Following the computation which led to (197) we also obtain the identity
σˆT 0ˆR d ×S d−1 (Id−s ⊗ s) : ∇w dV t (x, s) dt = −σˆT 0ˆR d n u · (n u · ∇)w d|∇χ u | dt − σˆT 0ˆR d ×S d−1 (s−ξ) · (s−ξ) · ∇ w dV t (x, s) dt + σˆT 0ˆR d (1 − θ t ) ξ · (ξ · ∇)w d|V t | S d−1 (x) dt + σˆT 0ˆR d (n u − ξ) · (n u − ξ) · ∇ w d|∇χ u | dt.
Using the fact that w is divergence-free, we may also rewrite − σˆT 0ˆR d n u · (n u · ∇)w d|∇χ u | dt
= −σˆT 0ˆR d n u · (n u − ξ) · ∇ w d|∇χ u | dt + σˆT 0ˆR d χ u ∇ · (ξ · ∇)w dx dt = −σˆT 0ˆR d ξ · (n u − ξ) · ∇ w d|∇χ u | dt − σˆT 0ˆR d (n u − ξ) · (n u − ξ) · ∇ w d|∇χ u | dt + σˆT 0ˆR d χ u ∇w : ∇ξ T dx dt.
Appealing once more to the fact that ξ = n v on the interface I v of the strong solution (see Definition 12) and ∇ · w = 0, we obtain σˆT 0ˆIv(t)
H · w dS dt = −σˆT 0ˆR d (Id −n v ⊗ n v ) : ∇w dS dt = σˆT 0ˆR d n v · (ξ · ∇)w dS dt = −σˆT 0ˆR d χ v ∇ · (ξ · ∇)w dx dt = −σˆT 0ˆR d χ v ∇w : ∇ξ T dx dt.
The last three identities together with (197) and (196) in total finally yield the following representation of the surface tension terms on the right hand side of (185) and (188)
RHS surT en = −σˆT 0ˆR d n u · (n u · ∇)v d|∇χ u | dt (198) + σˆT 0ˆR d (n u − ξ) · (n u − ξ) · ∇ v d|∇χ u | dt − σˆT 0ˆR d χ v (u − v) · ∇ (∇ · ξ) dx dt − σˆT 0ˆR d ×S d−1 (s−ξ) · (s−ξ) · ∇ v dV t (x, s) dt + σˆT 0ˆR d (1 − θ t ) ξ · (ξ · ∇)v d|V t | S d−1 (x) dt − σˆT 0ˆR d ×S d−1 (s−ξ) · (s−ξ) · ∇ w dV t (x, s) dt + σˆT 0ˆR d (1 − θ t ) ξ · (ξ · ∇)w d|V t | S d−1 (x) dt − σˆT 0ˆR d ξ · (n u − ξ) · ∇ w d|∇χ u | dt + σˆT 0ˆR d (χ u − χ v )∇w : ∇ξ T dx dt.
It remains to collect the viscosity terms from the left hand side of (185), (187) and (188). Adding also zero, we obtain
LHS visc =ˆT 0ˆR d 2 µ(χ u ) − µ(χ v ) D sym v : D sym (u − v − w) dx dt −ˆT 0ˆR d 2µ(χ u )D sym v : D sym (u − v − w) dx dt +ˆT 0ˆR d 2µ(χ u )D sym u : D sym u dx dt −ˆT 0ˆR d 2µ(χ u )D sym u : D sym (v + w) dx dt =ˆT 0ˆR d 2µ(χ u )|D sym (u − v − w)| 2 dx dt (199) +ˆT 0ˆR d 2 µ(χ u ) − µ(χ v ) D sym v : D sym (u − v − w) dx dt +ˆT 0ˆR d 2µ(χ u )D sym w : D sym (u − v − w) dx dt.
In particular, as an intermediate summary we obtain the following bound making already use of the notation of Proposition 9: Taking the bound (189) together with the identities from (190) to (195) as well as (198) and (199) yieldŝ
R d 1 2 ρ(χ u (·, T ))|u − v − w| 2 (·, T ) dx +ˆT 0ˆR d 2µ(χ u )|D sym (u − v − w)| 2 dx dt + σ|∇χ u (·, T )|(R d ) + σˆR d 1 − θ T d|V T | S d−1 ≤ˆR d 1 2 ρ(χ 0 u )|u 0 − v 0 − w(·, 0)| 2 dx + σ|∇χ 0 u |(R d )(200)+ R dt + R visc + R adv + A visc + A dt + A adv + A surT en − σˆT 0ˆR d n u · (n u · ∇)v d|∇χ u | dt + σˆT 0ˆR d (n u − ξ) · (n u − ξ) · ∇ v d|∇χ u | dt − σˆT 0ˆR d χ v (u − v − w) · ∇ (∇ · ξ) dx dt − σˆT 0ˆR d χ u w · ∇ (∇ · ξ) dx dt − σˆT 0ˆR d ×S d−1 (s−ξ) · (s−ξ) · ∇ v dV t (x, s) dt + σˆT 0ˆR d (1 − θ t ) ξ · (ξ · ∇)v d|V t | S d−1 (x) dt.
The aim of the next step is to use σ(∇ · ξ) (see Definition 12) as a test function in the transport equation (6b) for the indicator function χ u of the varifold solution. For the sake of brevity, we will write again n u = ∇χu |∇χu| . Plugging in σ(∇ · ξ) and integrating by parts yields −σˆR d n u (·, T ) · ξ(·, T ) d|∇χ u (·, T )| + σˆR
d n 0 u · ξ(·, 0) d|∇χ 0 u | = −σˆT 0ˆR d n u · ∂ t ξ d|∇χ u | dt + σˆT
= σˆT 0ˆR d n u · Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇v) T ξ d|∇χ u | dt
+ σˆT 0ˆR d n u · (v · ∇)ξ d|∇χ u | dt + σˆT 0ˆR d χ u (u · ∇)(∇ · ξ ) dx dt + σˆT 0ˆR d n u · Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇V n − ∇v) T ξ d|∇χ u | dt + σˆT 0ˆR d n u · (V n − v) · ∇ ξ d|∇χ u | dt
which holds for almost every T ∈ [0, T strong ). Next, we study the quantity RHS tilt := σˆT 0ˆR d n u · (v · ∇)ξ d|∇χ u | dt + σˆT 0ˆR d χ u (u · ∇)(∇ · ξ ) dx dt (202) + σˆT 0ˆR d n u · Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇v) T · ξ d|∇χ u | dt.
Due to the regularity of v resp. ξ as well as the incompressibility of the velocity field v we get
σˆT 0ˆR d n u · (v · ∇)ξ d|∇χ u | dt = −σˆT 0ˆR d χ u ∇ · (v · ∇)ξ dx dt = −σˆT 0ˆR d χ u ∇ 2 : v ⊗ ξ dx dt = −σˆT 0ˆR d χ u ∇ · (ξ · ∇)v dx dt − σˆT 0ˆR d χ u ∇ · v(∇ · ξ) dx dt = −σˆT 0ˆR d χ u (v · ∇)(∇ · ξ ) dx dt (203) + σˆT 0ˆR d n u · (ξ · ∇)v d|∇χ u | dt.
Exploiting the fact that ξ(x) = n v (P Iv(t) x)ζ(x) and n v (P Iv(t) x) only differ by a scalar prefactor, namely the cut-off multiplier ζ(x) which one can shift around, it turns out to be helpful to rewrite σˆT 0ˆR d n u · Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x) (∇v) T · ξ d|∇χ u | dt = σˆT 0ˆR d ξ · (Id−n v (P Iv(t) x) ⊗ n v (P Iv(t) x))n u · ∇)v d|∇χ u | dt (204) = σˆT 0ˆR d ξ · n u − (n v (P Iv(t) x) · n u )n v (P Iv(t) x) · ∇ v d|∇χ u | dt
= σˆT 0ˆR d ξ · (n u − ξ) · ∇ v d|∇χ u | dt − σˆT 0ˆR d (ξ · n u ) n v (P Iv(t) x) · n v (P Iv(t) x) · ∇ v d|∇χ u | dt + σˆT 0ˆR d ξ · (ξ · ∇)v d|∇χ u | dt.
Hence, by using (203) and (204) we obtain
RHS tilt = σˆT 0ˆR d n u · (n u · ∇)v d|∇χ u | dt (205) − σˆT 0ˆR d (n u − ξ) · (n u − ξ) · ∇ v d|∇χ u | dt + σˆT 0ˆR d χ u (u − v) · ∇ (∇ · ξ) dx dt − σˆT 0ˆR d (ξ · n u ) n v (P Iv(t) x) · n v (P Iv(t) x) · ∇ v d|∇χ u | dt + σˆT 0ˆR d ξ · (ξ · ∇)v d|∇χ u | dt.
Appendix
Theorem 34 (Boundedness of singular integral operators of convolution type in L p ). Let d ≥ 2, p ∈ (1, ∞), and let K : S d−1 → R be a function of class C 1 with vanishing average. Let f ∈ L p (R d ) and define
Kf (x) :=ˆR d K x−x |x−x| |x −x| d f (x) dx,
where the integral is understood in the Cauchy principal value sense. Then there exists a constant C > 0 depending only on d, p, and K such that
Kf L p (R d ) ≤ C f L p (R d ) .
We also state a non-trivial result from geometric measure theory on properties of one-dimensional sections of Caccioppoli sets.
Theorem 35 ([30, Theorem G]). Consider a set G of finite perimeter in R d , denote by ν G = (ν G x1 , . . . , ν G x d−1 , ν G y ) ∈ R d the associated measure theoretic inner unit normal vector field of the reduced boundary ∂ * G, and let χ * G be the precise representative of the bounded variation function χ G . Then for Lebesgue almost every x ∈ R d−1 the one-dimensional sections G x := {y ∈ R : (x, y) ∈ G} satisfy the following properties: i) G x is a set of finite perimeter in R, χ G (x, ·) = χ * G (x, ·) Lebesgue almost everywhere in G x , ii) (∂ * G) x = ∂ * G x , iii) ν G y (x, y) = 0 for all y ∈ R such that (x, y) ∈ ∂ * G, and iv) lim y→y + 0 χ * G (x, y) = 1 and lim y→y − 0 χ * G (x, y) = 0 whenever ν G y (x, y 0 ) > 0, and vice versa if ν G y (x, y 0 ) < 0. In particular, for every Lebesgue measurable set M ⊂ R d−1 there exists a Borel measurable subset M G ⊂ M such that L d−1 (M \ M G ) = 0 and the four properties stated above are satisfied for all y ∈ M G .
To bound the L 4 -norm of the interface error heights h ± in the case of a twodimensional interface, we employ the following optimal Orlicz-Sobolev embedding. Then for any weakly differentiable function u decaying to 0 at infinity in the sense {|u(x)| > s} < ∞ for all s > 0, the following estimate holds true:
R d B |u(x)| K ´R d A(|∇u(x)|) dx 1/d dx ≤ˆR d A(|∇u(x)|) dx.(209)
The application of the optimal Orlicz-Sobolev embedding to our setting is stated and proved next. for e(t) ≤ s ≤ 1, 2s − 1 for s ≥ 1.
We also set A e(t) (Du(t)) :=´I (t) A e(t) (|∇u(t)|) dS + |D s u(t)|(I(t)). Then the following estimate holds truê
I(t) |u(x, t)| 4 dS ≤ C r 12 c 1+ log 1 e(t)(210)
× e(t) 4 + 1 e(t) 2 u(t) 6 L 2 (I(t)) +A 3 e(t) (Du(t)) + u(t) 4 L 2 (I(t)) + A 2 e(t) (Du(t))
for almost every t ∈ [0, T ] and a constant C > 0.
Proof. Let U ⊂ R 2 be an open and bounded set and consider u ∈ C 1 cpt (U ) such that u L ∞ ≤ 1. For the sake of brevity, let us suppress for the moment the dependence on the variable t ∈ [0, T ). The idea is to apply the optimal Orlicz-Sobolev embedding provided by the preceding theorem with respect to the convex function A e . Observe first that A e indeed satisfies all the assumptions. Moreover, since d = 2 we compute (H(r)) 2 =ˆr for r ≤ e, 1 + log r e for e ≤ r ≤ 1, 1 + log 1 e + r−1 2 + 1 4 log(2r − 1) for r ≥ 1. As a consequence, we get
H −1 (y) = = ey 2
for y ≤ 1, = e exp(y 2 − 1) for 1 ≤ y ≤ 1 + log 1 e , ≥ (y 2 − 1 − log 1 e ) + 1 for y ≥ 1 + log 1 e , ≤ 2(y 2 − 1 − log 1 e ) + 1 for y ≥ 1 + log 1 e . This in turn entails
We then deduce from Theorem 36, d = 2, u L ∞ ≤ 1, the bound exp(s 2 ) ≥ 1 2 s 4 for all s ≥ 0 as well as the bound s 2 − log 1 e ≥ which is precisely what is claimed. Note that since u is continuously differentiable, the singular part in the definition of A e (Du) vanishes. In a next step, we want to extend to smooth functions u on the manifold I(t). By assumption, we may cover I(t) with a finite family of open sets of the form U (x i ) := I(t) ∩ B 2rc (x i ), x i ∈ I(t), such that U (x i ) can be represented as the graph of a function g : B 1 (0) ⊂ R 2 → R with |∇g| ≤ 1 and |∇ 2 g| ≤ r −1 c . We fix a partition of unity {ϕ i } i subordinate to this finite cover of I(t). Note that |∇ϕ i | ≤ Cr −1 c . Note also that the cardinality of the open cover is uniformly bounded in t. Hence, we proceed with deriving the desired bound only for one uϕ, where ϕ = ϕ i is supported in U = U (x i ). Abbreviatingũ = u • g andφ = ϕ • g, we obtain from the previous step Using the bounds A e (t +t) ≤ CA e (t) + CA e (t) and A e (λt) ≤ C(λ + λ 2 )A e (t), which hold for all λ > 0 and all t,t ≥ 0, as well as the product and chain rule we compute A e |u|(g(x)) dx + CˆB
1(0)
A e |∇u|(g(x)) dx.
By definition of A e we can further estimatê
B1(0)
A e |u|(g(x)) dx ≤ Ce 2 +ˆB 1(0) |u| 2 (g(x)) dx.
Definition 5 implies a lower bound of cr c for the length of any connected component of I(t)) and such that at any point x ∈ I(t) there are at most two i with η i (x) > 0. Treating by abuse of notation the function η i u as if defined on a real interval I = (a, b), we then write η i (x)u(x) =ˆx a η i (y)u (y) + η i (y)u(y) dy From this we infer the desired estimate by approximation.
This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie Grant Agreement No.665385 .
Figure 1 .
1Pinchoff of a liquid droplet driven by surface tension. bubbles in water): The flow of each single fluid is described by the incompressible Navier-Stokes equation, while the fluid-fluid interface evolves by pure transport along the fluid flow and a surface tension force acts at the fluid-fluid interface.
Figure 2 .
2An illustration of the interface error. The red and the blue region (separated by the black solid curve) correspond to the regions occupied by the two fluids in the strong solution. The shaded area corresponds to the region occupied by the blue fluid in the varifold solution, the interface in the varifold solution corresponds to the dotted curve.
Figure 3 .
3An illustration of the approximation of the interface error by the mollified height function h + e(t) .
6. 1 .
1The evolution of the local height of the interface error. Consider a strong solution (χ v , v) to the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) in the sense of Definition 6 on some time interval [0, T strong ). For the sake of better readability, let us recall some definitions and constructions related to the associated family of evolving interfaces I v (t) of the strong solution.
is a family of smoothly evolving domains and I v (t) := ∂Ω + t is a family of smoothly evolving surfaces in the sense of Definition 5. Let ξ be the extension of the unit normal vector field n v from Definition 12. Let χ u ∈ L ∞ ([0, T strong ); BV(R d ; {0, 1})) be another indicator function and let then h + resp. h − be as defined in Proposition 25. Let θ : R + → [0, 1] be a smooth cutoff with θ(s) = 1 for s ∈ [0, 1 4 ] and θ(s) = 0 for s ≥ 1 2 . Let e : [0, T strong ) → (0, r c ] be a C 1 -function and define the regularized height of the local interface error
Proposition 27 .
27Let (χ u , u, V ) be a varifold solution to the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) in the sense of Definition 2 on some time interval [0, T vari ). Let (χ v , v) be a strong solution to (1a)-(1c) in the sense of Definition 6 on some time interval [0, T strong ) with T strong ≤ T vari . Let ξ be the extension of the inner unit normal vector field n v of the interface I v (t) from Definition 12. Let e : [0, T strong ) → (0, r c ] be a C 1function and assume that the relative entropy is bounded by E[χ u , u, V |χ v , v](t) ≤ e(t) 2 . Let the regularized local interface error heights h + e(t) and h − e(t) be defined as in Proposition 26.
For example, one may choose η(x, t) := θ( dist(x,Iv(t)) rc ) where θ : R + → [0, 1] is the smooth cutoff already used in the definition of the regularized local interface error heights in Proposition 26.
(76b) the definition of χ v,h + e(t) ,h − e(t) , we may estimate the term (IV ) b from (149) by a change of variables Φ t , (17), Hölder's and Young's inequality, (96)
, (52), (53), (54), (146), (159), (164), (171) and (173). Proposition 33 (Post-processed relative entropy inequality). Let d ≤ 3. Let (χ u , u, V ) be a varifold solution to the free boundary problem for the incompressible Navier-Stokes equation for two fluids (1a)-(1c) in the sense of Definition 2 on some time interval [0, T vari ). Let (χ v , v) be a strong solution to (1a)-(1c) in the sense of Definition 6 on some time interval [0, T strong ) with T strong ≤ T vari .
0ˆR d ×S d− 1 ((
1Id−s ⊗ s) : ∇v dV t (x, s) dt − σˆT Id−s ⊗ s) : ∇w dV t (x, s) dt + σˆT 0ˆIv(t)
(u · ∇)(∇ · ξ ) dx dt for almost every T ∈ [0, T strong ). Making use of the evolution equation(31) for ξ and the fact that ξ is supported in the space-time domain {dist(x, I v (t)) < r c }, we get by adding zero − σˆR d n u (·, T ) · ξ(·, T ) d|∇χ u (·, T )| + σˆR d n 0 u · ξ(·, 0) d|∇χ 0 u |
Theorem 36 (
36Optimal Orlicz-Sobolev embedding, [31, Theorem 1]). For every d ≥ 2, there exists a constant K depending only on d such that the following holds true: Let A : [0, ∞) → [0, ∞) be a convex function with A(0) = 0, A(t) → ∞ for t → ∞
s) := A(H −1 (s)).
Proposition 37 .
37Let T > 0 and (I(t)) t∈[0,T ] be a family of smoothly evolving surfaces in R 3 in the sense of Definition 5. Consider u ∈ L ∞ ([0, T ]; BV(I(t))) such that |u| ≤ 1. Let e : [0, T ] → (0, ∞) be a measurable function. We define A e(t)
B
(s) = A e (H −1 (s)) = 2 exp(2s 2 − 2) for 1 ≤ s ≤ 1 + log 1 e , ≥ s 2 − log 1 e for s ≥ 1 + log 1 e .
D(ũφ) + A 2 e D(ũφ) .
(y) u (y) − 1 + − u (y) − (−1) − dy.Hence, we may estimate using Jensen's inequalityη i (x)|u(x)| ≤ |I(t)| 1/2 ˆI (t) η i | max min{|∇ tan u|, 1}, −1 | 2 dS
(|∇ tan u| − 1) + dS + Cr −1 cˆI (t)∩supp ηi |u| dSfor any x ∈ I(t). Taking the fourth power, integrating over x, and summing over i, we deducêI(t) |u(x)| 4 dS ≤ C|I(t)| 3 ˆI (t)| max min{|∇ tan u|, 1},
E[u|v] := E[u] − DE[v](u − v) − E[v]is a measure for the error between u and v, as it is nonzero if and only if u = v.At the same time, to evaluate the time evolution d dt E[u|v] of the relative entropy, it is sufficient to exploit the entropy dissipation inequality d dt E[u] ≤ −D[u] for the weak solution u and test the weak formulation of the evolution equation for u by the typically more regular test function DE[v]. Having derived an explicit expression for the time derivative d dt E[u|v] of the relative entropy, it is often possible to derive a Gronwall-type estimate like d dt E[u|v] ≤ CEor energy) E[u] subject
to a dissipation estimate d
dt E[u] ≤ −D[u]: For strictly convex entropies E[u], the
relative entropy
{t} the third spatial derivatives of the velocity field exist and satisfysup
t∈[0,Tstrong)
with α = 1 2 for d = 2 and α = 1 4 for d = 3, as well as the assumption E[χ u , u, V |χ v , v](t) ≤e(t) 2 we obtain
s 2 1+log 1 e
21for all s ≥ 1 + log 1ê
U
|u(x)| 4 dx
=ˆU
∩ |u|≤K
√
Ae(Du)
|u(x)| 4 dx
+ˆU
∩ K
√
Ae(Du)≤|u|≤K
√
Ae(Du)
√
1+log 1
e
|u(x)| 4 dx
+ˆU
∩ |u|≥K
√
Ae(Du)
√
1+log 1
e
|u(x)| 4 dx
≤ K 4 A 2
e (Du)
e 2ˆU
∩ |u|≤K
√
Ae(Du)
e 2 |u(x)| 2
K 2 A e (Du)
dx
+ K 4 A 2
e (Du)
e 2ˆU
∩ K
√
Ae(Du)≤|u|≤K
√
Ae(Du)
√
1+log 1
e
e 2 |u(x)| 4
K 4 A 2
e (Du)
dx
+ K 2 1 + log
1
e
A e (Du)ˆU
∩ |u|≥K
√
Ae(Du)
√
1+log 1
e
|u(x)| 4
K 2 1 + log 1
e A e (Du)
dx
≤ C 1 + log
1
e
1
e 2 A 3
e (Du) + A 2
e (Du) ,
− n v (P Iv(t) x) · n u d|∇χ u (·, t)|.
This entailsfor almost every T ∈ [0, T strong ). The next step consists of summing (185), (186), (187) and (188). We represent this sum as follows:LHS kin (T ) + LHS visc + LHS surEn (T ) (189) ≤ RHS kin (0) + RHS surEn (0) + RHS dt + RHS adv + RHS surT en , where each individual term is obtained in the following way. The terms related to kinetic energy at time T on the left hand side of (186), (187) and (188) in total yield the contributionThe same computation may be carried out for the initial kinetic energy termsNote that because of (8) it holdsThe terms in the energy dissipation inequality related to surface energy are therefore given byas well asThis in turn finally entailswhich holds for almost every T ∈ [0, T strong ). In a last step, we use the truncation of the identity β from Proposition 9 composed with the signed distance to the interface of the strong solution as a test function in the transport equations (6b) resp. (10b) for the indicator functions χ v resp. χ u of the two solutions. However, observe first that by the precise choice of the weight function β it holdsHence, when testing the equation (6b) for the indicator function of the varifold solution and then subtracting the corresponding result from testing the equation (10b) for the indicator function of the strong solution, we obtainwhich holds for almost every T ∈ [0, T strong ). Note that testing with the function) in our definition of solutions) and due to the fact that β( dist ± (x,Iv(t)) rc ) is of class C 1 . Indeed, one first multiplies β by a cutoff θR ∈ C ∞ cpt (R d ) on a scaleR, i.e. θ ≡ 1 on {x ∈ R d : |x| ≤R}, θ ≡ 0 outside of {x ∈ R d : |x| ≥ 2R} and ∇θ R L ∞ (R d ) ≤ CR −1 for some universalWEAK-STRONG UNIQUENESS FOR TWO-PHASE FLOW WITH SHARP INTERFACE 95constant C > 0. Then, one can use θRβ in the transport equations as test functions and pass to the limitR → ∞ because of the integrability of χ v and χ u . From this, one obtains the above equation.Since the weight β vanishes at r = 0, we may infer from the incompressibility of the velocity fields thatHence, we can rewrite (207) aŝfor almost every T ∈ [0, T strong ). It remains to make use of the evolution equation for β composed with the signed distance function to the interface of the strong solution. But before we do so, let us remark that because of (21)where the vector field V n is the projection of the velocity field v of the strong solution onto the subspace spanned by the unit normal n v (P Iv(t) x):for all (x, t) such that dist(x, I v (t)) < r c . Thus, using the evolution equation(33)we finally obtain the identitŷwhich holds true for almost every T ∈ [0, T strong ). The asserted relative entropy inequality now follows from a combination of the bounds (200), (206) as well as (208). This concludes the proof.Changing back to the local coordinates on the manifold I(t) we deducêThis yields the claim in the case of a smooth function u :In a last step, we extend this estimate by mollification to u ∈ BV(I(t)) with u L ∞ ≤ 1. To this end, let θ : R + → [0, 1] be a smooth cutoff with θ(s) = 1 for s ∈ [0,1 4] and θ(s) = 0 for s ≥ 1 2 . We then define for each n ∈ N u n (x, t) :.Since the analogous bound to(79)holds true, we infer u n L ∞ ≤ 1 as well as u n − u L 1 (I(t)) → 0 as n → ∞. In particular, we have pointwise almost everywhere convergence at least for a subsequence. This in turn implies by Lebesgue's dominated convergence theorem that u n − u L 4 (I(t)) → 0 as n → ∞ at least for a subsequence. Moreover, the exact same computation which led to(78)shows.Integrating this bound over the manifold and then using Fubini shows that A e(t) (Du n (t)) ≤ Cr −2 c A e(t) (Du(t)) holds true uniformly over all n ∈ N. By applying the bound (212) from the second step, we may conclude the proof.In the case where the interface I v is a curve in R 2 , a much more elementary argument yields the following bound.Lemma 38. Let T > 0 and let (I(t)) t∈[0,T ] be a family of smoothly evolving curves in R 2 in the sense of Definition 5. Let u ∈ L ∞ ([0, T ]; BV(I(t))) such that |u| ≤ 1. Consider the convex function G(s) := s 2 , |s| ≤ 1, 2s − 1, |s| > 1.We also define |Du(t)| G :=´I (t) G(|∇u(x, t)|) dS + |D s u(t)|(Γ). Then,|u(x, t)| 4 dS ≤ C(1 + H 1 (I(t))) 3 r 4 c |Du(t)| 2 G + |Du(t)| 4 G + ||u|| 4holds true for almost every t ∈ [0, T ] with some universal constant C > 0.Proof. Fix t > 0. First, observe that I(t) essentially consists of a finite number of nonintersecting curves. By approximation, we may assume u(t) ∈ W 1,1 (I(t)). Let η i be a partition of unity on I(t) with |∇ tan η i (x)| ≤ Cr −1 c such that the support of each η i is isometrically equivalent to a bounded interval (note that the
The initial-value problem for the Navier-Stokes equations with a free surface in L q -Sobolev spaces. H Abels, Adv. Differential Equations. 101H. Abels. The initial-value problem for the Navier-Stokes equations with a free surface in L q -Sobolev spaces. Adv. Differential Equations, 10(1):45-64, 2005.
On generalized solutions of two-phase flows for viscous incompressible fluids. H Abels, terfaces Free Bound. 9H. Abels. On generalized solutions of two-phase flows for viscous incompressible fluids. In- terfaces Free Bound., 9(1):31-65, 2007.
Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. H Abels, Comm. Math. Phys. 2891H. Abels. Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Comm. Math. Phys., 289(1):45-73, 2009.
Pressure reconstruction for weak solutions of the two-phase incompressible Navier-Stokes equations with surface tension. H Abels, J Daube, C Kraus, arxiv:1801.04840PreprintH. Abels, J. Daube, and C. Kraus. Pressure reconstruction for weak solutions of the two-phase incompressible Navier-Stokes equations with surface tension. Preprint, 2018. arxiv:1801.04840.
Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities. H Abels, D Depner, H Garcke, J. Math. Fluid Mech. 153H. Abels, D. Depner, and H. Garcke. Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities. J. Math. Fluid Mech., 15(3):453-480, 2013.
Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. H Abels, H Garcke, G Grün, Math. Models Methods Appl. Sci. 22340H. Abels, H. Garcke, and G. Grün. Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci., 22(3):1150013, 40, 2012.
Sharp Interface Limit for a Stokes/Allen-Cahn System. H Abels, Y Liu, Arch. Ration. Mech. Anal. 2291H. Abels and Y. Liu. Sharp Interface Limit for a Stokes/Allen-Cahn System. Arch. Ration. Mech. Anal., 229(1):417-502, 2018.
Sharp interface limits for diffuse interface models for twophase flows of viscous incompressible fluids. H Abels, Y Liu, A Schöttl, Transport processes at fluidic interfaces. ChamBirkhäuser/SpringerH. Abels, Y. Liu, and A. Schöttl. Sharp interface limits for diffuse interface models for two- phase flows of viscous incompressible fluids. In Transport processes at fluidic interfaces, Adv. Math. Fluid Mech., pages 231-253. Birkhäuser/Springer, Cham, 2017.
Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids. H Abels, M Röger, Ann. Inst. H. Poincaré Anal. Non Linéaire. 266H. Abels and M. Röger. Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire, 26(6):2403-2424, 2009.
Functions of Bounded Variation and Free Discontinuity Problems (Oxford Mathematical Monographs). L Ambrosio, N Fusco, D Pallara, Oxford University PressL. Ambrosio, N. Fusco, and D. Pallara. Functions of Bounded Variation and Free Disconti- nuity Problems (Oxford Mathematical Monographs). Oxford University Press, 2000.
Level set approach to mean curvature flow in arbitrary codimension. L Ambrosio, H M Soner, J. Differential Geom. 434L. Ambrosio and H. M. Soner. Level set approach to mean curvature flow in arbitrary codi- mension. J. Differential Geom., 43(4):693-737, 1996.
Diffuse-interface methods in fluid mechanics. D M Anderson, G B Mcfadden, A A Wheeler, Annual review of fluid mechanics. Palo Alto, CAAnnual Reviews30D. M. Anderson, G. B. McFadden, and A. A. Wheeler. Diffuse-interface methods in fluid mechanics. In Annual review of fluid mechanics, Vol. 30, volume 30 of Annu. Rev. Fluid Mech., pages 139-165. Annual Reviews, Palo Alto, CA, 1998.
A computed example of nonuniqueness of mean curvature flow in R 3. S Angenent, T Ilmanen, D L Chopp, Comm. Partial Differential Equations. 20S. Angenent, T. Ilmanen, and D. L. Chopp. A computed example of nonuniqueness of mean curvature flow in R 3 . Comm. Partial Differential Equations, 20(11-12):1937-1958, 1995.
Fourier Analysis and Nonlinear Partial Differential Equations. H Bahouri, J.-Y Chemin, R Danchin, SpringerBerlin HeidelbergH. Bahouri, J.-Y. Chemin, and R. Danchin. Fourier Analysis and Nonlinear Partial Differ- ential Equations. Springer Berlin Heidelberg, 2011.
Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. C Bardos, F Golse, C D Levermore, Comm. Pure Appl. Math. 465C. Bardos, F. Golse, and C. D. Levermore. Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math., 46(5):667-753, 1993.
Front propagation and phase field theory. G Barles, H M Soner, P E Souganidis, SIAM J. Control Optim. 312G. Barles, H. M. Soner, and P. E. Souganidis. Front propagation and phase field theory. SIAM J. Control Optim., 31(2):439-469, 1993.
The initial value problem for the Navier-Stokes equations with a free surface. J T Beale, Comm. Pure Appl. Math. 343J. T. Beale. The initial value problem for the Navier-Stokes equations with a free surface. Comm. Pure Appl. Math., 34(3):359-392, 1981.
Large-time regularity of viscous surface waves. J T Beale, Arch. Rational Mech. Anal. 84484J. T. Beale. Large-time regularity of viscous surface waves. Arch. Rational Mech. Anal., 84(4):307-352, 1983/84.
The motion of a surface by its mean curvature. K A Brakke, of Mathematical Notes. Princeton, N.J.Princeton University Press20K. A. Brakke. The motion of a surface by its mean curvature, volume 20 of Mathematical Notes. Princeton University Press, Princeton, N.J., 1978.
Y Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations. 25Y. Brenier. Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations, 25(3-4):737-754, 2000.
Weak-strong uniqueness for measure-valued solutions. Y Brenier, C De Lellis, L SzékelyhidiJr, Comm. Math. Phys. 3052Y. Brenier, C. De Lellis, and L. Székelyhidi, Jr. Weak-strong uniqueness for measure-valued solutions. Comm. Math. Phys., 305(2):351-361, 2011.
Onsager's conjecture almost everywhere in time. T Buckmaster, Comm. Math. Phys. 3333T. Buckmaster. Onsager's conjecture almost everywhere in time. Comm. Math. Phys., 333(3):1175-1198, 2015.
Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1. T Buckmaster, M Colombo, V Vicol, arXiv:1809.00600PreprintT. Buckmaster, M. Colombo, and V. Vicol. Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1. Preprint, 2018. arXiv:1809.00600.
Anomalous dissipation for 1/5-Hölder Euler flows. T Buckmaster, C De Lellis, P Isett, L SzékelyhidiJr, Ann. of Math. 1822T. Buckmaster, C. De Lellis, P. Isett, and L. Székelyhidi, Jr. Anomalous dissipation for 1/5-Hölder Euler flows. Ann. of Math. (2), 182(1):127-172, 2015.
Nonuniqueness of weak solutions to the Navier-Stokes equation. T Buckmaster, V Vicol, arXiv:1709.10033PreprintT. Buckmaster and V. Vicol. Nonuniqueness of weak solutions to the Navier-Stokes equation. Preprint, 2017. arXiv:1709.10033.
. A Castro, D Córdoba, C Fefferman, F Gancedo, J Gómez-Serrano, Ann. of Math. 1782Finite time singularities for the free boundary incompressible Euler equationsA. Castro, D. Córdoba, C. Fefferman, F. Gancedo, and J. Gómez-Serrano. Finite time singu- larities for the free boundary incompressible Euler equations. Ann. of Math. (2), 178(3):1061- 1134, 2013.
Splash singularities for the free boundary Navier-Stokes equations. A Castro, D Córdoba, C Fefferman, F Gancedo, J Gómez-Serrano, arXiv:1504.02775PreprintA. Castro, D. Córdoba, C. Fefferman, F. Gancedo, and J. Gómez-Serrano. Splash singularities for the free boundary Navier-Stokes equations. Preprint, 2015. arXiv:1504.02775.
Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition. N Chemetov, Š Nečasová, B Muha, arXiv:1710.01382PreprintN. Chemetov,Š. Nečasová, and B. Muha. Weak-strong uniqueness for fluid-rigid body inter- action problem with slip boundary condition. Preprint, 2017. arXiv:1710.01382.
Weak-strong uniqueness of renormalized solutions to reaction-crossdiffusion systems. X Chen, A , arXiv:1805.02950Nonlinear Anal. X. Chen and A. Jüngel. Weak-strong uniqueness of renormalized solutions to reaction-cross- diffusion systems. Nonlinear Anal., 2018. arXiv:1805.02950.
The perimeter inequality under Steiner symmetrization: Cases of equality. M Chlebík, A Cianchi, N Fusco, Ann. of Math. 1622M. Chlebík, A. Cianchi, and N. Fusco. The perimeter inequality under Steiner symmetriza- tion: Cases of equality. Ann. of Math. (2), 162:525-555, 2005.
A fully anisotropic Sobolev inequality. A Cianchi, Pacific J. Math. 1962A. Cianchi. A fully anisotropic Sobolev inequality. Pacific J. Math., 196(2):283-295, 2000.
Ill-posedness of Leray solutions for the hypodissipative Navier-Stokes equations. M Colombo, C De Lellis, L. De Rosa, Comm. Math. Phys. 3622M. Colombo, C. De Lellis, and L. De Rosa. Ill-posedness of Leray solutions for the hypodis- sipative Navier-Stokes equations. Comm. Math. Phys., 362(2):659-688, 2018.
Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. P Constantin, C Fefferman, Indiana Univ. Math. J. 423P. Constantin and C. Fefferman. Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J., 42(3):775-789, 1993.
Unique solvability of the free-boundary Navier-Stokes equations with surface tension. D Coutand, S Shkoller, arXiv:math/0212116PreprintD. Coutand and S. Shkoller. Unique solvability of the free-boundary Navier-Stokes equations with surface tension. Preprint, 2002. arXiv:math/0212116.
On the impossibility of finite-time splash singularities for vortex sheets. D Coutand, S Shkoller, arXiv:1407.1479PreprintD. Coutand and S. Shkoller. On the impossibility of finite-time splash singularities for vortex sheets. Preprint, 2014. arXiv:1407.1479.
On the splash singularity for the free-surface of a Navier-Stokes fluid. D Coutand, S Shkoller, Ann. Inst. H. Poincaré Anal. Non Linéaire. D. Coutand and S. Shkoller. On the splash singularity for the free-surface of a Navier-Stokes fluid. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2018.
The second law of thermodynamics and stability. C M Dafermos, Arch. Rational Mech. Anal. 702C. M. Dafermos. The second law of thermodynamics and stability. Arch. Rational Mech. Anal., 70(2):167-179, 1979.
Entropy and the stability of classical solutions of hyperbolic systems of conservation laws. C M Dafermos, Recent mathematical methods in nonlinear wave propagation. Montecatini Terme; BerlinSpringer1640C. M. Dafermos. Entropy and the stability of classical solutions of hyperbolic systems of con- servation laws. In Recent mathematical methods in nonlinear wave propagation (Montecatini Terme, 1994), volume 1640 of Lecture Notes in Math., pages 48-69. Springer, Berlin, 1996.
Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations. S Daneri, L Székelyhidi, Arch. Ration. Mech. Anal. 2242S. Daneri and L. Székelyhidi. Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations. Arch. Ration. Mech. Anal., 224(2):471-514, 2017.
The Euler equations as a differential inclusion. C De Lellis, L SzékelyhidiJr, Ann. of Math. 1702C. De Lellis and L. Székelyhidi, Jr. The Euler equations as a differential inclusion. Ann. of Math. (2), 170(3):1417-1436, 2009.
Dissipative continuous Euler flows. C De Lellis, L SzékelyhidiJr, Invent. Math. 1932C. De Lellis and L. Székelyhidi, Jr. Dissipative continuous Euler flows. Invent. Math., 193(2):377-407, 2013.
Weak-strong uniqueness of dissipative measure-valued solutions for polyconvex elastodynamics. S Demoulini, D M A Stuart, A E Tzavaras, Arch. Ration. Mech. Anal. 2053S. Demoulini, D. M. A. Stuart, and A. E. Tzavaras. Weak-strong uniqueness of dissipa- tive measure-valued solutions for polyconvex elastodynamics. Arch. Ration. Mech. Anal., 205(3):927-961, 2012.
Problem of the motion of two viscous incompressible fluids separated by a closed free interface. I V Denisova, Mathematical problems for Navier-Stokes equations. Centro37I. V. Denisova. Problem of the motion of two viscous incompressible fluids separated by a closed free interface. Acta Appl. Math., 37(1-2):31-40, 1994. Mathematical problems for Navier-Stokes equations (Centro, 1993).
Uniqueness of solutions to hyperbolic conservation laws. R J Diperna, Indiana Univ. Math. J. 281R. J. DiPerna. Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math. J., 28(1):137-188, 1979.
Mean-field limits for some Riesz interaction gradient flows. M Duerinckx, SIAM J. Math. Anal. 483M. Duerinckx. Mean-field limits for some Riesz interaction gradient flows. SIAM J. Math. Anal., 48(3):2269-2300, 2016.
The role of self-similarity in singularities of partial differential equations. J Eggers, M A Fontelos, Nonlinearity. 221J. Eggers and M. A. Fontelos. The role of self-similarity in singularities of partial differential equations. Nonlinearity, 22(1):R1-R44, 2009.
Splash singularity for a free-boundary incompressible viscoelastic fluid model. S S Elena Di Iorio, Pierangelo Marcati, arXiv:1806.11089PreprintS. S. Elena Di Iorio, Pierangelo Marcati. Splash singularity for a free-boundary incompressible viscoelastic fluid model. Preprint, 2018. arXiv:1806.11089.
Weak-strong uniqueness for the general Ericksen-Leslie system in three dimensions. E Emmrich, R Lasarzik, Discrete Contin. Dyn. Syst. 389E. Emmrich and R. Lasarzik. Weak-strong uniqueness for the general Ericksen-Leslie system in three dimensions. Discrete Contin. Dyn. Syst., 38(9):4617-4635, 2018.
Colloquium lectures on geometric measure theory. H Federer, Bull. Amer. Math. Soc. 84H. Federer. Colloquium lectures on geometric measure theory. Bull. Amer. Math. Soc., 84:291-339, 1978.
On the absence of splash singularities in the case of two-fluid interfaces. C Fefferman, A D Ionescu, V Lie, Duke Math. J. 1653C. Fefferman, A. D. Ionescu, and V. Lie. On the absence of splash singularities in the case of two-fluid interfaces. Duke Math. J., 165(3):417-462, 2016.
Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. E Feireisl, B J Jin, A Novotný, J. Math. Fluid Mech. 144E. Feireisl, B. J. Jin, and A. Novotný. Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech., 14(4):717-730, 2012.
Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. E Feireisl, A Novotný, Arch. Ration. Mech. Anal. 2042E. Feireisl and A. Novotný. Weak-strong uniqueness property for the full Navier-Stokes- Fourier system. Arch. Ration. Mech. Anal., 204(2):683-706, 2012.
A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation. J Fischer, SIAM J. Numer. Anal. 535J. Fischer. A posteriori modeling error estimates for the assumption of perfect incompress- ibility in the Navier-Stokes equation. SIAM J. Numer. Anal., 53(5):2178-2205, 2015.
Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations. J Fischer, Nonlinear Anal. 159J. Fischer. Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equa- tions. Nonlinear Anal., 159:181-207, 2017.
Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. P Germain, J. Math. Fluid Mech. 131P. Germain. Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J. Math. Fluid Mech., 13(1):137-146, 2011.
A posteriori analysis for dynamic model adaptation in convection-dominated problems. J Giesselmann, T Pryer, Math. Models Methods Appl. Sci. 2713J. Giesselmann and T. Pryer. A posteriori analysis for dynamic model adaptation in convection-dominated problems. Math. Models Methods Appl. Sci., 27(13):2381-2423, 2017.
On global weak solutions of the nonstationary two-phase Stokes flow. Y Giga, S Takahashi, SIAM J. Math. Anal. 253Y. Giga and S. Takahashi. On global weak solutions of the nonstationary two-phase Stokes flow. SIAM J. Math. Anal., 25(3):876-893, 1994.
The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. F Golse, L Saint-Raymond, Invent. Math. 1551F. Golse and L. Saint-Raymond. The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels. Invent. Math., 155(1):81-161, 2004.
Two-phase binary fluids and immiscible fluids described by an order parameter. M E Gurtin, D Polignone, J Viñals, Math. Models Methods Appl. Sci. 66M. E. Gurtin, D. Polignone, and J. Viñals. Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci., 6(6):815-831, 1996.
Weak-strong uniqueness for measure-valued solutions of some compressible fluid models. P Gwiazda, A Świerczewska-Gwiazda, E Wiedemann, Nonlinearity. 2811P. Gwiazda, A.Świerczewska-Gwiazda, and E. Wiedemann. Weak-strong uniqueness for measure-valued solutions of some compressible fluid models. Nonlinearity, 28(11):3873-3890, 2015.
Theory of dynamic critical phenomena. P C Hohenberg, B I Halperin, Rev. Mod. Phys. 49P. C. Hohenberg and B. I. Halperin. Theory of dynamic critical phenomena. Rev. Mod. Phys., 49:435-479, Jul 1977.
Elliptic regularization and partial regularity for motion by mean curvature. T Ilmanen, Mem. Amer. Math. Soc. 10852090T. Ilmanen. Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc., 108(520):x+90, 1994.
A proof of Onsager's conjecture. P Isett, Ann. of Math. 1882P. Isett. A proof of Onsager's conjecture. Ann. of Math. (2), 188(3):871-963, 2018.
On the vortex filament conjecture for Euler flows. R L Jerrard, C Seis, Arch. Ration. Mech. Anal. 2241R. L. Jerrard and C. Seis. On the vortex filament conjecture for Euler flows. Arch. Ration. Mech. Anal., 224(1):135-172, 2017.
On the motion of a curve by its binormal curvature. R L Jerrard, D Smets, J. Eur. Math. Soc. 176R. L. Jerrard and D. Smets. On the motion of a curve by its binormal curvature. J. Eur. Math. Soc., 17(6):1487-1515, 2015.
L 2 -contraction of large planar shock waves for multidimensional scalar viscous conservation laws. M.-J Kang, A Vasseur, Y Wang, arXiv:1609.01825PreprintM.-J. Kang, A. Vasseur, and Y. Wang. L 2 -contraction of large planar shock waves for multi- dimensional scalar viscous conservation laws. Preprint, 2016. arXiv:1609.01825.
Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations. N Leger, A Vasseur, Arch. Ration. Mech. Anal. 2011N. Leger and A. Vasseur. Relative entropy and the stability of shocks and contact discon- tinuities for systems of conservation laws with non-BV perturbations. Arch. Ration. Mech. Anal., 201(1):271-302, 2011.
Sur le mouvement d'un liquide visqueux emplissant l'espace. J Leray, Acta Math. 631J. Leray. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math., 63(1):193- 248, 1934.
A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. C Liu, J Shen, Phys. D. 1793-4C. Liu and J. Shen. A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D, 179(3-4):211-228, 2003.
Quasi-incompressible Cahn-Hilliard fluids and topological transitions. J Lowengrub, L Truskinovsky, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454J. Lowengrub and L. Truskinovsky. Quasi-incompressible Cahn-Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454(1978):2617-2654, 1998.
An existence theorem for the multifluid Navier-Stokes problem. A Nouri, F Poupaud, J. Differential Equations. 1221A. Nouri and F. Poupaud. An existence theorem for the multifluid Navier-Stokes problem. J. Differential Equations, 122(1):71-88, 1995.
Generalized solutions of a problem on the motion of a non-Newtonian fluid with a free boundary. P I Plotnikov, Sibirsk. Mat. Zh. 344P. I. Plotnikov. Generalized solutions of a problem on the motion of a non-Newtonian fluid with a free boundary. Sibirsk. Mat. Zh., 34(4):127-141, iii, ix, 1993.
Un teorema di unicità per le equazioni di Navier-Stokes. G Prodi, Ann. Mat. Pura Appl. 484G. Prodi. Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. (4), 48:173-182, 1959.
On the Rayleigh-Taylor instability for the two-phase Navier-Stokes equations. J Prüss, G Simonett, Indiana Univ. Math. J. 596J. Prüss and G. Simonett. On the Rayleigh-Taylor instability for the two-phase Navier-Stokes equations. Indiana Univ. Math. J., 59(6):1853-1871, 2010.
On the two-phase Navier-Stokes equations with surface tension. J Prüss, G Simonett, Interfaces Free Bound. 123J. Prüss and G. Simonett. On the two-phase Navier-Stokes equations with surface tension. Interfaces Free Bound., 12(3):311-345, 2010.
Analytic solutions for the two-phase Navier-Stokes equations with surface tension and gravity. J Prüss, G Simonett, Parabolic problems. Basel AG, BaselBirkhäuser/Springer80J. Prüss and G. Simonett. Analytic solutions for the two-phase Navier-Stokes equations with surface tension and gravity. In Parabolic problems, volume 80 of Progr. Nonlinear Differential Equations Appl., pages 507-540. Birkhäuser/Springer Basel AG, Basel, 2011.
Moving Interfaces and Quasilinear Parabolic Evolution Equations. J Prüss, G Simonett, Springer International PublishingJ. Prüss and G. Simonett. Moving Interfaces and Quasilinear Parabolic Evolution Equations. Springer International Publishing, 2016.
Infinitely many leray-hopf solutions for the fractional navier-stokes equations. L D Rosa, to appear in Comm. Partial Differential EquationsL. D. Rosa. Infinitely many leray-hopf solutions for the fractional navier-stokes equations. to appear in Comm. Partial Differential Equations, 2018.
Convergence of solutions to the Boltzmann equation in the incompressible Euler limit. L Saint-Raymond, Arch. Ration. Mech. Anal. 1661L. Saint-Raymond. Convergence of solutions to the Boltzmann equation in the incompressible Euler limit. Arch. Ration. Mech. Anal., 166(1):47-80, 2003.
Hydrodynamic limits: some improvements of the relative entropy method. L Saint-Raymond, Ann. Inst. H. Poincaré Anal. Non Linéaire. 263L. Saint-Raymond. Hydrodynamic limits: some improvements of the relative entropy method. Ann. Inst. H. Poincaré Anal. Non Linéaire, 26(3):705-744, 2009.
An inviscid flow with compact support in space-time. V Scheffer, J. Geom. Anal. 34V. Scheffer. An inviscid flow with compact support in space-time. J. Geom. Anal., 3(4):343- 401, 1993.
Navier-Stokes equations with lower bounds on the pressure. G Seregin, V Šverák, Arch. Ration. Mech. Anal. 1631G. Seregin and V.Šverák. Navier-Stokes equations with lower bounds on the pressure. Arch. Ration. Mech. Anal., 163(1):65-86, 2002.
Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations. S Serfaty, J. Amer. Math. Soc. 303S. Serfaty. Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations. J. Amer. Math. Soc., 30(3):713-768, 2017.
On the interior regularity of weak solutions of the Navier-Stokes equations. J Serrin, Arch. Rational Mech. Anal. 9J. Serrin. On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rational Mech. Anal., 9:187-195, 1962.
The initial value problem for the Navier-Stokes equations. J Serrin, Nonlinear Problems (Proc. Sympos. Madison, Wis; Madison, WisUniv. of Wisconsin PressJ. Serrin. The initial value problem for the Navier-Stokes equations. In Nonlinear Problems (Proc. Sympos., Madison, Wis., 1962), pages 69-98. Univ. of Wisconsin Press, Madison, Wis., 1963.
On a free boundary problem for the Navier-Stokes equations. Y Shibata, S Shimizu, Differential Integral Equations. 203Y. Shibata and S. Shimizu. On a free boundary problem for the Navier-Stokes equations. Differential Integral Equations, 20(3):241-276, 2007.
Solvability of the problem of the motion of a viscous incompressible fluid that is bounded by a free surface. V A Solonnikov, Izv. Akad. Nauk SSSR Ser. Mat. 41V. A. Solonnikov. Solvability of the problem of the motion of a viscous incompressible fluid that is bounded by a free surface. Izv. Akad. Nauk SSSR Ser. Mat., 41:1388-1424, 1977.
Unsteady flow of a finite mass of a fluid bounded by a free surface. V A Solonnikov, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI). 152Kraev. Zadachi Mat. Fiz. i Smezhnye Vopr. Teor. FunktsiȋV. A. Solonnikov. Unsteady flow of a finite mass of a fluid bounded by a free surface. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152(Kraev. Zadachi Mat. Fiz. i Smezhnye Vopr. Teor. Funktsiȋ 18):137-157, 183-184, 1986.
Solvability of a problem on the evolution of a viscous incompressible fluid. V A Solonnikov, Algebra i Analiz. 31V. A. Solonnikov. Solvability of a problem on the evolution of a viscous incompressible fluid, bounded by a free surface, on a finite time interval. Algebra i Analiz, 3(1):222-257, 1991.
Relative entropy and hydrodynamics of Ginzburg-Landau models. H.-T Yau, Lett. Math. Phys. 221H.-T. Yau. Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys., 22(1):63-80, 1991.
| [] |
[
"Beam splitting and Hong-Ou-Mandel interference for stored light",
"Beam splitting and Hong-Ou-Mandel interference for stored light"
] | [
"A Raczyński \nInstytut Fizyki\nInstytut Matematyki i Fizyki\nUniwersytet Miko laja Kopernika\nulica Grudziadzka 5, Akademia Techniczno-Rolnicza, Aleja Prof. S. Kaliskiego 787-100, 85-796Toruń, BydgoszczPoland, Poland\n",
"J Zaremba \nInstytut Fizyki\nInstytut Matematyki i Fizyki\nUniwersytet Miko laja Kopernika\nulica Grudziadzka 5, Akademia Techniczno-Rolnicza, Aleja Prof. S. Kaliskiego 787-100, 85-796Toruń, BydgoszczPoland, Poland\n",
"S Zielińska-Kaniasty \nInstytut Fizyki\nInstytut Matematyki i Fizyki\nUniwersytet Miko laja Kopernika\nulica Grudziadzka 5, Akademia Techniczno-Rolnicza, Aleja Prof. S. Kaliskiego 787-100, 85-796Toruń, BydgoszczPoland, Poland\n"
] | [
"Instytut Fizyki\nInstytut Matematyki i Fizyki\nUniwersytet Miko laja Kopernika\nulica Grudziadzka 5, Akademia Techniczno-Rolnicza, Aleja Prof. S. Kaliskiego 787-100, 85-796Toruń, BydgoszczPoland, Poland",
"Instytut Fizyki\nInstytut Matematyki i Fizyki\nUniwersytet Miko laja Kopernika\nulica Grudziadzka 5, Akademia Techniczno-Rolnicza, Aleja Prof. S. Kaliskiego 787-100, 85-796Toruń, BydgoszczPoland, Poland",
"Instytut Fizyki\nInstytut Matematyki i Fizyki\nUniwersytet Miko laja Kopernika\nulica Grudziadzka 5, Akademia Techniczno-Rolnicza, Aleja Prof. S. Kaliskiego 787-100, 85-796Toruń, BydgoszczPoland, Poland"
] | [] | Storing and release of a quantum light pulse in a medium of atoms in the tripod configuration are studied. Two complementary sets of control fields are defined, which lead to independent and complete photon release at two stages. The system constitutes a new kind of a flexible beam splitter in which the input and output ports concern photons of the same direction but well separated in time. A new version of Hong-Ou-Mandel interference is discussed. | 10.1103/physreva.75.013810 | [
"https://arxiv.org/pdf/quant-ph/0606184v1.pdf"
] | 119,422,013 | quant-ph/0606184 | efb41d9592580639f15721f3b541b496454b12e0 |
Beam splitting and Hong-Ou-Mandel interference for stored light
22 Jun 2006
A Raczyński
Instytut Fizyki
Instytut Matematyki i Fizyki
Uniwersytet Miko laja Kopernika
ulica Grudziadzka 5, Akademia Techniczno-Rolnicza, Aleja Prof. S. Kaliskiego 787-100, 85-796Toruń, BydgoszczPoland, Poland
J Zaremba
Instytut Fizyki
Instytut Matematyki i Fizyki
Uniwersytet Miko laja Kopernika
ulica Grudziadzka 5, Akademia Techniczno-Rolnicza, Aleja Prof. S. Kaliskiego 787-100, 85-796Toruń, BydgoszczPoland, Poland
S Zielińska-Kaniasty
Instytut Fizyki
Instytut Matematyki i Fizyki
Uniwersytet Miko laja Kopernika
ulica Grudziadzka 5, Akademia Techniczno-Rolnicza, Aleja Prof. S. Kaliskiego 787-100, 85-796Toruń, BydgoszczPoland, Poland
Beam splitting and Hong-Ou-Mandel interference for stored light
22 Jun 2006arXiv:quant-ph/0606184v1PACS numbers: 4250Gy, 0367-a * Electronic address: raczyn@physumkpl 2
Storing and release of a quantum light pulse in a medium of atoms in the tripod configuration are studied. Two complementary sets of control fields are defined, which lead to independent and complete photon release at two stages. The system constitutes a new kind of a flexible beam splitter in which the input and output ports concern photons of the same direction but well separated in time. A new version of Hong-Ou-Mandel interference is discussed.
I. INTRODUCTION
Light propagation and storing in laser driven atomic media have recently become a subject of numerous studies, stimulated both by a fundamental character of the discussed problems and by possible applications. A generic system for such studies is a medium of three-level atoms interacting with two laser beams (signal and control) in the Λ configuration. The essence of the process of light storing is a coherent mapping of the signal pulse into an atomic excitation, described by an atomic coherence, due to a switch-off of a control laser beam. By switching the control field again one restores the trapped pulse, preserving all the phase relations. This means in fact first writing down an information conveyed by the photons and later reading it back. The whole process can be effectively interpreted in terms of a joint medium+field excitation called dark state polariton. For recent reviews on light storage see, e.g., Refs [1,2,3].
Admitting additional fields and thus additional active atomic states enriches the dynamics of the process and gives new possibilities of its control. In particular by adding another upper state one obtains a double Λ system [4] in which it is possible to make the medium transparent simultaneously for two signal pulses or to split a signal pulse into two ones of different frequencies [5,6]. Channelization of information is also possible in the case of inverted Y systems [7]. Coupling the Λ system with a side level makes it possible to release a pulse of a frequency different from the original one or to temporarily prevent the signal from being released [8]. Recently the dynamics of light propagation in 5-level atomic systems in the M configuration has also been discussed [9].
A special example is the so-called tripod atomic system in which three low-lying levels are coupled with an upper level by a signal field and two control ones. By a manipulation on control fields one can steer the propagation, storing and release of the signal, which may be stored in the form of two atomic coherences. Light propagation and slowdown in such a medium have been studied in Refs [10,11] while the problem of light storing has been discussed in detail in our recent paper [12]. An attempt of releasing the trapped pulse by an arbitrary combination of two control fields results in general in splitting it into two parts one of which was leaving the sample while the other one remained trapped and could be released by switching on a new set of control fields. The evolution of such a system could be effectively described in terms of a couple of polaritons.
The previously used classical description of dividing the pulse into two or more parts may be essentially enriched by treating the pulse quantum-mechanically and by asking new questions about the photon statistics, especially important in the case of nonclassical light states, e.g., Fock states with a small photon number. Such an approach is also necessary in the consideration of quantum information storage in an atomic memory. Wang et al. [13] demonstrated a possibility of time splitting of a single photon by first storing it in one coherence, than by a sequence of transferring a part of the excitation into the other coherence and of a photon release from the latter. An additional transfer procedure based on a fractional adiabatic population transfer (F-STIRAP) had to be performed before each release procedure.
In this paper we investigate light propagation and storage of a quantum signal field in a medium of atoms in the tripod configuration. We discuss mapping of a quantum wave packet into two atomic coherences which means in particular a channelization of a single photon. In our approach, being more general than that of Ref. [13], at the release stage the two control fields may be twice switched on and off simultaneously. If the control pulses are complementary, which means that their amplitude and phase relations are properly chosen, the whole of the stored signal is divided into two parts, which are released at two separated time instants. In the case of a single trapped photon this leads to its time-entangled state, but without a need of an additional transfer stage of Ref. [13]. We discuss in detail the case in which two time-separated photons are trapped at the same place of the sample due to an action of two complementary sets of control pulses. They are later released by a different pair of sets of such pulses. We examine the possibility of photon coalescence, i.e. a release of both photons at the same release stage. By changing localizations and shapes of the stored photon wavepackets and the details of the control fields we obtain an analogue of the Hong-Ou-Mandel interferometer [14,15,16,17] working on stored light. In contradistinction to the original version of such an interferometer we have to do with photons propagating always in the same spatial direction but well resolved in time.
II. QUANTUM POLARITONS IN THE TRIPOD CASE
In order to investigate quantum effects in light storing and propagation both the signal and the medium have to be described quantum-mechanically. Some formulae of this section, pertaining to quantum operators, resemble those presented in our previous paper [12] for classical pulse. However, they are rederived below not only to make the presentation complete but also because they will serve to a discussion of physical effects connected with a quantum field, in particular with single photon states. In particular polaritons which in the present formulation are quantummechanical operators will play a central role in describing photon correlation effects.
Consider a one-dimensional medium of atoms in the tripod configuration ( Fig. 1) including an upper state a and three stable lower states b (initial), c and d. The sample is irradiated with three collinear laser beams. The quantized signal field couples the states b and a and is written as:
ǫ(z, t) = ǫ (+) (z, t) + ǫ (−) (z, t) = Σ k g k a k exp[i(kz − ωt)] exp[−i(k ab z − ω ab t)] + h.c.,(1)
where a k is the annihilation operator of a photon with wave number k, g k = ω 2ǫ0V , V is the quantization volume, ǫ 0 -the vacuum electric permittivity and
k αβ = ω αβ /c = (E α − E β )/( c), α, β = a, b, c, d.
The two control fields ǫ 2,3 (t) cos(k 2,3 z − ω 2,3 t) are treated classically and are supposed to be strong enough for the propagation effects to be neglected; they couple respectively the states c and d with a. The medium excitation is described in the formalism of the second quantization by the flip operators σ αβ = |α >< β| exp(ik αβ z), k αβ being the corresponding wave vector. The medium is treated in a continuous way; this is why N L dz in the above equation has replaced the sum over N atoms (L is the length of the sample).
The interaction hamiltonian in the rotating-wave approximation reads
H = N L dz(−d ab σ ab exp[−i(k ab z − ω ab t)]Σa k g k exp[i(kz − ωt)] + h.c. − 1 2 d ac σ ac exp[−i(k ac z − ω ac t)]ǫ 2 (t) exp[i(k 2 z − ω 2 t)] + h.c. (2) − 1 2 d ad σ ad exp[−i(k ad z − ω ad t)]ǫ 3 (t) exp[i(k 3 z − ω 3 t)] + h.c.),
where d αβ are the dipole moment's matrix elements, assumed for simplicity to be real and positive. In our analysis in terms of polaritons it is enough to restrict oneself to the deterministic part of the Heisenberg evolution (relaxation-and noise-free). The equations of motion for the flip and signal field operators in the first-order approximation with respect to the signal field, in resonance conditions read
iσ ba (z, t) = − d ab ǫ (+) (z, t) − Ω 2 (t)σ bc (z, t) − Ω 3 (t)σ bd (z, t), iσ bc (z, t) = −Ω * 2 (t)σ ba (z, t), (3) iσ bd (z, t) = −Ω * 3 (t)σ ba (z, t), ( ∂ ∂t + c ∂ ∂z )ǫ (+) (z, t) = iN g 2 d ba σ ba (z, t),
where g is the value of g k for the central field frequency of the signal field, and Ω 2 (t) = d ac ǫ 2 (t)/ (2 ) and Ω 3 (t) = d ad ǫ 3 (t)/(2 ) are the Rabi frequencies for the control fields. It is convenient to express the solutions of Eqs (3) in terms of two field+medium excitations called polaritons: Ψ(z, t) (the dark-state polariton) and Z(z, t)
Ψ(z, t) = exp(−iχ){ǫ (+) (z, t) cos θ − κ d ab sin θ[exp(iχ 2 ) cos φσ bc (z, t) + exp(iχ 3 ) sin φσ bd (z, t)]}, Z(z, t) = κ d ab [exp(iχ 2 ) sin φσ bc (z, t) − exp(iχ 3 ) cos φσ bd (z, t)],(4)
where the mixing angles θ and φ are defined as tan θ = κ/Ω,
tan φ = |Ω 3 /Ω 2 | with χ 2 = arg(Ω 2 ), χ 3 = arg(Ω 3 ), Ω = |Ω 2 | 2 + |Ω 3 | 2 , κ 2 = N g 2 |d ab | ]2 / 2 ; χ satisfies the equationχ = sin 2 θχ 2 .
For simplicity we assume thatχ 2 =χ 3 . The field and medium components of the polariton Ψ are
ǫ (+) = exp(iχ)Ψ cos θ,(5)exp(iχ 2 ) cos φσ bc + exp(iχ 3 ) sin φσ bd = − d ab κ exp(iχ)Ψ sin θ.(6)
The evolution equations for the two polaritons read
( ∂ ∂t + c cos 2 θ ∂ ∂z )Ψ = exp(−iχ) sin θφZ, ∂ ∂t Z = iχ 2 Z − exp(iχ) sin θφΨ.(7)
The operators Ψ(z, t)/(g √ L) and Z(z, t)/(g √ L) fulfil typical bosonic commutation relations in the first-order approximation, in which σ bb = 1,
σ cc = σ dd = σ cd = 0, ([Ψ(z), Ψ † (z ′ )] = [Z(z), Z † (z ′ )] = g 2 Lδ(z−z ′ ), [Ψ(z), Z † (z ′ )] = 0)
, so those operators may be considered annihilation operators of the joint field+medium excitation. The essence of the approach used in this paper will be treating a medium excitation due to the stored photon as a superposition of two excitations, properly suited to the pulse releasing control fields (see Eq. (9) below).
If the control fields are proportional, i.e.φ = 0 andχ 2 =χ 3 = 0 the evolution equations for the polaritons are decoupled. The polariton Ψ propagates through the medium without changing its "shape", i.e. the solution is
Ψ(z, t) = Ψ(z − c t 0 cos 2 θ(t ′ )dt ′ , t = 0),(8)
while the polariton Z = const keeps both its shape and localization. The above equations constitute a generalization of the approach of Fleischhauer et al.
[? ] for a four-level system and at the same time of our previous results [12], here formulated in the language of a quantum signal field. Due to such an approach we will be able to discuss the effect of the process in terms of photons rather than in the language of splitting a classical pulse.
If at the beginning the control fields corresponding to a given constant mixing angle φ 0 and given constant χ 0 2 and χ 0 3 are strong enough (θ ≈ 0), the medium is transparent and an incoming photon enters the medium. Switching the control field off (θ = π 2 ) results in the medium becoming opaque and the photon becomes trapped in a coherent superposition of the atomic coherences. The propagation and storing are described in terms of the polaritons Ψ 0 and Z 0 . The former polariton has evolved from being almost purely electromagnetic to become purely atomic while the latter one has not taken part in the evolution. The probability amplitudes that the photon has actually been trapped in the coherence σ bc and σ bd are respectively exp(iχ 0 2 ) cos φ 0 and exp(iχ 0 3 ) sin φ 0 . If now the control fields of the same mixing angle and the same phases are switched on again the photon will be released with certainty in the same state as the initial one. This is mutatis mutandis a repetition of the case of a usual Λ system discussed before.
If the control fields to be used at the release stage are characterized by the angles φ 1 , χ 1 2 and χ 1 3 it is proper to describe the further evolution in terms of the corresponding new polaritons Ψ 1 and Z 1 . The latter pair of polaritons is expressed in terms of the former ones as
Ψ 1 = {cos φ 1 cos φ 0 exp[i(χ 1 2 − χ 0 2 )] + sin φ 1 sin φ 0 exp[i(χ 1 3 − χ 0 3 )]}Ψ 0 + {cos φ 1 sin φ 0 exp[i(χ 1 2 − χ 0 2 )] − sin φ 1 cos φ 0 exp[i(χ 1 3 − χ 0 3 )]}Z 0 ,(9)Z 1 = {sin φ 1 cos φ 0 exp[i(χ 1 2 − χ 0 2 )] − cos φ 1 sin φ 0 exp[i(χ 1 3 − χ 0 3 )]}Ψ 0 + {sin φ 1 sin φ 0 exp[i(χ 1 2 − χ 0 2 )] + cos φ 1 cos φ 0 exp[i(χ 1 3 − χ 0 3 )]}Z 0 .
The polariton Ψ 1 will be released at turn into a photon, while the Z 1 will remain trapped. It follows from the form of the polaritons (Eq. (4)) that applying another pair of the control fields with φ 2 = π 2 − φ 1 , χ 2 2 = χ 1 2 + π and χ 2 3 = χ 1 3 leads to an exchange of the role of the polaritons: Z 1 becomes Ψ 2 which will be released and turn into a photon. The sets of control fields marked with (1) and(2) can be considered complementary in the sense that their subsequent application implies a certain photon release.
The above results are much more general than those of Ref. [13] in which the authors proposed a time splitting of a photon by first storing the pulse in one coherence σ bc (here this corresponds to φ 0 = 0), than by pumping a part of the excitation into the other coherence σ bd and finally by a pulse release from the latter coherence (here φ 1 = π 2 ). The pumping procedure (F-STIRAP) was to be separated from the release stages which required using light pulses transverse to both signal and control fields; the latter should be realized in ultracold gases. Here, due to using simultaneously pairs of control fields of the same shape pumping and releasing stages coincide.
III. BEAM SPLITTER AND HONG-OU-MANDEL -TYPE INTERFEROMETER
The above-described behavior of the atom+field allows one to use the medium and the system of control fields as an effective and flexible beam splitter working on stored light. The technical difference between the usual device and that of ours is that incoming photons may arrive from only one direction but at one or two different time instants. Also the released photons have the same direction but are separated in time. By changing the mixing angle φ 1 and the phases χ 0,1 2 and χ 0,1 3 of the control fields one can smoothly regulate the amplitudes and phases of the pulse components released at the two stages, which would correspond to changing the transmission and reflection rate of a usual beam splitter.
The operation of photon storing can be performed at two stages (input ports). First we trap a first portion of incoming photons by switching on and off a pair of control fields characterized by the parameters φ 0 ,χ 0 2 and χ 0 3 . After the this part of the trapping operation has been finished we may trap a second portion of photons by applying the control fields of the mixing angle π 2 − φ 0 , χ 0 2 + π and χ 0 3 ). Note that the second trapping operation does not affect the coherences due to the first one. Thus a photon from the first portion is described by the polariton field Ψ 0 and that from the second portion -by Z 0 . A special case is assumed in which φ 0 = 0 which means that photons have been first trapped in the coherence σ bc and next -in σ bd .
The release operation consists also of two stages (output ports) connected with two pairs of control fields separated in time, the first of which is characterized by the mixing angle φ 1 and χ 1 2 and χ 1 3 and the second by π 2 − φ 1 , χ 1 2 + π and χ 1 3 . The output includes thus two portions of photons, separated in time, and the photon release is complete. In the case of trapping and releasing of two photons we have a new realization of the Hong-Ou-Mandel effect. In its original formulation it concerns a 50-50 photon beam splitter with exactly one incoming photon in each of the two input ports (incoming photon directions). Due to interference phenomena the probability amplitudes for obtaining exactly one photon in each of the two output ports (outgoing photon directions) cancel out.
In our realization of the input ports single photons, separated in time, are trapped, one in each of the two superpositions of the atomic coherences, connected with complementary sets of control fields. In the output ports photons are released, again at two stages, from two other superpositions of the two coherences. The analogue of photon incoming from two orthogonal directions is their storing at two storing stages. The analogues of reflection and transmission on a traditional beam splitter are photon release at the first or second release stage while the parameters characterizing the control fields and the atomic system determine the analogues of the reflection and transmission coefficient and phase relations. Using the above-mentioned transformations of polaritons at the storing stage and having in mind that the incoming polaritons Ψ 0 , Z 0 (outgoing polaritons Ψ 1 , Z 1 ) become almost purely electric field operators ǫ 1 , ǫ 2 , (ǫ 3 , ǫ 4 ) for t → −∞ (t → ∞) we may give the net result for the transformation of the field operators
R 31 R 32 R 41 R 42 ǫ 1 ǫ 2 → ǫ 3 ǫ 4 .(10)
where
R 31 = cos φ 1 cos φ 0 exp[i(χ 1 2 − χ 0 2 )] + sin φ 1 sin φ 0 exp[i(χ 1 3 − χ 0 3 )], R 32 = cos φ 1 sin φ 0 exp[i(χ 1 2 − χ 0 2 )] − sin φ 1 cos φ 0 exp[i(χ 1 3 − χ 0 3 )], R 41 = sin φ 1 cos φ 0 exp[i(χ 1 2 − χ 0 2 )] − cos φ 1 sin φ 0 exp[iχ 1 3 − χ 0 3 )], R 42 = sin φ 1 sin φ 0 exp[i(χ 1 2 − χ 0 2 )] + cos φ 1 cos φ 0 exp[iχ 1 3 − χ 0 3 )].(11)
Let the respective wave packets of the stored excitation be f 1 (z) and f 2 (z), their shape being identical with the shape of the initial photon wavepackets and their localization depending on the time instants of the switch-off of the control fields. The maximum value (unity) of the overlap of the packets s ≡ f * j (z)f k (z)dz corresponds to the situation in which two photons with wavepackets of the same shape have been stored exactly at the same place inside the sample.
The field operators corresponding to the input ports are Ψ 0 (j) = f * j (z)Ψ 0 (z)dz, Z 0 (j) = f * j (z)Z 0 (z)dz. The corresponding operators in the output ports are Ψ 1 (j) = f * j (z)Ψ 1 (z)dz, Z 1 (j) = f * j (z)Z 1 (z)dz. The key commutation relations are [Ψ 0 (j), Ψ 0 † (j)] = [Z 0 (j), Z 0 † (j)] = g 2 L, [Ψ 0 (1), Ψ 0 † (2)] = [Z 0 (1), Z 0 † (2)] = g 2 Ls, with analogous relations for Ψ 1 and Z 1 .
The quantum state of the medium after trapping has been accomplished is constructed as due to a creation of two excitations due to complementary control fields and characterized by two possibly different wave packets
|ζ >= 1 g 2 L Ψ 0 † (1)Z 0 † (2)|0 >,(12)
where the "vacuum" state |0 > means all atoms in the state |b >). The state corresponding to two photons being released at the first stage (photon coalescence) is constructed as due to a creation of two excitations corresponding to the first release stage characterized by possibly different wave packets
|ζ 1 >= 1 g 2 L 1 1 + |s| 2 Ψ 1 † (1)Ψ 1 † (2)|0 >,(13)
where a normalizing factor for the two-particle states has been included. Note that the state corresponding to the photon coalescence at the second stage is constructed in an analogous way, using the Z operators instead of Ψ.
Calculating the projections with the use of the above mentioned commutation relations yields the probability amplitude of photon coalescence at the first release stage
< ζ 1 |ζ >= (1 + |s| 2 ){cos φ 0 cos φ 1 exp[−i(χ 0 2 − χ 1 2 )] + sin φ 0 sin φ 1 exp[−i(χ 0 3 − χ 1 3 )]} {sin φ 0 cos φ 1 exp[−i(χ 0 2 − χ 1 2 )] − cos φ 0 sin φ 1 exp[−i(χ 0 3 − χ 1 3 )]}(14)
If in particular φ 0 = 0, which means that exactly one photon has been trapped in each coherence, then the photon coalescence probability at the first release stage is
P coal (1) = 1 4 (1 + |s| 2 ) sin 2 2φ 1 .(15)
If the wave packets overlap (s = 1) we obtain for φ 1 = π 4 (equal amplitudes of the releasing control fields) that P coal (1) = 1 2 . The same value is obtained for the coalescence probability at the second stage. This means that the situation is impossible in which exactly one photon is released at each stage. This is analogue of a symmetric beam splitter with exactly one photon in each input port. A reduction of the overlap integral leads in a continuous way to the situation in which the probability of the latter situation grows to 1 2 ; this is an analogue of changing the length of the arms of the standard HOM interferometer. For example if the wave packets f 1,2 are Gaussians with standard deviations δ 1,2 separated by a distance a, the probability of releasing a single photon at each stage (photon noncoalescence), being our analogue of coincident registration of single photons in the usual realization, is
P noncoal = 1 2 [1 − 2 δ2 δ1 + δ1 δ2 exp(− a 2 2(δ 2 1 + δ 2 2 ) )].(16)
This probability as a function of the distance a exhibits a minimum called the Mandel dip, reaching zero for equal packet widths (see Fig. 2), which corresponds to an ideal photon coalescence. On the other hand in the case of coinciding packet centers (a = 0) the probability equals (δ 1 − δ 2 ) 2 /[2(δ 2 1 + δ 2 2 )] (see Fig. 3). Note that it is only for equal packets' widths that the coalescence probability drops to zero.
The effect can also be controlled by choosing the phases of the control field. From Eq. (14) it follows for example that for φ 0 = φ 1 = π 4 the coalescence probability at the first stage is
P coal (1) = 1 4 (1 + |s| 2 ) sin 2 (χ 0 2 − χ 1 2 − χ 0 3 + χ 1 3 ).(17)
Thus the probability of the photons' coalescence strongly depends not only on the shape of the wave packets and the relative amplitudes of the control fields but also, in an oscillatory way, on the relative phases of the latter.
IV. CONCLUSIONS
We have analyzed the propagation and storing of photons in an atomic medium in the tripod configuration. An adiabatic evolution of the system can be described in terms of a couple of polaritons. Using two complementary sets of control fields at the storing stage one can stop two photons arriving at different time instants. Applying two other complementary sets of the control fields at the release stage one retrieves two time-entangled photons. The field operators of the outgoing photons are expressed by those of the incoming ones, depending on the parameters of the control fields. We have thus obtained a kind of a beam splitter, operating on a stored light, in which the input and output ports correspond to photons arriving and leaving at different time instants. Our analogues of the transmission and reflection coefficients as well phase relations can be regulated on demand. We have demonstrated the effect of the Hong-Ou-Mandel -type interference and analyzed its dependence on the shapes and positions of the trapped photon wavepackets and on the parameters of the control fields.
FiguresFIG. 1 :
1Level and coupling scheme of the tripod system. FIG. 2: The probability of photon noncoalescence as a function of the packet's separation a for δ1 = δ2 solid line, and δ1 = 3δ2, dashed line. FIG. 3: The probability of photon noncoalescence as a function of the ratio δ2δ1 of the packets' widths for a = 0, solid line, and a = 2δ1, dashed line.
AcknowledgmentsThe authors thank Mr. Wojciech Wasilewski for stimulating discussions. The work has been supported by Polish budget funds allocated for the years 2005-2007 as a research grant No. 1 P03B 010 28. The subject belongs to the scientific program of the National Laboratory of AMO Physics in Toruń, Poland.
. M D Lukin, Rev. Mod. Phys. 75457M. D. Lukin, Rev. Mod. Phys. 75, 457 (2003).
. A André, M D Eisaman, R L Walsworth, A S Zibrov, M D Lukin, J. Phys. B: At. Mol. Opt. Phys. 38589A. André, M. D. Eisaman, R. L. Walsworth, A. S. Zibrov and M. D. Lukin, J. Phys. B: At. Mol. Opt. Phys. 38, S589 (2005).
. M Fleischhauer, A Imamoglu, J P Marangos, Rev. Mod. Phys. 77M. Fleischhauer, A. Imamoglu J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005)).
. M D Lukin, P R Hemmer, M Scully, Adv. At. Mol. Opt. Phys. 42M. D. Lukin, P.R. Hemmer, and M. O Scully, Adv. At. Mol. Opt. Phys. 42, 347,(2000).
. A Raczyński, J Zaremba, S Zielińska-Kaniasty, Phys. Rev. A. 6943801A.Raczyński, J. Zaremba, and S. Zielińska-Kaniasty, Phys. Rev. A 69, 043801 (2004).
. Xiong-Jun Liu, Hui Jing, Mo-Lin Ge, arXiv-quant-ph/040317Xiong-Jun Liu, Hui Jing, and Mo-Lin Ge, arXiv-quant-ph/040317 (2004).
. A Joshi, Min Xiao, Phys. Rev. A. 7141801A. Joshi and Min Xiao, Phys. Rev. A 71, 041801(R) (2005).
. A Raczyński, J Zaremba, S Zielińska-Kaniasty, Opt Commun. 217275A. Raczyński, J. Zaremba, and S. Zielińska-Kaniasty, Opt Commun. 217, 275 (2003).
. Pengbo Li, Ying Gu, Kai Wang, Qihuang Gong, Phys. Rev. A. 7332343Pengbo Li, Ying Gu, Kai Wang, and Qihuang Gong, Phys. Rev. A 73 032343 (2006).
. E Paspalakis, P L Knight, J. Opt. B: Quantum Semiclass. Opt. 4372E. Paspalakis and P. L. Knight, J. Opt. B: Quantum Semiclass. Opt., 4, S372 (2002).
. I E Mazets, arXiv:quant-ph/04071731I. E. Mazets, arXiv:quant-ph/0407173 v1 (2004).
. A Raczyński, M Rzepecka, J Zaremba, S Zielińska-Kaniasty, Opt. Commun. 26073A. Raczyński, M. Rzepecka, J. Zaremba, and S. Zielińska-Kaniasty. Opt. Commun. 260, 73 (2006).
. Tun Wang, M Koštrun, S F Yelin, Phys. Rev. A. 7053822Tun Wang, M. Koštrun, and S. F. Yelin, Phys. Rev. A 70 053822 (2004).
C C Gerry, P L Knight, Introductory Quantum Optics. CambridgeCambridge University PressC. C. Gerry and P. L. Knight, Introductory Quantum Optics, (Cambridge University Press, Cambridge, 2005).
. P Grangier, Nature. 419577P. Grangier, Nature 419, 577 (2002).
. C K Hong, Z Y Ou, L Mandel, Phys. Rev. Lett. 952044C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 95 2044 (1987).
. C Olindo, M A Sagioro, C H Monken, S Padua, Phys. Rev. A. 7343806C. Olindo, M. A. Sagioro, C. H. Monken and S. Padua, Phys. Rev. A 73, 043806 (2006).
. M Fleischhauer, M D Lukin, Phys. Rev. Lett. 845094M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094 (2000).
| [] |
[
"Residual Squeeze-and-Excitation Network for Fast Image Deraining",
"Residual Squeeze-and-Excitation Network for Fast Image Deraining"
] | [
"Jun Fu [email protected] \nUniversity of Science\nTechnology of China\n",
"Jianfeng Xu \nKDDI Research, Inc\n\n",
"Kazuyuki Tasaka [email protected] \nKDDI Research, Inc\n\n",
"Zhibo Chen [email protected] \nUniversity of Science\nTechnology of China\n"
] | [
"University of Science\nTechnology of China",
"KDDI Research, Inc\n",
"KDDI Research, Inc\n",
"University of Science\nTechnology of China"
] | [] | Image deraining is an important image processing task as rain streaks not only severely degrade the visual quality of images but also significantly affect the performance of high-level vision tasks. Traditional methods progressively remove rain streaks via different recurrent neural networks. However, these methods fail to yield plausible rain-free images in an efficient manner. In this paper, we propose a residual squeeze-and-excitation network called RSEN for fast image deraining as well as superior deraining performance compared with stateof-the-art approaches. Specifically, RSEN adopts a lightweight encoder-decoder architecture to conduct rain removal in one stage. Besides, both encoder and decoder adopt a novel residual squeezeand-excitation block as the core of feature extraction, which contains a residual block for producing hierarchical features, followed by a squeezeand-excitation block for channel-wisely enhancing the resulted hierarchical features. Experimental results demonstrate that our method can not only considerably reduce the computational complexity but also significantly improve the deraining performance compared with state-of-the-art methods. arXiv:2006.00757v1 [eess.IV] 1 Jun 2020 | null | null | 219,177,528 | 2006.00757 | 340898e3326b74478849f8373fd124fa3abcb42d |
Residual Squeeze-and-Excitation Network for Fast Image Deraining
Jun Fu [email protected]
University of Science
Technology of China
Jianfeng Xu
KDDI Research, Inc
Kazuyuki Tasaka [email protected]
KDDI Research, Inc
Zhibo Chen [email protected]
University of Science
Technology of China
Residual Squeeze-and-Excitation Network for Fast Image Deraining
Image deraining is an important image processing task as rain streaks not only severely degrade the visual quality of images but also significantly affect the performance of high-level vision tasks. Traditional methods progressively remove rain streaks via different recurrent neural networks. However, these methods fail to yield plausible rain-free images in an efficient manner. In this paper, we propose a residual squeeze-and-excitation network called RSEN for fast image deraining as well as superior deraining performance compared with stateof-the-art approaches. Specifically, RSEN adopts a lightweight encoder-decoder architecture to conduct rain removal in one stage. Besides, both encoder and decoder adopt a novel residual squeezeand-excitation block as the core of feature extraction, which contains a residual block for producing hierarchical features, followed by a squeezeand-excitation block for channel-wisely enhancing the resulted hierarchical features. Experimental results demonstrate that our method can not only considerably reduce the computational complexity but also significantly improve the deraining performance compared with state-of-the-art methods. arXiv:2006.00757v1 [eess.IV] 1 Jun 2020
Introduction
Outdoor images taken on rainy days often contain various rain streaks. These rain streaks not only cause noticeable degradation in scene visibility but also significantly impair the performance of advanced visual tasks, such as pedestrian detection [Liu et al., 2019], object tracking [Redmon and Farhadi, 2018], and autonomous vehicles [Zang et al., 2019]. Therefore, it is important and necessary to develop image deraining algorithms.
In general, a rainy image (I) can be regarded as a linear combination of a clean background (B) and a rain streak layer (R): I = B + R.
(1) Image deraining aims to recover the clean background from the observed rainy image. However, due to lacking information on the rain streak layer, rain removal is a serious ill-posed problem. In the past few decades, image deraining has received considerable attention from industry and academia. Existing methods can be divided into two categories, including model-driven methods and data-driven methods. Modeldriven methods can be further divided into filter based ones [Xu et al., 2012;Zheng et al., 2013;Ding et al., 2016;Kim et al., 2013] and prior based ones [Luo et al., 2015;Li et al., 2016;Chang et al., 2017]. Considering rain removal as a task of signal filtering, filter based ones utilize physical properties of rain streaks and edge-preserving filters to obtain rain-free images. However, prior based ones formulate rain removal as an optimization problem and utilize various handcrafted image priors to regularize the solution space. Different from model-driven methods, datadriven methods regard rain removal as a task of learning a non-linear function mapping the observed rainy image into the clean background. Motivated by the unprecedented success of deep learning, they model the mapping function with various convolution neural networks (CNNs). Most of them progressively remove rain streaks via different re-current neural networks [Li et al., 2018;Yang and Lu, 2019]. Additionally, adversarial learning-based methods are proposed to prevent derained images from the blur artifact.
However, existing methods suffer from two key limitations despite achieving deraining performance boost. On the one hand, model-driven methods tend to leave some rain streaks or introduce the blur artifact in derained images. This is because physical properties of rain streaks and handcrafted image priors are easily violated on real-world examples where rain streaks are far more complex than modeled. Furthermore, these methods involve heuristic parameter tuning and expensive computation. Data-driven methods, on the other hand, fail to yield plausible rain-free images in an efficient manner (as shown in Fig. 1). Although some lightweight methods [Fan et al., 2018;Fu et al., 2019] have been proposed to improve the computational efficiency, they result in a significant decrease in the deraining performance.
Motivated by addressing above two issues, we propose a residual squeeze-and-excitation network called RSEN for fast image deraining as well as superior deraining performance compared with state-of-the-art approaches. Unlike prevalent recurrent networks reusing flat network structures, RSEN adopts a lightweight encoder-decoder architecture to conduct rain removal in one stage. Besides, both encoder and decoder adopt a novel residual squeeze-and-excitation block as the core of feature extraction, which contains a residual block for producing hierarchical features, followed by a squeeze-andexcitation block for channel-wisely enhancing the resulted hierarchical features.
Main contributions of this paper are listed as follows:
• We propose a novel residual squeeze-and-excitation network called RSEN for image rain removal. It adopts a lightweight encoder-decoder architecture and is capable to effectively remove rain streaks while well preserving texture details. • We propose to incorporate a residual squeeze-andexcitation block in our network, which can not only generate channel-wisely enhanced hierarchical features but also well benefit gradient propagation. • Experimental results show that our proposed method can not only considerably reduce the computational complexity but also significantly improve the deraining performance compared with state-of-the-art methods. The remainder of this paper is organized as follows. Section 2 discusses the related works. Section 3 introduces the proposed network. Then, performance evaluation and comparison are presented in Section 4. Finally, Section 5 concludes the paper and discusses future work.
2 Related Work 2.1 Image Deraining Image deraining, a highly ill-posed problem, has drawn increasingly more attention from industry and academia over the past few decades. Existing works can be categorized into two classes, i.e., model-driven methods and data-driven methods.
Model-driven methods can be further divided into filter based ones and prior based ones. Considering rain removal as a task of signal filtering, filter based ones utilize physical properties of rain streaks and edge-preserving filters to obtain rain-free images. Specifically, [Xu et al., 2012] utilize guided filter [He et al., 2010] to remove rain streaks based on the chromatic and brightness property of rain streaks. [Zheng et al., 2013], [Kim et al., 2013], and [Ding et al., 2016] boost the deraining performance via multi-guided filter, non-local means filtering, and guided L 0 smoothing filter, respectively. However, prior based ones formulate rain removal as an optimization problem and employ various handcrafted image priors to regularize the solution space. These image priors include sparse-coding prior [Luo et al., 2015], Gaussian prior [Li et al., 2016] and low-rank prior [Chang et al., 2017].
Different from model-driven methods, data-driven approaches employ various convolution neural networks to automatically learn a non-linear mapping function between the rainy image and the rain-free image from data. More specifically, [Fu et al., 2017a] design a shallow convolution neural network to address rain removal and improve the deraining performance by a deeper network [Fu et al., 2017b]. ] present a multi-task framework, which simultaneously deals with rain detection and rain removal. Unlike onestage deraining methods, [Li et al., 2018] remove rain streaks stage by stage via a recurrent squeeze-and-excitation context aggregation network. Recently, [Wang et al., 2019] present a spatial attentive single-image deraining approach and [Yang and Lu, 2019] introduce a recurrent hierarchy enhancement network. To preserve more texture details of derained images, adversarial learning-based methods are proposed. In addition, some lightweight methods [Fu et al., 2019;Fan et al., 2018] are dedicated to reduce the computational complexity.
However, existing methods suffer from two key limitations despite achieving deraining performance boost. On the one hand, model-driven methods suffer from under-/overderaining on real-world examples where rain streaks are far more complex than modeled. On the other hand, existing neural networks fail to produce plausible rain-free images in an efficient manner due to the complex framework. Some lightweight networks attempt to improve computational efficiency but at the cost of obvious performance degradation.
Convolution Neural Network
Recent years have witnessed the convolution neural network goes increasingly deeper (e.g., VGGNet [Simonyan and Zisserman, 2014]). Deep neural networks typically are superior to shallow networks while meeting more challenges in convergence and generalization. To address this problem, [He et al., 2016] first propose residual learning, which can benefit gradient propagation and accelerate convergence. In addition, diverse attention mechanisms are proposed to improve the capability of neural networks, such as channel-wise attention , self-attention mechanism [Vaswani et al., 2017] and spatial attention [Woo et al., 2018]. 3 Proposed Method
Problem Formulation
Considering rain streak accumulation and overlapping, [Yang et al., 2017] proposes a rain model as follows:
I = α(B + n t=1 S t M ) + (1 − α)A,(2)
where each S t denotes a layer of rain streaks that have the same direction, n denotes the total number of rain streak layers, M records the locations of S t , A represents the global atmospheric light, and α is the atmospheric transmission. Unlike the above model requiring rain detection, [Li et al., 2018] present a simpler model, i.e., dividing the captured rainy scene into the combination of several rain streak layers and a rain-free background. Thus, the rain model can be reformulated as follows:
I = B + n t=1 R t ,(3)
where R t and n represent the t-th rain streak layer that consists of one kind of rain streaks and the maximum number of rain streak layers, respectively. In this paper, we further simplify the rain model as follows:
I = B + R.(4)
As Eq. 4 implied, we can obtain the rain-free background by subtracting rain streaks R from the rainy image I. Therefore, we formulate the rain removal as a task of learning a nonlinear mapping function between the rainy image and rain streaks. Inspired by the recent success of deep learning, we propose a residual squeeze-and-excitation network call RSEN to model the non-linear mapping function. As illustrated in Fig. 2, RSEN adopts a commonly used encoder-decoder architecture. The detailed design of this architecture is introduced and explained next.
Residual Squeeze-and-Excitation Network
The encoder-decoder network typically adopts a symmetric convolution neural network architecture composed of an encoder and a decoder. The encoder transforms the input data into feature maps with smaller spatial sizes and more channels while the decoder transforms the resulted feature maps back to the shape of the input. In addition, skip connections are widely used in this architecture because they can aggregate features at multiple levels and accelerate convergence. Compared with flat architectures, the encoder-decoder network has shown its superiority in many low-level vision tasks, such as image deblurring [Tao et al., 2018] and image inpainting [Nazeri et al., 2019]. However, to accommodate specific tasks, such a network needs to be carefully designed.
For the task of image deraining, we take the following three key aspects into account in our design. First, the receptive field needs to be large enough to handle heavy rain streaks. To this end, a naive approach is to stack more levels or adding more convolution layers at each level for encoder/decoder modules. However, this strategy will result in a sharp increase in computational complexity and parametric size. Furthermore, such a deep neural architecture generally suffers from a low speed of convergence. Second, aside from global skip connections, local skip connections are also beneficial to gradient propagation and accelerate convergence. Third, according to the experimental results of [Li et al., 2018] and [Yang et al., 2017], the channel-wise attention mechanism is a promising alternative to improve the deraining performance.
Based on the aforementioned analysis, we adapt the encoder-decoder network into our task as follows. First, we propose a residual squeeze-and-excitation block (RSEBlock), which contains a residual block for producing hierarchical features via local skip connections, followed by a squeezeand-excitation block for channel-wisely enhancing the resulted hierarchical features. It is worth noting that we remove the batch normalization [Ioffe and Szegedy, 2015] from the original residual blocks in ResNet [He et al., 2016] accord-ing to [Tao et al., 2018] and our experimental results. Second, both encoder and decoder employ the RSEBlock as their core of feature extraction instead of conventional convolution layers. Third, considering that the spatial size of the middle feature map needs to be large enough to keep sufficient spatial information for reconstruction, we only stack two levels in the encoder and the decoder. Finally, to enlarge the size of the receptive field, we stack three RSEBlocks after the encoder.
The proposed network can be mathematically expressed as The implementation details of our proposed RSEN are specified as follows. In our proposed architecture, there are 1 InBlock, 2 EBlocks, followed by 1 Bottleneck, 2 DBlocks, and 1 OutBlock, as illustrated in Fig. 2. InBlock composed of a convolution layer and a RSEBlock transforms the input 3-channel rainy image into a 64-channel feature map. Each EBlock adopts the same structure as InBlock while doubling the number of kernels in the previous layer and downsampling feature maps by half. DBlocks is symmetric to EBlock. It is designed to double the spatial size of feature maps and halve channels, composed of a RSEBlock and a upsampling block. In the upsampling block, a point-wise convolution is used to increase the channel dimension of the input 4 times, followed by a pix-shuffle layer [Shi et al., 2016] whose scale factor is set to 2. Additionally, 2 point-wise convolutions are designed for skip connections. The Bottleneck contains 3 RESBlocks that have the same number of channels. OutBlock takes previous feature maps as input and generates estimated rain streaks. All squeeze-and-excitation blocks squeeze the channel dimension of the input feature map to 6. The stride size for the convolution layer in EBlocks is 2, while all others are 1. Rectified Linear Units (ReLU) are used as the activation function for all layers, and all kernel sizes are set to 3.
f = N et E (I; θ E ), h = N et B (f ; θ B ), B = I − N et D (h; θ D ),(5)
Loss Function
The loss function is defined as the mean square error (MSE) between the derained image and its corresponding groundtruth, which can be formulated as follows:
L = 1 2HW C i j k ||B i,j,k − B i,j,k || 2 2 ,(6)
where H, W , and C represent the height, width, and channel number of the rain-free image.B and B denote the restored rain-free image and the groundtruth, respectively.
Experiments
Dataset and Evaluation Metrics
We choose four synthetic datasets and a real-world dataset for the deraining experiment. The synthetic datasets include Rain100H, Rain800, Rain1200 and Rain1400. Rain100H is synthesized by [Yang et al., 2017]. Rain800 is obtained from . Rain1200 is provided by [Zhang and Patel, 2018] while Rain1400 comes from [Fu et al., 2017b]. The real-world dataset is constructed by [Yang et al., 2017]. Rainy images in these datasets are diverse in terms of content as well as the type of rain streaks. More details of these datasets are listed in Table 1.
Deraining performances on the synthetic datasets are evaluated in terms of peak signal-to-noise (PSNR) [Huynh-Thu and Ghanbari, 2008] and structural similarity (SSIM) [Wang et al., 2004]. Due to lacking the groundtruth of real-world rainy images, performances of different methods on the realworld dataset are evaluated visually. We compare RSEN with state-of-the-art methods including DSC [Luo et al., 2015], LP [Li et al., 2016], JCAS [Gu et al., 2017], DDN [Fu et al., 2017b], JORDER [Yang et al., 2017]
Training Details
We implement our model using the PyTorch [Paszke et al., 2017] framework. During training, 4 rain-free/rainy patch pairs with a size of 256 × 256 are randomly generated from input image pairs per iteration. All trainable variables of the proposed RSEN are initialized by the default initializer and optimized via the Adam optimizer [Kingma and Ba, 2014]. β 1 , β 2 and of the Adam optimizer are set to 0.9, 0.999, and 10 −8 respectively. We train RSEN on a NVIDIA Geforce GTX 1080Ti GPU for 700 epochs. The learning rate is initialized as 10 −4 and decayed in half every 150 epochs. For fair comparison with existing methods, we only use rainfree/rainy image pairs for training without other additional labels and any data augmentation. Table 2 presents quantitative results of different methods on four synthetic datasets. We can observe that the proposed method achieves substantial improvements over state-of-theart approaches in terms of both PSNR and SSIM. Specifically, the proposed method obtains significant improvements by PSNR of 2.80 dB, 0.62 dB, 0.55 dB, 0.31 dB and SSIM of 0.014, 0.002, 0.09, 0.06 on Rain100H, Rain800, Rain1200, and Rain1400, respectively. Average running time for a rainy image with a size of 512 × 512 and the parametric size of each method are also chosen for comparison. It can be observed that the proposed method significantly reduces the computational complexity in spite of using relatively more parameters compared with other methods. For instance, RSEN reaches around 4, 10 and 50 times faster than ReHEN, PReNET, and SPANet, respectively. This confirms the superiority of our proposed architecture. Fig. 4 visually compares the proposed method with three recent state-of-the-art methods. We can observe that the proposed method achieves the best visual quality. Specifically, compared with other methods, our proposed method avoids leaving obvious artifacts (e.g., rain streaks and the blur artifact) in derained images as well as preserving more structural information.
Results on Synthetic Dataset
Results on Real-World Dataset
To assess the generalization of the proposed method, we also evaluate the proposed method on a real-world dataset. For fair comparison, all the methods employ the pre-trained model trained on the Rain100H dataset to remove rain streaks from real-world rainy images. As demonstrated in Fig. 5, the proposed method produces more natural and pleasant derained images compared with four state-of-the-art methods. Specifically, in our experiment, DDN fails to remove rain streaks from real-world rainy images in most cases while JORDER is prone to dim and blur the details of the derained results. Compared with PReNET and ReHEN, the proposed method can more effectively remove rain streaks from real-world rainy images and restore more texture details. Table 3 and Fig. 3 show the ablation investigation on the effects of the skip connection (SKip), the residual connection (RES), and the squeeze-and-excitation block (SE). As can be observed, both Skip and RES are beneficial for convergence and boosting deraining performance. Specifically, the Skip brings gains by PSNR of 1.74 dB, 5.02 dB, 0.24 dB and 0.09 dB on Rain100H, Rain800, Rain1200, and Rain1400, respectively, while the RES obtains 0.84 dB, 0.43 dB, 0.23 dB, and 0.09 dB gains. The SE is useful for most datasets except for Rain100H. This may be because synthesized rainy images with five streak directions in Rain100H are more complex than other datasets with one streak direction. Hence, we disable the SE in all the RSEBlocks when performing rain removal on Rain100H.
Ablation Study
Conclusion
In this paper, we propose a novel residual squeeze-andexcitation network called RSEN for fast image deraining as well as superior deraining performance compared with existing approaches. Specifically, RSEN adopts a lightweight encoder-decoder architecture to conduct rain removal in one stage. Besides, both encoder and decoder adopt a novel residual squeeze-and-excitation block as the core of feature extraction, where a residual block and a squeeze-andexcitation block are used to produce and channel-wisely enhance hierarchical features, respectively. Experimental results demonstrate that our method can not only considerably reduce the computational complexity but also achieve significant improvements in the deraining performance compared with state-of-the-art methods. In the future, we plan to extend our work to deal with video deraining, e.g., incorporating an extra module in our network to capture temporal information.
Figure 1 :
1The PSNR-Runtime trade-off plot on the Rain1200 dataset. Compared to five state-of-the-art competitors including DDN, DID MDN, JORDER, RESCAN, and ReHEN, RSEN achieves superior quality and lower computational complexity.
Figure 2 :
2The whole framework of the proposed residual squeeze-and-excitation network. I, B, Conv, and SE represent the input rainy image, the clean background, the conventional convolution and the squeeze-and-excitation block, respectively.
, DID-MDN [Zhang and Patel, 2018], DualCNN [Pan et al., 2018], RESCAN [Li et al., 2018], ID CGAN [Zhang et al., 2019], ReHEN [Yang and Lu, 2019], SPANET [Wang et al., 2019], and PReNET [Ren et al., 2019].
Figure 3 :
3Training convergence analysis on PSNR of RSEN with different modules.
Figure 4 :Figure 5 :
45Derained results of RESCAN, JORDER, PReNET, and the proposed method on synthetic rainy images. Derained results of DDN, JORDER, PReNET, ReHEN, and the proposed method on real-world rainy images.
where N et E , N et B and N et D are encoder, bottleneck, and decoder CNNs with parameters θ E , θ B and θ D . f and h are the intermediate feature maps.B is the recovered rain-free image.
Table 1 :
1Details of synthetic and real-world datasets. Values in each
column of the training set and testing set indicate the number of rain-
free/rainy image pairs with the exception of the real-world set with
rainy images only.
Datasets
Training Set Testing Set
Label
Rain100H
1800
100 rain mask/rain map
Rain800
700
100
-
Rain1200
12000
1200 rain mask/rain map
Rain1400
12600
1400
-
Real-world set
-
13
-
Table 2 :
2Parametric size, running time, and average PSNR and SSIM values on four synthetic datasets. The value with red bold font denotes ranking the first place in this column while value with blue font is the second place. It is worth noting that both PSNR and SSIM are calculated in the RGB color space.Time (s)
Rain100H
Rain800
Rain1200
Rain1400
Methods
Params 512x512 PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM
Rainy
-
-
12.13 0.349 21.16 0.652 21.15 0.778 23.69 0.757
DSC(ICCV'15)
-
-
15.66 0.544 18.56 0.599 21.44 0.789 22.03 0.799
LP(CVPR'16)
-
-
14.26 0.423 22.27 0.741 22.75 0.835 25.64 0.836
JCAS(ICCV'17)
-
-
13.65 0.459 22.19 0.766 27.91 0.778 28.77 0.819
DDN(CVPR'17)
57,369
0.547
24.95 0.781 21.16 0.732 27.33 0.898 27.61 0.901
JORDER(CVPR'17)
369,792
0.268
22.15 0.674 22.24 0.776 24.32 0.862 27.55 0.853
DID-MDN(CVPR'18)
372,839
0.532
17.39 0.612 21.89 0.795 27.95 0.908 27.99 0.869
DualCNN(CVPR'18)
687,008
20.19
14.23 0.468 24.11 0.821 23.38 0.787 24.98 0.838
RESCAN(ECCV'18)
134,424
0.281
26.45 0.846 24.09 0.841 29.95 0.884 28.57 0.891
ID CGAN(TCSVT'19)
817,824
0.286
14.16 0.607 22.73 0.817 23.32 0.803 21.93 0.784
ReHEN(MM'19)
298,263
0.181
27.97 0.864 26.96 0.854 32.64 0.914 31.33 0.918
SPANet(CVPR'19)
283,716
2.301
-
-
-
-
28.64
0.91
-
-
PReNET(CVPR'19)
168,693
0.461
28.06 0.888
-
-
-
-
30.73 0.918
Ours
4,851,373
0.040
30.86 0.902 27.58 0.856 33.19 0.923 31.64 0.924
Table 3 :
3Ablation study on different modules of the proposed RSEN. The CoarseNet denotes the architecture without any enhancement tricks. CoarseNet + Skip + RES 30.86 0.902 26.90 0.854 33.11 0.923 31.54 0.924 CoarseNet + Skip + RES + SE 30.30 0.893 27.58 0.856 33.19 0.923 31.64 0.924Rain100H
Rain800
Rain1200
Rain1400
Method
PSNR SSIM PSNR SSIM PSNR SSIM PSNR SSIM
CoarseNet
28.28 0.822 21.42 0.647 32.64 0.908 31.36 0.917
CoarseNet + Skip
30.02 0.888 26.47 0.845 32.88 0.919 31.45 0.922
(c) Rain1200
(d) Rain1400
(a) Rain800
(b) Rain100H
Transformed low-rank model for line pattern noise removal. Chang , Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer VisionReferences [Chang et al., 2017] Yi Chang, Luxin Yan, and Sheng Zhong. Transformed low-rank model for line pattern noise removal. In Proceedings of the IEEE International Conference on Computer Vision, pages 1726-1734, 2017.
Single image rain and snow removal via guided l0 smoothing filter. [ Ding, Multimedia Tools and Applications. 75[Ding et al., 2016] Xinghao Ding, Liqin Chen, Xianhui Zheng, Yue Huang, and Delu Zeng. Single image rain and snow removal via guided l0 smoothing filter. Multime- dia Tools and Applications, 75(5):2697-2712, 2016.
Residual-guide feature fusion network for single image deraining. [ Fan, arXiv:1804.07493arXiv preprint[Fan et al., 2018] Zhiwen Fan, Huafeng Wu, Xueyang Fu, Yue Hunag, and Xinghao Ding. Residual-guide feature fusion network for single image deraining. arXiv preprint arXiv:1804.07493, 2018.
Clearing the skies: A deep network architecture for single-image rain removal. [ Fu, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition26Removing rain from single images via a deep detail network[Fu et al., 2017a] Xueyang Fu, Jiabin Huang, Xinghao Ding, Yinghao Liao, and John Paisley. Clearing the skies: A deep network architecture for single-image rain re- moval. IEEE Transactions on Image Processing, 26(6):2944-2956, 2017. [Fu et al., 2017b] Xueyang Fu, Jiabin Huang, Delu Zeng, Yue Huang, Xinghao Ding, and John Paisley. Removing rain from single images via a deep detail network. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3855-3863, 2017.
Lightweight pyramid networks for image deraining. [ Fu, IEEE transactions on neural networks and learning systems. [Fu et al., 2019] Xueyang Fu, Borong Liang, Yue Huang, Xinghao Ding, and John Paisley. Lightweight pyramid networks for image deraining. IEEE transactions on neural networks and learning systems, 2019.
Joint convolutional analysis and synthesis sparse representation for single image layer separation. [ Gu, Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer Vision[Gu et al., 2017] Shuhang Gu, Deyu Meng, Wangmeng Zuo, and Lei Zhang. Joint con- volutional analysis and synthesis sparse representation for single image layer sepa- ration. In Proceedings of the IEEE International Conference on Computer Vision, pages 1708-1716, 2017.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionSpringerEuropean conference on computer visionet al., 2010] Kaiming He, Jian Sun, and Xiaoou Tang. Guided image filtering. In European conference on computer vision, pages 1-14. Springer, 2010. [He et al., 2016] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770-778, 2016.
Squeeze-and-excitation networks. Hu, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionHu et al., 2018] Jie Hu, Li Shen, and Gang Sun. Squeeze-and-excitation networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7132-7141, 2018.
Scope of validity of psnr in image/video quality assessment. -Thu , Ghanbari ; Quan Huynh, - Thu, Mohammed Ghanbari, Electronics letters. 4413-Thu and Ghanbari, 2008] Quan Huynh-Thu and Mohammed Ghanbari. Scope of validity of psnr in image/video quality assessment. Electronics letters, 44(13):800-801, 2008.
Batch normalization: Accelerating deep network training by reducing internal covariate shift. Sergey Ioffe, Christian Szegedy, arXiv:1502.03167arXiv preprintand Szegedy, 2015] Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015.
Single-image deraining using an adaptive nonlocal means filter. [ Kim, arXiv:1412.69802013 IEEE International Conference on Image Processing. BaIEEEarXiv preprintAdam: A method for stochastic optimization[Kim et al., 2013] Jin-Hwan Kim, Chul Lee, Jae-Young Sim, and Chang-Su Kim. Single-image deraining using an adaptive nonlocal means filter. In 2013 IEEE In- ternational Conference on Image Processing, pages 914-917. IEEE, 2013. [Kingma and Ba, 2014] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
Rain streak removal using layer priors. [ Li, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognition[Li et al., 2016] Yu Li, Robby T Tan, Xiaojie Guo, Jiangbo Lu, and Michael S Brown. Rain streak removal using layer priors. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2736-2744, 2016.
Recurrent squeeze-and-excitation context aggregation net for single image deraining. [ Li, Proceedings of the European Conference on Computer Vision (ECCV). the European Conference on Computer Vision (ECCV)[Li et al., 2018] Xia Li, Jianlong Wu, Zhouchen Lin, Hong Liu, and Hongbin Zha. Re- current squeeze-and-excitation context aggregation net for single image deraining. In Proceedings of the European Conference on Computer Vision (ECCV), pages 254-269, 2018.
High-level semantic feature detection: A new perspective for pedestrian detection. [ Liu, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition[Liu et al., 2019] Wei Liu, Shengcai Liao, Weiqiang Ren, Weidong Hu, and Yinan Yu. High-level semantic feature detection: A new perspective for pedestrian detection. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5187-5196, 2019.
Removing rain from a single image via discriminative sparse coding. Proceedings of the IEEE International Conference on Computer Vision. the IEEE International Conference on Computer Visionet al., 2015] Yu Luo, Yong Xu, and Hui Ji. Removing rain from a single image via discriminative sparse coding. In Proceedings of the IEEE International Conference on Computer Vision, pages 3397-3405, 2015.
[ Nazeri, arXiv:1901.00212Generative image inpainting with adversarial edge learning. arXiv preprint[Nazeri et al., 2019] Kamyar Nazeri, Eric Ng, Tony Joseph, Faisal Qureshi, and Mehran Ebrahimi. Edgeconnect: Generative image inpainting with adversarial edge learning. arXiv preprint arXiv:1901.00212, 2019.
Learning dual convolutional neural networks for low-level vision. Pan, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognition[Pan et al., 2018] Jinshan Pan, Sifei Liu, Deqing Sun, Jiawei Zhang, Yang Liu, Jimmy Ren, Zechao Li, Jinhui Tang, Huchuan Lu, Yu-Wing Tai, et al. Learning dual convo- lutional neural networks for low-level vision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3070-3079, 2018.
Automatic differentiation in pytorch. [ Paszke, [Paszke et al., 2017] Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017.
Progressive image deraining networks: a better and simpler baseline. Farhadi ; Joseph Redmon, Ali Redmon, Farhadi ; Ren, arXiv:1804.02767Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern RecognitionarXiv preprintYolov3: An incremental improvement[Redmon and Farhadi, 2018] Joseph Redmon and Ali Farhadi. Yolov3: An incremen- tal improvement. arXiv preprint arXiv:1804.02767, 2018. [Ren et al., 2019] Dongwei Ren, Wangmeng Zuo, Qinghua Hu, Pengfei Zhu, and Deyu Meng. Progressive image deraining networks: a better and simpler baseline. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3937-3946, 2019.
Real-time single image and video super-resolution using an efficient sub-pixel convolutional neural network. Shi, Proceedings of the IEEE conference on computer vision and pattern recognition. the IEEE conference on computer vision and pattern recognitionShi et al., 2016] Wenzhe Shi, Jose Caballero, Ferenc Huszár, Johannes Totz, An- drew P Aitken, Rob Bishop, Daniel Rueckert, and Zehan Wang. Real-time single image and video super-resolution using an efficient sub-pixel convolutional neural network. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1874-1883, 2016.
Very deep convolutional networks for large-scale image recognition. Zisserman ; Karen Simonyan, Andrew Simonyan, Zisserman, arXiv:1409.1556arXiv preprint[Simonyan and Zisserman, 2014] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.
Scale-recurrent network for deep image deblurring. [ Tao, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. Vaswani et al., 2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhinthe IEEE Conference on Computer Vision and Pattern RecognitionAdvances in neural information processing systems[Tao et al., 2018] Xin Tao, Hongyun Gao, Xiaoyong Shen, Jue Wang, and Jiaya Jia. Scale-recurrent network for deep image deblurring. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 8174-8182, 2018. [Vaswani et al., 2017] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998-6008, 2017.
Image quality assessment: from error visibility to structural similarity. [ Wang, IEEE transactions on image processing. 134[Wang et al., 2004] Zhou Wang, Alan C Bovik, Hamid R Sheikh, Eero P Simoncelli, et al. Image quality assessment: from error visibility to structural similarity. IEEE transactions on image processing, 13(4):600-612, 2004.
Spatial attentive single-image deraining with a high quality real rain dataset. [ Wang, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition[Wang et al., 2019] Tianyu Wang, Xin Yang, Ke Xu, Shaozhe Chen, Qiang Zhang, and Rynson WH Lau. Spatial attentive single-image deraining with a high quality real rain dataset. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 12270-12279, 2019.
Joon-Young Lee, and In So Kweon. Cbam: Convolutional block attention module. [ Woo, Proceedings of the European Conference on Computer Vision (ECCV). the European Conference on Computer Vision (ECCV)[Woo et al., 2018] Sanghyun Woo, Jongchan Park, Joon-Young Lee, and In So Kweon. Cbam: Convolutional block attention module. In Proceedings of the European Con- ference on Computer Vision (ECCV), pages 3-19, 2018.
Removing rain and snow in a single image using guided filter. [ Xu, IEEE International Conference on Computer Science and Automation Engineering. 2IEEE[Xu et al., 2012] Jing Xu, Wei Zhao, Peng Liu, and Xianglong Tang. Removing rain and snow in a single image using guided filter. In 2012 IEEE International Confer- ence on Computer Science and Automation Engineering (CSAE), volume 2, pages 304-307. IEEE, 2012.
Single image deraining via recurrent hierarchy enhancement network. ; Lu, Hong Yang, Lu, Proceedings of the 27th ACM International Conference on Multimedia. the 27th ACM International Conference on MultimediaACMand Lu, 2019] Youzhao Yang and Hong Lu. Single image deraining via recur- rent hierarchy enhancement network. In Proceedings of the 27th ACM International Conference on Multimedia, pages 1814-1822. ACM, 2019.
Deep joint rain detection and removal from a single image. [ Yang, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. the IEEE Conference on Computer Vision and Pattern Recognition[Yang et al., 2017] Wenhan Yang, Robby T Tan, Jiashi Feng, Jiaying Liu, Zongming Guo, and Shuicheng Yan. Deep joint rain detection and removal from a single image. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recogni- tion, pages 1357-1366, 2017.
The impact of adverse weather conditions on autonomous vehicles: How rain, snow, fog, and hail affect the performance of a self-driving car. [ Zang, Vishwanath Sindagi, and Vishal M Patel. Image deraining using a conditional generative adversarial network. IEEE transactions on circuits and systems for video technology. 14Proceedings of the IEEE conference on computer vision and pattern recognition[Zang et al., 2019] Shizhe Zang, Ming Ding, David Smith, Paul Tyler, Thierry Rako- toarivelo, and Mohamed Ali Kaafar. The impact of adverse weather conditions on autonomous vehicles: How rain, snow, fog, and hail affect the performance of a self-driving car. IEEE Vehicular Technology Magazine, 14(2):103-111, 2019. [Zhang and Patel, 2018] He Zhang and Vishal M Patel. Density-aware single image de-raining using a multi-stream dense network. In Proceedings of the IEEE confer- ence on computer vision and pattern recognition, pages 695-704, 2018. [Zhang et al., 2019] He Zhang, Vishwanath Sindagi, and Vishal M Patel. Image de- raining using a conditional generative adversarial network. IEEE transactions on circuits and systems for video technology, 2019.
Single-image-based rain and snow removal using multi-guided filter. [ Zheng, International Conference on Neural Information Processing. [Zheng et al., 2013] Xianhui Zheng, Yinghao Liao, Wei Guo, Xueyang Fu, and Xing- hao Ding. Single-image-based rain and snow removal using multi-guided filter. In International Conference on Neural Information Processing, pages 258-265.
. Springer, Springer, 2013.
| [] |
[] | [
"Emilio Santos \nDepartamento de Física\nUniversidad de Cantabria. Santander\nSpain\n"
] | [
"Departamento de Física\nUniversidad de Cantabria. Santander\nSpain"
] | [] | Quartic gravity theory is considered with the Einstein-Hilbert Lagrangean R + aR 2 + bR µν R µν , R µν being Ricci´s tensor and R the curvature scalar. The parameters a and b are taken of order 1 km 2 . Arguments are give which suggest that the effective theory so obtained may be a plausible approximation of a viable theory. A numerical integration is performed of the field equations for a free neutron gas. The result is that the star mass increases with increasing central density until about 1 solar mass and then decreases. The baryon number increases monotonically, which suggests that the theory allows stars in equilibrium with arbitrary baryon number, no matter how large. | 10.1007/s10509-012-1049-y | [
"https://arxiv.org/pdf/1104.2140v4.pdf"
] | 119,240,358 | 1104.2140 | e68bf944d3f31468b1d94fc53e4db20af44832c7 |
3 Mar 2012 February, 7, 2012
Emilio Santos
Departamento de Física
Universidad de Cantabria. Santander
Spain
3 Mar 2012 February, 7, 2012Neutron stars in generalized f(R) gravitynumbers: 0440Dg0460-m0450Kd
Quartic gravity theory is considered with the Einstein-Hilbert Lagrangean R + aR 2 + bR µν R µν , R µν being Ricci´s tensor and R the curvature scalar. The parameters a and b are taken of order 1 km 2 . Arguments are give which suggest that the effective theory so obtained may be a plausible approximation of a viable theory. A numerical integration is performed of the field equations for a free neutron gas. The result is that the star mass increases with increasing central density until about 1 solar mass and then decreases. The baryon number increases monotonically, which suggests that the theory allows stars in equilibrium with arbitrary baryon number, no matter how large.
Introduction
General relativity has passed all observation tests so far, but the real theory of gravity may well differ significantly from it in strong field regions. In fact conceptual difficulties in quantizing Einstein's theory and astrophysical observations suggest that general relativity may require modifications. In recent years a great effort has been devoted to the study of extended gravity theories, mainly with the goal of finding physical explanations to the accelerated expansion of the universe and other astrophysical observations, like the flat rotation curves in galaxies [1], [2]. Compact stars are an ideal natural laboratory to look for possible modifications of Einsteins theory and their observational signatures. A rather general class of alternative theories of gravity has been considered recently [3] to study slowly rotating compact stars with the purpose of investigating constraints on alternative theories. Several studies of compact stars, in particular neutron stars, have been made within extended gravity theories [4], [6], [7], [5]. There are also theories which prevent the appareance of singularities like "gravastars" [8] and Eddington inspired gravity [9].
The most popular modification of general relativity, since the early days of general relativity [10], derives from an extension of the Einstein-Hilbert action of the form
S = 1 2k d 4 x √ −g(R + F ) + S mat ,(1)
where k is 8π times Newton´s constant and I use units c = 1 throughout, and F is a function of the scalars which may be obtained by combining the Riemann tensor, R µνλσ , and its derivatives, with the metric tensor, g µν . In particular the theory derived from the choice F (R), where R is the Ricci scalar, has been extensively explored under the name of f(R)-gravity [11], [12].
More general is fourth order gravity [13], which derives from the choice
F = F R, R µν R µν , R µνλσ R µνλσ ,(2)
R µν being the Ricci tensor. A particular example of eq.(2) is the quadratic Lagrangian which may be written without loss of generality
F = aR 2 + bR µν R µν .(3)
(Riemann square does not appear because it may be eliminated using the the Gauss-Bonnett combination
R 2 GB ≡ R 2 − 4R µν R µν + R µνλσ R µνλσ ,
which does not contribute to the field equations in a quadratic Lagrangian.) The Newtonian limit of the field equations derived from the Lagrangian eq.(3) has been studied elsewhere [14], [15]. In this paper I report on a calculation of neutron stars using the theory derived from eq.(3) with the particular choice of parameters b = −2a, √ a = 1 km. That theory is apparently not viable for two reasons. Firstly, in order not contradicting Solar System and terrestrial observations the parameters a and b should be not greater than a few millimeters. Secondly the weak field limit of the theory should not present ghosts [16], [17]. A solution to both problems is to assume that eq.(3) is an approximation, valid for the strong fields appearing in neutron stars, of another function F which is extremely small in the weak field limit. This would be the case, for instance, if F has the form
F = aR 2 + bR µν R µν − c log 1 + (a/c)R 2 + (b/c)R µν R µν .(4)
with a ≃ 10 6 m 2 , b − 2a, c = 1/(10 26 m 2 ). Thus eq.(4) may be approximated by
F ≃ 1 2c aR 2 + bR µν R µν 2 10 −32 R ,(5)
for the Solar System and the relative error due to the terms neglected in going from eq.(4) to eq.(5) is smaller than 10 −12 . I have taken into account that R 2 ∼ R µν R µν ∼ (kρ) 2 and that the typical density ρ ∼ 10 4 kg/m 3 . The inequality in (5) shows that the correction to GR due to the function F , eq.(4) , is neglible in Solar System or terrestrial calculations. Also the problem of the ghosts in the weak field limit disappears with that choice. Indeed the theory is fine in the context of low-energy effective actions because the contribution of R µν R µν is so small that it never dominates the dynamics of the background. On the other hand the R 2 term does not introduce extra graviton modes. In contrast in neutron stars, where ρ ∼ 10 18 kg/m 3 , the latter (logaritmic) term of eq.(4) is about 10 −12 times the former terms and it may be ignored in the calculation.
Field equations
The tensor field equation derived from the functional eqs.(1) and the latter eq. (5) may be taken from the literature [18], [19]. I shall write it in terms of the Einstein tensor, G µν , rather than the Ricci tensor, R µν , and in a form that looks like the standard Einstein equation of general relativity eq.(6). That is
G µν ≡ R µν − 1 2 g µν R = k T mat µν + T ef µν ,(6)kT ef µν ≡ −(2a + b) [∇ µ ∇ ν G − g µν G] − 2 (a + b) −GG µν + 1 4 g µν G 2 −b 2G σ µ G σν − 1 2 g µν G λσ G λσ − ∇ σ ∇ ν G σ µ − ∇ σ ∇ µ G σ ν + G µν .(7)
We are interested in static problems of spherical symmetry and will use the standard metric
ds 2 = − exp (β (r)) dt 2 + exp (α (r)) dr 2 + r 2 dΩ 2 .(8)
Thus G µν (r) and T mat µν (r) have 3 independent components each, so that including α (r) and β (r) there are 8 variables. On the other hand there are 8 equations, namely 3 eqs (7) , 3 more equations giving the independent components of G µν in terms of α and β and 2 equations of state relating the 3 independent components of T mat µν . I shall assume local isotropy for matter so that one of the latter will be the equality T mat
11 = T mat 22 (= T mat 33
in spherical symmetry.) In principle the remaining 7 coupled non-linear equations may be solved exactly by numerical methods, as will be explained in Section 4.
Before proceeding, a note about the signs convention is in order. As is well known different authors use different signs in the definition of the relevant quantities. Here I shall make a choice which essentially agrees with the one of Ref. [11]. It may be summarized as follows
g 00 = − exp β, G µν = R µν − 1 2 g µν R = kT µν , T 0 0 = −ρ.(9)
After that I shall write the three independent components of eq.(7)using the notation
T 0 0 = −ρ, T 1 1 = p, T 2 2 = q, T µ µ = T = p + 2q − ρ, (T mat ) 0 0 = −ρ mat , (T mat ) 1 1 = (T mat ) 2 2 = (T mat ) 3 3 = p mat .(10)
In the following I will name ρ, p and q the total density, radial pressure and transverse pressure respectively, whilst ρ mat and p mat will be named matter density and pressure respectively (remember that we assume local isotropy for matter, that is the equality of radial and transverse matter pressures.) The differences ρ − ρ mat , p − p mat and q − p mat will be named effective density, radial pressure and transverse pressure respectively.
After some algebra I get for the components of the tensor eq.(7)
− ρ mat = −ρ + (2a + b)e −α − d 2 T dr 2 − 2 r − 1 2 α ′ dT dr + (a + b)k( 1 2 T 2 + 2T ρ) +b exp(−α) − 2β ′ r (q − p) + 1 2 α ′ β ′ − β ′′ − 2β ′ r (ρ + p) +b −∆ρ + 2kρ 2 − 1 2 k ρ 2 + p 2 + 2q 2 ,(11)p mat = p − (2a + b)e −α 2 r + 1 2 β ′ dT dr + (a + b)k( 1 2 T 2 − 2T p) +b ∆p + 2kp 2 − 1 2 k ρ 2 + p 2 + 2q 2 +b exp(−α) 2α ′ r + 4 r 2 (q − p) + − 1 2 α ′ β ′ + β ′′ (ρ + p) ,(12)p mat = q − (2a + b)e −α d 2 T dr 2 + 1 r + 1 2 β ′ − 1 2 α ′ dT dr +(a + b)k( 1 2 T 2 − 2T q) + b ∆q + 2kq 2 − 1 2 k ρ 2 + p 2 + 2q 2 +b exp(−α) − α ′ r + β ′ r − 2 r 2 (q − p) + β ′ r (ρ + p) .(13)
Addition of these 3 equations gives the trace equation, that is
T mat ≡ 3p mat − ρ mat = T − (6a + 2b) ∆T,(14)
where ∆ is the Laplacean operator in curved space-time
∆ ≡ exp(−α) d 2 dr 2 + 2 r + 1 2 β ′ − 1 2 α ′ d dr .(15)
The quantities G ν µ are related to the metric coefficients α and β and their derivatives (see e.g. [20]), hence to ρ, p and q, that is
exp(−α) = 1 − 2m r , α ′ 2 = m − 4πρr 3 r 2 − 2mr , β ′ = 2 m + 4πr 3 p r 2 − 2mr , β ′′ = 8πr 2 (rρ + rp + 3p ′ ) r 2 − 2mr − 4 (m + 4πr 3 p) (r − m − 4πr 3 ρ) (r 2 − 2mr) 2 ,(16)
where I have used units k = 8π, c = 1 and the radial derivative of α (β ′ ) is labelled α ′ (β ′′ ). The mass parameter m is defined by
m = r 0 4πx 2 ρ(x)dx.(17)
The condition that Einstein tensor, G µν , is divergence free leads to the hydrostatic equilibrium equation, that is
dp dr = 2(q − p) r − 1 2 β ′ (ρ + p) .(18)
Application to neutron stars
For neutron stars, when are quadratic gravity corrections relevant?.
In order to answer this question we should estimate the conditions where T ef µν , eq. (7) , is comparable toT mat µν . Terms with derivatives are of order
a G ∼ (a/R 2 0 )G,
R 0 being the radius of the hypothetical star. Thus these terms are relevant if the dimensionless quantity a/R 2 0 is of order unity, which implies that a and b should be of order the star radius, that is a few kilometers. Terms without derivatives are of order aG 2 ∼ (akρ/c 2 )G, similar to those with derivatives. In order to solve eqs. (11) to (18) we need an equation of state (eos), that is a relation between p mat and ρ mat , appropriate for a system of neutrons. For the calculation here reported I shall choose the eos of a free (non-interacting) neutron gas. In order to make easier the rather involved numerical integration of the equations, I will simplify the said eos writing
ρ mat = 3p mat + Cp 3/5 mat , C = 2.34,(19)
where ρ mat and p mat are in units of 7.2×10 18 kg m −3 . This equation is correct in the limit of high density, where ρ mat ≃ 3p mat , and has the same dependence
p mat ∝ ρ 5/3
mat as the eos of the free neutron gas in the nonrelativistic limit of low density. The constant C is so chosen that we get the same result as Oppenheimer and Volkoff [21] for the maximum mass stable star in their general relativistic calculation.
A relevant quantity is the baryon number of the star, N, which may be calculated from the baryon number density n(r) via
N = R 0 n(r) 1 − 2m(r)/r 4πr 2 dr,(20)
in our units. A relation between the number density and the matter density (or pressure) may be got from the solution of the equation
p mat = n dρ mat dn − ρ mat ,
which follows from the first law of thermodynamics. Inserting eq.(19) here we get a differential equation which may be easily solved with the condition ρ mat /n → µ for ρ → 0, µ being the neutron mass. I get
n = C 5/8 p 3/5 mat 4p 2/5 mat + C 3/8 ,(21)
where the unit of baryon number density is µ −1 7.2 × 10 18 kg m −3 .
Neutron stars in extended gravity
In order to derive the structure of neutron stars in generalized f(R) gravity theory, as defined by eqs. (7) , we should solve the coupled eqs. (11) to (20) plus the hydrostatic equilibium eq.(18) with our choice of the parameters a and b, namely b = −2a, √ a = 0.96 km. This choose of a and b makes the calculation specially simple. We need just 3 amongst the 4 eqs. (11) to (14) , because only 3 are independent. I choose eqs. (14) ,the difference eq.(13) minus eq.(12) , and eq.(12), which may be rewritten
dT dr = T ′ , dT ′ dr = − 2 r + 1 2 β ′ − 1 2 α ′ dT dr + T − T mat 2a ,(22)dh dr = h ′ , h ≡ q − p, dh ′ dr = − 2 r + 1 2 β ′ − 1 2 α ′ h ′ + exp α h 2a + kT h − 2k(h + 2p)h − − 3α ′ r + β ′ r − 6 r 2 h + ( β ′ r − β ′′ + 1 2 α ′ β ′ )(ρ + p) ,(23)p mat = p + ak(2T p − 1 2 T 2 − 3p 2 + ρ 2 + 2q 2 ) −2a exp(−α) ∆p + 2α ′ r + 4 r 2 h + β ′′ − 1 2 α ′ β ′ (ρ + p) ,(24)
where the Laplacean operator ∆ was defined in eq. (15) . Finally we need the hydrostatic equilibrium eq.(18). The numerical calculation goes as follows. From the values of all variables at a given radial coordinate r, integration of the linear differential eqs. (22) , (23) and (18) , taking eq.(17) into account, provides the values of m, T, T ′ , h, h ′ and p at r + dr. Hence the relation (see eq.(10))
ρ = p + 2q − T = 2h + 3p − T,
gives ρ (r + dr) , whence eq.(18) gives p ′ (r + dr) which allows obtaining ρ ′ (r + dr) . After that we have all quantities needed to get d 2 p/dr 2 from the derivative of eq. (18) , that is
d 2 p dr 2 = 2h ′ r − 2h r 2 − 1 2 β ′′ (ρ + p) − 1 2 β ′ (ρ ′ + p ′ ) .
Hence we get p mat from eq.(24) taking eqs. (15) and (16) into account, which allows obtaining ρ mat via the eos eq. (19) , whence T mat = 3p mat − ρ mat follows (remember that we assume local isotropy for matter, that is p mat = q mat .) In this way we obtain all the quantities at r + dr and the process may be repeated in order to get the quantities at r + 2dr , and so on. This shows that our equations form a consistent system. As initial conditions for the differential equations we need the values of the following variables at the origin: T (0) , h (0) , p (0) , T ′ (0) , h ′ (0) . The latter 2 should be taken equal to zero in order that there is no singularity, and h (0) = 0 because there is no distinction between radial, p, and transverse pressure, q at the origin. We are left with just two free parameters, namely p (0) and T (0), but there is a constraint, that is the condition that T → 0 for r → ∞. Indeed the matter density and pressure are positive within the star, so that p mat (r) = 0 for any r > R, R being the star radius (incidentally, there is some contribution to the star mass outside the star surface due to the effective density.) For r > R (r) the quantity T (r) (and the density ρ (r)) should decrease rapidly with increasing r. As a consequence only the value of p (0) may be chosen at will, whilst the value of T (0) should be so chosen as to guarantee the rapid decrease of T (r) for r > R. Consequently I have been lead to perform the integration several times for each choice of p (0), with a different value of T (0) each time, until I got a value of T (r) sufficiently small for large enough r (that is greater then the star radius). This procedure presents the practical difficulty that requires a fine tuning of T (0) due to the fact that for large r > R the solution of eqs. (22) is approximately of the form
T (0) ∼ A r exp r √ 2a + B r exp − r √ 2a .
Thus the parameter A should be very accurately nil in order that the first term does not surpasses the second one at large r. This is specially so if the parameter a is small, and this is why I have chosen to study the case of a relatively large value of a. Also in order to alleviate the problem I have substituted a differential equation for a new variable f for the eqs. (22) where
T = f r exp − r √ 2a .
Thus the condition that f remains bounded for r → ∞ replaces the stronger condition that T → 0 and the numerical procedure is less unstable. In summary we obtain a one-parameter family of equilibrium stars, one for each value of the central total pressure p (0). Table 1 reports the results of the calculation. As in the standard (GR) theory of neutron stars [21] the radius decreases with increasing central density, whilst the mass increases until a maximum value and then decreases. Therefore our theory also predicts a maximum mass for equilibrium neutron stars. However there is a dramatic difference in the behaviour of the baryon number, which here is always increasing with increasing central density. Of course in stars with very large central density, matter will not be in the form of neutrons but will consists of a mixture of different particles but I will assume that the total baryon number is well defined. Although I have not made a rigorous proof, the results of the calculation suggest that there may be equilibrium configurations of neutron stars for any baryon number no matter how large. A consequence of the strong increase of the baryon number with a decrease of the mass implies that the binding energy becomes very large, about 90% of the mass in the stars with the highest central density amongst those studied here. Table 1 also shows that both the baryon number density, n, and the matter density, ρ mat , become very large for moderately large central total density. This implies that the effective density, ρ ef f = ρ − ρ mat is negative in the central region of the star although becoming positive near and beyond the surface. However neither ρ ef f nor ρ mat have a real physical meaning, only the total density ρ being meaningful, and it remains always positive. A similar thing happens with the pressure. The surface relative red shift is higher in our theory than in the standard (GR) theory, but the difference is not dramatic. Table 1. Our calculation. Central total pressure, p (0), central total density, ρ (0) , and central matter density, ρ mat (0), are in units ρ c ≡ 7.
Discussion
The calculations of this paper show that, if there are corrections to general relativiy of the form of eqs. (1) and (3), then the structure of neutron stars would be dramatically different from the one predicted by the general relativity. In particular a new scenario would emerge for the evolution of the central region of massive white dwarfs stars after the supernova explosion. Indeed the said central region might contract strongly by emitting an amount of energy corresponding to a very large fraction of the total mass. The final result will be a neutron star with a mass maybe surpassing the believed (Oppenheimer-Volkoff) limit. It is not possible to know how large is the new mass limit until calculations with more realistic equations of state are performed. In addition the predictions of the theory here considered may be quite different for other choices of the parameters a and b.
Appendix. Neutron stars in general relativity
For the sake of comparison with the results of our calculation using eqs. (11) to (21) , I summarize in Table 2 the results of a calculation similar to the one performed by the Oppenheimer and Volkoff calculation [21] using general relativity. It corresponds to taking a = b = 0 in eqs. (11) to (14), so that ρ = ρ mat , p = p mat , and I use the eos eq. (19) . I have extended the calculation to quite high values of the central pressure because for those values the corrections to GR in our generalized f (R) gravity theory are most relevant. Table 2. General relativistic calculation. Central pressure, p (0) , and central density, ρ (0) , are in units 7. Table 2 shows that both the mass, M, and the baryon number, N, increase with increasing central density until ρ (0) ≃ 0.46, where M ≃ 0.72 M • (the OV mass limit), and both M and N decrease after that. There are no equilibrium configurations, either stable or unstable, with baryon number above N ≃ 0.74. Actually for every baryon number N < 0.74 there are two equilibrium configurations, one of them with ρ (0) < 0.46 and another one with ρ (0) > 0.46, the latter having higher mass than the former. Furthermore, as is shown in Table 2, stars with large central density have a negative binding energy and therefore cannot be stable.
2×10 18 kg/m 3 . Central baryon number density, n(0), in units ρ c /µ, µ being the neutron mass. Star radius, R, is in kilometers, mass, M, in solar masses and baryon number, N, in units of solar baryon number. I report also the dimensionless fractional surface red shift, ∆λ/λ = 1/ 1 − 2M/R − 1, and percent binding energy, BE = 100(N − M)/N.
2 × 10 18 kg m −3 , star radius, R, in kilometers, mass, M, in solar masses and baryon number, N, in units of solar baryon number. I report also the percent binding energy, BE, defined by the ratio 100(N − M)/N and the fractional surface red shift, ∆λ/
An expressions like 1.6E2 means 1.6 × 10 2 mat (0) 1.6E2 2.5E3 4.3E4 7.8E5 1.6E7 3.2E8 6.3E9p (0)
0.01
0.1
1
10
100
1000 10000
ρ (0)
0.18
0.82
4.5
34
3.1E2 3.0E3 3.0E5
ρ n (0)
56
4.5E2 3.7E3 3.3E4 3.2E5 3.0E6 2.8E7
R
10.7
6.7
4.0
2.7
2.1
2.2
2.2
M
0.67
0.80
0.60
0.39
0.264 0.268 0.292
N
0.73
0.94
1.03
1.13
1.44
2.00
2.63
BE
8.9% 15%
41%
65%
82%
87%
89%
∆λ/λ
0.106 0.22
0.34
0.31
0.23
0.23
0.26
. S Nojiri, S D Odintsov, arXiv:1011.0544S Nojiri and S D Odintsov, arXiv:1011.0544 (2011).
. S Capozziello, M De Laurentis, arXiv:1108.6266S. Capozziello and M. De Laurentis, arXiv:1108.6266.
. P Pani, E Berti, V Cardoso, J Read, arXiv:1109.0928P. Pani, E. Berti, V. Cardoso and J. Read, arXiv:1109.0928 (2011).
. S Capozziello, M De Laurentis, S D Odintsov, A Stabile, Phys. Rev. 8364004S. Capozziello, M. De Laurentis, S.D. Odintsov, A. Stabile, Phys. Rev. D83:064004 (2011).
. M De Laurentis, S Capozziello, arXiv:1202.0394M. De Laurentis, S. Capozziello, arXiv:1202.0394
. T Kobayashi, K I Maeda, Phys. Rev. 7864019T. Kobayashi and K. I. Maeda, Phys. Rev. D78, 064019 (2008);
. Phys. Rev. 7924009Phys. Rev. D79, 024009 (2009).
. E Babichev, D Langlois, Phys.Rev. 80121501E. Babichev and D. Langlois, Phys.Rev. D80, 121501 (2009) ;
. Phys. Rev. 81124051Phys. Rev. D81, 124051 (2010).
For more recent papers on the subject see. P O Mazur, E Mottola, Phys. Rev. R. Chan, M. F. A da Rocha and A. Wang641013JCAP. and references thereinP. O. Mazur and E. Mottola, Phys. Rev. D64, 104022 (2001). For more recent papers on the subject see R. Chan, M. F. A da Rocha and A. Wang, JCAP, 1013 (2011) and references therein.
. P Pani, E Berti, V Cardoso, T Delsate, arXiv: gr-qc/1106.3569P.Pani, E. Berti, V. Cardoso and T. Delsate, arXiv: gr-qc/1106.3569 (2011).
. H J Schmidt, Int. J. Geom. Mech. Mod. Phys. 4H. J. Schmidt, Int. J. Geom. Mech. Mod. Phys. 4, 209-248 (2007).
. T P Sotiriou, V Faraoni, Rev. Mod. Phys. 82451T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010).
. A De Felice, S Tsujikawa, Living Rev. Rel. 133A. de Felice and S. Tsujikawa, Living Rev. Rel. 13, 3 (2010).
. S Capozziello, A Stabile, arXiv:1009.3441S. Capozziello, A. Stabile, arXiv:1009.3441.
. E Santos, Phys. Rev. 8164030E. Santos, Phys. Rev. D81, 064030 (2010).
. A Stabile, Phys. Rev. 82124026A. Stabile, Phys. Rev. D82, 124026 (2010);
. S Capozziello, A Stabile, arXiv: gr-qc/1009.3441v1S. Capozziello and A. Sta- bile, arXiv: gr-qc/1009.3441v1 (2010).
. A Hindawi, B A Ovrut, D Waldram, Phys. Rev. 535583A. Hindawi, B. A. Ovrut and D. Waldram, Phys. Rev. D53, 5583 (1996);
. Phys. Rev. 535597Phys. Rev. D53, 5597(1996).
. A De Felice, J. Phys. A: Math. Theor. 407061A. de Felice, J. Phys. A: Math. Theor. 40, 7061 (2007);
. A De Felice, M Hindmarsh, J. Cosmol. Astropart. Phys. 070628A. de Felice and M. Hindmarsh, J. Cosmol. Astropart. Phys. 0706, 028 (2007).
. S Nojiri, S D Odintsov, Int.J.Geom. Meth. Mod. Phys. 4S. Nojiri and S. D. Odintsov, Int.J.Geom. Meth. Mod. Phys.4:115-146 (2007).
. S M Carroll, A De Felice, V Duvvuri, D A Easson, M Trodden, M S Turner, Phys. Rev. 7163513S. M. Carroll, A. De Felice, V. Duvvuri, D. A. Easson, M. Trodden and M. S. Turner, Phys. Rev. D71, 063513 (2005).
Page 272. I must point out that in the book Einstein´s tensor G µν is defined with a sign opposite to the one used in this paper. J L Synge, Interscience PublishersNew YorkRelativity: The general theoryJ. L. Synge, Relativity: The general theory. Interscience Publishers, New York, 1960. Page 272. I must point out that in the book Einstein´s tensor G µν is defined with a sign opposite to the one used in this paper.
. J R Oppenheimer, G M Volkoff, Phys. Rev. 55J. R. Oppenheimer, and G. M. Volkoff, Phys. Rev. 55, 374 (1935).
Compact objects in astrophysics. M Camenzind, SpringerBerlinM. Camenzind, Compact objects in astrophysics. Springer, Berlin, 2007.
| [] |
[
"SUPERMASSIVE BLACK HOLES IN GALACTIC NUCLEI WITH TIDAL DISRUPTION OF STARS: PAPER II -AXISYMMETRIC NUCLEI",
"SUPERMASSIVE BLACK HOLES IN GALACTIC NUCLEI WITH TIDAL DISRUPTION OF STARS: PAPER II -AXISYMMETRIC NUCLEI"
] | [
"Shiyan Zhong ",
"Peter Berczik ",
"Rainer Spurzem "
] | [] | [] | Tidal Disruption of stars by supermassive central black holes from dense rotating star clusters is modelled by high-accuracy direct N -body simulation. As in a previous paper on spherical star clusters we study the time evolution of the stellar tidal disruption rate and the origin of tidally disrupted stars, now according to several classes of orbits which only occur in axisymmetric systems (short axis tube and saucer). Compared with that in spherical systems, we found a higher TD rate in axisymmetric systems. The enhancement can be explained by an enlarged loss-cone in phase space which is raised from the fact that total angular momentum J is not conserved. As in the case of spherical systems, the distribution of the last apocenter distance of tidally accreted stars peaks at the classical critical radius. However, the angular distribution of the origin of the accreted stars reveals interesting features. Inside the influence radius of the supermassive black hole the angular distribution of disrupted stars has a conspicuous bimodal structure with a local minimum near the equatorial plane. Outside the influence radius this dependence is weak. We show that the bimodal structure of orbital parameters can be explained by the presence of two families of regular orbits, namely short axis tube and saucer orbits. Also the consequences of our results for the loss cone in axisymmetric galactic nuclei are presented. | 10.1088/0004-637x/811/1/22 | [
"https://arxiv.org/pdf/1508.02838v1.pdf"
] | 119,292,234 | 1508.02838 | 50e5ffe14a63540551115a02020acf10ca733a77 |
SUPERMASSIVE BLACK HOLES IN GALACTIC NUCLEI WITH TIDAL DISRUPTION OF STARS: PAPER II -AXISYMMETRIC NUCLEI
12 Aug 2015 Draft version July 8, 2017 Draft version July 8, 2017
Shiyan Zhong
Peter Berczik
Rainer Spurzem
SUPERMASSIVE BLACK HOLES IN GALACTIC NUCLEI WITH TIDAL DISRUPTION OF STARS: PAPER II -AXISYMMETRIC NUCLEI
12 Aug 2015 Draft version July 8, 2017 Draft version July 8, 2017Preprint typeset using L A T E X style emulateapj v. 5/2/11Subject headings: black holes -galactic nuclei -stellar dynamics
Tidal Disruption of stars by supermassive central black holes from dense rotating star clusters is modelled by high-accuracy direct N -body simulation. As in a previous paper on spherical star clusters we study the time evolution of the stellar tidal disruption rate and the origin of tidally disrupted stars, now according to several classes of orbits which only occur in axisymmetric systems (short axis tube and saucer). Compared with that in spherical systems, we found a higher TD rate in axisymmetric systems. The enhancement can be explained by an enlarged loss-cone in phase space which is raised from the fact that total angular momentum J is not conserved. As in the case of spherical systems, the distribution of the last apocenter distance of tidally accreted stars peaks at the classical critical radius. However, the angular distribution of the origin of the accreted stars reveals interesting features. Inside the influence radius of the supermassive black hole the angular distribution of disrupted stars has a conspicuous bimodal structure with a local minimum near the equatorial plane. Outside the influence radius this dependence is weak. We show that the bimodal structure of orbital parameters can be explained by the presence of two families of regular orbits, namely short axis tube and saucer orbits. Also the consequences of our results for the loss cone in axisymmetric galactic nuclei are presented.
INTRODUCTION
A large fraction of galaxies show evidence of supermassive black holes (henceforth SMBH) residing in their center. They are typically embedded in nuclear star clusters (NSC); if resolution allows to observe the NSCs, they are among the densest clusters known. Their size is similar to galactic globular clusters, but they are much heavier and brighter (Böker et al. 2002(Böker et al. , 2004. In massive galaxies NSCs may not be significant or even do not exist, however, the SMBHs are still surrounded by enormous number of stars. SMBH residing in these NSCs will tidally disrupt stars that come close to its tidal radius and eventually accrete the gaseous debris, which can light up the central SMBH for a period of time (Rees 1988;Evans & Kochanek 1989). This kind of event is a useful tool to examine the relativistic physics near SMBH since the disruption occurs at a place very close to the BH's Schwarzschild radius. Also it can help us to investigate SMBH in non-active galactic center. Although tidal disruption of stars has been proposed for almost half a century, only until last decade do people realize the importance of such events, after the discovery of a dozens of tidal disruption candidates (Komossa 2002;Komossa & Merritt 2008). Liu et al. (2014) a candidate of binary SMBH system by analyzing the break in the light curve of TD event, demonstrate it as a promising tool for searching hidden SMBH binaries in quiescent galactic center. In order to compute the tidal disruption event rate, many theoretic works have been done in the past few decades (Frank & Rees 1976;Lightman & Shapiro 1977;Magorrian & Tremaine 1999;Wang & Merritt 2004). The core of the story is loss cone theory, which was first established in the case of spherical symmetric systems.
Stars with orbital pericenter smaller than the tidal radius r t are defined to be inside the loss cone, with r t be expressed by
r t = αr ⋆ 3 5 − n M bh m ⋆ 1 3 ,(1)
where r ⋆ , m ⋆ are the radius and mass of a star, n is its polytropic index (assuming the stellar structure can be approximated by a polytropic sphere) and α is a free parameter used by us for scaling. Stars with angular momentum J < J lc ≈ √ 2GM • r t are inside the loss cone. Typically, loss cone stars are consumed in dynamical time scales. If no new star is supplied to loss cone, there will be no more tidal disruption event. Based on the status of loss cone, it can be divided into two regime, namely empty and full loss cone. Due to the short "lifetime" of the loss cone stars, loss cone will become empty quickly. Refilling of loss cone happens in relaxation timescales and is often referred to as diffusion process in angular momentum space. Thus in empty loss cone regime it is the refilling rate which controls the disruption rate. Note that throughout this paper, and like in most if not all of the cited papers on tidal accretion of stars onto SMBH, we assume that a star is disrupted completely at r t and all its mass, energy and angular momentum absorbed momentarily by the SMBH. We know that this is not realistic, and more detailed numerical models of the process of disruption, possible disk formation and accretion show that only fractions of the material are absorbed into the SMBH after a number of orbits (Guillochon & Ramirez-Ruiz 2013;Hayasaki et al. 2013). However, the assumption that the process is fast is reasonable compared to the orbital time scales of stars further out in the cluster.
In a previous paper (Zhong et al. (2014), henceforth Paper I) we have shown that the classical loss cone approximation, for a spherically symmetric system in the diffusive empty loss cone regime, can be well reproduced by large direct N -body models with tidal accretion of stars onto SMBH. Now we are focusing on the generalization to axisymmetric galactic nuclei and compare our new results in an otherwise very similar study to those of Paper I.
Tidal disruption of stars is one possible way of growth for SMBH, especially in quiescent galactic nuclei. Since most models assumed spherical stellar clusters, SMBH growth rates by tidal disruption are very low, limited by the very long relaxation time to refill the loss cone, and the contribution of the process to the overall growth of SMBH is considered as relatively insignificant. However, the stellar distribution in real galactic nuclei might not be spherically symmetric. Many galactic nuclei show evidence of rotation in their centers, even very close to the SMBH (Miyoshi et al. 1995;Neufeld & Maloney 1995;Greenhill et al. 1995).
According to the current standard model of structure formation massive galaxies have undergone quite significant mergers (in number and mass ratio). Numerical models of the merging process of galaxies show that the merger remnant shows rotation, axial symmetry or even triaxiality in the central regions (Khan et al. 2011;Preto et al. 2011;Gualandris & Merritt 2012;Bois et al. 2013).
In the center of our own Milky Way the NSC can be observed in unparalleled high resolution Schödel et al. 2014). It consists of 1.4 × 10 7 M ⊙ within its effective radius (4.2 pc); kinematic data indicate that it possesses bulk rotation . The formation mechanism of NSCs is still under debate. There are two scenarios, in situ formation (Milosavljević 2004) and a sinking scenario (globular cluster sink to the center and merge) (Tremaine et al. 1975;Lotz et al. 2001). NSCs in a sample of nearby galaxies observed by Seth et al. (2006Seth et al. ( , 2008 show that these objects are non-spherical and even contain multicomponent (younger disk plus older spherical component),which favor the in situ scenario. However, Antonini et al. (2012) have performed a series of N -body simulations to study the formation of NSCs, which support the sinking scenario. The model NSCs formed in their simulations by merging between infalling globular clusters initially have mildly triaxial shape. After the final infall, the shape of the NSC will gradually become axisymmetric in following dynamical evolution.
Despite the debate between different formation scenarios, we think that it is quite likely that NSC are nonspherical. This provides a good motivation to study the tidal disruption rate in axisymmetric (and triaxial) clusters. Some works have already been done but the mission is not over. Fiestas & Spurzem (2010) used 2D Fokker-Plank model (Einsel & Spurzem 1999;Kim et al. 2002) to study rotating dense stellar clusters with BHs and cross checked with N -body models (Fiestas et al. 2012). Both works find that BH embedded in rotating model have higher tidal disruption rate (hereafter TDR) compare to spherical models. BH mass at the end of simulation is roughly 20% higher in rotating case. They find an excess of accreted prograde rotating stars which are originated mainly outside the influence radius r h and call for a further investigation of the roles of stars with nonconserved J x , J y angular momentum. As shown by the works of Merritt & Poon (2004), in non-spherical systems chaotic orbits (existing in regions outside r h ) can keep the loss cone full for sufficient long time, thus tidal disruption can contribute a lot of mass within Hubble time and could play an important role in the BH growth across cosmic time.
On the other hand, the loss cone itself might be enlarged as pointed out by Magorrian & Tremaine (1999), due to the fact that angular momentum J is not conserved in axisymmetric potential. Vasiliev & Merritt (2013) confirmed this picture in a detailed analysis of the loss cone problem in axisymmetric galactic nuclei. They analyzed the depletion and refilling of loss cone orbits and found that tidal disruption rates could be increased by a moderate factor due to axisymmetry as compared to spherical symmetry. In their work chaotic orbits with low angular momentum, which can reach just outside the influence radius at apocenter, but also get close to the central SMBH at pericenter, cause some difficulty in comparison with Fokker-Planck models, as was already found by Malkov et al. (1993) (Note that the last author of this paper is the same person than the last author of Malkov et al. (1993), there was a mistake in retranslating the name from Russian language).
In this work we follow an experimental numerical approach to the problem, following Paper I for the case of spherically symmetric systems. We treat particle number and tidal radius as free parameters and analyze the tidal accretion rate of the system as a function of the strength of deviation from spherical symmetry. We measure the shape of the loss cone in axisymmetric potential and and characterize the characteristic orbits of stars in the loss cone. We find that it is indeed enlarged and can account for the higher TDR as compared to spherically symmetric galactic nuclei. This paper is organized as follows: we describe the model setup of the simulation in Section 2 and present the result of TDR measurement in Section 3. Section 4 is devoted to the measurement of loss cone shape in axisymmetric potential and we demonstrate the enlargement of loss cone. In Section 5, we present the result for the origin and orbital classification of disrupted stars. In Section 6, we discuss the potential application of our results.
2. N -BODY MODEL We adopt the standard N -body unit definitions from Heggie & Mathieu (1986), namely G = M = 1 and E = −1/4, where G is the gravitational constant, M is the total mass of the model cluster and E is the total energy. In our N -body models we assume that all the particles have the same mass, so m = 1/N , where m is the particle mass and N is the total particle number. To preserve the scale invariance of our N -body simulations we fix the initial black hole mass relative to the total mass of the star cluster (0.01) and use the particle number and the tidal radius r t in N -body units (which is a dimensionless number) as free parameters. We have shown in Paper I that the method of scaling to realistic parameters for N and r t can be used to obtain astrophysically meaningful results from the collection of our models. In order to support our scaling procedure we even do not change the tidal radius during the simulations -since the BH mass changes within one order of magnitude only during the simulation, relative changes in tidal radius are small (notice that r t ∝ (M • ) 1/3 ).
The initial distribution of particles follows a generalized King model with rotation. The distribution function is (Einsel & Spurzem 1999;Ernst et al. 2007)
f (E, J z ) = C · [exp(− E σ 2 K ) − 1] · exp(− Ω 0 J z σ 2 K ),(2)
where σ K is the King velocity dispersion and Ω 0 is a characteristic angular velocity. Since we are considering an isolated system, the Φ t is set to 0. This rotating King model has two dimensionless parameters: W 0 and ω 0 . The King parameter W 0 = −Φ 0 /σ 2 K , where Φ 0 is the central potential, controls the degree of central concentration. And the rotation parameter ω 0 = 9/(4πGρ 0 )·Ω 0 , where ρ 0 is the central density, controls the degree of rotation. ω 0 = 0 will reduce the model to a usual nonrotating spherically symmetric King model.
We limit our current study to only one concentration parameter W 0 = 6 and two rotation parameters ω 0 = 0.3, 0.6; the density profile of King model with this concentration is similar to that of the Plummer model used in Paper I, so it is possible to compare with the previous results and focus on the effects of rotation and axial symmetry only. The rotation is moderate (cf. e.g. Einsel & Spurzem (1999)) and resembles that of Milky Way globular clusters.
For completeness we also employ non-rotating King model with W 0 = 6 and ω 0 = 0.0, which is used as a fiducial model and also a bridge to the results of Paper I, confirming our claim that it indeed closely resembles the results for the Plummer model used in Paper I (e.g. in the evolution of the TDR). In another test run we used a larger rotation with ω 0 = 0.9 -it experienced an unstable stage during which a bar formed but quickly disappeared. This bar formation could probably be identified with the radial orbit instability of Aguilar & Merritt (1990). We note that our standard models with ω 0 = 0.3, 0.6 remain fully axisymmetric during the entire simulation; to study tidal disruption in triaxial systems with bars is beyond the scope of our current paper. Fig. 1 shows the axial ratio (c/a) of the model clusters as a function of radius up to r = 2.0 (within which most of stars are located). We estimate the axial ratio for both rotating models, using the moment of inertia tensor measured in concentric shells. One can see c/a is close to 1 at the innermost part and decreases outward: ω 0 = 0.3 model decreases slowly to its minimum value 0.9; ω 0 = 0.6 model decreases faster and has a minimum value 0.71. If we measure the c/a for the whole cluster, the results for the two models are 0.9 (ω 0 = 0.3) and 0.75 (ω 0 = 0.6). Fig. 1 also shows that c/a is almost unchanged during long time evolution, except for the inner part of ω 0 = 0.6 model, which exhibits slight decrease. In rotating systems, there is a phenomenon called gravo-gyro instability, which is caused by the negative specific moment of inertia (Inagaki & Hachisu 1978;Hachisu 1979Hachisu , 1982. This kind of instability happens in long term evolution of rotating cluster which is much longer than our integration time (Ernst et al. 2007).
The model set is summarized in Table 1. We run the simulation for more than one initial halfmass relaxation time (t rh ), which is estimated using the same formula in Paper I and the values can be found there as well ( Table 2).
All simulations are running with the ϕGRAPE code , which runs with high performance (up to 350 Gflop/s per GPU) on our GPU clusters in Beijing (NAOC/CAS). This code is a direct N -body sim-ulation package, with a high order Hermite integration scheme and individual block time steps. A direct Nbody code evaluates in principle all pairwise forces between the gravitating particles, and its computational complexity per crossing time scales asymptotically with N 2 ; however, it is not to be confused with a simple brute force shared time step code, due to the block time steps. We refer more interested readers to a general discussion about N -body codes and their implementation in Spurzem et al. (2011a,b). The present code is well tested and already used to obtain important results in our earlier large scale few million body simulation (Khan et al. 2012).
TIDAL DISRUPTION RATE (TDR)
3.1. Results of our work In this section, we present the TDR measured in simulations with our rotating King models and compare it with the TDR of the non-rotating model of Paper I. In Fig. 2, we show the TDR (both in terms of mass and particle number) as it evolves with time for two different tidal radii; in each panel two different rotation parameters are plotted together with the data of the nonrotating system. The time is given in units of initial half mass relaxation time t rh , which is convenient for comparison of simulations with different particle numbers. To smooth out fluctuations due to particle noise we have plotted in the figure the TDR averaged over a time interval (here 1/4t rh ). . TDR as a function of time in units of initial half mass relaxation time; averaged over intervals of 1/4t rh . Top panels: mass accretion rate; bottom panels: particle accretion rate; left and right panels for two different tidal radii as indicated. Curves with symbols are stand for 128K models, those without symbols are stand for 64K models. Line thickness indicate different rotating parameters.
The TDR with a large tidal radius (i.e. r t = 10 −3 ) initially quickly rises in the N = 64K model to its peak value, and then decreases; for the N = 128K model the TDR almost decreases from the beginning. The initial phase is connected with the formation of a central density cusp in the surrounding stellar system and with the process of transition from initially full to empty loss cone. The BH gains mass from the accreted stars, thus the mass ratio between stars and the BH (γ := m/M • ) de-creases with time, and as a result the BH's random motion damps. We have discussed in Paper I that the status of the loss-cone is connected with the BH's Brownian motion in the sense that once the amplitude of Brownian motion is smaller than 10r t the system enters the empty loss-cone regime, during which the cusp and central density are still growing but TDR begins to fall. In the N = 128K model, the mass ratio γ is smaller, so the initial loss cone depletion is very short, practically invisible in the plots, and the subsequent evolution is determined by cusp formation and damping of BH motion.
In the models with small tidal radius (r t = 10 −4 ), there is always an initial growth phase of TDR, followed by the convergent approach to a stationary state. Due to the small r t their BH growth is slow, thus they need more time to achieve the mass required to limit their Brownian motion. Fig. 2 also shows the TDR dependence on rotation parameter ω 0 as a new result compared to Paper I. For large tidal radius (r t = 10 −3 ), faster rotation will result in a higher TDR, note that these models are in empty losscone regime. Table 2 list out the numbers for TDR measurement. One can see ω 0 = 0.3 model has a TDR on average 13 percent higher than ω 0 = 0.0 model. And TDR in ω 0 = 0.6 model is on average 35 percent higher than that in ω 0 = 0.0 model. BH mass of these 3 models measured at T = 1500 are 0.131, 0.143 and 0.167. The fractional increase of final BH mass with increasing degree of rotation is consistent with the result of Fiestas et al. (2012). The reason for this dependence of ω 0 is that in these systems the effective loss-cone is larger than classic one in spherical system. We will investigate such an enlarged loss-cone in more detail in the next section. For small tidal radius (r t = 10 −4 ), however, we observe a different behavior of TDR. From beginning to about ∼ 1.5t rh , faster rotation result in a smaller TDR! The argument presented by Magorrian & Tremaine (1999) may provide some hints: if BH's wandering time-scale is shorter than dynamical time-scale, a decrease in TDR will happen. We note in the simulation at this early stage the BH is quickly wandering due to its small mass and slow growth. Furthermore, in axisymmetric systems a star's pericenter distance changes with time (even ignoring irregular perturbations from other stars). So when the BH comes back to the place where it was, it may still miss the star which is supposed to be disrupted shortly before. Note. -Measured TDR for models with same N (128K) and rt (10 −3 ) but different rotating parameters, at different evolution time.Ṅ 0 is TDR in classic King model (ω 0 = 0);Ṅ 3 andṄ 6 are results for ω 0 = 0.3 and ω 0 = 0.6 models. We also give the boost factorṄ 3 /Ṅ 0 andṄ 6 /Ṅ 0 .
Afterwards the system begins to enter the empty loss cone regime, and all TDR curves converge to each other; for small tidal radius more tidally disrupted stars originate from inside the BH influence radius, where the system is approximately spherically symmetric. Any deviation from spherical symmetry in our rotating models prevails near and outside the influence radius. Convergence of TDR reflects the original results obtained in Paper I for spherical systems. , ω 0 = 0.0) with that in Plummer model. In r t = 10 −3 models, except the initial higher accretion rate in King model, the two models have similar TDR in following evolution. While in the case of r t = 10 −4 , King model have a higher accretion rate during most of the time, but later on they gradually come to the same level as the Plummer model. The higher rate in King model could be explained by the slightly higher density in the core region at beginning. In the following evolution of r t = 10 −3 models, the two models form cusp similar to each other so they have roughly same accretion rate. In the case of r t = 10 −4 , the initial accretion rate ratiȯ N King /Ṅ P lum is higher than those in r t = 10 −3 . BH inside King cluster growing faster and also the growth of cusp, in the following evolution King model always have a higher density in the cusp which in return gives a higher accretion rate. Only after the BH gain enough mass and become a "static" object, the accretion rate slowly reaches a maximum and begins to drop afterward.
Up to this point, all results were presented in model units (N -body units). As in Paper I (see Sect. 5 and Appendix therein) we will discuss now any conclusions which can be made for the case of real galactic nuclei and environments from our results. This will be useful for observational programmes on TDR. To predict the TDR in real galactic nuclei, we use the method of scaling. The TDR obtained in our simulations has to be scaled up in two ways: first from relatively low (N ∼ 10 5 ) to more realistic high particle numbers (N ∼ 10 8 ). Second, our accretion radius r t has been chosen very large compared to any realistic tidal radius (for smaller N simulations it has to be done in order to get any meaningful results on TDR). So, we also have to discuss how to scale down the TDR from our simulated values of r t (10 −3 , 10 −4 ) to the small more realistic regime of r t (10 −7 ). This can be done by applying scaling relations from known scaling laws (obtained e.g. from Paper I and other literature) for N , and an empirically determined one for r t . We have shown above that the TDR in King (W 0 = 6) and Plummer models is very similar, so we expect the scaling formula (A10) derived in Paper I for a Plummer model to be also valid for our King model used here. So, we apply the same boost factor of TDR with respect to the axisymmetry of a galactic nucleus for the real galactic nucleus as we find here in this paper for our simulated systems. For example, in Paper I we estimated the TDR of the Milky Way SMBH to be 1.09 × 10 −5 yr −1 after a scaling procedure with respect to N and r t . By fitting surface brightness profile to mid-infrared images of the nuclear cluster in our Milky Way, Schödel et al. (2014) reported the mean ratio between minor and major axes is 0.71, which is close to our rotating ω 0 = 0.6 model. For this model we find a boost of TDR by 35% in our simulations, and we apply the same factor here for the case of axisymmetry, to get a higher TDR of 1.47 × 10 −5 yr −1 .
Relation to other current papers in the field
With regard to the enhancement of TDR in axisymmetric systems we have shown that our results are in agreement with Fiestas et al. (2012); but recently numerical simulations published by Vasiliev & Merritt (2013) and Vasiliev (2014) seem to contradict our findings. They claimed that the TDR in axisymmeric nuclei can be a few times larger than in the spherical case. Also analyzed the distribution of stellar orbits in an axisymmetric galaxy and found that total number of stars that can interact with the central SMBH binary is six times larger than in the spherical system. In this subsection we will discuss why there is such a discrepancy to our results -we find a much smaller enhancement of TDR in axisymmetric systems.
The main difference between the cited papers and our work is the initial model. In all of the above mentioned papers, a flattened Dehnen model is used (their density profile, given the parameters they chose, is identical also to the Hernquist model). Their models possess a fixed axial ratio (c/a = 0.75) throughout the entire cluster and an initial central cusp, while our rotating initial model has initially a core density distribution in the center, and we have a radial variation of c/a from nearly spherical (c/a ≈ 1) to about c/a ≈ 0.7 in the outskirts (see Fig. 1). However, in the radius range where most of the disrupted stars originate from (c.f. Fig. 9), the system deviates significantly from spherical symmetry, thus we can confirm that the enhancement of TDR is connected with the non-spherical geometry. But in the relevant region of our ω 0 = 0.6 deviation from spherical symmetry is less than in the other cited papers, which may be an explanation for the weaker effect in our case.
We also notice that even within the cited other papers there are some discrepancies in the results even for models with the same initial density profile. For exam-ple, the enhancement of the number of accreted stars in Vasiliev & Merritt (2013) was smaller than 100% (see Table 2 in their paper), while found a factor of six. On the other hand some of the models in Vasiliev & Merritt (2013) only show mild enhancement which is in the same level as ours. Another example comes from the debate about the "final parsec problem" in SMBH binary evolution. Based on their simulation results, Khan et al. (2013) claimed that the "final parsec problem" is not a problem in axisymmetric host galaxies, while Vasiliev et al. (2014) reached an opposite conclusion according to their simulation. We notice that both of these work employs similar flattened galaxies model, however, they used a different method to generate the initial model. Vasiliev & Merritt (2013), Vasiliev (2014) and Vasiliev et al. (2014) utilized the orbital superposition method of (Schwarzschild 1979) to construct their model. On the other hand, Khan et al. (2013) and used another method called "adiabatic squeeze technique" developed by Holley-Bockelmann et al. (2001). We notice that in the process of adiabatic squeeze, which contains a step which applies a slow and smooth velocity change on the stars in the z direction. This step may artificially reduce the energy and angular momentum of the stars in the model cluster. Although the radius and velocity vectors of the stars are rescaled after the squeeze, it is not clear how the rescaling affects the phase space distribution. Thus it might be possible that the process produces more stars of low energy and low angular momentum. Another evidence of a similar effect can be derived from Vasiliev (2014); while they still use the orbital superposition method they changed the generation of their initial model so that it creates more low energy and low angular momentum stars. In their test run (Fig. 2 in their paper) we see a much larger enhancement of the number of accreted stars compared to Vasiliev & Merritt (2013). So to add more low energy and angular momentum stars seems to be promising in abridging the different enhancement factors between Vasiliev & Merritt (2013) and . We suggest that a detailed comparison between models constructed with these two methods (and their phase space distribution) should be performed in order to explain the discrepancy.
According to the central two parsecs of their model galaxy exhibit a slight triaxiality, which could also introduce some additional centrophilic orbits, thus increase the number of stars that can interact with the central SMBH binary.
Before finishing this section, we want to make a final remark on the result of . Their model integrates individual orbits in a fixed model potential with one SMBH in the center, in a static way. So the number of stars that can interact with the central SMBH binary according to their results should be considered as an upper limit. Once two-body relaxation is turned on, some of the stars that are supposed to be inside the loss cone might be scattered out. And the presence of a SMBH binary in an evolving system may also affect the result of how many stars can interact with them.
LOSS CONE IN AXISYMMETRIC POTENTIAL
First, we summarize the loss cone theory for stellar orbits in a spherically symmetric gravitational potential, in order to discuss different behavior in an axisymmetric potential later. If a stellar orbit has a pericenter distance less than the tidal radius it is considered to be in the loss cone. In spherical symmetry the boundary of the loss cone can be expressed in terms of a critical loss cone angular momentum J lc ≈ √ 2GM • r t (if r ≫ r t ; cf. e.g. Amaro-Seoane et al. (2004)). The loss cone is then defined as the region in phase space where the angular momentum J of a star fulfils J < J lc . All stars inside the loss cone will reach the tidal radius within a dynamical (orbit) time scale. As a consequence the loss cone would become empty in that relatively short time. Once a star is inside the loss cone and reaches the tidal radius, we assume that it will be destroyed by the BH's tidal force instantaneously and add its total mass to the black hole at the same moment. Most authors studying stellar dynamics and TDR of star clusters around a BH used similar approximations. Rees (1988) already argued that the stellar debris after tidal disruption will make several orbits until it is finally accreted by the BH; nevertheless the orbital time near the BH is very short compared to the original orbital time of the star before its disruption. Recent detailed simulations on tidal disruptions (Guillochon & Ramirez-Ruiz 2013;Hayasaki et al. 2013Hayasaki et al. , 2015 show that in some case not all material of the star may be accreted and that general assumptions about the tidal fallback rate are not correct; for example in a longer lived accretion disk may form, which would delay the black hole growth. In a spherical system, without interactions between the stars, angular momentum J would be strictly conserved. So, without any repopulation of the loss cone, the accretion process would stop after a few dynamical times. But stars do interact with each other while moving inside the star cluster by two body relaxation through mutual encounters; in this process they can exchange angular momentum and energy and so the loss cone will be repopulated in the two body relaxation time scale, which is generally long compared to the dynamical time (Cohn & Kulsrud 1978;Amaro-Seoane et al. 2004). The repopulation of the loss cone is modelled in these papers as a diffusive process using the Fokker-Planck approximation.
In an axisymmetric potential, the situation is more complex since J is not a conserved quantity. It changes continuously due to the non-central force resulting from the geometry of the potential. In this case, stars with J > J lc may have a chance to drift into the loss-cone and get disrupted. In other words, the loss cone is enlarged in the J dimension in axisymmetric potential. However, the z component of angular momentum J z is still conserved, so a solid boundary of the loss-cone is J z ≤ J lc . Magorrian & Tremaine (1999) investigated this topic using a symplectic map introduced by Touma & Tremaine (1997). In this work, we analyze the enlarged loss cone in phase space in terms of energy E, modulus of angular momentum J and the z component of angular momentum J z for stellar orbits near the BH. We use a different approach as Magorrian & Tremaine (1999) here, which is based on a numerical particle scattering experiment. In what follows, we first describe the method we used in this experiment, then present our results.
First step, we need to know the smooth gravi-tational potential as a function of position without the fluctuations due to the discrete particle structure. We use a so-called self-consistent field code (SCF, Hernquist & Ostriker (1992)) to generate the analytical function for the gravitational potential. The expansion coefficients C lm , D lm , E lm , F lm used in computing forces (Eq.(3.21)-(3.23) in Hernquist & Ostriker (1992)) are computed based on snapshot data generated during the direct N -body simulation. By default, the code uses radial basis functions labeled from n = 0 to n max = 14, and spherical harmonic function truncated at l max = 10. Our particle distribution is self consistently achieved as a consequence of the co-evolution of stars and BH. Using the SCF code means that all two-body interactions are smoothed out in the experiment. Because we assume that most of the two-body interactions happens during the apocenter passage, which is also used in Touma & Tremaine (1997). After getting the coefficients, we can calculate the acceleration, jerk and do orbit integration using a Hermite 4 th integrator with variable time steps, developed by ourselves. This code works very well and the energy and angular momentum errors of the test particle stays in the level of 10 −9 over long time integration. In an axisymmetric system all coefficient with m = 0 should be 0. But in practice one will get some small numbers very close to 0 due to particle noise. We just ignore these terms, otherwise J z would no longer be conserved. We also ignore coefficients with odd l, because the rotating system should be symmetric about the equatorial plane and do not have pear-like shape.
Next step is to generate initial positions and velocities for test particles. The basic idea of this experiment is to do parameter space scanning. We uniformly sample E, J and J z , all test particles are initially put at their apocenter. Firstly, we choose a particular energy and calculate J lc through equation J lc = r t 2(Φ(r t ) − E). Then we choose a pair of (J, J z ), J can be a few times larger than J lc but J z keeps smaller than J lc . Given the combination of (E, J, J z ) and the potential distribution we can find the apocenter position given by (r, θ). Here r is distance to center and θ is the angle between position vector and z-axis. We note that there are actually four parameters (E, J, J z , θ) to define the initial conditions for a particular orbit. So we further sampled 100 data points in θ dimension. In order to plot the result in a 2D plane, we introduce a filling factor P for every (E, J, J z ) combination to describe this θ dependence, which is the fraction of stars in the loss cone for a given combination of (E, J, J z ) (number of data points in loss cone divided by total sample size, e.g. 100), meaning that among all stars with same (E, J, J z ) only a fraction of P are inside the loss cone.
By our definition a star in the loss cone will be disrupted by the BH within one dynamical time, so for every test particle we only integrate their orbits for one orbital cycle. If a particle comes back to its apocenter, we consider it as out of the loss cone and move to the next integration with new initial orbital data. Fig. 4 shows results from the experiment in a slowly rotating model (ω 0 = 0.3), it represents the loss cone shape in phase space. Since J is not conserved we use its initial value at apocenter for the figure; at the time of disruption J must be less than J lc . From panel a) to d), the energy of the test particles are in descending sequence, so their position of apocenters are getting closer and closer to central BH. One can see that the whole plane comprises 3 regions: 1) inner region where P equals 1, meaning particles with these (E, J, J z ) can hit the BH within one dynamical time scale; 2) transition region where P is non-zero but less than 1, particles with these (E, J, J z ) have a chance to hit the BH depending on their apocenter position (θ value); 3) outer region where P = 0, none of particle in this region can hit the BH. In panel a) one can see only a few points are red and a lot of points are located in transition region. From a) to d), the fraction of P = 1 points in the (J, J z )-plane increases and the transition region is compressed by the inner and outer region in horizontal direction (J dimension). This is because test particles with high energy (loosely bound or unbound with respect to the BH) can go beyond the BH's influence radius to the intermediate and outer regions of the cluster, where the axisymmetric stellar potential dominates. The angular momentum of these test particles will have large variations. So a wide transition region exists in high energy cases. But in the low energy case (stars strongly bound to the BH), e.g. panel d), test particles are moving inside the BH's influence sphere where the potential is dominated by the BH and thus approaches spherical symmetry. All loss cone stars following the classical loss cone approximation, should have both J and J z to be smaller than J lc . In all panels of Fig. 4, on the contrary, we see how stars with J > J lc could be still in the new, extended loss cone of an axisymmetric system with a certain non-zero probability.
For faster rotating models (ω 0 = 0.6) the results are similar. Three regions are presented on the (J, J z ) plane, however, the extent of each region is different from the counterpart of same energy in slow rotating model. Fig. 5 gives an example, in both left (ω 0 = 0.3) and right (ω 0 = 0.6) panel the test particle have same energy, however, the resulting appearances are quite different. In the left panel we see the the outer border extended to J = 0.024, while in the right one the outer border goes to J = 0.04 and is not as clear as that in the left panel. Also in the right panel the red region is almost disappeared. These results show how rotation modifies the loss cone shape in phase space. In both of these plots, the maximum radius stars can achieve are roughly the same. However, faster rotation means we have a more flattened cluster shape, which enhances the torque acting on stars, thus the variation in J becomes larger. So, the higher the degree of rotation in the stellar system, the larger is the extension of the loss cone in J direction.
On first glance at Fig. 4 (also Fig. 5) one might think that the loss cone is generally enlarged by a significant factor. However, as we pointed out above, there is a filling factor P for every point on the (J, J z ) plane. To find the net enlargement of the loss cone in axisymmetric potentials we introduce an effective area S of the loss cone in these plots by integrating the filling factor P over the (J, J z ) plane. For example, the effective area of the classical loss cone is just given by the size of the triangle J, J z < J lc in our plots, since in the classical case P is unity everywhere in this triangle region. Now we compare the loss-cone size comparing the integrals S with each other. We define the quotient α lc = S ef f /S lc , where S lc = J 2 lc /2 is the classical loss cone integral. α lc is plotted in Fig. 6 as a function of binding energy |E|. In the plot we show both slow and fast rotating models at two different evolution times. For the slow rotating model the ratio α lc is even smaller than unity for binding energies larger than 1.4 -1.5, meaning that at large |E| the loss cone is smaller compared to the classical one. This is caused by the reduction of the probability P at the boundary and inside the classical loss cone region J ≤ J lc . P is decreasing from inside toward outside. While for intermediate |E| case, although P is still decreasing function of J, the large number of valid points overwhelms, so the net effect is increasing the effective area. However, if one goes further toward small |E| the ratio will drop again, like the case of fast rotating model. This is just because P is sufficiently small in this case and win the game. For fast rotating model, another interesting feature is the ratio drops below 1 at |E| = 1.8. From this figure we see that the enlargement of loss cone, quantified by the ratio α lc as a function of binding energy. Interestingly, the change of the effective loss cone size in every energy slice is less than 5-10%. These mild changes seem unable to raise TDR with the amount observed in the simulation, to address this it will be useful if we can estimate the TDR based on the effec-tive loss cone measurement and compare with simulation. However, the knowledge of how stars are distributed in energy and angular momentum is required. With the limited particle numbers of the model cluster, it is difficult to get an accurate and reliable distribution function. Also in current work, we sample the energy space with large intervals (∆E = 0.1), which may cause large errors in the estimated TDR. So we did not make the estimation. There are still plenty of works could be done with this topic. w030, T = 500 1000 w060, T = 500 1000 Figure 6. Ratio between effective area S ef f of loss-cone in axisymmetric system and S lc in spherical system.
ORBITAL PROPERTIES OF DISRUPTED STARS
In this section, we investigate the origin of disrupted stars. Under the assumption that stars in loss cone can survive for only one orbital period, the origin of these stars can be examined by looking at their energy and angular momentum, as well as their origin (apocenter) in spatial coordinates (radius and angle θ). In spherical systems one can use effective potential to compute the apocenter of orbit, but in the axisymmetric case we do not have such convenient solutions except to run the simulation twice. In the first run we find out the ID for those stars that will be disrupted by BH. Then, in the second run we make records for these stars more frequently than other stars, in order to catch their last apocenter position.
We found in the beginning that the total TDR, especially for small r t does only marginally depend on the rotation of the system; consistent with this we found in the previous chapter that the loss cone structure does change significantly, but the total integral over the loss cone space for axisymmetric systems yields only relatively small changes. Still it is interesting to study how the orbital properties of stars, which are tidally disrupted, change with the rotation of the system. In order to address this, we now turn back to our full N body simulations and study the distribution of |E| (Fig. 7) and J (Fig. 8) of the disrupted stars at their apocenter passage in three time intervals. From Fig. 7 one can see that most of the tidally disrupted stars have a binding energy between 1 and 2, coincident with the small bumps in Fig. 6 where α lc > 1. Another evidence comes from the distribution of J as shown in Fig. 8, where one can see the peaks are lying outside of the J lc which is roughly 0.015. The peaks are moving toward larger J, which is caused by the increase of BH mass (recall the expression for J lc ). A significant fraction of stars comes from places outside of the classical loss cone in (J, J z ) plane. In spherical systems it is usually sufficient to describe the apocenter of an orbit by its radial distance from the center (the BH); the orientation of the orbit does not play any role for the orbital time and the nature of the encounter with the central BH. However, in axisymmetric systems, orbits with different angle θ (the angle between position vector of the star at apocenter and the z-axis) will differ from each other significantly. Therefore we have to describe the distribution of apocenters of tidally disrupted stars in terms of both the r (Fig. 9) and θ (Fig. 10) dimension. From Fig. 9 one can see the peaks are quite far from the BH, in a region comparable to the BH influence radius, which is similar to the apocenter distribution in spherical systems (Paper I). The difference turns out to be in the θ dimension, as shown in Fig. 10. We compare the θ distribution between spherical and axisymmetric systems. Imagine we project all the apocenter points onto a sphere with radii equals 1. The measured number counts in each θ bin ∆N (θ) are computed by 2π · Σ(θ) · sin(θ)∆θ, where Σ(θ) is the surface density of projected points on the unit sphere. If apocenters are uniformly distributed with θ, Σ(θ) is constant, then ∆N (θ) ∝ sin(θ)∆θ. Here we choose an equal bin size, so the measured number count should follow a sin(θ) curve. The right panel of Fig. 10 plots θ distribution for spherical model, which is taken from our last work (Paper I). In left panel we see the last apocenter distribution have deficit at polar region comparing to sin(θ) curve, and excess at places beyond and below the equatorial plane, showing a double peak feature. The deficit at the polar region may have something to do with the flattening of the cluster, however, this is not the only reason. The double peak feature around the equatorial plane obviously does not relate to a geometrical origin, otherwise the peak should be placed at the equatorial plane. In Fig. 11 we compare the θ distribution between slow and fast rotating models. One can see that in fast rotating model, the double peaks are more significant, accompanied by a further deficit in the angle range 0.2 − 0.4π and 0.6 − 0.8π. In order to understand the double peak feature, we turn to the orbit structure of these disrupted stars. In non-spherical symmetric stellar system with a SMBH in its center, the space populated by stars can be divided into three parts depending on the distance to the BH, namely the regular, chaotic and mixing region (Poon & Merritt 2001). Inside the BH's influence radius r h , the potential felt by the star is dominated by the BH plus a small perturbation from the non-spherical stellar potential. In this region, the motion of stars is essentially regular, as in a spherical potential. Outside of r h , stars passing the center will suffer a large angle deflection by the BH, which in conjunction with the non-spherical potential near and outside r h , could make their orbits stochastic.
We are interested in stellar orbits in an axisymmetric stellar potential, which can get close to the central BH. These are typically two classes of orbits, short-axis tube (SAT) and saucer (see Vasiliev (2014) for example); they can be distinguished by their third integral of motion I 3 . Although I 3 may help us quickly distinguish orbit families, finding the functional form of I 3 is difficult (see Lupton & Gunn (1987) and discussion in Sridhar & Touma (1999)) and is beyond the scope of this paper. We choose alternative ways to do orbit classification, such as Surface of Section (SoS) plot and Fourier analysis of J x (see Appendix). Figure 12. Examples of orbital structure for SAT (left) and saucer (right) orbit. Stars achieve maximum J when their instant orbital plane coincide with B plane, while minimum when coincide with A plane. Fig. 12 gives examples of SAT and saucer orbits in configuration space. The plot is made in cylindrical coordinates so that one can catch the main point easily. For SAT orbit, one can see its apocenter can go both above and below the equatorial plane. While apocenter of saucer orbit can only exist on one side of mid-plane, due to restrictions by the 3rd integral. We also check the value of J at each apocenter passage. We find that SAT orbit achieve its minimum J at the equatorial plane; a saucer orbit cannot reach the equatorial plane, but its minimum J is achieved at the place which is next to the equatorial plane as marked in the plot by A plane. Recall in the last section we said no matter what J one star has at the apocenter, at the time of disruption it must be smaller than J lc . So the last apocenter place should be around the A plane. This seems to be promising to explain the double peak in θ distribution, however, need to be confirmed. In order to see this we try to do orbit classification for the disrupted stars, which is computationally expensive. So we just randomly select a subsample of disrupted stars and divide them into 3 orbit families: SAT, saucer and others (here "others" means they do not belong to the former two families, and may be chaotic orbits). Among the 2943 sample stars, 1719 are classified as "others", 757 as saucer and 467 as SAT. Then we re-plot the r and θ distribution for different orbit families in Fig. 13. The results show that the apocenter distribution of different orbit families not only differs in θ but also in r.
One can see the innermost region is dominated by SAT orbits, and concentrated to the equatorial plane. Intermediate radius is mostly occupied by saucer orbits, and the distribution in θ shows double peaks as expected. Further out is the region dominated by orbits marked as others. These orbits can go outside of the influence radius and are basically chaotic orbits. From Fig. 13 one can also find out the fractions of each orbit family contributing to the budget of disrupted stars: the largest fraction comes from chaotic orbits; SAT orbits contribute least to the budget because they are deeply buried in the cluster center where the total star number is small; the intermediate contribution is from saucer orbits which create the two peaks in the θ distribution.
6. CONCLUSIONS AND DISCUSSIONS Tidal Disruption (TD) of stars by supermassive central black holes (SMBH) from dense rotating star clusters is modelled by high-accuracy direct N -body simulation. As in a previous paper on spherical star clusters we study the time evolution of the stellar tidal disruption rate and the origin of tidally disrupted stars, now according to several classes of orbits which only occur in rotating axisymmetric systems (short axis tube and saucer). In empty loss cone regime, comparing spherically symmetric and axisymmetric systems we find a higher TD rate in large r t models in axisymmetric case, but for small r t casesomewhat surprisingly -there is virtually no difference in the TD rate, maybe a small increase due to axisymmetry.
We define an extended loss cone by the condition that stars in the axisymmetric potential reach the BH within one orbit. A detailed analysis shows that the structure of the loss cone significantly differs from the spherical case; if J lc is the critical angular momentum to be in the loss cone in a spherical system, and J, J z are the total and z-component of the angular momentum of a stellar orbit, there are many stars with J > J lc in the loss cone; since, however, there are also some stars with J > J lc , which are now not in the loss cone. In the total balance the number of loss cone stars is only very moderately increased.
In the experiment of measuring the shape of loss cone, we assume the test star can survive only one dynamical time in collisional system, after one orbit it will be "kicked" to another place in phase space due to interactions with other stars. However, in collisionless limit, if we allow the test star to survive more orbit cycles, test star with much higher J will also have chance to get rid of its angular momentum and be disrupted by BH. Then it is possible that an even larger loss cone region in phase space than what we presented here may exist, and result in a higher TDR. In order to check this, simulations with much more particles are needed and we would like to leave this task for future work.
The orbit type of disrupted stars strongly depends on energy as we discuss in detail in the previous sections. TD of stars most strongly bound to the BH are dominated by short-axis tube (SAT) orbits. In intermediate regions saucer orbits dominate, which create a characteristic double peak structure in the last apocenter position of their orbit relative to the equatorial plane. And further out chaotic orbits.
It is known for almost half a century that tidal disruption of stars should occur near SMBH, but only much more recently the X-ray emission of tidal disruption events has been detected (Komossa 2002;Komossa & Merritt 2008). A simple argument on the fallback time for tidal debris by Rees (1988) has led to the prediction of a characteristic power law of the light curve with time, which can be used to distinguish TD events from other transients. It is interesting that a SMBH binary can cause characteristic disruptions in such an otherwise standard TD light curve (Liu et al. 2014). Hayasaki and collaborators claim that eccentric TD events lead to somewhat longer lived central accretion disks (Hayasaki et al. 2013(Hayasaki et al. , 2015. It will be very interesting to see whether and how the evolution of tidal debris and the fallback rate are affected by different orbits of the disrupted stars as discussed here.
It has been claimed that rotation may help to quickly refill loss cones around binary supermassive black holes, which helps significantly to accelerate shrinking and final coalescence of SMBH binaries in cosmologically short time scales (Berczik et al. 2006;Preto et al. 2011;Khan et al. 2013;Khan 2014). In our paper we study by direct N -body simulation the tidal accretion of stars and their orbital parameters in rotating axisymmetric systems. We confirm the result of Vasiliev & Merritt (2013) that there is an increase in the rate of refilling of the loss cone, but it is moderate. However, the situation deserves more detailed study, because a SMBH binary creates a much stronger deviation from spherical symmetry than the one used in our models with single SMBH. And second the detailed structure of the rotation in a central nuclear star cluster could affect the enhancement of the loss cone.
APPENDIX
ORBIT CLASSIFICATION
Surface of Section
From Fig. 14 we can see the whole accessible region on (R, v R ) plane is divided into two parts (note that points with opposite v R actually belongs to same orbit, so this plot is symmetric with horizontal axis). Each part represents a family of orbit. Curves that intersect with R-axis are footprints of short axis tube (SAT) orbits, others are of saucer orbits. x axis is distance R to origin on equatorial plane. y axis is v R ≡ dR/dt when star go across the equatorial plane.
Fourier Analysis of J x evolution In axisymmetric potential, force is not centripetal hence exerted a torque on the star which will change the x and y components of its angular momentum. Fig 15 show the time evolution of J x for both SAT and saucer orbits. The pattern of J x and J y are the same but shifted with a phase of π/2, so in the following discussion we only focus on J x . Furthermore, the evolution of J x shows some quasi-periodicity. From eye inspection, one can guess the mathematical expressions for the curves.
As shown in Fig. 15, the curve for SAT orbits seems to be represented by sin(f 1 t)(1 + cos(f 2 t)) (f 1 < f 2 ), which can be further converted to sin(f a t) + sin(f b t) + sin(f c t) (ignore coefficients before the trigonometric functions), with f a = f 1 , f b = f 2 − f 1 and f c = f 2 + f 1 . If we perform a Fourier analysis on this curve, we expect to find 3 principal frequencies: f a , f b , f c in ascending sequence. And these 3 frequencies satisfy the equation f a = (f c − f b )/2.
For saucer orbits, the curve seems to be represented by sin(f 1 t)cos(f 2 t) (f 1 < f 2 ), following the same procedure we expected to find 2 principal frequencies: f a , f b , with f a = f 2 − f 1 and f b = f 2 + f 1 .
A demonstration is shown in Fig. 16, one can clearly see the 3 principal frequencies for SAT orbits and the 2 for saucer orbits. Some of the small peaks appeared at higher frequencies which is the order harmonics and some are produced from other components.
We use both methods to cross check the validity of orbit classification for the tidally disrupted stars.
Figure 1 .
1Axial ratio for rotating models as a function of radius. For each model we show the axial ratio measured at different evolution stage: T=0 (red); T=500 (green); T=1000 (blue). Lines with symbols are results for ω 0 = 0.3 model, lines without symbols are for ω 0 = 0.6 models.
Figure 2
2Figure 2. TDR as a function of time in units of initial half mass relaxation time; averaged over intervals of 1/4t rh . Top panels: mass accretion rate; bottom panels: particle accretion rate; left and right panels for two different tidal radii as indicated. Curves with symbols are stand for 128K models, those without symbols are stand for 64K models. Line thickness indicate different rotating parameters.
Figure 3 .
3x axis is time in unit of initial half-mass relaxation time. y axis for panel a) and b) is the averaged mass accretion rate in given time range (i.e. 1/4 t rh ); y axis for panel c) and d) is the number accretion rate. Panel a) and c) show the result for rt = 10 −3 . Panel b) and d) show the result for rt = 10 −4 .
Fig. 3
3compares the TDR of classic King model (W = 6
Figure 4 .
4Filling factor P of the loss coneas a function of J and Jz for different energies; the x axis is the modulus of the total angular momentum J, while the y axis is itsz component Jz. Panel a) correspond to E = −1.3, b) E = −1.5, c) E = −1.7, d) E = −1.9.Colors represent the filling factor P in percentage. All data are given in standard N -body units.
Figure 5 .
5Comparison between 2 models with same test particle energy −1.5. x axis is module of angular momentum in N -body unit. y axis is z component of angular momentum. Left panel correspond to ω = 0.3; right panel ω = 0.6. Colors indicate the filling factor P in percentage.
Figure 7 .Figure 8 .
78Panel a) and b) show normalized distribution of binding energy |E| of tidally disrupted stars, for different rotating models and for three different time intervals (indicated by color) in the full N -body simulation with rt = 10 −3 . The distribution is normalized to the total number of disrupted stars in each time interval. Panel c) and d) show cumulative fraction profile corresponding to aSame asFig. 7, but here for the distribution of total angular momentum of the disrupted stars.
Figure 9 .Figure 10 .
910Distribution of last apocenter distance of disrupted stars in 3 different time interval. Each curve are normalized to the total number of disrupted stars in given time interval. Normalized distribution of zenith angle θ of last apocenter of disrupted stars. Left panel is axisymmetric model, right panel is spherical model.
Figure 11 .
11Normalized distribution of zenith angle θ of last apocenter of disrupted stars. Compare between slow (w030) and fast (w060) rotating models.
Figure 13 .
13Normalized distribution of apocenter distance r and zenith angle θ of last apocenter of disrupted stars for different orbit families. The distribution is normalized to the number of stars in each orbit family.
Figure 14 .
14Surface of section plot.
Figure 15 .Figure 16 .
1516Time evolution of Jx for SAT orbit (left) and saucer orbit (right). Fourier frequency distribution of Jx. Horizontal axis is frequency in unit of [T ] −1 . Vertical axis is amplitude of corresponding component. Left panel represents SAT orbit, right panel represents saucer orbit.
discovered National Astronomical Observatories of China and Key Lab for Computational Astrophysics, Chinese Academy of Sciences, 20A Datun Rd., Chaoyang District, 100012, Beijing, China Key Lab of Frontiers in Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, P.R. China1 2 Astronomisches Rechen-Institut, Zentrum für Astronomie,
University of Heidelberg, Mönchhofstrasse 12-14, 69120, Heidel-
berg, Germany
3 Main Astronomical Observatory, National Academy of Sci-
ences of Ukraine, 27 Akademika Zabolotnoho St., 03680, Kyiv,
Ukraine
4 Kavli Institute for Astronomy and Astrophysics, Peking Uni-
versity, Beijing, China
5
Table 1
1Full set of our model runs. Column 4 : black hole's tidal radius. Column 5 : total integration time. rt and T are in model unit.Model
N/K ω 0
rt
T
R20w00
64
0.0 10 −3 1500
R30w00
128
0.0 10 −3 1600
R21w00
64
0.0 10 −4 1500
R31w00
128
0.0 10 −4 1300
R20w03
64
0.3 10 −3 1500
R30w03
128
0.3 10 −3 1500
R21w03
64
0.3 10 −4 2600
R31w03
128
0.3 10 −4 2000
R20w06
64
0.6 10 −3 1500
R30w06
128
0.6 10 −3 1500
R21w06
64
0.6 10 −4 1600
R31w06
128
0.6 10 −4 2000
Note. -Column 1 : Model codename. Column 2 : Particle
number in the unit of K(=1024). Column 3 : dimensionless rota-
tion parameter.
Table 2
2TDR results for rt = 10 −3 models.t/t rhṄ0Ṅ3Ṅ3 /Ṅ 0Ṅ6Ṅ6 /Ṅ 0
0.25
14.96 15.41
1.03
16.30
1.09
0.50
12.89 13.67
1.06
17.30
1.34
0.75
10.44 12.07
1.16
14.52
1.39
1.00
8.73
10.32
1.18
12.65
1.45
1.25
7.97
9.73
1.22
11.09
1.39
1.50
6.99
7.93
1.13
10.04
1.44
ACKNOWLEDGEMENTSWe acknowledge support by Chinese Academy of Sciences through the Silk Road Project at NAOC, through the Chinese Academy of Sciences Visiting Professorship for Senior International Scientists, Grant Number 2009S1-5 (RS), and through the "Qianren" special foreign experts program of China.The special GPU accelerated supercomputer laohu at the Center of Information and Computing at National Astronomical Observatories, Chinese Academy of Sciences, funded by Ministry of Finance of People's Republic of China under the grant ZDY Z2008-2, has been used for the simulations.PB acknowledge the special support by the NAS Ukraine under the Main Astronomical Observatory GPU/GRID computing cluster project.SZ thank Yohai Meiron for providing the SCF source code which is used in this work.
. L A Aguilar, D Merritt, ApJ. 35433Aguilar, L. A. & Merritt, D. 1990, ApJ, 354, 33
. P Amaro-Seoane, M Freitag, R Spurzem, MNRAS. 352655Amaro-Seoane, P., Freitag, M., & Spurzem, R. 2004, MNRAS, 352, 655
. F Antonini, R Capuzzo-Dolcetta, A Mastrobuono-Battisti, D Merritt, ApJ. 750111Antonini, F., Capuzzo-Dolcetta, R., Mastrobuono-Battisti, A., & Merritt, D. 2012, ApJ, 750, 111
. P Berczik, D Merritt, R Spurzem, H.-P Bischof, ApJ. 64221Berczik, P., Merritt, D., Spurzem, R., & Bischof, H.-P. 2006, ApJ, 642, L21
P Berczik, K Nitadori, S Zhong, R Spurzem, T Hamada, X Wang, I Berentzen, A Veles, W Ge, International conference on High Performance Computing. Kyiv, UkraineBerczik, P., Nitadori, K., Zhong, S., Spurzem, R., Hamada, T., Wang, X., Berentzen, I., Veles, A., & Ge, W. 2011, in International conference on High Performance Computing, Kyiv, Ukraine, October 8-10, 2011., p. 8-18, 8-18
M Bois, E Emsellem, F Bournaud, K Alatalo, L Blitz, M Bureau, M Cappellari, R L Davies, T A Davis, P T De Zeeuw, P.-A Duc, S Khochfar, D Krajnović, H Kuntschner, P.-Y Lablanche, R M Mcdermid, R Morganti, T Naab, T Oosterloo, M Sarzi, N Scott, P Serra, A.-M Weijmans, L M Young, Astronomical Society of the Pacific Conference Series. W.-H. Sun, C. K. Xu, N. Z. Scoville, & D. B. Sanders47797Galaxy Mergers in an Evolving UniverseBois, M., Emsellem, E., Bournaud, F., Alatalo, K., Blitz, L., Bureau, M., Cappellari, M., Davies, R. L., Davis, T. A., de Zeeuw, P. T., Duc, P.-A., Khochfar, S., Krajnović, D., Kuntschner, H., Lablanche, P.-Y., McDermid, R. M., Morganti, R., Naab, T., Oosterloo, T., Sarzi, M., Scott, N., Serra, P., Weijmans, A.-M., & Young, L. M. 2013, in Astronomical Society of the Pacific Conference Series, Vol. 477, Galaxy Mergers in an Evolving Universe, ed. W.-H. Sun, C. K. Xu, N. Z. Scoville, & D. B. Sanders, 97
. T Böker, S Laine, R P Van Der Marel, M Sarzi, H.-W Rix, L C Ho, J C Shields, AJ. 1231389Böker, T., Laine, S., van der Marel, R. P., Sarzi, M., Rix, H.-W., Ho, L. C., & Shields, J. C. 2002, AJ, 123, 1389
. T Böker, M Sarzi, D E Mclaughlin, R P Van Der Marel, H.-W Rix, L C Ho, J C Shields, AJ. 127105Böker, T., Sarzi, M., McLaughlin, D. E., van der Marel, R. P., Rix, H.-W., Ho, L. C., & Shields, J. C. 2004, AJ, 127, 105
. H Cohn, R M Kulsrud, ApJ. 2261087Cohn, H. & Kulsrud, R. M. 1978, ApJ, 226, 1087
. C Einsel, R Spurzem, MNRAS. 30281Einsel, C. & Spurzem, R. 1999, MNRAS, 302, 81
. A Ernst, P Glaschke, J Fiestas, A Just, R Spurzem, MNRAS. 377465Ernst, A., Glaschke, P., Fiestas, J., Just, A., & Spurzem, R. 2007, MNRAS, 377, 465
. C R Evans, C S Kochanek, ApJ. 34613Evans, C. R. & Kochanek, C. S. 1989, ApJ, 346, L13
. A Feldmeier, N Neumayer, A Seth, R Schödel, N Lützgendorf, P T De Zeeuw, M Kissler-Patig, S Nishiyama, C J Walcher, A&A. 5702Feldmeier, A., Neumayer, N., Seth, A., Schödel, R., Lützgendorf, N., de Zeeuw, P. T., Kissler-Patig, M., Nishiyama, S., & Walcher, C. J. 2014, A&A, 570, A2
. J Fiestas, O Porth, P Berczik, R Spurzem, MNRAS. 41957Fiestas, J., Porth, O., Berczik, P., & Spurzem, R. 2012, MNRAS, 419, 57
. J Fiestas, R Spurzem, MNRAS. 405194Fiestas, J. & Spurzem, R. 2010, MNRAS, 405, 194
. J Frank, M J Rees, MNRAS. 176633Frank, J. & Rees, M. J. 1976, MNRAS, 176, 633
. L J Greenhill, C Henkel, R Becker, T L Wilson, J G A Wouterloot, A&A. 30421Greenhill, L. J., Henkel, C., Becker, R., Wilson, T. L., & Wouterloot, J. G. A. 1995, A&A, 304, 21
. A Gualandris, D Merritt, ApJ. 74474Gualandris, A. & Merritt, D. 2012, ApJ, 744, 74
. J Guillochon, E Ramirez-Ruiz, ApJ. 76725Guillochon, J. & Ramirez-Ruiz, E. 2013, ApJ, 767, 25
. I Hachisu, PASJ. 34313PASJHachisu, I. 1979, PASJ, 31, 523 -. 1982, PASJ, 34, 313
. K Hayasaki, N Stone, A Loeb, MNRAS. 434909Hayasaki, K., Stone, N., & Loeb, A. 2013, MNRAS, 434, 909
. K Hayasaki, N C Stone, A Loeb, Hayasaki, K., Stone, N. C., & Loeb, A. 2015, ArXiv e-prints
The Use of Supercomputers in Stellar Dynamics. D C Heggie, R D Mathieu, Lecture Notes in Physics. P. Hut & S. L. W. McMillanBerlin Springer Verlag267233Heggie, D. C. & Mathieu, R. D. 1986, in Lecture Notes in Physics, Berlin Springer Verlag, Vol. 267, The Use of Supercomputers in Stellar Dynamics, ed. P. Hut & S. L. W. McMillan, 233
. L Hernquist, J P Ostriker, ApJ. 386375Hernquist, L. & Ostriker, J. P. 1992, ApJ, 386, 375
. K Holley-Bockelmann, J C Mihos, S Sigurdsson, L Hernquist, ApJ. 549862Holley-Bockelmann, K., Mihos, J. C., Sigurdsson, S., & Hernquist, L. 2001, ApJ, 549, 862
. S Inagaki, I Hachisu, PASJ. 3039Inagaki, S. & Hachisu, I. 1978, PASJ, 30, 39
F M Khan, 2-1040th COSPAR Scientific Assembly. Moscow, Russia40COSPAR MeetingKhan, F. M. 2014, in COSPAR Meeting, Vol. 40, 40th COSPAR Scientific Assembly. Held 2-10 August 2014, in Moscow, Russia, Abstract E1.20-5-14., 1462
. F M Khan, I Berentzen, P Berczik, A Just, L Mayer, K Nitadori, S Callegari, ApJ. 75630Khan, F. M., Berentzen, I., Berczik, P., Just, A., Mayer, L., Nitadori, K., & Callegari, S. 2012, ApJ, 756, 30
. F M Khan, K Holley-Bockelmann, P Berczik, A Just, ApJ. 773100Khan, F. M., Holley-Bockelmann, K., Berczik, P., & Just, A. 2013, ApJ, 773, 100
. F M Khan, A Just, D Merritt, ApJ. 73289Khan, F. M., Just, A., & Merritt, D. 2011, ApJ, 732, 89
. E Kim, C Einsel, H M Lee, R Spurzem, M G Lee, MNRAS. 334310Kim, E., Einsel, C., Lee, H. M., Spurzem, R., & Lee, M. G. 2002, MNRAS, 334, 310
S Komossa, Reviews in Modern Astronomy. R. E. Schielicke1527Reviews in Modern AstronomyKomossa, S. 2002, in Reviews in Modern Astronomy, Vol. 15, Reviews in Modern Astronomy, ed. R. E. Schielicke, 27
. S Komossa, D Merritt, ApJ. 68321Komossa, S. & Merritt, D. 2008, ApJ, 683, L21
. B Li, K Holley-Bockelmann, F Khan, A P Lightman, S L Shapiro, ApJ. 211244Li, B., Holley-Bockelmann, K., & Khan, F. 2014, ArXiv e-prints Lightman, A. P. & Shapiro, S. L. 1977, ApJ, 211, 244
. F K Liu, S Li, S Komossa, ApJ. 786103Liu, F. K., Li, S., & Komossa, S. 2014, ApJ, 786, 103
. J M Lotz, R Telford, H C Ferguson, B W Miller, M Stiavelli, J Mack, ApJ. 552572Lotz, J. M., Telford, R., Ferguson, H. C., Miller, B. W., Stiavelli, M., & Mack, J. 2001, ApJ, 552, 572
. R H Lupton, J E Gunn, AJ. 931106Lupton, R. H. & Gunn, J. E. 1987, AJ, 93, 1106
. J Magorrian, S Tremaine, MNRAS. 309447Magorrian, J. & Tremaine, S. 1999, MNRAS, 309, 447
E A Malkov, E Y Vil'koviskij, T N Nuzhnova, R Shpurtsem, Problems of Physics of Stars and Extragalactic Astronomy. Malkov, E. A., Vil'Koviskij, E. Y., Nuzhnova, T. N., & Shpurtsem, R. 1993, in Problems of Physics of Stars and Extragalactic Astronomy, ed. A. V. Kurchakov, 139-152
. D Merritt, M Y Poon, ApJ. 606788Merritt, D. & Poon, M. Y. 2004, ApJ, 606, 788
. M Milosavljević, ApJ. 60513Milosavljević, M. 2004, ApJ, 605, L13
. M Miyoshi, J Moran, J Herrnstein, L Greenhill, N Nakai, P Diamond, M Inoue, Nature. 373127Miyoshi, M., Moran, J., Herrnstein, J., Greenhill, L., Nakai, N., Diamond, P., & Inoue, M. 1995, Nature, 373, 127
. D A Neufeld, P R Maloney, ApJ. 44717Neufeld, D. A. & Maloney, P. R. 1995, ApJ, 447, L17
. M Y Poon, D Merritt, ApJ. 549192Poon, M. Y. & Merritt, D. 2001, ApJ, 549, 192
. M Preto, I Berentzen, P Berczik, R Spurzem, ApJ. 73226Preto, M., Berentzen, I., Berczik, P., & Spurzem, R. 2011, ApJ, 732, L26
. M J Rees, Nature. 333523Rees, M. J. 1988, Nature, 333, 523
. R Schödel, A Feldmeier, D Kunneriath, S Stolovy, N Neumayer, P Amaro-Seoane, S Nishiyama, A&A. 566236ApJSchödel, R., Feldmeier, A., Kunneriath, D., Stolovy, S., Neumayer, N., Amaro-Seoane, P., & Nishiyama, S. 2014, A&A, 566, A47 Schwarzschild, M. 1979, ApJ, 232, 236
. A C Seth, R D Blum, N Bastian, N Caldwell, V P Debattista, ApJ. 687997Seth, A. C., Blum, R. D., Bastian, N., Caldwell, N., & Debattista, V. P. 2008, ApJ, 687, 997
. A C Seth, J J Dalcanton, P W Hodge, V P Debattista, AJ. 1322539Seth, A. C., Dalcanton, J. J., Hodge, P. W., & Debattista, V. P. 2006, AJ, 132, 2539
R Spurzem, P Berczik, I Berentzen, W Ge, X Wang, H.-Y Schive, K Nitadori, T Hamada, Large Scale Computing Techniques for Complex Systems and Simulations. W. Dubitzky, K. Kurowski, & B. SchottWiley PublishersSpurzem, R., Berczik, P., Berentzen, I., Ge, W., Wang, X., Schive, H.-Y., Nitadori, K., & Hamada, T. 2011a, in Large Scale Computing Techniques for Complex Systems and Simulations, ed. W. Dubitzky, K. Kurowski, & B. Schott, Wiley Publishers, 35-58
. R Spurzem, P Berczik, T Hamada, K Nitadori, G Marcus, A Kugel, R Männer, I Berentzen, J Fiestas, R Banerjee, R Klessen, 26145Computer Science -Research and Development (CSRD)Spurzem, R., Berczik, P., Hamada, T., Nitadori, K., Marcus, G., Kugel, A., Männer, R., Berentzen, I., Fiestas, J., Banerjee, R., & Klessen, R. 2011b, Computer Science -Research and Development (CSRD), 26, 145
. S Sridhar, J Touma, MNRAS. 303483Sridhar, S. & Touma, J. 1999, MNRAS, 303, 483
. J Touma, S Tremaine, MNRAS. 292905Touma, J. & Tremaine, S. 1997, MNRAS, 292, 905
. S D Tremaine, J P Ostriker, Spitzer, L Jr, ApJ. 196407Tremaine, S. D., Ostriker, J. P., & Spitzer, Jr., L. 1975, ApJ, 196, 407
. E Vasiliev, Classical and Quantum Gravity. 31244002Vasiliev, E. 2014, Classical and Quantum Gravity, 31, 244002
. E Vasiliev, F Antonini, D Merritt, ApJ. 785163Vasiliev, E., Antonini, F., & Merritt, D. 2014, ApJ, 785, 163
. E Vasiliev, D Merritt, ApJ. 77487Vasiliev, E. & Merritt, D. 2013, ApJ, 774, 87
. J Wang, D Merritt, ApJ. 600149Wang, J. & Merritt, D. 2004, ApJ, 600, 149
. S Zhong, P Berczik, R Spurzem, ApJ. 792137Zhong, S., Berczik, P., & Spurzem, R. 2014, ApJ, 792, 137
| [] |
[
"Mathematical Subject Classification. Primary 17B10, 17B20; Secondary 35C05",
"Mathematical Subject Classification. Primary 17B10, 17B20; Secondary 35C05"
] | [
"Xiaoping Xu \nInstitute of Mathematics\nAcademy of Mathematics & System Sciences\nHua Loo-Keng Mathematical Laboratory\nChinese Academy of Sciences\n100190BeijingP.R.China\n"
] | [
"Institute of Mathematics\nAcademy of Mathematics & System Sciences\nHua Loo-Keng Mathematical Laboratory\nChinese Academy of Sciences\n100190BeijingP.R.China"
] | [] | Given a weight of sl(n, C), we derive a system of variable-coefficient secondorder linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group S n on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exactly spanned by {σ(1) | σ ∈ S n }. Moreover, the singular vectors of sl(n, C) in the Verma module are given by those σ(1) that are polynomials. The well-known results of Verma, Bernstein-Gel'fand-Gel'fand and Jantzen for the case of sl(n, C) are naturally included in our almost elementary approach of partial differential equations.semisimple Lie algebra to determining the embeddings of the other Verma modules into 1 | 10.1007/s10468-010-9239-1 | [
"https://arxiv.org/pdf/0903.4239v1.pdf"
] | 18,610,311 | 0903.4239 | 79095313c6b54478d346380fc149820598e13ef6 |
Mathematical Subject Classification. Primary 17B10, 17B20; Secondary 35C05
2000
Xiaoping Xu
Institute of Mathematics
Academy of Mathematics & System Sciences
Hua Loo-Keng Mathematical Laboratory
Chinese Academy of Sciences
100190BeijingP.R.China
Mathematical Subject Classification. Primary 17B10, 17B20; Secondary 35C05
2000Differential-Operator Representations of S n and Singular Vectors in Verma Modules 1
Given a weight of sl(n, C), we derive a system of variable-coefficient secondorder linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group S n on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exactly spanned by {σ(1) | σ ∈ S n }. Moreover, the singular vectors of sl(n, C) in the Verma module are given by those σ(1) that are polynomials. The well-known results of Verma, Bernstein-Gel'fand-Gel'fand and Jantzen for the case of sl(n, C) are naturally included in our almost elementary approach of partial differential equations.semisimple Lie algebra to determining the embeddings of the other Verma modules into 1
Introduction
One of the most beautiful things in Lie algebras is the highest weight representation theory. It was established based on the induced modules of a Lie algebra with respect to a Cartan decomposition from one-dimensional modules of the Borel subalgebra associated with a linear function (weight) on the Cartan subalgebra. These modules are now known as Verma modules [V1]. A singular vector (or canonical vector) in a Verma module is a weight vector annihilated by positive root vectors. It is well known that the structure of a Verma module of a finite-dimensional simple Lie algebra is completely determined by its singular vectors (cf. [V1]). In this paper, we find explicit formulas for singular vectors in Verma modules for the Lie algebra sl(n, C) in terms of a differential-operator representation of the symmetric group S n on a certain space of truncated power series.
The structure of Verma module was first studied by Verma [V1]. Verma reduced the problem of determining all submodules of a Verma module of a finite-dimensional the objective module. He proved that the multipicity of the embedding is at most one.
Bernstein, Gel'fand and Gel'fand [BGG] introduced the well-known useful notion of category O of representations, and found a necessary and sufficient condition for the existence of such a embedding in terms of the action of Weyl group on weights. Sapovolov [S] introduced a certain bilinear form on a universal envelopping algebra. Lepowsky [L1-L4] studied analogous induced modules with respect to Iwasawa decomposition that is more general than Cartan decomposition, and obtained similar results as those in [V1] and [BGG]. These modules are now known as generalized Verma modules.
Jantzen [J1, J2] introduced his famous "Jantzen filtrations" on Verma modules and used Sapovolov form to determine weights of singular vectors in Verma modules. Verma modules of infinite-dimensional Lie algebras were first studied by Kac [Kv1]. Kac and Kazhdan [KK] generalized the results of Verma [V1] and Bernstein-Gel'fand-Gel'fand [BGG] to the contragredient Lie algebra corresponding a symmetrizable generalized Cartan matrix. Deodhar, Gabber and Kac [DGK] generalized the results further to more general matrix Lie algebras. Rocha-Caridi and Wallach [RW1,RW2] generalized the results of Verma [V1] and Bernstein-Gel'fand-Gel'fand [BGG] to a class of graded Lie algebras possessing a Cartan decomposition and obtained Jantzen's character formula corresponding to the quotient of two Verma modules. The resolutions of irreducible highest weight modules over rank-2 Kac-Moody algebras were constructed.
One of the fundamental and difficult remaining problems in this direction is how to determine the singular vectors explicitly. Malikov, Feigin and Fuchs [MFF] introduced a formal manipulation on products of several general powers of negative simple root vectors and used free Lie algebras to give a rough condition when such product is well defined.
It seems to us that their condition can not be verified in general and their method can practically be applied only to finding very special singular vectors.
In this paper, we introduce an almost elementary partial differential equation approach of determining the singular vectors in any Verma module of sl(n, C). First, we identify the Verma modules with a space of polynomials, and the action of sl(n, C) on the Verma module is identified with a differential operator action of sl(n, C) on the polynomials.
Any singular vector in the Verma module becomes a polynomial solution of a system of variable-coefficient second-order linear partial differential equations. Thus we have changed a difficult problem in a noncommutative space to a problem in commutative space. However, it is in general impossible to solve the system in the space of polynomials.
So we extend the action of sl(n, C) on the polynomial space to a larger space of certain truncated formal power series. On this larger space, the negative simple root vectors become differential operators whose arbitrary complex powers are well defined (so are their products). In this way, we overcome the difficulty of determining whether a product of several general powers of negative simple root vectors is well defined in the work [MFF] of Malikov, Feigin and Fuchs. Next we define a differential-operator representation of the symmetric group S n on the space of truncated power series. Using commutator relations among root vectors and a certain substitution-of-variable technique that we developed in [X], we prove that the solution space of the system of partial differential equations in the space of truncated power series is exactly spanned by {σ(1) | σ ∈ S n }. Moreover, the singular vectors of sl(n, C) in the Verma module are given by those σ(1) that are polynomials. In particular, there are exactly n! singular vectors up to scalar multiples in the Verma module when the weight is dominant integral.
In Section 2, we derive the system of partial differential equations and a differentialoperator representation of the symmetric group S n on the space of certain truncated formal power series. Moreover, we prove that {σ(1) | σ ∈ S n } are the solutions of the system. In Section 3, we completely solve the system in the space of power series.
Differential Equations and Representations
In this section, we first derive a system of variable-coefficient second-order partial differential equations that determines the singular vectors in the Verma modules over the special linear Lie algebra sl(n, C). Then we construct a differential operator representation of the symmetric group S n on the related space of truncated formal power series and prove that {σ(1) | σ ∈ S n } are the solutions of the system in the space.
Denote by E i,j the square matrix with 1 as its (i, j)-entry and 0 as the others. The special linear Lie algebra
sl(n, C) = 1≤i<j≤n (CE i,j + CE j,i ) + n−1 r=1 C(E r,r − E r+1,r+1 ) (2.1)
with the Lie bracket:
[A, B] = AB − BA for A, B ∈ sl(n, C). (2.2) Set h i = E i,i − E i+1,i+1 , i = 1, 2, ..., n − 1. (2.3)
The subspace
H = n−1 i=1 Ch i (2.4)
forms a Cartan subalgebra of sl(n, C). We choose be the torsion-free additive semigroup of rank n(n − 1)/2 with ǫ i,j as base elements. Let G − be the Lie subalgebra spanned by (2.7) and let U(G − ) be its universal enveloping algebra. For
{E i,j | 1 ≤ i < j ≤α = 1≤j<i≤n α i,j ǫ i,j ∈ Γ, (2.10)
we denote
E α = E α 2,1 2,1 E α 3,1 3,1 E α 3,2
3,2 E α 4,1 4,1 · · · E α n,1 n,1 · · · E α n,n−1 n,n−1 ∈ U(G − ).
(2.11)
Then {E α | α ∈ Γ} forms a basis of U(G − ).
(2.12)
Let λ be a weight, which is a linear function on H, such that λ(h i ) = λ i for i = 1, 2, ..., n − 1.
(2.13)
Recall that sl(n, C) is generated by {E i,i+1 , E i+1,i | i = 1, 2, ..., n − 1} as a Lie algebra.
The Verma sl(n, C)-module with the highest-weight vector v λ of weight λ is given by
M λ = Span{E α v λ | α ∈ Γ},(2.14)
with the action determined by
E i,i+1 (E α v λ ) = ( i−1 j=1 α i+1,j E α+ǫ i,j −ǫ i+1,j − n j=i+2 α j,i E α+ǫ j,i+1 −ǫ j,i +α i+1,i (λ i + 1 − n j=i+1 α j,i + n j=i+2 α j,i+1 )E α−ǫ i+1,i )v λ , (2.15) E i+1,i (E α v λ ) = (E α+ǫ i+1,i + i−1 j=1 α i,j E α+ǫ i+1,j −ǫ i,j )v λ (2.16)
for i = 1, ..., n − 1. For any α ∈ Γ, we define the weight of E α v λ by
(wt E α v λ )(h i ) = (λ i + i−1 p=1 (α i,p − α i+1,p ) + n j=i+2 (α j,i+1 − α j,i ) − 2α i+1,i )h i (2.17)
for i = 1, ..., n − 1. Then the Verma module M λ is a space graded by weights. A singular vector is a homogeneous nonzero vector u in M λ such that
E i,i+1 (u) = 0 for i = 1, ..., n − 1. (2.18)
Here we have used the fact that all positive root vectors are generated by simple positive root vectors. The Verma module is irreducible if and only if any singular vector is a scalar multiple of v λ .
Consider the polynomial algebra
A = C[x i,j | 1 ≤ j < i ≤ n] (2.19) in n(n − 1)/2 variables. Set x α = 1≤j<i≤n x α i,j i,j for α ∈ Γ.
(2.20)
Then {x α | α ∈ Γ} forms a basis of A.
(2.21)
Thus we have a linear isomorphism τ :
M λ → A determined by τ (E α v λ ) = x α for α ∈ Γ. (2.22)
The algebra A becomes sl(n, C)-module by the action
A(f ) = τ (A(τ −1 (f ))) for A ∈ sl(n, C), f ∈ A. (2.23)
For convenience, we denote the partial derivatives
∂ i,j = ∂ x i,j for 1 ≤ j < i ≤ n.
(2.24)
In particular,
d i = E i,i+1 | A = (λ i − n j=i+1 x j,i ∂ j,i + n j=i+2 x j,i+1 ∂ j,i+1 )∂ i+1,i + i−1 j=1 x i,j ∂ i+1,j − n j=i+2 x j,i+1 ∂ j,i (2.25) for i = 1, 2, ..., n − 1 by (2.15).
Proposition 2.1. A homogeneous vector u ∈ M λ is a singular vector if and only if d i (τ (u)) = 0 for i = 1, 2, ..., n − 1.
(2.26)
The system of partial differential equations
(λ i − n j=i+1 x j,i ∂ j,i + n j=i+2 x j,i+1 ∂ j,i+1 )∂ i+1,i (z) + i−1 j=1 x i,j ∂ i+1,j (z) − n j=i+2 x j,i+1 ∂ j,i (z) = 0 (2.27)
for i = 1, 2, ..., n − 1 and unknown function z in {x i,j | 1 ≤ j < i ≤ n}, is called the system of partial differential equations for the singular vectors of sl(n, C).
Next we want to construct a differential operator representation of the symmetric group S n on the related space of truncated series and prove that {σ(1) | σ ∈ S n } are the solutions of the system. First, we have
η i = E i+1,i | A = x i+1,i + i−1 j=1 x i+1,j ∂ i,j (2.28)
for i = 1, 2, ..., n−1 by (2.16). Now we view {d i , η i | i = 1, 2, ..., n−1} purely as differential operators acting on functions of {x i,j | 1 ≤ j < i ≤ n}. In this way, we get a Lie algebra
action on functions of {x i,j | 1 ≤ j < i ≤ n} through E i,i+1 = d i and E i+1,i = η i because sl(n, C) is generated by {E i,i+1 , E i+1,i | i = 1, 2, ..., n − 1} as a Lie algebra. Note that h i (E α v λ ) = (λ i + i−1 p=1 (α i,p − α i+1,p ) + n j=i+2 (α j,i+1 − α j,i ) − 2α i+1,i )E α v λ (2.29)
for i = 1, 2, ..., n − 1 and α ∈ Γ. Accordingly, we set
ζ i = h i | A = λ i + i−1 p=1 (x i,p ∂ i,p −x i+1,p ∂ i+1,p )+ n j=i+2 (x j,i+1 ∂ j,i+1 −x j,i ∂ j,i )−2x i+1,i ∂ i+1,i (2.30) for i = 1, 2, ..., n − 1. The elements h i act on functions of {x i,j | 1 ≤ j < i ≤ n} through ζ i . A function f of {x i,j | 1 ≤ j < i ≤ n} is called weighted if there exist constants µ 1 , µ 2 , ..., µ n−1 such that ζ i (f ) = µ i f for i = 1, 2, ..., n − 1. (2.31)
Since d i maps weighted functions to weighted functions, the system (2.27) is a weighted system. Any nonzero weighted solution of the system (2.27) is a singular vector of sl(n, C).
In particular, any nonzero weighted polynomial solution f of the system (2.27) gives a singular vector τ −1 (f ) in the Verma module M λ .
Let
A 0 = C[x i,j | 1 ≤ j ≤ i − 2 ≤ n − 2] (2.32) be the polynomial algebra in {x i,j | 1 ≤ j ≤ i − 2 ≤ n − 2}. We denote x a = n−1 i=1 x a i i+1,i
for a = (a 1 , a 2 , ..., a n−1 ) ∈ C n−1 .
(2.33)
Let
A 1 = { j∈N n−1 p i=1 f a i − j x a i − j | 1 ≤ p ∈ N, a i ∈ C n−1 , f a i − j ∈ A 0 } (2.34) be the space of truncated-up formal power series in {x 2,1 , x 3,2 , ..., x n,n−1 } over A 0 . Then A is a subspace of A 1 . Since A 1 is invariant under the action of {E i,i+1 | A 1 = d i , E i+1,i | A 1 = η i | i = 1, 2, ..., n − 1}, A 1 becomes an sl(n, C)-module.
For a ∈ C and p ∈ N, we denote
a p = a(a − 1)(a − 2) · · · (a − p + 1). (2.35)
Moreover, by (2.28), we define
η a i = (x i+1,i + i−1 j=1 x i+1,j ∂ i,j ) a = ∞ p=0 a p p! x a−p i+1,i ( i−1 j=1 x i+1,j ∂ i,j ) p (2.36)
as differential operators on A 1 , for i = 1, 2, ..., n − 1 and a ∈ C. If a ∈ N, then the above summation is infinite and the positions of
x i+1,i and ( i−1 j=1 x i+1,j ∂ i,j ) are not symmetric. Since x i+1,i and ( i−1 j=1 x i+1,j ∂ i,j ) commute, we have η a 1 i η a 2 i = η a 1 +a 2 i for a 1 , a 2 ∈ C.
(2.37)
In particular, the inverse of the differential operator η a i is exactly η −a i . Given two differential operators d andd, we define the commutator
[d,d] = dd −dd.
(2.38)
For any element f ∈ A 1 and r ∈ C, we have Denote the Cartan matrix of sl(n, C) by a 1,1 a 1,2 · · · a 1,n−1 a 2,1 a 2,2 · · · a 2,n−1 . . . . . . . . . a n−1,1 a n−1,2 · · · a n−1,n−1 In order to construct a differential-operator representation of the symmetric group S n on A 1 , we need the following result.
[∂ i+1,i , x r i+1,i ](f ) = ∂ i+1,i (x r i+1,i f ) − x r i+1,i ∂ i+1,i (f ) = rx r−1 i+1,i f, (2.39) that is, [∂ i+1,i , x r i+1,i ] = rx r−1 i+1,i as operators. (2.40) Note that if (r, s) ∈ {(i + 1, j) | j = 1, ..., i} and (p, q) ∈ {(i, j) | j = 1, ..., i − 1}, then [∂ r,s , η a i ] = [x p,q , η a i ] = 0 (2.41) directly by (2.36). Now [∂ i+1,i , η a i ] = ∞ p=0 a p p! [∂ i+1,i , x a−p i+1,i ( i−1 j=1 x i+1,j ∂ i,j ) p ] = ∞ p=0 a p p! (a − p)x a−p−1 i+1,i ( i−1 j=1 x i+1,j ∂ i,j ) p = ∞ p=0 a a − 1 p p! x a−p−1 i+1,i ( i−1 j=1 x i+1,j ∂ i,j ) p = aη a−1 i (2.42) by (2.40). Moreover, for j = 1, 2, ..., i − 1, [∂ i+1,j , η a i ] = ∞ p=0 a p p! [∂ i+1,j , x a−p i+1,i ( i−1 s=1 x i+1,s ∂ i,s ) p ] = ∞ p=0 a p p! px a−p i+1,i ( i−1 s=1 x i+1,s ∂ i,s ) p−1 ∂ i,j = ∞ p=0 a a − 1 p−1 (p − 1)! x a−p i+1,i ( i−1 s=1 x i+1,s ∂ i,s ) p−1 ∂ i,j = aη a−1 i ∂ i,j(2[d l , η a i ] = aδ i,l η a−1 i (1 − a + ζ i ). (2.45) Proof. Note that [E l,l+1 , E m i+1,i ] = mδ i,l E m−1 i+1,i (1 − m + h i ) for m ∈ N( = 2 −1 −1 2 . . . . . . . . . −1 −1 2 .(
Lemma 2.4. For any a 1 , a 2 ∈ C and 1 ≤ i < n − 1, we have
η a 1 i η a 1 +a 2 i+1 η a 2 i = η a 2 i+1 η a 1 +a 2 i η a 1 i+1 . (2.50) Proof. Note that for a ∈ C, we have [ i p=1 x i+2,p ∂ i+1,p , x a i+1,i ] = ax a−1 i+1,i x i+2,i (2.51) by (2.40). Moreover, [ i p=1 x i+2,p ∂ i+1,p , i−1 j=1 x i+1,j ∂ i,j ] = i−1 j=1 x i+2,j ∂ i,j . (2.52) Hence η a 1 i η a 1 +a 2 i+1 = ∞ p,q=0 a 1 p a 1 + a 2 q p!q! x a 1 +a 2 −q i+2,i+1 x a 1 −p i+1,i ( i−1 j 1 =1 x i+1,j 1 ∂ i,j 1 ) p ( i j 2 =1 x i+2,j 2 ∂ i+1,j 2 ) q = ∞ p,q,r,s=0 (−1) r+s a 1 p+r a 1 + a 2 q p s q r+s r!s!p!q! x a 1 +a 2 −q i+2,i+1 ( i j 2 =1 x i+2,j 2 ∂ i+1,j 2 ) q−r−s x a 1 −p−r i+1,i ×( i−1 j 1 =1 x i+1,j 1 ∂ i,j 1 ) p−s x r i+2,i ( i−1 j=1 x i+2,j ∂ i,j ) s = ∞ p,q,r,s=0 (−1) r+s a 1 p+r a 1 + a 2 q r!s!(p − s)!(q − r − s)! x a 1 +a 2 −q i+2,i+1 ( i j 2 =1 x i+2,j 2 ∂ i+1,j 2 ) q−r−s x a 1 −p−r i+1,i ×( i−1 j 1 =1 x i+1,j 1 ∂ i,j 1 ) p−s x r i+2,i ( i−1 j=1 x i+2,j ∂ i,j ) s = ∞ q,k,s=0 ∞ p=0 (−1) k a 1 k a 1 − k p−s a 1 + a 2 q (k − s)!s!(p − s)!(q − k)! x a 1 +a 2 −q i+2,i+1 ( i j 2 =1 x i+2,j 2 ∂ i+1,j 2 ) q−k ×x a 1 −k−(p−s) i+1,i ( i−1 j 1 =1 x i+1,j 1 ∂ i,j 1 ) p−s x k−s i+2,i ( i−1 j=1 x i+2,j ∂ i,j ) s = ∞ q,k,s=0 (−1) k a 1 k a 1 + a 2 q (k − s)!s!(q − k)! x a 1 +a 2 −q i+2,i+1 ( i j 2 =1 x i+2,j 2 ∂ i+1,j 2 ) q−k ×η a 1 −k i x k−s i+2,i ( i−1 j=1 x i+2,j ∂ i,j ) s = ∞ q,k=0 ∞ s=0 (−1) k a 1 k a 1 + a 2 q (k − s)!s!(q − k)! x a 1 +a 2 −q i+2,i+1 ( i j 2 =1 x i+2,j 2 ∂ i+1,j 2 ) q−k ×η a 1 −k i x k−s i+2,i ( i−1 j=1 x i+2,j ∂ i,j ) s = ∞ q,k=0 (−1) k a 1 k a 1 + a 2 q k!(q − k)! x a 1 +a 2 −q i+2,i+1 ( i j 2 =1 x i+2,j 2 ∂ i+1,j 2 ) q−k ×η a 1 −k i (x i+2,i + i−1 j=1 x i+2,j ∂ i,j ) k = ∞ k=0 ∞ q=0 (−1) k a 1 k a 1 + a 2 k a 1 + a 2 − k q−k k!(q − k)! x a 1 +a 2 −k−(q−k) i+2,i+1 ( i j 2 =1 x i+2,j 2 ∂ i+1,j 2 ) q−k ×η a 1 −k i (x i+2,i + i−1 j=1 x i+2,j ∂ i,j ) k = ∞ k=0 (−1) k a 1 k a 1 + a 2 k k! η a 1 +a 2 −k i+1 η a 1 −k i (x i+2,i + i−1 j=1 x i+2,j ∂ i,j ) k(i η a 1 i+1 = ∞ k=0 (−1) k a 1 k a 1 + a 2 k k! η a 1 −k i+1 η a 1 +a 2 −k i (x i+2,i + i−1 j=1 x i+2,j ∂ i,j ) k . (2.54) Thus η a 1 i η a 1 +a 2 i+1 η a 2 i = ∞ k=0 (−1) k a 1 k a 1 + a 2 k k! η a 1 +a 2 −k i+1 η a 1 +a 2 −k i (x i+2,i + i−1 j=1 x i+2,j ∂ i,j ) k = η a 2 i+1 η a 1 +a 2 i η a 1 i+1 . 2 (2.55)
It is well known that the symmetric group S n is a group generated by {σ 1 , ..., σ n−1 } with the defining relations:
σ i σ i+1 σ i = σ i+1 σ i σ i+1 , σ r σ s = σ s σ r , σ 2 r = 1 (2.56)
for i = 1, 2, ..., n − 2 and r, s = 1, 2, ..., n − 1 such that |r − s| ≥ 2. According to (2.31) and (2.34), any element f ∈ A 1 can be written as f = j∈Z f j such that
ζ i (f j ) = µ (j) (h i )f j , µ (j) ∈ H * .
(2.57)
We define an action of {σ 1 , ..., σ n−1 } on A 1 by
σ i (f ) = j∈Z η µ (j) (h i )+1 i (f j ), i = 1, 2, ..., n − 1.
(2.58)
Theorem 2.5. Expression (2.58) gives a representation of the symmetric group S n .
Moreover, {σ(1) | σ ∈ S n } are weighted solutions of the system (2.27) of partial differential equations.
Proof. Let f ∈ A 1 be a weight function with weight µ, that is, ζ i (f ) = µ(h i )f for i = 1, 2, ..., n − 1. Then
(σ i | A 1 ) 2 (f ) = σ i (η µ(h i )+1 i (f )) = η −µ(h i )−1 i (η µ(h i )+1 i (f )) = f((σ i | A 1 ) 2 = Id A 1 for i = 1, 2, ..., n − 1. (2.60) Note [(σ i | A 1 )(σ i+1 | A 1 )(σ i | A 1 )](f ) = σ i [σ i+1 (η µ(h i )+1 i (f ))] = σ i [η µ(h i+1 )+µ(h i )+2 i+1 (η µ(h i )+1 i (f ))] = η µ(h i+1 )+1 i [η µ(h i+1 )+µ(h i )+2 i+1 (η µ(h i )+1 i (f ))] = (η µ(h i+1 )+1 i η µ(h i+1 )+µ(h i )+2 i+1 η µ(h i )+1 i )(f ) = (η µ(h i )+1 i+1 η µ(h i+1 )+µ(h i )+2 i η µ(h i+1 )+1 i+1 )(f ) = η µ(h i )+1 i+1 [η µ(h i+1 )+µ(h i )+2 i (η µ(h i+1 )+1 i+1 (f ))] = [(σ i+1 | A 1 )(σ i | A 1 )(σ i+1 | A 1 )](f(σ i | A 1 )(σ i+1 | A 1 )(σ i | A 1 ) = (σ i+1 | A 1 )(σ i | A 1 )(σ i+1 | A 1 ). (2.62)
For |r − s| ≥ 2, we have
η a r η b s = (x r+1,r + r−1 j=1 x r+1,j ∂ r,j ) a (x s+1,s + s−1 j=1 x s+1,j ∂ s,j ) b = η b s η a r (2.63) for a, b ∈ C by (2.36). So [(σ r | A 1 )(σ s | A 1 )](f ) = (η µ(hr)+1 r η µ(hs)+1 s )(f ) = [(σ s | A 1 )(σ r | A 1 )](f ), (2.64) which implies (σ r | A 1 )(σ s | A 1 ) = (σ s | A 1 )(σ r | A 1 ). (2.65)
According to (2.56), this proves that (2.58) defines a representation of S n on A 1 .
Now we assume that f ∈ A 1 is a weighted solution of (2.27) with weight µ, that is,
d i (f ) = 0 for i = 1, 2, ..., n − 1. Given r ∈ {1, 2, ..., n − 1}, d i (σ r (f )) = d i η µ(hr)+1 r (f ) = [d i , η µ(hr)+1 r ](f ) = (µ(h r ) + 1)δ i,r η µ(hr) r (−µ(h r ) + ζ r )(f ) = 0 (2.66)
by (2.45). So σ r (f ) is also a weighted solution of (2.27). Recall that S n is generated by σ 1 , σ 2 , ..., σ n−1 . For any σ ∈ S n , we write σ = σ i 1 σ i 2 · · · σ ir and σ(1) = σ i 1 (σ i 2 (· · · (σ ir (1)) · · · )) (2.67)
is a weighted solution of (2.27) by (2.66) and induction on r. 2
Completeness
In this section, we want to prove the following theorem of completeness: Proof. For any a = (a 1 , a 2 , ..., a n−1 ) ∈ C n−1 ,
(3.1) we define φ a = η a 2 2 η a 2 −λ n−1 −1 1 · · · η a i i η a i −λ n−1 −1 i−1 · · · η a i −λ n−1 −···−λ n−(i−1) −(i−1) 1
· · · η a n−1 n−1 η a n−1 −λ n−1 −1 n−2 · · · η a n−1 +2−λ 2 −···−λ n−1 −n 1
(1). Claim. An element z in A 1 is a solution of the system (3.4) if and only if it can be written as
z = j∈N n−1 p i=1 c a i − j x a i 1 −j 1 2,1 φ a i − j with c a i − j ∈ C (3.5)
for some a 1 , ..., a p ∈ C n−1 .
Recall η 1 = x 2,1 . By Lemma 2.2, the sufficiency holds. Now we want to prove the necessity. Recall that
E i+1,i = d i , E i,i+1 = η i as operators on A 1 (3.6) (cf.
(2.25) and (2.28)) for i = 1, 2, ..., n − 1. Note
d n−1 = (λ n−1 − x n,n−1 ∂ n,n−1 )∂ n,n−1 + n−2 i=1 x n−1,i ∂ n,i . (3.7)
Moreover, (2.11) tells us that
d n−2,n = E n−2,n | A 1 = (λ n−1 + λ n−2 − x n,n−2 ∂ n,n−2 − x n,n−1 ∂ n,n−1 )∂ n,n−2 −d n−1 ∂ n−1,n−2 + n−3 i=1
x n−2,i ∂ n,i = (λ n−1 + λ n−2 + 1 − x n,n−2 ∂ n,n−2 − x n,n−1 ∂ n,n−1 )∂ n,n−2 −∂ n−1,n−2 d n−1 + n−3 i=1
x n−2,i ∂ n,i . Furthermore, d n−3,n = E n−3,n | A 1 = (λ n−3 − 2 − x n,n−3 ∂ n,n−3 − x n,n−2 ∂ n,n−2 − x n,n−1 ∂ n,n−1 )∂ n,n−3 −d n−2,n ∂ n−2,n−3 − d n−1 ∂ n−1,n−3 +
n−4 i=1 x n−3,i ∂ n,i = (λ n−2 − 2 − 3 p=1
x n,n−p ∂ n,n−p ))∂ n,n−3 − ∂ n−2,n−3 d n−2,n − ∂ n−1,n−3 d n−1 −[d n−2,n , ∂ n−2,n−3 ] − [d n−1 , ∂ n−1,n−3 ] +
n−4 i=1 x n−3,i ∂ n,i = (λ n−3 − 3 p=1 x n,n−p ∂ n,n−p )∂ n,n−3 + n−4 i=1 x n−3,i ∂ n,i −∂ n−2,n−3 d n−2,n − ∂ n−1,n−3 d n−1 .
(3.10)
By induction, we can prove that
d i,n = E i,n | A 1 = (λ i − n−1 p=i x n,p ∂ n,p )∂ n,i + i−1 q=1 x i,q ∂ n,q − n−1 j=i+1 ∂ j,i d j,n (3.11)
for i = 2, 3, ..., n − 2, where we take d n−1,n = d n−1 .
(3.12)
Suppose that
z = f x a 1 2,1 φ a (3.13)
is a solution of the system (3.4) for some a ∈ C n−1 and f ∈ A 0 (cf. (2.32) and (3.2)). We want to prove that f is a constant. Denote
ǫ i = (0, ..., 0, i 1, 0, ..., 0) ∈ C n−1 (3.14)
and
ι i,j = a i+j − j p=1 (λ n−p + 1) for 2 ≤ i ≤ n − 1, 0 ≤ j ≤ n − i − 1. (3.15)
We define
U i = { j∈N n−1 g j x a 1 −j 1 2,1 φ a−ǫ i − j | g j ∈ A 0 } (3.16)
for i = 2, ..., n − 1, and
U = n−1 i=2 U i . (3.17)
Moreover, for fixed i ≥ 2,
{η ι i,j i | 0 ≤ j ≤ n − i − 1} (3.18)
are all the factors in the righthand side of (3.2) that contain x i+1,p or ∂ i,q with p = 1, ..., i and q = 1, 2, ...., i − 1 by (2.36). Besides,
[∂ i+1,i , η ι i,j i ] = ι i,j η ι i,j −1 i , (3.19) [∂ i+1,p , η ι i,j i ] = ι i,j η ι i,j −1 i ∂ i,p for p = 1, ..., i − 1, (3.20) [ i p=r x i+1,p ∂ i+1,p , η ι i,j i ] = ι i,j (η ι i,j i − r−1 p=1 x i+1,p η ι i,j −1 i ∂ i,p ), (3.21) [ i−1 p=q x i,p ∂ i,p , η ι i,j i ] = −ι i,j i−1 p=q x i+1,p η ι i,j −1 i ∂ i,p ,(3.22)
where 2 ≤ r ≤ i and 2 ≤ q ≤ i − 1.
By (3.18)-(3.22), we have (3.24) for 2 ≤ i ≤ n − 1 and 1 ≤ r ≤ i. Since for 2 ≤ i ≤ n − 2,
∂ i+1,r (φ a ) ∈ U i (3.23) and ( i p=r x i+1,p ∂ i+1,p )(φ a ) ≡ c i,r φ a (mod i s=1 U s ), c i,r ∈ C,E i,n = [E i,i+1 , [E i+1,i+2 , · · · , [E n−2,n−1 , E n−1,n ] · · · ]],(3.25)
we have
d i,n = [d i , [d i+1 , · · · [d n−2 , d n−1 ] · · · ]].d i,n (z) = [(λ i −c n−1,i )∂ n,i (f )+ i−1 q=1 x i,q ∂ n,q (f )]x a 1 2,1 φ a − n−1 j=i+1 ∂ j,i d j,n (z) ≡ 0 (mod U) (3.28)
for i = 2, 3, ..., n − 2 and d n−1 (z) = ( n−2 q=1
x n−1,q ∂ n,q (f ))x a 1 2,1 φ a ≡ 0 (mod U).
(3.29)
Since the constraint on d r,n (z) ≡ 0 (mod U) for r ≥ 2 implies ∂ r,s d r,n (z) ≡ 0 (mod U) for s = 1, 2, ..., r − 1 by (3.23), (2.28) is equivalent to x n−1,q ∂ n,q (f ) = 0.
d i,n (z) = [(λ i − c n−1,i )∂ n,i (f ) + i−1 q=1 x i,q ∂ n,q (f )]x a 1 2,1 φ a ≡ 0 (mod U)
(3.32)
We view ∂ n,1 (f ), ∂ n,2 (f ), · · · , ∂ n,n−2 (f ) as unknowns.
(3.33)
Then the coefficient determinant of the system (3.31) and (3.32) is
x 2,1 λ 2 − c n−1,2
x 3,1 x 3,2 . . .
. . . . . . . . . λ n−2 − c n−1,n−2 x n−1,a · · · x n−1,n−3 x n−1,n−2 = n−1 p=2
x p,p−1 + g(x 2,1 , x 3,2 , ..., x n−1,n−2 ) ≡ 0, (3.34)
where g(x 2,1 , x 3,2 , ..., x n−1,n−2 ) is a polynomial of degree n − 3 in {x 2,1 , x 3,2 , ..., x n−1,n−2 } over A 0 (cf. (2.32)). Therefore, ∂ n,q (f ) = 0 for q = 1, 2, ...., n − 2.
(3.35)
Based on our calculations in (3.23)-(3.25), we can prove by induction that
∂ q+r,q (f ) = 0 for 1 ≤ q ≤ n − 2. 2 ≤ r ≤ n − q. (3.36) So f is a constant.
Suppose that z is any solution of the system (3.4) in A 1 . By (2.34) and (3.2), it can be written as
z = j∈N n−1 p i=1 f a i − j x a i 1 −j 1 2,1 φ a i − j with f j ∈ A 0 . (3.37) Let S = { b ∈ C n−1 | f b = 0; f b+ j = 0 for all 0 = j ∈ N n−1 }. (3.38)
The above arguments show that
{f b | b ∈ S} are constants (3.39) (cf.
the key equations (3.29) and (3.30)).
Since b∈S f b x b 1 2,1 φ b is a solution of the system (3.4), so is z − b∈S f b x b 1 2,1 φ b .
By induction, we prove the Claim To solve the system (2.27) in A 1 , we only need to consider the solutions of the form z = x a 1 2,1 φ a with a ∈ C n−1 by the above claim, because (2.27) is a weighted system. Note
d 1 = (λ 1 − n j=2
x j,1 ∂ j,1 + n j=3
x j,2 ∂ j,2 )∂ 2,1 − n j=3
x j,2 ∂ j,1 . (λ q + 1),â 1,1 = a 1 +â 1,2 (3.41) for r = 2, 3, ..., n − 1, a 2,r = a r − r−2 p=1 (λ n−p + 1) for r = 2, 3, ..., n − 1.
(3.42)
andã = n−1 i=2 a i − n−1 p=3 (p − 2)(λ p + 1). (3.43) Letting x p,q = 0 for 1 ≤ q ≤ p − 2 ≤ n − 2 in d 1 (z) = [(λ 1 − n j=2
x j,1 ∂ j,1 + n j=3
x j,2 ∂ j,2 )∂ 2,1 − n j=3
x j,2 ∂ j,1 ](x a 1 2,1 φ a ) = 0, (3.44)
we getâ 1,1 (λ 1 + 1 −â 1,1 +ã) − n−1 r=2â 2,râ1,r = 0 (3.45) by (2.36) and (3.2).
Suppose n > 3. We take x p, q = 0 for 1 ≤ q ≤ p − 2 ≤ n − 2 in
∂ n,1 d 1 (z) = ∂ n,1 [(λ 1 − n j=2
x j,1 ∂ j,1 + n j=3
x j,2 ∂ j,2 )∂ 2,1 − n j=3
x j,2 ∂ j,1 ](x a 1 2,1 φ a ) = 0, (3.46) and obtain a n−1
n−2 i=1 (a n−1 − i − i p=1 λ n−p ) [(â 1,1 − 1)(λ 1 −â 1,1 +ã) − n−2 r=2â
2,r (â 1,r − 1) − (â 2,n−1 − 1)(â 1,n−1 − 1)] = 0.
(3.47)
Note that
[â 1,1 (λ 1 + 1 −â 1,1 +ã) − n−1 r=2â 2,râ1,r ] − [(â 1,1 − 1)(λ 1 −â 1,1 +ã) − n−2 r=2â
2,r (â 1,r − 1) − (â 2,n−1 − 1)(â 1,n−1 − 1)] = λ 1 +ã − n−1 r=2â 2,r + 1 −â 1,n−1 = λ 1 + 1 −â 1,n−1 = (n − 1) + n−1 i=1 λ i − a n−1 .
(3.48) By (3.45), (3.47) and (3.48), we have a n−1
n−1 i=1 (a n−1 − i − i p=1 λ n−p ) = 0. (3.49)
Therefore, a n−1 ∈ {0, i + i p=1 λ n−p | i = 1, 2, ..., n − 1}.
(3.50) Assume n = 3. Then
d 1 = (λ 1 − x 2,1 ∂ 2,1 − x 3,1 ∂ 3,1 + x 3,2 ∂ 3,2 )∂ 2,1 − x 3,2 ∂ 3,1 , (3.51) z = x a 1 2,1 φ a = x a 1 2,1 (x 3,2 + x 3,1 ∂ 2,1 ) a 2 (x a 2 −λ 2 −1 2,1 ),(3.52)
and (3.45) becomes (a 1 + a 2 − λ 2 − 1)(λ 1 + λ 2 + 2 − a 1 ) − a 2 (a 2 − λ 2 − 1) = 0 (3.53) Letting x 3,1 = 0 in
∂ 3,1 d 1 (z) = ∂ 3,1 [(λ 1 − x 2,1 ∂ 2,1 − x 3,1 ∂ 3,1 + x 3,2 ∂ 3,2 )∂ 2,1 −x 3,2 ∂ 3,1 ][x a 1 2, (x 3,2 + x 3,1 ∂ 2,1 ) a 2 (x a 2 −λ 2 −1 2,1 )] = 0, (3.54) we get a 2 (a 2 − λ 2 − 1)(a 1 + a 2 − λ 2 − 2)(λ 1 + λ 2 + 1 − a 1 ) −a 2 (a 2 − 1)(a 2 − λ 2 − 1)(a 2 − λ 2 − 2) = 0, (3.55)
equivalently a 2 (a 2 − λ 2 − 1)[(a 1 + a 2 − λ 2 − 2)(λ 1 + λ 2 + 1 − a 1 ) − (a 2 − 1)(a 2 − λ 2 − 2)] = 0. (3.56) By (3.53), we have (a 1 + a 2 − λ 2 − 2)(λ 1 + λ 2 + 1 − a 1 ) − (a 2 − 1)(a 2 − λ 2 − 2)
= −(λ 1 + a 2 ) + a 2 (a 2 − λ 2 − 1) − (a 2 − 1)(a 2 − λ 2 − 2) = −(λ 1 + a 2 ) + 2a 2 − λ 2 − 2 = a 2 − λ 1 − λ 2 − 2.
(3.57) Thus (3.56) and (3.57) give a 2 (a 2 − λ 2 − 1)(a 2 − λ 1 − λ 2 − 2) = 0, (3.58) which implies that (3.50) holds for any n ≥ 2.
When n = 2, the solution space of (2.27) is C + Cσ 1 (1) by (2.66). In general, we can use (3.50) to reduce the problem of solving (2.27) to sl(n − 1) as follows. Denote Ψ i = x a 1 2,1 η a 2 2 η a 2 −λ n−1 −1 1 · · · η ar r η ar−λ n−1 −1 r−1 · · · η ar−λ n−1 −···−λ n−(r−1) −(r−1) 1 · · · η a n−2 n−2 η a n−2 −λ n−1 −1 n−3 · · · η a n−2 +3−λ 3 −···−λ n−1 −n 1
η −λ i−1 −1 i−2 η −λ i−1 −λ i−2 −2 i−3 · · · η −λ 3 −···−λ i−1 −(i−2) 1
(3.59) for i = 1, ..., n − 1, n, where we treat
η −λ i−1 −1 i−2 η −λ i−1 −λ i−2 −2 i−3 · · · η −λ i −···−λ i−1 −(i−2) 1 = 1 if i = 1, 2.
(3.60) Moreover, we set ψ n = 1, ψ i = σ n−1 σ n−2 · · · σ i (1) = η n−i+ P n−i p=1 λ n−p n−1 η n−i−1+ P n−i p=2 λ n−p n−2 · · · η λ i +1 i (1), (3.61) for i = 1, 2, ..., n − 1. Then {ψ i | i = 1, ..., n − 1, n} are solutions of (2.27) by Theorem 2.5. Denote λ n−1,n = 0, λ n−1,i = n − i + n−i p=1 λ n−p for i = 1, 2, ..., n − 1.
(3.62) According to (3.50), a n−1 = λ n−1,i n−1 for some i n−1 ∈ {1, ..., n − 1, n}. for j = 1, 2, ..., n − 2. By (2.29) and (2.30), h j (ψ i n−1 ) = ζ j (ψ i n−1 ) = λ (n−2) j ψ i n−1 for j = 1, 2, ..., n − 2.
(3.66)
Defined i = (λ (n−2) i − n−1 j=i+1 x j,i ∂ j,i + n−1 j=i+2 x j,i+1 ∂ j,i+1 )∂ i+1,i + i−1 j=1 x i,j ∂ i+1,j − n−1 j=i+2
x j,i+1 ∂ j,i (3.67) for i = 1, 2, ..., n − 2 by (2.25). Then the system d i (Ψ i n−1 ψ i n−1 ) = [d i , Ψ i n−1 ](ψ i n−1 ) = 0 for i = 1, 2, ..., n − 2 (3.68)
is equivalent to the system d i (Ψ i n−1 (1)) = 0 for i = 1, 2, ..., n − 2 (3.68) by Lemma 2.2, (3.64) and (3.66). The system (3.68) is a version (2.27) for sl(n − 1). By induction on n, there exist a σ ′ ∈ S n−1 and a constant c ∈ C such that Ψ i n−1 ψ i n−1 = cσ ′ (ψ i n−1 ). Thus z = cσ ′ (1) or z = cσ ′ σ n−1 σ n−1 · · · σ i n−1 (1) for some i n−1 ≤ n − 1. The last conclusion follows from (2.58) and induction. 2
If n − 1 + n−1 p=1 λ p ∈ N + 1, (3.69) then φ = σ 1 · · · σ n−2 σ n−1 σ n−2 · · · σ 1 (1) (3.70)
is a polynomial and so τ −1 (φ) (cf. (2.22)) is a nontrivial singular vector in the Verma module M λ (cf. (2.14)), which was obtained by Malikov, Feigin and Fuchs [MFF]. In general, for 1 ≤ i < j ≤ n − 1, φ i,j = σ i · · · σ j−1 σ j σ j−1 · · · σ i (1) is a polynomial if j − i + 1 + j r=i λ r ∈ N.
(3.71) By (3.50) and induction on n, we have:
Corollary 3.2. The Verma module M λ is irreducible if and only if
j + j−1 p=0 λ i+p ∈ N for 1 ≤ i ≤ n − 1, 0 ≤ j ≤ n − i.
(3.72)
Theorem 3. 1 .
1The solution space of the system (2.27) in A 1 is spanned by {σ(1) | σ ∈ S n }. Moreover, {σ(1) | σ ∈ S n } are all the weighted solutions of the system (2.27) in A 1 up to scalar multiples. In particular, there are exactly n! singular vectors up to scalar multiples in the Verma module M λ when the weight λ is dominant integral.
. (2.32)-(2.34)) and is a solution of the system d 2 (z) = d 3 (z) = · · · = d n−1 (z) = 0 (3.4) by Lemmas 2.2, 2.3 and (2.66).
,q ∂ n,q (f ) + (λ i − c n−1,i )∂ n,i (f ) = 0 for i = 2, ..., n
if j < i n−1 − 1, λ i n−1 + λ i n−1 −1 + 1 if j = i n−1 − 1, λ j+1 if i n−1 ≤ j ≤ n − 2(3.65)
Lemma 2.2. For i, l ∈ {1, 2, ..., n − 1} and a ∈ C, we have:.43)
and similarly,
[x i,j , η a
i ] = −aη a−1
i
x i+1,j
for j = 1, 2, ..., i − 1.
(2.44)
Gel'fand, Structure of representations generated by vectors of highest weight. I N Bernstein, I M , S I , Russian) Funktsional. Anal. i Prilozhen. 81I. N. Bernstein, I. M. Gel'fand and S. I. Gel'fand, Structure of representations generated by vectors of highest weight, (Russian) Funktsional. Anal. i Prilozhen. 8 (1971), no. 1, 1-9.
Structure of some categories of representations of infinite-dimensional Lie algebras. V V Deodhar, O Gabber, V G Kac, Adv. in Math. 451V. V. Deodhar, O. Gabber and V. G. Kac, Structure of some categories of rep- resentations of infinite-dimensional Lie algebras, Adv. in Math. 45 (1982), no. 1, 92-116.
On multipicity in the Jordan-Hölder series of Verma modules. V V Deodhar, J Lepowsky, J. Algebra. 492V. V. Deodhar and J. Lepowsky, On multipicity in the Jordan-Hölder series of Verma modules, J. Algebra 49 (1977), no. 2, 512-524.
J E Humphreys, Introduction to Lie Algebras and Representation Theory. Springer-Verlag New York IncJ. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer- Verlag New York Inc., 1972.
Zur charakterformel gewisser darstellungen halbeinfacher grunppen und Lie-algebrun. J C Jantzen, Math. Z. 140J. C. Jantzen, Zur charakterformel gewisser darstellungen halbeinfacher grunppen und Lie-algebrun, Math. Z. 140 (1974), 127-149.
Kontravariante formen auf induzierten Darstellungen habeinfacher Lie-algebren. J C Jantzen, Math. Ann. 2261J. C. Jantzen, Kontravariante formen auf induzierten Darstellungen habeinfacher Lie-algebren, Math. Ann. 226 (1977), no. 1, 53-65.
Moduln mit einem höchsten gewicht, Lecture Note in Math. J C Jantzen, Springer750BerlinJ. C. Jantzen, Moduln mit einem höchsten gewicht, Lecture Note in Math. 750, Springer, Berlin, 1979.
Highest weight representations of infinite-dimensional Lie algebras. V G Kac, Proc. Intern. Congr. Math. Acad. Sci. FennicaV. G. Kac, Highest weight representations of infinite-dimensional Lie algebras, Proc. Intern. Congr. Math. (Helsinki, 1978), pp. 299-304, Acad. Sci. Fennica, Helsinki, 1980.
Some problems on infite-dimensional Lie algebras and their representations. V G Kac, Lie algebras and related topics. New BrunswickV. G. Kac, Some problems on infite-dimensional Lie algebras and their represen- tations, Lie algebras and related topics (New Brunswick, N.J., 1981), pp. 117-126.
. Lecture Note in Math. 933Lecture Note in Math. 933, Springe, Berlin-New York, 1982.
Infinite-Dimensional Lie algebras. V G Kac, Cambridge University Press3rd EditionV. G. Kac, Infinite-Dimensional Lie algebras, 3rd Edition, Cambridge University Press, 1990.
Structure of representations with highest weight of infinite-dimensional Lie algebras. V G Kac, D A Kazhdan, Adv. in Math. 341V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite-dimensional Lie algebras, Adv. in Math. 34 (1979), no. 1, 97-108.
V G Kac, D H Peterson, Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA78V. G. Kac and D. H. Peterson, Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Natl. Acad. Sci. USA 78 (1981), 3308-3312.
Representation theory of the vertex algebra W 1+∞. V G Kac, A , Trans. Groups. 1V. G. Kac and A. Radul, Representation theory of the vertex algebra W 1+∞ , Trans. Groups 1 (1996), 41-70.
Canonical vectors in induced modules. J Lepowsky, Trans. Amer. Math. Soc. 208J. Lepowsky, Canonical vectors in induced modules, Trans. Amer. Math. Soc. 208 (1975), 219-272.
Existence of canonical vectors in induced modules. J Lepowsky, Ann. of Math. 2J. Lepowsky, Existence of canonical vectors in induced modules, Ann. of Math. (2) 102 (1975), no. 1, 17-40.
On the uniqueness of canonical vectors. J Lepowsky, Proc. Amer. Math. Soc. 572J. Lepowsky, On the uniqueness of canonical vectors, Proc. Amer. Math. Soc. 57 (1976), no. 2, 217-220.
Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism. J Lepowsky, J. Algebra. 492J. Lepowsky, Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism, J. Algebra 49 (1977), no. 2, 470-495.
Singular vectors in Verma modules over Kac-Moody algebras. F G Malikov, B L Feigin, D B Fuchs, Russian) Funktsional. Anal. i Prilozhen. 202F. G. Malikov, B. L. Feigin and D. B. Fuchs, Singular vectors in Verma modules over Kac-Moody algebras, (Russian) Funktsional. Anal. i Prilozhen. 20 (1986), no. 2, 25-37.
Projective modules over graded Lie algebras, I, Math. A Rocha-Caridi, N R Walllach, Z. 1082A. Rocha-Caridi and N. R. Walllach, Projective modules over graded Lie algebras, I, Math. Z. 108 (1982), no. 2, 151-177.
Highest weight modules over graded Lie algebras, resolutions, filtrations and character formulas. A Rocha-Caridi, N R Walllach, Trans. Amer. Math. Soc. 2771A. Rocha-Caridi and N. R. Walllach, Highest weight modules over graded Lie algebras, resolutions, filtrations and character formulas, Trans. Amer. Math. Soc. 277 (1983), no. 1, 133-162.
A certain bilinear form on the universal enveloping algebra of a complex semisimple Lie algebras. N N Sapovolov, Russian) Funktsional. Anal. i Prilozhen. 44N. N. Sapovolov, A certain bilinear form on the universal enveloping algebra of a complex semisimple Lie algebras, (Russian) Funktsional. Anal. i Prilozhen. 4 (1972), no. 4, 65-70.
Structure of certain induced representations of complex semisimple Lie algebras, thesis. D , - N Verma, Yale UniversityD,-N. Verma, Structure of certain induced representations of complex semisimple Lie algebras, thesis, Yale University, 1966.
Structure of certain induced representations of complex semisimple Lie algebras. D , - N Verma, Bull. Amer. Math. Soc. 74D,-N. Verma, Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74(1968), 160-166.
Differential invariants of classical groups. X Xu, Duke Math. J. 94X. Xu, Differential invariants of classical groups, Duke Math. J. 94 (1998), 543-572.
| [] |
[
"Comment on \"Nonmagnetic Impurity Resonances as a Signature of Sign-Reversal Pairing in Fe-As-Based Superconductors\"",
"Comment on \"Nonmagnetic Impurity Resonances as a Signature of Sign-Reversal Pairing in Fe-As-Based Superconductors\""
] | [
"Maria Daghofer ",
"Adriana Moreo ",
"[ ",
"] M Daghofer ",
"A Moreo ",
"J A Riera ",
"E Arrigoni ",
"D J Scalapino ",
"E Dagotto ",
"\nDepartment of Physics and Astronomy\nIFW Dresden\nP.O. Box 27 01 16D-01171DresdenGermany\n",
"\n-1200 and Oak Ridge National Laboratory\nUniversity of Tennessee\n37966, 37831-6032Knoxville, Oak RidgeTN, TNUSA\n"
] | [
"Department of Physics and Astronomy\nIFW Dresden\nP.O. Box 27 01 16D-01171DresdenGermany",
"-1200 and Oak Ridge National Laboratory\nUniversity of Tennessee\n37966, 37831-6032Knoxville, Oak RidgeTN, TNUSA"
] | [
"Phys. Rev. Lett"
] | PACS numbers:In a recent Letter[1], the energy band structure of Fe-As-based superconductors is fitted with a tight-binding model with two Fe ions per unit cell and two degenerate d xz and d yz orbitals per Fe ion. The author claims that the proposed model, which differs markedly from a model previously used by other authors for the same two orbitals in the same compounds [2, 3], possesses the symmetry required to describe the Fe-As planes in ironpnictide superconductors. In this comment we argue that this is not the case.As discussed in Ref. 1, the unit cell of the Fe-As planes contains two iron ions, and the Hamiltonian Eq. (1) employed there reflects this. However, each two-iron unit cell also has an internal symmetry that the Hamiltonian needs to obey. To see this, consider the square lattice made up by all iron ions, i.e., with one iron per unit cell, and the Fe-Fe distance as a new basis vector. The Fe-As planes are invariant under (a) translation by one unit along the Fe-Fe direction, followed by (b) a reflexion on the Fe-Fe plane. At first sight, the Hamiltonian of Ref. 1 appears to take this into account, because a translation by one Fe-Fe distance and an additional exchange of hoppings t 2 (mediated by an As above the plane) and t 3 (via an As below the plane) indeed leaves it invariant. But since the Fe-As-Fe distances and angles are the same for As ions above and below the plane, these two hopping paths are very symmetric, which induces additional restrictions [4] and the two paths actually give the same hopping[5].This missing symmetry of the Hamiltonian also reveals itself in the momentum-dependent band structure. The d xz and d yz orbitals should be degenerate at the Γ point[4,6], as it can also be seen in band structures obtained from the local density approximation[7][8][9][10][11]or by a Slater-Koster approach[6,12]. Another consequence of the internal symmetry of the two-iron unit cell is that all bands have to be two-fold degenerate at the boundaries of the Brillouin zone (BZ) corresponding to the two-Fe unit cell[4], which is clearly violated in the Hamiltonian presented in Ref. 1, see the data for the (π, 0)-(π, π) path inFig. (2).[13]In fact, any tight-binding Hamiltonian for the Fe-As planes can be written in block form, where each block is expressed in terms of the orbitals of one single Fe per unit cell[4,6]. Instead of studying both blocks for the BZ of the original two-iron unit cell, one can then consider just one block, but in the extended BZ corresponding to a unit cell with just one Fe ion, because the two blocks correspond to momenta k and k + (π, π) in the extended BZ. In this description, the system has D 4h symmetry which means that all the eigenstates at the center of the BZ should transform according to irreducible representations of D 4h . In particular the orbitals d xz and d yz transform according to the two dimensional representation E g leading to the degeneracy at the center of the BZ, which is missing in the Hamiltonian of Ref. 1.Finally, t 3 is five times larger than t 2 in Ref. 1. This difference is extreme even for the orthorhombic phase of the pnictides, where a slight oxygen distortion of 1% (6%) in bulk (on surfaces) [14] might in principle justify a small difference. | 10.1103/physrevlett.104.089701 | [
"https://arxiv.org/pdf/1002.4939v1.pdf"
] | 28,120,488 | 1002.4939 | 8795f8af345902569f6cc54cb23d74cb5f01f1e0 |
Comment on "Nonmagnetic Impurity Resonances as a Signature of Sign-Reversal Pairing in Fe-As-Based Superconductors"
2009. 2008
Maria Daghofer
Adriana Moreo
[
] M Daghofer
A Moreo
J A Riera
E Arrigoni
D J Scalapino
E Dagotto
Department of Physics and Astronomy
IFW Dresden
P.O. Box 27 01 16D-01171DresdenGermany
-1200 and Oak Ridge National Laboratory
University of Tennessee
37966, 37831-6032Knoxville, Oak RidgeTN, TNUSA
Comment on "Nonmagnetic Impurity Resonances as a Signature of Sign-Reversal Pairing in Fe-As-Based Superconductors"
Phys. Rev. Lett
1032205032009. 2008
PACS numbers:In a recent Letter[1], the energy band structure of Fe-As-based superconductors is fitted with a tight-binding model with two Fe ions per unit cell and two degenerate d xz and d yz orbitals per Fe ion. The author claims that the proposed model, which differs markedly from a model previously used by other authors for the same two orbitals in the same compounds [2, 3], possesses the symmetry required to describe the Fe-As planes in ironpnictide superconductors. In this comment we argue that this is not the case.As discussed in Ref. 1, the unit cell of the Fe-As planes contains two iron ions, and the Hamiltonian Eq. (1) employed there reflects this. However, each two-iron unit cell also has an internal symmetry that the Hamiltonian needs to obey. To see this, consider the square lattice made up by all iron ions, i.e., with one iron per unit cell, and the Fe-Fe distance as a new basis vector. The Fe-As planes are invariant under (a) translation by one unit along the Fe-Fe direction, followed by (b) a reflexion on the Fe-Fe plane. At first sight, the Hamiltonian of Ref. 1 appears to take this into account, because a translation by one Fe-Fe distance and an additional exchange of hoppings t 2 (mediated by an As above the plane) and t 3 (via an As below the plane) indeed leaves it invariant. But since the Fe-As-Fe distances and angles are the same for As ions above and below the plane, these two hopping paths are very symmetric, which induces additional restrictions [4] and the two paths actually give the same hopping[5].This missing symmetry of the Hamiltonian also reveals itself in the momentum-dependent band structure. The d xz and d yz orbitals should be degenerate at the Γ point[4,6], as it can also be seen in band structures obtained from the local density approximation[7][8][9][10][11]or by a Slater-Koster approach[6,12]. Another consequence of the internal symmetry of the two-iron unit cell is that all bands have to be two-fold degenerate at the boundaries of the Brillouin zone (BZ) corresponding to the two-Fe unit cell[4], which is clearly violated in the Hamiltonian presented in Ref. 1, see the data for the (π, 0)-(π, π) path inFig. (2).[13]In fact, any tight-binding Hamiltonian for the Fe-As planes can be written in block form, where each block is expressed in terms of the orbitals of one single Fe per unit cell[4,6]. Instead of studying both blocks for the BZ of the original two-iron unit cell, one can then consider just one block, but in the extended BZ corresponding to a unit cell with just one Fe ion, because the two blocks correspond to momenta k and k + (π, π) in the extended BZ. In this description, the system has D 4h symmetry which means that all the eigenstates at the center of the BZ should transform according to irreducible representations of D 4h . In particular the orbitals d xz and d yz transform according to the two dimensional representation E g leading to the degeneracy at the center of the BZ, which is missing in the Hamiltonian of Ref. 1.Finally, t 3 is five times larger than t 2 in Ref. 1. This difference is extreme even for the orthorhombic phase of the pnictides, where a slight oxygen distortion of 1% (6%) in bulk (on surfaces) [14] might in principle justify a small difference.
In a recent Letter [1], the energy band structure of Fe-As-based superconductors is fitted with a tight-binding model with two Fe ions per unit cell and two degenerate d xz and d yz orbitals per Fe ion. The author claims that the proposed model, which differs markedly from a model previously used by other authors for the same two orbitals in the same compounds [2,3], possesses the symmetry required to describe the Fe-As planes in ironpnictide superconductors. In this comment we argue that this is not the case.
As discussed in Ref. 1, the unit cell of the Fe-As planes contains two iron ions, and the Hamiltonian Eq. (1) employed there reflects this. However, each two-iron unit cell also has an internal symmetry that the Hamiltonian needs to obey. To see this, consider the square lattice made up by all iron ions, i.e., with one iron per unit cell, and the Fe-Fe distance as a new basis vector. The Fe-As planes are invariant under (a) translation by one unit along the Fe-Fe direction, followed by (b) a reflexion on the Fe-Fe plane. At first sight, the Hamiltonian of Ref. 1 appears to take this into account, because a translation by one Fe-Fe distance and an additional exchange of hoppings t 2 (mediated by an As above the plane) and t 3 (via an As below the plane) indeed leaves it invariant. But since the Fe-As-Fe distances and angles are the same for As ions above and below the plane, these two hopping paths are very symmetric, which induces additional restrictions [4] and the two paths actually give the same hopping [5].
This missing symmetry of the Hamiltonian also reveals itself in the momentum-dependent band structure. The d xz and d yz orbitals should be degenerate at the Γ point [4,6], as it can also be seen in band structures obtained from the local density approximation [7][8][9][10][11] or by a Slater-Koster approach [6,12]. Another consequence of the internal symmetry of the two-iron unit cell is that all bands have to be two-fold degenerate at the boundaries of the Brillouin zone (BZ) corresponding to the two-Fe unit cell [4], which is clearly violated in the Hamiltonian presented in Ref. 1, see the data for the (π, 0)-(π, π) path in Fig. (2). [13] In fact, any tight-binding Hamiltonian for the Fe-As planes can be written in block form, where each block is expressed in terms of the orbitals of one single Fe per unit cell [4,6]. Instead of studying both blocks for the BZ of the original two-iron unit cell, one can then consider just one block, but in the extended BZ corresponding to a unit cell with just one Fe ion, because the two blocks correspond to momenta k and k + (π, π) in the extended BZ. In this description, the system has D 4h symmetry which means that all the eigenstates at the center of the BZ should transform according to irreducible representations of D 4h . In particular the orbitals d xz and d yz transform according to the two dimensional representation E g leading to the degeneracy at the center of the BZ, which is missing in the Hamiltonian of Ref. 1. Finally, t 3 is five times larger than t 2 in Ref. 1. This difference is extreme even for the orthorhombic phase of the pnictides, where a slight oxygen distortion of 1% (6%) in bulk (on surfaces) [14] might in principle justify a small difference. This work was supported by the NSF grant DMR-0706020 and the Division of Materials Science and Engineering, U.S. DOE, under contract with UT-Battelle, LLC.
. D Zhang, Phys. Rev. Lett. 103186402D. Zhang, Phys. Rev. Lett. 103, 186402 (2009).
. S Raghu, X.-L Qi, C.-X Liu, D J Scalapino, S.-C Zhang, Phys. Rev. B. 77220503S. Raghu, X.-L. Qi, C.-X. Liu, D. J. Scalapino, and S.-C. Zhang, Phys. Rev. B 77, 220503 (2008).
. M Daghofer, A Moreo, J A Riera, E Arrigoni, D J Scalapino, E Dagotto, Phys. Rev. Lett. 101237004M. Daghofer, A. Moreo, J. A. Riera, E. Arrigoni, D. J. Scalapino, and E. Dagotto, Phys. Rev. Lett. 101, 237004 (2008).
. H Eschrig, K Koepernik, Phys. Rev. B. 80104503H. Eschrig and K. Koepernik, Phys. Rev. B 80, 104503 (2009).
It may be worth noting that going to a different orbital basis does not restore the symmetry of the Hamiltonian, as can be seen from the discussion of the momentum-dependent bands. Additionally, the sign of t4 should depend on the hopping direction. which do not depend on the orbital basisAdditionally, the sign of t4 should depend on the hop- ping direction. It may be worth noting that going to a different orbital basis does not restore the symmetry of the Hamiltonian, as can be seen from the discussion of the momentum-dependent bands, which do not depend on the orbital basis.
. A Moreo, M Daghofer, J A Riera, E Dagotto, Phys. Rev. B. 79134502A. Moreo, M. Daghofer, J. A. Riera, and E. Dagotto, Phys. Rev. B 79, 134502 (2009).
. S Lebegue, Phys. Rev. B. 7535110S. Lebegue, Phys. Rev. B 75, 035110 (2007).
. D J Singh, M.-H Du, Phys. Rev. Lett. 100237003D. J. Singh and M.-H. Du, Phys. Rev. Lett. 100, 237003 (2008).
. G Xu, W Ming, Y Yao, X Dai, S.-C Zhang, Z Fang, EPL. 8267002G. Xu, W. Ming, Y. Yao, X. Dai, S.-C. Zhang, and Z. Fang, EPL 82, 67002 (2008).
. C Cao, P J Hirschfeld, H.-P Cheng, Phys. Rev. B. 77220506C. Cao, P. J. Hirschfeld, and H.-P. Cheng, Phys. Rev. B 77, 220506 (2008).
. H.-J Zhang, G Xu, X Dai, Z Fang, Chin. Phys. Lett. 2617401H.-J. Zhang, G. Xu, X. Dai, and Z. Fang, Chin. Phys. Lett. 26, 017401 (2009).
. M J Calderón, B Valenzuela, E Bascones, Phys. Rev. B. 8094531M. J. Calderón, B. Valenzuela, and E. Bascones, Phys. Rev. B 80, 094531 (2009).
This applies to 122 compounds when a two-dimensional description is used. This applies to 122 compounds when a two-dimensional description is used.[4]
. V B Nascimento, Phys. Rev. Lett. 10376104V. B. Nascimento et al., Phys. Rev. Lett. 103, 076104 (2009).
| [] |
[
"Gauge-Invariant Variables Reveal the Quantum Geometry of Fractional Quantum Hall States",
"Gauge-Invariant Variables Reveal the Quantum Geometry of Fractional Quantum Hall States"
] | [
"Yingkang Chen \nDepartment of Physics and Astronomy\nPurdue University\n47907West LafayetteIN\n",
"Rudro R Biswas \nDepartment of Physics and Astronomy\nPurdue University\n47907West LafayetteIN\n"
] | [
"Department of Physics and Astronomy\nPurdue University\n47907West LafayetteIN",
"Department of Physics and Astronomy\nPurdue University\n47907West LafayetteIN"
] | [] | Herein, we introduce the framework of gauge invariant variables to describe fractional quantum Hall (FQH) states, and prove that the wavefunction can always be represented by a unique holomorphic multi-variable complex function. As a special case, within the lowest Landau level, this function reduces to the well-known holomorphic coordinate representation of wavefunctions in the symmetric gauge. Using this framework, we derive an analytic guiding center Schrödinger's equation governing FQH states; it has a novel structure. We show how the electronic interaction is parametrized by generalized pseudopotentials, which depend on the Landau level occupancy pattern; they reduce to the Haldane pseudopotentials when only one Landau level is considered. Our formulation is apt for incorporating a new combination of techniques, from symmetric functions, Galois theory and complex analysis, to accurately predict the physics of FQH states using first principles. arXiv:1807.03306v2 [cond-mat.str-el] | 10.1103/physrevb.102.165313 | [
"https://arxiv.org/pdf/1807.03306v2.pdf"
] | 52,522,810 | 1807.03306 | 9e5f8cbe5fd9f79bb2a684a1e5117e231e3ab490 |
Gauge-Invariant Variables Reveal the Quantum Geometry of Fractional Quantum Hall States
13 Aug 2018
Yingkang Chen
Department of Physics and Astronomy
Purdue University
47907West LafayetteIN
Rudro R Biswas
Department of Physics and Astronomy
Purdue University
47907West LafayetteIN
Gauge-Invariant Variables Reveal the Quantum Geometry of Fractional Quantum Hall States
13 Aug 2018
Herein, we introduce the framework of gauge invariant variables to describe fractional quantum Hall (FQH) states, and prove that the wavefunction can always be represented by a unique holomorphic multi-variable complex function. As a special case, within the lowest Landau level, this function reduces to the well-known holomorphic coordinate representation of wavefunctions in the symmetric gauge. Using this framework, we derive an analytic guiding center Schrödinger's equation governing FQH states; it has a novel structure. We show how the electronic interaction is parametrized by generalized pseudopotentials, which depend on the Landau level occupancy pattern; they reduce to the Haldane pseudopotentials when only one Landau level is considered. Our formulation is apt for incorporating a new combination of techniques, from symmetric functions, Galois theory and complex analysis, to accurately predict the physics of FQH states using first principles. arXiv:1807.03306v2 [cond-mat.str-el]
Since the discovery of the quantum Hall effect [1,2], substantial progress has been made in our understanding of topological quantum phases of matter, of which quantum Hall states are prototypical examples [3][4][5][6][7][8][9][10][11]. However, current theories have limited scope in reliably predicting the existence and properties of these states in experimentally achievable scenarios [12][13][14]. In turn, such predictions are critical for establishing the viability and performance of topological quantum computation setups [15].
Specifically, there is a critical need for developing a universal theoretical approach for deriving the properties of strongly correlated topological quantum states, starting from realistic, experimentally-relevant microscopic Hamiltonians. Addressing this need, herein we establish a universal approach to formulating the quantum problem of calculating the properties of fractional quantum Hall (FQH) states at any filling fraction, involving any Landau level, by using the language of gauge-invariant variables (GIVs). The GIV representation [16][17][18] exploits the characteristic quantum geometry of topological phases [19] and allows us to isolate the essential onedimensional physics of electronic motion inside a Landau level [20][21][22].
In the absence of interactions, planar electrons in a perpendicular magnetic field can only have discrete energy values, corresponding to the well-known Landau levels [23]. Landau levels have macroscopic degeneracy, corresponding to the freedom of motion of the guiding center coordinates, which determine the spatial location of the electron's cyclotron orbit [10,13]. The presence of interactions splits this macroscopic degeneracy, giving rise to the FQH ground states at rational filling fractions. Since degenerate perturbation theory requires an exact manybody diagonalization of the interaction potential, finding the energy eigenstates corresponding to FQH ground and excited states, starting from the microscopic Hamiltonian, is a challenging problem.
It is widely accepted that the exact form of the interaction potential does not affect the large-scale wavefunc- , namely, the kinetic momenta, π, and the guiding center coordinates, R. In (b), adding interactions is the only modification to the simple 'free' picture necessary, for describing strongly correlated fractional quantum Hall phases. In contrast, in (a), first the magnetic field needs to be incorporated and then interactions are added in. tion structure of many FQH states, such as those at the simplest Laughlin fractions [13]. However, it is increasingly apparent that these details are important for determining the bulk structure of the technologically-relevant non-abelian FQH states (e.g., ν = 5/2) [24][25][26][27][28][29]. They are also important for determining the details of edge reconstruction, critical for interpreting interferometric experiments probing the braiding statistics of quasiparticles [12,15,30].
In this work we use the method of coherent states [31][32][33][34][35] to show that the many-body guiding-center wavefunction, which encapsulates the physics of FQH behavior, has a one-to-one correspondence with complex multivariate holomorphic functions, with one complex variable per particle. Our result holds for states with any filling fraction, irrespective of which Landau levels are involved. As a special case, our approach yields the holomorphic function representation of real-space wavefunctions in the lowest Landau level [10,13]. It also accounts for the success of similar representations for higher Landau level states, which arise from the use of conformal blocks [12,24]. Further, we derive the energy eigenvalue equation in this framework, Eq. (12), a qualitatively new form of Schrödinger's equation. Our formulation is apt for incorporating a new combination of techniques, from symmetric functions, Galois theory and complex analysis, to accurately predict the many-body energy eigenstates and eigenvalues corresponding to any FQH phase, starting from a microscopic Hamiltonian.
Gauge-Invariant Variables (GIV). Our approach utilizes Gauge-Invariant Variables (GIVs) [16][17][18]. To motivate their use, we contrast with the conventional kinematic description of two-dimensional motion of particles using real-space coordinates and linear momenta (see Figure 1). The intuition underlying the conventional description is that free particles should move in straight lines. Upon introducing a perpendicular magnetic field, the Lorentz force causes these trajectories to bend into cyclotron orbits. Interactions add a second layer of complexity to the already-modified picture. Thus, the description of interacting particles in a magnetic field, namely, the physical situation where FQH states arise, is laborious in the conventional framework.
Our proposal to remedy this situation is to instead use a language which naturally includes the magnetic field in the 'free' picture, thus leaving us to deal only with the addition of interactions to the problem. This is accomplished by using GIVs. In this framework, the 'free' dynamical units are not the particles that move in straight lines, but rather, entire cyclotron orbits which are static in the absence of external fields (other than the background magnetic field). These orbits exhibit nonintuitive responses such as drifting perpendicularly to an in-plane electric field with a universal drift velocity E/B [10,36]. This universal drift velocity is fundamentally related to the quantization of Hall conductance.
In what follows we consider electrons (with charge −e) moving in an infinite flat plane. For this scenario, the GIVs are the well-known kinetic momenta and guiding center coordinates [16], respectively,
π = p + eA(r), R = r + 2 / ẑ × π.
(1) = /eB is the magnetic length and the magnetic field is B = Bẑ. These GIVs satisfy the commutation relations:
[R x , R y ] = i 2 , [π y , π x ] = i 2 / 2 ,(2)
while π and R commute among themselves. The commutation relation of the guiding center coordinates captures the quantum geometry characterizing topological quantum systems. This is clear from analogous results in lattice systems [19] where this commutator has been related to the Chern number. For brevity, in what follows we have set the magnetic length ( ), electronic charge (e) and to unity. In these units, the commutation relations between the GIVs are analogous to the canonical commutation relations between the 2D coordinates and canonical momenta:
[x, p x ] = i, [y, p y ] = i, [(x, p x ), (y, p y )] = 0.(3)
Thus, the GIVs can be obtained from canonical coordinates and momenta via a canonical transformation. Therefore, a unitary transformation relates the orthonormal quantum Hilbert space basis labeled by the coordinates, {|x, y }, to another labeled by {|R x , π y }, the values of one operator from each of the canonical pairs in Eq. (2). Consequently, the quantum wavefunction expressed in the GIV basis is a function of one component from each canonical pair in Eq. (2), for e.g. ψ(R x , π y ). For more details see [17,18]. By straightforward generalization, the form of the many-body electronic wavefunc-
tion in the GIV language is Ψ({R x , π y } 1 , {R x , π y } 2 . . .),
where the numerical subscripts label particles. This wavefunction is completely antisymmetric under the permutation of every particle pair.
The Hamiltonian in terms of GIVs. We consider the following interacting electronic many-body Hamiltonian in a magnetic field:
H = α K(p α + A(r α )) + αβ U (|r α − r β |).(4)
The Greek indices label particles, and the prime on the second sum denotes a summation over distinct pairs. K and V are respectively the single-particle kinetic energy and the pairwise isotropic interaction potential. The kinetic energy K(p + A(r)) ≡ K(π) is a function only of the kinetic momenta and has a discrete spectrum. These discrete energies correspond to the wellknown Landau levels, whose specification fixes the kinetic momentum part of the wave function. The exact form of K and the presence of spin (or pseudospin) structure in the Hamiltonian do not alter our narrative. Therefore we ignore them in this presentation. In the absence of interactions, the single electron Hilbert space within a Landau level has macroscopic degeneracy, arising from the free guiding center part of the wavefunction, which does not affect the energy, since R commutes with π.
When the topmost Landau level is partially filled, weak interactions split the macroscopic Landau-level degeneracy and give rise to FQH physics. In this regime, we can renormalize the interaction by averaging over the fast motion of kinetic momenta:
U (|r α − r β |) π = U (|R α − R β +ẑ × (π α − π β )|) π ≡ V (|R α − R β |).(5)
The renormalized interaction, V , depends on the Landau level structure and is a function only of the guiding center coordinates. This renormalization procedure involves all Landau levels and incorporates inter-Landau-level correlations, critical for accurately capturing the physics at higher fillings [37]. (See Supplementary Section 1 and definition of generalized pseudopotentials below.) For the simple case when inter-Landau-level correlations are neglected, and only the physics of the topmost Landau level is considered, this reduces to the Landau level projection technique [38]. Additional techniques exist for incorporating the effects of Landau level mixing [17,18], which we will explore elsewhere. Here we focus on the renormalized Hamiltonian:
H ren = α K(π α ) + αβ V (|R α − R β |).(6)
Since the kinetic and potential parts of this Hamiltonian commute, the energy eigenstates are products of Landau level kinetic momentum eigenfunctions of individual particles and a many-body wavefunction in guiding center space. When multiple Landau levels are involved, antisymmetrization entangles both GIV spaces in straightforward but complex ways, leading to interesting physics in states with filling fractions greater than one [13,39,40]. For brevity, here we consider a single fractionally-filled Landau level and focus on the FQH physics induced by orbital interactions. We can show that the essential physics is unchanged when multiple Landau levels are occupied. (See Supplementary Section 1 for details.) The low-lying many-body energy eigenstates of such a partly filled Landau level are of the form Ψ m ({(R x , π y )}) = η(π y,1 )η(π y,2 ) . . . × ψ m ({R x }). (7) In this expression the braces denote the set of all particle coordinates, η is the single particle Landau level kinetic momentum eigenfunction (it is a simple harmonic oscillator eigenfunction for isotropic dispersion [17,18]) and ψ m is a completely antisymmetric eigenfunction of the effective interaction:
αβ V (|R α − R β |) ψ m ≡ U eff ψ m = m ψ m .(8)
It is straightforward to incorporate the non-interacting energy contributed by the kinetic part, E K . The manybody energies corresponding to Ψ m are simply:
E m = E K + m .(9)
The set {ψ m ({R x }), m } encodes the FQH physics arising due to interactions. This form also explicitly demonstrates that FQH physics is of a 1+1 dimensional nature, arising from guiding center dynamics [20][21][22]. Finding solutions to Eq. (8) is a difficult problem. First, it is a many-body equation and hence a complex multivariate problem. The second impediment is that the two canonically conjugate variables, R x and R y , appear with equal prominence in the operator U eff in Eq. (8). This feature is distinct from familiar physical problems, in which the canonical momentum and coordinate variables appear in distinct additive parts of the Hamiltonian, with different energy scales. (A notable exception is the simple harmonic oscillator, which is satisfactorily solved only through an exact technique.) This feature of Eq. (8) thus renders ineffective the usual approximation techniques for finding solutions. We propose to overcome this impediment by utilizing the language of coherent states, developed in the context of quantum optics [31][32][33][34][35], wherein a similar challenge arises.
GIV coherent states. Consider indexing the singleparticle one-dimensional guiding center Hilbert space by an orthonormal basis {|n }, where n is a non-negative integer. Since the renormalized interaction potential, V , is rotationally invariant, we will choose the particular basis given by the eigenstates of R 2 : R 2 |n = (2n + 1) |n . These states are simple harmonic oscillator states in guiding center space. They are also eigenstates of the projected interaction, V ( √ 2|R|) |n = V n |n . The {V n } are generalized pseudopotentials. (See Supplementary Section 1 for a general formula.) For the special case when only one Landau level is considered, they reduce to the standard Haldane pseudopotentials [38]. If V has other symmetries, other choices for {|n } may be useful. Any quantum state in this Hilbert space can be uniquely expressed as a complex vector sum: |ψ = n ψ n |n , with n |ψ n | 2 = 1. The overcomplete basis of 'coherent states', labelled by the complex variable z, is defined as follows [31][32][33][34][35]:
|z = e −|z| 2 /2 n φ n (z) |n , φ n (z) = z n √ n! .(10)
The {φ n } are holomorphic functions whose choice above is motivated by the fact that we are considering motion in an infinite flat plane. e −|z| 2 /2 is a normalizing factor. Due to the orthonormality of the φ n states, C d 2 ze −|z| 2 φ * m (z)φ n (z) = πδ mn , the coherent states satisfy the well-known completeness relation:
C d 2 z |z z| = πI.
For other useful properties of coherent states we refer the reader to [31][32][33][34][35]. Using the definition of coherent states, we can map any quantum state in guiding center space, |ψ = n ψ n |n , to a unique holomorphic function:
ψ(z) ≡ e |z| 2 /2 ψ|z = n ψ * n φ n (z).(11)
We have thus shown that the guiding center part of the wavefunction, whose properties are critical for uncovering FQH physics, can be described using holomorphic functions. This is true irrespective of the filling fraction and which Landau levels are occupied.
Our results have connections with the following known results. In the symmetric gauge, the quantum wavefunction in the lowest Landau level can be identified with a holomorphic function, ψ 0 (z), where z = x+iy (ignoring a fixed Gaussian factor) [10,13]. This mathematical representation played a crucial role in identifying the Laughlin and related trial wavefunctions for FQH states in the lowest Landau level. An independent approach for generating real-space wavefunctions with desirable ground state characteristics involves using conformal blocks, which also give rise to approximately holomorphic functions [12,24]. Our analysis demonstrates that holomorphic functions can be used for describing states with multiple occupied Landau levels, due to the quantum geometry encapsulated by the commutation properties of the guiding center GIVs. We have also identified the precise universal relationship between these holomorphic functions and the microscopic many-body wavefunction in the coordinate basis. As a special case, in the lowest Landau level, our holomorphic function in the coherent state representation, ψ, has a straightforward relation to the holomorphic function ψ 0 characterizing the coordinate representation in the symmetric gauge:
ψ(z) = ψ 0 (−i √ 2z * ) * .
The GIV Schrödinger equation. Thus far we have established that the many-body electronic wavefunction is a product of a set of non-interacting kinetic momenta wavefunctions (which correspond to fast high energy cyclotron orbit motion) and a many-body guiding center wavefunction (which captures the physics of FQH states).
When multiple Landau levels and spins are involved, the permutation symmetries of the kinetic momentum and guiding center wavefunctions can be complicated, but are straightforward to incorporate [39,40]. For the simple case of spinless electrons in a single Landau level, the kinetic momentum wavefunction is symmetric and the guiding center wavefunction completely antisymmetric under particle permutation. The same simple symmetry structure is also applicable when multiple Landau levels are populated. (See Supplementary Section 1.) Once the permutation symmetry structure is fixed, the guiding center wavefunction corresponds to a many-variable holomorphic function ψ({z}) with the same permutation properties.
We now present how the GIV holomorphic representation can be put to practical use, by deriving the corresponding Schrödinger equation for determining the energy eigenstates. This is achieved by expressing Eq.
(R + , R − ), where R ± = (R 1 ± R 2 )/ √ 2.
The interaction is a function only of R − with eigenvalues that are the generalized pseudopotentials {V n }. (See Supplementary Section 1.) With this insight, we use well-known properties of coherent states to obtain the GIV Schrödinger equation:
∞ n=0 V n n! i<j z i − z j 2 n ∂ ∂ξ i − ∂ ∂ξ j n ψ ({ξ}) ξ k =z k ,ξi=ξj = z i +z j 2 = ψ({z}).(12)
This equation is valid for any filling fraction. It is derived directly from first principles and takes into account interactions between different Landau levels, i.e., filled Landau levels are not discarded as inert. To reiterate, ψ({z}) describes electrons from all occupied Landau levels.
Due to its novel form, the solution of the GIV Schrödinger equation, Eq. (12), requires the development of new techniques, to be presented in future works. As previously noted, the symmetric gauge real space wavefunction in the lowest Landau level, ψ 0 , is related to ψ by the transformation ψ(z) = ψ 0 (−i √ 2z * ) * . This transformation leaves Eq. (12) intact; thus the same equation should also hold for real space wavefunctions in the lowest Landau level. In the context of coordinate basis wavefunctions, applicable only to the lowest Landau level, such an operator representation has appeared pre-viously in [41]. The GIV Schrödinger equation, Eq. (12), is however applicable to any Landau level configuration and uses GIV wavefunctions.
To illustrate how Eq. (12) can be utilized, we derive the following well known result: the Laughlin state at filling fraction ν = 1/m is the exact unique ground state when the first m−2 pseudopotentials (V 1 , V 2 . . . V m−2 ) are positive and the rest are zero. (For details see Supplementary Section 4.) Briefly, this follows from the fact that each pairwise operator multiplying the pseudopotentials is a projection operator. Thus, the eigenstates of a positive linear combination of these, the operator on the left side of Eq. (12), cannot have negative eigenvalues. At filling fraction 1/m, when the first m − 2 pseudopotentials are positive and the rest zero, the unique state with zero energy is the corresponding Laughlin state, thus proving that the Laughlin state must be the unique ground state.
Concluding remarks. We have used the language of GIVs to derive a holomorphic representation of FQH physics, which is valid for any Landau level filling pattern and for arbitrary forms of the kinetic energy. The framework that we have developed can be generalized to accomodate a variety of scenarios involving different real space manifolds and symmetries. We have demonstrated its use by formulating the FQH problem in this language, on an infinite plane with arbitrary isotropic pairwise interactions. Putting these insights and results together, we have derived the analyic GIV Schrödinger equation, Eq. (12), which naturally discards irrelevant high energy physics. The FQH many-body ground and excited state wavefunctions and energies correspond to the eigenstates and eigenvalues of this novel differential equation.
Eq. (12) provides a new route for deriving the properties of FQH states from microscopic Hamiltonians, by recasting the quantum many-body calculation in the GIV representation. Since the wavefunction corresponds to a holomorphic function with certain permutation symmetries under particle exchange, our formulation also provides an avenue to exploit insights from diverse mathematical fields like symmetric polynomials, Galois theory, complex analysis, etc. We hope that a synthesis of our formalism with these techniques will allow for first-principles based predictions of FQH state properties, starting from realistic microscopic Hamiltonians. Apparently distinct descriptions of FQH physics, such as trial holomorphic wavefunctions in the lowest Landau level [5,13]; the conformal block picture from conformal field theory [12,24]; the composite fermion approach [13]; topological quantum field theory [42]; and matrix product states [43] may be naturally unified in our coherent state GIV formulation.
Author contributions: This research was conceived of and designed by RRB. YK and RRB performed calculations and wrote the paper. We acknowledge useful discussions with Mike Manfra, Chris Greene, Gabor Csathy, Leonid Rokhinson and Srividya Iyer-Biswas. This research was supported by Purdue University Startup Funds and the Purdue Research Foundation. This work was completed and presented to limited audiences before [44] was made available on arXiv.
FIG. 1 .
1Comparing kinematic formulations of planar quantum mechanics in the presence of a magnetic flux, B, using (a) conventional position and linear momentum, vs. (b) gaugeinvariant variables (GIVs)
(8) in the coherent state representation, ψ|U eff | {z} = ψ| {z} = ψ({z}). (Details are provided in the Supplementary Sections 2 and 3.) Briefly, the central challenge is to calculate matrix elements of the form ψ|V (|R 1 − R 2 |) | {z 1 , z 2 } . This is achieved by rotating from the commuting pair of canonical GIVs (R 1 , R 2 ) to another canonical commuting GIV pair:
. K Klitzing, G Dorda, M Pepper, 10.1103/PhysRevLett.45.494Phys. Rev. Lett. 45494K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
. D C Tsui, H L Stormer, A C Gossard, 10.1103/PhysRevLett.48.1559Phys. Rev. Lett. 481559D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).
. R B Laughlin, 10.1103/PhysRevB.23.5632Phys. Rev. B. 235632R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).
. B I Halperin, 10.1103/PhysRevB.25.2185Phys. Rev. B. 252185B. I. Halperin, Phys. Rev. B 25, 2185 (1982).
. R B Laughlin, 10.1103/PhysRevLett.50.1395Phys. Rev. Lett. 501395R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).
. A M M Pruisken, 10.1016/0550-3213(84)90101-9Nuclear Physics B. 235277A. M. M. Pruisken, Nuclear Physics B 235, 277 (1984).
R E Prange, S M Girvin, The Quantum Hall effect. New YorkSpringer-Verlag2nd ed.R. E. Prange and S. M. Girvin, The Quantum Hall effect, 2nd ed. (Springer-Verlag, New York, 1990).
. X.-G Wen, 10.1080/00018739500101566Advances in Physics. 44405X.-G. Wen, Advances in Physics 44, 405 (1995).
. H L Stormer, D C Tsui, A C Gossard, 10.1103/RevModPhys.71.S298Rev. Mod. Phys. 71298H. L. Stormer, D. C. Tsui, and A. C. Gossard, Rev. Mod. Phys. 71, S298 (1999).
The quantum Hall effect. D Yoshioka, SpringerBerlinD. Yoshioka, The quantum Hall effect (Springer, Berlin, 2002).
. G Murthy, R Shankar, 10.1103/RevModPhys.75.1101Rev. Mod. Phys. 751101G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101 (2003).
. T H Hansson, M Hermanns, S H Simon, S F Viefers, 10.1103/RevModPhys.89.025005Rev. Mod. Phys. 8925005T. H. Hansson, M. Hermanns, S. H. Simon, and S. F. Viefers, Rev. Mod. Phys. 89, 025005 (2017).
J K Jain, Composite fermions. Cambridge University PressJ. K. Jain, Composite fermions (Cambridge University Press, 2007).
. M O Goerbig, 10.1103/RevModPhys.83.1193Rev. Mod. Phys. 831193M. O. Goerbig, Rev. Mod. Phys. 83, 1193 (2011).
. C Nayak, S H Simon, A Stern, M Freedman, S. Das Sarma, 10.1103/RevModPhys.80.1083Rev. Mod. Phys. 801083C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).
. M H Johnson, B A Lippmann, 10.1103/PhysRev.76.828Phys. Rev. 76828M. H. Johnson and B. A. Lippmann, Phys. Rev. 76, 828 (1949).
. R R Biswas, arXiv:1311.7149R. R. Biswas, (2013), arXiv:1311.7149.
. Y Chen, G Jiang, R R Biswas, Y. Chen, G. Jiang, and R. R. Biswas, (2018), unpub- lished.
. T A Loring, M B Hastings, 10.1209/0295-5075/92/67004EPL. 9267004T. A. Loring and M. B. Hastings, EPL 92, 67004 (2010).
. E J Bergholtz, A Karlhede, 10.1103/PhysRevLett.94.026802Phys. Rev. Lett. 9426802E. J. Bergholtz and A. Karlhede, Phys. Rev. Lett. 94, 026802 (2005).
. A G Abanov, P B Wiegmann, 10.1103/PhysRevLett.95.076402Phys. Rev. Lett. 9576402A. G. Abanov and P. B. Wiegmann, Phys. Rev. Lett. 95, 076402 (2005).
. M Horsdal, J M Leinaas, 10.1103/PhysRevB.76.195321Phys. Rev. B. 76195321M. Horsdal and J. M. Leinaas, Phys. Rev. B 76, 195321 (2007).
. L D Landau, Z. Phys. 64629L. D. Landau, Z. Phys. 64, 629 (1930).
. G Moore, N Read, 10.1016/0550-3213(91)90407-ONuclear Physics B. 360362G. Moore and N. Read, Nuclear Physics B 360, 362 (1991).
. S.-S Lee, S Ryu, C Nayak, M P A Fisher, 10.1103/PhysRevLett.99.236807Phys. Rev. Lett. 99236807S.-S. Lee, S. Ryu, C. Nayak, and M. P. A. Fisher, Phys. Rev. Lett. 99, 236807 (2007).
. M Levin, B I Halperin, B Rosenow, 10.1103/PhysRevLett.99.236806Phys. Rev. Lett. 99236806M. Levin, B. I. Halperin, and B. Rosenow, Phys. Rev. Lett. 99, 236806 (2007).
. A C Balram, Y.-H Wu, G J Sreejith, A Wójs, J K Jain, 10.1103/PhysRevLett.110.186801Phys. Rev. Lett. 110186801A. C. Balram, Y.-H. Wu, G. J. Sreejith, A. Wójs, and J. K. Jain, Phys. Rev. Lett. 110, 186801 (2013).
. E H Rezayi, 10.1103/PhysRevLett.119.026801Phys. Rev. Lett. 11926801E. H. Rezayi, Phys. Rev. Lett. 119, 026801 (2017).
. R S K Mong, M P Zaletel, F Pollmann, Z Papić, 10.1103/PhysRevB.95.115136Phys. Rev. B. 95115136R. S. K. Mong, M. P. Zaletel, F. Pollmann, and Z. Papić, Phys. Rev. B 95, 115136 (2017).
. R Sabo, I Gurman, A Rosenblatt, F Lafont, D Banitt, J Park, M Heiblum, Y Gefen, V Umansky, D Mahalu, 10.1038/nphys4010Nature Physics. 13491R. Sabo, I. Gurman, A. Rosenblatt, F. Lafont, D. Banitt, J. Park, M. Heiblum, Y. Gefen, V. Umansky, and D. Ma- halu, Nature Physics 13, 491 (2017).
. E C G Sudarshan, 10.1103/PhysRevLett.10.277Phys. Rev. Lett. 10277E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
. R J Glauber, 10.1103/PhysRev.131.2766Phys. Rev. 1312766R. J. Glauber, Phys. Rev. 131, 2766 (1963).
. R J Glauber, 10.1103/PhysRev.130.2529Phys. Rev. 1302529R. J. Glauber, Phys. Rev. 130, 2529 (1963).
J Klauder, E Sudarshan, Fundamentals of Quantum Optics. Dover PublicationsJ. Klauder and E. Sudarshan, Fundamentals of Quantum Optics (Dover Publications, 2006).
Generalized Coherent States and Their Applications. A Perelomov, Theoretical and Mathematical Physics. BerlinSpringerA. Perelomov, Generalized Coherent States and Their Applications, Theoretical and Mathematical Physics (Springer, Berlin, 2012).
J Jackson, Classical electrodynamics. WileyJ. Jackson, Classical electrodynamics (Wiley, 1975).
. A A Koulakov, M M Fogler, B I Shklovskii, 10.1103/PhysRevLett.76.499Phys. Rev. Lett. 76499A. A. Koulakov, M. M. Fogler, and B. I. Shklovskii, Phys. Rev. Lett. 76, 499 (1996).
The hierarchy of fractional states and numerical studies. F D M Haldane, 10.1007/978-1-4612-3350-3{_}8The Quantum Hall Effect. R. E. Prange and S. M. GirvinNew York, New York, NYSpringerF. D. M. Haldane, "The hierarchy of fractional states and numerical studies," in The Quantum Hall Effect, edited by R. E. Prange and S. M. Girvin (Springer New York, New York, NY, 1990) Chap. 8, pp. 303-352.
. B I Halperin, Helv. Phys. Acta. 5675B. I. Halperin, Helv. Phys. Acta 56, 75 (1983).
. M O Goerbig, N Regnault, 10.1103/PhysRevB.75.241405Phys. Rev. B. 75241405M. O. Goerbig and N. Regnault, Phys. Rev. B 75, 241405 (2007).
. A Cappelli, C A Trugenberger, G R Zemba, 10.1016/0550-3213(93)90660-HNuclear Physics B. 396465A. Cappelli, C. A. Trugenberger, and G. R. Zemba, Nu- clear Physics B 396, 465 (1993).
X.-G Wen, 10.1093/acprof:oso/9780199227259.001.0001Quantum field theory of many-body systems. Oxford University PressX.-G. Wen, Quantum field theory of many-body systems (Oxford University Press, 2004).
. M P Zaletel, R S K Mong, 10.1103/PhysRevB.86.245305Phys. Rev. B. 86245305M. P. Zaletel and R. S. K. Mong, Phys. Rev. B 86, 245305 (2012).
. F D M Haldane, arXiv:1806.10106F. D. M. Haldane, (2018), arXiv:1806.10106.
| [] |
[
"Gravitational recoils of supermassive black holes in hydrodynamical simulations of gas rich galaxies",
"Gravitational recoils of supermassive black holes in hydrodynamical simulations of gas rich galaxies"
] | [
"Debora Sijacki \nKavli Institute for Cosmology, Cambridge and Institute of Astronomy\nMadingley RoadCB3 0HACambridge\n",
"Volker 2⋆ \nHarvard-Smithsonian Center for Astrophysics\n60 Garden Street02138CambridgeMAUSA\n",
"Springel \nHeidelberg Institute for Theoretical Studies\nSchloss-Wolfsbrunnenweg 3569118HeidelbergGermany\n\nZentrum für Astronomie\nUniversität Heidelberg\nARI\nMönchhofstr. 12-1469120HeidelbergGermany\n",
"Martin G Haehnelt \nKavli Institute for Cosmology, Cambridge and Institute of Astronomy\nMadingley RoadCB3 0HACambridge\n"
] | [
"Kavli Institute for Cosmology, Cambridge and Institute of Astronomy\nMadingley RoadCB3 0HACambridge",
"Harvard-Smithsonian Center for Astrophysics\n60 Garden Street02138CambridgeMAUSA",
"Heidelberg Institute for Theoretical Studies\nSchloss-Wolfsbrunnenweg 3569118HeidelbergGermany",
"Zentrum für Astronomie\nUniversität Heidelberg\nARI\nMönchhofstr. 12-1469120HeidelbergGermany",
"Kavli Institute for Cosmology, Cambridge and Institute of Astronomy\nMadingley RoadCB3 0HACambridge"
] | [
"Mon. Not. R. Astron. Soc"
] | We study the evolution of gravitationally recoiled supermassive black holes (BHs) in massive gas-rich galaxies by means of high-resolution hydrodynamical simulations. We find that the presence of a massive gaseous disc allows recoiled BHs to return to the centre on a much shorter timescale than for purely stellar discs. Also, BH accretion and feedback can strongly modify the orbit of recoiled BHs and hence their return timescale, besides affecting the distribution of gas and stars in the galactic centre. However, the dynamical interaction of kicked BHs with the surrounding medium is in general complex and can facilitate both a fast return to the centre as well as a significant delay. The Bondi-Hoyle-Lyttleton accretion rates of the recoiling BHs in our simulated galaxies are favourably high for the detection of off-centred AGN if kicked into gas-rich discs -up to a few per cent of the Eddington accretion rateand are highly variable on timescales of a few 10 7 yrs. In major merger simulations of gas-rich galaxies, we find that gravitational recoils increase the scatter in the BH mass -host galaxy relationships compared to simulations without kicks, with the BH mass being more sensitive to recoil kicks than the bulge mass. The BH mass can be lowered by a factor of a few due to a recoil, even for a relatively short return timescale, but the exact magnitude of the effect depends strongly on the BH binary hardening timescale and on the efficiency of star formation in the central regions. A generic result of our numerical models is that the clumpy massive discs suggested by recent high-redshift observations, as well as the remnants of gas-rich mergers, exhibit a gravitational potential that falls steeply in the central regions, due to the dissipative concentration of baryons. As a result, supermassive BHs should only rarely be able to escape from massive galaxies at high redshifts, which is the epoch where the bulk of BH recoils is expected to occur. | 10.1111/j.1365-2966.2011.18666.x | [
"https://arxiv.org/pdf/1008.3313v2.pdf"
] | 119,181,119 | 1008.3313 | d215112ea820bd8452e0a8291a49633ac5658a5e |
Gravitational recoils of supermassive black holes in hydrodynamical simulations of gas rich galaxies
6 Apr 2011 8 April 2011
Debora Sijacki
Kavli Institute for Cosmology, Cambridge and Institute of Astronomy
Madingley RoadCB3 0HACambridge
Volker 2⋆
Harvard-Smithsonian Center for Astrophysics
60 Garden Street02138CambridgeMAUSA
Springel
Heidelberg Institute for Theoretical Studies
Schloss-Wolfsbrunnenweg 3569118HeidelbergGermany
Zentrum für Astronomie
Universität Heidelberg
ARI
Mönchhofstr. 12-1469120HeidelbergGermany
Martin G Haehnelt
Kavli Institute for Cosmology, Cambridge and Institute of Astronomy
Madingley RoadCB3 0HACambridge
Gravitational recoils of supermassive black holes in hydrodynamical simulations of gas rich galaxies
Mon. Not. R. Astron. Soc
00000006 Apr 2011 8 April 2011Printed 8 April 2011arXiv:1008.3313v2 [astro-ph.CO] (MN L A T E X style file v2.2)methods: numerical -black hole physics -cosmology: theory
We study the evolution of gravitationally recoiled supermassive black holes (BHs) in massive gas-rich galaxies by means of high-resolution hydrodynamical simulations. We find that the presence of a massive gaseous disc allows recoiled BHs to return to the centre on a much shorter timescale than for purely stellar discs. Also, BH accretion and feedback can strongly modify the orbit of recoiled BHs and hence their return timescale, besides affecting the distribution of gas and stars in the galactic centre. However, the dynamical interaction of kicked BHs with the surrounding medium is in general complex and can facilitate both a fast return to the centre as well as a significant delay. The Bondi-Hoyle-Lyttleton accretion rates of the recoiling BHs in our simulated galaxies are favourably high for the detection of off-centred AGN if kicked into gas-rich discs -up to a few per cent of the Eddington accretion rateand are highly variable on timescales of a few 10 7 yrs. In major merger simulations of gas-rich galaxies, we find that gravitational recoils increase the scatter in the BH mass -host galaxy relationships compared to simulations without kicks, with the BH mass being more sensitive to recoil kicks than the bulge mass. The BH mass can be lowered by a factor of a few due to a recoil, even for a relatively short return timescale, but the exact magnitude of the effect depends strongly on the BH binary hardening timescale and on the efficiency of star formation in the central regions. A generic result of our numerical models is that the clumpy massive discs suggested by recent high-redshift observations, as well as the remnants of gas-rich mergers, exhibit a gravitational potential that falls steeply in the central regions, due to the dissipative concentration of baryons. As a result, supermassive BHs should only rarely be able to escape from massive galaxies at high redshifts, which is the epoch where the bulk of BH recoils is expected to occur.
April 2011
ABSTRACT
We study the evolution of gravitationally recoiled supermassive black holes (BHs) in massive gas-rich galaxies by means of high-resolution hydrodynamical simulations. We find that the presence of a massive gaseous disc allows recoiled BHs to return to the centre on a much shorter timescale than for purely stellar discs. Also, BH accretion and feedback can strongly modify the orbit of recoiled BHs and hence their return timescale, besides affecting the distribution of gas and stars in the galactic centre. However, the dynamical interaction of kicked BHs with the surrounding medium is in general complex and can facilitate both a fast return to the centre as well as a significant delay. The Bondi-Hoyle-Lyttleton accretion rates of the recoiling BHs in our simulated galaxies are favourably high for the detection of off-centred AGN if kicked into gas-rich discs -up to a few per cent of the Eddington accretion rateand are highly variable on timescales of a few 10 7 yrs. In major merger simulations of gas-rich galaxies, we find that gravitational recoils increase the scatter in the BH mass -host galaxy relationships compared to simulations without kicks, with the BH mass being more sensitive to recoil kicks than the bulge mass. The BH mass can be lowered by a factor of a few due to a recoil, even for a relatively short return timescale, but the exact magnitude of the effect depends strongly on the BH binary hardening timescale and on the efficiency of star formation in the central regions. A generic result of our numerical models is that the clumpy massive discs suggested by recent high-redshift observations, as well as the remnants of gas-rich mergers, exhibit a gravitational potential that falls steeply in the central regions, due to the dissipative concentration of baryons. As a result, supermassive BHs should only rarely be able to escape from massive galaxies at high redshifts, which is the epoch where the bulk of BH recoils is expected to occur.
INTRODUCTION
In the last couple of years, numerical relativity simulations of coalescing black hole (BH) binaries have opened a whole new window in our understanding of the BH properties in the merger aftermath (e.g. Pretorius 2007;González et al. 2007;Herrmann et al. 2007;Koppitz et al. 2007;Campanelli et al. 2007;Baker et al. 2008). These computations have established that for asymmetries in the mass and/or spin of the two merging BHs, net linear momentum ⋆ Hubble Fellow. E-mail: [email protected] will be carried away by the asymmetric emission of gravitational waves, imparting a gravitational recoil to the remnant BH. In the case of the spinning BHs, the recoil velocities can become very large -comparable to or even larger than the virial velocities of the most massive host systems. This is what makes this phenomenon astrophysically very interesting and relevant, and poses important questions to which we still do not have definite answers: How often do the recoils occur within the hierarchical structure formation scenario? What is the fraction of expelled BHs, and how fast can they sink back to the centre? What are the imprints of gravita-tionally recoiled BHs and why has it been difficult to detect them so far?
The importance of BH recoils in the astrophysical context has been realised in pioneering works early on (e.g. Bekenstein 1973;Blandford 1979;Kapoor 1985). More recently, analytical and numerical studies (e.g. Merritt et al. 2004;Boylan-Kolchin et al. 2004;Madau & Quataert 2004;Micic et al. 2006) have found that globular clusters and dwarf galaxies are the prime targets for getting depleted of their central BHs. Moreover, recoiled BHs which do not escape their host galaxies are likely to induce stellar cores through repeated passages close to the centre.
In the case of purely stellar systems, accurate N-body simulations (for a recent study see Gualandris & Merritt 2008) have led to a good understanding how gravitationally recoiled BHs orbit in spherical systems and what their typical return timescales for different kick velocities are. For systems with gas, however, only a handful of numerical studies (e.g. Kornreich & Lovelace 2008;Devecchi et al. 2009;Guedes et al. 2011) are available. These studies computed the trajectories of recoiled BHs and accounted for their possible interaction with the surrounding gas. Adopting a semi-analytical approach, BH luminosities during the wandering phase have nevertheless been estimated (e.g. Volonteri & Perna 2005;Blecha & Loeb 2008;Fujita 2009;Guedes et al. 2011). Recently, Dotti et al. (2010) performed simulations of BH binaries embedded in a circum-nuclear disc allowing for gas accretion, and by analytically tracing BH spin evolution, found that high recoil velocities should not be very likely. While observationally several candidates for recoiled BHs have been proposed (Komossa et al. (2008); Civano et al. (2010), but see Shields et al. (2009);Heckman et al. (2009)), there is scarce evidence for luminous recoiled quasars, at least in the case of large kinematic offsets (Bonning et al. 2007).
For galaxies containing gas, the evolution of recoiled BHs can possibly vary significantly depending on the mass fraction of the gas, and its spatial distribution and thermodynamic state. This situation presents a much more complex problem than for purely stellar systems. For example, recoiled BHs might accrete some of the interstellar gas which could in return affect their trajectory. Also, if BH feedback effects associated with accretion are not negligible, they could provide distributed heating throughout the galaxy. It is clear that a full understanding of these possibilities will require the exploration of a large parameter space.
At redshifts of z ∼ 2 − 3, galaxy mergers are much more common in the hierarchical structure formation scenario than at the present day, and it is believed that many galaxies are gas-rich, with gas fractions of up to 50% (Förster Schreiber et al. 2009, and references therein). Massive compact ellipticals at z ∼ 2 with much smaller sizes and higher stellar densities than their local counterparts (e.g. van Dokkum et al. 2008) are likely to be the end products of gas-rich mergers where remnant BHs would reside. This epoch also corresponds to the peak of the quasar space density, indicating that BHs are accreting gas supplied by their hosts efficiently. This is hence an extremely important and interesting regime, where BH mergers and recoils are expected to occur frequently and at the same time the gas component in the host systems cannot be neglected.
In this paper, we present an exploratory study of the properties of recoiled BHs in gas-rich galaxies, where BH accretion and feedback processes are followed self-consistently. We first focus on studying an isolated spiral galaxy, simulated at a high resolution to understand how the inclusion of gas accretion or additional BH feedback affects the orbits of recoiled BHs, their return timescale to the centre, and their imprints on the host galaxy. We simulate kicks of different magnitude, both in the plane and perpendicular to the plane of the galaxy, to gauge the dependence of the AGN luminosity on the kick orientation and on the assumed equation of state for the interstellar gas. We then perform a major merger simulation of two gas-rich galaxies, each containing a supermassive BH in its centre. After BH coalescence, we follow the evolution of the gravitationally recoiled BH and study how its growth is affected by the kick, and which consequences this has for the scatter in the BH mass-host galaxy scaling relations. Note that a systematic study of recoiled BHs in merging galaxies is underway (Blecha et al. 2011, private communication).
The paper is organised as follows. In Section 2, we outline the numerical methods we adopted. Most of our results are presented in Section 3, where we discuss isolated spiral galaxies with uniform or clumpy discs (Section 3.1), and major mergers of two gas-rich galaxies (Section 3.2). In Section 4, we discuss our findings and draw our conclusions.
METHODOLOGY
The numerical code
In this work we perform hydrodynamical simulations with the massively parallel Tree-SPH code GADGET-3 (last described in Springel 2005). The code computes gravitational forces acting on dark matter, gas, star and BH particles, as well as the hydrodynamical forces that affect the baryons. Gas is modelled as an optically thin plasma of hydrogen . BH distance from the minimum of the potential as a function of time for simulations where the BH is (a) not subject to dynamical friction forces (blue line), (b) experiences dynamical friction from stars only (green line), and (c) additionally is subject to dynamical friction from the gas in the disc (red line). and helium, which can radiatively cool and heat. Star formation and supernova feedback is implemented adopting a subresolution multi-phase model (Springel & Hernquist 2003), while BH growth and feedback is modelled as described in Di ; Springel et al. (2005); Sijacki et al. (2007Sijacki et al. ( , 2009. For completeness, we briefly summarise this model below.
BH model
In the simulation code, BHs are treated as collisionless sink particles. Starting out with a certain seed mass, BHs can grow in time by gas accretion, or by merging with other BHs that happen to be sufficiently close (within each others smoothing lengths) and that have small relative velocities (less than the local gas sound speed). The accretion rate is parametrised in terms of a spherically symmetric Bondi-Hoyle-Lyttleton type accretion flow (Hoyle & Lyttleton 1939;Bondi & Hoyle 1944;Bondi 1952)
, i.e. MBH = 4 π α G 2 M 2 BH ρ c 2 s + v 2 3/2 ,(1)
where α is a dimensionless parameter, ρ is the gas density, cs the sound speed of the gas, and v is the velocity of the BH relative to the gas. Here, as a default value we fix α to 100, as in all of our previous work, because the multiphase model for star formation gives a comparatively high mean gas temperature as a result of supernova feedback. A volume-average of the Bondi-rates for the sub-grid cold and hot phases of the interstellar medium recovers a value of α close to 100. We have however found that our results are relatively insensitive to the adopted value for the α parameter (see the tests in Appendix A). The maximum BH accretion Table 1. Gas fraction, equation of state parameter for the interstellar medium, virial velocity, central escape velocity, BH recoil velocity and initial BH mass for the isolated spiral galaxy with a smooth and a clumpy disc, and for the gas-rich major merger. As a standard choice, the gravitational recoil direction is within the galactic plane, and the BH is allowed to accrete gas and exert feedback. †In this case we also simulate kicks perpendicular to the galactic plane, and for the recoil in the galactic plane we run extra simulations where: i) the BH is massless, ii) the BH does not accrete, iii) the BH accretes, but there is no feedback, iv) BH accretion and feedback is allowed, but α = 1 in equation (1). ‡This simulation is analogous to the one listed in the second row, but the galaxy has been simulated at twice as high spatial resolution, resulting in a somewhat deeper central potential and thus higher central escape velocity.
fgas q EOS v 200 vesc(0) v kick M BH [km s −1 ] [km s −1 ] [km s −1 ] [h −1 M ⊙ ] UNIFORM
rate is limited to the Eddington rate. BH feedback associated with the accretion is injected in thermal form, where a small fraction (5%) of the bolometric luminosity is isotropically coupled to the surrounding gas.
Gravitational recoils of BHs
BH merger remnants are subject to a gravitational recoil due to the anisotropic emission of gravitational waves, which carry away net linear momentum. In Sijacki et al. (2009), we have implemented a simple method to represent this effect in our simulations. At the moment of a BH binary merger, the involved BH masses and spins ⋆ are known in the simulation. By making assumptions about the orientation of the spins with respect to the orbital angular momentum of the BH binary, it is then possible to compute the expected gravitational kick velocities accurately based on fitting formulae that are calibrated against numerical relativity simulations (for a review see Pretorius 2007). In the case of the isolated galaxies with a single BH, where the primary goal is to understand the interaction between a recoiling BH and its host, we do not use these fitting formulae but consider a wide range of possible kick velocities in the plane and perpendicular to the plane of the galaxy. In the case of the major merger discussed in Section 3.2 we again consider a range of physically interesting kick velocities (comparable to the central escape velocity), and by assuming that the two BHs are close to maximally spinning we adopt the relations from Campanelli et al. (2007) to estimate possible kick magnitudes. In our original model (Sijacki et al. 2009), the BH recoil direction is assumed to be random with respect to the gas distribution of the host, given that there is no firm observational evidence for any special alignment of BH jets with the host galaxy structure. Here however we focus on the case where the BH is recoiled in the plane of the forming circum-nuclear disc. This turns out to be a physically more interesting case for studying recoiled BH -host interactions without compromising the general validity of our findings.
Galaxy models
We first consider an isolated, massive, gas-rich galaxy, that we simulate at high numerical resolution. The initial conditions of this galaxy are set-up as described in detail in Springel et al. (2005). Specifically, the total mass of the galaxy is M200 = 6.28×10 12 h −1 M⊙ (with h = 0.7), its virial velocity is v200 = 300 km s −1 , the dark matter halo concentration is c = 9, and the spin parameter is λ = 0.033. The fraction of the total mass in the disc is m d = 0.041, while the bulge mass fraction amounts to m b = 0.008. The galactic disc scale length is 4.8 h −1 kpc. The disc is gas-rich, with a gas fraction of fgas = 0.6, while the remaining 40% are in stars. Circular velocity curves for the different galactic components are shown in Figure 1.
We select the numerical parameters of the multiphase model for star formation such that the gas consumption timescale is long, specifically we adopt t 0 * = 12.6 Gyrs, A0 = 6000, TSN = 6 × 10 8 K (in the notation of Springel & Hernquist 2003), and we assume a softer equation of state with qEOS = 0.5. This choice of parameters assures that during the simulated time-span (which can be a considerable fraction of the Hubble time) the galaxy stays comparatively gas-rich and the gaseous disc is stable. We additionally perform runs where we keep all numerical and physical parameters the same, except for the equation of state parameter which we set to 0.05, corresponding to a nearly isothermal equation of state and resulting in a very clumpy disc. The galaxy is simulated with 300000 dark matter particles in the halo, 200000 star particles in the disc, 200000 gas particles in the disc and 50000 star particles belonging to the bulge. The gravitational softening of the disc and bulge is set to 60 h −1 pc, and of the dark matter halo to 2 h −1 kpc.
Numerical setup
We first evolve the isolated gas-rich galaxy for 2.8 × 10 8 yrs without a central BH, until the gas disc develops a well defined spiral structure which is long lived. Thus, during the simulated time of the central BH the gaseous disc will be in a stable, quasi stationary state. We then introduce a BH particle in the centre of the galaxy, assuming that such a gas-rich galaxy should already contain a supermassive BH. To be confident that the dynamical friction force acting on the BH is numerically well resolved we set the initial mass of the BH to 5 × 10 8 h −1 M⊙ in most of our calculations † . We have also investigated a lower mass BH of 5 × 10 7 h −1 M⊙ (corresponding to ∼ 10 −3 M bulge ), and a simulation where the host galaxy is simulated at twice as high spatial resolution to verify the validity of our findings (see Appendix A). Given that we want to track dynamical friction forces acting on the BH as accurately as possible, we do not perform simulations with a lower mass BHs here, because then the mass ratio between the BH and the gas or star particles would not be sufficiently large. Also, by simulating a more massive BH we are in a more advantageous regime for spatially resolving regions close to the Bondi radius.
Once the BH particle is introduced in the centre we let it evolve for an additional 1.4 × 10 7 yrs ‡ after which the BH is imparted a certain recoil velocity, with the direction either parallel to the galactic disc or perpendicular to it. Table 1 summarises the main parameters of our simulated galaxies, as well as initial BH masses and recoil velocities. For comparison, we also carried out runs where the BH stays in the centre of the galaxy, and is not subject to any gravitational recoil. The simulations with the stationary BH permit us to verify that the BH position always coincides with the minimum of the gravitational potential within the spatial resolution of the simulation, hence the resolution is sufficiently high to reduce two-body noise acting on the BH to negligible levels.
Most of our simulations were performed with BH accretion and feedback included, but we also considered cases where BH accretion, or selectively only BH feedback, was switched-off. This allows us to disentangle purely dynamical processes acting on a recoiled BH from the additional effects due to BH accretion and feedback, which can affect the properties of the surrounding gas and thus possibly alter the BH dynamics.
RESULTS
Isolated galaxies
Importance of the stellar and gas dynamical friction
We first want to assess the importance of dynamical friction forces acting on the BH as it moves trough the simulated galaxy. For this purpose, we consider the BH to be a collisionless particle which does not accrete, and has a constant mass of 5 × 10 8 h −1 M⊙ . Just before the BH is imparted the kick, we estimate the central escape velocity from the halo based on the minimum of the gravitational potential, which yields vesc(0) = 2 Φ(r = 0) ∼ 1450 km s −1 . Note that the central escape velocity is rather high, vesc(0) ∼ 4.8 v200, due to the presence of a central stellar bulge. Using instead the often adopted equation vesc = 2 f (c) v200, where f (c) = c (ln(1 + c) − c/(1 + c)) −1 is only a function of the concentration parameter, one would obtain a significantly † Note that we have deliberately selected a galaxy model with a somewhat smaller bulge mass fraction to simulate the long term evolution in a stable state, reducing the occurrence of bulgedriven bar instabilities. ‡ This delay time is introduced such that the BH orbit is not artificially altered by initial accretion caused by the introduction of the BH into very gas-rich and dense material. lower estimate of vesc(0) ∼ 3.6 v200, as this considers only the distribution of the dark matter.
[kpc / h] b) -3 -2 -1 0 1 2 3 x [kpc / h] -3 -2 -1 0 1 2 3 z [kpc / h] c) -3 -2 -1 0 1 2 3 x [kpc / h] -3 -2 -1 0 1 2 3 y [kpc / h] d)
We then impart a gravitational recoil velocity to the BH of 700 km s −1 (∼ 0.5 vesc) along the x-axis in the plane of the disc. Figure 2 shows the resulting distance of the BH from the minimum of the potential as a function of time for three different runs. The blue line is for a simulation where the BH particle is massless, and thus is not subject to dynamical friction forces. The red line is for our default case where the galactic disc is gas-rich, while the green line is for a simulation where we have set-up a galaxy with exactly the same structure, only that the galactic disc is made up entirely of stars. In the case where dynamical friction is not acting on the BH particle, the apocentric distances reached during the simulated time-span stay roughly constant. The BH simple simply oscillates through the nearly static potential of the galaxy. However, as expected, dynamical friction from stars, and even more so from the gas, leads to the systematic reduction of the apocentric distance with time. Dynamical friction already reduces the first apocentric distance reached, and the effect becomes more pronounced with further passages of the BH through the innermost regions. From Figure 2 we infer that the BH returns to the centre § after ∼ 3.3 × 10 8 yrs due to the dynamical friction from a fully stellar disc. When the host galaxy has instead a substantial fraction of the disc mass in gas, but otherwise the same structure and the same initial potential shape, the BH return time is almost halved. The BH trajectory is sensitive to the composition of the galactic disc in our simulations for three reasons: i) given that the BH is moving supersonically, dynamical friction is more efficient if the BH is embedded in a gaseous rather than a stellar background (Ostriker 1999); ii) unlike stars, gas is collisional and through radiative cooling it can radiate away energy transferred by the moving BH and thus form a more concentrated wake behind it, exerting a larger dynamical friction; iii) radiatively cooling gas is deepening the gravitational potential with time which also results in shorter return times to the centre.
Properties of recoiled BHs
In Figure 3, we show three illustrative examples of recoiled BH orbiting in the gas-rich galactic disc after t = 3.92 × 10 7 yrs. In all three simulations, the initial BH mass is 5 × 10 8 h −1 M⊙ and the recoil velocity is 700 km s −1 ∼ 0.5 vesc along the x-axis. In the left-hand panel we show the case where the BH is not accreting, in the central panel we present the case where BH accretion is switched-on but there is no feedback, while in the right-hand panel we illustrate the case where both BH accretion and feedback are followed. As can been seen from the left-hand panel, the motion of the BH through the high density gas creates a density enhancement in its wake (Ostriker 1999) that propagates outwards, as has been previously reported in numerical simulations of a single BH moving through a uniform gas cloud (Escala et al. 2004), as well as in simulations of BH binaries in gaseous discs during the initial evolutionary phase (Escala et al. 2005). We further found that the local potential of the stellar disc at the position of the BH becomes notably deeper, over a typical size of order ≤ 500 h −1 pc. The shape of the stellar potential deformation is roughly spherical, but shows a tendency to be more elongated behind the BH for a fraction of the time. The reason for a more prominent gaseous wake in respect to the stellar wake behind the BH lies in the collisional nature of the gas which is allowed to radiatively cool.
When we allow the BH to accrete, the BH orbit is considerably different (central panel). Even though the BH is moving with a high relative velocity with respect to the surrounding gas, it does manage to accrete a substantial amount of gas, allowing it to almost double its mass to MBH = 9.1 × 10 8 h −1 M⊙ at t = 3.92 × 10 7 yrs after the kick. The larger dynamical mass and the considerable drag forces from gas accretion act in the same direction and prevent the BH from reaching similar apocentric distances as in the case without accretion. The orbit also circularises more efficiently in the direction of galaxy rotation when BH accretion occurs ¶ . While initially the accreting BH also generates a high density wake, after it has gained enough mass some of the gas particles actually become bound to it and create a high density blob around the BH.
If in addition to BH accretion we include BH feedback, the BH orbit is again different (right-hand panel). Instead of a high density wake, we now observe a low density wake, which similarly propagates outwards from the BH orbit. This low density wake is due to the BH feedback, which creates a hot expanding gas plume. Whereas the gas drag from accretion tends to bend the BH orbit in the prograde direction from the initially radial orbit, the BH orbit is here deflected in the retrograde direction after reaching the apocentre (opposite to the rotation of the galaxy). Note also that due to the BH feedback the BH does not grow much in mass, reaching only MBH = 5.1 × 10 8 h −1 M⊙ at t = 3.92 × 10 7 yrs. As a result, the gas drag from accretion is much smaller than in the previous case. The effect of the hot, expanding density plume on the evolution of the BH orbit turns out to be very important, as we discuss in detail below.
In Figure 4, we show the BH orbits and distances from the centre of the host galaxy during the simulated time-span for four different runs. In panel (a) the case with v kick = 700 km s −1 in the galactic plane without BH accretion is shown. The BH orbit is initially radial, and after the BH reaches the apocentre, its trajectory is bent in the prograde direction due to gravitational drag. The BH describes four precessing loops with decreasing apocentric distances and then quickly spirals inwards reaching the minimum of the gravitational potential at ∼ 1.7 × 10 8 yrs after the kick. The orbit is essentially completely contained in the galactic disc, with excursions in the z-direction of less than 100 h −1 pc.
When BH accretion but no feedback is included [panel (d)], the BH returns to the galactic centre on an even shorter timescale, in only ∼ 1.2 × 10 8 yrs. Without self-regulating feedback the BH mass grows rapidly, making dynamical friction more effective. Moreover, the drag force the BH is experiencing from accreted gas particles is efficiently circularising the BH orbit and causing it to co-rotate with the galactic disc. These two processes acting together lead to rapid spiralling in.
BH feedback can nonetheless have an impeding impact on the return timescale of gravitationally recoiled BHs [panel ¶ While dynamical friction tends to circularise a BH orbiting within a galactic disc -as it happens in the case where the BH is not accreting, see panel (a) of Figure 4 (see also Dotti et al. 2007, for BHs on initially eccentric orbits) -BH accretion can cause much more efficient circularisation. (b)]. As anticipated above, the BH heats the surrounding gas, which then forms a hot plume that propagates outwards and expands, compressing the gas in a thin rim and generating a low density wake within. To understand in more detail the nature of the resulting BH orbit, Figure 5 shows unsharp masked maps of the gravitational potential of the gas at the moment when the BH reaches the apocentre for the first time. The unsharp masking has been performed by subtracting from the initial potential a version smoothed on a scale of ∼ 600 h −1 pc. As can be seen in the top panel, for the case without BH accretion, the high density wake exerts gravitational drag on the BH and the net gravitational force from the surrounding gas acting on the BH is directed prograde along the y-axis. In the case with BH accretion and feedback, the situation is more complex, as shown in the bottom panel. Here the BH is located within the dip associated with high density compressed gas, while above the BH a hot low density plume is expanding away from the BH. The net gravitational force acting on the BH from this gas distribution is pointing retrograde along the y-axis, and thus the BH experiences a gravitational drag in the opposite direction than in the top panel. Moreover, cold gas in the galactic disc, which approaches the hot gas plume due to the rotation of the galaxy, is compressed and transfers its angular momentum to the gas in the plume. This effect further reduces the effective drag force onto the BH from gas accretion.
The BH describes a precessing elliptical trajectory, and with each passage the ellipses widen such that after five relatively close passages to the galactic centre (after about 3×10 8 yrs) the orbit starts to circularise, but with a BH that is counter-rotating, even though it started out with the same radial motion. Due to the feedback a ring of hot, low density gas forms, within which the BH orbits, partially decoupled from the rest of the galaxy. The BH trajectory is loop-like, with ±500 h −1 pc excursions from the plane. For the rest of the simulated time of ∼ 1.7 × 10 9 yrs, the BH remains at a considerable distance from the galactic centre, in the range of 3.5 − 6.5 h −1 kpc, even though ∼ 7 × 10 8 yrs after the kick, the apocentric distance starts to decay, indicating that the BH will eventually return to the centre (see also Appendix A).
The case where the BH is kicked perpendicular to the plane of the galaxy is shown in panel (c) of Figure 4. During the first ∼ 3×10 8 yrs, the BH describes about a dozen orbits before it returns to the galactic centre. In this case, the BH feedback does not significantly delay the return timescale because the BH accretes only during short passages through the galactic plane, due to the mostly radial orbit along the z-axis. Note also that the maximum distance reached by the BH at the first apocentre is lower than for the BH kicked in the plane of the galaxy, as a result of the larger gravitational force exerted from the disc of the galaxy in this direction (see panel on the right).
Finally, in the right-hand panel of Figure 4, we also show a case where the kick velocity is much lower, equal to 375 km s −1 (blue line). Here the BH briefly leaves the minimum of the potential and reaches a radial distance of 500 h −1 pc, but then returns to the centre within 7 × 10 6 yrs.
In Figure 6 we show the BH mass (top panel) and bolometric luminosity (middle panel) as a function of time, as well as the time evolution of the total star formation rate (SFR) of the host galaxy (bottom panel) for different cases. The green lines are for the case where the BH does not experience a gravitational recoil, while the red and orange lines are for v kick = 700 km s −1 parallel and perpendicular to the plane of the galaxy, respectively. From the top panel it can be seen that the BH which does not experience any recoil increases its mass moderately, by ∼ 8.5 × 10 7 h −1 M⊙ over 2 Gyrs. This is due to secular processes which gradually transport some gas towards the inner regions, fuelling BH accretion. The bolometric luminosity is initially around For the computation of the bolometric luminosity a radiative efficiency of 0.1 is assumed throughout the paper. 10 45 erg s −1 and slowly decreases with time, but we note that after ∼ 0.5 Gyrs the BH accretion falls below 0.01 of the Eddington rate, as indicated by the continuous lines (becoming probably radiatively inefficient). Thus, for most of the time this AGN would be optically dim, exhibiting only very few, brief luminous episodes.
In contrast, the BH kicked in the plane of the galaxy initially grows more as it encounters a larger supply of material on its orbit through the gas-rich galactic disc. Consequently, its bolometric luminosity is up to an order of magnitude higher than that of the BH which never leaves the centre. Once the BH orbit circularises in a ring of low density material formed by feedback, its accretion is even more sub-Eddington than that of the BH which stays in the galactic centre. Note however that the AGN bolometric luminosity obtained in this case should be an upper limit for several reasons. First, it is probably not very common that the BH is gravitationally recoiled exactly within the disc, as assumed here, given that at present there is no evidence for a correlation between the spin orientations of the BH and of the host galaxy. Second, during a galaxy merging event, which is a much more realistic setting for the occurrence of a gravitationally recoiled BH, the largest amount of gas available for accretion will be in central regions, meaning that a kicked BH will be biased towards accreting less gas (see Section 3.2). Finally, the BH accretion rate estimated from equation (1) should be considered as an upper limit if the gas surrounding the BH is not multiphase, and the BH feedback is not strong enough to self-regulate the BH growth. While for a stationary BH in the centre of the host galaxy this is unlikely to occur, for a recoiled BH the actual accretion rate may well be lower if it leaves the dense multiphase medium. We explore this issue in detail in Appendix A. Nonetheless, our findings suggest that the recoiled AGN could have accretion rates up to a few percent of the Eddington rate on timescales of a few 10 7 yrs, if their orbits are approximately contained within the gas-rich galactic disc.
Gravitational recoil of the BH perpendicular to the galactic disc significantly suppresses BH accretion, but it does not truncate it all together. As the BH orbit decays towards the centre, the accretion rate increases as the BH experiences more and more passages through the disc. Eventually, once back in the centre the accretion rate is very similar to the case of the stationary BH, and the difference between the final masses is not very large, i.e. ∼ 10 7 h −1 M⊙ . This is, however, very likely a lower limit to the mass difference between a recoiled and a stationary BH, given the quiescent nature of the host galaxy. In a more realistic scenario, where the progenitor BHs merge during a merger of two galaxies, a large amount of gas will be funnelled towards the central regions. This gas will form a copious reservoir for BH accretion and it will thus make a much bigger difference for BH growth whether the remnant BH stays in the centre or is gravitationally recoiled, as we discuss in Section 3.2.
Impact of recoiled BH on the host galaxy
In the bottom panel of Figure 6, we show the total SFR of the simulated galaxy, where the blue line denotes the simulation result without a BH, for comparison. The feedback from the stationary BH reduces the SFR of the host galaxy in the central regions during the simulated time span. In the Figure 7. Mass density profiles of the stellar bulge (continuous lines), stellar disc (dashed lines) and gaseous disc (triple dot-dashed lines) are shown in the upper panels for simulations with recoiled BHs (red colour) and without BHs (green colour). The ratio of these density profiles for each galactic component (same line styles) is plotted in the bottom panels. The BH is subject to an initial recoil of 700 km s −1 in the galactic plane. Results for a BH without (with) gas accretion are shown in the left-hand panel (right-hand panel). Blue diamond symbols denote the BH distance from the minimum of the potential, while the pink crosses indicate those distances where the BH velocity is less than 400 km s −1 .
case where the BH undergoes a gravitational recoil perpendicular to the galactic disc, the SFR of the host galaxy is similarly diminished as in the case where the BH is stationary, because the recoiled BH returns relatively quickly back to the centre. On the other hand, a BH which is kicked in the plane of the galaxy is found to have a sizable impact on the amount of stars produced, both because the BH then grows more in mass (feedback effects are hence stronger) and also because the BH can affect larger portions of the star-forming disc on its orbit.
We do however find that the motion of the BH through the galaxy triggers brief local bursts of star formation. These are highlighted in the inset plot, where we show the evolution of the SFR during the first 0.6 Gyrs after the kick. Feedback from recoiling BHs generates expanding plumes of hot gas, which push the surrounding material and induce localised bursts of star formation with rates of several M⊙ yr −1 .
In Figure 7, we show radial density profiles of stars in the bulge (continuous lines), stars in the disc (dashed lines) and gas in the disc (triple dot-dashed lines). The results for the simulations without BH and for the gravitationally recoiled BH are denoted with green and red colour, respectively. In the left-hand panels we illustrate the case where the recoiled BH was not allowed to accrete, and the profiles are computed once the BH has returned to the centre of the galaxy. During the simulated time-span, the density of stars in the bulge decreases systematically in the case with recoiled BH, as can be seen more clearly in the lower panel, where the ratio of the profiles is shown. This result agrees well with those from N-body simulations of recoiled BHs in stellar cores (see Gualandris & Merritt 2008, and references therein), where the BH scatters the surrounding stars transferring some of its kinetic energy. In the particular case studied here, the central bulge mass deficit (evaluated within 1 h −1 kpc) is of the order of the BH mass once the BH returns to the centre. Moreover, we find that the stars in the galactic disc are also perturbed by the orbiting BH, but there is no clear systematic trend with the radial distance from the centre as a function of time. Interestingly, however, the high density gas in the wake of the BH follows the BH all the way to the centre, creating a large density enhancement.
In the right-hand panels of Figure 7, we show density profiles for the case where BH accretion and feedback have been switched-on. Here, as discussed above, the BH does not return to the centre, but instead describes a loop-like orbit, counter-rotating with respect to the disc. The BH spends considerable amount of time away from the central regions, as indicated by the coloured symbols, and transfers kinetic energy to the stars which are not in the centre. This causes a clear dip in the density of stars in the bulge for 0.5−3 h −1 kpc (see bottom panel). Stars which are scatteredaway from this region create density enhancements on both sides, towards the centre and away from it. Thus, instead of a mass deficit the bulge mass within 1 h −1 kpc is increased by ∼ 0.7×MBH. In this case there is also a systematic reduction of the density of stars in the galactic disc in the region of the disc where the BH spends considerable amount of time (pink symbols). The bulge density declines more steeply than the disc density, and thus for radii ≥ 1 h −1 kpc the BH is more likely to transfer its kinetic energy to the stars in the disc. Additionally, due to the BH feedback in this region a ring of hot, low density gas creates a local perturbation in the gravitational potential, which is felt by the stars in the disc. A dip in the density distribution of the disc stars is formed.
While we have seen that the recoiling BH which returns to the centre (left-hand panels of Figure 7) leads to a moderate mass deficit of the bulge, of the order of the BH mass, in some cases the mass deficit can be substantially bigger. In the simulation where the BH is recoiled perpendicular to the galactic disc, and passes many more times close to the galactic centre, the mass deficit becomes of the order of ∼ 3.5 × MBH when the BH finally returns to the centre.
Clumpy discs
At present the spatial distribution of stars and gas in high redshift galaxies are still rather uncertain, yet this is the regime where BH mergers and thus gravitational recoils should be most frequent. Recent observational studies suggest that there is a significant population of high redshift galaxies that have gas-rich, thick and clumpy discs (Förster Schreiber et al. 2009, and references therein). To numerically explore this scenario, we now discuss cases where the galactic disc is very clumpy instead of stable and quasi-stationary.
Even though in the model we consider the total galaxy mass and virial radius are the same, the clumpy disc has a deeper central potential with a central escape velocity vesc ∼ 1980 km s −1 ∼ 6.6 v200, as a result of more efficient radiative cooling in the innermost regions. While in the case of a smooth disc a BH with kick velocity ≥ 0.3 vesc can leave the innermost regions, here a larger kick velocity of > 0.5 vesc is needed to displace the BH from the central region at all. Also, the maximum distance reached at the first apocentre is affected by the structure of the gaseous disc: if the disc is relatively smooth, a BH recoiled with ∼ 0.5 vesc reaches ∼ 4.6 h −1 kpc, while if the disc is clumpy a BH with initial kick velocity of ∼ 0.8 vesc does not reach a distance farther than 3 h −1 kpc from the centre. To the extent that clumpy discs are accompanied by elevated deposition rates of baryons in the centre, as it happens in our simulation, BHs are more likely to stay in the centre, or leave only for brief periods of time, when they experience merger kicks. This may contribute to explain the very scarce observational evidence for off-centred quasars (Bonning et al. 2007).
An example of a gravitationally recoiled BH with v kick = 1600 km s −1 (∼ 0.8 vesc) in the plane of the clumpy disc is shown in Figure 8. The BH orbit (denoted with a white line) is plotted on top of the projected density map, where the clumpy nature of the disc can be clearly seen. Regardless of gas accretion and feedback processes, the BH returns to the minimum of the potential in 3 × 10 8 yrs. The BH trajectory is completely contained in the disc, with |z| < 100 h −1 pc, and soon after reaching the first apocentre the BH starts co-rotating with the galactic disc. During its orbit through the clumpy disc, the BH accretes a relatively small amount of gas. By the time the BH returns to the centre, the BH mass has increased by ∼ 3.6 × 10 7 h −1 M⊙. Most of the mass gain happens when the BH is on its way back to the innermost regions, passing through a dense central lump of gas. The bolometric luminosity of the BH moving through a clumpy disc is on average lower than that of the BH passing though a more uniform gas distribution: in fact, the bolometric luminosity is reduced by at least one order of magnitude with respect to the values shown in Figure 6. During most of the 3 × 10 8 yrs while displaced from the centre, the BH should be in the radiatively inefficient accretion regime with only a few brief bursts of typical duration < 10 7 yrs when passing through a dense gas lump.
When the BH finally approaches the centre, its accre-tion rate increases, making the AGN optically bright for ∼ 5 × 10 7 yrs, after which the accretion rate drops again due to ensuing feedback. This suggests an interesting possibility for spectroscopically detecting a recoiled BH on its way back to the centre of a galaxy, given that its relative velocity during the luminosity maximum is still sufficiently large (∼ 500km s −1 ) to discriminate it from a stationary BH. However, it is possible that the BH accretes the gas with a significant time delay compared to our model estimates, in which case it might become optically bright only when it has already settled into the minimum of the potential.
In the case of a clumpy disc we do not see any clear evidence for the impact of BH feedback on the surrounding gas, e.g. in the form of a low density wake in the gas as we witnessed for the smooth disc. We conclude that while BH feedback can alter the local gas density distribution and thus influence the BH dynamics, these feedback effects are not universal and depend sensitively on the thermodynamic state and the spatial distribution of the local gas.
Merging galaxies
The simulations presented in Section 3.1 have been instructive for understanding the complex interaction of gravitationally recoiled BHs with host galaxies that have pure stellar disc or gas-rich discs with different equations of state. However, these models considered isolated, unperturbed systems, which is not very realistic for the most commonly expected BH mergers, and this simplification may hide important aspects of the evolution of the recoiled BHs.
In order to illustrate the changes expected in a more realistic case, we now consider a simulation of the major merger of two equal mass galaxies. The two merging galaxies have been set-up initially in exactly the same way as described for the isolated gas-rich galaxies, with each of them containing a central BH with a mass of 5 × 10 7 h −1 M⊙. The galaxies collide on a prograde parabolic orbit, break due to dynamical friction of their dark matter halos, and eventually coalesce to form a spheroidal remnant system. When the galaxy cores merge, their central BHs merge as well, leading to the recoil of the BH merger remnant.
In Figure 9, we show the time evolution of the gas density during the major merger of the two gas-rich galaxies. The left-hand panels correspond to times prior to the BH coalescence, while in the right-hand panels we illustrate snapshots after the remnant BH has been gravitationally kicked with v kick ∼ 3535 km s −1 . In this particular case, we deliberately selected an initial BH trajectory that should maximise the possible interaction with the dense gaseous arms which are falling towards the core. The top right-hand panel shows a wake of low density, hot gas behind the BH, similarly to the case of the isolated galaxy with a smooth disc. However, the occurrence of feedback-generated hot gas is short lived and is present for only ∼ 5×10 7 yrs, as a result of the rapidly changing gas distribution in the significantly disturbed system. After reaching the apocentre, the BH passes trough the dense disc-like structure which is forming around the galactic core (bottom right-hand panel). Here BH feedback leads again to the formation of a low density wake. Thus, if BH accretion and feedback processes are not significantly suppressed compared to our numerical model, and if the gas discs are sufficiently smooth, such wakes of hot, low den-sity gas may be ubiquitous in post merger systems, albeit short-lived.
We note that the recoil velocity of v kick ∼ 3535 km s −1 imparted on the remnant BH is extremely high. At present it is unclear whether such kicks are attainable (Campanelli et al. 2007;Baker et al. 2008), but it seems certain that they should be very rare. The estimated central escape velocity from our merging system at the moment of BH coalescence is vesc ∼ 3510 km s −1 . This is also a strikingly large value, caused here by the fact that we consider comparably massive galaxies in which substantial baryonic dissipation has created a massive central stellar bulge. For recoil velocities ≤ 0.8 vesc, the remnant BH does not leave the innermost regions, making it highly unlikely for the BH to ever get kicked out from the centre. To illustrate why recoil velocities ≤ 0.8 vesc are not sufficient for the BH to leave the galactic core, in Figure 10 we show radial profiles of the gravitational potential for our merging system as well as for isolated galaxies with smooth and clumpy discs, for comparison. Clearly, already in the case of the clumpy disc, the gravitational potential is not only almost twice as deep as the gravitational potential of the galaxy with a smooth disc, but it is also much steeper in the inner parts, which explains why the BH needs to have a larger fraction of the central escape velocity to leave the innermost regions. For the merging system, the central potential is almost six times deeper. Much of this depth arises in the steep part within r < 1 h −1 kpc, so that the BH indeed has to have a recoil velocity comparable with the escape speed to leave the core. Similar results have been found for a variety of gasrich mergers, extending to lower mass systems Blecha et al. (2011).
In order to explore the imprints of strong gravitational recoils in this system, we hence consider very high kick velocities: v kick ∼ 3253 km s −1 ∼ 0.9 vesc and v kick ∼ 3535 km s −1 ∼ vesc, which could be achieved if the BHs are close to maximally spinning, are of comparable mass, and have anti-aligned spins in the orbital plane, as we assume here. Note that the probability of such a high recoil velocity is very low for a typical merger, highlighting that even maximal recoil velocities currently proposed might not be sufficient for expelling the BHs from massive, gas-rich hosts. Nonetheless, by adopting these very high kick velocities we can explore the properties of recoiled BHs in a realistic merging setting and draw firm qualitative conclusions without loss of generality. The BH orbits for the two kick velocities turn out to be rather different due to the shape of the gravitational potential. For v kick ∼ 0.9 vesc the BH reaches only 1.5 h −1 kpc at the first apocentre. After experiencing several mostly radial-type passages close to the centre, it returns to the centre in 10 8 yrs. Instead, the BH kicked with v kick ∼ vesc leaves the central region all together. Along its way towards the outskirts it experiences drag forces from the infalling dense gaseous arms, such that it only reaches an apocentric distance of ∼ 27 h −1 kpc. Thereafter, the BH describes a precessing elliptical trajectory, and after reaching each apocentre, the BH moves retrograde with respect to the forming circum-nuclear disc (similar to the simulation findings in the case of the uniform isolated disc). This will likely increase the time needed for the BH to return to the centre. In fact, at time ∼ 10 9 yrs after the recoil event the apocentric distances are still very large, ≥ 20 h −1 kpc, Figure 9. Projected mass-weighted gas density maps of two gas-rich merging galaxies, each containing a supermassive BH in the centre. The left-hand panels show a time sequence at t = 1.55, 1.62 and 1.69 Gyrs from the beginning of the simulation. At 1.69 Gyrs, the two BHs coalesce. In the right-hand panels we show the orbit of the gravitationally recoiled BH at times t = 1.75 and 1.89 Gyrs, which at the moment of the merger was kicked with a recoil velocity of 2500 km s −1 along the x-axis and −2500 km s −1 along the y-axis. . Radial profiles of the gravitational potential for the isolated galaxy with a smooth disc (blue continuous line), for the isolated galaxy with a clumpy disc (green dashed line), and for the system undergoing a major merger of two gas-rich galaxies (red triple dot-dashed line). The potential in the central regions is progressively both deeper and steeper in these three cases.
implying that this BH will be wandering within the host galaxy for several Gyrs before returning to the centre.
In Figure 11, we show the time evolution of BH mass, bolometric luminosity and SFR of the merging system. The selected time sequence starts before the two BHs merge, which happens at ∼ 1.69 Gyrs. In each panel, the blue continuous lines denote the case where the remnant BH does not experience a gravitational recoil, while the green dashed lines and the red triple dot-dashed lines are for the simulations where the remnant BH is kicked with ∼ 0.9 vesc and ∼ vesc, respectively. A number of interesting features can be seen in this figure: i) Prior to coalescence, both BHs grow rapidly, reaching the Eddington limit; ii) During this period, the merging system also enters into a starburst phase, with a SFR peaking at 4000 M⊙ yr −1 ; iii) Once the remnant BH is gravitationally recoiled its accretion rate drops, especially in the case where the BH leaves the dense central regions (i.e. for v kick ∼ vesc), implying that the AGN will have a much lower bolometric luminosity and will grow less in mass; iv) Because the remnant BH is wandering away from the centre, it becomes much less efficient in regulating the central properties of the host galaxy. Thus, in the case of recoiled BHs, the central starburst activity will be prolonged, with more young stars formed in situ; v) The extended star formation activity has a direct impact on the BH growth once it returns to the centre: more stars have already formed, leaving less gas to fuel further BH accretion. This explains why the BH mass in the case of the ∼ 0.9 vesc kick stays much lower and cannot catch-up with that of the stationary BH, even though it returns to the centre already after 10 8 yrs; vi) While the amount of stars formed has a large impact on the BH growth, it does not significantly affect the total stellar mass, which changes by a few percent at most. We hence find that the gravitational recoil for a single merger could in principle introduce more scatter in the BH mass (by up to factor of ∼ 2) than in the bulge mass when considering BH mass -bulge mass scaling relation (see Blecha et al. (2011) for a comprehensive study of the scatter in the BH mass -host galaxy scaling relations for different simulated merging systems and e.g. Volonteri (2007) who reaches simular conclusions adopting semi-analytical techniques).
While the findings discussed above should reflect the general characteristics of gas-rich mergers of galaxies, we would however like to stress that the quantitative details will be very sensitive to the physics of BH binary hardening, as well as to star formation and feedback. In our simulations, we cannot follow the BH binary hardening due to insufficient spatial resolution. Instead, we simply assume that the BH coalescence happens rapidly in a gas-rich environment. This is a plausible assumption, but obviously not guaranteed to be the case. If the final BH binary hardening should take longer, this would have a significant impact on the results. From the left-hand panel of Figure 11, we can infer that during a very short time interval of the order of ≤ 5 × 10 7 yrs, from the merger of the galactic cores to the moment where the BH growth becomes self-regulated, the BH grows rapidly. Thus, a delay in the BH coalescence relative to what we have assumed here, and hence a delay in the moment the recoil occurs, can significantly reduce the mass difference between a stationary and a recoiled BH. Similarly, the efficiency of star formation in the innermost regions will affect the amount of gas that is still available for accretion once the BH returns to the centre.
DISCUSSION AND CONCLUSIONS
In this study, we have used numerical simulations to discuss the complex interplay between the baryonic component of gas-rich galaxies and the dynamics of supermassive BHs recoiling due to a gravitational wave induced binary merger. Our analysis has focused on understanding how BH accretion and feedback processes can possibly modify the orbit and return timescale of recoiled BHs. This question has important implications both for the assembly history of the population of supermassive BHs as well as possible detections of displaced AGN in galaxies.
In our simulations, we have deliberately chosen massive and gas-rich systems, which could be representative of high redshift progenitors of present day ellipticals and brightest cluster galaxies. Using isolated disc galaxies, we were able to study the orbital evolution of kicked BHs for a variety of different assumptions about BH accretion and feedback, and about the treatment of the interstellar gas. We have then extended the analysis with simulation models of major galaxy mergers, yielding a more realistic accounting of the perturbed state of the systems that are expected to host merger recoil events of BHs. Our main conclusions from these simulations are as follows:
• At the centre of galaxies, the gravitational potential is strongly dominated by the baryonic component. The expected escape velocity will thus generally be significantly larger than that from the dark matter alone. It will depend on the detailed assumptions for the spatial distribution and dynamical evolution of the baryonic component at the host galaxy centre, which in turn can be affected by BH feedback as well as dynamical heating by the motion of the BH. For the compact remnants of galaxy mergers the escape velocity will be larger by a factor of a few, and the kick velocities required for a significant displacement of the supermassive BHs will need to be comparable to the central escape velocity, due to the steepness of the potential.
• Recent claims that massive high-redshift galaxy spheroids and discs are significantly more compact than their low-redshift counterparts should thus have a large effect on the expected trajectory of recoiling BH merger remnants. For all but the most extreme and rather unlikely values of the kick velocities, supermassive BHs should thus not be removed from the rather massive galaxies which appear to host the bulk of the supermassive BH population and the displacements should be mostly moderate (less than a few kpc) and rather short-lived.
• During mergers of gas-rich systems, where it is very likely that the BHs will grow rapidly due to large amounts of gas being funnelled towards the centre, gravitational recoils increase the scatter in BH mass -host galaxy scaling relationships predicted by the simulations. In particular, the BH mass is very sensitive to the occurrence of recoil, and can be lower by up to a factor of a few in respect to the mass of a BH which does not experience any kick. We note however that the exact amount of BH mass reduction is very dependent on the BH binary hardening timescale and also on the efficiency of the starburst to consume the central gas supplies.
• Unfortunately, the strong sensitivity of the dynamics of a recoiling BH on the detailed spatial distribution and thermal state of the baryonic component, which in turn depends on the details of the feedback of the accreting BHs, adds another level of complexity to predictions of the expected distribution of the luminosity and the displacement of off-centred AGN.
• The overall amount of accretion and therefore also the luminosity of a recoiling BH depends sensitively on the distribution and thermal state of gas down to scales smaller than can be resolved by our simulations. As in previous work, we have thus parametrised the accretion rate as Bondi-Hoyle-Lyttleton accretion with a parameter α to take into account that the simulations cannot fully resolve the multiphase medium of the interstellar gas. If the recoiling BH leaves the dense multi-phase gas α = 100 probably overestimates the accretion rate somewhat. We have thus also tested what happens if we assume α = 1, i.e. Bondi-Hoyle-Lyttleton accretion as derived from the actual density and temperature distribution of the gas in the simulation. This should now underestimate the actual accretion rate. The corresponding reduction of the accretion rate and luminosity is about a factor five -much weaker than linear because of the self-regulating effect of the feedback on the accretion rate. In reality, the accretion rates and the corresponding luminosities should lie somewhere in between. The generally rather large accretion rates of BHs if recoiled within gas-rich discs predicted by our simulations would bode well for the possibility to detect off-centred AGN, and may give added confidence in the recent claims of such detections. Figure A1. Bolometric luminosity as a function of time for simulations with two different values for the α parameter in equation (1): α = 100 (blue line) and α = 1 (green line). The same galaxy has also been simulated with twice as high spatial resolution and α = 100 (red line). Straight lines denote an accretion rate equal to 0.01 of the Eddington rate. (1) (blue continuous line) and where α = 1 (green dashed line). The same galaxy has also been simulated with twice as high spatial resolution and α = 100 (red triple dot dashed line). The time evolution of the BH distance for a simulation where the initial BH mass is set to 5×10 7 h −1 M ⊙ and the kick velocity is 500 km s −1 is shown as well (orange dotted line), confirming that BH feedback delays the return timescale also for less massive BHs.
BH growth and feedback gives numerically robust and convergent results. Figure A2 shows the BH distance from the minimum of the potential for the same set of simulations as in Figure A1, and additionally for a simulation with a smaller initial BH mass. At first the BH orbits for α = 1 and α = 100 are very similar, but after 10 8 yrs the recoiled BH in the case of α = 1 reaches smaller apocentric distances. Nonetheless, we observe that also in the case of α = 1 a plume of hot, low density gas develops in the wake of the BH, which forces the BH into a retrograde orbit with respect to the galactic disc. Given that the BH accretion rate is lower, the magnitude of this effect is somewhat diminished and thus the BH returns to the centre on a shorter timescale, which is however still rather long ∼ 1.6 × 10 9 yrs. Note that this findings means that our qualitative findings are very robust regardless of the exact value of the α parameter, which is very encouraging. We furthermore confirm that also for a BH with an initial mass ten times smaller, BH feedback causes the BH to follow a retrograde orbit and thus results in an analogous delay in the BH return timescale.
The BH orbit for the higher resolution simulation is more different than perhaps may have been expected, given that the BH mass growth is very similar. The reason for this can be attributed to the detailed shape of the gravitational potential, which is resolved down to smaller scales in the higher resolution run. As a result, the central potential is about 10% deeper than in the lower resolution simulation. This difference is sufficient for the recoiled BH to reach only ∼ 3.2 h −1 kpc instead of ∼ 4.6 h −1 kpc at the first apocentre.
Figure 1 .
1Circular velocity curves of the isolated galaxy with v 200 = 300 km s −1 for different components: bulge (dot-dashed line), disc (dotted line), dark matter halo (dashed line), and total (continuous line).
Figure 2
2Figure 2. BH distance from the minimum of the potential as a function of time for simulations where the BH is (a) not subject to dynamical friction forces (blue line), (b) experiences dynamical friction from stars only (green line), and (c) additionally is subject to dynamical friction from the gas in the disc (red line).
⋆
Note that in our simulations BH spins change only when BHs merge, and any possible spin-up/down due to the gas accretion is neglected (King & Pringle 2006; Kesden et al. 2010; Dotti et al. 2010).
Figure 3 .
3Projected mass-weighted gas density maps of the isolated galaxy at time t = 3.92 × 10 7 yrs after BH recoil. The three panels show simulation results where the BH does not accrete nor exerts feedback (left-hand panel), where the BH accretes but has zero feedback efficiency (central panel), and where the BH is allowed both to accrete and inject feedback energy (right-hand panel). The dotted lines show the BH orbits from the start of the simulation. Note that the colour maps display different density ranges, as given by the individual colour bars.
Figure 4 .
4Four left-hand panels: Orbits of BHs over the course of the simulated time-span, for different cases. Panel a) v kick = 700 km s −1 (∼ 0.5 vesc) in the galactic plane without BH accretion; Panel b) v kick = 700 km s −1 in the galactic plane with BH accretion and feedback; Panel c) v kick = 700 km s −1 perpendicular to the galactic plane with BH accretion and feedback; Panel d) v kick = 700 km s −1 in the galactic plane with BH accretion but no feedback. Right-hand panel: BH distance from the minimum of the potential as a function of time for the same simulations as in left-hand panels. The additional blue line is for an initial recoil of 375 km s −1 (∼ 0.26 vesc) in the galactic plane. The arrows indicate when the BHs return to the centre of the galaxy.
Figure 5 .
5Projected unsharp masked map of the gravitational potential of the gas (absolute value), for a simulation without BH accretion (top panel), and with BH accretion and feedback (bottom panel). The displayed snapshots correspond to the moment when the BHs reach the apocentres for the first time.
Figure 6 .
6BH mass (top panel), bolometric luminosity (central panel) and total SFR (bottom panel) as a function of time. The green lines are for the run where the BH is always at the centre of the galaxy. Cases where the BH experiences a gravitational recoil of 700 km s −1 in the plane of the galaxy or perpendicular to the galactic plane are denoted with red and orange lines, respectively. The blue line (only in the bottom panel) is for the simulation without a BH. Continuous lines in the middle panel denote an accretion rate equal to 0.01 of the Eddington rate.
Figure 8 .
8Projected mass-weighted gas density map of the isolated galaxy with a very clumpy disc. The over-plotted white line shows the BH orbit from the beginning of the simulation to the moment when it returns to the galaxy centre.
Figure 10
10Figure 10. Radial profiles of the gravitational potential for the isolated galaxy with a smooth disc (blue continuous line), for the isolated galaxy with a clumpy disc (green dashed line), and for the system undergoing a major merger of two gas-rich galaxies (red triple dot-dashed line). The potential in the central regions is progressively both deeper and steeper in these three cases.
Figure A2 .
A2BH distance from the minimum of the potential as a function of time for the simulations where α = 100 in equation
1.60 1.65 1.70 1.75 1.80 1.85 1.90 t [Gyr] Figure 11. BH mass (left-hand panel), bolometric luminosity (central panel) and total SFR (right-hand panel) as a function of time during a major merger of two gas-rich galaxies. The blue continuous lines are for the run where after the BH coalescence no recoil is imparted on the remnant BH. Green dashed and red triple dot-dashed lines are for the simulations where the remnant BH is kicked with v kick ∼ 3253 km s −1 ∼ 0.9 vesc and v kick ∼ 3535 km s −1 ∼ vesc, respectively. The green arrows indicate when the gravitationally recoiled BH returns to the centre in the case of lower kick velocity.2.0•10 8
4.0•10 8
6.0•10 8
8.0•10 8
1.0•10 9
1.2•10 9
M
BH [
M
O •
/ h ]
no recoil
v kick ~ 0.9 v esc (0)
v kick ~ v esc (0)
no recoil
v kick ~ 0.9 v esc (0)
v kick ~ v esc (0)
no recoil
v kick ~ 0.9 v esc (0)
v kick ~ v esc (0)
c 0000 RAS, MNRAS 000, 000-000
§ Throughout the paper we define that the BH has returned to the centre if its position coincides with the minimum of the potential within the gravitational softening length of the simulation. Note that due to the limited spatial resolution achieved we cannot track BH-galaxy core oscillations(Gualandris & Merritt 2008) which may further prolong BH settling to the centre, at least in the case with only stars and no gas at the centre.
ACKNOWLEDGEMENTSWe would like to thank Jim Pringle for very useful discussions and comments on the manuscript and Giuseppe Lodato for enthusiastic discussions on the topic. DS acknowledges a Postdoctoral Fellowship from the UK Science and Technology Funding Council (STFC) and NASA Hubble Fellowship through grant HST-HF-51282.01-A. MH was partially supported by STFC grant LGAG 092/ RG43335. Simulations were performed on the Cambridge High Performance Computing Cluster DARWIN in Cambridge (http://www.hpc.cam.ac.uk).APPENDIX A: NUMERICAL ISSUESIf the gravitationally recoiled BH leaves the dense multiphase medium while on its orbit through the galaxy, the accretion rate might be overpredicted by equation(1)when α = 100 is used (see alsoBooth & Schaye 2009). A lower gas accretion rate and hence a lower associated BH feedback may then possibly change the impact of the recoiled BH on its surroundings, which in turn then leads to a different BH trajectory.To explore these issues and test the robustness of our results, we have performed three additional simulation studies. In one run we reduce the α parameter to 1, thereby providing a lower limit to the expected accretion. In the second we increase the numerical resolution by a factor of 8 in particle number for each galactic component, resulting in twice as high spatial resolution per dimension, equal to 30 h −1 pc for the bulge and the disc, and to 1 h −1 kpc for the dark matter halo. In the third run we choose a smaller initial BH mass of 5 × 10 7 h −1 M⊙ (corresponding to ∼ 10 −3 M bulge ) and the kick velocity of 500 km s −1 .InFigure A1we show the bolometric luminosity of a BH which has been kicked in the plane of the galaxy with 700 km s −1 . The blue line is for our default simulation, as shown in the central panel ofFigure 6. The green line is for a test run performed with α = 1, and the red line is the result of our higher resolution simulation. For the case of α = 1 the accretion rate and thus the bolometric luminosity is lower than for the simulation with α = 100, with the bolometric luminosity being reduced by a factor of ∼ 5 on average. We note, however, that α = 1 provides clearly a lower limit of the expected Bondi-Hoyle-Lyttleton rate, given that the recoiled BH trajectory is confined mostly within the dense gaseous disc, which has a multi-phase structure. In either case we expect that the recoiled BH can still have several radiatively efficient episodes while it travels through the gasrich disc, but these will occur during shorter time intervals of a few 10 7 yrs. FromFigure A1it can also be seen that the bolometric luminosity for our lower resolution run is in excellent agreement with the finding of the higher resolution simulation. This indicates that our subresolution model for
. J G Baker, W D Boggs, J Centrella, B J Kelly, S T Mcwilliams, M C Miller, J R Van Meter, ApJ. 68229Baker J. G., Boggs W. D., Centrella J., Kelly B. J., McWilliams S. T., Miller M. C., van Meter J. R., 2008, ApJ, 682, L29
. J D Bekenstein, ApJ. 183657Bekenstein J. D., 1973, ApJ, 183, 657
R D Blandford, Sources of Gravitational Radiation Massive black holes and gravitational radiation. L. L. SmarrBlandford R. D., 1979, in L. L. Smarr ed., Sources of Grav- itational Radiation Massive black holes and gravitational radiation. pp 191-210
. L Blecha, T J Cox, A Loeb, L Hernquist, MNRAS. Blecha L., Loeb A.3901311MNRASBlecha L., Cox T. J., Loeb A., Hernquist L., 2011, MNRAS, pp 38-+ Blecha L., Loeb A., 2008, MNRAS, 390, 1311
. H Bondi, MNRAS. 112195Bondi H., 1952, MNRAS, 112, 195
. H Bondi, F Hoyle, MNRAS. 104273Bondi H., Hoyle F., 1944, MNRAS, 104, 273
. E W Bonning, G A Shields, S Salviander, ApJ. 66613Bonning E. W., Shields G. A., Salviander S., 2007, ApJ, 666, L13
. C M Booth, J Schaye, MNRAS. 39853Booth C. M., Schaye J., 2009, MNRAS, 398, 53
. M Boylan-Kolchin, C Ma, E Quataert, ApJ. 61337Boylan-Kolchin M., Ma C., Quataert E., 2004, ApJ, 613, L37
. M Campanelli, C Lousto, Y Zlochower, D Merritt, ApJ. 6595Campanelli M., Lousto C., Zlochower Y., Merritt D., 2007, ApJ, 659, L5
. F Civano, M Elvis, G Lanzuisi, K Jahnke, G Zamorani, L Blecha, A Bongiorno, M Brusa, ApJ. 717209Civano F., Elvis M., Lanzuisi G., Jahnke K., Zamorani G., Blecha L., Bongiorno A., Brusa M. e. a., 2010, ApJ, 717, 209
. B Devecchi, E Rasia, M Dotti, M Volonteri, M Colpi, MNRAS. 394633Devecchi B., Rasia E., Dotti M., Volonteri M., Colpi M., 2009, MNRAS, 394, 633
. Di Matteo, T Springel, V Hernquist, L , Nature. 433604Di Matteo T., Springel V., Hernquist L., 2005, Nature, 433, 604
. M Dotti, M Colpi, F Haardt, L Mayer, MNRAS. 379956Dotti M., Colpi M., Haardt F., Mayer L., 2007, MNRAS, 379, 956
. M Dotti, M Volonteri, A Perego, M Colpi, M Ruszkowski, F Haardt, MNRAS. 402682Dotti M., Volonteri M., Perego A., Colpi M., Ruszkowski M., Haardt F., 2010, MNRAS, 402, 682
. A Escala, R B Larson, P S Coppi, D Mardones, ApJ. 607765Escala A., Larson R. B., Coppi P. S., Mardones D., 2004, ApJ, 607, 765
. A Escala, R B Larson, P S Coppi, D Mardones, ApJ. 630152Escala A., Larson R. B., Coppi P. S., Mardones D., 2005, ApJ, 630, 152
. Förster Schreiber, N M Genzel, R Bouché, N Cresci, G Davies, R Buschkamp, P Shapiro, K Tacconi, L , ApJ. 7061364Förster Schreiber N. M., Genzel R., Bouché N., Cresci G., Davies R., Buschkamp P., Shapiro K., Tacconi L. J. e. a., 2009, ApJ, 706, 1364
. Y Fujita, ApJ. 6911050Fujita Y., 2009, ApJ, 691, 1050
. J A González, U Sperhake, B Brügmann, M Hannam, S Husa, Physical Review Letters. 9891101González J. A., Sperhake U., Brügmann B., Hannam M., Husa S., 2007, Physical Review Letters, 98, 091101
. A Gualandris, D Merritt, ApJ. 678780Gualandris A., Merritt D., 2008, ApJ, 678, 780
. J Guedes, P Madau, L Mayer, S Callegari, ApJ. 729125Guedes J., Madau P., Mayer L., Callegari S., 2011, ApJ, 729, 125
. T M Heckman, J H Krolik, S M Moran, J Schnittman, S Gezari, ApJ. 695363Heckman T. M., Krolik J. H., Moran S. M., Schnittman J., Gezari S., 2009, ApJ, 695, 363
. F Herrmann, I Hinder, D Shoemaker, P Laguna, R A Matzner, ApJ. 661430Herrmann F., Hinder I., Shoemaker D., Laguna P., Matzner R. A., 2007, ApJ, 661, 430
The effect of interstellar matter on climatic variation. F Hoyle, R A Lyttleton, Proceedings of the Cambridge Philisophical Society. the Cambridge Philisophical Society35405Hoyle F., Lyttleton R. A., 1939, in Proceedings of the Cam- bridge Philisophical Society Vol. 35, The effect of inter- stellar matter on climatic variation. p. 405
. R C Kapoor, Ap&SS. 112347Kapoor R. C., 1985, Ap&SS, 112, 347
. M Kesden, U Sperhake, E Berti, ApJ. 7151006Kesden M., Sperhake U., Berti E., 2010, ApJ, 715, 1006
. A R King, J E Pringle, MNRAS. 37390King A. R., Pringle J. E., 2006, MNRAS, 373, L90
. S Komossa, H Zhou, H Lu, ApJ. 67881Komossa S., Zhou H., Lu H., 2008, ApJ, 678, L81
. M Koppitz, D Pollney, C Reisswig, L Rezzolla, J Thornburg, P Diener, E Schnetter, Physical Review Letters. 9941102Koppitz M., Pollney D., Reisswig C., Rezzolla L., Thorn- burg J., Diener P., Schnetter E., 2007, Physical Review Letters, 99, 041102
. D A Kornreich, R V E Lovelace, ApJ. 681104Kornreich D. A., Lovelace R. V. E., 2008, ApJ, 681, 104
. P Madau, E Quataert, ApJ. 60617Madau P., Quataert E., 2004, ApJ, 606, L17
. D Merritt, M Milosavljević, M Favata, S A Hughes, D E Holz, ApJ. 6079Merritt D., Milosavljević M., Favata M., Hughes S. A., Holz D. E., 2004, ApJ, 607, L9
. M Micic, T Abel, S Sigurdsson, MNRAS. 3721540Micic M., Abel T., Sigurdsson S., 2006, MNRAS, 372, 1540
. E C Ostriker, ApJ. 513252Ostriker E. C., 1999, ApJ, 513, 252
Relativistic Objects in Compact Binaries: From Birth to Coalescence. F Pretorius, Colpi, arXiv:0710.1338Canopus Publishing LimitedPretorius F., 2007, "Relativistic Objects in Compact Bi- naries: From Birth to Coalescence", Editor: Colpi et al., Pulisher: Springer Verlag, Canopus Publishing Limited, eprint arXiv:0710.1338
. G A Shields, E W Bonning, S Salviander, ApJ. 6961367Shields G. A., Bonning E. W., Salviander S., 2009, ApJ, 696, 1367
. D Sijacki, V Springel, T Di Matteo, L Hernquist, MNRAS. 380877Sijacki D., Springel V., di Matteo T., Hernquist L., 2007, MNRAS, 380, 877
. D Sijacki, V Springel, M G Haehnelt, MNRAS. 400100Sijacki D., Springel V., Haehnelt M. G., 2009, MNRAS, 400, 100
. V Springel, MNRAS. 3641105Springel V., 2005, MNRAS, 364, 1105
. V Springel, Di Matteo, T Hernquist, L , MNRAS. 361776Springel V., Di Matteo T., Hernquist L., 2005, MNRAS, 361, 776
. V Springel, L Hernquist, MNRAS. 339289Springel V., Hernquist L., 2003, MNRAS, 339, 289
. P G Van Dokkum, M Franx, M Kriek, B Holden, G D Illingworth, D Magee, R Bouwens, D Marchesini, R Quadri, G Rudnick, E N Taylor, S Toft, ApJ. 6775van Dokkum P. G., Franx M., Kriek M., Holden B., Illing- worth G. D., Magee D., Bouwens R., Marchesini D., Quadri R., Rudnick G., Taylor E. N., Toft S., 2008, ApJ, 677, L5
. M Volonteri, ApJ. 6635Volonteri M., 2007, ApJ, 663, L5
. M Volonteri, R Perna, MNRAS. 358913Volonteri M., Perna R., 2005, MNRAS, 358, 913
| [] |
[
"Comparison of EEG based epilepsy diagnosis using neural networks and wavelet transform",
"Comparison of EEG based epilepsy diagnosis using neural networks and wavelet transform"
] | [
"Mohammad Reza Yousefi [email protected] ",
"Melika Mohammad Hosseini ",
"\nDepartment of Electrical Engineering, Najafabad Branch\nIslamic Azad University\nNajafabadIran\n",
"\nIslamic Azad University\nNajafabadIran\n",
"\nIslamic Azad University\nNajafabadIran\n"
] | [
"Department of Electrical Engineering, Najafabad Branch\nIslamic Azad University\nNajafabadIran",
"Islamic Azad University\nNajafabadIran",
"Islamic Azad University\nNajafabadIran"
] | [] | Epilepsy is one of the common neurological disorders characterized by recurrent and uncontrollable seizures, which seriously affect the life of patients. In many cases, electroencephalograms signal can provide important physiological information about the activity of the human brain which can be used to diagnose epilepsy. However, visual inspection of a large number of electroencephalogram signals is very time-consuming and can often lead to inconsistencies in physicians' diagnoses. Quantification of abnormalities in brain signals can indicate brain conditions and pathology so the electroencephalogram (EEG) signal plays a key role in the diagnosis of epilepsy. In this article, an attempt has been made to create a single instruction for diagnosing epilepsy, which consists of two steps. In the first step, a low-pass filter was used to preprocess the data and three separate mid-pass filters for different frequency bands and a multilayer neural network were designed. In the second step, the wavelet transform technique was used to process data. In particular, this paper proposes a multilayer perceptron neural network classifier for the diagnosis of epilepsy, that requires normal data and epilepsy data for education, but this classifier can recognize normal disorders, epilepsy, and even other disorders taught in educational examples. Also, the value of using electroencephalogram signal has been evaluated in two ways: using wavelet transform and non-using wavelet transform. Finally, the evaluation results indicate a relatively uniform impact factor on the use or non-use of wavelet transform on the improvement of epilepsy data functions, but in the end, it was shown that the use of perceptron multilayer neural network can provide a higher accuracy coefficient for experts. | null | [
"https://arxiv.org/pdf/2204.04488v1.pdf"
] | 248,085,477 | 2204.04488 | 0754939c28100b46def244793a6da5886bfab74f |
Comparison of EEG based epilepsy diagnosis using neural networks and wavelet transform
Mohammad Reza Yousefi [email protected]
Melika Mohammad Hosseini
Department of Electrical Engineering, Najafabad Branch
Islamic Azad University
NajafabadIran
Islamic Azad University
NajafabadIran
Islamic Azad University
NajafabadIran
Comparison of EEG based epilepsy diagnosis using neural networks and wavelet transform
Saina Golnejad Digital processing and machine vision research center, Najafabad Branch, Digital processing and machine vision research center, Najafabad Branch,EpilepsyElectroencephalogramWavelet transformNeural network Introduction
Epilepsy is one of the common neurological disorders characterized by recurrent and uncontrollable seizures, which seriously affect the life of patients. In many cases, electroencephalograms signal can provide important physiological information about the activity of the human brain which can be used to diagnose epilepsy. However, visual inspection of a large number of electroencephalogram signals is very time-consuming and can often lead to inconsistencies in physicians' diagnoses. Quantification of abnormalities in brain signals can indicate brain conditions and pathology so the electroencephalogram (EEG) signal plays a key role in the diagnosis of epilepsy. In this article, an attempt has been made to create a single instruction for diagnosing epilepsy, which consists of two steps. In the first step, a low-pass filter was used to preprocess the data and three separate mid-pass filters for different frequency bands and a multilayer neural network were designed. In the second step, the wavelet transform technique was used to process data. In particular, this paper proposes a multilayer perceptron neural network classifier for the diagnosis of epilepsy, that requires normal data and epilepsy data for education, but this classifier can recognize normal disorders, epilepsy, and even other disorders taught in educational examples. Also, the value of using electroencephalogram signal has been evaluated in two ways: using wavelet transform and non-using wavelet transform. Finally, the evaluation results indicate a relatively uniform impact factor on the use or non-use of wavelet transform on the improvement of epilepsy data functions, but in the end, it was shown that the use of perceptron multilayer neural network can provide a higher accuracy coefficient for experts.
Introduction
Epilepsy is a neurological disorder that affects about 50 million people worldwide and affects almost two out of every three new symptoms in developing countries. There are around 2.4 million new cases of Epilepsy around the world. Taking into account the aging society, according to forecasts, mental and neurological diseases, including epilepsy will be the main health problem in the world in the near future. For this reason, it is extremely important to develop appropriate tools (computer-aided diagnosis) that support the diagnosis of neurologists and psychiatrists. Also, epilepsy can be classified into two types: primary epilepsy and focal epilepsy. The patients who are affected by primary epilepsy are 20% and focal epilepsy is 80% as per the report of the World Health Organization [1]. The signals from the brain area are classified into either focal or nonfocal. The signal that comes from the abnormal brain cells is called a focal signal, and the signal that comes from the normal brain cells is called a nonfocal signal. The epilepsy disease is detected by differentiating and detecting the focal signal from nonfocal signal. Certain signal processing techniques have been developed for detecting epilepsy disease by differentiating the focal from nonfocal signals. The severity analysis of the focal signal is also required for the patient before surgery. The severity may either be "early" or "advanced" and surgery is needed for those patients with "advanced" severity levels. The decomposition of EEG signal will help the radiologist to detect the focal behavior of the signal through the extracted features from each of the decomposed sub-band layers [2]. Automatic identification and detection of Epilepsy or brain-related disorders are the major challenges for both clinicians and research scientists. Generally, neurological experts or brain surgeons inspect EEG recordings to detect epileptic activity. Therefore, it is essential to develop a reliable and automated technique to detect Epileptic seizures is essential. An automatic interpretation of EEG signals in the diagnosis and treatment of brain diseases is one of the most important areas of study. There are different techniques for reading brain activity including electroencephalography (EEG), magnetoencephalography (MEG), functional magnetic resonance imaging (fMRI), and functional near-infrared spectroscopy (fNIRS). Because neurons mutually communicate via electrical signals, which eventually reach the brain surface, EEG is used to capture brain activity through sensors (called electrodes) [3][4][5]. Lately, researchers in the multidisciplinary fields of engineering, neuroscience, microelectronics, bioengineering, and neurophysiology have made considerable efforts to utilize all of the information provided by EEG signals for many applications, such as external device control, communications, and medical diagnosis. EEG-based signalprocessing techniques are vital for diagnosing and monitoring neurological brain disorders because they help reflect the electrical activities or disorders of neurons in the human brain. Electroencephalogram (EEG) is an effective and non-invasive method that is commonly used for brain activity monitoring and epilepsy diagnosing [6]. As shown in Table 1, the brain has 5 frequency sub-bands that can be measured with an EEG signal, and each is related to different mental states of the brain [7]. Learning, cognitive processing
Another definition of epilepsy is that it is a recurrence of two or more attacks of unstimulated seizures with an interval of more than 24 hours. Research has also shown that approximately 4% of people worldwide suffer from epilepsy in their lifetime, while only 1% of them develop epilepsy. Epilepsy is an association with the central nervous system (CNS) in which there is abnormal brain activity [8]. This unusual activity may occur in one part of the brain or all over the brain. The result of this irregular brain activity includes seizures, loss of consciousness, overexposure to emotions, and abnormal behavior. Epilepsy can happen to anyone, regardless of gender, age or ethnicity. The main symptoms of epilepsy are staring, mental confusion, uncontrollable body shaking, loss of consciousness, psychological symptoms such as fear, etc [9]. The classification of epileptic seizures is examined in the part of the brain where abnormal brain activity begins. According to the International League Against Epilepsy (ILAE), epilepsy diagnosing requires at least one seizure without prior stimulation, or considering the risk of recurrent seizures or diagnosing epilepsy syndrome. Pediatric epilepsy is a highly variable condition due to the age-related expression of syndromes that require specific diagnosis, evaluations, and treatments. Children with epilepsy differ from their adult counterparts in many important ways, mostly related to the agerelated expression of specific epilepsy syndromes. This results in many important considerations related to the epilepsy diagnosis, classification, evaluations to determine an etiology, as well as treatment guidelines. A good understanding of these factors will help to establish an accurate epilepsy diagnosis, which in turn will guide appropriate testing and treatment decisions [8]. This is a serious disorder with recurrent epileptic seizures caused by sudden dysfunction of the brain. Seizures 1 Electroencephalogram occur accidentally to disrupt normal brain function. Most patients also suffer from numerous other unpredictable side effects of epilepsy, such as memory loss, depression, and other mental disorders. Therefore, it is important to identify epilepsy in the early stages to help the physician take appropriate action to prevent accidental consequences and ensure the patient's health. As a result, the automatic detection of seizures and epilepsy detection of EEG 1 signals has become an active research topic in recent decades [11]. EEG is a commonly used non-invasive auxiliary method in the clinical diagnosis of epilepsy. However, it is a highly tedious, laborious, timeconsuming, and costly task for neurologists to identify seizures from EEG for a long time. Therefore, it is necessary to develop a reliable epilepsy automatic detection system, which can significantly improve the quality of life of epilepsy patients [12]. Epilepsy according to the type of symptoms and the degree of involvement of the brain is divided into two general categories: general and local. The general category includes those in which epileptic waves cover the entire brain and involve the whole brain, while in the local category, epileptic waves affect only a part of the brain [13]. In previous studies, it was noted that executive functions are the most important brain-cognitive activities that are affected in people with various diseases such as Alzheimer's, Multiple sclerosis (MS), attention deficit hyperactivity disorder (ADHD), and epilepsy, and unfortunately, impair their function. The results of some studies indicate that surgery in the frontal region in patients with epilepsy can help to improve some of their functions such as, increasing awareness and reducing the distance between their attacks. Ideally, the designed model in this study can improve the diagnosis of patients who have recurrent seizures and have difficulty diagnosing these seizures [14]. Epilepsy predictors (in terms of sensitivity, specificity, accuracy, etc.) will be evaluated separately in both methods, such as the use of wavelet transform and the use of data without wavelet transform, and finally, a system will be designed, which can detect epilepsy by receiving EEG signals. Wavelet transform is one of the important mathematical transformation that is used in the field of various sciences and follows the following general equation [1]:
( ) = 0 2 + ∑( cos + sin ) ∞ =0
(1)
The properties of a wavelet (Ψx) function include: A) Must be appropriately limited in time. B) Must have a mean of zero.
∫ Ψt dt = 0 +∞ −∞ C) has a non-zero state:
(2)
0 < ∫ |Ψ | 2 < ∞ +∞ −∞
1-1. OVERVIEW OF THE PROPOSED METHOD
This study summarize as the following steps. First, previouse studies are desribed. Then, the datasets used in this study are decribed. In the next step, the our proposed menthod are explained including preprocessing, signal processing, feature extraction, and classification. Finally, the results of our method will be compared with counterpart methods.
. RELATED WORKS
Tesimorta et al. Have proposed a method for automatic detection of discrete wavelet transform (DWT) seizures to divide EEG recordings into specific sub-bands and extract several features. These properties are then given as an input vector for training a classifier called support vector machines (SVM). Their Materials and methods contain four stages: segmentation, wavelet analysis, feature extraction, and classification. In the first step, the long-term EEG recording of each patient's activity is divided into 2-second windows. Then, 5-level wavelet analysis is applied to each part of the EEG, which divides each signal into several frequency sub-bands. In the next step, 5 attributes that create an attribute vector are calculated from each sub-band and Finally, the attribute vector is used to teach an SVM 2 classifier [14].
In the Kumar and Hyderbank study, two techniques were used using SVM to classify EEG convulsive and non-convulsive signals. In both techniques, the input patterns are subdivided into sets of sub-patterns. In the case of sub-pattern-based principal component analysis (SpPCA), attributes are extracted by applying PCA 3 to each subset. When attribute extraction from these subsets is complete, the extracted attributes are combined according to the pattern partition sequence to form the final attribute vectors secondly, PCA is performed on the features extracted in the previous step to further reduce the dimensions and extract the overall features [17]. Schubb and Guttag presented a paper in which the machine learning algorithm 'SVM' was applied to the scalp EEG dataset, to diagnose epileptic seizures, and this approach achieved 96% accuracy in terms of experimental data [18]. [29]. Anuragi and Sisodia presented approach-based machine learning methods and wavelet transform for alcohol use disorder. The aim of this study is to classify alcoholic and healthy individuals automatically. SVM and Naïve Bayes methods were utilized in the study [30]. Mutlu proposed an approach using Hilbert vibration decomposition for epilepsy. In the study, the least squares SVM was selected as the classifier. The aim of the study is to distinguish the signals between normal and epileptic 3 Principal component analysis
individuals. Time, accuracy, and receiver operating characteristics were selected as the evaluation parameter [31]. The proposed technique also includes a genetic algorithm approach for selecting more effective features and finally, classification is performed by two strategies as artificial neural network (ANN) and support vector machine (SVM). The performance of two classifiers is compared where the simulation results show that the proposed strategy accuracy in detecting epilepsy seizures is better than other similar approaches in the literature [35].
DATABASE
For this method, the epilepsy data from the Bonn University database [36] has been used. This database includes 5 different registration models that are set in 500 pieces and each piece has 100 points, which are: A)Non-epileptic recording with open eyes B)Non-epileptic recording with closed eyes C,D) Recording using placement of brain scalp electrode E) Data recorded during an epileptic seizure. The length of each recording is 23.6 seconds and the sampling frequency is 173.6 Hz. In this paper, models B and E were used. first without using the wavelet transform function and then the mother wavelet transform functions with the decomposition level of 8 were used. Table2 summarizes the details of the database:
Proposed Method
After receiving the data from the mentioned database, data preprocessing was started. At this step, a 70 Hz low-pass filter was designed and implemented for both methods. Then, in the data processing step, alpha, beta, and gamma frequency bands were extracted by designing a separate mid-pass filter. In the feature extraction step, features such as Frequency median (FMD), frequency mean (FMN), frequency ratio (FR), and Waveform length (WL) were extracted. Finally, in the last step, a multilayer perceptron neural network (MLP) classification was designed. After performing experiments with this classifier and obtaining the desired results, the wavelet transform method was applied separately. In Fig. 1, diagram of the suggested method is presented. Fig. 1 Diagram of the proposed method
4-1. Signal Preprocessing
The preprocessing step was performed by applying a 70 Hz low-pass filter to the raw EEG signal collected in both stages using and without the use of wavelet transform. The FDATOOL toolbox is used to design this filter, and the wave menu tool in MATLAB 2020 software is used for the wavelet transform. To remove the noise by using the wavelet transform function, we use the de-noise section in the same wavelet transform toolbox. In this section, the wavelet can independently guess the noise by matching the daughter signal (EEG signal) with the mother signal (wavelet signal) and if they do not match with each other, remove the details from the lower levels while preserving the generalities .
4-2. Signal Processing
In this step, on the alpha, gamma and beta bands, a bandpass filter with : Fpass1 = 25HZ, Fstop2 = 71HZ, Fstop1 = 24HZ and separately for the gamma band, a band-pass filter with the minimum order values : Fpass2 = 70 and Fstop = 71HZ was used. The passband and the stopband ripple are 0.1 and 0.01, respectively. To process the signal using wavelet transform, the filter coefficients, which include approximation and detail, are obtained, and then the detail coefficient and general coefficient of different levels are calculated separately for the alpha-beta and gamma bands. The output signal for these coefficients is calculated using the ''wrcoef'' command in the time domain in MATLAB.
4-3. Feature extraction
Feature extraction is an essential step in epileptic seizure detection, which is used to establish an epilepsy detection model via standard epilepsy data, and epilepsy detection from actual collected EEG signal data. The wavelet transform decomposes a signal in terms of scaled and translated versions of a mother wavelet and a scaling function. The discrete wavelet transform (DWT) has been frequently used in recent epileptic spike and seizure detection approaches that have shown promising results The effect of feature extraction is closely related to the accuracy of epilepsy detection, so it is imperative to improve feature extraction. To record the epilepsy signal separately, the EEG recording takes about 20 minutes and includes photostimulation and rest periods with open and closed eyes. In this paper, an attempt has been made to extract four important features in the field of frequency and time for epileptic and non-epileptic data, separately in each alpha, gamma, and beta frequency band, which are: FMD4, FMN5, FR6, and WL7. The extracted features are shown in Table.3. 3.
-
In a person with epilepsy, because of the higher brain frequency, the median frequency and mean frequency increase, and the frequency ratio decreases. In the Waveform length features, the changes are measured and the absolute value of the next sample is reduced from the previous sample, and if the signal is smooth without too much of changes, this property becomes zero. Therefore, in epilepsy disease where the entropy of the disorder is low and the brain oscillations are the same, the Waveform length decreases. Shannon entropy was used to 4 Frequency median 5 Frequency mean calculate the entropy using wavelet transform, which calculates the entropy based on bit changes [37].
4-4 Classification
To classify the epilepsy signal, a multilayer neural network with 10 input layers and 5% validation was used and this classifier achieved 95.5% accuracy in multilayer neural network which validation results are shown in Figure 2.
Discussion
As mentioned before, the causes of epileptic seizures include epidemiology, etiology, acute stroke, brain infection, and the destructive effects of some medications. In previous studies, it was shown that some researchers had used a combination of EEG signals with FMRI to detect physiological values [8] and some Others preferred to use neurofeedback to treat epileptic seizures [36]. In recent years, advanced methods focusing on functional dynamics have been proposed in studies to help people with epilepsy. For example, in various studies, the SL parameter has been studied in the theta frequency band of people with epilepsy, which can classify epilepsy with a sensitivity of 62% and a specificity of 76%. Recently, EEG signal-analysis techniques have been improved because EEG signals reflect neurological brain activities, and they have also become an important tool for diagnosing neurological brain disorders. The purpose of this study was to demonstrate the predictive value of physiological signals measured using EEG, which was performed in two different experimental stages. And also by several tests on the Bonn datasets, it was shown competitive results in comparison to other recent methods that how a predictive model with EEG characteristics makes significant progress compared to clinical diagnosis and helps us to diagnose the presence or absence of epileptic symptoms in the EEG signal. In the first step, using a multilayer perceptron neural network (MLP) classifier, epilepsy, and control signals in alpha, beta, and gamma frequency bands were examined separately and 95% accuracy was obtained to detect the epileptic signal. Then using a wavelet transform function, the signal for the alpha, beta, and gamma frequency bands was re-examined and about 91% accuracy was obtained to detect the epileptic signal. to detect the epileptic signal [38], Compared to other studies, this study succeeded in obtaining a simpler method for assessing epilepsy from EEG signals and also achieved appropriate parameters such as accuracy.
Conclusion
An epileptic seizure evaluation by expert physicians, which is performed by visual examination of long-term EEG recordings, is a very costly and time-consuming method. To address these problems, the current study proposed a new seizure detection algorithm using EEG signals. The main strength of this study in epilepsy is the description of the available sample with a multilayer perceptron neural network classifier and its comparison with a wavelet transform function.
Future studies with larger sample sizes are needed to confirm our findings and advance the results In this study, the diagnostic value of physiological signals measured using EEG was evaluated . This study also succeeded in extracting features such as (FMD), (FMN), (FR), and (WL). )FMD( and )FMN(, which are the median and mean frequencies, are obtained by using the Fourier transform of the desired signal, and in a person with epilepsy, these two characteristics increase because the brain frequency is higher than a normal person. (FR) or frequency rate refers to the high-to-low frequency ratio of the brain, and because the brain frequency of a person with epilepsy is higher than a normal person, the frequency rate decreases. But (WL) or Waveform length, which is a temporal feature, subtracts the absolute magnitude of subsequent samples of the signal from the previous samples and adds them together, actually measuring the changes and complexity of the signal, and in a person with epilepsy due to Decreased entropy and similar fluctuations in the brain (WL) are reduced. Calculation of these features provides good information for physicians to diagnose and use this method in patients with epilepsy.
According to the studies performed in this paper, although not much difference was observed between each of the two methods used in this paper, as you have seen in Table.4 the experiment performed in the first step and using multilayer perceptron neural network (MLP) achieved more accuracy than the experiment performed with using wavelet transform function. In recent years, more and more new methods have begun to be applied to the automatic detection of epilepsy. The development of faster and more accurate epilepsy detection models will contribute to epilepsy detection techniques in clinical diagnosis and the development of portable and integrated epilepsy detection equipment. Therefore, a concise and efficient epilepsy detection model will become an inevitable development trend in the future. [2] Srinath, R. and Gayathri, R., 2021. Detection and classification of electroencephalogram signals for epilepsy disease using machine learning methods. international Journal of imaging Systems and technology, 31(2), pp.729-740.
[3] Chen, Z., Lu, G., Xie, Z. and Shang, W., 2020. A unified framework and method for EEG-based early epileptic seizure detection and epilepsy diagnosis. IEEE Access, 8, pp.20080-20092. [4] Dehghani, A., Soltanian-Zadeh, H. and Hossein-Zadeh, G.A., 2020. Probing fMRI brain connectivity and activity changes during emotion regulation by EEG neurofeedback. arXiv preprint arXiv:2006.06829.
[5] Dehghani, A., Soltanian-Zadeh, H. and Hossein-Zadeh, G.A., 2020. Global data-driven analysis of brain connectivity during emotion regulation by electroencephalography neurofeedback. Brain Connectivity, 10(6), pp.302-315.
[6] Alturki, F.A., Aljalal, M., Abdurraqeeb, A.M., Alsharabi, K. and Al-Shamma'a, A.A., 2021. Common spatial pattern technique with EEG signals for diagnosis of autism and epilepsy disorders. IEEE Access, 9, pp.24334-24349.
Figure ( 2
2): The validation value and accuracy of perceptron multilayer neural network Then wavelet transform with level 8 and decomposition level 4 and Shannon entropy for calculating the entropy was used. 91% accuracy was obtained for the diagnosis of epilepsy with this method and the validation results are also shown inFigure 3.
Figure
Figure (3): The validation value and accuracy after wavelet transform conversion
Table1.Brain sub-bandsSub-band
Frequency
Range(Hz)
Related
mental states
Waveform
Delta
Up to 4
Sleep,coma
THETA
4-8
Creative
thinking
ALPHA
8-13
Being relaxed
BETA
13-25
Concentration,
thinking
GAMMA
ABOVE 25
Rahul Sharma et al utilized a third-order cumulant function for the automatic detection of focal EEG signals. The features were extracted from the EEG signals using locality sensitive discriminant analysis (LSDA), method, and then these features were classified using the support vector machine (SVM) classification technique. The authors obtained a maximum classification accuracy of 99% on the Bern-Barcelona EEG data set[19]. Siddharth et al developed a method for discriminating the focal signals from nonfocal EEG signals using the sliding mode singular spectrum analysis method. The reconstructed component features were computed from EEG signals and those features were classified using a radial basis function neural network. The authors tested their proposed method on the EEG signals of the Barcelona EEG data set and obtained an average 2 support vector machines accuracy of 99.11%, an average sensitivity of 98.52%, and an average specificity of 99.7%[20]. Rahul Sharma et al presented a bispectrum method in order to extract 25 magnitude features from EEG signals. These features were obtained by using LSDA method and then the computed features were classified using SVM classification approach. The authors used 10-fold cross validation approach on the EEG signals of the Barcelona EEG data set and obtained a classification accuracy of 96.2%[21]. the scaling levels. This method was tested by using the Kruskal-Wallis statistical test, and the authors had obtained an average classification accuracy of 89.1% on the EEG signals of the Barcelona EEG data set[25].Nigam and Graupe suggested EEG-based computer-aided
diagnosis to diagnose epilepsy using a multistage nonlinear pre-
processing filter in combination with an artificial neural network
(ANN); their proposed technique achieved accuracy of 97.2%
[22].
Kannathal et al. compared different entropy algorithms and
suggested that entropy values can distinguish between normal
EEG and epileptic EEG; they used an adaptive neuro-fuzzy
inference system (ANFIS) for classification and achieved
accuracy of 92.2% [23]. Moreover, Sadati et al. used an adaptive
neural fuzzy network instead of ANFIS for epilepsy diagnosis;
they used the energy of discrete wavelet transform (DWT) sub-
bands for feature extraction. However, their proposed method
achieved low accuracy (about 85.9%) [24]. Dalal et al had
explored the flexible analytic wavelet transform (FAWT) for
obtaining the decomposition coefficients of EEG signals. This
nonstationary transform produced fractal dimension features at
each of Deivasigamani et al analyzed a soft computing-based adaptive
neuro-Fuzzy inference system (ANFIS) classification method
for differentiating the focal signal and nonfocal signals using
their feature values. It was applied to 700 EEG signals out of
which 694 signals were classified correctly. The authors had
obtained a detection rate of 99.1% [26]. Abhinaya et al22 had
dealt with a methodology that extracted entropy-based features
from the EEG signals. Those extracted features were optimized
using the sequential forward feature selection method. Then,
SVM in the linear regression model was applied to the optimized
features for differentiating the focal and nonfocal EEG signals.
The authors had obtained a classification rate of 92.8% by
applying it to the open-access EEG data set [27]. Ibrahim et al.
analyzed a method-based wavelet and Shannon entropy for
epilepsy. In their study, SVM, LDA, artificial neural networks,
and k-nearest neighbor were used as a classifier [28].
Gruszczyńska et al. suggested a study-based recurrence
quantification analysis to classify EEG signals. The principal
component analysis was used to visualize the results. SVM was
utilized for the classification of results obtained
Al-Salman et al. proposed a study using wavelet Fourier analysis for the detection of sleep spindles detection from EEG signals.The least-square SVM was used for the classifier [32]. Asghar
Zarei et al. used the discrete wavelet transform (DWT) and
orthogonal matching pursuit (OMP) techniques. d to extract
different coefficients from the EEG signals. Then, some non-
linear features, such as fuzzy/approximate/sample/alphabet and
correct conditional entropy, along with some statistical features
are calculated using the DWT and OMP coefficients. Three
widely-used EEG datasets were utilized to assess the
performance of the proposed techniques [33]. Recently, Yavus
et al. explored the utility of Mel frequency cepstral coefficients
(MFCCs) and generalized regression neural networks for
distinguishing normal and seizure EEG recordings [34].
Mehdi Omidvar et al. used discrete wavelet transform (DWT) to
extract features of EEG signals by dividing them into five sub-
bands.
Table . 2
.Overview of Bonn datasetSet
Patients
Setup
Phase
A
healthy
Surface EEG
Open eyes
B
healthy
Surface EEG
Open eyes
C
epilepsy
intracranial EEG
interictal
D
epilepsy
intracranial EEG
interictal
E
epilepsy
intracranial EEG
seizure
The database used in this research is open access on the
website of Bonn university.
Table . 3
.Extracted featureNumber
Frequency
Domain Features
Number
Time Domain
Features
1.
Frequency median
(FMD)
1.
Waveform
length (WL)
2. Frequency mean
(FMN)
2.
-
3. Frequency
rate
(FR)
Table . 4
.Results of the performed method References [1] Liu, G., Xiao, R., Xu, L. and Cai, J., 2021. Minireview ofMethods
Accuracy
wavelet transform
91.0%
Multilayer
perceptron
neural network
95.5%
7. Epilepsy
Detection
Techniques
Based
on
Electroencephalogram
Signals. Frontiers
in
systems
neuroscience, 15, p.44.
Early detection of Alzheimer's disease from EEG signals using Hjorth parameters. M S Safi, S M M Safi, Biomedical Signal Processing and Control. 65102338Safi, M.S. and Safi, S.M.M., 2021. Early detection of Alzheimer's disease from EEG signals using Hjorth parameters. Biomedical Signal Processing and Control, 65, p.102338.
Classification of EEG signals for epileptic seizures using hybrid artificial neural networks based wavelet transforms and fuzzy relations. O Kocadagli, R Langari, Expert Systems with Applications. 88Kocadagli, O. and Langari, R., 2017. Classification of EEG signals for epileptic seizures using hybrid artificial neural networks based wavelet transforms and fuzzy relations. Expert Systems with Applications, 88, pp.419-434.
Predictive value of functional MRI and EEG in epilepsy diagnosis after a first seizure. G S Drenthen, J F Jansen, E Gommer, L Gupta, P A Hofman, V H Van Kranen-Mastenbroek, D M Hilkman, M C Vlooswijk, R P Rouhl, W H Backes, Epilepsy & Behavior. 115107651Drenthen, G.S., Jansen, J.F., Gommer, E., Gupta, L., Hofman, P.A., van Kranen-Mastenbroek, V.H., Hilkman, D.M., Vlooswijk, M.C., Rouhl, R.P. and Backes, W.H., 2021. Predictive value of functional MRI and EEG in epilepsy diagnosis after a first seizure. Epilepsy & Behavior, 115, p.107651.
Predictive value of functional MRI and EEG in epilepsy diagnosis after a first seizure. G S Drenthen, J F Jansen, E Gommer, L Gupta, P A Hofman, V H Van Kranen-Mastenbroek, D M Hilkman, M C Vlooswijk, R P Rouhl, W H Backes, Epilepsy & Behavior. 115107651Drenthen, G.S., Jansen, J.F., Gommer, E., Gupta, L., Hofman, P.A., van Kranen-Mastenbroek, V.H., Hilkman, D.M., Vlooswijk, M.C., Rouhl, R.P. and Backes, W.H., 2021. Predictive value of functional MRI and EEG in epilepsy diagnosis after a first seizure. Epilepsy & Behavior, 115, p.107651.
Epilepsy attacks recognition based on 1D octal pattern, wavelet transform and EEG signals. T Tuncer, S Dogan, G R Naik, P Pławiak, Multimedia Tools and Applications. 8016Tuncer, T., Dogan, S., Naik, G.R. and Pławiak, P., 2021. Epilepsy attacks recognition based on 1D octal pattern, wavelet transform and EEG signals. Multimedia Tools and Applications, 80(16), pp.25197-25218
Automated detection of abnormal EEG signals using localized wavelet filter banks. M Sharma, S Patel, U R Acharya, Pattern Recognition Letters. 133Sharma, M., Patel, S. and Acharya, U.R., 2020. Automated detection of abnormal EEG signals using localized wavelet filter banks. Pattern Recognition Letters, 133, pp.188-194.
An automated system for epilepsy detection using EEG brain signals based on deep learning approach. I 5-Ullah, M Hussain, H Aboalsamh, Expert Systems with Applications. 1075-Ullah, I., Hussain, M. and Aboalsamh, H., 2018. An automated system for epilepsy detection using EEG brain signals based on deep learning approach. Expert Systems with Applications, 107, pp.61-71.
Application of machine learning to epileptic seizure detection. A H Shoeb, J V Guttag, Shoeb, A.H. and Guttag, J.V., 2010, January. Application of machine learning to epileptic seizure detection. In ICML
An efficient epileptic seizure detection based on tunable Q-wavelet transform and DCVAE-stacked Bi-LSTM model using electroencephalogram. S Sivasaravanababu, V Prabhu, V Parthasarathy, R K Mahendran, The European Physical Journal Special Topics. Sivasaravanababu, S., Prabhu, V., Parthasarathy, V. and Mahendran, R.K., 2021. An efficient epileptic seizure detection based on tunable Q-wavelet transform and DCVAE-stacked Bi- LSTM model using electroencephalogram. The European Physical Journal Special Topics, pp.1-13.
Epileptic seizures classification based on long-term EEG signal wavelet analysis. K D Tzimourta, A T Tzallas, N Giannakeas, L G Astrakas, D G Tsalikakis, M G Tsipouras, International Conference on Biomedical and Health Informatics. SingaporeSpringerTzimourta, K.D., Tzallas, A.T., Giannakeas, N., Astrakas, L.G., Tsalikakis, D.G. and Tsipouras, M.G., 2017, November. Epileptic seizures classification based on long-term EEG signal wavelet analysis. In International Conference on Biomedical and Health Informatics (pp. 165-169). Springer, Singapore.
Epileptic seizure detection in EEG signal using machine learning techniques. Australasian physical & engineering sciences in medicine. A K Jaiswal, H Banka, 41Jaiswal, A.K. and Banka, H., 2018. Epileptic seizure detection in EEG signal using machine learning techniques. Australasian physical & engineering sciences in medicine, 41(1), pp.81-94.
Application of machine learning to epileptic seizure detection. A H Shoeb, J V Guttag, Shoeb, A.H. and Guttag, J.V., 2010, January. Application of machine learning to epileptic seizure detection. In ICML
A new technique for classification of focal and non-focal EEG signals using higher order spectra. R Sharma, P Sircar, R B Pachori, J Mech Med Biol. 19011940010Sharma R, Sircar P, Pachori RB. A new technique for classification of focal and non-focal EEG signals using higher order spectra. J Mech Med Biol. 2019;19(01):1940010.
Discrimination of focal and non-focal seizures from EEG signals using sliding mode singular Spectrum analysis. T Siddharth, R K Tripathy, R B Pachori, IEEE Sensors J. 1924Siddharth T, Tripathy RK, Pachori RB. Discrimination of focal and non-focal seizures from EEG signals using sliding mode singular Spectrum analysis. IEEE Sensors J. 2019;19(24):12286-12296.
Empirical mode decomposition based classification of focal and non-focal EEG signals. R Sharma, R B Pachori, S Gautam, Proc Int Conf Med Biometrics. 1Sharma R, Pachori RB, Gautam S. Empirical mode decomposition based classification of focal and non-focal EEG signals. Proc Int Conf Med Biometrics. 2015;1:135-140.
A neural-network-based detection of epilepsy. V P Nigam, D Graupe, doi: 10. 1179/016164104773026534Neurolog. Res. 261V. P. Nigam and D. Graupe, ''A neural-network-based detection of epilepsy,'' Neurolog. Res., vol. 26, no. 1, pp. 55- 60, Jan. 2004, doi: 10. 1179/016164104773026534.
Entropies for detection of epilepsy in EEG. N Kannathal, M L Choo, U R Acharya, P K Sadasivan, 10.1016/j.cmpb.2005.06.012Comput. Methods Programs Biomed. 803N. Kannathal, M. L. Choo, U. R. Acharya, and P. K. Sadasivan, ''Entropies for detection of epilepsy in EEG,'' Comput. Methods Programs Biomed., vol. 80, no. 3, pp. 187- 194, Dec. 2005, doi: 10.1016/j.cmpb.2005.06.012.
Epileptic seizure detection using neural fuzzy networks. N Sadati, H R Mohseni, A Maghsoudi, Proc. IEEE Int. Conf. Fuzzy Syst. IEEE Int. Conf. Fuzzy SystN. Sadati, H. R. Mohseni, and A. Maghsoudi, ''Epileptic seizure detection using neural fuzzy networks,'' in Proc. IEEE Int. Conf. Fuzzy Syst., Jul. 2006, pp. 596-600.
Automated identification system for focal EEG signals using fractal dimension of FAWT based sub-bands signals. M Dalal, M Tanveer, P Ram, IEEE Int Conf Mach Intell Signal Process. 748Dalal M, Tanveer M, Ram P. Automated identification system for focal EEG signals using fractal dimension of FAWT based sub-bands signals. IEEE Int Conf Mach Intell Signal Process. 2017;748:22-24.
Classification of focal and nonfocal EEG signals using ANFIS classifier for epilepsy detection. S Deivasigamani, C Senthilpari, W H Yong, Int J Imaging Syst Technol. 264Deivasigamani S, Senthilpari C, Yong WH. Classification of focal and nonfocal EEG signals using ANFIS classifier for epilepsy detection. Int J Imaging Syst Technol. 2016;26(4): 277-283.
Feature extraction and selection of a combination of entropy features for real-time epilepsy detection. B Abhinaya, D Charanya, K Palanithanaraj, Intl J Eng Comput Sci. 54Abhinaya B, Charanya D, PalaniThanaraj K. Feature extraction and selection of a combination of entropy features for real-time epilepsy detection. Intl J Eng Comput Sci. 2016;5(4):16073-16078..
. S Ibrahim, R Djemal, A Alsuwailem, Ibrahim S, Djemal R, Alsuwailem A (2018)
EEG) signal processing for epilepsy and autism spectrum disorder diagnosis. Electroencephalography, Biocybernetics and Biomedical Engineering. 38Electroencephalography (EEG) signal processing for epilepsy and autism spectrum disorder diagnosis. Biocybernetics and Biomedical Engineering 38:16-26.
Epilepsy identification based on EEG signal using RQA method. Advances in medical sciences. I Gruszczyńska, R Mosdorf, P Sobaniec, M Żochowska-Sobaniec, M Borowska, Gruszczyńska I, Mosdorf R, Sobaniec P, Żochowska- Sobaniec M, Borowska M (2019) Epilepsy identification based on EEG signal using RQA method. Advances in medical sciences.
presented an approach based machine learning methods and wavelet transform for alcohol use disorder. The aim of this study is to classify alcoholic and healthy individuals automatically. SVM and Naïve Bayes methods were utilized in the study. Sisodia Anuragi, Anuragi and Sisodia (2018) [9] presented an approach based machine learning methods and wavelet transform for alcohol use disorder. The aim of this study is to classify alcoholic and healthy individuals automatically. SVM and Naïve Bayes methods were utilized in the study.
Detection of epileptic dysfunctions in EEG signals using Hilbert vibration decomposition. Biomedical Signal Processing and Control. 40Mutlu AYMutlu AY (2018) Detection of epileptic dysfunctions in EEG signals using Hilbert vibration decomposition. Biomedical Signal Processing and Control. 40:33-40.
Detecting sleep spindles in EEGs using wavelet fourier analysis and statistical features. W Al-Salman, Y Li, P Wen, Biomedical Signal Processing and Control. 48Al-Salman W, Li Y, Wen P (2019) Detecting sleep spindles in EEGs using wavelet fourier analysis and statistical features. Biomedical Signal Processing and Control. 48:80-92
Automatic seizure detection using orthogonal matching pursuit, discrete wavelet transform, and entropy based features of EEG signals. A Zarei, B M Asl, Computers in Biology and Medicine. 131104250Zarei, A. and Asl, B.M., 2021. Automatic seizure detection using orthogonal matching pursuit, discrete wavelet transform, and entropy based features of EEG signals. Computers in Biology and Medicine, 131, p.104250.
An epileptic seizure detection system based on cepstral analysis and generalized regression neural network. E Yavuz, M C Kasapbas¸ı, C Eyüpoglu, ˘ R Yazıcı, Biocybernetics and Biomedical Engineering. E. Yavuz, M.C. Kasapbas¸ı, C. Eyüpoglu, ˘ R. Yazıcı, An epileptic seizure detection system based on cepstral analysis and generalized regression neural network, Biocybernetics and Biomedical Engineering (2018) 201-216.
EEG signal processing for epilepsy seizure detection using 5-level Db4 discrete wavelet transform, GA-based feature selection and ANN/SVM classifiers. M Omidvar, A Zahedi, H Bakhshi, Journal of Ambient Intelligence and Humanized Computing. 1211Omidvar, M., Zahedi, A. and Bakhshi, H., 2021. EEG signal processing for epilepsy seizure detection using 5-level Db4 discrete wavelet transform, GA-based feature selection and ANN/SVM classifiers. Journal of Ambient Intelligence and Humanized Computing, 12(11), pp.10395-10403.
J Tanke, O H Kwon, P Stotko, R A Rosu, M Weinmann, H Errami, S Behnke, M Bennewitz, R Klein, A Weber, A Yao, arXiv:1912.06354Bonn activity maps: Dataset description. arXiv preprintTanke, J., Kwon, O.H., Stotko, P., Rosu, R.A., Weinmann, M., Errami, H., Behnke, S., Bennewitz, M., Klein, R., Weber, A. and Yao, A., 2019. Bonn activity maps: Dataset description. arXiv preprint arXiv:1912.06354.
Comparison of the effectiveness of neurofeedback therapy in improving the executive functions of patients with hyperactivity and epilepsy. Khaksarian, Hassanvandi, Piri Saba, Sohrabi Fard, Found. 221Khaksarian, Hassanvandi, Saba, Piri and Sohrabi Fard, 2020. Comparison of the effectiveness of neurofeedback therapy in improving the executive functions of patients with hyperactivity and epilepsy. Found, 22 (1), pp.13-24 .
Electroencephalography (EEG) signal processing for epilepsy and autism spectrum disorder diagnosis. S Ibrahim, R Djemal, A Alsuwailem, Biocybernetics and Biomedical Engineering. 381Ibrahim, S., Djemal, R. and Alsuwailem, A., 2018. Electroencephalography (EEG) signal processing for epilepsy and autism spectrum disorder diagnosis. Biocybernetics and Biomedical Engineering, 38(1), pp.16-26.
| [] |
[
"DIFFERENCE TONES IN \"NON-PYTHAGOREAN\" SCALES BASED ON LOGARITHMS",
"DIFFERENCE TONES IN \"NON-PYTHAGOREAN\" SCALES BASED ON LOGARITHMS"
] | [
"Thomas Morrill "
] | [] | [] | In order to explore tonality outside of the "Pythagorean" paradigm of integer ratios, Robert Schneider introduced a musical scale based on the logarithm function. We seek to refine Schneider's scale so that the difference tones generated by different degrees of the scale are themselves octave equivalents of notes in the scale. In doing so, we prove that a scale which contains all its difference tones in this way must consist solely of integer ratios. With this in mind, we present some methods for producing logarithmic scales which contain many, but not all, of the difference tones they generate. | 10.1080/17459737.2019.1704084 | [
"https://arxiv.org/pdf/1804.08067v5.pdf"
] | 108,284,379 | 1804.08067 | 12dfc0515132fd0c6ba1e92c7dcf1c4da2c4e0ca |
DIFFERENCE TONES IN "NON-PYTHAGOREAN" SCALES BASED ON LOGARITHMS
28 Oct 2019
Thomas Morrill
DIFFERENCE TONES IN "NON-PYTHAGOREAN" SCALES BASED ON LOGARITHMS
28 Oct 2019arXiv:1804.08067v5 [math.HO]
In order to explore tonality outside of the "Pythagorean" paradigm of integer ratios, Robert Schneider introduced a musical scale based on the logarithm function. We seek to refine Schneider's scale so that the difference tones generated by different degrees of the scale are themselves octave equivalents of notes in the scale. In doing so, we prove that a scale which contains all its difference tones in this way must consist solely of integer ratios. With this in mind, we present some methods for producing logarithmic scales which contain many, but not all, of the difference tones they generate.
In 1754, the Italian violinist Giuseppe Tartini described a musical phenomenon which he called terzi suoni, or third sounds [Jon35,Tar54]. He found that when two notes were played simultaneously, a third note could be perceived, whose frequency was the difference of the frequencies of the two played notes. When listening to two tones of frequencies f 1 < f 2 , more tones may be perceived in addition to Tartini's terzi suoni, whose frequencies are other linear combinations of the frequencies being played [RP99]. Hence, the frequencies mf 1 + nf 2 , are called the combination tones of frequencies f 1 and f 2 , where n, m ∈ Z.
Perception of combination tones in humans has been qualitatively studied [RP99], and combination tones have been empirically measured in string instruments [LCC11]. Combination tones are thought to be physically produced by nonlinear resonance in the recieving auditory system, such as the human eardrum [Sch05,LCC11].
Empirical evidence shows that the combination tones f 2 − f 1 , 2f 1 − f 2 , and 3f 1 −2f 2 are the most commonly perceived, and the amplitude of combination tones in general varies depending on the frequencies f 1 and f 2 and their intensities [RP99]. Being the most commonly perceived combination tone, we will focus exclusively on the first order difference tone f 2 − f 1 .
Since Tartini's discovery, combination tones have been used to develop novel musical scales. A particularly deep example is Wilson's use of sum-diagonals of the Meru Prastala, perhaps more familiar to the reader as Pascal's triangle, of which Wilson documented 192 recurrent sequences [Bur02]. Another example is Bohlen's use of the combination tone f 1 + f 2 in his so-called 833 cents scale [Boh05, S + 16].
In 2012, Schneider [Sch12] introduced an infinite scale based over the frequency f in order to explore tonality which does not seek to approximate integer ratios, whose frequencies are given by
f ln(3), f ln(4), f ln(5), . . . .(1)
Here and throughout the paper, we use ln(x) to denote the natural logarithm of x, and log b (x) = ln(x)/ ln(b) to denote the base b logarithm. As a consequence of our methodology, we refer to (1) as the logarithmic series. As opposed to the harmonic seres f, 2f, 3f, . . ., the frequencies of the logarithmic series grow closer together as one proceeds further up the series. A piece using the logarithmic series may be heard in the audio example "Cantor.wav".
Schneider also derived a scale which divides the octave into 12 pitches from the logarithmic series, given by the frequencies f ln(4), f ln(5), . . ., f ln(16) = 2f log(4).
In private correspondence, Schneider raised the following question: Which scales contain all of their difference tones up to octave equivalence? We will show that any such scale must consist solely of integer ratios. Since the motivation of the logarithmic scale was to explore harmony outside of integer ratios, we offer some methods for constructing logarithmic scales which contain some, but not all of their difference frequencies.
The rest of the paper is laid out as follows. In Section 1, we lay out the methodology and notation necessary to prove our main theorem in Section 2. In Sections 3 and 4 we offer constructions of frequency series and scales, respectively, which contain many of their difference frequencies. Section 5 describes a chordal technique for these scales which may be of interest to composers. Audio examples may be found at https://github.com/tsmorrill/Non-Pythagorean-Examples. Finally, in Section 6 we give our closing remarks.
Methodology
We make some simplifying assumptions on the perception of combination tones. The first is that combination tones are only generated by the fundamental frequencies of tones, and not their higher partials. We also assume that only the first order difference tone f 2 − f 1 is perceived. These assumptions are justified based on the empirical evidence regarding perception of combination tones [RP99].
With this in mind, a frequency will always be taken to mean a positive real number measured in Hertz. A frequency series is an increasing sequence of frequencies f 1 < f 2 < . . . , which tends to infinity. A scale is a finite increasing set of frequencies S = {f 1 , f 2 , . . . , f n }. The latter definition is compatible with the restriction that a scale consists of frequencies spanning a single octave, but we do not require this for our work.
Transposing a frequency series or scale is accomplished by replacing the reference f 1 by some other frequency f ′ 1 , and replacing each f i by a f ′ i satisfying f i :
f 1 = f ′ i : f ′
1 . This is equivalent to multiplying each of the frequencies f i by the constant f ′ 1 /f 1 . We will therefore normalize frequency series and scales by dividing all of their frequencies by f 1 . This has the effect that all scales and series we present satisfy f 1 = 1. When these frequencies are defined by logarithms, this also has the effect of changing the base of the logarithm. For example, the scale S = {ln(4), ln(5)} normalizes to {1, log 4 (5)}. Note that the normalization preserves ratios between frequencies, here ln(5) ln(4) = log 4 (5) 1 .
From this point, normalized frequencies will be represented by a closed form expression of f i /f 1 , a decimal approximation of f i /f 1 , and the interval f i : f 1 measured in cents. All decimal expansions will be rounded to the third place. For example, here is the normalization of Schneider's octave-dividing scale. This may be heard in the audio example "Schneider Scale.wav".
Complete Difference Tone Scales
In addition to the terminology of the previous section, we will call a scale S a complete difference tone scale if for every pair x, y ∈ S with x > y, there exists a z ∈ S and an integer t such that
x − y = 2 t z.
We now prove our main result.
Theorem 2.1. A complete difference tone scale consists solely of rational intervals.
More precisely, any complete difference scale S must be contained in the set f Q = {f r|r ∈ Q} for some frequency f . Thus, any ratio of frequencies in the scale is a rational number.
Proof of Theorem 2.1. Suppose that S = {f 1 , f 2 , . . . , f n } is a complete difference tone scale. Then for each index i < n, there is an index j with 1 ≤ j ≤ n and an integer t i so that f n − f i = 2 ti f j . We use these relations to define a function h so that h(i) = j. As this would be undefined for i = n, we additionally define h(n) = n.
We call i a periodic point under h if h(h(· · · (h(s))) = h s (i) = i for some s ≥ 1. First, we claim that periodic points under h are the indices of rational multiples of f n . This is trivial for the periodic point i = n. For periodic points i = n, we may write
f n − f i1 = 2 ti 1 f i2 f n − f i2 = 2 ti 2 f i3 . . . f n − f is−1 = 2 ti s f i1 , which implies f n − f i1 = 2 ti 1 (f n − 2 ti 2 (f n − 2 ti 3 (f n − · · · 2 ti s f i1 ))).
By rearranging, we have αf i1 = βf n , where α and β are rational numbers.
We call i a preperiodic point under h if h s (i) is a periodic point under h for some s ≥ 1. We claim that preperiodic points under h are also the indices of rational multiples of f n . We have
f n − f i1 = 2 ti 1 f i2 f n − f i2 = 2 ti 2 f i3 . . . f n − f is 1 = 2 ti s f is = 2 ti s αf n ,
where α is some rational number. This implies that
f n − f i1 = 2 ti 1 (f n − 2 ti 2 (f n − 2 ti 3 (f n − · · · 2 ti s αf n ))).
By rearranging, we find that f i1 is a rational multiple of f n .
Finally, note that each i is either periodic or preperiodic under h. Thus, every f i ∈ S is a rational multiple of f n , which is to say, S consists solely of rational intervals.
We pause to reflect on the consequences of Theorem 2.1. Consider its converse: We observe that S is a complete difference tone scale. However, S contains no irrational intervals, which may be seen from its normalization, S ′ = {1, 5/4, 3/2}. In keeping with Schneider's stated goal of exploring harmony outside of integer ratios, we will now construct logarithmic series and scales which feature irrational intervals and contain some of their difference frequencies. However, as a consequence of Corollary 2.2, these scales cannot contain all of their difference frequencies.
Construction of Logarithmic Frequency Series
Consider the difference frequencies generated by a frequency series. For example, it's easy to check that the harmonic series f, 2f, 3f, . . . , contains all of its difference frequencies: Any two frequencies mf < nf have a difference frequency of (n − m)f , which also occurs in this series. However, this is not true of Schneider's logarithmic series f log(3), f log(4), f log(5) . . . , as the first two frequencies have a difference of f log(4/3). In general, the difference frequency f log(m/n) occurs in the logarithmic series if and only if n/m reduces to an integer. How then can we restrict the logarithmic series such that m/n will always reduce to an integer?
One option is to take logarithms of the factorial function [Inc19], which gives what we call the logarithmic factorial series. The differences of this series are of the form f log(n!/m!) = f log((m + 1)(m + 2) · · · n), which falls in the logarithmic series. Another method is to take logarithms of the primorial numbers, where each entry n k is the product of the first k prime numbers p 1 , p 2 , . . . p k . Suppose that x is a real number and p k is the largest prime no greater than x. Then we have ln(n k ) = ln p 1 p 2 · · · p k = pi≤x ln(p i ).
Closed Form
which is the definition of the Chebyshev theta function, ϑ(x). Hence, we call the corresponding frequency series the Chebyshev series. Similarly to the logarithmic factorial series, the difference frequencies of this series are of the form f ln(p i p i+1 · · · p j ), which falls in the logarithmic series. However, the difference frequencies of both these subseries fall in the full logarithmic series, not in the subseries themselves.
We could instead choose a sequence of integers d 1 , d 2 , . . . , d k and specify that the sequence of difference frequencies is periodic:
f 2 − f 1 = ln d 1 f 3 − f 2 = ln d 2 . . . f k−1 − f k = ln d k f k − f k+1 = ln d 1 f k+2 − f k+1 = ln d 2 . . .
This correseponds to the frequency series
ln(d 1 ), ln(d 1 d 2 ) . . . , ln(d 1 d 2 · · · d k ), ln(d 2 1 d 2 · · · d k ), ln(d 2 1 d 2 2 · · · d k ),
Construction of Logarithmic Scales
Here we seek to produce logarithmic scales which contain many, but not all, of their difference tones. Occasionally, these methods will produce a rational interval. This was also true for Schneider's original scale: note the perfect fifth between f log(4) and f log(8). Thus, we do not consider the occasional rational interval a flaw of our method, so long as we also produce irrational intervals. Throughout this section we take f to be an arbitrary reference frequency.
We start with two families of scales which divide the octave. For the first family, choose integers n and m so that for some positive integer k, n 2 k is approximately equal to m. Then the root approximation scale is given by {f log(m), f log(nm), f log(n 2 m), . . . , f log(n 2 k−1 m), f log(n 2 2k ), f log(m 2 )}.
In this scale, many of the difference frequencies take the form 2 s f log(n), which is an octave equivalent of f log(n 2 k This may be heard in the audio example "Root Approximation Scale.wav". Note that if n 2 k = m, then the scale reduces to {f log(n 2 k ), f log(n 2 k +1 ), f log(n 2 k +2 ), . . . , f log(n 2 2k−1 ), f log(n 2 2k )}, As in Section 2, we see that this scale consists exclusively of rational intervals, which runs counter to our goal. For the second family, we choose a composite integer N with the prime factorization p a1 1 p a2 2 · · · p at t , whose positive divisors we list as 1 = n 1 < n 2 < . . . < n k = N.
The factorization scale is given by This may be heard in the audio example "Factorization Scale.wav". Here, the difference frequencies take the form f ln(p b1
{f ln(1 × N ), f ln(n 2 N ), . . . , f ln(n k−1 N ), f ln(N 2 )} ∪ {f ln(p 2 b i i )|1 ≤ i ≤ t}, where b i = ⌈log 2 (log pi (N ))⌉.1 p b2 2 · · · p bt t ), where −a i ≤ b i ≤ a i . If m = p b1
1 p b2 2 · · · p bt t reduces to an integer and satisfies m = a 2 t j for some j and t, then the difference tone ln(m) occurs in the scale, up to octave equivalence. However, if m does not reduce to an integer, then ln(m) will not appear in the logarithmic series, let alone the factorization scale.
This raises the question, can we develop scales whose difference frequencies are the logarithms of rational numbers, rather than integers? We close with a such a family, whose scales do not contain redundant intervals, and do not span one octave. Non-octave scales are not unheard of in musical practice. For example, none of Carlos's α, β, and γ scales contain the octave [Car].
Choose bases b 1 , b 2 , . . . , b k and heights h b1 , h b2 , . . . , h b k . Let T be the set of rational numbers of the form t = b a1 gcd(a 1 , a 2 , . . . , a k ) = 1, and t > 1. The projective scale is then given by S = {f log(t)|t ∈ T }. For example, choosing bases 2, 3 and heights h 2 = 2, h 3 = 1 produces the set T = {4/3, 3/2, 2, 3, 6, 12}, and the projective scale is given by Closed This may be heard in the audio example "Projective Scale.wav". In this example, the difference frequencies take the form log 4/3 (2 i 3 j ), which are octave equivalents of frequencies in the scale with the exception of log 4/3 (3/2) − log 4/3 (4/3) = log 4/3 (9/8) log 4/3 (6) − log 4/3 (4/3) = log 4/3 (9/2).
1 b a1 1 · · · b a k k , where we require |a i | ≤ h i ,
Logarithmic Composition
It is natural to ask what benefit these scales offer to a composer. Certainly the main appeal is a systematic method to incorporate difference frequencies while excluding rational intervals from composition.
Controlling when rational and irrational intervals occur may be achieved using the factorization of integers into primes. Let A be a positive integer with the unique prime factorization
A = k i=1 p ai i .
Because of the property
ln(A) = k i=1 a i log(p i ),
each positive integer corresponds to a unique pitch set in the logarithmic series,
C A = {a 1 log(p 1 ), a 2 log(p 2 ), . . . , a k log(p k )}.
So long as A has at least three distinct prime divisors, playing the frequencies of C A simultaneously will produce a chord of the logarithmic series. Under this assumption, we call C A the factored chord corresponding to A. Two factored chords C A1 and C A2 relate to each other harmonically depending on which primes occur in both the factorizations of A 1 and A 2 . A short piece using factored chords may be heard in the audio example "Factored Chords.wav". For example, the factored chords C 2016 and C 4752 are given by This reflects the fact that 2016 and 4752 are both divisible by 2 and by 3. Recordings demonstrating this technique, as well as recordings of each of the constructions in Section 4 are available online at https://github.com/tsmorrill/Non-Pythagorean-Examples.
Conclusion
Our main result is that complete difference tone scales consist solely of rational intervals. Hence, a scale constructed using logarithms with the intention to feature irrational intervals cannot be a complete difference tone scale. With this in mind, we have given three refinements of Schneider's logarithmic series, and several families of scales, parameterized by one or more positive integers, which aim to contain some of their difference frequencies, but not all.
Owing to our methodology, these scales do not account for combination tones generated between higher partials of notes, or combination tones besides the first order difference f 2 − f 1 . In his experimental pieces, Schneider primarily uses sine waves, which do not have any partials above the fundamental frequency [Sch12]. We think it would be interesting to use non-harmonic tones with these logarithmic scales, whose partials fall in the logarithmic series, or one of the refinements given in Section 3.
Funding
This work was supported by Australian Research Council Discovery Project DP160100932.
Corollary 2.2. A scale which contains an irrational interval is not a complete difference tone scale. Thus, while it is possible to construct a complete difference tone scale from the logarithmic series, the normalization of such a scale would reduce to rational freqeuncies. Consider the scale S = {ln(16), ln(32), ln(64)}. The first order differences of this scale are ln(64) − ln(32) = ln(
C 2016 =
2016{5 ln(2), 2 ln(3), ln(7)} C 4752 = {4 ln(2), 3 ln(3), ln(11)}.DIFFERENCE TONES IN "NON-PYTHAGOREAN" SCALES BASED ON LOGARITHMS 9Transitioning between C 2016 and C 4752 involves two rational
. . . .This series contains all of its difference frequencies, with the exception of ln(d 2 ),
ln(d 3 ), . . . , ln(d k ). Appending the missing frequencies ln(d i ) may introduce addi-
tional difference frequencies depending on their divisibility properties. For example,
with k = 2 and d 1 = 3, d 2 = 5, we have
Closed Form Decimal
Cents
1
1
0
log 3 (5)
1.465
661.050
log 3 (15)
2.465
1561.887
log 3 (45)
3.465
2151.413
log 3 (225)
4.930
2761.887
log 3 (675)
5.930
3081.623
log 3 (3375)
7.395
3463.842
log 3 (16875)
8.860
3776.747
. . .
. . .
. . .
An 833 cents scale: An experiment on harmony. Heinz Bohlen, Heinz Bohlen. An 833 cents scale: An experiment on harmony, 2005.
Developing and composing with scales based on recurrent sequences. Proceedings ACMC. Warren Burt, Warren Burt. Developing and composing with scales based on recurrent sequences. Pro- ceedings ACMC, pages 123-132, 2002.
Three asymmetric divisions of the octave. Wendy Carlos, Wendy Carlos. Three asymmetric divisions of the octave.
OEIS Foundation Inc. The on-line encyclopedia of integer sequences. OEIS Foundation Inc. The on-line encyclopedia of integer sequences, 2019.
The discovery of difference tones. Arthur Taber, Jones , American Journal of Physics. 32Arthur Taber Jones. The discovery of difference tones. American Journal of Physics, 3(2):49-51, 1935.
Combination tones in violins. Angela Lohri, Sandra Carral, Vasileios Chatziioannou, Archives of Acoustics. 364Angela Lohri, Sandra Carral, and Vasileios Chatziioannou. Combination tones in violins. Archives of Acoustics, 36(4):727-740, 2011.
The perception of musical tones. Rudolf Rasch, Reinier Plomp, The Psychology of Music. ElsevierSecond EditionRudolf Rasch and Reinier Plomp. The perception of musical tones. In The Psychology of Music (Second Edition), pages 89-112. Elsevier, 1999.
Two non-octave tunings by heinz bohlen: A practical proposal. + 16] Reilly Smethurst, Proceedings of Bridges 2016: Mathematics. Bridges 2016: MathematicsMusic, Art, Architecture, Education, CultureTessellations Publishing+ 16] Reilly Smethurst et al. Two non-octave tunings by heinz bohlen: A practical proposal. In Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture, pages 519-522. Tessellations Publishing, 2016.
Eric Schwitzgebel. Difference tone training. Eric Schwitzgebel. Difference tone training, 2005.
A Non-Pythagorean Musical Scale Based on Logarithms. Robert P Schneider, Proceedings of Bridges: Mathematics, Music, Art, Architecture, Culture Conference. Bridges: Mathematics, Music, Art, Architecture, Culture ConferenceRobert P. Schneider. A Non-Pythagorean Musical Scale Based on Logarithms. Proceed- ings of Bridges: Mathematics, Music, Art, Architecture, Culture Conference, pages 549-552, 2012.
Trattato di musica secondo la vera scienza dell'armonia. Nella stamperia del Seminario, appresso G. Manfrè. Giuseppe Tartini, 1754Giuseppe Tartini. Trattato di musica secondo la vera scienza dell'armonia. Nella stam- peria del Seminario, appresso G. Manfrè, 1754.
| [
"https://github.com/tsmorrill/Non-Pythagorean-Examples.",
"https://github.com/tsmorrill/Non-Pythagorean-Examples."
] |
[
"Gaussian phase-space representation of Fermion dynamics; Beyond the time-dependent-Hartree-Fock approximation",
"Gaussian phase-space representation of Fermion dynamics; Beyond the time-dependent-Hartree-Fock approximation"
] | [
"Saar Rahav \nDepartment of Chemistry\nUniversity of California\n92697IrvineCA\n",
"Shaul Mukamel \nDepartment of Chemistry\nUniversity of California\n92697IrvineCA\n"
] | [
"Department of Chemistry\nUniversity of California\n92697IrvineCA",
"Department of Chemistry\nUniversity of California\n92697IrvineCA"
] | [] | A Gaussian operator representation for the many body density matrix of fermionic systems, developed by Corney and Drummond [Phys. Rev. Lett, 93, 260401 (2004)], is used to derive approximate decoupling schemes for their dynamics. In this approach the reduced single electron density matrix elements serve as stochastic variables which satisfy an exact Fokker-Planck equation. The number of variables scales as ∼ N 2 rather than ∼ exp(N ) with the basis set size, and the time dependent Hartree Fock approximation (TDHF) is recovered in the "classical" limit. An approximate closed set of equations of motion for the one and two-particle reduced density matrices, provides a direct generalization of the TDHF. | 10.1103/physrevb.79.165103 | [
"https://arxiv.org/pdf/0902.3227v2.pdf"
] | 18,628,505 | 0902.3227 | 2bb63622189dd1c2eb0e6e918655cb7415321eae |
Gaussian phase-space representation of Fermion dynamics; Beyond the time-dependent-Hartree-Fock approximation
1 Jun 2009
Saar Rahav
Department of Chemistry
University of California
92697IrvineCA
Shaul Mukamel
Department of Chemistry
University of California
92697IrvineCA
Gaussian phase-space representation of Fermion dynamics; Beyond the time-dependent-Hartree-Fock approximation
1 Jun 2009arXiv:0902.3227v2 [cond-mat.other]
A Gaussian operator representation for the many body density matrix of fermionic systems, developed by Corney and Drummond [Phys. Rev. Lett, 93, 260401 (2004)], is used to derive approximate decoupling schemes for their dynamics. In this approach the reduced single electron density matrix elements serve as stochastic variables which satisfy an exact Fokker-Planck equation. The number of variables scales as ∼ N 2 rather than ∼ exp(N ) with the basis set size, and the time dependent Hartree Fock approximation (TDHF) is recovered in the "classical" limit. An approximate closed set of equations of motion for the one and two-particle reduced density matrices, provides a direct generalization of the TDHF.
I. INTRODUCTION
The study of interacting many-body systems remains one of the most active research fields in physics 1,2,3,4 . The main computational challenge in electronic structure calculations is that a basis of many-body states, spanning the Hilbert space, becomes exponentially large with system size 5 . In many cases, however, one is not interested in all the information contained in the full many-body wave-function (or density matrix). Experimental observations are typically related to expectation values of certain one and two-body operators. This allows for various approximation schemes, which focus on correlation functions to retain reduced information on the state of the system. This is the basis for the density-functional-theory (DFT) and its time-dependent extension (TDDFT) 6,7,8,9,10 .
The many body hierarchy of reduced density matrices can be truncated systematically by making a cumulant expansion of the wavefunction. Extensive work had been devoted to reduced description in terms of the coupled one and two-body density matrices 11,12 . The main difficulty in the direct computation of reduced density matrices has been the N-representability problem, namely the lack of exact conditions that guarantee that a given reduced density matrix can be obtained from an N electron wavefunction.
In the TDHF method 1,4,7,8,13,14,15,16 the state of the system is assumed to be given by a single Slater determinant, resulting in a closed system of equations for the occupied single-particle orbitals. This is equivalent to truncation at the level of the reduced single-particle density matrix 14 . TDHF is commonly used as a simple, affordable, approximation for the electronic excitations and the optical responce of a system 15 . TDDFT response functions have the same formal structure and simplicity as TDHF ones, except that the burden of the many body problem is shifted into the construction of the functional 17,18 .
The TDHF method can be recasted as a system of equations for a set of coordinates, ρ αβ = ĉ † αĉ β 13,19 , which describe the reduced, single particle density matrix. These can be used to calculate expectation values of any single body operator, such as the optical polarization. Note that if the orbital space contain n occupied and m unoccupied orbitals, only nm out of the (n + m) 2 elements of this reduced density matrix, namely the electron-hole excitations, are needed to represent the full density matrix, and calculate the response 14 . The TDHF equations of motion are approximate, and systematic extentions are not obvious due to the absence of a simple small parameter.
Recently, an exact phase space representation for the many-body fermionic density matrix was developed by Corney and Drummond 20,21,22 . This method is similar in spirit to the coherent states in Hilbert space approach of Cahill and Glauber 23 , since both employ an overcomplete basis set. In Corney and Drummond's approach the many body density matrix is expanded as an ensemble of Gaussian operatorsΛ(n) [Eq. (10)] in the many-electron phase space,
ρ = dnP (n, t)Λ(n),(1)
where P (n, t) is a classical probability distribution for the matrix elements n αβ . This density matrix is assumed to be block diagonal in Fock space, so that coherences are only allowed between states with the same number of particles, which is adequate for most applications. Coherences between states differing by an even number of particles, resulting in anomalous correlations, can be included as well. 20,21,22 Grassmann variables, which are essential for the coherent state representation in Hilbert space, are avoided, thus providing a more intuitive physical interpretation of various quantities. The exact time evolution of the probability distribution P (n, t) for fermions with two body interactions is described by a Fokker-Planck equation. The dynamics of the many-body system is thus mapped onto an ensemble of stochastic trajectories n αβ (t) in real (or imaginary) time. Gaussian phase space operator representation have been also used to derive the Hartree Fock Bogoliubov equations for Bose Einstein condensates 24 .
So far the Gaussian phase-space representation (GPSR) for fermions has mostly been used to study the ground states of Hubbard-like models 25,26,27 . Exceptions are the study of a mixed boson-fermion model of molecular dissociation 20 and a non interacting system 22 . The GPSR can be used to study the time evolution of a system simply by using the Fokker-Planck equation to propagate P (n, t) in time. Therefore, this method maps the many body dynamics onto an evolution in a classical parameter space. The goal of the current paper is to show that the GPSR can be used to derive a new approximation scheme, which naturally connect to, and go beyond the TDHF. The possibility of using GSPR to solve the sign problem in imaginary time calculations is under debate. 28 However, in the real time dynamics considered here complex phases are unavoidable.
The Gaussian parameters n αβ play a role similar to the reduced single electron density matrix elements ρ αβ in TDHF. P (n) can be viewed as a distribution of the matrix elements of the reduced single electron density matrix. We show that there exist a form of the stochastic equations for n αβ (t) whose deterministic part, the non random drift terms, coincides with the TDHF equations. It should be noted that the Gaussian operator basis is overcomplete, and thus allows for many equivalent forms of the stochastic equations. This freedom may be used to simplify the implementation of the method.
A practical, exact, numerical scheme for computing the stochastic trajectories in real time is yet to be developed. We use the GPSR to construct new types of approximations for the dynamics of excitations. By assuming that the probability distribution of the parameters n αβ is Gaussian, we obtain a closed system of equations for the single-particle and two-particle reduced density matrices, which is a direct extension of TDHF. All approaches for performing many body computations by using single and two particles density matrices suffer from the N representability problem, i.e., it is not guaranteed that the approximate reduced density matrices can be derived from an N electron wavefunction 11 . The GPSR of the density matrix in Fock space implies that N has a distribution. Developing constraints that restrict Eq. (1) to a pure state described by a wavefunction with N electrons will be of interest for molecular applications. This limitation is less severe for large systems where a distribution of N makes a minimal effect. or for open systems like molecules in junctions.
In Sec. II we present the TDHF equations of motion of the reduced density matrix. This sets the stage for the other methods. In Sec. III we present the GPSR of Corney and Drummond 20,21,22 and derive the Fokker-Planck equation. In Sec IV we derive an expression for the probability distribution of the number of electrons, and examine the conditions which ensure that a representation correspond to a state with a given particle number. In Sec. V we develop an approximate truncated hierarchy, which goes beyond TDHF, by assuming that the probability distribution P (n, t) in Eq. (1) has a Gaussian form. In Sec. VI we present stochastic equations of motion, which are equivalent to the Fokker-Planck equation, and compare them to the TDHF equation derived in Sec. II. Our results are summarized in Sec. VII.
II. THE TIME DEPENDENT HARTREE-FOCK APPROXIMATION
We consider a many-fermion system with two-body interactions, whose Hamiltonian is given bŷ
H = αβ t αβĉ † αĉ β + αβγδ V αβγδĉ † αĉ † βĉ γĉδ .(2)
The indices α, β, γ, δ denote an orthogonal one particle basis of spin orbitals. The creation and annihilation operators satisfy the Fermi anti-commutation ruleĉ †
αĉ β +ĉ βĉ † α = δ αβ .(3)
Without restricting the generality, the two-body interaction V αβγδ is taken to be anti-symmetric with respect to permutation of the indices α and β or γ and δ. Our goal is to derive an equation of motion for a reduced single particle density matrix
ρ αβ ≡ ĉ † αĉ β = Φ(t)|ĉ † αĉ β |Φ(t) ,(4)
where |Φ(t) is the many body wavefunction. We start from the Heisenberg equatioṅ
ρ ǫζ = i Φ(t)| Ĥ ,ĉ † ǫĉ ζ |Φ(t) ,(5)
where we work in units such that = 1.
The commutator in Eq. (5) is easily evaluated, leading tȯ
ρ ǫζ = −i γ (t ζγ ρ ǫγ − t γǫ ρ γζ ) − i αβγδ V αβγδ Φ(t)|ĉ † αĉ † βĉ δĉζ |Φ(t) δ ǫγ − Φ(t)|ĉ † αĉ † βĉ γĉζ |Φ(t) δ ǫδ + Φ(t)|ĉ † ǫĉ † βĉ γĉδ |Φ(t) δ ζα − Φ(t)|ĉ † ǫĉ † αĉγĉδ |Φ(t) δ ζβ .(6)
The TDHF approximation assumes that the state of the system is given by a single slater determinant |Φ(t) , at all times. Using Wick's theorem, we replace the two-body matrix elements with products of single particle matrix elements,
Φ(t)|ĉ † αĉ † βĉ γĉδ |Φ(t) = ρ αδ ρ βγ − ρ αγ ρ βδ .(7)
Substituting Eq. (7) into Eq. (6) results in the TDHF equations for the reduced density matrix,
ρ ǫζ = −i γ (t ζγ ρ ǫγ − t γǫ ρ γζ ) − i αβγ ρ αβ ρ γζ (V αγβǫ − V γαβǫ + V γαǫβ − V αγǫβ ) − i αβδ ρ αβ ρ ǫδ (V αζδβ − V αζβδ + V ζαβδ − V ζαδβ ) .(8)
Eq. (8) can be further simplified using the antisymmetry of V αβγδ to permutations of α and β, or γ and δ. This giveṡ
ρ ǫζ = −i γ (t ζγ ρ ǫγ − t γǫ ρ γζ ) − 4i αβγ ρ αβ ρ γζ V αγβǫ − 4i αβδ ρ αβ ρ ǫδ V αζδβ .(9)
Eq. (9) implies that ρ αβ can be viewed as classical oscillator coordinates, which follow a deterministic trajectory 13,19 . The TDHF equations will be generalized in Sec. V using the phase-space representation of a fermionic system, which retains the same number of variables, N 2 , as the TDHF but treats them as stochastic coordinates. We must then work with their distribution rather than with deterministic trajectories in that space.
III. GAUSSIAN PHASE-SPACE REPRESENTATION FOR FERMIONS
In this section we briefly present the main results of the GPSR, without proofs, following Refs. 21,22. We start by introducing the Gaussian operatorsΛ
(n) ≡ detñ : exp −ĉ † 2I −ñ −T ĉ :,(10)
whereñ = I − n is a square matrix of parameters whose size is given by the spin orbitals basis set, and : · · · : denotes normal ordering. A normal-ordered product of creation and annihilation operators is one where the creation operators are to the left of the annihilation operators. The sign must be changed when two operators are interchanged during reordering. (For instance, :ĉ αĉ † β := −ĉ † βĉ α . ) The operatorsΛ have the following useful properties
TrΛ = 1,(11)Tr Λĉ † αĉ β = n αβ ,(12)Tr Λĉ † αĉ † βĉ γĉδ = n αδ n βγ − n αγ n βδ ,(13)c † αĉβΛ = n αβΛ + γδñ αγ n δβ ∂Λ ∂n δγ ,(14)Λĉ † αĉ β = n αβΛ + γδ n αγñδβ ∂Λ ∂n δγ ,(15)
where Tr denotes the trace is in the many-body Hilbert space. Generally, the Gaussian operators also obey Wick's theorem,
Tr :â µ1 · · ·â µ2rΛ : = P (−1) P Tr :â ν1âν2Λ : × · · · × Tr :â ν2r−1âν2rΛ : ,
whereâ is a vector composed of all creation and annihilation operators, and ν j = µ P (j) . The sum runs over all the distinct pair permutations. It should be noted that the normal ordering convention used in Eq. (16) is that only the operators which do not belong toΛ are ordered so that all the creation operators are to its left, and the annihilation operators to its right. Eq. (13) is a special case of Eq. (16). Proofs, as well as several other similar relations, can be found in Ref. 21. Equations (14) and (15) allow to replace the action of creation and annihilation operators onΛ by derivatives with respect to the parameters n αβ . With the help of Eqs. (1) and (12) we see that the reduced single particle density matrix is given by the first moment of the distribution P (n),
ρ αβ = n αβ P (n, t)dn = n αβ P .(17)
We introduce the following notation, for a fermion operatorÂ
 ≡ dnP (n, t) Tr ÂΛ (n) ≡ TrÂΛ(n) P .(18)
Note that there is a double averaging here. The Tr takes care of the quantum average for a given set of parameters n. We then apply a classical average . . . P over the phase space distribution P (n). The reduced p-particle density matrix depends on the lowest p moments of P (n, t). We will use the expressions for expectation values of two and three body operators
M αβ,γδ ≡ ĉ † αĉ † βĉ γĉδ = n βγ n αδ − n αγ n βδ P ,(19)Y αβγ,δǫζ ≡ ĉ † αĉ † βĉ † γĉ δĉǫĉζ
= n αζ n βǫ n γδ − n αδ n βǫ n γζ + n αδ n βζ n γǫ − n αǫ n βζ n γδ + n αǫ n γζ n βδ − n αζ n βδ n γǫ P .
Before turning to derive the equation of motion for P (n, t), it is important to note that the representation Eq. (1) is not unique. More than one probability distribution P (n, t) may represent the same many-particle density matrix, due to the overcompleteness of the phase space representation. This is easily demonstrated by the fact that the many body density matrix of a system with M orbitals only depends on the lowest M 'th moments of P (n). The higher moments can thus be arbitrarily chosen. In the following we present the simplest derivation of the Fokker-Planck equation, which is naturally related to the TDHF. Alternative Fokker-Planck equations, whose solutions constitute different representations of the same many body density matrix, are discussed in App. A.
Substitution of Eq. (1) into the Liouville equation
∂ρ ∂t = −i Ĥ ,ρ ,(21)
gives
∂P (n, t) ∂tΛ (n)dn = −i P (n, t) Ĥ ,Λ(n) dn.(22)
The identities (14) and (15) can now be used to replace the commutator in Eq. (22) with a differential operator. To that end, we decompose the matrix V αβγδ in the form
V αβγδ = i 2 c Q αγ,c Q βδ,c .(23)
It should be always possible to find such a decomposition. This decomposition and the number of components c are not unique, and the latter can be chosen to be arbitrarily large. This decomposition eventually leads to a Fokker-Planck equation whose diffusion matrix is manifestly positive definite. For Hamiltonians with two-body interactions Eqs (14) and (15) this leads to a differential operator with second order derivatives
− i Ĥ ,Λ = αβ A αβ ∂Λ ∂n αβ + 1 2 αβγδ c B (1) αβ,c B (1) γδ,c ∂ 2Λ ∂n αβ ∂n γδ + 1 2 αβγδ c B (2) αβ,c B (2) γδ,c ∂ 2Λ ∂n αβ ∂n γδ .(24)
Closed expressions for A αβ and B (1,2) αβ , starting with the Hamiltonian Eq. (2), are obtained by a straightforward but tedious calculation. The decomposition (23) allows to write the terms with second order derivatives in the form of Eq. (24), with
B (1) ǫζ,c = αβ Q αβ,c n αζñǫβ (25) B (2) ǫζ,c = i αβ Q αβ,cñαζ n ǫβ ,(26)
whereñ was defined after Eq. (10). Furthermore, a lengthy calculation results in
A ǫζ = −i γ (t ζγ n ǫγ − t γǫ n γζ ) − i αβγ n αβ n γζ (V αγβǫ − V γαβǫ + V γαǫβ − V αγǫβ ) − i αβδ n αβ n ǫδ (V αζδβ − V αζβδ + V ζαβδ − V ζαδβ ) .(27)
By substituting (24) in Eq. (22), and integrating by parts, one obtains a Fokker-Planck-type equation for P (n, t). However, one should note that the coefficients A αβ and B
(1,2) αβ are complex valued rather than real, and it is not clear at this point that the term with the second order derivatives is positive definite. A representation with real parameters can be obtained by using the analyticity ofΛ(n), and treating the real and the imaginary parts of n αβ = n
(x) α,β + in (y) αβ
as independent variables. The identity ∂Λ/∂n αβ = ∂Λ/∂n
(x) αβ = −i∂Λ/∂n (y)
αβ can then be used to write
A αβ ∂Λ ∂n αβ = A (x) αβ ∂Λ ∂n (x) αβ + A (y) αβ ∂Λ ∂n (y) αβ .(28)
Here A (x) (A (y) ) denotes the real (imaginary) parts of A respectively. A similar relation holds for the terms involving second derivatives and for B (1,2) . After separating the real and imaginary parts of A and B, and performing an integration by parts, one obtains
∂P ∂t = − αβ ∂ ∂n (x) αβ A (x) αβ P + ∂ ∂n (y) αβ A (y) αβ P(29)+ 1 2 αβγδ c ∂ 2 ∂n (x) αβ ∂n (x) γδ B (1,x) αβ,c B (1,x) γδ,c P + ∂ 2 ∂n (y) αβ ∂n (y) γδ B (1,y) αβ,c B (1,y) γδ,c P + 2 ∂ 2 ∂n (x) αβ ∂n (y) γδ B (1,x) αβ,c B (1,y) γδ,c P + 1 2 αβγδ c ∂ 2 ∂n (x) αβ ∂n (x) γδ B (2,x) αβ,c B (2,x) γδ,c P + ∂ 2 ∂n (y) αβ ∂n (y) γδ B (2,y) αβ,c B (2,y) γδ,c P + 2 ∂ 2 ∂n (x) αβ ∂n (y) γδ B (2,x) αβ,c B (2,y)
γδ,c P .
Eq. (29) is a positive definite Fokker-Planck equation, and the probability distribution P (n, t) is guaranteed to remain non-negative at all times. In the derivation of Eq. (29) boundary terms in the integration by parts were neglected. When boundary terms are finite, the methods discussed in App. A can be used to derive equivalent Fokker-Planck equations with vanishing boundary terms.
IV. STATISTICAL PROPERTIES OF THE NUMBER OF PARTICLES IN THE GAUSSIAN PHASE-SPACE REPRESENTATION
The Hamiltonian (2) conserves the number of electrons. However the distribution function P (n) may represent a statistical mixture of states with different numbers of electrons. It is of interest to find conditions which ensure that P (n) represent a fixed number a system with N electrons. So far the GPSR was used to study systems where knowledge about the distribution of the number of electrons was not crucial. In this section we derive conditions which guarantee that P (n) represents a state with a given number of electrons. This will be needed for application to isolated molecules. Furthermore, for general states represented by P (n), we derive expressions for the probability to find k electrons in the system,P (k).
We start by calculating the distribution of values ofN = αĉ † αĉ α , the number operator,
f (N ) ≡ δ N −N .(30)
It will be convenient to compute the generating function
G(s) = dN e −isN f (N ) = e −isN .(31)
With the help of Eq. (14), it is easy to show that
NΛ = α n αα + αµνñ αµ n να ∂ ∂n νµ Λ .(32)
As a result
G(s) = Tr dnP (n) exp −is α n αα + αµνñ αµ n να ∂ ∂n νµ Λ .(33)
By using the Baker-Campbell-Hausdorff formula we then obtain
G(s) = exp −isA(n) − 1 2 s 2 B(n) P(34)
where A(n) = tr(n) and B(n) = tr(n) − tr(n 2 ), where tr denotes the trace in the single particle Hilbert space. (Recall that a trace in the many body space is denoted by Tr.) Assuming that the integrations over n converge, we obtain the probability distribution (30) by the inverse Fourier transform of Eq. (31)
f (N ) = 1 √ 2π 1 B(n) exp − (N − A(n)) 2 2B(n) P .(35)
The distribution P (n) will represent a state with a given number of electrons, N 0 , when f (N ) = δ(N − N 0 ). This requires that P (n) > 0 only for points n which satisfy tr(n) = tr(n 2 ) = N 0 .
Distributions P (n) which do not satisfy this conditions represent statistical mixtures of different number of electrons. To calculate the probability to find k electrons,P (k), we note that the Gaussian operators have a block structure in Fock space, with coherences only between states with the same number of particles. For instance, the Gaussian operator of a system with two orbitals is 21 Λ(n) = detñ |00 00| + (n 11ñ22 + n 12 n 21 ) |10 10| + (ñ 11 n 22 + n 12 n 21 ) |01 01| + det n |11 11| (37) + n 21 |10 01| + n 12 |01 10| .
The probability to find the system with any number of particles can be found by taking the coefficients of the corresponding populations and averaging over P (n). This can be done by defining
C l ≡ Trρ α1,··· ,α lĉ † α1 · · ·ĉ † α lĉ α l · · ·ĉ α1 ,(38)
for l = 1, 2, · · · , M , where M denotes the basis size (i.e. the number of orbitals). Only populations of states with at least l electrons contribute to C l . A direct calculation gives
C l = M k=l k! (k − l)!P (k).(39)
This relation can be inverted, leading toP
(k) = 1 k! M l=k (−1) l−k 1 (l − k)! C l ,(40)
which is valid for k = 1, 2, · · · , M . Wick's theorem (16) can be used to write
C l = l! (α1,··· ,α l ) det n [α 1 , · · · , α l ] P ,(41)
where sum is over all ordered sets of indices (α 1 , · · · , α l ), while n [α 1 , · · · , α l ] is the minor of n obtained by deleting all rows and columns except the ones whose indices are α 1 , · · · , α l . Equations (40) and (41) combine to give the probability to find k electrons in the system
P (k) = M l=k (−1) l−k l k (α1,··· ,α l ) det n [α 1 , · · · , α l ] P ,(42)
which is valid for k = 1, 2, · · · , M . Equation (42) is especially simple for the probability that all the orbitals are filled
P (M ) = det n P .(43)
We obtain the probability that there are no electrons (all orbitals are unoccupied) by using the normalization conditioñ
P (0) = 1 − M k=1P (k) = 1 + M l=1 (−1) l (α1,··· ,α l ) det n [α 1 , · · · , α l ] P = detñ P .(44)
In this section we have investigated the correspondence between the phase space distribution P (n) and the number of particles in the system. This is one aspect of a broader problem, namely, how to constrain P (n) and its phase space evolution, so that it will correspond to a density matrix which exhibits a certain physical property. For instance, what are the conditions on P (n) so that it corresponds to a pure state, or to a single slater determinant? The identification of such constraints is an open question.
V. TRUNCATING THE HIERARCHY FOR ONE AND TWO-BODY DENSITY MATRICES
In this section we use the exact GPSR of the many-body dynamics to develop new approximation schemes which provide a natural extension of the TDHF approximation.
The time evolution of expectation values of any operator follows the Heisenberg equation
d dt TrρÔ = i Trρ Ĥ ,Ô .(45)
For Hamiltonians of the form (2), substitution of normally ordered products of equal numbers of creation and annihilation operators in Eq. (45) results in the usual many body hierarchy of equations of motion. The N 'th equation in the hierarchy gives the time derivative of an N -particle reduced density matrix in terms the reduced density matrices of up to N + 1 particles. The first two members in this hierarchy are given by
d dt ρ ǫζ = i αβ t αβ (ρ αζ δ βǫ − ρ ǫβ δ ζα ) + αβγδ V αβγδ (M αβ,+ i αβγδ V αβγδ [Y αβǫ,γµν δ δζ − Y αβǫ,δµν δ γζ + Y αβζ,δµν δ γǫ − Y αβζ,γµν δ δǫ + Y ǫζα,νγδ δ µβ − Y ǫζα,µγδ δ νβ + Y ǫζβ,µγδ δ να − Y ǫζβ,νγδ δ µα + M αβ,µν (δ δǫ δ γζ − δ γǫ δ δζ ) + M ǫζ,γδ (δ µα δ νβ − δ να δ µβ )] ,(47)
where ρ, M and Y were defined in Eqs. (17), (19) and (20). The hierarchy for the reduced single-particle and two-particle density matrices will be closed by assuming that the probability distribution P (n) is Gaussian. Such an approximation is similar in spirit to the Hartree-Fock approximation, since it makes an ansatz on the time dependent state of the system and it includes a single Slater determinant as a special case. However, it is more general, since it takes some two body correlations into account. Due to the lack of a variational principle, it is not guaranteed that the Gaussian approximation in phase space improves upon the Hartree-Fock approximation. However, it is likely to do so.
When P (n) has a Gaussian form, averages which are cubic in n αβ , such as the ones appearing in Eq. (20), can be expressed in terms of the first and second moments of the distribution. For instance n αζ n βǫ n γδ P = n αζ P n βǫ n γδ P + n βǫ P n αζ n γδ P + n γδ P n αζ n βǫ P − 2 n αζ P n βǫ P n γδ P .
(48)
Substitution of (48) in Eq. (20), with the help of Eqs. (19) and (17), leads to
Y αβγ,δǫζ = ρ βǫ M αγ,δζ + ρ αδ M βγ,ǫζ + ρ βζ M αγ,ǫδ + ρ γδ M αβ,ǫζ + ρ αǫ M βγ,ζδ + ρ γζ M αβ,δǫ (49) + ρ βδ M αγ,ζǫ + ρ αζ M βγ,δǫ + ρ γǫ M αβ,ζδ − 2 {ρ αζ ρ βǫ ρ γδ − ρ αδ ρ βǫ ρ γζ + ρ αδ ρ βζ ρ γǫ − ρ αǫ ρ βζ ρ γδ + ρ αǫ ρ γζ ρ βδ − ρ αζ ρ βδ ρ γǫ } .
Equation (49) represent one out of many possible decoupling schemes which can be used to truncate the hierarchy of N -body reduced density matrices. It was derived by making an assumption on the form of P (n). Since it is based on an approximate ansatz on the many body density matrix it has an important advantage over other decoupling schemes: it ensures that the approximate density matrix is a physically allowed one, and therefore guarantees that the decoupling (49) will not result in unphysical expectation values of operators.
We can thus truncate the hierarchy at the ρ and M level. Before doing that, we define the deviation of the two body correlation functions from its TDHF values ∆M αβ,γδ ≡ M αβ,γδ − ρ αδ ρ βγ + ρ αγ ρ βδ .
(50)
By substitution of Eqs. (17), (19) and (49) in Eqs. (46) and (47) we obtaiṅ
ρ ǫζ = −i γ (t ζγ ρ ǫγ − t γǫ ρ γζ )−4i αβγ V αγβǫ ρ αβ ρ γζ −4i αβδ V αζδβ ρ αβ ρ ǫδ −2i αβδ V αβǫδ ∆M αβ,δζ −2i βγδ V ζβγδ ∆M ǫβ,γδ ,(51)+ ρ αν ρ βµ − ρ αµ ρ βν )] − 2i βγδ V µβγδ [2ρ ζδ ∆M ǫβ,γν + 2ρ ǫδ ∆M βζ,γν + 2ρ βγ ∆M ǫζδν + ρ βν (∆M ǫζ,γδ + ρ ǫδ ρ ζγ − ρ ǫγ ρ ζδ )] − 2i αγδ V ανγδ [2ρ ζγ ∆M ǫα,µδ + 2ρ ǫγ ∆M αζ,µδ + 2ρ αδ ∆M ǫζ,µγ + ρ αµ (∆M ǫζ,γδ + ρ ǫδ ρ ζγ − ρ ǫγ ρ ζδ )] .
Equations (51) and (52) constitute an approximate closed set of equations for the one particle and two particle reduced density matrices which extend TDHF equations to include two-body correlations. The TDHF equation (9) can be recovered from Eq. (51) by setting ∆M αβ,γδ = 0. The exact solution of the Fokker-Planck equation (29) does not necessarily satisfy our Gaussian ansatz for P (n), which is why Eqs. (51) and (52) are approximate. One consequence is that they do not conserve the number of particles. This can be seen by calculating the time derivatives of the moments of the number operatorN , as evaluated using Eqs. (51) and (52). We find that the first two moments are indeed conserved
d dt N = d dt N 2 = 0.(53)
However, this is not the case for the third moment
d dt N 3 = 0.(54)
We conclude that the probability distribution f (N ), derived from Eqs. (51) and (52), is not conserved. It should be emphasized that the approximations leading to the TDHF equations and to the hierarchy derived in this section are of a different nature. The TDHF equations are obtained under the assumption that the state of the system is a single Slater determinant at all times. In contrast, here we assumed that the system can be represented by a Gaussian operator expansion whose probability distribution is Gaussian at all times. This provides a generalization of the TDHF method, suggesting that Eqs. (51) and (52) should lead to more accurate results than the TDHF equations. This approximation scheme should be useful for large systems with many particles, which are likely to be less sensitive to changes in the distribution of number of particles.
VI. UNRAVELLING THE FOKKER-PLANCK EQUATION IN TERMS OF ITO STOCHASTIC TRAJECTORIES
A probability density evolving according to a Fokker-Planck equation can be computed numerically by simulating an ensemble of trajectories following a stochastic equation of motion 29 . This is commonly used to trade off computer memory by time. This approach was used to study the ground states of Hubbard-type models 20,22,25,26,27 . Here we present the form of the stochastic equations of motion that naturally connect with the TDHF equations.
Stochastic equations of motion which are equivalent to the Fokker-Planck equation (29) are given by
dn αβ = A αβ dt + c B (1) αβ,c dW (1) c + B (2) αβ,c dW (2) c ,(55)
where we have combined the equations for the real and imaginary parts of n αβ into a single complex equation.
The noise Wiener increments are real, with Gaussian probability distribution, satisfying dW (55) should be integrated using Ito stochastic calculus 30 . The many-body dynamics can be simulated by repeatedly integrating Eq. (55) to create an ensemble of stochastic trajectories. This ensemble, together with Eq. (17), or similar identities, can be used to calculate expectation values.
(i) c (t)dW (j) c ′ (t ′ ) = δ cc ′ δ ij δ (t − t ′ ) dt. Equation
The non-random, or drift, component of the stochastic equation of motion (55), given by (27), coincides with the right hand side of Eq. (8), as long as we identify n αβ with ρ αβ . Such identification can naturally be made when the distribution P (n) is narrow, or when the noise part is absent (for instance when there are no two-body interactions).
It is possible to derive an equivalent form of the stochastic differential equations by using a different stochastic calculus 30 . For instance, the Stratonovich calculus results in a similar stochastic equation with A (strat) αβ = A αβ . This breaks the simple correspondence with TDHF. For completeness, we give an explicit form of the drift terms in the Stratonovich scheme in App. B.
The coefficients A αβ , and B (1,2) αβ,c , appearing in Eq.(55), depend quadratically on n. As a result, the nonlinear stochastic trajectories generated from Eq.(55) may escape to infinity. This is expected to happen in the vicinity of certain phase space regions. This problem may be solved in various ways. When calculating the ground state using an imaginary time equation, one can force the stochastic equations of motion to be real 20,22,25 . This limits the dynamics to a small subset of phase space, which is not likely include the most unstable regions. Alternatively, some combination of the gauge freedoms, described in App. A, can be used to obtain an equivalent, more stable, form of the stochastic equations. A simple systematic way of obtaining stable stochastic equations for the time evolution of general fermionic models is yet to be developed.
VII. DISCUSSION
In this paper we have used a Gaussian phase-space representation to generalize a common Hilbert space approximation for many-body dynamics, the TDHF, which assumes that the wavefunction of the system is given by a single Slater determinant at all times. Here the many body density matrix is represented by an ensemble of Gaussian operators whose distribution of parameters satisfies a Fokker-Planck equation. We have shown that the drift terms of this Fokker-Planck equation have similar form to the TDHF equations [compare Eqs. (27) and (8)]. This suggests that the phase-space representation can serve as a good basis for new approximation schemes, which systematically go beyond TDHF.
The Fokker Planck equation (29) maps the original fermion system rigorously into a equivalent set of classical oscillators n αβ with stochastic dynamics. The number of oscillators is the same as the number of elements of the reduced single electron density matrix and it thus scales as ∼ N 2 with the basis set size. The TDHF is a mean field approximation to this stochastic dynamics whereby the same classical coordinates satisfy a deterministic equation. Generalizations of the type given here were conjectured in ref. 19 but the GPSR puts it on a firm theoretical basis.
Linear and nonlinear response functions e.g. optical spectra have a very different formal structure for quantum and classical systems. To calculate the response function to a classical field E(t) one needs to add a coupling term E(t)A to the Hamiltonian where A is a dynamical variable. nth order quantum response functions are then given by n nested commutations such as [[A(τ 1 ), A(τ 2 )] , A(τ 3 )] which gives 2 n terms (Liouville space pathways). Classical response functions have a very different form and require to run groups of several nearby trajectories and carefully monitor how they diverge. The responce functions can be expressed in terms of the stability matrices of the stochastic trajectories of the free system (without external driving) 31,32 , or by solving the distribution of the driven system and combining terms with different orders of interactions (finite field techniques) 33 . An intriguing aspect of the present approach is that the quantum response functions are recast rigorously in a classical form. This could result in new insights and could suggest new numerical simulation techniques for fermions.
Studying the merits of this new representation will be an interesting future direction. Moreover, the quantum response function is one specific combination of the Liouville space pathways. Other combinations represent spontaneous fluctuations and how they are affected by external driving 34,35 . Many attempts have been made to connect nonlinear response and fluctuations through generalized fluctuation-dissipation relations 36 . The GPSR may shed a new light into this issue and provide a classical picture for the connection between quantum response and fluctuations.
We have constructed an approximate, simple, numerical scheme by further assuming that the probability distribution P (n) has a Gaussian form at all times. The hierarchy of equations for reduced many-body density matrices can be truncated at the single and two particle level. This yields a coupled set of equations, (51) and (52), which can be solved to calculate the time dependence of expectation values of all the one and two-body operators.
The exact time evolution of the phase-space distribution function, which represents the many body density matrix, conserves the distribution of the number of electrons. This is not true for the approximate dynamics of Eqs. (51) and (52). While the mean number of electrons, and its variance, do not vary with time, higher moments are not conserved. This approximation scheme thus suffers from the old N representability problem 12 common to other reduced density matrix approaches. Nevertheless, this scheme may be a useful description, most likely for systems with many electrons, where expectation values are less sensitive to small variations in the distribution of the number of particles.
The phase-space representation allows the development of new approximation schemes, by making other assumptions on the form of P (n). It may be possible to include constraints that ensure that approximations on P (n) conserve the distribution of number of particles, while still allowing to truncate the hierarchy. More work is needed in order to gain understanding on the way in which various assumptions on P (n) manifest themeselves on the many body dynamics.
Acknowledgments
We gratefully acknowledge the support of the National Science Foundation through grant No. CHE-0745892.
APPENDIX A: GAUGE FREEDOM IN THE FOKKER-PLANCK EQUATION
As is the case for Bosonic coherent states 37 , the Gaussian operator basis for fermions is overcomplete. This means that there exist an infinite number of equivalent forms of the time evolution equations for P (n) which give the same expectation values of all physical observables. Below we give a brief description of the various ways of obtaining such equivalent representations, following Ref. 22, and examine whether they retain the correspondence between the drift terms and the TDHF equations.
A way of changing the Fokker-Planck equation, without affecting the physical observables, is to add formally vanishing operators, such asĉ † αĉ † αĉ βĉγ , to the Hamiltonian, and then use Eqs (14) and (15) to replace these terms by a differential operator. This method, which is only applicable for systems of identical fermions, was termed Fermi gauge in Refs. 20,22. A direct calculation shows that such Fermi gauges do not change the form of the drift terms A αβ in Eq. (29), but do affect the noise terms. Fermi gauges were used to obtain imaginary-time equations for P where all the parameters are real rather than complex valued.
A different method of obtaining equivalent forms of the equations of motion is termed drift gauge 22,38 . Here one adds a new parameter, which corresponds to a weight given to each stochastic trajectory, thus replacing the (uniform) average over trajectories by a weighted average. As a result, a new distribution function is defined in a larger parameter space, which includes the weight as a variable, and a Fokker-Planck equation is derived for this new distribution functions. At the same time, the evaluation of averages using the distribution P is replaced by weighted averages involving the new weight parameter. It is then possible to modify the drift terms A αβ , while at the same time adding suitable noise terms to part of the Fokker-Planck equation related to the new weight parameter, in such a way that will not change the values of weighted averages 22,25 . Drift gauges have been used to change the drift terms in order to avoid trajectories which run to infinity.
An equivalent, but different form of the stochastic equations of motion can be obtained by using different forms for the decomposition of Q in Eq. (23). This will not affect Eq.(29), but will change its stochastic trajectory simulation, for instance using Eq. (55). However, when combined with drift gauges the choice of solution of Eq. (23) may result in a different Fokker-Planck equation in the generalized phase space. The freedom to choose different solutions of Eq. (23) was termed diffusion gauge in Refs. 25,38. The gauge freedoms may be useful for practical implementation of the GSPR. If a solution of Eq. (29) leads to a probability distribution with long tails, and therefore for the appearance of boundary terms, due to the integration by parts, the gauge freedom can be used to generate an equivalent representation whose probability distribution is more localized. One can hope to use the many ways to generate equivalent representations, by combining the various gauges, to find a representation which does not suffer from boundary terms.
APPENDIX B: THE STRATONOVICH STOCHASTIC EQUATIONS
The Stratonovich calculus is an alternative to the Ito calculus used in eq. (55). In this scheme the terms in the stochastic differential equation are evaluated at the midpoint of the interval, as opposed to the initial point used in the Ito scheme.
In practice, one solves the finite differences equation
αβ,c (n mid )dW (2) c ,
with n mid ≡ (n (i+1) + n (i) )/2, in order to get n (i+1) . This implicit equation can be solved by iteration. It is of interest to note that implicit methods tend to show better numerical stability then explicit ones, see Ref. 39 for a review. This stochastic integration should lead to the Fokker-Planck equation (29). However, this means that the coefficients A (strat) can not be equal to the ones appearing in Eq. (27). This is due to correlations in the noise terms, which lead to the second order derivatives with the form 1 2 ∂ ∂n B ∂ ∂n BP , rather than 1 2 ∂ ∂n ∂ ∂n BBP which would appeared when using the Ito scheme. (We suppressed subscripts for brevity.)
To compensate for these correlations, the drift terms of the Ito and Stratonovich schemes are related by
We have just shown that the Stratonovich form of the drift term differ from the one obtained using the Ito scheme, and therefore, differs from the terms appearing in the TDHF equations of motion. Nevertheless, by construction, both stochastic schemes give the same ensemble of trajectories.
γζ δ δǫ − M αβ,δζ δ γǫ + M ǫα,γδ δ ζβ − M ǫβ,γδ δ ζα ) M ǫζ,µν = i αβ t αβ [M αζ,µν δ βǫ − M αǫ,µν δ βζ + M ǫζ,νβ δ µα − M ǫζ,µβ δ να ]
βγ n αγ − t γα n γβ ) − i γδ (V βδδγ n αγ − V γδδα n γβ ) + i γδǫ n γδ n αǫ (V βγǫδ + V γβδǫ ) − i γδǫ n γδ n ǫβ (V ǫγαδ + V γǫδα ) .
and ∆M ǫζ,µν = i γ [t γǫ ∆M γζ,µν + t γζ ∆M ǫγ,µν − t µγ ∆M ǫζ,γν − t νγ ∆M ǫζ,µγ ] (52) αβǫζ [ρ αν ρ βµ − ρ αµ ρ βν + ∆M αβ,µν ] + 2i γδ V µνγδ [ρ ǫδ ρ ζγ − ρ ǫγ ρ ζδ + ∆M ǫζ,γδ ] + 2i αβγ V αβγζ [2ρ αµ ∆M ǫβ,γν + 2ρ βγ ∆M ǫα,µν + 2ρ αν ∆M ǫβ,µγ + ρ ǫγ (∆M αβ,µν + ρ αν ρ βµ − ρ αµ ρ βν )] αβǫδ [2ρ βµ ∆M αζ,δν + 2ρ αδ ∆M βζ,µν + 2ρ βν ∆M αζ,µδ + ρ ζδ (∆M αβ,µν− 2i
αβ
V + 2i
αβδ
V
The Nuclear Many-Body Problem. P Ring, P Schuck, SpringerNew YorkP. Ring and P. Schuck, The Nuclear Many-Body Problem (Springer, New York, 1980).
J W Negele, H Orland, Quantum Many-Particle Systems. BoulderWestview pressJ. W. Negele and H. Orland, Quantum Many-Particle Systems, (Westview press, Boulder, 1998).
J Rammer, Quantum Field Theory of Non-equilibrium States. CambridgeCambridge University PressJ. Rammer, Quantum Field Theory of Non-equilibrium States, (Cambridge University Press, Cambridge, 2007)
J.-P Blaizot, G Ripka, Quantum Theory of Finite Systems. Cambridge MassachusettsMIT PressBlaizot, J.-P. and Ripka, G., Quantum Theory of Finite Systems (MIT Press, Cambridge Massachusetts, 1986).
. R J Bartlett, M , Rev. Mod. Phys. 79291R. J. Bartlett and M. Musial, Rev. Mod. Phys., 79, 291 (2007).
M E Casida, Recent Advances in Density-Functional Methods. D. A. ChongSingaporeWorld Scientific3M. E. Casida, in Recent Advances in Density-Functional Methods, Vol. 3 of Part I, edited by D. A. Chong, World Scientific, Singapore, 1995.
. G F Bertsch, J I Iwata, A Rubio, K Yabana, Phys. Rev. B. 627998G. F. Bertsch, J. I. Iwata, A. Rubio, and K. Yabana, Phys. Rev. B, 62, 7998 (2000).
. Y Takimoto, F D Vila, J J Rehr, J Chem. Phys. 127154114Y. Takimoto, F. D. Vila, and J. J. Rehr, J Chem. Phys., 127, 154114 (2007).
M A L Marques, C A Ullrich, F Nogueira, A Rubio, K Burke, E K U Gross, Time-Dependent Density Functional Theory. BerlinSpringerM. A. L. Marques, C. A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E. K. U. Gross, Eds., Time-Dependent Density Functional Theory, (Springer, Berlin, 2006).
. G Onida, L Reining, A Rubio, Rev. Mod. Phys. 74601G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys., 74, 601 (2002)
Reduced-Density-Matrix Mechanics. D A Mazziotti, Ed , John Wiley & SonsHobokenD. A. Mazziotti, Ed., Reduced-Density-Matrix Mechanics (John Wiley & Sons, Hoboken 2007).
A J Coleman, V I Yukalov, Reduced Density Matrices: Coulson's Challenge. BerlinSpringer-VerlagA.J. Coleman, V.I. Yukalov, Reduced Density Matrices: Coulson's Challenge, (Springer-Verlag, Berlin 2000).
. V Chernyak, S Mukamel, J. Chem. Phys. 104444V. Chernyak and S. Mukamel, J. Chem. Phys., 104, 444 (1996).
The Quantum Mechanics of Many-Body Systems. D J Thouless, Academic PressNew YorkThouless, D. J. The Quantum Mechanics of Many-Body Systems, (Academic Press, New York, 1972).
. S Tretiak, S Mukamel, Chem. Rev. 102S. Tretiak and S. Mukamel, Chem. Rev., 102, 3171-3212 (2002).
. S Mukamel, S Tretiak, T Wagersreiter, V Chernyak, Science. 277S. Mukamel, S. Tretiak, T. Wagersreiter, and V. Chernyak, Science, 277, 781-787 (1997).
. O Berman, S Mukamel, Phys. Rev. A. 6742503O. Berman and S. Mukamel, Phys. Rev. A, 67, 042503 (2003).
. S Tretiak, V Chernyak, J. Chem. Phys. 1198809S. Tretiak and V. Chernyak, J. Chem. Phys., 119, 8809 (2003).
. S Tretiak, V Chernyak, S Mukamel, Int. J. Quant. Chem. 70711S. Tretiak, V. Chernyak, and S. Mukamel, Int. J. Quant. Chem., 70, 711 (1998).
. J F Corney, P D Drummond, Phys. Rev. Lett. 93260401J. F. Corney and P. D. Drummond, Phys. Rev. Lett., 93, 260401 (2004).
. J F Corney, P D Drummond, J. Phys. A: Math. Gen. 39269J. F. Corney and P. D. Drummond, J. Phys. A: Math. Gen., 39, 269 (2006).
. J F Corney, P D Drummond, Phys. rev. B. 73125112J. F. Corney and P. D. Drummond, Phys. rev. B, 73, 125112 (2006).
. K E Cahill, R J Glauber, Phys. Rev. A. 591538K. E. Cahill and R. J. Glauber, Phys. Rev. A, 59, 1538 (1999).
. V Chernyak, S Choi, S Mukamel, Phys. Rev. A. 6753604V. Chernyak, S. Choi, and S. Mukamel, Phys. Rev. A, 67, 053604 (2003).
. F F Assaad, P Werner, P Corboz, E Gull, M Troyer, Phys. Rev. B. 72224518F. F. Assaad, P. werner, P. Corboz, E. Gull, and M. Troyer, Phys. Rev. B, 72, 224518 (2005).
. T Aimi, M Imada, J. Phys. Soc. Japan. 7684709T. Aimi and M. Imada, J. Phys. Soc. Japan, 76, 084709 (2007).
. P Corboz, M Troyer, A Kleine, I P Mcculloch, U Schollw'ock, F F Assaad, Phys. Rev. B. 85108P. Corboz, M. Troyer, A. Kleine, I. P. McCulloch, U. Schollw'ock, and F. F. Assaad, Phys. Rev. B, 085108 (2008).
. S M A Rombouts, Phys. Rev. Lett. 96188901S. M. A. Rombouts, Phys. Rev. Lett., 96, 188901 (2006);
. J F Corney, P D Drummond, Phys. Rev. Lett. 96188902J. F. Corney and P. D. Drummond, Phys. Rev. Lett., 96, 188902 (2006).
H-P Breuer, F Pettruccione, The theory of open quantum systems. LondonOxford University PressH-P. Breuer and F. Pettruccione, The theory of open quantum systems (Oxford University Press, London, 2002).
. C W Gardiner, Handbook of Stochastic Methods. Springer-VerlagC. W. Gardiner, Handbook of Stochastic Methods, (Springer-Verlag, Berlin, 1985).
. S Mukamel, V Khidekel, V Chernyak, Phys. Rev. E. 531S. Mukamel, V. Khidekel, and V. Chernyak, Phys. Rev. E, 53, R1 (1996).
. C Dellago, S Mukamel, Phys. Rev. E. 6735205C. Dellago and S. Mukamel, Phys. Rev. E, 67, 035205 (2003).
. S Mukamel, J B Maddox, J. Chem. Phys. 12136S. Mukamel and J. B. Maddox, J. Chem. Phys., 121, 36 (2004).
. U Harbola, S Mukamel, Phys. Rev. A. 7052506U. Harbola, and S. Mukamel, Phys. Rev. A, 70, 052506 (2004).
. A E Cohen, S Mukamel, Phys. Rev. Lett. 91233202A. E. Cohen and S. Mukamel, Phys. Rev. Lett, 91, 233202 (2003).
. E Wang, U Heinz, Phys. Rev. D. 6625008E. Wang and U. Heinz, Phys. Rev. D, 66, 025008 (2002).
Quamtum Theory of Optical Coherence. R J Glauber, Wiley-VCHWeinheimR. J. Glauber, Quamtum Theory of Optical Coherence, (Wiley-VCH, Weinheim 2007).
. P Deuar, P D Drummond, Phys. Rev. A. 6633812P. Deuar and P. D. Drummond, Phys. Rev. A, 66, 033812 (2002).
. P D Drummond, I K Mortimer, J. Comp. Phys. 93144P. D. Drummond and I. K. Mortimer, J. Comp. Phys., 93, 144 (1991).
| [] |
[
"Ring-theoretical methods in topology especially for compactifications",
"Ring-theoretical methods in topology especially for compactifications"
] | [
"Abolfazl Tarizadeh ",
"Mohammad Reza Rezaee "
] | [] | [] | In this paper, using commutative algebra, new advances on the compactifications of topological spaces, especially on the Stone-Čech and Alexandroff compactifications have been made. Especially among them, it is proved that the minimal spectrum of the direct product of a family of integral domains indexed by a set X is the Stone-Čech compactification of the discrete space X. Dually, it is proved that the maximal spectrum of the direct product of a family of local rings indexed by X is also the Stone-Čech compactification of the discrete space X. The Alexandroff (one-point) compactification of a discrete space is constructed by a new method. Finally, we proceed to give a natural and quite simple way to construct ultra-rings. Then this new approach is used to obtain several new results on the Stone-Čech compactification. | null | [
"https://arxiv.org/pdf/1910.09884v5.pdf"
] | 247,518,539 | 1910.09884 | 48d5d634cf99e3057faa161cf40293003a122d36 |
Ring-theoretical methods in topology especially for compactifications
17 Mar 2022
Abolfazl Tarizadeh
Mohammad Reza Rezaee
Ring-theoretical methods in topology especially for compactifications
17 Mar 2022
In this paper, using commutative algebra, new advances on the compactifications of topological spaces, especially on the Stone-Čech and Alexandroff compactifications have been made. Especially among them, it is proved that the minimal spectrum of the direct product of a family of integral domains indexed by a set X is the Stone-Čech compactification of the discrete space X. Dually, it is proved that the maximal spectrum of the direct product of a family of local rings indexed by X is also the Stone-Čech compactification of the discrete space X. The Alexandroff (one-point) compactification of a discrete space is constructed by a new method. Finally, we proceed to give a natural and quite simple way to construct ultra-rings. Then this new approach is used to obtain several new results on the Stone-Čech compactification.
Introduction
Compactification is one of the main topics which is investigated in this paper from new points of view. Amongst various compactifications, the Stone-Čech compactification of a discrete space X is particularly important. One of the main reasons of its importance is that it admits a semigroup structure whenever X is a semigroup, and this semigroup structure has vast and interesting applications in diverse fields of mathematics specially in combinatorial number theory, Ramsey theory, topological dynamics and Ergodic theory. An accessible concrete description of this compactification often remains elusive. For instance the semigroup βN, the Stone-Čech compactification of the natural numbers, is amazingly complicated and there are some unanswered questions about its semigroup structure. For example, whether or not βN contains any elements of finite order which are not idempotent still remains a challenging open problem. See [12] and [27] and their rich bibliography for further studies. Perhaps as another main reason for the importance of the Stone-Čech compactification of a discrete space is its vital role in proving Theorem 4.4 which asserts that every topological space admits the Stone-Čech compactification.
Classically, the Stone-Čech compactification of a discrete space is usually constructed via the ultrafilters of that space. In this paper, we find two new and interesting ways to construct this compactification using only the standard methods of commutative algebra. In fact in Theorem 3.5, we prove that the minimal spectrum of the direct product of a family of integral domains indexed by a set X is the Stone-Čech compactification of the discrete space X. In Theorem 5.4, it is shown that the maximal spectrum of the direct product of a family of local rings indexed by X is the Stone-Čech compactification of the discrete space X. These results improve all of the former constructions of the Stone-Čech compactification of a discrete space, and also show that this compactification is independent of choosing of integral domains and local rings. In particular, we get that βX = Spec P(X). The classical construction is also recovered (see Remark 3.10). Throughout this paper, βX denotes the Stone-Čech compactification of the discrete space X. These results allow us to understand the number of prime ideals of the infinite direct products of integral domains and local rings more precisely. As another application, the Stone-Čech compactification of an arbitrary topological space X is deduced from the Stone-Čech compactification of the discrete space X by passing to a certain quotient (see Theorem 4.4). It is worth mentioning that our results generalize several related results in the literature (see e.g. [2]).
In §6, using ultra-rings and Theorems 3.5 and 5.4, then we obtain powerful results on the Stone-Čech compactification, see Theorems 6.2 and 6.3.
We introduce a new way to build the Alexandroff (one-point) compactification of a discrete space, see Corollary 7.3. This result tells us that for any set X, then αX = Spec(R).
Here αX denotes the Alexandroff compactification of the discrete space X and R is a certain subring of P(X). Then in Theorem 8.2, we show that every totally disconnected compactification of a discrete space X is precisely of the form Spec(R ′ ) where the ring R ′ satisfies in the extensions of rings R ⊆ R ′ ⊆ P(X). After establishing this result, we were informed that it is also proved in [19, Theorems 2.2 and 2.3] by another approach using Boolean algebras. Therefore all of the totally disconnected compactifications of a discrete space are in the scope of the Zariski topology. In particular this class, up to isomorphisms, forms a set and the extensions of the corresponding rings put a partial order over this set in a way that the Alexandroff compactification is the minimal one and the Stone-Čech compactification is the maximal one.
It is well known that if the discrete space X is also a (commutative) semigroup then its operation can be extended uniquely to an operation on βX which forms a semigroup structure as well, see [12,Theorems 4.1 and 4.4]. This result opens new horizons to explore the basic and also sophisticated properties of the semigroup βX. Although some of them have been done in the literature over the years (see [12] and its bibliography), there is a pressing need for new constructions to aid the development and the understanding the algebraic structure of this semigroup specially βN more deeply. We have made very little contributions to this subject but the results are general and including Theorems 9.1, 9.2 and 9.3. Indeed, in Theorem 9.1, we reformulate this important result into a more standard form and then it is proven by a new approach. Then in Theorems 9.2 and 9.3, various aspects of the semigroup βX are investigated, specially it is shown that this semigroup structure is actually functorial. Finally, in Section 10, the absolutely flatness of the total ring of fractions is investigated.
Preliminaries
In this paper, all rings are commutative. If ϕ : R → R ′ is a morphism of rings then the induced map Spec(R ′ ) → Spec(R) given by p ϕ −1 (p) is denoted by Spec(ϕ), (sometimes it is also denoted by ϕ * ). Let X be a set and I an ideal of the power set ring P(X). If
A, B ∈ I then A ∪ B = A + B + A ∩ B ∈ I. Also if A ∈ I and B ⊆ A, then B ∈ I.
For the definition of power set ring see e.g. [24, §2]. By Fin(X) we mean the set of all finite subsets of X, it is an ideal of P(X). For the definition of Clop(X) see [24, §3]. By a compact space we mean a quasi-compact and Hausdorff topological space.
Definition 2.1. By a compactification of a topological space X we mean a compact space X together with a continuous open embedding (an injective open map) η : X → X such that η(X) is a dense subspace of X. If moreover X \ η(X) consisting only a single point then X is called the one-point or the Alexandroff compactification of X and it is often denoted by αX, and this single point is called the point at infinity.
Let R be a ring. The set of minimal primes of R is denoted by Min(R) and the set of maximal ideals of R is denoted by Max(R). Note that Min(R) is not necessarily Zariski quasi-compact. The Jacobson radical of R is denoted by J.
Definition 2.2. The Stone-Čech compactification of a topological space X is the pair (βX, η) where βX is a compact space and η : X → βX is a continuous map such that the following universal property holds. For each such pair (Y, ϕ), i.e. Y is a compact space and ϕ : X → Y is a continuous map, then there exists a unique continuous map ϕ : βX → Y such that ϕ = ϕ • η.
It is important to notice that the Stone-Čech compactification need not to be a compactification in the sense of Definition 2.1.
All of the remaining undefined notions such as flat topology, retraction, mp-ring and etc can be found in [1], [20], [22], [23], [25] and [26].
Minimal spectrum as Stone-Čech compactification
The main result of this section (see Theorem 3.5) asserts that the minimal spectrum of the direct product of a family of integral domains indexed by a set X is the Stone-Čech compactification of the discrete space X.
We start with the following result which generalizes [7, p. 460].
Proposition 3.1. Consider the canonical ring map π : R → S −1 R where S is a multiplicative subset of a ring R, and let f ∈ R. Then f ∈ p∈Im π * p if and only if there exists some g ∈ S such that f g is nilpotent.
Proof. If f ∈ p∈Im π * p then:
f /1 ∈ p∈Im π * S −1 p = q∈Spec(S −1 R) q = √ 0.
Thus there exist a nutural number n ≥ 1 and some g ∈ S such that f n g = 0. Hence, f g is nilpotent. The reverse implication is easy. Then p is a minimal prime of R if and only if for each f ∈ p there exists some g ∈ R \ p such that f g is nilpotent.
Proof. It is an immediate consequence of Proposition 3.1. Proof. Let f ∈ R. If p ∈ W = Min(R) ∩ V (f ) then by Corollary 3.2, there exists some g ∈ R \ p such that f g is nilpotent. This yields that p ∈ Min(R) ∩ D(g) ⊆ W . Therefore ∈ S(f ). Then clearly ef = 0 and g = ge for all g ∈ Ann(f ). Hence Ann(f ) is generated by the sequence e. Now let p ∈ Min(Λ) ∩ V (e). If f ∈ p then by Corollary 3.2, there exists some h ∈ Λ \ p such that f h is nilpotent. But Λ is a reduced ring. Hence h ∈ Ann(f ). Thus h = he ∈ p. But this is a contradiction. This shows that U = Min(Λ) ∩ V (e) is a flat open of Min(Λ).
For each x ∈ X then p x := Ker π x is a minimal prime of Λ and it is generated by the sequence 1 − ∆ x where π x : Λ → R x is the canonical projection, ∆ x = (δ x,y ) y∈X and δ x,y is the Kronecker delta.
In the following result and also in Theorems 4.4, 5.4, 8.2 and Corollary 7.3, we study compactifications, especially the Stone-Čech compactification by purely algebraic methods.
Theorem 3.5. The space Min(Λ) together with the canonical map η : X → Min(Λ) given by x p x is the Stone-Čech compactification of the discrete space X.
Proof. By Corollary 3.4, the space Min(Λ) is compact. It remains to check out the universal property of the Stone-Čech compactification. Let Y be a compact topological space and ϕ : X → Y a function. We shall find a continuous function ϕ : Min(Λ) → Y such that ϕ = ϕ • η and then we show that such function is unique. If p ∈ Min(Λ) then the subsets S(f ) with f ∈ Λ \ p have the finite intersection property. It follows that the subsets ϕ S(f ) and so their closures ϕ S(f ) with f ∈ Λ \ p have the finite intersection property. This yields that f ∈Λ\p ϕ S(f ) = ∅ because Y is quasi-compact. We claim that this intersection has exactly one point. If y and y ′ are two distinct points of the intersection then there exist disjoint opens U and V in Y such that y ∈ U and y ′ ∈ V . Then consider the sequence f ∈ Λ where f x is either 0 or 1, according as x ∈ ϕ −1 (U) or x / ∈ ϕ −1 (U). Then we have either f ∈ p or 1−f ∈ p since f is an idempotent. If f ∈ p then ϕ −1 (V )∩S(1−f ) = ∅. So we may choose some x in this intersection. Thus x / ∈ ϕ −1 (U), hence f x = 1. But this is a contradiction since x ∈ S(1 − f ). If 1 − f ∈ p then ϕ −1 (U) ∩ S(f ) = ∅, but this is again a contradiction. Hence, there exists a unique point y p ∈ Y such that
f ∈Λ\p ϕ S(f ) = {y p }.
This establishes the claim. Then we define the map ϕ :
Min(Λ) → Y as p y p . It is easy to see that ϕ(x) ∈ f ∈Λ\px ϕ S(f ) for all x ∈ X. Therefore ϕ = ϕ • η. Now we show that ϕ is continuous. Let U be an open of Y and let p ∈ ( ϕ) −1 (U).
There exists an open neighborhood V of y p such that V ⊆ U, because it is well known that every compact space is a normal space. Let h ∈ Λ be a sequence which is defined as h
x = 1 or h x = 0, according as x ∈ ϕ −1 (V ) or x / ∈ ϕ −1 (V ). Then p ∈ D(h), since if h ∈ p then 1 − h / ∈ p and so ϕ −1 (V ) ∩ S(1 − h) = ∅, which is impossible. To conclude the continuity of ϕ we show that Min(Λ) ∩ D(h) ⊆ ( ϕ) −1 (U). Suppose there exists some q ∈ Min(Λ) ∩ D(h) such that y q / ∈ U. Thus y q ∈ W := Y \ V . It follows that W ∩ ϕ S(h) = ∅. But this is impossible since S(h) = ϕ −1 (V ) and so W ∩ ϕ S(h) ⊆ W ∩ V = ∅. Therefore ϕ is continuous. If Min(Λ) ∩ D(f )
is non-empty then f = 0 and so there exists some x ∈ X such that p x ∈ D(f ). This shows that η(X) is a dense subspace of Min(Λ), hence the uniqueness of ϕ is deduced from the basic fact that if two continuous maps into a Hausdorff space agree on a dense subspace of the domain, they are equal.
since {p x } = Min(Λ) ∩ D(∆ x ) for all x ∈ X.
Lemma 3.7. If each R x is a field, then every prime ideal of Λ is a maximal ideal.
Proof. Let p be a prime ideal of Λ and f ∈ Λ \ p. Then consider the sequence
g = (g x ) ∈ Λ where g x is 1 or 1/f (x), according as f x = 0 or f x = 0. Then it is obvious that f (1 − f g) = 0 ∈ p.
This yields that 1 − f g ∈ p. Therefore Λ/p is a field. As a second proof, the assertion is also deduced from the fact that Λ is an absolutely flat ring. Proof. If each R x is a field then the assertion is deduced from Theorem 3.5 and Lemma
3.7. Conversely, if m is a maximal ideal of R x then π −1 x (m) = π −1 x (0) because Spec(Λ)
is Hausdorff and so every prime ideal of Λ is a maximal ideal. But π x is surjective and so the induced map π *
x is injective. Therefore the zero ideal of R x is a maximal ideal and so it is a field.
Corollary 3.9. The space Spec P(X) together with the canonical map η :
X → Spec P(X) given by x m x = P(X \ {x}) is the Stone-Čech compactification of the discrete space X.
Proof. The map P(X) → x∈X Z 2 given by A χ A is an isomorphism of rings where χ A is the characteristic function of A and Z 2 = {0, 1}. Then apply Corollary 3.8.
Remark 3.10.
Here we establish a bridge that allows us to translate all of the theory of Boolean algebras into the standard language of commutative algebra (and vice versa). For instance, the classical approach to construct the Stone-Čech compactification of a discrete space X is easily recovered. Indeed, if X is a set then one can easily check that the map M P(X) \ M = {A ∈ P(X) : A c ∈ M} is a homeomorphism from Spec P(X) onto F (X), the space of ultrafilters on X equipped with the Stone topology. Recall that the collection of d(A) = {F ∈ F (X) : A ∈ F } with A ∈ P(X) forms a base for the opens of the Stone topology. The space F (X) is called the Stone space of the Boolean algebra P(X). Note that the above identification can be generalized to any Boolean ring R. In fact, the map M R \ M is a homeomorphism from Spec(R) onto the Stone space of the corresponding Boolean algebra of R.
If X is a set with the cardinality κ and X is a compactification of the discrete space X, then by [5, Tag 0909] and assuming the generalized continuum hypothesis, we have | X| ∈ {κ, 2 κ , 2 2 κ }.
Corollary 3.11. If X and Y are two sets then we have the following canonical bijections:
Mor Set (X, βY ) ≃ Mor Top (βX, βY ) ≃ Mor Ring P(Y ), P(X) .
Proof. The first bijection follows form Corollary 3.9, and the second bijection is an immediate consequence of [24,Theorem 5.6].
The Stone-Čech compactification of an arbitrary space
It is well known that every completely regular topological space admits the Stone-Čech compactification. In this section, we prove "every topological space admits the Stone-Čech compactification". To realize this, first we prove some results which are interesting in their own right. These results are also proved in [18] by using the theory of ultrafilters. In fact this work [18] was the main motivation in emerging the ideas of this section. We will proceed by using the ring-theoretical methods. This approach is so much simpler than the theory of ultrafilters.
Let X be a topological space, x ∈ X and M a maximal ideal of P(X). Then we say that M is convergent (or, Zariski convergent) to the point x if whenever U is an open subset of X containing x, then M ∈ D(U). Proof. Take x ∈ A and let S be the set of all opens of X which are containing x. Then by the hypothesis, the ideal of P(X) generated by A and the elements U c = X \ U with U ∈ S is the whole ring. Thus we may find a finite number U 1 , ..., U n of elements
Lemma 4.1. Let X be a set. If M is a maximal ideal of P(X) then P(η) * (M) is convergent to the point M ∈ βX = Spec P(X). Proof. Let U be an open of βX such that M ∈ U. If U ∈ P(η) * (M) then η −1 (U) ∈ M. But there exists some A ∈ P(X) such that M ∈ D(A) ⊆ U. If x ∈ A then η(x) = m x ∈ D(A) and so x ∈ η −1 (U). This shows that A ⊆ η −1 (U). Thus A ∈ M. But this is a contradiction.of S such that X = A∪( n i=1 U c i ). It follows that x ∈ n i=1 U i ⊆ A. Hence, A is an open of X.
Note that the converse of the above lemma holds trivially.
Let ϕ : X → Y be a continuous map of topological spaces. If a maximal ideal M of P(X) converges to some point x ∈ X, then clearly P(ϕ) * (M) is convergent to ϕ(x). In the following result we establish its converse. Proof. Consider the equivalence relation ∼ on βX = Spec P(X) defined as M ∼ N if ϕ : X → Y is a continuous function to a compact space Y then ϕ(M) = ϕ(N) where ϕ : βX → Y is the unique continuous function such that ϕ = ϕ•η, see the proof of Theorem 3.5. Now to prove that the pair (X ′ , π • η) is the Stone-Čech compactification of the space X it suffices to show that π • η : X → X ′ is continuous where π : βX → X ′ = βX/ ∼ is the canonical map and X ′ is equipped with the quotient topology. To prove the continuity of π • η, by Corollary 4.3, it will be enough to show that if a maximal ideal M of P(X) converges to some point
x ∈ X then P(π • η) * (M) is convergent to the point (π • η)(x). We have P(π • η) * (M) = P(π) * P(η) * (M) . By Lemma 4.1, N := P(η) * (M) is convergent to the point M ∈ βX. Thus P(π) * (N) is convergent to the point π(M) since π is continuous. Then we show that M ∼ m x . Because take A ∈ P(X) \ M and let V be an open of a compact space Y such that ϕ(x) ∈ V where ϕ : X → Y is a continuous map. Then ϕ −1 (V ) / ∈ M. Note that S(A) = A. Now if V ∩ ϕ(A) = ∅ then A ∈ M, a contradiction. Hence, ϕ(x) ∈ ϕ S(A) .
Thus by the definition of ϕ, see the proof of Theorem 3.5, we get that ϕ(x) = ϕ(M) and so M ∼ m x . Therefore P(π • η) * (M) is convergent to the point π(M) = (π • η)(x). Note that during to verify the universal property of the Stone-Čech compactification for the pair (X ′ , π • η), the uniqueness is deduced from the fact that (π • η)(X) is a dense subspace of X ′ .
Maximal spectrum as Stone-Čech compactification
The main result of this section (see Theorem 5.4) asserts that the maximal spectrum of the direct product of a family of local rings indexed by a set X is the Stone-Čech compactification of the discrete space X. Then some applications are also given.
Let R be a ring and f ∈ R. If m ∈ U = Max(R) ∩ D(f ) then there exist some g ∈ m and h ∈ R such that 1 = f h + g. This yields that m ∈ Max(R) ∩ V (g) ⊆ U. Thus U is a flat open of Max(R). Therefore the induced flat topology over Max(R) is finer than the induced Zariski topology. Similarly we get that U c = Max(R) \ U = Max(R) ∩ V (J) where J is a (finitely generated) ideal of R. It follows that I + J = R. Thus there exist some f ∈ I and g ∈ J such that f + g = 1. This implies that U = Max(R) ∩ V (f ).
Throughout this paper, Γ = x∈X R x where each R x is a local ring with the maximal ideal
m x . For each x ∈ X then M x := π −1 x (m x ) is a maximal ideal of Γ, because the ring map Γ/M x → R x /m x induced by the canonical projection π x : Γ → R x is an isomorphism. If f = (f x ) ∈ Γ then we define Ω(f ) = {x ∈ X : f x / ∈ m x }.
It is obvious that f is invertible in Γ if and only if Ω(f ) = X. It is also easy to see that Ω(f g) = Ω(f ) ∩ Ω(g) for all f, g ∈ Γ.
Lemma 5.3. Let f ∈ Γ.
Then Ω(f ) = ∅ if and only if f ∈ J.
Proof.
If Ω(f ) = ∅ then f x ∈ m x for all x. This yields that Ω(1 + f g) = X for all g ∈ Γ.
Thus f ∈ J. Conversely, if f ∈ J then f ∈ M x for all x.
So Ω(f ) is empty. Proof. If f ∈ Γ then consider the sequence g = (g x ) ∈ Γ such that g x is either 0 or Corollary 5.6. There exists a unique homeomorphism:
f −1 x , according as f x ∈ m x or f x / ∈ m x . Then Ω(1 + f h(1 − f g)) = X for all h ∈ Γ. Hence, f (1−f g) ∈ J.Min(Λ) ≃ / / Max(Γ) such that p x is mapped into M x for all x ∈ X.
Proof. It is deduced from the universal property of the Stone-Čech compactification by taking into account Theorems 3.5 and 5.4.
In the next section, we will precisely determine the rule of isomorphism of Corollary 5.6.
Corollary 5.7. Let R be a ring and let X be a subset of Spec(R). Then the following spaces are canonically isomorphic (up to a unique isomorphism). Proof. It is an immediate consequence of Corollary 5.6. Corollary 5.9. Let X and Y be two sets with the cardinalities κ and λ, respectively. Then the number of all ring maps P(X) → P(Y ) is either κ λ or 2 λ2 κ , according as X is finite or infinite.
Proof. It is deduced from Corollaries 3.11 and 5.8.
If κ is an infinite cardinal, then λ2 κ = max{λ, 2 κ }. To see its proof apply Cantor's theorem and [8, p. 162, Lemma 6R] which states that κκ = κ.
Corollary 5.10. Let X be a set with the cardinality κ. Then the number of all ring maps P(X) → P(X) is either κ κ or 2 2 κ , according as X is finite or infinite.
Proof. It is an immediate consequence of Corollary 5.9.
Ultra-rings and their applications in compactification
In this section, we introduce a new way to construct the ultraproduct of rings which greatly simplifies the general method in the literature. For the general method and its applications see e.g. [4], [6], [9] and [17]. Then we use this new approach to determine precisely the isomorphisms whose rules are already obtained in an implicit way, see e.g. Corollary 5.6.
Let (R x ) be a family of rings indexed by a set X and R = x∈X R x . Let M be a maximal ideal of P(X). Then it can be easily seen that M * = {f ∈ R : S(f ) ∈ M} is an ideal of R, because clearly S(0) = ∅ ∈ M and so 0 ∈ M * , also S(f + g) ⊆ S(f ) ∪ S(g) and S(f g) ⊆ S(f ) ∩ S(g) for all f, g ∈ R. We call the quotient ring R/M * the ultraproduct (or, ultra-ring) of the family (R x ) with respect to M.
It is interesting to notice the map ϕ : R → P(X) given by f S(f ) is not a morphism of rings, since it is not additive, in fact S(f ) + S(g) ⊆ S(f + g). In spite of this, the inverse image of each ideal of P(X) under ϕ is an ideal of R. In particular, M * = ϕ −1 (M).
If each R x is a non-zero ring, then M * is a proper ideal of R. We also have the following result which shows that some properties (statements in first-order logic) of rings are preserved under the formation of ultraproducts.
(f ) = {x ∈ X : f x / ∈ m x } ⊆ S(f ) for all f ∈ R where m x is the maximal ideal of R x . If f ∈ R \ M ♭ then S(1 − f g) ⊆ Ω(f ) c ∈ M where g = (g x ) and each g x is either f −1 x or 1, according as x ∈ Ω(f ) or x /
∈ Ω(f ). Therefore S(1 − f g) ∈ M and so 1 − f g ∈ M * . Hence, M ♭ /M * is the only maximal ideal of R/M * . The proof of (iv) is easy and left as an exercise to the reader. with m x is the maximal ideal of R x . If f ∈ M ♭ then S(f ) ⊆ Ω(f ) ∈ M and so S(f ) ∈ M. Hence, the above map is well-defined. Clearly it is also an isomorphism. Proof. First we need to show that M * is a minimal prime of Λ. By Theorem 6.1 (ii), M * is a prime ideal of Λ. Suppose there exists a prime ideal p of Λ such that p ⊆ M * . If f ∈ M * then consider the sequence g = (g x ) ∈ Λ where each g x is either 1 or 0, according as x ∈ S(f ) or x / ∈ S(f ). Then clearly S(g) = S(f ) ∈ M and so g ∈ M * . Moreover f (1 − g) = 0. This yields that f ∈ p. Hence, M * is a minimal prime of Λ. The map ϕ is continuous, since ϕ −1 Min(Λ) ∩ D(f ) = D S(f ) for all f ∈ Λ. Clearly m * x = p x for all x ∈ X, where m x = P(X \ {x}) and for p x see the above of Theorem 3.5. This shows that η = ϕ • η ′ where η : X → Min(Λ) and η ′ : X → Spec P(X) are the canonical maps. Therefore, by the universal property of the Stone-Čech compactification and by taking into account Theorem 3.5 and Corollary 3.9, we deduce that ϕ is a homeomorphism. Proof. By the proof of Theorem 6.1(iii), M ♭ is a maximal ideal of Γ. Hence, the above map is well-defined. It is also continuous, since
ψ −1 Max(Γ) ∩ D(f ) = D Ω(f ) for all f ∈ Γ. Moreover m ♭ x = M x for all x ∈ X,
where m x = P(X \ {x}) and for M x see Theorem 5.4. This shows that η = ψ • η ′ where η : X → Max(Γ) and η ′ : X → Spec P(X) are the canonical maps. Thus, by the universal property of the Stone-Čech compactification and by taking into account Corollary 3.9 and Theorem 5.4, we deduce that ψ is a homeomorphism.
Alexandroff compactification
Let R be the set of all subsets of a set X which are either finite or cofinite (i.e. its complement is finite). Then clearly R is a subring of the power set ring P(X). Recall that if x ∈ X then m x = P(X \ {x}) is a maximal ideal of P(X). In the following result the maximal ideals of R are characterized.
Theorem 7.1. Let X be an infinite set. Then the maximal ideals of R are precisely Fin(X) or of the form m x ∩ R where x ∈ X.
Proof. First we have to show that Fin(X) is a maximal ideal of R. Clearly Fin(X) = R since X is infinite. If there exists an ideal I of R strictly containing Fin(X) then we may choose some A ∈ I such that A / ∈ Fin(X). It follows that A c ∈ Fin(X) and so 1 = A + A c ∈ I. Hence, Fin(X) is a maximal ideal of R. Conversely, let M be a maximal ideal of R such that M = m x ∩ R for all x ∈ X. It follows that A x := X \ {x} ∈ R \ M for all x ∈ X. But {x}.A x = 0 ∈ M. Therefore {x} ∈ M for all x ∈ X. This yields that Fin(X) ⊆ M and so Fin(X) = M.
Remark 7.2. Let X be an infinite set. Here we give a second proof to show that Fin(X) is a maximal ideal of R. There exists a maximal ideal M of P(X) such that Fin(X) ⊆ M since Fin(X) = P(X). We have then Fin(X) = M ∩ R. Thus Fin(X) is a maximal ideal of R. Clearly Fin(X) is the point at infinity of the Alexandroff compactification of the infinite discrete space X.
Totally disconnected compactifications
In this section it is shown that every totally disconnected compactification of a discrete space X is precisely of the form Spec(R ′ ) where the ring R ′ satisfies in the extensions of rings R ⊆ R ′ ⊆ P(X). For R see §7. Proof. Let A be a clopen of Y such that f −1 (A) = ∅. If A is non-empty then A ∩ f (X) is non-empty. But this is a contradiction. Here is a second proof. Let D 1 and D 2 be two
clopens of Y such that f −1 (D 1 ) = f −1 (D 2 ). Suppose there exists some y ∈ D 1 such that y / ∈ D 2 . It follows that (D 1 ∩ D c 2 ) ∩ f (X) = ∅. Hence there exists some x ∈ X such that f (x) ∈ D 1 ∩ D c 2 .
But this is a contradiction. Therefore D 1 = D 2 . Theorem 8.2. Every totally disconnected compactification of a discrete space X is precisely of the form Spec(R ′ ) where the ring R ′ satisfies in the extensions of rings R ⊆ R ′ ⊆ P(X).
Proof. It is easy to see that for any such ring R ′ then Spec(R ′ ) together with the canonical open embedding η : X → Spec(R ′ ) which sends each point x ∈ X into m x ∩ R ′ is a totally disconnected compactification of the discrete space X. Conversely, let ( X, η) be a totally disconnected compactification of a discrete space X. By [24,Corollary 5.4], the space X is homeomorphic to Spec(R) where R = Clop( X). By Lemma 8.1, the induced map Clop(η) : R → Clop(X) = P(X) is an injective ring map. So the ring R is isomorphic to R ′ , the image of Clop(η). It remains to show that R ⊆ R ′ . Take A ∈ R. If A is finite then D := η(A) = x∈A {η(x)} is a closed subset of X and so D ∈ Clop( X). Therefore
A = η −1 (D) ∈ R ′ .
But if A is cofinite then the above argument shows that A c ∈ R ′ , and so A = 1 − A c ∈ R ′ . Remark 8.3. If ( X, η) is an arbitrary compactification of a discrete space X then by [24,Theorem 5.2], the space of connected components π 0 ( X) is homeomorphic to Spec(R ′ ) where R ′ = Clop( X). Also R ′ , via the ring map Clop(η), can be viewed as a subring of P(X) and containing R. Note that there are compactifications of a discrete space which are not totally disconnected.
Semigroup structure on βX
In this section, βX = Spec P(X) together with the canonical map η : X → βX denotes the Stone-Čech compactification of the discrete space X. If f : X → Y is a function then by Corollary 3.9, there exists a unique continuous function βf : βX → βY such that (βf )(m x ) = m f (x) for all x ∈ X. This yields that βf = P(f ) * . In particular, if f : X → Y is injective then βf : βX → βY is as well.
Let (S, * ) be a semigroup such that S is a topological space. Then equip S × S with the product topology. If the operation * : S × S → S is continuous, then (S, * ) is called a topological semigroup. If the operation * is not continuous, then this leads us to a weaker notion. Indeed, the pair (S, * ) is called a left topological semigroup if the operation * is left semi-continuous. That is, for each p ∈ S then the map ℓ p : S → S given by x p * x is continuous. The right topological semigroup is defined dually. Obviously every topological semigroup is both right topological and left topological semigroup. We have then the following interesting result.
Theorem 9.1. The operation of every commutative semigroup (X, .) can be extended uniquely to an operation * on βX such that: (βX, * ) is a left topological semigroup, the canonical map η : X → βX is a morphism of semigroups and m x * M = M * m x for all M ∈ βX and x ∈ X. If moreover e is the identity of X, then m e is the identity of βX.
Proof. If x ∈ X then by Theorem 3.8, there exists a unique continuous function ϕ x : βX → βX such that ϕ x (m y ) = m x.y for all y ∈ X. For a fixed M ∈ βX, again by Theorem 3.8, there exists a unique continuous map θ M : βX → βX such that θ M (m x ) = ϕ x (M) for all x ∈ X. Now we define the operation * on βX as M * N = θ M (N). Then we show this operation is associative. To prove this it suffices to show that θ M • θ N = θ L for every M, N ∈ βX with L = θ M (N). To see this it will be enough to show that θ M • ϕ x = ϕ x • θ M for all M ∈ βX and x ∈ X. But to see the latter it suffices to show that θ M • ϕ x and ϕ x • θ M agree on η(X), (recall that if two continuous maps into a Hausdorff space agree on a dense subspace of the domain, they are equal). This reduces to show that ϕ x • ϕ y = ϕ x.y for all x, y ∈ X. Finally, to see this it suffices to show that (ϕ x • ϕ y )(m z ) = ϕ xy (m z ) for all z ∈ X. But the latter obviously holds since the operation of X is associative. Clearly ℓ M = θ M for all M ∈ βX. Hence, (βX, * ) is a left topological semigroup. The map η is a morphism of semigroups since ϕ x = θ mx for all x ∈ X. This also yields that m x * M = M * m x for all M ∈ βX and x ∈ X. To see the uniqueness of * , suppose there is another operation * ′ on βX such that (βX, * ′ ) is a left topological semigroup, the canonical map η : X → (βX, * ′ ) is a morphism of semigroups and m x * ′ M = M * ′ m x for all M ∈ βX and x ∈ X. Then clearly for each x ∈ X, the maps ℓ mx and ℓ ′ mx agree on η(X), hence they are equal. It follows that for each M ∈ βX, then ℓ M and ℓ ′ M agree on η(X), hence they are equal. The latter implies that * = * ′ . Finally, if e is the identity element of X then ϕ e is the identity map. It follows that m e is the identity element of βX.
Note that the operation * of Theorem 9.1 is not necessarily commutative. Hence, we may define a new operation on βX as M * ′ N := θ N (M) = N * M. Then it is easy to see that (βX, * ′ ) is a right topological semigroup. Therefore we may consider βX as left topological or right topological semigroup, depending on the preferred construction, but never both (specially when X is an infinite set).
In the proof of Theorem 9.1, we have ϕ x = P(f x ) * for all x ∈ X where the function f x : X → X is defined by f x (y) = x.y. By [24,Theorem 5.6 ], there exists a (unique) morphism of rings h M : P(X) → P(X) such that θ M = Spec(h M ). In the following result, the rule of this morphism is determined explicitly. Proof. It is not hard to see that the map ζ M is actually a morphism of rings. To see
θ M = Spec(ζ M ) it suffices to show that θ M (m x ) = ζ −1 M (m x ) for all x ∈ X.
By the category of left topological monoids we mean a category whose objects are the left topological monoids and whose morphisms are the continuous morphisms of monoids. Proof. By the universal property of the Stone-Čech compactification, it is a functor provided that we could prove that if f : X → Y is a morphism of commutative monoids then βf : βX → βY is a morphism of monoids. Clearly βf preserves the identities. It remains to show that (βf ) • θ M = θ M ′ • (βf ) for all M ∈ Spec P(X) with M ′ = (βf )(M), for the notations see the proof of Theorem 9.1. To see this it suffices to show that these functions agree on η(X). To see the latter it will be enough to show that (βf ) • ϕ x = ϕ f (x) • (βf ) for all x ∈ X. Clearly these maps agree on η(X), hence they are equal. Proof. Assume T (R) is absolutely flat. Then R is reduced, since every absolutely flat ring and so each subring are reduced. By [25,Lemma 3.4], Min(R) is Zariski compact. If I = (f 1 , ..., f n ) is a finitely generated and faithful ideal of R then for each i, there exists a non zero-divisor g i of R such that f i (g i − f i h i ) = 0 for some h i ∈ R. It follows that (g 1 − f 1 h 1 )...(g n − f n h n ) = 0 and so g 1 ...g n ∈ I. Conversely, if f ∈ R then it will be enough to find a non zero-divisor g of R such that f g = f 2 h for some h ∈ R. Setting X = {p ∈ Min(R) : f ∈ p}. If p ∈ X then there exists some x p ∈ R \ p such that f x p = 0. It follows that Min(R) ⊆ D(f ) ∪ p∈X D(x p ) . Using the quasi-compactness of Min(R),
then we may write Min(R) ⊆ D(f ) ∪ n i=1
D(x i ) and that f x i = 0 for all i. Therefore I = (f, x 1 , ..., x n ) is a faithful ideal of R, because suppose rI = 0, if p ∈ Min(R) then r ∈ p and so r ∈ p∈Min(R) p = √ 0 = 0. Hence, I contains a non zero-divisor g of R. Thus we may
write g = f h + n i=1
r i x i where h, r 1 , ..., r n ∈ R. This yields that f g = f 2 h. p ∈ Min(R). So for each i, there exists some g i ∈ R \ p such that f i g i = 0. Therefore gI = 0 where g = g 1 . . . g n . But this is a contradiction. Hence, I admits a non zero-divisor of R. Therefore by Theorem 10.1, T (R) is absolutely flat.
Corollary 10.4. Let R be a reduced ring such that Spec(R) is a noetherian space with respect to the Zariski topology. Then T (R) is absolutely flat. Remark 10.8. Here we prove the reverse implication of Theorem 10.1 by an alternative approach. Clearly a ring R is reduced if and only if T (R) is reduced. To prove that T (R) is zero dimensional it will be enough to show that every prime ideal q of R which does not meet R \ Z(R), then it is a minimal prime. To see this it suffices to show that q ⊆ p for some p ∈ Min(R). Suppose q p for all p ∈ Min(R), then there exists some x p ∈ q such that x p / ∈ p. So Min(R) ⊆ (ii) If f ∈ R there exists some g ∈ R such that f g = Ann(Rf + Rg) = 0. (iii) If an ideal I of R is contained in Z(R), then I ⊆ p for some p ∈ Min(R).
Proof. (i) ⇒ (ii) : There exists a non zero-divisor s ∈ R such that f s = f 2 h for some h ∈ R. Then g := f h − s is the desired element. (ii) ⇒ (i) : It suffices to show that h := f − g is a non zero-divisor of R. Suppose rh = 0 and p is a minimal prime ideal of R such that r / ∈ p. Then f, g ∈ p and so there exist f ′ , g ′ ∈ R \ p such that f f ′ = gg ′ = 0. This yields that f ′ g ′ ∈ Ann(Rf + Rg) = 0 which is a contradiction. Hence, r ∈ p∈Min(R) p = √ 0 = 0. the ideal (f i : i ∈ S) admits a non zero-divisor g of R. So there exists a finite subset S ′ of S such that g = i∈S ′ r i f i where r i ∈ R for all i ∈ S ′ . This yields that Min(R) ⊆ i∈S ′ D(f i ), since otherwise we may find some p ∈ Min(R) such that g ∈ p, but this is impossible since p ⊆ Z(R). Hence, Min(R) is quasi-compact. Now let I be a finitely generated and faithful ideal of R. If I ⊆ Z(R) then I is contained in a minimal prime ideal p of R. Thus we may find some s ∈ R \ p such that sI = 0, which is a contradiction. So I admits a non zero-divisor of R. Therefore by Theorem 10.1, T (R) is absolutely flat.
Corollary 3. 2 .
2([11, Lemma 1.1] and [14, Lemma 3.1]) Let p be a prime ideal of a ring R.
Theorem 3. 3 .
3Let R be a ring. Then the induced Zariski topology over Min(R) is finer than the induced flat topology. These two topologies over Min(R) are the same if and only if Min(R) is Zariski compact.
W is a Zariski open of Min(R). Hence the Zariski topology over Min(R) is finer than the flat topology. For any ring R, then by [1, Lemma 3.2], Min(R) is Zariski Hausdorff. Also Min(R) is flat quasi-compact. Therefore if these two topologies over Min(R) are the same then Min(R) is Zariski compact. Conversely, suppose Min(R) is Zariski compact. In the above we observed that U = Min(R) ∩ D(f ) is a Zariski clopen of Min(R). Every closed subspace of a quasi-compact space is quasi-compact. Thus there exist a finitely many elements g 1 , ..., g n ∈ R such that U c = Min(R) \ U = n i=1 Min(R) ∩ D(f i ). It follows that U = Min(R) ∩ V (I) where I = (g 1 , ..., g n ) is a finitely generated ideal of R. Thus U is a flat open of Min(R). Throughout this paper, Λ = x∈X R x where each R x is an integral domain. For each f = (f x ) ∈ Λ, the set Supp(f ) = {x ∈ X : f x = 0} is simply denoted by S(f ). Clearly S(f g) = S(f ) ∩ S(g) for all f, g ∈ Λ.
Corollary 3. 4 .
4The space Min(Λ) is Zariski compact. Proof. By Theorem 3.3, it suffices to show that for each f ∈ Λ then U = Min(Λ) ∩D(f ) is a flat open of Min(Λ). Consider the sequence e = (e x ) ∈ Λ where e x is either 0 or 1, according as x ∈ S(f ) or x /
Remark 3 . 6 .
36The space Min(Λ) is the compactification of the discrete space X in the sense of Definition 2.1. Because the map η : X → Min(Λ) given by x p x is an open embedding
Corollary 3 . 8 .
38The space Spec(Λ) together with the canonical map η : X → Spec(Λ) is the Stone-Čech compactification of the discrete space X if and only if each R x is a field.
Lemma 4 . 2 .
42Let X be a topological space and let A be a subset of X with the property that D(A) contains every maximal ideal of P(X) which is convergent to a point of A. Then A is an open subset of X.
Corollary 4. 3 .
3Let ϕ : X → Y be a function between topological spaces with the property that P(ϕ) * (M) is convergent to ϕ(x) whenever a maximal ideal M of P(X) converges to some point x ∈ X. Then ϕ is continuous.Proof. It is easily deduced from Lemma 4.2.
Theorem 4 . 4 .
44Every topological space X admits the Stone-Čech compactification.
Proposition 5. 1 .
1For a ring R the following statements are equivalent. (i) R/J is a zero dimensional ring. (ii) The induced Zariski and flat topologies over Max(R) are the same. (iii) Max(R) is flat compact. Proof. (i) ⇒ (ii) : If f ∈ R then there exists some g ∈ R such that f (1 − f g) ∈ J, because R/J is reduced and so it is absolutely flat. It follows that Max(R) ∩ V (f ) = Max(R) ∩ D(1 − f g). (ii) ⇒ (iii) : The subset Max(R) is Zariski quasi-compact and flat Hausdorff. (iii) ⇒ (i) : See [23, Theorem 4.5].
Lemma 5. 2 .
2Let R be a ring such that R/J is a zero dimensional ring. Then the clopens of Max(R) are precisely of the formMax(R) ∩ V (f ) where f ∈ R.Proof. By Proposition 5.1, the Zariski and flat topologies over Max(R) are the same. If f ∈ R then we observed that Max(R) ∩ V (f ) is a clopen of Max(R). Conversely, let U be a clopen of Max(R). It is easy to see that every closed subspace of a quasi-compact space is quasi-compact. Hence, we may write U = n k=1 Max(R) ∩ V (I k ) where each I k is a (finitely generated) ideal of R. This yields that U = Max(R) ∩ V (I) where I = I 1 ...I n .
Theorem 5. 4 .
4The space Max(Γ) together with the canonical map η : X → Max(Γ) given by x M x is the Stone-Čech compactification of the discrete space X.
Thus Γ/J is absolutely flat. Therefore by Proposition 5.1, the space Max(Γ) is compact. Then we verify the universal property of the Stone-Čech compactification. Let Y be a compact topological space and ϕ : X → Y a function. If M ∈ Max(Γ) then by Lemma 5.3, the subsets Ω(f ) with f ∈ Γ \ M have the finite intersection property. Thus by a similar argument as applied in the proof of Theorem 3.5, there exists a unique point y M ∈ Y such that f ∈Γ\M ϕ Ω(f ) = {y M }. Then we define the map ϕ : Max(Γ) → Y as M y M . Again exactly like the proof of Theorem 3.5, it is shown that ϕ = ϕ • η and ϕ is continuous. Finally, to prove the uniqueness of ϕ it suffices to show that η(X) is a dense subspace of Max(Γ). The space Max(Γ) is totally disconnected, see[23, Proposition 4.4]. It is well known that in a compact totally disconnected space, the collection of clopens is a base for the opens. Using this and Lemma 5.2, then the collection of Max(Γ) ∩ V (f ) with f ∈ Γ forms a base for the opens of Max(Γ). Now if Max(Γ) ∩ V (f ) is non-empty then Ω(f ) = X. Hence there exists some x ∈ X such that M x ∈ Max(Γ) ∩ V (f ). Therefore η(X) is a dense subspace of Max(Γ).
Remark 5 . 5 .
55The canonical map η : X → Max(Γ) given by x M x is an open embedding. In fact, {M x } = Max(Γ) ∩D(∆ x ) for all x ∈ X. To see this let M ∈ Max(Γ) ∩D(∆ x ) and f ∈ M. If f / ∈ M x then Ω(1 − ∆ x + ∆ x f ) = X and so 1 − ∆ x + ∆ x f is invertible in the ring Γ. But this is a contradiction because 1 − ∆ x + ∆ x f ∈ M. Therefore M ⊆ M x and so M = M x .
Corollary 5 . 8 .
58If X is an infinite set with the cardinality κ, then | Min(Λ)| = | Max(Γ)| = | Spec P(X)| = 2 2 κ .Proof. It follows from Corollaries 3.9 and 5.6 and the fact that the cardinality of the Stone-Čech compactification of the infinite discrete space X is equal to 2 2 κ . To see the proof of this fact please consider[12, Theorem 3.58] or [27, Theorem on page 71].
Theorem 6 . 1 .
61The following statements hold. (i) If each R x is a field, then R/M * is a field.(ii) If each R x is an integral domain, then R/M * is an integral domain. (iii) If each R x is a local ring, then R/M * is a local ring. (iv) If each K x is the fraction field of an integral domain R x , then the ultra-ring of the family (K x ) with respect to M is the fraction field of R/M * . (v) If each R x isa local ring with the residue field K x , then the ultra-ring of the family (K x ) with respect to M is the residue field of R/M * . Proof. (i) : Take f ∈ R \ M * and consider the sequence g = (g x ) ∈ R where each g x is either f −1 x or 1, according as x ∈ S(f ) or x / ∈ S(f ). Then clearly S(1 − f g) ⊆ S(f ) c ∈ M. Thus S(1 − f g) ∈ M and so 1 − f g ∈ M * . (ii) : Suppose f g ∈ M * for some f, g ∈ R. Then clearly S(f ) ∩ S(g) ⊆ S(f g) ∈ M. Thus S(f ) ∩ S(g) ∈ M. It follows that either f ∈ M * or g ∈ M * . (iii) : Clearly M ♭ = {f ∈ R : Ω(f ) ∈ M} is a proper ideal of R and M * ⊆ M ♭ , since Ω
(v) : It suffices to show that the map R/M ♭ → R ′ /M * given by f + M ♭ f + M * is an isomorphism of rings where R ′ = x∈X K x , M * = {g ∈ R ′ : S(g) ∈ M} and f = (f x + m x )
Theorem 6 . 2 .
62The map ϕ : Spec P(X) → Min(Λ) given by M M * is a homeomorphism.
Theorem 6 . 3 .
63The map ψ : Spec P(X) → Max(Γ) given by M M ♭ is a homeomorphism.
Corollary 7 . 3 .
73If X is an infinite set, then Spec(R) is the Alexandroff compactification of the discrete space X.Proof. The space Spec(R) is compact. The map η : X → Spec(R) given by x m x ∩R is an open embedding. Because by Theorem 7.1, D({x}) = {m x ∩ R} for all x ∈ X. Now if A is a subset of X then η(A) = x∈A D({x}). If U is an open neighborhood of Fin(X) in Spec(R) then U c is a finite set. Hence, η(X) is a dense subspace of Spec(R).
Lemma 8 . 1 .
81Let f : X → Y be a continuous map of topological spaces such that f (X) is a dense subspace of Y . Then the induced map Clop(f ) : Clop(Y ) → Clop(X) is injective.
Theorem 9 . 2 .
92Let (X, .) be a commutative semigroup and M a maximal ideal of P(X). Then the map ζ M : P(X) → P(X) given by A {x ∈ X : f −1 x (A) / ∈ M} is a morphism of rings and θ M = Spec(ζ M ).
Theorem 9 . 3 .
93The assignments X βX and h βf form a faithful covariant functor from the category of commutative monoids to the category of left topological monoids.
10 .
10On absolutely flatness of the total ring of fractions Theorems 10.1 and 10.9 provide new and simple proofs to the main results of [10, Theorem 2.9], [13, Chap I, Theorem 4.5], [15, Proposition 1.4] and [16, Proposition 9]. In the following results, T (R) denotes the total ring of fractions of a ring R. Theorem 10.1. Let R be a ring. Then T (R) is absolutely flat if and only if the following two conditions hold. (i) R is reduced and Min(R) is Zariski compact.(ii) Every finitely generated and faithful ideal of R contains a non zero-divisor of R.
Corollary 10. 3 .
3Let R be a reduced ring such that Min(R) is a finite set. Then T (R) is absolutely flat. Proof. Let I = (f 1 , ..., f n ) be a faithful ideal of R and setting S := R \ Z(R). If I ∩ S = ∅ then there exists a prime ideal p of R such that I ⊆ p and p ∩ S = ∅. It follows that p ⊆ Z(R) = q∈Min(R) q. Thus by the Prime Avoidance Lemma (cf. [21, Theorem 2.2]),
Corollary 10 . 5 .
105Let R be a reduced and noetherian ring. Then T (R) is absolutely flat.Corollary 10.6. Let R be a ring. Then T (R[x]) is absolutely flat if and only if R is reduced and Min(R) is Zariski compact.Proof. It is interesting to notice that for any ring R, then every finitely generated and faithful ideal of R[x] contains a non zero-divisor of R[x]. Then apply Theorem 10.1.
Corollary 10 . 7 .
107Let R be a ring. If T (R) is absolutely flat, then T (R[x]) is as well.
.
x i ) where x i := x p i for all i. Then I = (x 1 , . . . , x n ) ⊆ q ⊆ Z(R). So I is not a faithful ideal, i.e., Ann(I) = 0. Thus we may choose some nonzero a in Ann(I). Now if p ∈ Min(R) then x i / ∈ p for some i. But ax i = 0. Thus a ∈ p. So a ∈ For a reduced ring R the following statements are equivalent.(i) T (R) is absolutely flat.
(i) ⇒ (iii) : Suppose I p for all p ∈ Min(R), then we may choose some x p ∈ I \ p. Usingthe quasi-compactness of Min(R), then we may write Min(R) ⊆ n i=1 D(x i ) where x i ∈ I for all i. It follows that J = (x 1 , ..., x n ) is a faithful ideal of R. Thus by Theorem 10.1, J admits a non zero-divisor which is a contradiction. (iii) ⇒ (i) : Suppose Min(R) ⊆ i∈S D(f i ) where f i ∈ R for all i. Then by the hypothesis,
Proposition 10 . 10 .
1010Let R be a ring. If T (R[x]) is a zero dimensional ring, then Min(R) is Zariski compact. Proof. For any ring R, the minimal prime ideals of R[x] are precisely of the form p[x] where p is a minimal prime ideal of R. Hence, the map ϕ : Min(R) → Min(R[x]) given by p p[x] is bijective. This map is continuous, because if f = n i=0 f i x i ∈ R[x] with the f i ∈ R, then ϕ −1 where U = Min(R[x]) ∩ D(f ) and U i = Min(R) ∩ D(f i ).
Remark 10.2. Let R be a ring. It is easy to see that if at least one of the coefficients of a polynomial f ∈ R[x] is a non zero-divisor of R, then f is a non zero-divisor of R[x]. But the converse does not hold. As an example, take R = Z/6Z then f = 2 + 3x is a non zero-divisor of R[x], but all of its coefficients are zero-divisors of R. This observation shows that if T (R[x]) is zero dimensional (or, an absolutely flat ring) then the same assertion does not necessarily hold for T (R).
The converse of ϕ is also continuous, because it is induced by the ring extension R ⊆ R[x]. Therefore ϕ is a homeomorphism. By[25, Lemma 3.4], Min(R[x]) is Zariski compact. Hence, Min(R) is Zariski compact.
Characterizations of Gelfand rings specially clean rings and their dual rings. M Aghajani, A Tarizadeh, Results Math. 753125M. Aghajani and A. Tarizadeh, Characterizations of Gelfand rings specially clean rings and their dual rings, Results Math. 75(3) (2020) 125.
On the structure space of a direct product of rings. K Aoyama, J. Sci. Hiroshima Univ. SER. A-I. 34K. Aoyama, On the structure space of a direct product of rings, J. Sci. Hiroshima Univ. SER. A-I 34 (1970) 339-353.
Rings of continuous functions with values in a topological field. G Bachman, Trans. Amer Math. Soc. 204G. Bachman et al., Rings of continuous functions with values in a topological field, Trans. Amer Math. Soc. 204 (1975) 91-112.
Ultraproducts and approximation in local rings I. J Becker, Invent. Math. 51J. Becker et al., Ultraproducts and approximation in local rings I, Invent. Math. 51 (1979) 189-203.
The Stacks Project, see http:stacks.math.columbia. A J De Jong, A.J. de Jong, et al., The Stacks Project, see http:stacks.math.columbia.edu., (2021).
Ultraproducts and approximation in local rings II. J Denef, L Lipshitz, Math. Ann. 253J. Denef and L. Lipshitz, Ultraproducts and approximation in local rings II, Math. Ann. 253 (1980) 1-28.
Commutative rings in which every prime ideal is contained in a unique maximal ideal. G , De Marco, A Orsatti, Proc. Amer. Math. Soc. 303G. De Marco and A. Orsatti, Commutative rings in which every prime ideal is contained in a unique maximal ideal, Proc. Amer. Math. Soc. 30 (3) (1971) 459-466.
Elements of set theory. H Enderton, Academic Press, IncH. Enderton, Elements of set theory, Academic Press, Inc. (1977).
Big polynomial rings and Stillman's conjecture. D Erman, Invent. Math. 2182D. Erman et al., Big polynomial rings and Stillman's conjecture, Invent. Math. 218(2) (2019) 413-439.
Controlling the zero divisors of a commutative ring. S Glaz, Marcel Dekker Lecture Notes in Pure Appl. Math. 231S. Glaz, Controlling the zero divisors of a commutative ring, Marcel Dekker Lecture Notes in Pure Appl. Math. 231 (2002) 191-212.
The space of minimal prime ideals of a commutative ring. M Henriksen, M Jerison, Trans. Amer. Math. Soc. 115M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965) 110-130.
N Hindman, D Strauss, Algebra in the Stone-Čech compactification, theory and applications. De Gruyter Textbooksecond editionN. Hindman and D. Strauss, Algebra in the Stone-Čech compactification, theory and applications, second edition, De Gruyter Textbook (2012).
Commutative rings with zero divisors. J A Huckaba, Marcel Dekker, IncJ.A. Huckaba, Commutative rings with zero divisors, Marcel Dekker, Inc. (1988).
Minimal prime ideals in commutative semigroups. J Kist, Proc. London Math. Soc. 133J. Kist, Minimal prime ideals in commutative semigroups, Proc. London Math. Soc. 13(3) (1963) 31-50.
The minimal prime spectrum of a reduced ring. E Matlis, Illinois J. Math. 27E. Matlis, The minimal prime spectrum of a reduced ring, Illinois J. Math. 27 (1983) 353-391.
Sur la compacité du spectre minimal d'un anneau. Y Quentel, Bull. Soc. Math. France. 99Y. Quentel, Sur la compacité du spectre minimal d'un anneau, Bull. Soc. Math. France 99 (1971) 265-272.
The use of ultraproducts in commutative algebra. H Schoutens, Lecture Notes in Mathematics. SpringerH. Schoutens, The use of ultraproducts in commutative algebra, Lecture Notes in Mathematics, Springer (2010).
M Stekelenburg, Ultrafilters and topology. Bachelor Project, Leiden Univ.M. Stekelenburg, Ultrafilters and topology, Bachelor Project, Leiden Univ. (2014).
Compactifications of discrete spaces. U M Swamy, Int. J. Contemp. Math. Sciences. 422U.M. Swamy and et al., Compactifications of discrete spaces, Int. J. Contemp. Math. Sciences, 4(22) (2009) 1079-1084.
A new approach to the patch and flat topologies on a spectral space with applications. A Tarizadeh, Commun. Algebra. 4912A. Tarizadeh, A new approach to the patch and flat topologies on a spectral space with applications, Commun. Algebra, 49(12) (2021) 5322-5328.
Avoidance and absorbance. A Tarizadeh, J Chen, J. Algebra. 582A. Tarizadeh and J. Chen, Avoidance and absorbance, J. Algebra, 582 (2021) 88-99.
Flat topology and its dual aspects. A Tarizadeh, Commun. Algebra. 471A. Tarizadeh, Flat topology and its dual aspects, Commun. Algebra 47(1) (2019) 195-205.
Zariski compactness of minimal spectrum and flat compactness of maximal spectrum. A Tarizadeh, J. Algebra Appl. 18111950202A. Tarizadeh, Zariski compactness of minimal spectrum and flat compactness of maximal spectrum, J. Algebra Appl. 18(11) (2019) 1950202.
Stone type representations and dualities by power set ring. A Tarizadeh, Z Taheri, J. Pure Appl. Algebra. 22511106737A. Tarizadeh and Z. Taheri, Stone type representations and dualities by power set ring, J. Pure Appl. Algebra, 225(11), (2021) 106737.
Structure theory of p.p. rings and their generalizations. A Tarizadeh, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM). 1154178A. Tarizadeh, Structure theory of p.p. rings and their generalizations, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM), 115(4), 178 (2021).
Structural results on lifting, orthogonality and finiteness of idempotents. A Tarizadeh, P K Sharma, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM). 116154A. Tarizadeh and P.K. Sharma, Structural results on lifting, orthogonality and finiteness of idempo- tents, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM), 116(1), 54 (2022).
The Stone-Čech compactification. R C Walker, SpringerR.C. Walker, The Stone-Čech compactification, Springer (1974).
| [] |
[
"NSVS 06507557; a low-mass double-lined eclipsing binary ⋆ †",
"NSVS 06507557; a low-mass double-lined eclipsing binary ⋆ †"
] | [
"O Ç Akırlı \nScience Faculty, Astronomy and Space Sciences Dept\nEge University\n35100Bornova, İzmirTurkey\n",
"C İbanoǧlu \nScience Faculty, Astronomy and Space Sciences Dept\nEge University\n35100Bornova, İzmirTurkey\n"
] | [
"Science Faculty, Astronomy and Space Sciences Dept\nEge University\n35100Bornova, İzmirTurkey",
"Science Faculty, Astronomy and Space Sciences Dept\nEge University\n35100Bornova, İzmirTurkey"
] | [
"Mon. Not. R. Astron. Soc"
] | In this paper we present the results of a detailed spectroscopic and photometric analysis of the V=13 m .4 low-mass eclipsing binary NSVS 06507557 with an orbital period of 0.515 d. We obtained a series of mid-resolution spectra covering nearly entire orbit of the system. In addition we obtained simultaneous VRI broadband photometry using a small aperture telescope. From these spectroscopic and photometric data we have derived the system's orbital parameters and determined the fundamental stellar parameters of the two components. Our results indicate that NSVS 06507557 consists of a K9 and an M3 pre-main-sequence stars with masses of 0.66±0.09 M ⊙ and 0.28±0.05 M ⊙ and radii of 0.60±0.03 and 0.44±0.02 R ⊙ , located at a distance of 111±9 pc. The radius of the less massive secondary component is larger than that of the zero-age main-sequnce star having the same mass. While the radius of the primary component is in agreement with ZAMS the secondary component appers to be larger by about 35 % with respect to its ZAMS counterpart. Night-to-night intrinsic light variations up to 0 m .2 have been observed. In addition, the H α , H β lines and the forbidden line of [Oi] are seen in emission. The Lii 6708Å absorption line is seen in most of the spectra. These features are taken to be the signs of the classic T Tauri stars' characteristics. The parameters we derived are consistent with an age of about 20 Myr according to the stellar evolutionary models. The spectroscopic and photometric results are in agreement with those obtained by theoretical predictions. | 10.1111/j.1365-2966.2009.15703.x | [
"https://arxiv.org/pdf/0909.1921v1.pdf"
] | 16,554,635 | 0909.1921 | 86cb19e10366a28564d96d0f42524f205cd13796 |
NSVS 06507557; a low-mass double-lined eclipsing binary ⋆ †
2008
O Ç Akırlı
Science Faculty, Astronomy and Space Sciences Dept
Ege University
35100Bornova, İzmirTurkey
C İbanoǧlu
Science Faculty, Astronomy and Space Sciences Dept
Ege University
35100Bornova, İzmirTurkey
NSVS 06507557; a low-mass double-lined eclipsing binary ⋆ †
Mon. Not. R. Astron. Soc
0002008Released 2009 Xxxxx XXPrinted (MN L A T E X style file v2.2)stars:activity-stars:fundamental parameters-stars:low mass-stars :binaries:eclipsing-stars:T-Tauri
In this paper we present the results of a detailed spectroscopic and photometric analysis of the V=13 m .4 low-mass eclipsing binary NSVS 06507557 with an orbital period of 0.515 d. We obtained a series of mid-resolution spectra covering nearly entire orbit of the system. In addition we obtained simultaneous VRI broadband photometry using a small aperture telescope. From these spectroscopic and photometric data we have derived the system's orbital parameters and determined the fundamental stellar parameters of the two components. Our results indicate that NSVS 06507557 consists of a K9 and an M3 pre-main-sequence stars with masses of 0.66±0.09 M ⊙ and 0.28±0.05 M ⊙ and radii of 0.60±0.03 and 0.44±0.02 R ⊙ , located at a distance of 111±9 pc. The radius of the less massive secondary component is larger than that of the zero-age main-sequnce star having the same mass. While the radius of the primary component is in agreement with ZAMS the secondary component appers to be larger by about 35 % with respect to its ZAMS counterpart. Night-to-night intrinsic light variations up to 0 m .2 have been observed. In addition, the H α , H β lines and the forbidden line of [Oi] are seen in emission. The Lii 6708Å absorption line is seen in most of the spectra. These features are taken to be the signs of the classic T Tauri stars' characteristics. The parameters we derived are consistent with an age of about 20 Myr according to the stellar evolutionary models. The spectroscopic and photometric results are in agreement with those obtained by theoretical predictions.
INTRODUCTION
Low-mass stars constitute the majority of stars by number in our Galaxy. Since their lower masses with respect to the Sun they have also very low intrinsic brightness. Although the intrinsic faintness of these stars many low-luminosity stars were discovered particularly by the near-infrared sky surveys: Deep Near Infrared Survey (Delfosse et al. 1997), Two Micron All Sky Survey (Skrutskie et al. 1997), Sloan Digital Sky Survey (York et al. 2000), Northern
Sky Variability Survey (Wozniak et al. 2004). Since their main-sequence lifetimes are considerably longer than the age of the universe both the young and old low-mass stars are located on the lower right part of the the HR diagram. Low-mass stars surround many important regions of stellar parameter space which include the onset of complete convection in the stellar interior, the onset of electron degeneracy in the core, and the formation of dust and depletion metals onto dust grains in the stellar atmosphere (West et al. 2004). Recent studies have shown that while the observed radii of the low-mass stars are significantly larger than those predicted by current stellar models, in contrast their effective temperatures are cooler (Ribas et al. 2008, Lopez-Morales andRibas 2005). Chabrier, Gallardo & Baraffe (2007) have put forward the hypothesis that the observed radius and temperature discrepancies are consequences of the convection due to rotation and/or magnetic field and the presence of large surface magnetic spots. Therefore low-mass stars are key interest in studies of both formation of the stars in star-forming regions and comparison their parameters with those predicted from theoretical stellar models.
The fundamental parameters such as mass, radius, effective temperature and luminos-hereafter NSVS 0650) was discovered by Shaw and Lopez-Morales (2006) using the database NSVS (Wozniak et al. 2004). The eclipse period was determined to be 0.515 d. Later on the first VRI light curves and preliminary models are presented by Coughlin and Shaw (2007).
Taking the BVRI magnitudes from the USNO NOMAD catalog and JHK from 2MASS catalog they estimated the effective temperature of 3860 K for the primary, corresponding to a spectral type of M0V. As they have noted a difficulty encountered in modeling was the high-level spot activity of the components. Not only the radii and effective temperatures of the component stars were determined but also the rough masses estimated by them. We have conducted a photometric and spectroscopic monitoring program of several low-mass eclipsing binaries. In this paper we present, the results of multi-wavelength optical photometry and spectroscopy for double-lined eclipsing binary NSVS 0650.
OBSERVATION
Photometry
NSVS 0650757 was first identified in the Northern Sky Variability Survey (NSVS; Wozniak et al. 2004) as a detached eclipsing binary system with a maximum, out-of-eclipse V-bandpass magnitude V =13 m .05 and a period of P=0.51509 day. The data from the NSVS, obtained with the Robotic Optical Transient Search Experiment telescopes (ROTSE), contains positions, light curves and V magnitudes for about 14 million objects ranging in magnitudes from 8 to 15.5. The B, V, R, and I magnitudes for NSVS 0650 were listed in the USNO NOMAD catalog as ( Naval Observatory Merged Astronomical Dataset, NOMAD-1.0, Zacharias et al. 2004), B=14 m .53, V =13 m .37, R=12 m .47; on the other hand the infra-red magnitudes in three bandpasses were given as J=10 m .918 H=10 m .267, and K=10 m .092 in the 2MASS catalog (Cutri et al. 2003 the target NSVS0650 was placed near to the center of the CCD and three nearby stars located on the same frame were taken for comparison. The stars GSC 01760-01860 and USNO A2.0 1125 638990 were selected as comparison and check, respectively. Therefore the target and comparison stars could be observed simultaneously with an exposure time of 10 seconds. Since the variable is very cool, red star the signal-to-noise ratio was highest in I-and lowest in the V-bandpass. The differential observations of the comparison stars showed that they are stable during time span of our observations. The data were processed with standard data reduction procedures including bias and over scan subtraction, flatfielding, and aperture photometry. A total of 743, 812 and 612 photometric measurements were obtained in each V, R and I bandpasses, respectively. The average uncertainity of each differential measurement was less than 0 m .030. The V-, R-and I-bandpass magnitude differences, in the sense of variable minus comparison, are listed in Table 1
Orbital period and ephemeris
The first orbital period for NSVS 0650 was determined as P=0.51509957 d by Shaw and Lopez-Morales (2006) days, and an initial epoch T 0 (HJD)=2453312.3722±0.0005 for the mid-primary eclipse were calculated using a least square fit. Partial primary and secondary eclipses which were de- tected in the time series photometric data were used in combination with the NSVS photometry to derive this ephemeris for the system.
We obtained three times of mid-primary and one secondary eclipse during our observing run. The mid-eclipse timings and their standard deviations are caculated using the method of Kwee & van Woerden (1956). These timings of the eclipses were listed in Table 1 together with two primary and a secondary eclipse collected from literature. The times for mid-eclipses are the average of times obtained in three bandpasses. We define the epoch of the system, T 0 , to be the midpoint of the most complete primary eclipse. For this reason we use the V-, R-, and I-bandpass data obtained on JD=2 454 746 which cover almost the whole primary eclipse. A linear least square fit to the data listed in Table 1 yields the new ephemeris as,
Min I (HJD) = 24 54746.3801(5) + 0.51508836(9) × E,
where E corresponds to the cycle number. The residuals in the last column of Table 2 are computed with the new ephemeris. While the orbital period is nearly the same with that determined by Coughlin & Shaw (2007) its uncertainty is now very smaller than estimated by them. In the computation of the orbital phase for individual observations we used this ephemeris.
Intrinsic light variations
The light curve of NSVS 0650 shows two well-separated eclipses, as a typical of detached eclipsing binaries. The phase difference between the eclipses is about 0.5 which indicates a nearly circular orbit. Since the depths of the eclipses are very different, indicating that the components have unequal effective temperatures. The light variation both in primary and out-of-eclipse is clearly seen in all bandpasses. This light variation of about 0.2 mag peakto-peak in the out-of-eclipse portions of the light curve reveals that there is an intrinsic variation in one or both components of the system. The amplitude of the intrinsic variations seems to larger with longer wavelengths. The light variations observed on JD 2454746 with long duration, just between primary and secondary eclipses, and also on JD 2454767 with very short duration, resemble a flare-like event which is common in M-type dwarf stars.
The data obtained by us are concentrated on seven nights ranging a time span of 56 days. The stars having masses smaller than that of the Sun are known to be heavely spotted.
Therefore the out-of-eclipse light variations may be attributed to large spots on the surface of one or both component stars. In addition, flares on the less massive star cannot be ignored.
However, it should be noted that the intrinsic light variations do not resemble to those observed in the spotted stars. A spot or spot groups on one or both components produces usually wave-like distortion on their light curves. However, the out-of-eclipse light variations in NSVS 0650 seem to not correlated with the orbital period.
Spectroscopy
Optical spectroscopic observations of NSVS 0650 were obtained with the Turkish Faint Object Spectrograph Camera (TFOSC) attached to the 1.5 m telescope on 3 nights (September 15, 16, and 17, 2008) under good seeing conditions. Further details on the telescope and the spectrograph can be found at http://www.tug.tubitak.gov.tr. The wavelength coverage of each spectrum was 4100-8100Å in 11 orders, with a resolving power of λ/∆λ 7 000 at 6563
A and an average signal-to-noise ratio (S/N) was ∼120. We also obtained a high S/N spectrum of the M dwarf GJ 740 (M0 V) and GJ 623 (M1.5 V) for use as templates in derivation of the radial velocities (Nidever et al. 2002).
The electronic bias was removed from each image and we used the 'crreject' option for cosmic ray removal. Thus, the resulting spectra were largely cleaned from the cosmic rays.
The echelle spectra were extracted and wavelength calibrated by using Fe-Ar lamp source with help of the IRAF echelle package.
The stability of the instrument was checked by cross correlating the spectra of the standard star against each other using the fxcor task in IRAF. The standard deviation of the differences between the velocities measured using fxcor and the velocities in Nidever et al.
(2002) was about 1.1 km s −1 .
Spectral classification
We have used our spectra to reveal the spectral type of the primary component of NSVS 0650.
For this purpose we have degraded the spectral resolution from 7 000 to 3 000, by convolving them with a Gaussian kernel of the appropriate width, and we have measured the equivalent widths (EW ) of photospheric absorption lines for the spectral classification. We have followed the procedures of Hernández et al. (2004) ment with that we derived by wide-band B-V and V-I photometric colors. We estimated a temperature of 3920±175 K from the calibrations of Tokunaga (2000). Temperature uncertainty of the primary component results from considerations of spectral type uncertainties, and calibration differences. The weighted mean of the effective temperature of the primary star is 3960±80 K. The effective temperature of the primary star what we derived from the photometric measurements is an a good agreement with that we estimated from the spectra alone.
ANALYSIS
Radial velocity curve
To derive the radial velocities for the components of binary system, the 16 TFOSC spectraof the eclipsing binary were cross-correlated against the spectrum of GJ 740, a single-lined M0V star, on an order-by-order basis using the fxcor package in IRAF. The majority of the spectra showed two distinct cross-correlation peaks in the quadrature, one for each component of the binary. Thus, both peaks were fit independently in the quadrature with a Gaussian profile to measure the velocity and errors of the individual components. If the two peaks appear blended, a double Gaussian was applied to the combined profile using de-blend function in the task. For each of the 16 observations we then determined a weighted-average radial velocity for each star from all orders without significant contamination by telluric absorption features. Here we used as weights the inverse of the variance of the radial velocity measurements in each order, as reported by fxcor. In these data, we find no evidence for a third component, since the cross-correlation function showed only two distinct peaks.
We adopted a two-Gaussian fit algorithm to resolve cross-correlation peaks near the first and second quadratures when spectral lines are visible separately. Figure 2 shows examples of cros-correlations obtained by using the largest FWHM at nearly first and second quadratures.
The two peaks, non-blended, correspond to each component of NSVS 0650. The stronger peaks in each CCF correspond to the more luminous component which has a larger weight into the observed spectrum.
The heliocentric RVs for the primary (V p ) and the secondary (V s ) components are listed in Table 3, along with the dates of observation and the corresponding orbital phases com- to the heliocentric reference system by adopting a radial velocity of 9.5 km s −1 for the template star GJ 740. The RVs listed in Table 3 are the weighted averages of the values obtained from the cross-correlation of orders #4, #5, #6 and #7 of the target spectra with the corresponding order of the standard star spectrum. The weight W i = 1/σ 2 i has been given to each measurement. The standard errors of the weighted means have been calculated on the basis of the errors (σ i ) in the RV values for each order according to the usual formula (e.g. Topping 1972). The σ i values are computed by fxcor according to the fitted peak height, as described by Tonry & Davis (1979).
First we analysed the radial velocities for the initial orbital parameters. We used the orbital period held fixed and computed the eccentricity of the orbit, systemic velocity and semi-amplitudes of the RVs. The results of the analysis are as follows: e=0.002±0.001, i.e. formally consistent with a circular orbit, γ=44±6 km s −1 , K 1 =77±3 and K 2 =181±12 km s −1 .
Using these values we estimate the projected orbital semi-major axis and mass ratio as: asini=2.63±0.12 R ⊙ and q = M 2 M 1 =0.425±0.044.
Light curve modeling
As we noted in Section 2.3 the light curve of the system is considerably distorted due to light fluctuations both at maxima and in the deeper primary minimum. The largest distortion with longest duration was observed on JD 24 54746. Neither the amplitude nor the period or cycle of these intrinsic variations are known at this step. Therefore, we take all the available V-, R-and I-bandpass data for the orbital parameter analysis. The differential magnitudes of 743 in V-, 812 in R-and 612 in I-bandpass were converted to intensities using the differential magnitudes at out-of-eclipses as ∆V=1 m . and mass of the star. Therefore we adopted the linear limb-darkening coefficients from Van Hamme (1993) as 0.39 and 0.28 for the primary and secondary components, respectively; the bolometric albedos from Lucy (1967) as 0.5, typical for a fully convective stellar envelope, the gravity brightening coefficients as 0.32 for the both components. The rotation of components is assumed to be synchronous with the orbital one. The mass-ratio of 0.425 was adopted from the semi-amplitudes of the radial velocities. We started the light curve analysis with an effective temperature of 3960 K for the primary star of NSVS 0650. The adjustable parameters in the light curves fitting were the orbital inclination, the surface potentials, the effective temperature of secondary, the luminosity of the primary.
Using a trial-and-error method we obtained a set of parameters, which represented the observed light curves. A detached configuration, Mode 2, with coupling between luminosity and temperature was chosen for solution. The iterations were carried out automatically until convergence, and solution was defined as the set of parameters for which the differential corrections were smaller than the probable errors. The orbital and stellar parameters from
stars (cTTSs) and the weak-lined T Tauri stars (wTTSs). A cTTS is surrounded by an
optically thick disk from which it accretes material. Whereas a wTTS represents the final stage of accretion and disc-clearing processes (Bertout et al. 2007, Schisano et al. 2009).
The equivalent width of H α emission is used as an empirical criterion to distinguish between cTTS and wTTS, being smaller in the latters. Due to possible varability, no clean cut can be defined between the cTTS and wTTS based on the H α emission alone.
Spectral and photometric properties and night-to-night light variability of NSVS 0650
indicate that the active star in the system resembles many characteristics of the TTSs as given above and discussed by Alcala et al. (1993), Covino et al. (1996), and Alencar & Basri (2000). As it is known the optical emission lines are definite characteristics of the many late type, main-sequence systems, including NSVS 0650. Another fundamental characteristic of TTSs is the variations of H-line profiles (Ferro & Giridhar 2003). NSVS 0650 is composed of low-mass stars which cover most of the properties of the T Tauri stars.
Line profiles
The most conspicuous line with dramatic profile variations in the system's spectrum appears to be the H α . The H α line is the most prominent feature in the spectra of TTSs. The presence of the Li absorption line at 6708Å (see Fig. 5, for an example) and weak H α in emission leads us to classify the star as weak-lined T Tauri star. In Figure 6 we display the H α line to UX Tau A (see Reipurth et al. 1996). The dramatic changes in the shape of the H α line, collected on JD 2454727, are clearly seen in the last five panels of Fig. 6. The H α line in the spectra of the NSVS 0650 taken at phases of about 0.2114, 0.3141 and 0.3772 displays blue-shifted absorption, similar to wTTS GG Tau (Folha & Emerson, 2001). It turns to be single absorption at orbital phases of 0.4610 and 0.5410.
The higher Balmer series, H β and H γ lines of NSVS 0650 generally appear to be in emission at all orbital phases. Again, dramatic line profile changes are evident. Inverse P Cygni profiles are also visible at some orbital phases.
The existence of blue-shifted absorption components in the Balmer lines of TTSs' spectra was first noted by Herbig (1962), who suggested that these absorption components are evidence for strong stellar winds. On the other hand Walker (1972) drew attention to the wTTSs which have red-shifted absorption in the higher-order Balmer lines. These inverse P Cygni profiles have generally been interpreted in terms of material accreting onto the young stars. The optical observations of unidentified Einstein Observatory X-ray satellite sources led to the the discovery of many TTSs with weak H α and IR excess emission (Strom et al. 1990). The wTTSs have also dark spots as in the case of cTTSs but stronger X-ray in emission than cTTSs. They have also shallow or no disks. If a wTTS has still a disk some winds are blown away from this disk. Most TTSs are members of close binaries wich may be born without a disk or have a short-lived disk (Neuhauser, 1997). Three types of binaries including TTS without disks, with circumstellar disks and with circumbinary disks exist.
Forbidden lines
One of the most important characteristics in the spectra of the cTTSs is the presence of forbidden emission lines. The forbidden neutral oxygen lines are not seen in the spectra of wTTSs. In the spectra of NSVS 0650 we observed forbidden [Oi] emission line at 6300Å. [Oi] 6300Å is highly variable and seems to correlate with the orbital phase. This forbidden emission line appears to slightly stronger at the first quadrature than at the second one.
In the optical spectrum of cTTSs the forbidden emission lines are dominated. These lines are usually patterns of shocked low-density regions of young stars (Fernandez & Cameron 2001
DISCUSSION
One of the goals of the present study is to derive the physical parameters of the low-mass stars in the eclipsing binary systems. As it is known eclipsing binaries are the most suitable laboratories for determining the fundamental properties of the stars and thus for testing the predictions of theoretical models. For this reason we started optical photometric and spectroscopic observations of some selected low-mass stars. We obtained multi-band light (Zacharias 2005) curves and spectra with a wide wavelength range. We analyzed the V-, R-, and I-bandpass light curves and the radial velocities separately using the modern codes. Then, we combined the photometric and spectroscopic solutions and derived the absolute parameters of the component stars. The standard deviations of the parameters have been determined by JKTABSDIM 1 code, which calculates distance and other physical parameters using sev-eral different sources of bolometric corrections (Southworth et al. 2005). The best fitting parameters are listed in Table 5 together with their formal standard deviations.
The luminosity and absolute bolometric magnitudes M bol of the stars were computed from their effective temperatures and their radii. Since low-mass stars radiates more energy at the longer wavelengths we used V RIJHK magnitudes given by Coughlin & Shaw (2007).
Applying BV RIJHKL magnitudes-T ef f relations given by Girardi (2002) we calculated the distance to NSVS 0650 as d=111±9 pc. Estimating distances to low-mass stars are strongly depended on the bolometric corrections. If we adopt the bolometric corrections given by Siess, Forrestini & Dougados (1997) Locations of the primary and secondary components on the theoretical mass-radius and T ef f -log L/L ⊙ diagrams are shown in Fig. 9. The mass tracks and isochrones are adopted from Siess, Forrestini & Dougados (1997) and Siess (2000). Since the stars appear to be in pre-main sequence evolution we adopted Z=0.03. These mass tracks are very close to those obtained for solar abundance. The radius of more massive primary component is in agreement with that zero-age main sequence star having the same mass. However, the secondary is about 35 % larger than the main-sequence counterpart. This result confirm the hypothesis proposed by Chabrier, Gallardo & Baraffe (2007) for the larger radius of the low-mass convective stars. The existence of Li 6708Å absorption line in the spectra and comparison the absolute dimensions of the components with the evolutionary tracks may be taken as an indicator of the pre-main-sequence stars. The components of NSVS 0650 lie on the isochrones between 15-30 Myr, still in contracting phase toward the main-sequence. If we use the isochrones plotted M v versus B − V we estimate an age of about 10-15 Myr. This difference arises from the bolometric corretions given by Siess, Forrestini & Dougados (1997).
We used the color-temperature calibrations given for the main-sequence stars for estimating the effective temperature of the more massive primary component. If we use the colortemperature relation given for luminosity class IV stars (de Jager and Nieuwenhuijzen, 1987)
we find even smaller effective temperature af about 3 700 K for the primary component. The difference of about 250 K in the effective temperature of the primary star shifts its location to the lower-right in the HR diagram which corresponds to a smaller age. We estimate an age of about 50 Myr using the pre-main-sequence models given by D' Antona & Mazzetelli (1997), and 63 Myr by Baraffe et al. (1998) slightly larger than that given by Siess et al.
model. The evolutionary models indicate that a star with a mass of 0.66 M ⊙ takes about 100 Myr to contract and reach its normal main-sequence radius.
The Lii 6708Å line is often used an age indicator. In the spectra of NSVS 0650 the Lii line is clearly seen. Moreover, in the optical spectrum of the system, we observed also H α and H β lines as in emission. In addition, strong emission of the [Oi] forbidden line is visible.
These features are signs of cTTS but, in contrary, the measured EW values point a wTTS.
The primary component of NSVS 0650 appears in the region of Li−poor stars located on the HR diagram (see Fig. 8 in Sestito, Palla & Randich, 2008). The measured EW of 0.3Å for Lii is in agreement with this classification. High-resolution spectra are urgently required to confirm our finding and to derive which sub-group, cTTS or wTTS, it belongs.
(available in the electronic form at the CDS). The light curve shows a deep primary eclipse with an amount of 0 m .70 in the V-bandpass and a shallow secondary eclipse with an amount of 0 m .23 which are clearly separated in phase, as is typical of fully detached binaries. The primary and secondary eclipses occur almost 0.5 phase interval, indicating nearly circular orbit. An inspection of the nightly light curves presented in Fig. 1 clearly indicates considerable out-of-eclipse light variations up to 0 m .2. This intrinsic variation of the binary system manifests itself in the deeper primary eclipse.
Figure 1 .
1The V-, R-, and I-bandpass nightly light curves for NSVS 0650 from top to bottom. The V-, R-, and I-bandpass light curves clearly show that the brightness of the variable significantly varies from night to night, particularly in out of eclipse.
Figure 2 .
2Sample of Cross Correlation Functions (CCFs) between NSVS 0650 and the radial velocity template spectrum around the first and second quadrature.
Figure 3 .
3Radial velocity curve folded on a period of 0.51508836 days, where phase zero is defined to be at primary mid-eclipse. Symbols with error bars show the RV measurements for two components of the system (primary: open circles, secondary: open squares).
648±0 m .003, ∆R=1 m .198±0 m .001, ∆I=0 m .575±0 m .002. We used the most recent version of the eclipsing binary light curve modeling algorithm of Wilson & Devinney (1971) (with updates), as implemented in the phoebe code of Prša & Zwitter (2005). The code needs some input parameters, which depend upon the physical structures of the component stars. In the light curve solution we fixed some parameters whose values can be estimated from global stellar properties, such as effective temperature
Figure 4 .
4The phase folded VRI light curves for NSVS 0650. The best fitting solutions represented by the solid lines are also plotted for comparison (see text).the V-, R-and I-bandpass light and radial velocity curves analysis are listed inTable 4. The uncertainties given in this table are taken directly from the out-put of the program. The computed light and velocity curves corresponding to the individual light-velocity solutions are compared with the observations in Figs. 3 and 4.4 THE SPECTRUMNSVS 0650 has a complex spectrum over the wavelength interval from ∼4100 to 8100Å. The spectrum is dominated by forbidden lines and to a smaller degree, permitted emission lines of neutral metals. Strong and broad double-peaked H α , H β and [Oi] lines are present, with the peak separation in H α larger than the higher Balmer lines. The presence of the strong Li
Figure 5 .
5The composite spectrum in the spectral region containing the Li I 6708Å line observed on JD 24 54727.4836. 6708Å absorption line can serve as a reliable youth indicator of a star, as evidenced in the case of NSVS 0650. Young, low-mass pre-main-sequence stars are called T Tauri stars (TTS). They present the following characteristics: 1) Emission line spectra, 2) Presence of forbidden narrow-lines such as [Oi], [Nii] and [Siii], 3) Photospheric continuum excesses (Barrado y Navascues and Martin, 2003). TTSs are classified into two sub-groups, the classical T Tauri
Figure 6 .
6Variation of the Hα line profiles of NSVS 0650. The normalized spectrum at the Hα ordered with the orbital phase. The vertical thick and thin lines show the rest wavelenghts corresponding to the primary and secondary component photospheres, respectively. region observed at various orbital phases in three consecutive nights. Each spectrum has been normalized to the continuum. Julian date and the orbital phase for each observation are given in each panel. On JD 2454725 the H α line appears to be a single, shallow absorption, i.e., filled-in by emission, at orbital phase of about 0.3058. At orbital phases of 0.4242, 0.5079 and 0.5939 the same line becomes single, emission above the continuum and at phase of 0.6770 it turns to be an absorption again. On JD 2454726, the following night, the H α line is seen as single absorption at phases of 0.1404 and 0.2237, whereas double-peaked emission profiles at phases of 0.3162 and 0.3996, but it turns to absorption in a short time interval
Figure 7 .
7Variation of the forbidden emission line profiles of the [Oi] at 6300Å. at phases of 0.4829 and 0.5973. The H α emission line profile at orbital phase of 0.3996 has an unexpected shape because it is very resemblance of inverse P Cygni profile, most similar
Figure 7
7displays the [Oi] emission line profiles at various orbital phases. [Oi] emission line shows single peaks, but the line centroid is shifted to the blue. The strength of emission in
Figure 8 .
8Correlations between equivalent witdh of the most prominent emission lines measured in the spectra.
the distance to NSVS 0650 reduces to about d=86±4 pc. The mean light contribution of the secondary star L 2 /(L 1 +L 2 )=0.22 obtained directly from the I-bandpass light curve analysis is in a good agreement with that estimated from the bolometric luminosities as 0.22. This result indicates that the light contribution of the less massive component is very small, indicating its effect on the color at outsite eclipse is very limited.
Figure 9 .
9Comparison between stellar models and the absolute dimensions of NSVS 0650 in the mass-radius (a) and T ef flog (L/L ⊙ ) (b) planes. The mass-radius relations in panel (a) were derived using the stellar models of Siess et al. (2000) for Z=0.03 with an age of 15 (dotted), 20 (dashed) and 30 Myr (dot-dashed). Panel (b) shows the locations of the components in the HR diagram. Evolutionary tracks for the masses of 0.25, 0.30, 0.60 and 0.70 M ⊙ are shown for comparison. The diagonal lines from left to right indicate isochrones with an age of 15 (dotted), 20 (dashed) and 30 Myr (dot-dashed) and zero-age main-sequence (continuous line). The filled-circle and square indicate the primary and secondary components of NSVS 0650, respectively.
) .
)In the NSVS survey, 262 V-bandpass measurements of the variable were obtained during the period June 1999 -March 2000 with a median sampling rate of 0.25 −1 . The resulting light curve exhibits periodic eclipses with a depth of ∼0 m .7 in the deeper eclipse and the mean standard deviation in the out-of-eclipse phases was about 0 m .073. The photometric observations of NSVS 0650 were carried out with the 0.4 m telescope at the Ege University Observatory. The 0.4 m telescope equipped with an Apogee 1kx1k CCD camera and standard Bessel VRI bandpasses. The observations were performed on seven nights between September 01 and November 30, 2008. To get the higher accuracyTable 1. Differential photometric measurements of NSVS 0650 in the V, R and I bandpasses.HJD(2 400 000+)
∆V
HJD(2 400 000+)
∆R
HJD(2 400 000+)
∆I
54725.35259
1.6025
54725.35296
1.1780
54725.35326
0.5799
54725.35370
1.6304
54725.35406
1.1572
54725.35437
0.5563
...
...
...
...
...
...
...
...
...
...
...
...
Table 2 .
2Times of minima measured from the V RI-bandpass light curves.HJD(2 400 000+)
E
Type
O-C
51537.1282**
-6230.5
II
0.0099
51581.1569**
-6145.0
I
-0.0015
53312.3722 †
-2784.0
I
-0.0006
54746.3809±0.0005
0.0
I
0.0000
54767.4978±0.0006
41.0
I
-0.0018
54771.3631±0.0004
48.5
II
0.0004
54781.4056±0.0002
68.0
I
-0.0014
* * From the NSVS database.
† Coughlin & Shaw (2007).
, choosing helium lines in the blue-wavelength region, where the contribution of the secondary component to the observed spectrum is almost negligible. From several spectra we measured EW HeI+FeIλ4922 = 1.18 ± 0.12Å.From the calibration relations EW -Spectral-Type ofHernández et al. (2004), we have derived a spectral type of K8 with an uncertainty of about 1 spectral subclass. The effective temperature deduced from the calibrations ofDrilling & Landolt (2000) orde Jager & Nieuwenhuijzen (1987) is about 4 050 K. The spectral-type uncertainty leads to a temperature error of ∆T eff ≈ 300 K.The catalogs USNO, NOMAD and 2MASS provides BVRIJHK magnitudes for NSVS 0650. Using the observed colors of B-V=1.36±0.02 and V-I=2.13±0.02 mag and colortemperature relationships given byDrilling & Landolt (2000) for the main sequence stars we estimate a spectral type K9±1 with an effective temperature of 3930±50 K for the primary star. The observed infrared colors of J-H=0.651±0.043 and H-K=0.175±0.038 given in the 2MASS catalog(Cutri et al. 2003) correspond to a spectral type of K9±2 is in a good agree-
Table 3 .
3Heliocentric radial velocities of NSVS 0650. The columns give the heliocentric Julian date, the orbital phase (according to the ephemeris in Eq. 1), the radial velocities of the two components with the corresponding standard deviations.puted with the new ephemeris given in §2.2. The velocities in this table have been correctedHJD 2400000+
Phase
Star 1
Star 2
Vp
σ
Vs
σ
54725.4190
0.3058
-41.3
11.1
218.0
13.6
54725.4800
0.4242
1.0
12.0
138.0
21.3
54725.5231
0.5079
41.0
11.1
-
-
54725.5675
0.5941
74.0
10.1
-50.0
17.8
54725.6102
0.6770
112.0
8.4
-108.0
12.7
54726.3640
0.1404
-12.0
11.5
172.5
14.7
54726.4069
0.2237
-44.4
2.4
231.1
14.2
54726.4545
0.3161
-33.0
9.8
216.6
27.3
54726.4975
0.3996
-9.0
11.4
140.1
16.7
54726.5404
0.4829
24.0
14.1
-
-
54726.5993
0.5973
90.0
9.8
-60.0
16.9
54727.4307
0.2114
-34.0
9.9
218.5
17.3
54727.4836
0.3141
-20.3
8.8
225.9
12.2
54727.5161
0.3771
-12.0
9.9
184.9
12.2
54727.5593
0.4610
22.0
12.6
-
-
54727.6005
0.5410
66.0
11.8
-
-
Table 4 .
4Results of the V-, R-, and I-bandpass light curve analysis for NSVS 0650. The adopted values are the weighted means of the values determined from the individual light curves.Parameters
V
R
I
Adopted
i o
83.5±0.2
86.5±1.3
81.7±0.6
83.3±0.6
T ef f 1 (K)
3 960[Fix]
3 960[Fix]
3 960[Fix]
3 960[Fix]
T ef f 2 (K)
3 269±51
3 401±43
3 412±48
3 365±48
Ω 1
4.847±0.091
4.738±0.90
4.982±0.145
4.886±0.090
Ω 2
3.735±0.035
4.151±0.061
3.740±0.090
3.830±0.067
r 1
0.228±0.005
0.231±0.005
0.224±0.007
0.227±0.006
r 2
0.176±0.003
0.148±0.004
0.171±0.006
0.167±0.005
L 1 /(L 1 + L 2 ) 0.889±0.014
0.866±0.010
0.785±0.020
--
χ 2
1.345
1.858
0.880
--
a See §2.1.2
). These shocks can be produced by the outflowing materials, winds, and/or jets. Strong H α and [Oi] emission lines in the optical spectra of NSVS 0650 are indicative of ongoing accretion. The strength of H β and its equivalent width (EW) seems to correlate well with that of H α , as shown inFig. 8. However the EWs of [Oi] do not correlate well with those of the H α , as is seen in the upper panel ofFig. 8. It appears that as if there is an anti-correlationbetween [Oi] and H α . The range of variation in the EWs of H α is between about 2Å and
about -3Å. Whereas the EWs of [Oi] vary from 0 to about -1Å. However, we observed the
most strong emission in [Oi] on JD 2454726.3640 at orbital phase of 0.1404 with an EW of
-2.2Å. We also measured the average EW of Lii as 0.3±0.2Å.
Table 5 .
5Fundamental parameters of the system. Ks (mag) * 10.918±0.032, 10.267±0.028, 10.092±0.025 µα, µ δ (mas yr −1 ) * * 23.70±6.10, 17.50±6.10 U, V, W (km s −1 ) -40±5, +19±4, -14±3 *2MASS All-Sky Point Source Catalogue (Cutri et al. 2003) **NOMAD CatalogParameter
Primary
Secondary
Mass (M ⊙ )
0.656±0.086
0.279±0.045
Radius (R ⊙ )
0.600±0.030
0.442±0.024
log g (cgs)
4.699±0.032
4.594±0.047
T ef f (K)
3 960±80
3 365±80
(vsin i) calc. (km s −1 )
59±3
43±2
log (L/L ⊙ )
-1.097±0.057
-1.647±0.062
d (pc)
111±9
J, H,
NSVS 06507557; a low-mass double-lined eclipsing binary
This can be obtained from http://http://www.astro.keele.ac.uk/∼jkt/codes.html
ACKNOWLEDGEMENTSThe authors acknowledge generous allotments of observing time at TUBITAK National Observatory (TUG) of Turkey. We also wish to thank the Turkish Scientific and TechnicalResearch Council for supporting this work through grant Nr. 108T210 and EBİLTEM Ege University Science Foundation Project No:08/BİL/0.27 . We have use of 2MASS USNO and NOMAD Catalogs as well as the Simbad, Visir, and ADS databases. The anonymous referee is gratefully acknowledged for useful and constructive suggestions.
. J M Alcala, E Covino, M Franchini, J Krautter, L Terranegra, R Wichmann, 272225Alcala J. M., Covino E., Franchini M., Krautter J., Terranegra L.; Wichmann R., 1993, 272, 225
. S H P Alencar, G Basri, AJ. 1191881Alencar S. H. P., Basri G., 2000, AJ, 119, 1881
. D Barrado Y Navascues, E L Martin, AJ. 1262997Barrado y Navascues D. & Martin E. L., 2003, AJ,126, 2997
. I Baraffe, G Chabrier, F Allad, P H Hauschildt, A&A. 337403Baraffe I., Chabrier G., Allad F., Hauschildt P. H., 1998, A&A, 337, 403
. C Bertout, AA. 47321Bertout, C. et al. 2007, AA, 473, L21
. G Chabrier, J Gallardo, I Baraffe, A&A. 47217Chabrier G., Gallardo J., Baraffe I., 2007, A&A, 472, 17
. J L Coughlin, J S Shaw, JSARA. 17Coughlin J. L., and Shaw J. S., 2007, JSARA, 1, 7C
. E Covino, L Terranegra, A Magazzu, J M Alcala, S Allain, J Bouvier, J Krautter, R Wichmann, Aspc, 109421Covino E., Terranegra L., Magazzu A., Alcala J. M., Allain S., Bouvier J., Krautter J., Wichmann R., ASPC, 109, 421
The IRSA 2MASS All-Sky Point Source Catalog. R M Cutri, NASA/IPAC Infrared Science Archive. 477519Cutri R. M., et al., 2003, The IRSA 2MASS All-Sky Point Source Catalog, NASA/IPAC Infrared Science Archive. http://irsa.ipac.caltech.edu/applications/Gator/ D'Antona C., Mazzitelli A., 1997, ApJ, 477, 519
. X Delfosse, C G Tinney, T Forveille, A&A. 32725Delfosse X., Tinney C. G., Forveille T. et al., 1997, A&A, 327L, 25D
Allen's astrophysical quantities. J S Drilling, A U Landolt, 4th ed. Edited byDrilling J. S., Landolt A. U., 2000, Allen's astrophysical quantities, 4th ed. Edited by
. Arthur N Cox, ISBN: 0-387-98746-0AIP PressNew YorkArthur N. Cox. ISBN: 0-387-98746-0. Publisher: New York: AIP Press;
. Springer, 381Springer, 2000, p.381
. C De Jager, H Nieuwenhuijzen, AAP. 177217de Jager C., Nieuwenhuijzen H., 1987, AAP, 177, 217
. M Fernandez, F Cameron, A&A. 380264Fernandez M., Cameron F., 2001, A&A, 380, 264
. A Ferro, S Giridhar, A&A. 40829Ferro A., Giridhar S., 2003, A&A, 408, 29
. D F M Folha, J P Emerson, A&A. 36590Folha D. F. M. & Emerson J. P., A&A, 365, 90
. L Girardi, A Bressan, G Bertelli, C Chiosi, A&AS. 141371Girardi L., Bressan A., Bertelli G., and Chiosi C., 2002, A&AS, 141, 371
. G H Herbig, Adv. in Astron. Astrphys. 147Herbig G. H., 1962, Adv. in Astron. Astrphys., 1, 47
. J Hernández, N Calvet, C Briceño, L Hartmann, P Berlind, AJ. 1271682Hernández J., Calvet N., Briceño C., Hartmann L., Berlind P., 2004, AJ, 127, 1682
. K K Kwee, H Van Woerden, BAN. 12327Kwee K. K., van Woerden H., (1956), BAN, 12, 327
. M Lopez-Morales, I Ribas, ApJ. 6311120Lopez-Morales M. and Ribas I., 2005, ApJ, 631, 1120
. L B Lucy, Z. Astrophys. 6589Lucy L. B., 1967, Z. Astrophys., 65, 89
. D L Nidever, G W Marcy, R P Butler, D A Fischer, S S Vogt, ApJS. 141503Nidever D. L., Marcy G. W., Butler R. P., Fischer D. A., and Vogt S. S., 2002, ApJS, 141, 503
. R Neuhauser, Science. 2761363Neuhauser R., 1997, Science, 276, 1363
. A Prsa, T Zwitter, ApJ. 628426Prsa A. and Zwitter T., 2005, ApJ 628, 426
. B Reipurth, L.-A Nyman, R Chini, A&A. 314258Reipurth B., Nyman L.-A., Chini R., 1996, A&A, 314, 258
. I Ribas, A&A. 398239Ribas I., 2003, A&A, 398, 239
. J S Shaw, M Lopez-Morales, ASPC. 36215Shaw J. S., Lopez-Morales M., 2006, ASPC, 362, 15
. E Schisano, AA. 5011013Schisano, E. et al. 2009, AA, 501, 1013
. P Sestito, F Palla, S Randich, A&A. 487965Sestito P., Palla F., & Randich S., 2008, A&A, 487, 965
. L Siess, M Forestini, C Dougados, A&A. 325556Siess L., Forestini M., Dougados C., 1997, A&A, 325, 556
. L Siess, E Dufour, M Forestini, A&A. 358593Siess L., Dufour E., Forestini M., 2000, A&A, 358, 593
. R G Strom, S K Croft, J M Boyce, Science. 250437Strom R. G., Croft S. K., Boyce J. M., 1990, Science, 250, 437
. J Southworth, J V Clausen, A&A. 4611077Southworth J., Clausen J. V., 2005a, A&A, 461, 1077
. M F Skrutskie, AJ. 1311163Skrutskie M. F. et al., 1999, AJ, 131, 1163
Allen's astrophysical quantities. A T Tokunaga, A.N.CoxSpringer143Fouth EditionTokunaga A. T., 2000, "Allen's astrophysical quantities", Fouth Edition, ed. A.N.Cox (Springer), p.143
Errors of Observation and Their Treatment. J Topping, Chapman and Hall Ltd89Topping J., 1972, "Errors of Observation and Their Treatment", (Chapman and Hall Ltd.), p.89
. J Tonry, M Davis, AJ. 841511Tonry J. and Davis M., 1979, AJ 84, 1511
. W Van Hamme, AJ. 1062096van Hamme, W., 1993, AJ, 106, 2096
. P R Wozniak, AJ. 1272436Wozniak P. R. et al., 2004, AJ, 127, 2436
. A A West, S L Hawley, L M Walkowicz, AJ. 128426West A. A., Hawley S. L., Walkowicz L. M., et al., 2004, AJ, 128, 426
. R E Wilson, E J Devinney, ApJ. 166605Wilson R.E. and Devinney E.J., 1971, ApJ, 166, 605
. M F Walker, ApJ. 17589Walker M. F., 1972, ApJ, 175, 89
. D G York, J Adelman, J E Anderson, AJ. 1201579York D. G., Adelman J., Anderson J. E., et al., 2000, AJ, 120, 1579
. N Zacharias, D G Monet, S E Levine, AAS. 2054815Zacharias N., Monet D. G., Levine S. E., 2004, AAS, 205, 4815Z
| [] |
[
"Study of the decay K + → π + π − π + γ in the OKA experiment The OKA collaboration",
"Study of the decay K + → π + π − π + γ in the OKA experiment The OKA collaboration"
] | [
"M M Shapkin ",
"S A Akimenko ",
"A V Artamonov ",
"A M Blik ",
"V S Burtovoy ",
"S V Donskov ",
"A P Filin ",
"A V Inyakin ",
"A M Gorin ",
"G V Khaustov ",
"S A Kholodenko ",
"V N Kolosov ",
"A S Konstantinov ",
"V F Kurshetsov ",
"V A Lishin ",
"M V Medynsky ",
"Yu V Mikhailov ",
"V F Obraztsov ",
"V A Polyakov ",
"A V Popov ",
"V I Romanovsky ",
"V I Rykalin ",
"A S Sadovsky ",
"V D Samoilenko ",
"V K Semenov ",
"O V Stenyakin ",
"O G Tchikilev ",
"V A Uvarov ",
"O P Yushchenko ",
"V A Duk \nNow at University of Birmingham\nBirminghamUnited Kingdom\n",
"S N Filippov ",
"E N Guschin ",
"A A Khudyakov ",
"V I Kravtsov ",
"Yu G Kudenko ",
"A Yu Polyarush ",
"V N Bychkov ",
"G D Kekelidze ",
"V M Lysan ",
"B Zh ",
"Zalikhanov ",
"\n(NRC \"Kurchatov Institute\"-IHEP\nProtvinoRussia\n",
"\nINR RAS\nMoscowRussia), Russia\n",
"\nAlso at Moscow Institute of Physics and Technology\nMoscowRussia\n",
"\nAlso at NRNU Moscow Engineering Physics Institute (MEPhI)\nMoscowRussia\n"
] | [
"Now at University of Birmingham\nBirminghamUnited Kingdom",
"(NRC \"Kurchatov Institute\"-IHEP\nProtvinoRussia",
"INR RAS\nMoscowRussia), Russia",
"Also at Moscow Institute of Physics and Technology\nMoscowRussia",
"Also at NRNU Moscow Engineering Physics Institute (MEPhI)\nMoscowRussia"
] | [] | A high statistics data sample of the decays of K + mesons to three charged particles was accumulated by the OKA experiment in 2012 and 2013. This allowed to select a clean sample of about 450 events with K + → π + π − π + γ decays with the energy of the photon in the kaon rest frame greater than 30 MeV. The measured branching fraction of the K + → π + π − π + γ, with E * γ > 30 MeV is (0.71 ± 0.05) × 10 −5 . The measured energy spectrum of the decay photon is compared with the prediction of the chiral perturbation theory to O(p 4 ). A search for an up-down asymmetry of the photon with respect to the hadronic system decay plane is also performed. arXiv:1808.09176v2 [hep-ex] | 10.1140/epjc/s10052-019-6797-1 | [
"https://arxiv.org/pdf/1808.09176v2.pdf"
] | 102,351,783 | 1808.09176 | 446160e417415be4cf9833a1bc8d2b4c6f30d237 |
Study of the decay K + → π + π − π + γ in the OKA experiment The OKA collaboration
M M Shapkin
S A Akimenko
A V Artamonov
A M Blik
V S Burtovoy
S V Donskov
A P Filin
A V Inyakin
A M Gorin
G V Khaustov
S A Kholodenko
V N Kolosov
A S Konstantinov
V F Kurshetsov
V A Lishin
M V Medynsky
Yu V Mikhailov
V F Obraztsov
V A Polyakov
A V Popov
V I Romanovsky
V I Rykalin
A S Sadovsky
V D Samoilenko
V K Semenov
O V Stenyakin
O G Tchikilev
V A Uvarov
O P Yushchenko
V A Duk
Now at University of Birmingham
BirminghamUnited Kingdom
S N Filippov
E N Guschin
A A Khudyakov
V I Kravtsov
Yu G Kudenko
A Yu Polyarush
V N Bychkov
G D Kekelidze
V M Lysan
B Zh
Zalikhanov
(NRC "Kurchatov Institute"-IHEP
ProtvinoRussia
INR RAS
MoscowRussia), Russia
Also at Moscow Institute of Physics and Technology
MoscowRussia
Also at NRNU Moscow Engineering Physics Institute (MEPhI)
MoscowRussia
Study of the decay K + → π + π − π + γ in the OKA experiment The OKA collaboration
To be submitted to EPJC
A high statistics data sample of the decays of K + mesons to three charged particles was accumulated by the OKA experiment in 2012 and 2013. This allowed to select a clean sample of about 450 events with K + → π + π − π + γ decays with the energy of the photon in the kaon rest frame greater than 30 MeV. The measured branching fraction of the K + → π + π − π + γ, with E * γ > 30 MeV is (0.71 ± 0.05) × 10 −5 . The measured energy spectrum of the decay photon is compared with the prediction of the chiral perturbation theory to O(p 4 ). A search for an up-down asymmetry of the photon with respect to the hadronic system decay plane is also performed. arXiv:1808.09176v2 [hep-ex]
Introduction
The present experimental status of K + → π + π − π + γ is rather meagre. It was observed in one experiment on statistics of 7 events [1]. The photon energies in these events were low, that did not allow to search for deviations from а simple QED process of photon emission. The measured value of the K + → π + π − π + γ branching ratio, with E * γ > 5M eV is (1.04 ± 0.31) × 10 −4 [2]. In the present analysis we have a possibility for more detailed study of this decay using larger data sample collected by the OKA experiment. The appropriate theory framework for such investigation is the chiral perturbation theory [3] (CHPT). To the lowest order in an expansion in momenta and meson masses, the radiative decays are completely determined [4] by the non-radiative amplitude for K + → π + π − π + . At next-to-leading order, a full-fledged CHPT calculation of nonleptonic weak amplitudes of O(p 4 ) is required. Such a calculation is done in [5].
Separated kaon beam and OKA experiment
The OKA experiment makes use of a secondary hadron beam at the U-70 Proton Synchrotron of NRC "Kurchatov Institute"-IHEP, Protvino, with enhanced fraction of kaons obtained by RF-separation with Panofsky scheme [6]. The OKA setup, Fig. 1, is a double magnetic spectrometer complemented by electromagnetic and hadron calorimeters and a Decay Volume. The first magnetic spectrometer, consisting of the magnet M 1 and surrounding 1 mm pitch PC's (BPC (1Y ) , BPC (2Y,2X) , BPC (3X,3Y ) , BPC (4X,4Y ) ) serves for the beam momentum measurement. It is supplemented by two threshold Cherenkov counters 1 ,Č 2 for kaon identification and by beam scintillation counters S 1 , S 2 , S 3 , S 4 . The 11 m long Decay Volume (DV) filled with helium contains 11 rings of guard system (GS). To reinforce GS, a gamma detector (BGD), made of lead glass blocks located behind the DV is used as a veto at large angles, while low angle particles pass through a central opening. The wide aperture spectrometric magnet, SM (SP 40A) , with a field integral of ∼ 1 Tm serves as a spectrometer for the charged decay products together with corresponding tracking chambers: 2 mm PC's (PC 1,...,8 ), 9 mm diameter straw tubes ST (1,2,3) and 30 mm diameter drift tubes DT 1,2 . The matrix hodoscope HODO (matrix) is composed of 252 scintillator tiles with WLS+SiPM readout. It is used in the trigger, improves time resolution and links x-y projections of a track. Two scintillator counters S bk1 , S bk2 serve to suppress undecayed beam particles. At the end of the OKA setup there are two calorimeters: electromagnetic (GAMS-2000) made of lead glass blocks and a hadron one (HCAL (GDA) )-100 iron-scintillator sandwiches. Finally, four partially overlapping muon counters are located downstream the HCAL. The data acquisition system of the OKA setup [7] operates at ∼ 25 kHz event rate with the mean event size of ∼ 4 kByte. The detailes of the setup and the beam can be found elswhere [8].
The analysis procedure
The study of the decay K + → π + π − π + γ is performed with the data set accumulated in 2012 and 2013 runs with 17.7 GeV/c beam momentum. Two triggers were used. The first one selects beam kaons decays inside the OKA setup: Tr Kdecay = S 1 ·S 2 ·S 3 ·S 4 ·Č 1 ·Č 2 ·S bk and, in addition, requires an energy deposition in GAMS-2000 e.m. calorimeter higher than MIP: Tr Gams = Tr Kdecay · (E Gams > MIP). The second one, Tr HODO = Tr Kdecay ·(2 ≤ Mult ≤ 4), includes additionally a requirement on multiplicity in the Matrix hodoscope. The beam intensity (S 1 ·S 2 ·S 3 ·S 4 ) was ∼ 2 · 10 6 per spill, the fraction of kaons in the beam was ∼ 12.5%, i.e. the kaon intensity was ∼250k/spill. The total number of kaons entering the DV corresponds to ∼ 3.4 × 10 10 . For the estimation of the background contribution to the selected data set, a sample of the Monte Carlo events with six main decay channels of charged kaon ( π + π 0 , π + π 0 π 0 , π + π − π + , µ + ν, π 0 µ + ν, π 0 e + ν) mixed accordingly to the branching fractions, with the total statistics about equal to that of the recorded data sample is used. The Monte Carlo events are passed through full OKA simulation and reconstruction procedures. Monte Carlo sample for the signal events is produced in the same way as for the background. As the input, the weight, proportional to the square of the absolute value of the matrix element for the decay K + → π + π − π + γ, given in [4] is used.
Event selection
To select K + → π + π − π + γ decay channel in off-line analysis a set of requirements is applied: -the momentum of the beam track is required to be measured; -the number of the secondary charged tracks is required to be equal to three and the net charge of them is equal to +1; -the decay vertex should have good χ 2 and should be with a margin incide the DV ; -the charged tracks are not identified as electrons in the GAMS-2000 electromagnetic calorimeter; -the missing mass squared to each positive pion M 2 miss (π + ) = (P K + − P π + ) 2 should be greater than 0.07 GeV 2 ; -the event should contain one and only one photon with the energy E γ > 0.5 GeV -the invariant mass of the photon with each pion (M (πγ))should be greater than 0.17 GeV ; -the square of the transverse momentum of the π + π − π + γ system is less than 0.001 GeV 2 ; -the ratio of the momentum of the π + π − π + γ system to the momentum of the beam track should be within the range 0.95-1.05.
The cut on the square of the missing mass to each positive pion M 2 miss (π + ) is used for the suppression of the background from the decay K + → π + π 0 with π 0 → e + e − γ.The main source of the background for the decay K + → π + π − π + γ is the decay of kaon to three charged pions when the pions produce hadron showers in the electromagnetic calorimeter and, because of fluctuations, a part of a shower is not associated with a charged track by the reconstruction program. To suppress such kind of background a cut on M (πγ) is done. The reconstructed momentum of the π + π − π + γ system should be equal to the momentum of the beam particle. This motivates the last two cuts listed above.
The invariant mass distribution of the π + π − π + γ system after application of all the cuts listed above is shown in Fig. 2. We see clear separation of the signal, peaking around the nominal value of the kaon PDG [2] mass and the background concentrating at higher masses. The number of events in the signal region is about 450. As almost all the photons in the selected K + → π + π − π + γ events have the energy in the kaon rest frame greater than 0.03 GeV, in the following analysis we apply an additional explicit cut E * γ > 0.03 GeV. Figure 2: The invariant mass distribution of the π + π − π + γ system for the data (black circles), main sources of background (histogram) and signal decays of the kaons (black triangles).
Measurement of the branching ratio of the
K + → π + π − π + γ decay
The invariant mass distribution of the π + π − π + γ system in the signal region is shown in Fig. 3. The Monte Carlo distributions for the main sources of the background and the signal events are shown together with the data. We define the mass range of 0.486-0.504 GeV as a signal region. The number of the decays of K + → π + π − π + γ is determined as the difference of the number of the data events in the signal region and the expected number of events from the background. The expected background contribution to the signal region is determined using Monte Carlo events for six main channels of the kaon decays described in the above section. To determine the branching fraction of the decay K + → π + π − π + γ we use the decay K + → π + π − π + as the normalization channel.
We expect the cancellation of many ambiguities in the ratio of the branching fractions Br(K + → π + π − π + γ)/Br(K + → π + π − π + ). The invariant mass spectrum of the system of three charged pions for the same selection criteria as for the charged part of the π + π − π + γ decay is shown in : Invariant mass distributions of the π + π − π + γ system for the data (black circles), signal decays (blue histogram) and main sources of the background (black histogram) for the different energy ranges of the photon in the decay K + → π + π − π + γ. The top left distribution is for the photon energy range 0.03-0.04 GeV, the top right one is for the range 0.04-0.05 GeV, the bottom left one is for the 0.05-0.06 GeV and the last one is for the 0.06-0.07 GeV. The arrows show the signal regions used in the analysis.
The branching fraction of the decay K + → π + π − π + γ is determined in the following way:
Br(3πγ) = Br(3π) P DG × (3π) × N (3πγ) D / (3πγ)/N (3π) D ,
where Br(3π) P DG is the branching fraction of the decay K + → π + π − π + from PDG [2]; (3π) and (3πγ) are the reconstruction efficiencies for the decays K + → π + π − π + and K + → π + π − π + γ, determined from the Monte Carlo; N (3πγ) D and N (3π) D are the numbers of the decays of K + → π + π − π + γ and K + → π + π − π + in the data. Efficiencies Figure 6: Measured branching fractions for the decay K + → π + π − π + γ in the photon energy ranges in the kaon rest frame for the data (black circles) and the CHPT prediction from [5] (histogram). . The histogram shows the Monte Carlo prediction for the decay K + → π + π − π + γ with the matrix element from [5]. The main source of the systematic error is the uncertainty in the estimate of the background contribution to the signal region (see Fig. 2,3). For the estimation of this uncertainty we varied the normalization of the background distribution.The variations are quadratically summed with that due to the changes of the cuts listed in the selection criteria. The estimated value of the systematic uncertainty for the branching fraction is 0.03 × 10 −5 . The final result is Br(3πγ) = (0.71 ± 0.04(stat.) ± 0.03(syst.)) × 10 −5 , E * γ > 0.03GeV.
The theory prediction in the framework of the CHPT is Br th = (0.665±0.005)×10 −5 [5].
Measurement of the photon energy spectrum
To perform a measurement of the photon energy spectrum, we split the data sample into four parts with photon energies in the rest frame of the kaon lying in the ranges 0.03-0.04, Table 1: Values of the branching fractions for the decay K + → π + π − π + γ in the intervals of the photon energy in the kaon rest frame. The uncertainties of the data given in the table are the statistical ones.
Energy interval (GeV) Branching fraction(data) Branching fraction(CHPT [5]) 0.03-0.04 (5.17 ± 0.34) · 10 −6 (4.93 ± 0.05) · 10 −6 0.04-0.05 (1.55 ± 0.12) · 10 −6 (1.44 ± 0.01) · 10 −6 0.05-0.06 (0.35 ± 0.05) · 10 −6 (0.269 ± 0.003) · 10 −6 0.06-0.07 (0.11 ± 0.06) · 10 −7 (0.136 ± 0.002) · 10 −7 0.04-0.05, 0.05-0.06 and 0.06-0.07 GeV. After that we apply the procedure of the previous section to each of the four data samples. The π + π − π + γ invariant mass spectra for the listed above ranges of the photon energy are shown in Fig 5 together with expected signal and background contributions.
The obtained values of the branching fractions for the given energy ranges are listed in the table 1 and shown in the Fig. 6. For comparison, we also give the O(p 4 ) CHPT predictions from [5].
Search for a photon up-down asymmetry
In a decay in the beauty sector B + → K + π + π − γ the LHCb experiment has found significant up-down asymmetry of the photon with respect to the hadronic system decay plane [9]. This observable was proposed in [10], it is both P and T-odd. We perform an analogous study for the radiative kaon decay to three charged pions and photon to search for a New Physics effects. In Fig 7 we show the distribution of the cosine of the angle of the photon direction with respect to the pion system decay plane: cos(θ) = n γ · [p f (π) × p s (π)]/|[p f (π) × p s (π)]|, where n γ is the unit vector of photon direction in the 3-pion rest frame, [p f (π) × p s (π)] is the vector product of the momenta of the fastest and slowest pions in the same frame. For comparison, we show the same distribution for the Monte Carlo signal events with the matrix element from [5]. The observed asymmetry in the data A = (N (cosθ > 0) − N (cosθ < 0))/N total = 0.03 ± 0.05 ± 0.03 is consistent with zero within statistical error.
Conclusions
The decay K + → π + π − π + γ is studied on statistics of 450 events. The measured branching fraction is Br(3πγ) = (0.71 ± 0.05) × 10 −5 for E * γ > 0.03 GeV. The photon energy spectrum for this decay is also determined. The measured branching fraction and the energy spectrum agree well within the errors with the calculations in the framework of the chiral perturbation theory. The measured up-down asymmetry of the photon with respect to the decay plane of the hadronic system is 0.03 ± 0.06. No sign of P and T-odd effects is observed.
Figure 1 :
1Schematic elevation view of the OKA setup, see the text for details.
C
Fig 4 .
4
Figure 3 :
3The invariant mass distribution of the π + π − π + γ system for the data (black circles), the signal decays (blue histogram), the main sources of the background (red histogram) and the sum of the contributions of the signal decays and the background (black histogram). The arrows show the signal region used in the analysis.
Figure 4 :
4The invariant mass distribution of three charged pions for the data (black circles) and for the Monte Carlo for the six main channels of the charged kaon decay (histogram).
Figure 5
5Figure 5: Invariant mass distributions of the π + π − π + γ system for the data (black circles), signal decays (blue histogram) and main sources of the background (black histogram) for the different energy ranges of the photon in the decay K + → π + π − π + γ. The top left distribution is for the photon energy range 0.03-0.04 GeV, the top right one is for the range 0.04-0.05 GeV, the bottom left one is for the 0.05-0.06 GeV and the last one is for the 0.06-0.07 GeV. The arrows show the signal regions used in the analysis.
Figure 7 :
7The distribution (black circles) of the cosine of the angle of the photon direction with respect to the hadronic system decay plane(see the text)
( 3π )
3πand (3πγ), determined from the Monte Carlo, are 0.12 ± 0.002 and 0.024 ± 0.001, respectively. The obtained result is Br(3πγ) = (0.71 ± 0.04(stat.)) × 10 −5 .
AcknowledgementsWe express our gratitude to our colleagues in the accelerator department for the good performance of the U-70 during data taking; to colleagues from the beam department for the stable operation of the 21K beam line, including RF-deflectors, and to colleagues from the engineering physics department for the operation of the cryogenic system of the RF-deflectors.The work is supported by the Russian Fund for Basic Research, grant N18-02-00179A.
. V V Barmin, Sov. J. Nucl. Phys. 50421V.V. Barmin et al., Sov. J. Nucl. Phys. 50, 421 (1989).
. C Patringnani, Particle Data GroupChin. Phys. C. 40100001C. Patringnani et al. (Particle Data Group), Chin. Phys. C. 40, 100001 (2016).
. S Weinberg, Physica. 96A. 327S. Weinberg, Physica. 96A, 327 (1979);
. J Gasser, H Leutwyler, Ann. Phys. (N.Y.). 158142J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158, 142 (1984);
. Nucl. Phys. 250465Nucl. Phys. B250, 465 (1985);
. H Leutwyler, Ann. Phys. (N.Y.). 235165H. Leutwyler, Ann. Phys. (N.Y.) 235, 165 (1994).
. G Ambrosio, G Ecker, G Isidori, H Neufeld, Phys. Lett. 380165G. D'Ambrosio, G. Ecker, G. Isidori and H Neufeld, Phys. Lett. B380, 165 (1996).
. G Ambrosio, G Ecker, G Isidori, H Neufeld, Z. Phys. C76 Issue. 2301G. D'Ambrosio, G. Ecker, G. Isidori and H Neufeld, Z. Phys. C76 Issue 2, 301 (1997).
. V I Garkusha, IHEP preprint. V. I. Garkusha et al., IHEP preprint, IHEP 2003-4.
. S V Donskov, Instrum. Exp. Tech. 594519S. V. Donskov et al., Instrum. Exp. Tech. 59 no.4, 519 (2016).
. A S Sadovsky, Eur.Phys.J.C. 7892A. S. Sadovsky et al., Eur.Phys.J.C(2018) 78:92.
. R Aaij, LHCb CollaborationPhys. Rev. Lett. 112161801R. Aaij et al. (LHCb Collaboration), Phys. Rev. Lett. 112, 161801 (2012).
. M Gronau, Y Grossman, D Pirjol, A Ryd, Phys. Rev. Lett. 8851802M. Gronau, Y. Grossman, D. Pirjol and A. Ryd, Phys. Rev. Lett. 88, 051802 (2002).
| [] |
[
"GFF: Gated Fully Fusion for Semantic Segmentation",
"GFF: Gated Fully Fusion for Semantic Segmentation"
] | [
"Xiangtai Li \nMOE\nSchool of EECS\nKey Laboratory of Machine Perception\nPeking University\n\n",
"Houlong Zhao \nDeepMotion\n\n",
"Lei Han \nTencent AI lab\n\n",
"Yunhai Tong [email protected] \nMOE\nSchool of EECS\nKey Laboratory of Machine Perception\nPeking University\n\n",
"Kuiyuan Yang [email protected] \nDeepMotion\n\n"
] | [
"MOE\nSchool of EECS\nKey Laboratory of Machine Perception\nPeking University\n",
"DeepMotion\n",
"Tencent AI lab\n",
"MOE\nSchool of EECS\nKey Laboratory of Machine Perception\nPeking University\n",
"DeepMotion\n"
] | [] | Semantic segmentation generates comprehensive understanding of scenes at a semantic level through densely predicting the category for each pixel. High-level features from Deep Convolutional Neural Networks already demonstrate their effectiveness in semantic segmentation tasks, however the coarse resolution of high-level features often leads to inferior results for small/thin objects where detailed information is important but missing. It is natural to consider importing low level features to compensate the lost detailed information in high level representations. Unfortunately, simply combining multi-level features is less effective due to the semantic gap existing among them. In this paper, we propose a new architecture, named Gated Fully Fusion(GFF), to selectively fuse features from multiple levels using gates in a fully connected way. Specifically, features at each level are enhanced by higher-level features with stronger semantics and lower-level features with more details, and gates are used to control the propagation of useful information which significantly reduces the noises during fusion. We achieve the state of the art results on two challenging scene understanding datasets, i.e., 82.3% mIoU on Cityscapes test set and 45.3% mIoU on ADE20K validation set. Codes and the trained models will be made publicly available. | null | [
"https://arxiv.org/pdf/1904.01803v2.pdf"
] | 102,480,348 | 1904.01803 | 335a5b2017ac057a7b0b7aa7ee7adcc196150612 |
GFF: Gated Fully Fusion for Semantic Segmentation
Xiangtai Li
MOE
School of EECS
Key Laboratory of Machine Perception
Peking University
Houlong Zhao
DeepMotion
Lei Han
Tencent AI lab
Yunhai Tong [email protected]
MOE
School of EECS
Key Laboratory of Machine Perception
Peking University
Kuiyuan Yang [email protected]
DeepMotion
GFF: Gated Fully Fusion for Semantic Segmentation
Semantic segmentation generates comprehensive understanding of scenes at a semantic level through densely predicting the category for each pixel. High-level features from Deep Convolutional Neural Networks already demonstrate their effectiveness in semantic segmentation tasks, however the coarse resolution of high-level features often leads to inferior results for small/thin objects where detailed information is important but missing. It is natural to consider importing low level features to compensate the lost detailed information in high level representations. Unfortunately, simply combining multi-level features is less effective due to the semantic gap existing among them. In this paper, we propose a new architecture, named Gated Fully Fusion(GFF), to selectively fuse features from multiple levels using gates in a fully connected way. Specifically, features at each level are enhanced by higher-level features with stronger semantics and lower-level features with more details, and gates are used to control the propagation of useful information which significantly reduces the noises during fusion. We achieve the state of the art results on two challenging scene understanding datasets, i.e., 82.3% mIoU on Cityscapes test set and 45.3% mIoU on ADE20K validation set. Codes and the trained models will be made publicly available.
Introduction
Semantic segmentation densely predicts the semantic category for every pixel in an image, such comprehensive image understanding is valuable for many vision-based applications such as medical image analysis [28], remote sensing [15] and autonomous driving [36]. However, precisely predicting label for every pixel is challenging as illustrated in Fig. 1, since these pixels can be from tiny or large objects, far or near objects, and inside object or object boundary.
As being a semantic prediction problem, the basic task of * This work is done when Xiangtai Li is an intern at DeepMotion. semantic segmentation is to generate high-level representation for each pixel, i.e., a high-level and high-resolution feature map.Given the ability of ConvNets in learning highlevel representation from data, semantic segmentation has made much progress by leveraging such high-level representation. However, high-level representation from Con-vNets is generated along lowering the resolution, thus highresolution and high-level feature maps are distributed in two ends in a ConvNet.
To get a feature map with both high-resolution and highlevel, which is not readily available in a ConvNet, it is natural to consider fusing high-level feature maps from higher layers and high-resolution feature maps from lower layers. These feature maps are with different properties, that highlevel feature map can correctly predict most of the pixels on large patterns in a coarse manner, which dominants the current semantic segmentation approaches, while low-level feature maps can only predict few pixels on small patterns.
Unfortunately, simply combining high-level feature maps and high-resolution feature maps will drown useful information in massive useless information, and cannot get a desired high-level and high-resolution feature map. Therefore, an advanced fusion mechanism is required to collect information selectively from different feature maps. To achieve this, we propose Gated Fully Fusion (GFF) which uses a gate, a kind of operation commonly used in recurrent networks, at each pixel to measure the usefulness of each corresponding feature vector, and thus to control the propagation of the information through the gate. The principle of the gate at each layer is designed to either send out useful information to other layers or receive information from other layers when the information in the current layer is useless. Using gate to control information propagation, redundancies can also be effectively minimized in the network, allowing us to fuse multi-level feature maps in a fully-connected manner. Fig 1 compares the results of GFF and PSPNet [44], where GFF handles fine-level details such as poles in a much better way.
On the other hand, considering contextual information in large receptive field is also very important for semantic segmentation as proved by PSPNet [44], ASPP [3] and DenseASPP [37]. Therefore, we also model contextual information after GFF to achieve further performance improvement. Specifically, we propose a dense feature pyramid (DFP) module to encode the context information into each feature map. DFP reuses the contextual information for each feature level and aims to enhance the context modeling part while GFF operates on the backbone of network to capture more detailed information. Combining both components in a single end-to-end network, we achieve the state-of-the-art results on both Cityscapes and ADE20K datasets.
The main contributions of our work can be summarized as follows:
• Gated Fully Fusion is proposed to generate highresolution and high-level feature map from multi-level feature maps.
• Detailed analysis with visualization of gates learned in different layers intuitively shows the information regulation mechanism in GFF.
• The proposed method is extensively verified on two standard semantic segmentation benchmarks including Cityscapes and ADE20K, and achieves new state-ofthe-art performance. In particular, our model achieves 82.3% mIoU on Cityscapes test set trained only on the fine labeled data.
The rest of the paper is organized as follows: Section 2 provides an overview of the related works. Section 3 introduces the proposed method. Section 4 presents the experiments and analysis, and conclusion is made in Section 5.
Related Work
In this section, we review deep learning based methods for semantic segmentation from three categories, i.e., context modeling, multi-level feature fusion and gating mechanism.
Context modeling Though high-level feature maps in ConvNets have shown promising results on semantic segmentation [25], their receptive field sizes are still not large enough to capture contextual information for large objects and regions. Thus, context modeling becomes a practical direction in semantic segmentation. ParseNet [24] utilizes global pooling to encode contextual information, and PSP-Net [44] uses spatial pyramid pooling to aggregate multiscale contextual information. Deeplab series [2,3,4] develop atrous spatial pyramid pooling (ASPP) to capture multi-scale contextual information by dilated convolutional layers with different dilation rates. Instead of parallel aggregation as adopted in PSPNet and Deeplab, Yang et al. [37] and Bilinski et al. [1] follow the idea of the dense connection [13] to encode contextual information in a dense way. In [27], factorized large filters are directly used to increase the receptive field size for context modeling. In PSANet [45], contextual information is collected from all positions according to the similarities defined in a projected feature space. Similarly, OCNet [40] and DANet [9] use non-local operator [32] to aggregate information from the whole image based on similarities.
Multi-level feature fusion Despite a loss of contextual information, the top layer also lacks of fine detailed information. To address this issue, in FCN [25], predictions from middle layers are used to improve segmentation for detailed structures, while hypercolumns [10] directly combines features from multiple layers for prediction. The U-Net [28] adds skip connections between the encoder and decoder to reuse low level features, [43] improves U-Net by fusing high-level features into low-level features. Feature Pyramid Network (FPN) [23] uses the structure of U-Net with predictions made on each level of the feature pyramid. DeepLabV3+ [5] refines the decoder of its previous version by combing low-level features. In [21], Di et al. proposed to locally fuse every two adjacent feature maps in the feature pyramid into one feature map until only one feature map is left. These fusion methods operate locally in the feature pyramid without awareness of the usefulness of all feature maps to be fused, which limits the propagation of useful features.
Gating mechanism In deep neural networks, especially for recurrent networks, gates are commonly utilized to control the information propagation. For example, LSTM [12] and GRU [6] are two typical cases using different gates to handle long-term memory and dependencies. The highway network [30] uses gates to make training very deep network possible. To improve multi-task learning for scene parsing (c) shows a simple way of addition for fusion; and (d) is the proposed gated fully fusion, where G l is the gate map generated from X l , and features corresponding high gate values are allowed to send out and regions with low gate values are allowed to receive. and depth estimation, PAD-Net [35] is proposed to use gates to fuse multi-modal features trained from multiple auxiliary tasks. DepthSeg [18] proposes depth-aware gating module which uses depth estimates to adaptively modify the pooling field size in high-level feature map.
+ ෨ −1 (b) (c) (d) 1 1 ⋯ ⋯ × × × × + + 1 − + 1 ෨ ⋯ ⋯ ෨ ෨ ෨ ෨ 2 ෨ 1 2 1 ⋯ ⋯ (a)
Our method is related and inspired by the above methods, and differs from them that we propose to fuse multilevel feature maps simultaneously through gating mechanism, and the resulting method surpasses the state-of-the-art approaches.
Method
In this section, we first present the basic setting of multilevel feature fusion and three baseline fusion strategies. Then, we introduce the proposed multi-level fusion module (GFF) and the whole network with the context modeling module (DFP).
Multi-level Feature Fusion
Given L feature maps {X i ∈ R Hi×Wi×Ci } L i=1 extracted from some backbone networks such as ResNet [11], where feature maps are ordered by their depth in the network with increasing semantics but decreasing details, H i , W i and C i are the height, width and number of channels of the ith feature map respectively, feature maps of higher levels are with lower resolution due to the downsampling operations, i.e.,
H i+1 ≤ H i , W i+1 ≤ W i .
In semantic segmentation, the top feature map X L with 1/8 resolution of the raw input image is mostly used for its rich semantics. The major limitation of X L is its low spatial resolution without detailed information, because the outputs need to be with the same resolution as the input image. In contrast, feature maps of low level from shallow layers are with high resolution, but with limited semantics. Intuitively, combining the complementary strengths of multiple level feature maps would achieve the goal of both high resolution and rich semantics, and this process can be abstracted as a fusion process f , i.e.,
{X 1 , X 2 · · · X L } f → {X 1 ,X 2 · · ·X L } (1)
whereX l is the fused feature map for the lth level. Fig 2 illustrates several fusion strategies to be discussed. To simplify the notations in following equations, bilinear sampling and 1 × 1 convolution are ignored which are used to reshape the feature maps at the right hand side to let the fused feature maps have the same size as those at the left hand side. Concatenation is a straightforward operation to aggregate all the information in multiple feature maps, but it mixes the useful information with large amount of noninformative features. Addition is another simple way to combine feature maps by adding features at each position, while it suffers from the similar problem as concatenation. FPN [23] conducts the fusion process through a top-down pathway with lateral connections, where semantic features in higher levels are gradually fused into lower levels. The three fusion strategies can be formulated as,
Concat:X l = concat(X 1 , ..., X L ),(2)
Addition:
X l = L i=1 X i ,(3)FPN:X l =X l+1 + X l , whereX L = X L . (4)
The problem of these basic fusion strategies is that feature maps are fused together without measuring the usefulness of each feature vector, and massive useless features are mixed with useful feature during fusion.
Gated Fully Fusion
The basic task in multi-level feature fusion is to aggregate useful information together under interference of massive useless information. Gating is a mature mechanism to GFF Module measure the usefulness of each feature vector in a feature map and aggregates information accordingly. In this paper, Gated Fully Fusion (GFF) is designed based on the simple addition-based fusion by controlling information flow with gates. Specifically, each level l is associated with a gate map G l ∈ [0, 1] H l ×W l . With these gate maps, the addition-based fusion is formally defined as
X l = (1 + G l ) · X l + (1 − G l ) · L i=1,i =l G i · X i ,(5)
where · denotes element-wise multiplication broadcasting in the channel dimension, each gate map G l = sigmoid(w i * X i ) is estimated by a convolutional layer parameterized with w i ∈ R 1×1×Ci . A feature vector at (x, y) from level i, i = l can be fused to l only when G i (x, y) is large and G l (x, y) is small, i.e., information is sent when level i has the useful information that level l is missing. Besides that useful information can be regulated to the right place through gates, useless information can also be effectively suppressed on both the sender and receiver sides, and information redundancy can be avoided because the information is only received when the current position has useless features.
Dense Feature Pyramid
Context modeling aims to encode more global information, and it is orthogonal to the proposed GFF. Therefore, we further design a module to encode more contextual information from outputs of both PSPNet [44] and GFF. Considering dense connections can strengthen feature propagation [13,37], we also densely connect the feature maps in a top-down manner starting from feature map outputted from the PSPNet, and high-level feature maps are reused multiple times to add more contextual information to low levels, which was found important in our experiments for correctly segmenting large objects. Since the feature pyramid is in a densely connected manner, we denote this module as Dense Feature Pyramid (DFP). Both GFF and DFP can be plugged into any existing FCNs for end-to-end training with only slightly extra computation cost.
Network Architecture
Our network architecture is designed based on previous state-of-the-art network PSPNet [44] with ResNet [11] as backbone for basic feature extraction, the last two stages in ResNet are modified with dilated convolution to make both strides to 1 and keep spatial information. Fig 3 shows the overall framework including both GFF and DFP. PSPNet forms the bottom-up pathway with backbone network and pyramid pooling module (PPM), where PPM is at the top to encode contextual information. Feature maps from last residual blocks in each stage of backbone are used as the input for GFF module, and all feature maps are reduced to 256 channels with 1 × 1 convolutional layers. The output feature maps from GFF are further fused with two 3×3 convolutional layers in each level before feeding into the DFP module. All convolutional layers are followed by batch normalization [14] and ReLU activation function. After DFP, all feature maps are concatenated for final semantic segmentation. Comparing with the basic PSPNet, the proposed method only slightly increases the number of parameters and computations. The entire network is trained in an endto-end manner driving by cross-entropy loss defined on the segmentation benchmarks. To facilitate the training process, an auxiliary loss together with the main loss are used to help optimization following [19,44], where the main loss is defined on the final output of the network and the auxiliary loss is defined on the output feature map at stage3 of ResNet with weight of 0.4 [44].
Experiment
In this section, we evaluate and analyze the proposed method on Cityscapes [7] and ADE20K [46].
Implementation Details
Our implementation is based on PyTorch [26], and uses ResNet50 and ResNet101 pre-trained from ImageNet [29] as backbones.
Training settings: the weight decay is set to 1e-4. Standard SGD is used for optimization, and "poly" learning rate scheduling policy is used to adjust learning rate, where initial learning rate is set to 1e-3 and decayed by (1− iter max iter ) power with power = 0.9. Synchronized batch normalization [41] is used for better mean/variance estimation due to the limited number of images that can be hosted in each GPU. For Cityscapes, crop size of 864 × 864 is used, 100K training iterations with mini-batch size of 8 is carried for training. For ADE20K, crop size of 512 × 512 is used (images with side smaller than the crop size are padded with zeros), 150K training iterations are used with mini-batch size of 16. As a common practice to avoid overfitting, data augmentation including random horizontal flipping, random cropping, random color jittering within the range of [−10, 10], and random scaling in the range of [0.75, 2] are used during training.
Cityscapes Dataset
Cityscapes is a large-scale dataset for semantic urban scene understanding. It contains 5000 fine pixel-level annotated images, which is divided into 2975, 500, and 1525 images for training, validation and testing respectively, where labels of training and validation are publicly released and labels of testing set are hold for online evaluation. It also [44] as our baseline model which achieved state-of-the-art performance for semantic segmentation. We re-implement PSPNet on Cityscapes and achieve similar performance with mIoU of 78.6% on validation set. All results are reported by using sliding window crop prediction.
Ablation Study on Feature Fusion Methods First, we compare several methods introduced in Section 3. To speed up the training process, we use weights from the trained PSPNet to initialize the parameters in each fusion method. We use train-fine data (2975 images) for training and report performance on validation set. For fair comparison with concatenation and addition, we also reduce the channel dimension of feature maps to be fused to 256 and use two 3 × 3 convolutional layers to refine the fused feature map. As for FPN, we implement the original FPN for semantic segmentation following [17] and we add it to PSPNet. Note that FPN based on PSPNet fuses 5 feature maps, where one is context feature map from pyramid pooling module and others are from the backbone.
All the results are shown in Table 1. As expected, concatenation and addition only slightly improve the baseline, and FPN achieves the best performance among the three base fusion methods, while the proposed GFF obtains even much more improvement with mIoU of 80.4%. Since GFF is a gated version of addition-based fusion, the results demonstrate the effectiveness of the used gating mechanism. For further comparison, we also add the proposed gating mechanism into top-down pathway of FPN and obtains slightly improvement, which is reasonable since most highlevel features are useful for low levels. This demonstrates the advantage of fully fusing multi-level feature maps, and the importance of gating mechanism especially during fusing low-level features to high levels. Fig 4 shows results after using GFF, where the accuracies of predictions for both distant objects and object boundaries are significantly improved.
Ablation Study for Improvement Strategies We per- form two strategies to further boost the performance of our model:, (1) DFP: Dense Feature Pyramid is used after the output of GFF module; and (2) MS: multi-scale inference is adopted, where the final segmentation map is averaged from segmentation probability maps with scales {0.75, 1, 1.25, 1.5, 1.75} for evaluation. Experimental results are shown in Table 2, and DFP further improves the performance by 0.8% mIoU. Fig. 5 shows several visual comparisons, where DFP generates more consistent segmentation inside large objects and demonstrates the effec- Table 3. Computational cost comparison, where PSPNet serves as the baseline with image of size 512 × 512 as input.
tiveness in using contextual information for resolving local ambiguities.
With multi-scale inference, our model achieves 81.8% mIoU, which significantly outperforms previous state-ofthe-art model DeepLabv3+ (79.55% on Cityscapes validation set) by 2.25%.
In Table 3, we also study the computational cost of using our modules, where our method spends 7.7% more computational cost and 6.3% more parameters comparing with the baseline PSPNet.
Comparison to the State-of-the-Art As a common practice toward best performance, we average the predic- tions of multi-scaled, left-right flipped and overlappingtiled images for inference. For fair comparison, all methods are only trained using fine annotated dataset and evaluated on test set by the evaluation server. Table 4 summarizes the comparisons, our method achieves 80.9% mIoU by only use train-fine dataset and outperforms PSANet [45] by 2.3%. By fine-tuning the model on both train-fine and val-fine datasets, our method achieves best mIoU of 82.3%. Detailed per-category comparisons are reported in Table 6, where our method achieves the highest IoU on 15 out of 19 categories, and large improvements are from small/thin categories such as pole, street light/sign, person and rider.
More detailed analysis will be given by gate visualization.
ADE20K Dataset
ADE20K is a challenging scene parsing dataset annotated with 150 classes, and it contains 20K/2K images for training and validation. Images in this dataset are from more different scenes with more small scale objects, and are with varied sizes including max side larger than 2000 and min side smaller than 100. Follow the standard protocol, both mIoU and pixel accuracy evaluated on validation set are used as the performance metrics. Both ResNet50 and ResNet101 are used as backbones.
As performance comparison listed in Table 5, with ResNet50, all methods achieve similar performance, and our method achieves comparable mIoU comparing with PSANet but with slightly better pixel accuracy. With stronger backbone ResNet101, our method outperforms state-of-the-art methods with considerable margin in terms of both mIoU and pixel accuracy. Several visual comparison results are shown in Fig 6, where our method performs much better at details and object boundaries.
Visualization of Gates
In this section, we visualize what gates have learned and analyze how gates control the information propagation. Fig 7 shows ------------------79.3 BiSeNet [38] - level. As expected, we find that the higher-level features (e.g., G 3 , G 4 ) are more useful for large structures with explicit semantics, while the lower-level features (e.g., G 1 and G 2 ) are mainly useful for local details and boundaries.
- - - - - - - - - - - - - - - - - - 78.9 PSANet [45] - - - - - - - - - - - - - - - - - - -
Functionally, we find that the higher level features always spread information to other layers and only receive sparse feature signals. For example, the gate from stage 4 (in G 4 of Fig 8) shows that almost all pixels are of highconfidence. Higher-level features cover large receptive field with fewer details, and they can provide a ground scope of the main semantics. In contrast, the lower level layers prefer to receive information while only spread a few sparse signals. This verifies that lower level representations generally vary frequently along the spatial dimension and they require additional features as semantic supplement, while a benefit is that lower features can provide precise information for details and ob-ject boundaries (G 2 in Fig 7 and G 1 in Fig 8(a)).
To further verify the effectiveness of the learned gates, we set the value of each gate G i to zero and compare the segmentation results with learned gate values. Fig 8 (b) shows the comparison results, where wrongly predicted pixels after setting G i to zero are highlighted. Information through G 1 and G 2 is mainly help for object boundaries, while information through G 3 and G 4 is mainly help for large patterns such as cars. Additional visualization examples for the gates can be found in the supplementary materials.
Conclusion
In this work, we propose Gated Fully Fusion (GFF) to fully fuse multi-level feature maps controlled by learned gate maps. The novel module bridges the gap between high resolution with low semantics and low resolution with high semantics. We explore the proposed GFF for the task of semantic segmentation and achieve new state-of-the-art results on Cityscapes and ADE20K datasets. In particular, we find that the missing low-level features can be fused into each feature level in the pyramid, which indicates that our module can well handle small and thin objects in the scene. In our future work, we will verify the effectiveness of GFF in object detection tasks where fine details are also important.
Figure 1 .
1Illustration of challenges in semantic segmentation. (a), Input Image. (b), Ground Truth. (c), PSPNet result (d), Our result. Our method performs much better on small patterns such as distant poles and traffic lights.
Figure 2 .
2(a) shows a general fusion process f ; (b) shows a fusing block in FPN;
Figure 3 .
3Illustration of the overall architecture. (a) Backbone Network(e.g. ResNet [11]) with pyramid pooling module (PPM) [44] on the top. The backbone provides a pyramid of features at different levels. (b), Feature pyramid through gated fully fusion (GFF) modules. The detail of GFF module is illustrated in Fig 2 . (c), Then the final features containing context information are obtained from a dense feature pyramid (DFP) module.
Figure 4 .
4Visualization of segmentation results on two images using GFF and PSPNet. First column shows two input images with zoomed in regions marked with red dash rectangles. The second column shows results of PSPNet, and the third column shows results of using GFF. Fourth column lists the ground truth. Last column shows the refine parts by GFF. It shows that GFF can well handle distant missing objects like poles, traffic lights and object boundaries. Best view in color.
Figure 5 .
5DFP enhances segmentation results on large scale objects and generates more consistent results. Best view in color and zoom in.
Figure 6 .
6Visualization results on ADE20K validation dataset (ResNet101 as backbone). Comparing with PSPNet, our method captures more detailed information, and finds missing small objects (e.g., lights in first two examples) and generates "smoother" on object boundaries (e.g., figures on the wall in last example). Best view in color.
Figure 7 .
7Visualization of learned gate maps on ADE20K dataset. Gi represents gate map of the ith layer. Best view in color and zoom in for detailed information.
Figure 8 .
8(a) Visualization of learned gate maps on Cityscapes dataset, where Gi represents the gate map of the ith layer. (b) Wrongly classified pixels are highlighted after setting Gi to 0 comparing with using original gate values. Best view in color and zoom in for detailed information.
Table 2 .
2Comparison experiments on Cityscapes validation set, where PSPNet serves as the baseline method.Method
mIoU(%)
PSPNet(Baseline)
78.6
PSPNet + GFF
80.4 (1.8↑)
PSPNet + GFF + DFP
81.2 (2.6↑)
PSPNet + GFF + DFP + MS 81.8 (3.2↑)
Method
mIoU(%) FLOPS(G) Params(M)
PSPNet(Baseline)
78.6
580.1
65.6
PSPNet + GFF
80.4
600.1
69.7
PSPNet + GFF + DFP 81.2
625.5
70.5
Table 4 .
4State-of-the-art comparison experiments on Cityscapes
test set. †means only using the train-fine dataset. ‡means both
the train-fine and val-fine data are used.
Method
Backbone mIoU(%) Pixel Acc.(%))
PSPNet [44]
ResNet50
42.78
80.76
PSANet [45]
ResNet50
42.97
80.92
EncNet [41]
ResNet50
41.11
79.73
GCUNet [20]
ResNet50
42.60
79.51
Ours
ResNet50
42.92
81.03
RefineNet [22] ResNet101
40.20
-
PSPNet [44]
ResNet101
43.29
81.39
PSANet [45]
ResNet101
43.77
81.51
UperNet [34] ResNet101
41.22
79.98
EncNet [41]
ResNet101
44.65
81.69
GCUNet [20] ResNet101
44.81
81.19
Ours
ResNet101
45.33
82.01
Table 5 .
5State-of-the-art comparison experiments on ADE20K validation set. Our models achieve top performance measued by both mIoU and pixel accuracy.
the gates learned from ADE20K and Fig 8(a) shows the gates learned from Cityscapes respectively. For each input image, we show the learned gate map of each Method road swalk build wall fence pole tlight sign veg. terrain sky person rider car truck bus train mbike bike mIoU DUC-HDC [31] 98.5 85.5 92.8 58.6 55.5 65.0 73.5 77.8 93.2 72.0 95.2 84.8 68.5 95.4 70.9 78.7 68.7 65.9 73.8 77.6ResNet38 [33]
98.5 85.7 93.0 55.5 59.1 67.1 74.8 78.7 93.7 72.6 95.5 86.6 69.2 95.7 64.5 78.8 74.1 69.0 76.7 78.4
PSPNet [44]
98.6 86.2 92.9 50.8 58.8 64.0 75.6 79.0 93.4 72.3 95.4 86.5 71.3 95.9 68.2 79.5 73.8 69.5 77.2 78.4
AAF [16]
98.5 85.6 93.0 53.8 58.9 65.9 75.0 78.4 93.7 72.4 95.6 86.4 70.5 95.9 73.9 82.7 76.9 68.7 76.4 79.1
SegModel [8]
98.6 86.4 92.8 52.4 59.7 59.6 72.5 78.3 93.3 72.8 95.5 85.4 70.1 95.6 75.4 84.1 75.1 68.7 75.0 78.5
DFN [39]
-
80.1 DenseASPP [37] 98.7 87.1 93.4 60.7 62.7 65.6 74.6 78.5 93.6 72.5 95.4 86.2 71.9 96.0 78.0 90.3 80.7 69.7 76.8 80.6Table 6. Per-category results on Cityscapes test set. Note that all the models are trained with only fine annotated data. Our method outperforms existing approaches on 15 out of 19 categories, and achieves 82.3% mIoU.DANet [9]
98.6 87.1 93.5 56.1 63.3 69.7 77.3 81.3 93.9 72.9 95.7 87.3 72.9 96.2 76.8 89.4 86.5 72.2 78.2 81.5
Ours
98.7 87.2 93.9 59.6 64.3 71.5 78.3 82.2 94.0 72.6 95.9 88.2 73.9 96.5 79.8 92.2 84.7 71.5 78.8 82.3
Dense decoder shortcut connections for single-pass semantic segmentation. P Bilinski, V Prisacariu, CVPR. P. Bilinski and V. Prisacariu. Dense decoder shortcut con- nections for single-pass semantic segmentation. In CVPR, 2018.
Semantic image segmentation with deep convolutional nets and fully connected CRFs. L.-C Chen, G Papandreou, I Kokkinos, K Murphy, A L Yuille, ICLRL.-C. Chen, G. Papandreou, I. Kokkinos, K. Murphy, and A. L. Yuille. Semantic image segmentation with deep con- volutional nets and fully connected CRFs. ICLR, 2015.
Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs. L.-C Chen, G Papandreou, I Kokkinos, K Murphy, A L Yuille, PAMIL.-C. Chen, G. Papandreou, I. Kokkinos, K. Murphy, and A. L. Yuille. Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully con- nected crfs. PAMI, 2018.
Rethinking atrous convolution for semantic image segmentation. L.-C Chen, G Papandreou, F Schroff, H Adam, arXiv:1706.05587arXiv preprintL.-C. Chen, G. Papandreou, F. Schroff, and H. Adam. Re- thinking atrous convolution for semantic image segmenta- tion. arXiv preprint arXiv:1706.05587, 2017.
Encoder-decoder with atrous separable convolution for semantic image segmentation. L.-C Chen, Y Zhu, G Papandreou, F Schroff, H Adam, ECCV. L.-C. Chen, Y. Zhu, G. Papandreou, F. Schroff, and H. Adam. Encoder-decoder with atrous separable convolution for se- mantic image segmentation. In ECCV, 2018.
Learning phrase representations using rnn encoder-decoder for statistical machine translation. K Cho, B Van Merriënboer, C Gulcehre, D Bahdanau, F Bougares, H Schwenk, Y Bengio, arXiv:1406.1078arXiv preprintK. Cho, B. Van Merriënboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio. Learning phrase representations using rnn encoder-decoder for statistical ma- chine translation. arXiv preprint arXiv:1406.1078, 2014.
The cityscapes dataset for semantic urban scene understanding. M Cordts, M Omran, S Ramos, T Rehfeld, M Enzweiler, R Benenson, U Franke, S Roth, B Schiele, CVPR. M. Cordts, M. Omran, S. Ramos, T. Rehfeld, M. Enzweiler, R. Benenson, U. Franke, S. Roth, and B. Schiele. The cityscapes dataset for semantic urban scene understanding. In CVPR, 2016.
Semantic segmentation via structured patch prediction, context crf and guidance crf. S Y Shen, Gan Rui, G Zeng, CVPR. S. Y. Falong Shen, Gan Rui and G. Zeng. Semantic segmen- tation via structured patch prediction, context crf and guid- ance crf. In CVPR, 2017.
Dual attention network for scene segmentation. J Fu, J Liu, H Tian, Z Fang, H Lu, arXiv:1809.02983arXiv preprintJ. Fu, J. Liu, H. Tian, Z. Fang, and H. Lu. Dual at- tention network for scene segmentation. arXiv preprint arXiv:1809.02983, 2018.
Hypercolumns for object segmentation and fine-grained localization. B Hariharan, P Arbelaez, R Girshick, J Malik, CVPR. B. Hariharan, P. Arbelaez, R. Girshick, and J. Malik. Hyper- columns for object segmentation and fine-grained localiza- tion. In CVPR, 2015.
Deep residual learning for image recognition. K He, X Zhang, S Ren, J Sun, CVPR. K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In CVPR, 2016.
Long short-term memory. S Hochreiter, J Schmidhuber, Neural computation. S. Hochreiter and J. Schmidhuber. Long short-term memory. Neural computation, 1997.
Densely connected convolutional networks. G Huang, Z Liu, L Van Der Maaten, K Q Weinberger, CVPR. G. Huang, Z. Liu, L. van der Maaten, and K. Q. Weinberger. Densely connected convolutional networks. In CVPR, 2017.
Batch normalization: Accelerating deep network training by reducing internal covariate shift. S Ioffe, C Szegedy, arXiv:1502.03167arXiv preprintS. Ioffe and C. Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167, 2015.
Semantic segmentation of small objects and modeling of uncertainty in urban remote sensing images using deep convolutional neural networks. M Kampffmeyer, A.-B Salberg, R Jenssen, CVPRWM. Kampffmeyer, A.-B. Salberg, and R. Jenssen. Semantic segmentation of small objects and modeling of uncertainty in urban remote sensing images using deep convolutional neu- ral networks. In CVPRW, 2016.
Adaptive affinity fields for semantic segmentation. T.-W Ke, J.-J Hwang, Z Liu, S X Yu, ECCV. T.-W. Ke, J.-J. Hwang, Z. Liu, and S. X. Yu. Adaptive affinity fields for semantic segmentation. In ECCV, 2018.
A unified architecture for instance and semantic segmentation. A Kirillov, K He, R B Girshick, P Dollr, A. Kirillov, K. He, R. B. Girshick, and P. Dollr. A unified architecture for instance and semantic segmentation, 2017.
Recurrent scene parsing with perspective understanding in the loop. S Kong, C C Fowlkes, CVPR. S. Kong and C. C. Fowlkes. Recurrent scene parsing with perspective understanding in the loop. In CVPR, 2018.
Deeplysupervised nets. C.-Y Lee, S Xie, P Gallagher, Z Zhang, Z Tu, Artificial Intelligence and Statistics. C.-Y. Lee, S. Xie, P. Gallagher, Z. Zhang, and Z. Tu. Deeply- supervised nets. In Artificial Intelligence and Statistics, 2015.
Beyond grids: Learning graph representations for visual recognition. Y Li, A Gupta, NeurIPS. Y. Li and A. Gupta. Beyond grids: Learning graph represen- tations for visual recognition. In NeurIPS. 2018.
Multi-scale context intertwining for semantic segmentation. D Lin, Y Ji, D Lischinski, D Cohen-Or, H Huang, In ECCV. D. Lin, Y. Ji, D. Lischinski, D. Cohen-Or, and H. Huang. Multi-scale context intertwining for semantic segmentation. In ECCV, 2018.
Refinenet: Multipath refinement networks for high-resolution semantic segmentation. G Lin, A Milan, C Shen, I D Reid, CVPR. G. Lin, A. Milan, C. Shen, and I. D. Reid. Refinenet: Multi- path refinement networks for high-resolution semantic seg- mentation. In CVPR, 2017.
Feature pyramid networks for object detection. T.-Y Lin, P Dollr, R B Girshick, K He, B Hariharan, S J Belongie, CVPR. T.-Y. Lin, P. Dollr, R. B. Girshick, K. He, B. Hariharan, and S. J. Belongie. Feature pyramid networks for object detec- tion. In CVPR, 2017.
W Liu, A Rabinovich, A C Berg, arXiv:1506.04579Parsenet: Looking wider to see better. arXiv preprintW. Liu, A. Rabinovich, and A. C. Berg. Parsenet: Looking wider to see better. arXiv preprint arXiv:1506.04579, 2015.
Fully convolutional networks for semantic segmentation. J Long, E Shelhamer, T Darrell, CVPR. J. Long, E. Shelhamer, and T. Darrell. Fully convolutional networks for semantic segmentation. In CVPR, 2015.
Automatic differentiation in pytorch. A Paszke, S Gross, S Chintala, G Chanan, E Yang, Z De-Vito, Z Lin, A Desmaison, L Antiga, A Lerer, A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. De- Vito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer. Auto- matic differentiation in pytorch. In NIPS-W, 2017.
Large kernel matters improve semantic segmentation by global convolutional network. C Peng, X Zhang, G Yu, G Luo, J Sun, CVPR. C. Peng, X. Zhang, G. Yu, G. Luo, and J. Sun. Large kernel matters improve semantic segmentation by global convolu- tional network. In CVPR, 2017.
U-net: Convolutional networks for biomedical image segmentation. O Ronneberger, P Fischer, T Brox, MIC-CAIO. Ronneberger, P. Fischer, and T. Brox. U-net: Convolu- tional networks for biomedical image segmentation. MIC- CAI, 2015.
. O Russakovsky, J Deng, H Su, J Krause, S Satheesh, S Ma, Z Huang, A Karpathy, A Khosla, M Bernstein, A C Berg, L Fei-Fei, Imagenet large scale visual recognition challenge. IJCVO. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, M. Bernstein, A. C. Berg, and L. Fei-Fei. Imagenet large scale visual recog- nition challenge. IJCV, 2015.
Training very deep networks. R K Srivastava, K Greff, J Schmidhuber, NIPS. R. K. Srivastava, K. Greff, and J. Schmidhuber. Training very deep networks. In NIPS, 2015.
Understanding convolution for semantic segmentation. P Wang, P Chen, Y Yuan, D Liu, Z Huang, X Hou, G Cottrell, In WACV. P. Wang, P. Chen, Y. Yuan, D. Liu, Z. Huang, X. Hou, and G. Cottrell. Understanding convolution for semantic seg- mentation. In WACV, 2018.
Non-local neural networks. X Wang, R Girshick, A Gupta, K He, CVPR. X. Wang, R. Girshick, A. Gupta, and K. He. Non-local neural networks. In CVPR, June 2018.
Z Wu, C Shen, A Van Den, Hengel, arXiv:1611.10080Wider or deeper: Revisiting the resnet model for visual recognition. arXiv preprintZ. Wu, C. Shen, and A. van den Hengel. Wider or deeper: Revisiting the resnet model for visual recognition. arXiv preprint arXiv:1611.10080, 2016.
Unified perceptual parsing for scene understanding. T Xiao, Y Liu, B Zhou, Y Jiang, J Sun, ECCV. T. Xiao, Y. Liu, B. Zhou, Y. Jiang, and J. Sun. Unified per- ceptual parsing for scene understanding. In ECCV, 2018.
Pad-net: Multitasks guided prediction-and-distillation network for simultaneous depth estimation and scene parsing. D Xu, W Ouyang, X Wang, N Sebe, CVPR. D. Xu, W. Ouyang, X. Wang, and N. Sebe. Pad-net: Multi- tasks guided prediction-and-distillation network for simulta- neous depth estimation and scene parsing. In CVPR, 2018.
End-to-end learning of driving models from large-scale video datasets. H Xu, Y Gao, F Yu, T Darrell, CVPR. H. Xu, Y. Gao, F. Yu, and T. Darrell. End-to-end learning of driving models from large-scale video datasets. In CVPR, 2017.
Denseaspp for semantic segmentation in street scenes. M Yang, K Yu, C Zhang, Z Li, K Yang, CVPR. M. Yang, K. Yu, C. Zhang, Z. Li, and K. Yang. Denseaspp for semantic segmentation in street scenes. In CVPR, 2018.
Bisenet: Bilateral segmentation network for real-time semantic segmentation. C Yu, J Wang, C Peng, C Gao, G Yu, N Sang, In ECCV. C. Yu, J. Wang, C. Peng, C. Gao, G. Yu, and N. Sang. Bisenet: Bilateral segmentation network for real-time se- mantic segmentation. In ECCV, 2018.
Learning a discriminative feature network for semantic segmentation. C Yu, J Wang, C Peng, C Gao, G Yu, N Sang, CVPR. C. Yu, J. Wang, C. Peng, C. Gao, G. Yu, and N. Sang. Learn- ing a discriminative feature network for semantic segmenta- tion. In CVPR, 2018.
Ocnet: Object context network for scene parsing. Y Yuan, J Wang, arXiv:1809.00916arXiv preprintY. Yuan and J. Wang. Ocnet: Object context network for scene parsing. arXiv preprint arXiv:1809.00916, 2018.
Context encoding for semantic segmentation. H Zhang, K Dana, J Shi, Z Zhang, X Wang, A Tyagi, A , CVPR. H. Zhang, K. Dana, J. Shi, Z. Zhang, X. Wang, A. Tyagi, and A. Agrawal. Context encoding for semantic segmentation. In CVPR, 2018.
Scaleadaptive convolutions for scene parsing. R Zhang, S Tang, Y Zhang, J Li, S Yan, R. Zhang, S. Tang, Y. Zhang, J. Li, and S. Yan. Scale- adaptive convolutions for scene parsing. In ICCV, 2017.
Exfuse: Enhancing feature fusion for semantic segmentation. Z Zhang, X Zhang, C Peng, X Xue, J Sun, In ECCV. Z. Zhang, X. Zhang, C. Peng, X. Xue, and J. Sun. Ex- fuse: Enhancing feature fusion for semantic segmentation. In ECCV, 2018.
Pyramid scene parsing network. H Zhao, J Shi, X Qi, X Wang, J Jia, CVPR. H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia. Pyramid scene parsing network. In CVPR, 2017.
Psanet: Point-wise spatial attention network for scene parsing. H Zhao, Y Zhang, S Liu, J Shi, C Loy, D Lin, J Jia, ECCV. H. Zhao, Y. Zhang, S. Liu, J. Shi, C. Change Loy, D. Lin, and J. Jia. Psanet: Point-wise spatial attention network for scene parsing. In ECCV, 2018.
B Zhou, H Zhao, X Puig, S Fidler, A Barriuso, A Torralba, arXiv:1608.05442Semantic understanding of scenes through the ADE20K dataset. arXiv preprintB. Zhou, H. Zhao, X. Puig, S. Fidler, A. Barriuso, and A. Torralba. Semantic understanding of scenes through the ADE20K dataset. arXiv preprint arXiv:1608.05442, 2016.
| [] |
[
"FLAT MEROMORPHIC CONNECTIONS OF FROBENIUS MANIFOLDS WITH TT*-STRUCTURE",
"FLAT MEROMORPHIC CONNECTIONS OF FROBENIUS MANIFOLDS WITH TT*-STRUCTURE"
] | [
"Jiezhu Lin ",
"Claude Sabbah "
] | [] | [] | The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDVstructure by C. Hertling. For any CDV-structure on a Frobenius manifold M , the pull-back bundle π * T (1,0) M by the projection π : C × M → M carries two natural holomorphic structures equipped with two flat meromorphic connections. We show that, for any semi-simple CDV-structure, there is a formal isomorphism between these two bundles compatible with connections. Moreover, if we assume that the super-symmetric index Q vanishes, we give a necessary and sufficient condition for such a formal isomorphism to be convergent, and we make it explicit for dim M = 2.(1,0) M . This connection is integrable, hence the (0, 1)-part of this connection, which has no pole on C × M , gives another holomorphic structure, denoted by H 2 , on π * T (1,0) M . Moreover, D is a meromorphic 2000 Mathematics Subject Classification. 53D45, 34M55, 32Q99. | 10.1016/j.geomphys.2011.09.006 | [
"https://arxiv.org/pdf/1105.1542v1.pdf"
] | 119,600,837 | 1105.1542 | ef10a060b22ac57c905bd46568d78a0d7d6e6dd6 |
FLAT MEROMORPHIC CONNECTIONS OF FROBENIUS MANIFOLDS WITH TT*-STRUCTURE
8 May 2011
Jiezhu Lin
Claude Sabbah
FLAT MEROMORPHIC CONNECTIONS OF FROBENIUS MANIFOLDS WITH TT*-STRUCTURE
8 May 2011
The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDVstructure by C. Hertling. For any CDV-structure on a Frobenius manifold M , the pull-back bundle π * T (1,0) M by the projection π : C × M → M carries two natural holomorphic structures equipped with two flat meromorphic connections. We show that, for any semi-simple CDV-structure, there is a formal isomorphism between these two bundles compatible with connections. Moreover, if we assume that the super-symmetric index Q vanishes, we give a necessary and sufficient condition for such a formal isomorphism to be convergent, and we make it explicit for dim M = 2.(1,0) M . This connection is integrable, hence the (0, 1)-part of this connection, which has no pole on C × M , gives another holomorphic structure, denoted by H 2 , on π * T (1,0) M . Moreover, D is a meromorphic 2000 Mathematics Subject Classification. 53D45, 34M55, 32Q99.
0.
Introduction tt* geometry, which appeared first in papers of Cecotti and Vafa ( [2], [3]), is now understood after the work of C. Hertling [6], as an enrichment of that of harmonic Higgs bundle (E, Φ, h) previously introduced by N. Hitchin and C. Simpson. The one-to-one correspondence between harmonic Higgs bundles and variations of polarized twistor structures of weight 0 can be extended, after the work of C. Hertling [6] (respectively, C. Sabbah [11,Chap. 7]) to that between CV-structures (respectively, integrable harmonic Higgs bundles) and variations of pure TERP structures, cf. [6] for the terminology (respectively, integrable variations of Hermitian pure twistor structures of weight 0, cf. [11] for the terminology). The important object of a variation of pure TERP structure (respectively, integrable variations of Hermitian pure twistor structures of weight 0) is the twistor bundle on P 1 × M with a flat C ∞ -connection whose holomorphic part has a pole with Poincaré rank one along {0} × M and whose anti-holomorphic part has a pole with Poincaré rank one along {∞} × M . Such a connection will be called the structure connection of the CV-structure.
Of particular interest for us is the case where a CV-structure exists on a Frobenius manifold (M, g, •, e, E) in a compatible way. Such a structure is called a CDVstructure, and is defined and studied by C. Hertling in [6]. In such a case, there are two natural holomorphic structures on the pull-back tangent bundle π * T (1,0) M by π : C × M → M , each of which carries a flat connection. More precisely, one is the holomorphic vector bundle H 1 := π * T M with the structure connection of the Frobenius manifold M , denoted by ∇, which is an integrable meromorphic connection and has Poincaré rank one along {0} × M (and is extended as a logarithmic connection along {∞} × M ); the other one is the structure connection of the CV structure, denoted by D, on π * T connection on H 2 having Poincaré rank one along {0} × M . We will say that the CDV-structure is strongly potential if both holomorphic bundles with connection (H 1 , ∇) and (H 2 , D) are isomorphic (and an isomorphism between both is called a potential). A CDV-structure is strongly potential if and only if there exists a holomorphic isomorphism φ :
H 1 −→ H 2 , such that (0.1) φ ∇ = Dφ.
In [6,Th. 5.15], C. Hertling gives a criterion to produce CDV-structures, and the resulting structures are strongly potential. The terminology used here comes from [11] where the notion of a potential harmonic Frobenius manifold is considered, and where the criterion of Hertling is shown to produce a potential harmonic Frobenius manifold with stronger properties. It should be noticed that the CDV-structure constructed by Hertling on the base space of the universal unfolding of a hypersurface singularity (cf. [6, §8]), and that constructed by Sabbah for the universal unfolding of a convenient and non-degenerate Laurent polynomial (cf. [11, §4.c]) both use Hertling's criterion, and therefore give rise to a strongly potential CDVstructure (however, in [6, §8] this is not mentioned).
The purposes of this article are to construct a formal isomorphism between these two bundles with connections for any semi-simple CDV-structures, and to analyze the strength of the potentiality property in simple examples of CDV-structures. As a part of the data of a CDV-structure is a self-adjoint operator Q on T (1,0) M (the "new super-symmetric index"). More generally, such an operator exists on the underlying bundle K of a CV-structure (cf. [6] and [11,Chap. 7]). If the CV-structure corresponds to a variation of polarized Hodge structures, then the eigenvalue decomposition of Q corresponds to the Hodge decomposition, and the eigenvalues correspond to the Hodge exponents. The simplest examples of variation of polarized Hodge structures are those of Tate type, of pure type (0, 0). They are nothing but flat Hermitian vector bundles. By analogy, we will call a Tate CV-structure (resp. a Tate CDV-structure) a CV-structure (resp. a CDV-structure) such that Q = 0. In this article, we will restrict our attention to Tate CDV-structures when analyzing the strength of the potentiality property. The existence of a Tate CDV-structure on any semi-simple Frobenius manifold has been discussed in [7], where the author gave explicitly the Hermitian metric and proved that such a CDV-structure is a harmonic Frobenius manifold (cf. [11]).
For many interesting examples of CDV-structures, the two meromorphic connections (H 1 , ∇) and (H 2 , D) have irregular singularities along {0} × M , hence one cannot hope in general that there exists a holomorphic isomorphism between these two bundles compatible with connections.
The main result (Theorem 1.2) of this article is to show that, for any semisimple CDV-structure on a 1-connected complex manifold M , a formal isomorphism between the holomorphic bundles H 1 and H 2 compatible with the meromorphic connections does exist, and we notice that a holomorphic lift of this isomorphism exists as soon as it exists for the restriction at one point of M of the bundles with connection. We are then able to give a necessary and sufficient condition for such a formal isomorphism to be convergent for a Tate CDV-structure, and in the case that dim M = 2 we give an explicit formula (Corollary 1.5) for such an isomorphism.
The proof of Theorem 1.2 consists in constructing a formal isomorphism φ o between two restricted bundles H i | C×o compatible with the corresponding restricted connections for every point o ∈ M . Since we assume that the underlying Frobenius manifold is semi-simple, we can apply a theorem of Malgrange [8] (cf. also [10, Th. II.2.10]) to extend it and get a formal isomorphism φ between the bundles H i compatible the meromorphic connections. According to the local constancy of the Stokes sheaf, we also obtain that φ lifts to a holomorphic isomorphism compatible with the connections if and only if φ o does so.
Notation and terminology. We usually refer to [4,5,9,10] for the notion of Frobenius manifold, that we denote by (M, •, g, e, E), where M is a complex manifold, g is a metric on M (that is, a symmetric, non-degenerate bilinear form, also denoted by , ) with associated Levi-Civita connection denoted by ∇, • is a commutative and associative product on T M which depends smoothly on M , e is the unit vector field for • and E is called the Euler vector field. These data are subject to conditions that we do not repeat here.
In the following, we will restrict to semi-simple Frobenius manifolds, i.e., the multiplication by E has pairwise distinct eigenvalues at each point of M , and we will assume the existence of canonical coordinates, that we will denote by u = (u 1 , . . . , u m ), such that E = k u k ∂ u k . We note that, by assumption, we have
u i = u j on M if i = j.
We will set e i := ∂ u i . Such coordinates exist (at least in theétale sense) whenever M is 1-connected (cf. [10, §VII.1.8]), and we will mainly restrict to this case. We then define V ij by
Ve i = ∇E + 2 − d 2 Id e i := j V ij e j , ∀i.
Recall also that there exists a metric potential η, that is, a function such that
η i := ∂η/∂u i = g(e i , e i ) for all i. The matrix (V ij ) can be expressed in terms of this potential as (cf. [7, Eq. (87)]) (0.2) V ij = 0 if i = j, (u j − u i )∂ u i ∂ u j (η)/2∂ u j (η) if i = j.
For the notion and notation relative to TERP structures and CDV-structures, we refer to [6]. For the deformations of connections with poles of Poincaré rank one and their formal decompositions, we refer to [10,§III.2] In particular, we will use the real structure κ on T (1,0) M , which determines, together with g, a Hermitian form h on T (1,0) M whose associated Chern connection is denoted by
D = D ′ + ∂. The self-adjoint operator Q on T (1,0) M is defined by Q := D E − L E − 2−d 2 · Id .
If the given CDV-structure is semi-simple, we define Q ij by
Qe i = D E − L E − 2 − d 2 · Id e i =: j Q ij e j , ∀i.
By straightforward computation, we get that
(0.3) Q ij = 0 if i = j, ω j i (E) if i = j.
Here ω j i are the connection forms of D ′ under the local frame e i . On the product C × M , we usually denote by z the coordinate on C.
Acknowledgements. The authors thank Claus Hertling for attracting their interest to the question considered in this article, and for useful discussions and comments.
Main results
Before we state the main theorem, we give some equivalent conditions for a semi-simple Tate CDV⊕-structure.
Proposition 1.1. Let (M, g,
•, e, E, κ) be a semi-simple CDV⊕-structure with canonical coordinates as above. Let η be the associated metric potential. Then the following statements are equivalent:
a) (h(e i , e j )) m×m = diag(|η 1 |, . . . , |η m |); b) D ′ is a holomorphic connection, i.e. D ′ ∂ + ∂D ′ = 0; c) Q = 0.dles with connections ( H 1 , ∇) and ( H 2 , D), where we set H = O M ⊗ O C×M H and O M = lim ← −k O C×M /z k . Moreover, φ(H o 1 , ∇ o ) is equal to identity.
The following theorem will explain what happens when the semi-simple Tate CDV⊕-structure is strongly potential.
(ψ o ij ) ∈ GL m×m (C[z]) such that (1.3)( * ) ∂ z (ψ o ij ) = 1 z 2 (u i o − u j o )ψ o ij − 1 z k V o ik ψ o kj , ∀i, j. The matrix (φ o ij ) of an isomorphism φ o satisfying (0.1) at o is then given by (φ o ij ) = (ψ o ij ) · diag exp(zu 1 o ), . . . , exp(zu m o ) .
Corollary 1.4. Any semi-simple CDV⊕-structure on a 1-connected complex analytic manifold M , such that D ′ = ∇ (or equivalently, such that the canonical coordinates are ∇-flat), is of Tate type and strongly potential. Moreover, in the frame e i , the isomorphism φ :
(H 1 , ∇) → (H 2 , D) is determined by the following matrix (φ ij (z, u)) m×m = diag(c 1 · exp(zu 1 ), . . . , c m · exp(zu m )),
for some nonzero constants c 1 , . . . , c m .
When dim M = 2, we will make explicit the necessary and sufficient condition for a semi-simple Tate CDV⊕-structure on M to be strongly potential, as given in Theorem 1.2.
Proof of the theorems
We will use a system of holomorphic canonical coordinates u 1 , u 2 , . . . , u m of the Frobenius manifold as above and we set e i = ∂ u i . We will then use the following notation:
κ(e i ) = k K ik e k , (2.1) e i • e j = −Φ ei (e j ) = k C (i) k j e k , (2.2) −Φ † ei (e j ) = k C (i) k j e k , Φ † is the h-adjoint of Φ. (2.3)
By definition of the canonical coordinates, we have C (i) k j = δ ik · δ jk . Because of h(X, Y ) = g(X, κY ) and Φ * = Φ, we have Φ † = κΦκ since, for all X, Y ,
h(X, Φ † Y ) = h(ΦX, Y ) = g(ΦX, κY ) = h(X, κΦκY ).
This is expressed by C
(i) = K · C (i) · K.
Proof of proposition 1.1. a) ⇒ b) and a) ⇒ c). This is proved in [7,Th. 2]. b) ⇒ a). If D ′ is holomorphic, then by the harmonicity condition (cf. [11, (1
.1)]), we have Φ ∧ Φ † + Φ † ∧ Φ = 0.
By straightforward computations we get
(2.4) [C (i) , KC (j) K] = 0, ∀i, j.
Computing (2.4) directly, we conclude that, for any i there exists a unique j i such that K iji = 0, K jii = 0.
However,
h ii = K ii · η i > 0.
Hence we get that (2.5) K = diag(K 11 , . . . , K mm ).
The relations κ 2 = Id and (2.5) imply
|K ii | = 1, ∀i.
So we can conclude that
h ii = |η i |, ∀i. c) ⇒ b)
. Assume that Q = 0. From Equations (1.12)-(1.16) in [11] together with the h-adjoint ones, we obtain Recall now that, for a CDV⊕ structure, the connection V D is flat, being the restriction to z = 1 of the flat connection D + (∂ + zΦ † ). This implies therefore that D is flat, in particular D ′ is a holomorphic connection.
Remark 2.7. If the properties of the proposition are satisfied, then
C (i) = C (i) = C (i) , ∀i.
Corollary 2.8. Let (M, g, •, e, E, κ) be a semi-simple Tate CDV⊕-structure with canonical coordinates as above and let η be the associated metric potential. Then H 2 , D)). We will firstly prove that there exists a formal isomorphism between (H 1 , ∇) and (H 2 , D). For simplicity, we denote by U (resp. V and Q) the restriction at o of U (resp. V, Q). We note that V (resp. Q) is in the image of ad U : this can be seen by considering the local frame s i := (π * e i )| C×o of H 1 (resp. H 2 ), according to the relation (0.2) (resp. (0.3)). Then the system (U − zV )dz/z 2 (resp. (U − zQ − z 2 U † )dz/z 2 ) is equivalent, by a holomorphic base change, to a system (U + z 2 C(z))dz/z 2 (resp. (U + z 2 C(z))dz/z 2 ), where C(z) (resp. C(z)) is a matrix whose entries are holomorphic functions. The following lemma will imply that (U + z 2 C(z))dz/z 2 and (U + z 2 C(z))dz/z 2 are formally isomorphic to U dz/z 2 , therefore we conclude that they are formally isomorphic. Lemma 2.9. Given any matrix C(z) whose entries are formal series w.r.t. z, then the system with matrix (U + z 2 C(z))dz/z 2 is equivalent, by formal base change, to a system U dz/z 2 if we assume that U is a regular semi-simple matrix.
1) (U † ij ) = U = diag(u 1 , . . . , u m ), where (U † ij ) is the matrix of U † in
Proof of lemma 2.9. According to [10, Th. III.2.15], since U is regular semi-simple, the connection matrix (U + z 2 C(z))dz/z 2 is equivalent to a unique diagonal matrix, each diagonal term being equal to (u i + µ i z)dz/z 2 , where u i are the eigenvalues of U and µ i ∈ C. However, the coefficient of dz/z in (U + z 2 C(z))dz/z 2 is zero, hence by straightforward computation, we conclude that all the constants µ i are zero. Therefore, (U + z 2 C(z))dz/z 2 is formally equivalent to U dz/z 2 .
Let us continue our proof. We have proved that for any point o ∈ M the restricted bundles with connections are formally isomorphic. The existence of a formal isomorphism between the bundles themselves is given by the following lemma. Proof of lemma 2.10. Since R ∇ 0 (x) (resp. R D 0 (x)) is regular semi-simple and has eigenvalues λ i (x), we know by [10, Th. III.2.15] that ( E, ∇) (resp. ( F , D)) can be decomposed in a unique way as a direct sum of subbundles with connections of rank one. Moreover, if we assume that M is 1-connected, then E (resp. F ) is trivializable and admits a basis in which the matrix of connection forms of ∇ (resp. D) can be written as diag(ω ∇ 1 , . . . , ω ∇ m ) (resp. diag(ω D 1 , . . . , ω D m )), where ω ∇ i (resp. ω D i ) takes the form −d(λ i (x)/z) + µ ∇ i dz/z (resp. −d(λ i (x)/z) + µ D i dz/z), and µ ∇ i (resp. µ D i ) are some complex number. If there exists a formal isomorphism between the restricted bundles with connections of ( E, ∇) and ( F , D) at one point o ∈ M , then the uniqueness of the decomposition implies that µ ∇ i = µ D i for every i. Therefore ( E, ∇) and ( F , D) are formally isomorphic.
Assume now that the restriction at a point o ∈ M with coordinates (u 1 o , . . . , u m o ) of the Saito connection is holomorphically isomorphic to the connection with matrix (U/z − Q − zU † )dz/z. Recall (cf. [10,§II.6]) that the meromorphic Saito connection D)) can be reconstructed, up to isomorphism, from its formalization along z = 0 together with a section σ 1 (resp. σ 2 ) of the Stokes sheaf. Since M is simply connected and since the Stokes sheaf is locally constant (cf. [10, Th. II.6.1]), such a section is constant and uniquely determined by its germ at o. According to the base change property [10, Prop. II. 6.9], this germ at o is the Stokes cocycle of the restricted bundle with connection. Our assumption is that the germs at o of σ 1 and σ 2 are equal. Therefore, the section σ 1 is equal to σ 2 , that is, the formal isomorphism between the meromorphic bundles (H 1 [1/z], ∇) and (H 2 [1/z], D) is convergent. Since it induces a formal isomorphism between the holomorphic bundles (H 1 , ∇) and (H 2 , D), it is indeed a holomorphic isomorphism between these bundles.
(H 1 [1/z], ∇) (resp. (H 2 [1/z],
Let us prove the last statement in the theorem. Given any Tate semi-simple CDV⊕-structure, we consider the bundle π * T (1,0) M with the holomorphic structure ∂ + zΦ † , equipped with the meromorphic connection
D = D ′ + d ′ z + 1 z Φ + 1 z U − zU † dz z .
We can write
(2.11) ∂ + zΦ † = ∂ − z∂(U † ) = e zU † • ∂ • e −zU † .
Therefore, this holomorphic bundle H 2 with z-meromorphic connection D is isomorphic to the holomorphic bundle H 1 := ker ∂ equipped with
e −zU † • D ′ + d ′ z + 1 z Φ + 1 z U − zU † dz z • e zU † ,
which is written as
(2.12) D ′ + d ′ z + 1 z Φ + e −zU † Ue zU † · dz z 2 .
However, by Corollary 2.8, we know that U † commutes with U, hence the connection in (2.12) can be written as
(2.13) D ′ + d ′ z + 1 z Φ + U · dz z 2
Let us begin with the connection given by (2.13). Choose canonical local coordinates u 1 , . . . , u m of the underlying semi-simple Frobenius manifold. Set S i := π * ∂ u i , for all i. Then (S i ) i=1,...,m is a holomorphic local frame for H 1 . Obviously, Φ is diagonal in this frame, and S j is the eigenvector of U with eigenvalue u j . It follows that the connection (2.13) is holomorphically isomorphic to the direct sum of the connections d ′ + d ′ z − (d ′ + d ′ z )(u j /z) for all j = 1, 2, . . . , m. In particular, for each such bundle, the monodromy around z = 0 is equal to the identity. Hence the existence of a holomorphic lift will imply that the monodromy of (U − zV )dz/z 2 is equal to identity. equal to S −1 +,diag S −,diag , is also equal to Id, and the Stokes data is equivalent to the data S + = Id, S − = Id).
(c) Since the Stokes data are equal to identity, the system is holomorphically equivalent to the associated formal system ( * ). Since the formal monodromy is the identity, V diag has integral entries. By a suitable rescaling of the basis by powers of z, the system is then meromorphically equivalent to a system with V diag = 0, that is, U dz/z 2 .
(d) If V diag is already zero, the rescaling is not necessary and the isomorphism can be chosen holomorphic.
Assume now that (1), equivalently (2), is satisfied. We already know from the previous proof that V diag has integral entries (resp. is zero). We notice that, in the neighbourhood of ∞ with coordinate z ′ = 1/z, the connection is written (V − z ′ U )dz ′ /z ′ , hence has a simple pole at z ′ = 0. The last assertion of the lemma is then a consequence of the following lemma.
Lemma 2.16. Let ∇ be a meromorphic connection with a simple pole at z ′ = 0. Let (V + z ′ C(z ′ ))dz ′ /z ′ be its matrix. Assume that the monodromy is equal to identity. Then V is semi-simple with integral eigenvalues.
Proof. We will consider the Levelt normal form (see e.g. [10,Ex. II.2.20]). Denote by D the diagonal matrix whose entries are the integral parts of the real parts of the eigenvalues of V , that we can assume to be ordered as δ 1 ≥ · · · ≥ δ n . We can assume that V is block-diagonal, the blocks corresponding to distinct eigenvalues of V . We first consider the block-indexing by the distinct eigenvalues of D. Then (see loc. cit.), the monodromy matrix can be written as exp − 2πi(V − D + T ) , where T is strictly block-lower triangular. That exp − 2πi(V − D + T ) = Id implies first that T = 0. Now, V − D is block-diagonal with respect to distinct eigenvalues of V , and each block B satisfies exp(−2πiB) = Id, which implies that B is semi-simple with integral eigenvalues. Hence so is V . Remark 2.17. As a consequence, a necessary condition to have a holomorphic isomorphism φ (provided that Q = 0) is that V is semi-simple with integral eigenvalues. With the notation of Lemma 2.15, if U is regular semi-simple and V is semi-simple with integral eigenvalues, there is an inductive procedure to check whether the monodromy is the identity or not. But the condition on U, V is not easy to formulate. In order to compute the monodromy, we consider the system at z = ∞, where it has a regular singularity. Setting z ′ = 1/z, the system has matrix (V + z ′ U )dz ′ /z ′ . We then try to find a meromorphic base change (locally with respect to the variable z ′ ) so that the matrix of the system is constant. Since the system has regular singularity, such a base change is known to exist, and an inductive procedure is known. Once the matrix is constant, it is easy to check whether the monodromy is the identity or not. In rank two, it reduces to the condition that the diagonal part of V is zero: this is the contents of the computation of Corollary 1.5.
Proof of theorem 1.3. In the chosen canonical coordinates, the matrix of the endomorphism U, denoted by (U ij ), is diagonal, and (U ij ) = diag(u 1 , . . . , u m ).
Let φ : H 1 → H 2 be an isomorphism of holomorphic vector bundles. Consider the local C ∞ frame S i := (π * e i ) | C×M of H 1 and H 2 . It is ∂-holomorphic, but not ∂ + zΦ † -holomorphic, according to (2.11) and Corollary 2.8(1). Set
φ(S i ) = j φ ij · S j . So (φ ij ) is a non-degenerate C ∞ matrix on C × M .
This amounts to the following, equivalent to (2.19): Proof of Corollary 1.4. By [7] (cf. also Corollary 2.8), we know that the canonical coordinates u 1 , . . . , u m are ∇-flat (so that ∇ = D ′ is expressed as d in the frame (e 1 , . . . , e m )) and Q = 0. We also have V ij = 0 for all i, j, as a consequence of (0.2) and [7,Eq. (63)]. According to Claim 1 in the proof of Theorem 1.3, it is a matter of showing that the holomorphic matrix (ψ ij ) is diagonal and constant. The condition in Claim 2 above now reads ∂ z (ψ ij ) = 1 z 2 (u i − u j )ψ ij , ∀i, j, and the only holomorphic solutions consist of diagonal matrices depending on u only. On the other hand, for every j, the condition φ• ∇ ej = D ej • φ is equivalent to
∂ z (φ ij ) − φ ij u j = 1 z 2 (u i − u j )φ ij − 1 z k V ik φ kj , ∀i, j.∂ u j (φ il ) = 1 z (δ jl − δ ij )φ il , ∀i, l,
or equivalently to ∂ u j (ψ il ) = 1 z (δ jl − δ ij )ψ il , ∀i, l, and, since ψ is diagonal, it reduces to ∂ u j (ψ ii ) = 0, i.e., ψ is constant.
Proof of Corollary 1.5. The condition d/2 ∈ Z is necessary, as we have seen in Remark 2.17. Let us show that it is sufficient. We will set n = d/2 ∈ Z. We have e = e 1 + e 2 and g(e 1 , e 2 ) = g(e 2 , e 1 ) = 0. Since we assume g(e, e) = 0, we obtain η 2 := g(e 2 , e 2 ) = g(e, e) − g(e 1 , e 1 ) = −η 1 .
Recall also that η 12 = ∂ u 2 ∂ u 1 η = η 21 . Therefore, η 12 = η 21 = −η 11 = −η 22 , and from (0.2) we obtain
V 12 = (u 2 − u 1 ) · η 12 2η 2 = Eη 1 2η 2 = d 2 = V 21 .
By Theorem 1.3, we are reduced to proving the existence of ψ o ∈ GL 2 (C[z]) such that Then (2.20) k determines ψ o k+1 in terms of ψ o k up to a diagonal term δ k+1 , which in turn is determined by (2.20) k+1 and the condition that [U o , ψ o k+1 ] has zeros on the diagonal. One finds, for 1 ≤ k ≤ |n|,
ψ o k = k−1 j=0 (j 2 − n 2 ) k! x k Id − k n A D k .
and ψ o k = 0 for k ≥ |n| + 1.
Theorem 1 . 2 .
12For any semi-simple CDV-structure on a 1-connected complex manifold M , there exists a formal isomorphism φ between the formalized bun-
Theorem 1.3. A semi-simple Tate CDV⊕-structure on a 1-connected complex manifold M with canonical coordinates as above is strongly potential if and only if, for some point o ∈ M with canonical coordinates (u 1 o , . . . , u m o ), there exists a matrix
Corollary 1 . 5 .
15Let M be a 1-connected complex analytic manifold M with dim M = 2. Let (M, •, e, E, κ) be a semi-simple Tate CDV⊕-structure on M such that g(e, e) = 0. Let d be the constant such that ∇E + (∇E) * = (2 − d) · Id. Then the CDV⊕-structure is strongly potential if and only if d ∈ 2Z.
U) = D ′ (U) = −Φ, [U, D(U)] = 0, [U, D(U † )] = 0 and adjoint equations. Then V D := D + Φ + Φ † can be written as D − D(U + U † ). Since D(U + U † ) commutes with U and U † by (2.6), it commutes with U + U † and we have V D = e U +U † De −(U +U † ) .
the local frame e i := ∂ u i . In particular, [U, U † ] = 0; 2) D ′ = ∇ if and only if the canonical local coordinates u i are ∇-flat, i.e., ∇∂ u i = 0, ∀i. Proof of Theorem 1.2. Given any semi-simple CDV-structures on M , given a point o ∈ M , let (H 1 , ∇) (resp. (H 2 , D)) denote by the restricted bundles with connections at o of ( H 1 , ∇) (resp. (
Lemma 2. 10 .
10Let (E, ∇) and (F, D) be two holomorphic bundles on C × M with meromorphic connections of Poincaré rank one along {0} × M whose "residues" R ∇ 0 (x) and R D 0 (x) are regular semi-simple and have the same eigenvalues λ i (x), where M is a 1-connected complex manifold. Let ( E, ∇) and ( F , D) denote by the formal bundles with connections associated to (E, ∇) and (F, D). If there exists a formal isomorphism between the restricted bundles with connections of ( E, ∇) and ( F , D) at one point o ∈ M , then we can extend it as a formal isomorphism between ( E, ∇) and ( F , D).
End of the proof of Theorem 1.3. From Theorem 1.2, the existence of φ is equivalent to the existence of φ o and, by the previous claims, to the existence of a holomorphic invertible matrix ψ o satisfying (1.3)( * ). Notice also the entries of ψ o are entire functions of z which have moderate growth at infinity, since (1.3)( * ) has a regular singularity at z = ∞. Therefore, the entries of ψ o belong to C[z]. The same argument applies to (ψ o ) −1 .
set ψ o (z) = k≥0 ψ o k z k . The previous relation reduces to a o k = 0 for k < 0 and k ≫ 0, and ψ o 0 invertible. Let us first notice that, if ψ o exists, then [U o , ψ o 0 ] = 0, that is, ψ o 0 is diagonal. We will show the existence and uniqueness of a solution ψ o with ψ o 0 = Id, and it will be clear that any solution will be of the form ψ o · δ, where δ is diagonal constant and invertible. Setting also u 1 o − u 2 o = x, we have [U o , A] = xB, [U o , B] = xA, AB = D, AD = B.
lifts as a holomorphic isomorphism if and only if its restriction φ o at one point o ∈ M induces a holomorphic isomorphism between the restricted holomorphic bundles with connection. Lastly, if we assume that the CDV-structure is Tate and positive, then such a holomorphic lift exists if and only if the monodromy of
On the other hand, the structure connection ∇ on the holomorphic bundle H 1 , under the above holomorphic frame S i , can be reduced to(2.14)∇ + d ′ z + 1 z Φ + (U − zV) · dz z 2 , where V is given by (0.2).We will use the following lemma to prove the other direction of the last statement.Lemma 2.15. Consider a system (U −zV )dz/z 2 , where U is diagonal with pairwise distinct eigenvalues. The following properties are equivalent:(1) the system is meromorphically (resp. holomorphically) isomorphic to the system with matrix U dz/z 2 , (2) the monodromy is equal to the identity (resp. and the diagonal part V diag is zero). If these properties are satisfied, V is semi-simple with integral eigenvalues and integral (resp. zero) diagonal part.To conclude the proof of Theorem 1.2, it is enough, according to (2.14), to apply the lemma with U = U o , V = V o , since we know by (0.2) that the diagonal part of V o is zero.Proof of Lemma 2.15.(1) ⇒ (2). If this system is meromorphically isomorphic to the system with matrix U dz/z 2 , then the monodromy is clearly equal to identity. One also remarks that the system is equivalent, by holomorphic base change, to a systemwhere V diag is the diagonal part of V and C(z) is holomorphic. If it is holomorphically equivalent to U dz/z 2 , let us denote by P 0 + zP 1 + · · · a base change between both systems (hence P 0 is invertible). Then P 0 is diagonal since it commutes with U , and therefore it commutes with V diag . We must then have(2) ⇒ (1). Assume that the monodromy is equal to identity. (a) After a base change by a matrix in GL n (C[[z]]), the system takes the normal formhence it is of "exponential type", and can be described by Stokes data consisting of two Stokes matrices S + , S − (cf. for instance[1]). More precisely, consider polar coordinates z = ρe iθ and denote by A the sheaf on S 1 consisting of germs f of C ∞ functions on R + × S 1 satisfying z∂ z f = 0 on {ρ = 0}. Then each base change in GL n (C[[z]]) as above can be lifted as a base change in Γ(I, GL n (A)) so that the new matrix is ( * ), if I is a small neighbourhood of a closed interval of S 1 of length π with "generic" boundary points. The Stokes matrices S + , S − compute the multiplicative difference between base changes corresponding to two opposite intervals at each of the boundary points. In particular, if S + = Id and S − = Id, then there is a base change in Γ(S 1 , GL n (A)). Since Γ(S 1 , A) = C{z}, the latter group is GL(C{z}). In other words, if S + = Id and S − = Id, then the system is holomorphically equivalent to a system ( * ).(b) On the other hand, the monodromy of the system can be presented as a product S −1 + S − , where, in a suitable basis, the matrix of S + is upper triangular, and that of S − is lower triangular. Therefore, if the monodromy is the identity, we have S + = S − , so both are diagonal, and thus the formal monodromy, which isOn the other hand, d ′′ z (S j ) = 0. Therefore this is equivalent to ∂ z (φ ij ) = 0, or equivalently to ∂ z (ψ ij ) = 0.On the other hand, the component of ∂ + zΦ † on e j is, according to Remark 2.7,and its vanishing is equivalent toProof. By straightforward computations, we getBy similar computations, we getThe third equality holds because of Corollary 2.8(1). So the ∂ z component of (0.1) is equivalent to
Birkhoff invariants and Stokes' multipliers for meromorphic linear differential equations. Werner Balser, Wolfgang B Jurkat, Donald A Lutz, J. Math. Anal. Appl. 711Werner Balser, Wolfgang B. Jurkat, and Donald A. Lutz, Birkhoff invariants and Stokes' multipliers for meromorphic linear differential equations, J. Math. Anal. Appl. 71 (1979), no. 1, 48-94.
Topological-anti-topological fusion. Sergio Cecotti, Cumrun Vafa, Nuclear Phys. B. 3672Sergio Cecotti and Cumrun Vafa, Topological-anti-topological fusion, Nuclear Phys. B 367 (1991), no. 2, 359-461.
On classification of N = 2 supersymmetric theories. Comm. Math. Phys. 1583, On classification of N = 2 supersymmetric theories, Comm. Math. Phys. 158 (1993), no. 3, 569-644.
Boris Dubrovin, Geometry of 2D topological field theories, Integrable systems and quantum groups. Montecatini Terme; BerlinSpringer1620Boris Dubrovin, Geometry of 2D topological field theories, Integrable systems and quantum groups (Montecatini Terme, 1993), Lecture Notes in Math., vol. 1620, Springer, Berlin, 1996, pp. 120-348.
Frobenius manifolds and moduli spaces for singularities. Claus Hertling, Cambridge Tracts in Mathematics. 151Cambridge University PressClaus Hertling, Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, vol. 151, Cambridge University Press, Cambridge, 2002.
Frobenius manifolds, their connections, and the construction for singularities. J. Reine Angew. Math. 555tt * geometry, tt * geometry, Frobenius manifolds, their connections, and the construction for sin- gularities, J. Reine Angew. Math. 555 (2003), 77-161.
Some constraints on Frobenius manifolds with a tt*-structure. Jiezhu Lin, DOI10.1007/s00209-009-0610-zMath. Z. 267Jiezhu Lin, Some constraints on Frobenius manifolds with a tt*-structure, Math. Z. 267 (2011), 81-108, online: 2009, DOI 10.1007/s00209-009-0610-z.
Bernard Malgrange, Sur les déformations isomonodromiques. I, II, Séminaire E.N.S. Mathématique et Physique (L. Boutet de Monvel, A. Douady, and J.-L. VerdierBirkhäuser, Basel, Boston37Bernard Malgrange, Sur les déformations isomonodromiques, I, II, Séminaire E.N.S. Mathématique et Physique (L. Boutet de Monvel, A. Douady, and J.-L. Verdier, eds.), Progress in Math., vol. 37, Birkhäuser, Basel, Boston, 1983, pp. 401-438.
Frobenius manifolds, quantum cohomology, and moduli spaces. Yuri I Manin, American Mathematical Society47Providence, RIYuri I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society Colloquium Publications, vol. 47, American Mathematical Society, Providence, RI, 1999.
Claude Sabbah, Déformations isomonodromiques et variétés de Frobenius, Savoirs Actuels. Paris, 2002, English TranslSpringer & EDP SciencesUniversitextClaude Sabbah, Déformations isomonodromiques et variétés de Frobenius, Savoirs Actuels, CNRSÉditions & EDP Sciences, Paris, 2002, English Transl.: Universitext, Springer & EDP Sciences, 2007.
Universal unfoldings of Laurent polynomials and tt* structures, From Hodge theory to integrability and TQFT: tt*-geometry. Proc. Symposia in Pure Math. R. Donagi and K. WendlandSymposia in Pure MathProvidence, RIAmerican Math. Society78, Universal unfoldings of Laurent polynomials and tt* structures, From Hodge theory to integrability and TQFT: tt*-geometry (R. Donagi and K. Wendland, eds.), Proc. Symposia in Pure Math., vol. 78, American Math. Society, Providence, RI, 2008, pp. 1-29.
) Lin, Guangzhou, China E-mail address: [email protected] (C. Sabbah) UMR 7640 du CNRS, Centre de Mathématiques Laurent Schwartz,École polytechnique, F-91128 Palaiseau cedex, France E-mail address: [email protected] URL. Lin) Guangzhou, China E-mail address: [email protected] (C. Sabbah) UMR 7640 du CNRS, Centre de Mathématiques Laurent Schwartz,École polytechnique, F-91128 Palaiseau cedex, France E-mail address: [email protected] URL: http://www.math.polytechnique.fr/~sabbah
| [] |
[
"Off-Policy Imitation Learning from Observations",
"Off-Policy Imitation Learning from Observations"
] | [
"Zhuangdi Zhu \nMichigan State University\nMichigan State University\nMichigan State University\n\n",
"Kaixiang Lin \nMichigan State University\nMichigan State University\nMichigan State University\n\n",
"Bo Dai [email protected] \nMichigan State University\nMichigan State University\nMichigan State University\n\n",
"Google Research \nMichigan State University\nMichigan State University\nMichigan State University\n\n",
"Jiayu Zhou [email protected] \nMichigan State University\nMichigan State University\nMichigan State University\n\n"
] | [
"Michigan State University\nMichigan State University\nMichigan State University\n",
"Michigan State University\nMichigan State University\nMichigan State University\n",
"Michigan State University\nMichigan State University\nMichigan State University\n",
"Michigan State University\nMichigan State University\nMichigan State University\n",
"Michigan State University\nMichigan State University\nMichigan State University\n"
] | [] | Learning from Observations (LfO) is a practical reinforcement learning scenario from which many applications can benefit through the reuse of incomplete resources. Compared to conventional imitation learning (IL), LfO is more challenging because of the lack of expert action guidance. In both conventional IL and LfO, distribution matching is at the heart of their foundation. Traditional distribution matching approaches are sample-costly which depend on on-policy transitions for policy learning. Towards sample-efficiency, some off-policy solutions have been proposed, which, however, either lack comprehensive theoretical justifications or depend on the guidance of expert actions. In this work, we propose a sample-efficient LfO approach which enables off-policy optimization in a principled manner. To further accelerate the learning procedure, we regulate the policy update with an inverse action model, which assists distribution matching from the perspective of mode-covering. Extensive empirical results on challenging locomotion tasks indicate that our approach is comparable with state-of-the-art in terms of both sample-efficiency and asymptotic performance. | null | [
"https://arxiv.org/pdf/2102.13185v1.pdf"
] | 227,276,236 | 2102.13185 | 603f8d0ae641ccc6ec44b581576e9504fbe347af |
Off-Policy Imitation Learning from Observations
Zhuangdi Zhu
Michigan State University
Michigan State University
Michigan State University
Kaixiang Lin
Michigan State University
Michigan State University
Michigan State University
Bo Dai [email protected]
Michigan State University
Michigan State University
Michigan State University
Google Research
Michigan State University
Michigan State University
Michigan State University
Jiayu Zhou [email protected]
Michigan State University
Michigan State University
Michigan State University
Off-Policy Imitation Learning from Observations
Learning from Observations (LfO) is a practical reinforcement learning scenario from which many applications can benefit through the reuse of incomplete resources. Compared to conventional imitation learning (IL), LfO is more challenging because of the lack of expert action guidance. In both conventional IL and LfO, distribution matching is at the heart of their foundation. Traditional distribution matching approaches are sample-costly which depend on on-policy transitions for policy learning. Towards sample-efficiency, some off-policy solutions have been proposed, which, however, either lack comprehensive theoretical justifications or depend on the guidance of expert actions. In this work, we propose a sample-efficient LfO approach which enables off-policy optimization in a principled manner. To further accelerate the learning procedure, we regulate the policy update with an inverse action model, which assists distribution matching from the perspective of mode-covering. Extensive empirical results on challenging locomotion tasks indicate that our approach is comparable with state-of-the-art in terms of both sample-efficiency and asymptotic performance.
stationary distributions: one generated by the expert, and the other generated by the learning agent. To correctly estimate the distribution discrepancy, traditional approaches require on-policy interactions with the environment whenever the agent policy gets updated. This inefficient sampling strategy impedes wide applications of IL to scenarios where accessing transitions are expensive [14,15]. The same challenge is aggravated in LfO, as more explorations by the agent are needed to cope with the lack of action guidance.
Towards sample-efficiency, some off-policy IL solutions have been proposed to leverage transitions cached in a replay buffer. Mostly designed for LfD, these methods either lack theoretical guarantee by ignoring a potential distribution drift [4,16,17], or hinge on the knowledge of expert actions to enable off-policy distribution matching [3], which makes their approach inapplicable to LfO.
To address the aforementioned limitations, in this work, we propose a LfO approach that improves sample-efficiency in a principled manner. Specifically, we derive an upper-bound of the LfO objective which dispenses with the need of knowing expert actions and can be fully optimized with off-policy learning. To further accelerate the learning procedure, we combine our objective with a regularization term, which is validated to pursue distribution matching between the expert and the agent from a mode-covering perspective. Under a mild assumption of a deterministic environment, we show that the regularization can be enforced by learning an inverse action model. We call our approach OPOLO (Off POlicy Learning from Observations). Extensive experiments on popular benchmarks show that OPOLO achieves state-of-the-art in terms of both asymptotic performance and sample-efficiency.
Background
We consider learning an agent in an environment of Markov Decision Process (MDP) [18], which can be defined as a tuple: M = (S, A, P, r, γ, p 0 ). Particularly, S and A are the state and action spaces; P is the state transition probability, with P (s |s, a) indicating the probability of transitioning from s to s upon action a; r is the reward function, with r(s, a) the immediate reward for taking action a on state s; Without ambiguity, we consider an MDP with infinite horizons, with 0 < γ < 1 as a discounted factor; p 0 is the initial state distribution. An agent follows its policy π : S → A to interact with this MDP with an objective of maximizing its expected return: max J RL (π) := E s0∼p0,ai∼π(·|si),si+1∼P (·|si,ai),∀0≤i≤t ∞ t=0 γ t r(s t , a t ) = E (s,a)∼µ π (s,a) r(s, a) , in which µ π (s, a) is the stationary state-action distribution induced by π, as defined in Table 1.
Learning from demonstrations (LfD) is a problem setting in which an agent is provided with a fixed dataset of expert demonstrations as guidance, without accessing the environment rewards . The demonstrations R E contain sequences of both states and actions generated by an expert policy π E : R E = {(s 0 , a 0 ), (s 1 , a 1 ), · · · |a i ∼ π E (·|s i ), s i+1 ∼ P (·|s i , a i )}. Without ambiguity, we assume that the expert and agent are from the same MDP.
Among LfD approaches, distribution matching has been a popular choice, which minimizes the discrepancy between two stationary state-action distributions: one is µ E (s, a) induced by the expert, and the other is µ π (s, a) induced by the agent. Without loss of generality, we consider KL-divergence as the discrepancy measure for distribution matching, although any f -divergences can serve as a legitimate choice [2,19,20] : min J LfD (π) := D KL [µ π (s, a)||µ E (s, a)].
(1) Learning from observations (LfO) is a more challenging scenario where expert guidance R E contains only states. Accordingly, applying distribution matching to solve LfO yields a different objective that involves state-transition distributions [10,21,9]: min JLfO(π) := D KL [µ π (s, s )||µ E (s, s )].
(
There exists a close connection between LfO and LfD objectives. In particular, the discrepancy between two objectives can be derived precisely as follows (see Sec 9.2 in the appendix) [10]:
D KL [µ π (a|s, s )||µ E (a|s, s )] = D KL [µ π (s, a)||µ E (s, a)] − D KL [µ π (s, s )||µ E (s, s )].(3)
Remark 1. In a non-injective MDP, the discrepancy of D KL [µ π (a|s, s )||µ E (a|s, s )] cannot be optimized without knowing expert actions. In a deterministic and injective MDP, it satisfies that ∀ π : S → A, D KL [µ π (a|s, s )||µ E (a|s, s )] = 0.
State Distribution
State-Action Distribution Joint Distribution Transition Distribution
Inverse-Action Distribution Notation µ π (s) µ π (s, a) µ π (s, a, s ) µ π (s, s ) µ π (a|s, s )
Support S S × A S × A × S S × S A × S × S Definition (1 − γ) ∞ t=1 γ t µ π
t (s) µ π (s)π(a|s) µ π (s, a)P (s |s, a) A µ π (s, a, s )da µ π (s,a)P (s |s,a) µ π (s,s ) Table 1: Summarization on different stationary distributions, with µ π t (s) = p(s t = s|s 0 ∼ p 0 (·), a i ∼ π(·|s i ), s i+1 ∼ P (·|s i , a i )), ∀i < t).
Despite the potential gap between these two objectives, the LfO objective in Eq (2) is still intuitive and valid, as it emphasizes on recovering the expert's influence on the environment by encouraging the agent to yield the desired state-transitions, regardless of the immediate behavior that leads to those transitions. In this work, we follow this rationale and consider Eq (2) as our learning objective, which has also been widely adopted by prior art [9,[22][23][24]. We will show later that pursuing this objective is sufficient to recover expertise for various challenging tasks.
A common limitation of existing LfO and LfD approaches relies in their inefficient optimization. Work along this line usually adopts a GAN-style strategy [25] to perform distribution matching. Take the representative work of GAIL [2] as an example, in which a discriminator x : S × A → R and a generator π : S → A are jointly learned to optimize a dual form of the original LfD objective:
min π max x JGAIL(π, x) := E µ E (s,a) [log(x(s, a))] + E µ π (s,a) [log(1 − x(s, a))].
During optimization, on-policy transitions in the MDP are used to estimate expectations over µ π . It requires new environment interactions whenever π gets updated and is thus sample inefficient. This inconvenience is echoed in the work of LfO, which inherits the same spirit of on-policy learning [10,9]. In pursuit of sample-efficiency, some off-policy solutions have been proposed. These methods, however, either lack theoretical guarantee [17,4], or rely on the expert actions [4,3], which makes them inapplicable to LfO. We will provide more explanations in Sec 9.8 in the appendix.
To improve the sample-efficiency of LfO with a principled solution, in the next section we show how we explicitly introduce an off-policy distribution into the LfO objective, from which we derive a feasible upper-bound that enables off-policy optimization without the need of accessing expert actions.
3 OPOLO: Off-Policy Learning from Observations
Surrogate Objective
The idea of re-using cached transitions to improve sample-efficiency has been adopted by many RL algorithms [7,[26][27][28]. In the same spirit, we start by introducing an off-policy distribution µ R (s, a), which is induced by a dataset R of historical transitions. Choosing KL-divergence as a discrepancy measure, we obtain an upper-bound of the LfO objective by involving µ R (s, a) (see Sec 9.1 in the appendix for proof):
D KL µ π (s, s )||µ E (s, s ) ≤ E µ π (s,s ) log µ R (s, s ) µ E (s, s ) + D KL µ π (s, a)||µ R (s, a) . (4)
As a result, the LfO objective can be optimized by minimizing the RHS of Eq (4). Although widely adopted for its interpretability, KL divergence can be tricky to estimate due to issues of biased gradients [29,3]. To avoid the potential difficulty in optimization, we further substitute the term D KL [µ π (s, a)||µ R (s, a)] in Eq (4) by a more aggressive f -divergence, with f (x) = 1 2 x 2 , which serves as an upper-bound of the KL-divergence (See Sec 9.4 in the appendix):
D KL [P ||Q] ≤ D f [P ||Q].(5)
Our choice of f -divergence can be considered as a variant of Pearson χ 2 -divergence with a constant shift, which has also been adopted as a valid measure of distribution discrepancies [30,31]. Compared with KL-divergence, this f -divergence enables unbiased estimation without deteriorating the optimality, whose advantages will become increasingly visible in Section 3.2.
Built upon the above transformations, we reach an objective that serves as an effective upper-bound of D KL [µ π (s, s )||µ E (s, s )]:
min π J opolo (π) := E µ π (s,s ) log µ R (s, s ) µ E (s, s ) + D f [µ π (s, a)||µ R (s, a)].(6)
Off-Policy Transformation
Optimization Eq (6) is still on-policy and induces additional challenges through the term D f [µ π (s, a)||µ R (s, a)]. However, we show that it can be readily transformed into off-policy learning. We first leverage the dual-form of an f -divergence [32]:
−D f [µ π (s, a)||µ R (s, a)] = inf x:S×A→R E (s,a)∼µ π [−x(s, a)] + E (s,a)∼µ R [f * (x(s, a))],
and use this dual transformation to rewrite Eq (6):
min π Jopolo(π) ≡ max π E µ π (s,s ) − log µ R (s, s ) µ E (s, s ) − D f [µ π (s, a)||µ R (s, a)] ≡ max π min x:S×A→R Jopolo(π, x) := E µ π (s,a,s ) log µ E (s, s ) µ R (s, s ) − x(s, a) + E µ R (s,a) [f * (x(s, a))]. (7)
If we consider a synthetic reward as r(s, a, s ) = log µ E (s,s ) µ R (s,s ) − x(s, a), the first term in Eq (7) resembles an RL return function:Ĵ(π) = E (s,a,s )∼µ π (s,a,s ) [r(s, a, s )]. Observing this similarity, we turn to learning a Q-function by applying a change of variables:
Q(s, a) = E s ∼P (·|s,a),a ∼π(·|s ) −x(s, a) + log µ E (s, s ) µ R (s, s ) + γQ(s , a ) .
Equivalently, this Q function is a fixed point of a variant Bellman operator B π Q:
Q(s, a) = −x(s, a) + E s ∼P (·|s,a),a ∼π(·|s ) log µ E (s, s ) µ R (s, s ) + γQ(s , a ) = −x(s, a) + B π Q(s, a).
Rewriting x(s, a) = (B π Q − Q)(s, a) and applying it back to Eq (7), we finally remove the on-policy expectation by a series of telescoping (see Sec 9.6 in the appendix for derivation):
A similar rationale has also been the key component of distribution error correction (DICE) [30,31,33]. Based on the above transformation, we propose our main objective:
Specifically, when
f (x) = f * (x) = 1 2 x 2 , the second term E µ R (s,a) [f * ((B π Q − Q)(s, a))
] is reminiscent of an Bellman error, for which we can have unbiased estimation by mini-batch gradients.
Given access to the off-policy distribution µ R (s, a) and the initial distribution p 0 , optimization (9) can be efficiently realized once we resolve the term log µ E (s,s ) µ R (s,s ) contained in B π Q(s, a).
Adversarial Training with Off-Policy Experience
We can take the advantage of GAN training [25] to estimate the term log µ E (s,s ) µ R (s,s ) inside B π Q(s, a), by learning a discriminator D: which upon training to optimality, satisfies log( µ E (s,s ) µ R (s,s ) ) = log D * (s, s ) − log(1 − D * (s, s )). Unlike prior art [2,9,4] that requires estimating the ratio of log µ E µ π , the discriminator in our case is designed to be off-policy in accordance with our proposed objective. Up to this step, optimization (9) can be achieved by interactively optimizing Q, π, and D with pure off-policy learning.
Policy Regularization as Forward Distribution Matching
Optimization 9 essentially minimizes an upper-bound of the inverse KL divergence D KL [µ π (s, s )||µ E (s, s )], which is known to encourage a mode-seeking behavior [34]. Although mode-seeking is more robust to covariate-drift than mode-covering (such as behavior cloning), it requires sufficient explorations to find a reasonable state-distribution, especially at early learning stages. On the other hand, a mode-covering strategy has merits in quickly minimizing discrepancies on the expert distribution, by optimizing a forward KL-divergence such as D KL [π E (a|s)||π(a|s)].
To combine the advantages of both, in this section we show how we further speed up the learning procedure from a mode-covering perspective, without deteriorating the efficacy of our main objective. To achieve this goal, we first derive an optimizable lower-bound from a mode-covering objective:
D KL [πE(a|s)||π(a|s)] = D KL [µ E (s |s)||µ π (s |s)] + D KL [µ E (a|s, s )||µ π (a|s, s )],(10)
in which we define µ π (s |s) = A π(a|s)P (s |s, a)da as the conditional state transition distribution induced by π, likewise for µ E (s |s) (see Sec 9.5 in the appendix).
Similar to Remark 1, the discrepancy D KL [µ E (a|s, s )||µ π (a|s, s )] is not optimizable without knowing expert actions. However, under some mild assumptions, we found it feasible to optimize the other term D KL [µ E (s |s)||µ π (s |s)] by enforcing a policy regularization:
Remark 2. In a deterministic MDP, assuming the support of µ E (s, s ) is covered by µ R (s, s ), s.t. µ E (s, s ) > 0 =⇒ µ R (s, s ) > 0, then regulating policy using µ R (·|s, s ) minimizes D KL [µ E (s |s)||µ π (s |s)] (See Sec 9.5.2 in supplementary for a detailed discussion): ∃π : S → A, s.t. ∀(s, s ) ∼ µ E (s, s ),π(·|s) ∝ µ R (·|s, s ) =⇒π = arg min π D KL [µ E (s |s)||µ π (s |s)].
Intuitively, when expert labels are unavailable, this regularization can be considered as performing states matching, by encouraging the policy to yield actions that lead to desired footprints. Given a transition s → s from the expert observations, a conditional distribution µ R (·|s, s ) only has support on actions that yield this transition s → s . Therefore, following this regularization avoids the policy from drifting to undesired states.
In practice, we can estimate µ R (·|s, s ) by learning an inverse action model P I using off-policy transitions from µ R (s, a, s ) to optimize the following (See Sec 9.5.3 in the appendix):
max P I :S×S→A −D KL [µ R (a|s, s )||PI (a|s, s )] ≡ max P I :S×S→A E (s,a,s )∼µ R (s,a,s ) [log PI (a|s, s )].(11)
Algorithm
Based on all the abovementioned building blocks, we now introduce OPOLO in Algorithm 1.
OPOLO involves learning a policy π, a critic Q, a discriminator D, and an inverse action regularizer P I , all of which can be done through off-policy training.
In particular, π and Q is jointly learned to find a saddle-point solution to optimization (9). The discriminator D assists this process by estimating a density ratio log µ E (s,s ) µ R (s,s ) . For better empirical performance, we adopt − log(1 − D(s, s )) as the discriminator's output, which corresponds to a constant shift inside the logarithm term, in that log
( µ E (s,s ) µ R (s,s ) + 1) = − log(1−D * (s, s ))
. The inverse action model P I serves as a regularizer to infer proper actions on the expert observation distribution to encourage mode-covering . We defer more implementation details to Sec 9.7 in the appendix.
Related Work
Recent development on imitation learning can be divided into two categories:
Learning from Demonstrations (LfD) traces back to behavior cloning (BC) [35], in which a policy is pre-trained to minimize the prediction error on expert demonstrations. This approach is inherent with issues such as distribution shift and regret propagations. To address these limitations, [1] proposed a no-regret IL approach called DAgger, which however requires online access to oracle corrections. More recent LfD approaches favor Inverse reinforcement learning (IRL) [8], which work by seeking a reward function that guarantees the superiority of expert demonstrations, based on which regular RL algorithms can be used to learn a policy [36,37]. A representative instantiation of IRL is Generative Adversarial Imitation Learning (GAIL) [2]. It defines IL as a distribution matching problem and leverages the GAN technique [25] to minimize the Jensen-Shannon divergence between distributions induced by the expert and the learning policy. The success of GAIL has inspired many other related
Algorithm 1 Off-POlicy Learning from Observations (OPOLO)
Input: expert observations R E , off-policy-transitions R, initial states S 0 , f -function, policy π θ , critic Q φ , discriminator D w , inverse action model P I ϕ , learning rate α.
for n = 1, . . . do sample trajectory τ ∼ π θ , R ← R ∪ τ update D w : w ← w + αÊ (s,s )∼R E [ w log(D w (s, s )] +Ê (s,s )∼R [ w log(1 − D w (s, s ))]. set r(s, s ) = − log(1 − D w (s, s )). update P I ϕ : ϕ ← ϕ + αÊ (s,a,s )∼R [ ϕ log(P I ϕ (a|s, s ))]
. update π θ and Q φ :
J(π θ , Q φ ) = (1−γ)Ês∼S 0 [Q φ (s, π θ (s))]+Ê (s,a,s )∼R f * r(s, s )+γQ φ (s , π θ (s ))−Q φ (s, a) . J Reg (π θ ) = E (s,s )∼R E ,a∼P I ϕ (·|s,s ) [log π θ (a|s)]. φ ← φ − αJ φ (π θ , Q φ ); θ ← θ + α J θ (π θ , Q φ ) + J θ J Reg (π θ )
. end for work, including adopting different RL frameworks [4], or choosing different divergence measures [13,5,38] to enhance the effectiveness of imitation learning. Most work along this line focuses on on-policy learning, which is a sample-costly strategy.
As an off-policy extension of GAIL , DAC [4] improves the sample-efficiency by re-using previous samples stored in a relay buffer rather than on-policy transitions. Similar ideas of reusing cached transitions can be found in [16]. One limitation of these approaches is that they neglected the discrepancy induced when replacing the on-policy distribution with off-policy approximations, which results in a deviation from their proposed objective. Another off-policy imitation learning approach is ValueDICE [3], which inherits the idea of DICE [30] to transform an on-policy LfD objective to an off-policy one. This approach, however, requires the information of expert actions, which otherwise makes off-policy estimation unreachable in a model-free setting. Therefore, their approach is not directly applicable to LfO. We have analyzed this dilemma in Sec 9.8 in the appendix.
Learning from Observations (LfO) tackles a more challenging scenario where expert actions are unavailable. Work alone this line falls into model-free and model-based approaches. GAIfO [9] is a model-free solution which applies the principle of GAIL to learn a discriminator with state-only inputs. IDDM [10] further analyzed the theoretical gap between the LfD and LfO objectives, and proved that a lower-bound of this gap can be somewhat alleviated by maximizing the mutual-information between (s, (a, s )), given an on-policy distribution µ π (s, a, s ). Its performance is comparable to GAIL. [24] assumed that the given observation sequences are ranked by superiority, based on which a reward function is designed for policy learning. Similar to GAIL, the sample efficiency of these approaches is suboptimal due to their on-policy strategy.
Model-based LfO can be further organized into learning a forward [23,39] dynamics model or an inverse action model [17,21]. Especially, [23] proposed a forward model solution to learn timedependent policies for finite-horizon tasks, in which the number of policies to be learned equals the number of transition steps. This approach may not be suitable for tasks with long or infinite horizons. Behavior cloning from observations (BCO) [17] learns an inverse model to infer actions missing from the expert dataset, after which behavior cloning is applied to learn a policy. Besides the common issues faced by BC, this strategy does not guarantee that the ground-truth expert actions can be recovered, unless is a deterministic and injective MDP is assumed. Some other recent work focused on different problem settings than ours, in which the expert observations are collected with different transition dynamics [40] or from different viewpoints [21,41,42]. Readers are referred to [11] for further discussions of LfO.
Experiments
We compare OPOLO against state-of-the-art LfD and LfO approaches on MuJuCo benchmarks, which are locomotion tasks in continuous state-action space. In accordance with our assumption in Sec 3.4, these tasks have deterministic dynamics. Original rewards are removed from all benchmarks to fit into an IL scenario. For each task, we collect 4 trajectories from a pre-trained expert policy. All illustrated results are evaluated across 5 random seeds.
Baselines: We compared SAIL against 7 baselines. We first selected 5 representative approaches from prior work: GAIL (on-policy LfD), DAC (off-policy LfD), ValueDICE (off-policy LfD), GAIfO (onpolicy LfO), and BCO (off-policy LfO). We further designed two strong off-policy approaches, Specifically, we built DACfO, which is a variation of DAC that learns the discriminator on (s, s ) instead of (s, a), and ValueDICEfO, which is built based on ValueDICE. Instead of using groundtruth expert actions, ValueDICEfO learns an inverse model by optimizing Eq (11), and uses the approximated actions generated by the inverse model to fit an LfO problem setting. To the best of our knowledge, DACfO and ValueDICEfO have not been investigated by any prior art. Among these baselines, GAIL, DAC, and ValueDICE are provided with both expert states and actions, while all other approaches only have access to expert states. More experimental details can be found in the supplementary material.
Our experiments focus on answering the following important questions:
1. Asymptotic performance: Is OPOLO able to achieve expert-level performance given a limited number of expert observations? 2. Sample efficiency: Can OPOLO recover expert policy using less interactions with the environment, compared with the state-of-the-art? 3. Effects of the inverse action regularization: Does the inverse action regularization useful in speeding up the imitation learning process? 4. Sensitivity of the choice of f -divergence: Can OPOLO perform well given different f functions?
Performance Comparison
OPOLO can recover expert performance given a fixed budget of expert observations. As shown in Figure 1, OPOLO reaches (near) optimal performance in all benchmarks. For simpler tasks such as Swimmer and InvertedPendulum, most baselines can successfully recover expertise. For other complex tasks with high state-action space, on-policy baselines, such as GAIL and GAIfO, are struggling to reach their asymptotic performance within a limited number of interactions, As shown in Figure 2, the off-policy baseline BCO is prone to sub-optimality due to its behavior cloning-like strategy, On the other hand, the performance of ValueDICEfO can be deteriorated by potential action-drifts, as the inferred actions are not guaranteed to recover expertise. For fair comparison, performance of all off-policy approaches are summarized in Table 2 given a fixed number of interaction steps.
The asymptotic performance of OPOLO is 1) superior to DACfO and ValueDICEfO, 2) comparable to DAC, and 3) is more robust against overfitting compared with ValueDICE, whereas both DAC and ValueDICE enjoy the advantage of off-policy learning and extra action guidance.
Sample Efficiency
OPOLO is comparable with and sometimes superior to DAC in all evaluated tasks, and is much more sample-efficient than on-policy baselines. As shown in Figure 1, the sample-efficiency of OPOLO is emphasized by benchmarks with high state-action dimensions. In particular, for tasks such as Ant or HalfCheetah, the performance curves of on-policy baselines are barely improved at early learning stages. One intuition is that they need more explorations to build the current support of the learning policy, which cannot benefit from cached transitions. For these challenging tasks, OPOLO is even more sample-efficient than DAC that has the guidance of expert actions. We ascribe this improvement to the mode-covering regularization of OPOLO enforced by its inverse action model, whose effect will be further analyzed in Sec 5.3. Meanwhile, other off-policy approaches such as BCO and
ValueDICEfO, are prone to overfitting and performance degradation (as shown in Figure 2), which indicates that the effect of the inverse model alone is not sufficient to recover expertise. On the other hand, the ValueDICE algorithm, although being sample-efficient, is not designed to address LfO and requires expert actions.
Returns
HalfCheetah-v2 Figure 2: Compared with strong off-policy baselines, OPOLO is the only approach that consistently achieves competitive performance regarding both sample-efficiency and asymptotic performance across all tasks, without accessing expert actions.
Ablation Study
In this section, we further analyze the effects of the inverse action regularization by a group of ablation studies. Especially, we implement a variant of OPOLO that does not learn an inverse action model to regulate the policy update. We compare this approach, dubbed as OPOLO-x, against our original approach as well as the DAC algorithm.
Effects on Sample efficiency: Performance curves in Figure 3 show that removing the inverse action regularization from OPOLO slightly affects its sample-efficiency, although the degraded version is still comparable to DAC. This impact is more visible in challenging tasks such as HalfCheetah and Ant. From another perspective, the same phenomenon indicates that an inverse action regularization is beneficial for accelerating the IL process, especially for games with high observation-space. An intuitive exploration is that, while our main objective serves as a driving force for mode-seeking, a regularization term assists by encouraging the policy to perform mode-covering. Combing these two motivations leads to a more efficient learning strategy.
Effects on Performance: Given a reasonable number of transition steps, the effects of an inverseaction model are less obvious regarding the asymptotic performance. As shown in Table 2, OPOLOx is mostly comparable to OPOLO and DAC. This implies that the effect of the state-covering regularization will gradually fade out once the policy learns a reasonable state distribution. From another perspective, it indicates that following our main objective alone is sufficient to recover expert-level performance. Comparing with BCO which uses the inverse model solely for behavior cloning, we find it more effective when serving as a regularization to assist distribution matching from a forward direction. To analyze the effects of different f -functions on the performance of the proposed approach, we explored a family of f -divergence where f (x) = 1 p |x| p , f * (y) = 1 q |y| q , s.t. 1 p + 1 q = 1, p, q > 1, as adopted by DualDICE [30]. Evaluation results show that OPOLO yields reasonable performance across different f -functions, although our choice (q = p = 2 ) turns out to be most stable. Results using the Ant task is illustrated in Figure 4.
Conclusions
Towards sample-efficient imitation learning from observations (LfO), we proposed a principled approach that performs imitation learning by accessing only a limited number of expert observations. We derived an upper-bound of the original LfO objective to enable efficient off-policy optimization, and augment the objective with an inverse action model regularization to speeds up the learning procedure. Extensive empirical studies are done to validate the proposed approach.
Acknowledgments
This research was jointly supported by the National Science Foundation IIS-1749940, and the Office of Naval Research N00014-20-1-2382. We would like to thank Dr. Boyang Liu and Dr. Junyuan Hong (Michigan State University) for providing insightful comments. We also appreciate Dr. Mengying Sun (Michigan State University) for her assistance in proofreading the manuscript.
Broader Impact
The success of Imitation Learning (IL) is crucial for realizing robotic intelligence. Serving as an effective solution to a practical IL setting, OPOLO has a promising future in various applications, including robotics control [43], game-playing [6], autonomous driving [14], algorithmic trading [44], to name just a few.
On one hand, OPOLO provides an working evidence of sample-efficient IL. OPOLO costs less environment interactions compared with conventional IL approaches. For tasks where taking real actions can be expensive (high-frequency trading) or dangerous (autonomous driving), using less interactions for imitation learning is a crucial requirement for successful applications.
On the other hand, OPOLO validates the feasibility of learning from incomplete guidance, and can enable IL in applications where expert demonstrations are costly to access. Moreover, OPOLO is more resemblant to human intelligence, as it can recover expertise simply by learning from expert observations. In general, OPOLO has a strong impact on the advancement of IL, from the perspective of both theoretical and empirical studies.
Appendix
For all the following derivations, we use D KL [P (X)||Q(X)] to denote the KL-divergence between two distributions P and Q:
D KL [P (X)||Q(X)] = E x∼p(x) log p(x) q(x) = X p(x) log p(x) q(x) dx.
Accordingly, when P (X|Z) and Q(X|Z) are conditional distributions, D KL [P ||Q] denotes their conditional KL-divergence:
D KL [P (X|Z)||Q(X|Z)] = Z×X p(z)p(x|z) log p(x|z) q(x|z) dxdz.
For simplicity, we will equivalently use E x∼p(x) [·] and E p(x) [·] to denote certain expectation in which x is sampled from the distribution P (X).
Derivation of Surrogate Objective
We first refer Lemma 1 from [10] for a complete presentation: Lemma 1.
D KL [µ π (s, a, s )||µ E (s, a, s )] = D KL [µ π (s, a)||µ E (s, a))].
Proof. Proof. As defined in Table 1, µ π (a|s, s ) is the inverse-action transition probability induced by policy π:
D KL [µ π (s, a, s )||µ E (s,
µ π (a|s, s ) = µ π (s, a, s ) µ π (s, s ) = H H H µ π (s) π(a|s)P (s |s, a)
A H H H µ π (s) π(ā|s)P (s |s,ā)dā = π(a|s)P (s |s, a)
A π(ā|s)P (s |s,ā)dā .
Based on this notion, we can derive:
D KL [µ π (s, a)||µ E (s, a)]
= D KL [µ π (s, a, s )||µ E (s, a, s )]
Lemma 1 = S×A×S µ π (s, a, s ) log µ π (s, a, s ) µ E (s, a, s ) ds dads = S×A×S µ π (s, s )µ π (a|s, s ) log µ π (s, s ) × µ π (a|s, s ) µ E (s, s ) × µ E (a|s, s ) ds dads = S×A×S µ π (s, s )µ π (a|s, s ) log µ π (s, s ) µ E (s, s ) ds dads + S×A×S µ π (s, s )µ π (a|s, s ) log µ π (a|s, s ) µ E (a|s, s ) ds dads
= S×A×S µ π (s, s ) log µ π (s, s ) µ E (s, s ) ds ds + D KL [µ π (a|s, s )||µ E (a|s, s )] =D KL [µ π (s, s )||µ E (s, s )] + D KL [µ π (a|s, s )||µ E (a|s, s )](12)≥D KL [µ π (s, s )||µ E (s, s )].
Based on Lemma2, we can derive the upper-bound of our original objective: Theorem 1 (Surrogate Objective as the Divergence Upper-bound).
D KL [µ π (s, s )||µ E (s, s )] ≤ E µ π (s,s ) [log µ R (s, s ) µ E (s, s ) ] + D KL [µ π (s, a)||µ R (s, a)].
Proof. Proof. We can refer Eq (12) from the proof of Lemma 2:
D KL [µ π (s, s )||µ E (s, s )] = S×S µ π (s, s ) log µ π (s, s ) µ E (s, s ) dsds = S×S µ π (s, s ) log µ R (s, s ) µ E (s, s ) × µ π (s, s ) µ R (s, s ) dsds = S×S µ π (s, s ) log µ R (s, s ) µ E (s, s ) dsds + S×A µ π (s, s ) log µ π (s, s ) µ R (s, s ) dsds = E µ π (s,s ) [log µ R (s, s ) µ E (s, s ) ] + D KL [µ π (s, s )||µ R (s, s )] ≤ E µ π (s,s ) [log µ R (s, s ) µ E (s, s ) ] + D KL [µ π (s,D KL [µ π (s, a)||µ E (s, a)] = D KL [µ π (s, s )||µ E (s, s )] + D KL [µ π (a|s, s )||µ E (a|s, s )].
An Unoptimizable Gap Between LfO and LfD
Remark 1: In a non-injective MDP, the discrepancy of D KL [µ π (a|s, s )||µ E (a|s, s )] cannot be optimized without knowing expert actions.
Proof. We provide proof with a counter-example. Consider a non-injective MDP in a tabular case, whose transition dynamics is shown in Table 3, with |S| = 3, and |A| = 4. Especially, there exists two actions which lead to the same deterministic transition, i.e. for s 1 , s 2 ∈ S, ∃ a 0 , a 2 ∈ A, s.t. P (s 2 |s 1 , a 2 ) = P (s 2 |s 1 , a 0 ) = 1, as illustrated in Figure 5.
In this MDP, there is an expert policy π E as listed in Table 5. Trajectories generated by this expert are illustrated as blue lines in Figure 5. In a LfO scenario, a learning agent only has access to sequences P a 0 a 1 a 2 a 3 P (s 1 |s 1 , ·) 0 1 0 0 P (s 2 |s 1 , ·) 1 0 1 0 P (s 3 |s 1 , ·) 0 0 0 1 P (s 1 |s 2 , ·) 0 1 0 0 P (s 2 |s 2 , ·) 0 0 1 0 P (s 3 |s 2 , ·) 0 0 0 1 P (s 1 |s 3 , ·) 0 1 0 0 P (s 2 |s 3 , ·) 0 0 1 0 P (s 3 |s 3 , ·) 0 0 0 1 Table 3: A deterministic but non-injective MDP. π s 1 s 2 s 3 a 0 0.5 0 0 a 1 0 0 1 a 2 0.5 0 0 a 3 0 1 0 Table 4: Learning Policy π. Based on the given observations R E , a policy π can only satisfy the state distribution matching with D KL [µ π (s, s )||µ E (s, s )] = 0, but unable to optimize D KL [µ π (a|s, s)||µ E (a|s, s)], as both a 0 and a 2 lead to a deterministic transition of s 1 → s 2 . In lack of expert actions, the best guess for a learning policy is to equally distribute action probabilities with π(a 0 |s 1 ) = (a 2 |s 1 ) = 0.5. which results in µ π (a 0 |s 1 , s 2 ) = µ π (a 2 |s 1 , s 2 ) = 0.5, whereas µ E (a 2 |s 1 , s 2 ) = 1, µ E (a 0 |s 0 , s 1 ) = 0. Consequently, we reach at D KL [µ π (a|s, s )||µ E (a|s, s )] > 0.
π E s 1 s 2 s 3 a 0 0 0 0 a 1 0 0 1 a 2 1 0 0 a 3 0 1 0
Remark:
In a deterministic and injective MDP, it satisfies that ∀ π : S → A, D KL [µ π (a|s, s )||µ E (a|s, s )] = 0.
We provide proof in a finite, discrete state-action space, although the conclusion is valid to extend to continuous cases.
Proof. In a deterministic and injective MDP, we can interpret the transition dynamics with a deterministic function g: since this MDP is also injective, given arbitrary policy π and a transition s → s , (s, s ) ∼ µ π (s, s ), there exists one and only action a which satisfies g(s, a) = s , P (s |s, a) = 1.
Accordingly, µ π (a|s, s ) = π(a|s)P (s |s,a) Proof. Given two distributions P and Q, their density ratio is denoted as w p|q , with w p|q = p(x) q(x) ≥ 0. If we consider a function g(w) = w log(w) − 1 2 w 2 , g(w) is constantly decreasing when w ∈ (0, ∞), as ∂g ∂w = log w + 1 − w ≤ 0 ∀w ≥ 0. Since KL-Divergence is a special case of f -divergence with f KL (x) = x log x, it is sufficient to show that: Proof. Based on the definition of µ π (a|s, s ) in Table 1:
D KL [P ||Q] − D f [P ||Q] = X q(x) w p/q log(w p/q ) − 1 2 (w p/q ) 2 dx ≤ X q(x) sup w∈(0,+∞) (w log(w) − 1 2 w 2 )dx = X q(x) lim w→0 + (w log(w) − 1 2 w 2 )dx = 0.
µ π (a|s, s ) = π(a|s)P (s |s, a)
A π(ā|s)P (s |s,ā)dā = π(a|s)P (s |s, a) µ π (s |s) ,
and similar for µ E (a|s, s ), we can derive at the following:
D KL [π E (a|s)||π(a|s)] = S×A µ E (s)π E (a|s) log π E (a|s) π(a|s) dads = S×A µ E (s, a) log π E (
Policy Regularization as A Forward Distribution Matching
Without loss of generality, in this section we provide proof based on a finite, discrete state-action space. Based on Assumption 1, we have the following: Corollary 1. In a deterministic MDP, ∀ π : S → A, µ π (a|s, s ) > 0 =⇒ P (a|s, s ) = 1.
Proof. µ π (a|s, s ) ∝ π(a|s)P (s |s, a) > 0 =⇒ P (s |s, a) > 0. Based on Assumption 1, it holds that g(s, a) = s , therefore P (s |s, a) = 1.
Assumption 2 (Support Coverage). The support of expert transition distribution µ E (s, s ) is covered by µ R (s, s ):
µ E (s, s ) > 0 =⇒ µ R (s, s ) > 0.
Combing Corollary 1 and Assumption 2, we can reach at the following: Proof. In a discrete state-action space, µ π (s |s) can be denoted as µ π (s |s) = E a∼π(·|s) [P (s |s, a)], and the similar for µπ(s |s): Table 1 [Q(s, a)] − (1 − γ)
Corollary 2. ∀(s, s ) ∼ µ E (s, s ), µ R (a|s, s ) > 0 =⇒ P (a|s, s ) = 1.D KL [µ E (= (1 − γ) ∞ t=0 γ t E s∼µ π t (s),a∼π(s) see
Implementation Details
Practical Considerations for Algorithm Implementation
We provide some practical considerations to effectively implement our algorithm:
Initial state sampling: To increase the diversity of initial samples, we use state samples from an off-policy buffer and treat them as virtual initial states. A similar strategy is adopted by [3].
Constant shift on synthetic rewards: In practice, we adopt the same strategy of prior art [10] to use r(s, s ) = − log(1 − D(s, s )), instead of log(D) − log(1 − D) as the discriminator output. A fully optimized discriminator D * satisfies − log(1 − D * (s, s )) = log(1 + µ E (s,s ) µ R (s,s ) ), which corresponds to a constant shift on µ E (s,s ) µ R (s,s ) before the log term. Q and π network update: We follow the advice of AlgeaDICE [31] by using a target Q network and policy gradient clipping. Especially, when taking the gradients of J opolo (π, Q, α) w.r.t.Q, we use the value from a target Q network to calculate B π Q(s, a) in order to stabilize training; on the other hand, since an optimal x * (s, a) = (B π Q * − Q * )(s, a) = µ π (s,a) µ R (s,a) represents a density ratio and should always be non-negative, we clip (B π Q − Q)(s, a) to above 0 when taking gradients w.r.t.π. Table 6 lists the hyper-parameters for GAIL [2], GAIfO [9], BCO [17], DAC [4], and our proposed approach OPOLO. Specifically, for off-policy approaches, each self-generated interaction will be stored the replay buffer in a FIFO manner, and update frequency is the number of interactions sampled from the MDP after which the module is updated. Moreover, considering the different scales for the gradients of J(π θ , Q φ ) and J Reg (π θ ) in Algorithm 1, we apply a coefficient λ for OPOLO to adjust the regularization strength when calculating the total policy loss: θ ← θ + α J θ (π θ , Q φ ) + λJ θ J Reg (π θ ) . Table 6: Hyper-parameters for Different Algorithms
Hyper-parameters
Hyper-parameters
Challenges of DICE without Expert Actions
In this section, we analyze the principle of offline imitation learning using DICE [30,33,31] and the reason that impedes its direct application to an LfO setting.
In a LfO setting where expert actions are unavailable, the learning objective is to minimize the discrepancy of state-only distributions induced by the agent and the expert. Without loss of generality, we consider an arbitrary f-divergence D f as the discrepancy measure:
in which f * (x) is the conjugate of f (x) for the f -divergence. To remove the on-policy dependence of µ π (s, s ), we follow the rationale of DICE and use a similar change-of-variable trick mentioned in This value function is a fixed point solution to an variant Bellman operator B π , which, however, is problematic in a model-free setting. To see this, we substitute x(s, s ) by (B π v − v)(s, s ) to transform Eq (14) into the following: where B π v(s, s ) = γE a ∼π(.|s ),s ∼P (.|s ,a ) [v(s , s )]. Optimizing this objective is troublesome, in that the B π v(s, s ) in term 2 requires knowledge of P (·|s, π(s)), ∀s ∼ µ E (s). In another word, for any state sampled from the expert distribution, we need to know what would be the next state if following policy π from this state. A similar issue is echoed in term 1, where s 1 is sampled from P (·|s 0 , π(s 0 )). Consequently, directly applying DICE loses its advantage in a LfO setting, as it incurs a dependence on a forward transition model, which is costly to estimate and may counteract the efficiency brought by off-policy learning.
J
opolo (π, Q) := E (s,a,s )∼µ π (s,a,s ) [log µ E (s, s ) µ R (s, s ) − (B π Q − Q)(s, a)] + E (s,a)∼µ R (s,a) [f * ((B π Q − Q)(s, a))] = (1 − γ)E s0∼p0,a0∼π(·|s0) [Q(s 0 , a 0 )] + E (s,a)∼µ R (s,a) [f * ((B π Q − Q)(s, a))].
π, Q) := (1 − γ)E s 0 ∼p 0 ,a 0 ∼π(·|s 0 ) [Q(s0, a0)] + E µ R (s,a) [f * ((B π Q − Q)(s, a))].
E
(s,s )∼µ E (s,s ) log(D(s, s )) + E (s,s )∼µ R (s,s ) log(1 − D(s, s )) ,
Figure 1 :
1Interaction steps (x-axis) versus learning performance (y-axis). Compared with GAIL, BCO, GAIfO, and DAC, our proposed approach (OPOLO) is the most sample-efficient to reach expert-level performance (Grey horizontal line).
Figure 3 :
3Removing the inverse action regularization (OPOLO-x) results in slight efficiency drop, although its performance is still comparable to OPOLO and DAC.
Figure 4 :
4Performance of SAIL given different ffunctions.
a, s )] = S×A×S µ π (s, a, s ) log µ π (s, a) · P (s |s, a) µ E (s, a) · P (s |s, a) ds dads = S×A×S µ π (s, a, s ) log µ π (s, a) µ E (s, a) ds dads = S×A µ π (s, a) log µ π (s, a) µ E (s, a) dads = D KL [µ π (s, a)||µ E (s, a)]. Lemma 2. D KL [µ π (s, s )||µ E (s, s )] ≤ D KL [µ π (s, a)||µ E (s, a)].
D
KL [µ π (a|s, s )||µ E (a|s, s )] = D KL [µ π (s, a)||µ E (s, a)] − D KL [µ π (s, s )||µ E (s, s )].
Figure 5 :
5Transition of an non-injective MDP. of states visited by the expert: R E = {s 1 , s 2 , s 3 , s 1 , s 2 , s 3 , · · · }, without knowing what actions have been taken by the expert.
∃g : S × A → S, s.t. ∀ (s, a, s ), g(s, a) = s ⇐⇒ P (s |s, a) = 1, and g(s, a) = s ⇐⇒ P (s |s, a) = 0.
.
Eā ∼π(·|s) [P (s |s,a)] = 1[g(s, a) = s ] depends only on the transition dynamics, where 1(x) is an indicator function. The same conclusion applies to µ E (a|s, s ) as well.Therefore, we reach at:∀ π : S → A, D KL [µ π (a|s, s )||µ E (a|s, s )] For two arbitrary distributions P and Q, and an f -divergence with f (x) = 1 2 x 2 , it satisfies that D KL [P ||Q] ≤ D f [P ||Q] .
D
KL [π E (a|s)||π(a|s)] = D KL [µ E (s |s)||µ π (s |s)] + D KL [µ E (a|s, s )||µ π (a|s, s )].
Assumption 1 (
1Deterministic MDP). ∃g : S × A → S a deterministic function, s.t. ∀ (s, a, s ), g(s, a) = s ⇐⇒ P (s |s, a) = 0, and g(s, a) = s ⇐⇒ P (s |s, a) = 1.
Lemma 3 .
3Given a policyπ, s.t. ∀(s, s ) ∼ µ E (s, s ),π(a|s) ∝ µ R (a|s, s ), then it satisfies that: ∀π : S → A, D KL [µ E (s |s)||µ π (s |s)] ≥ D KL [µ E (s |s)||µπ(s |s)].
9. 6
6Derivation of Eq (8): J opolo (π, Q) = E (s,a,s )∼µ π (s,a,s ) [r(s, s ) − (B π Q − Q)(s, a)] + E (s,a)∼µ R (s,a) [f * ((B π Q − Q)(s, a))], where B π Q(s, a) = E s ∼P (·|s,a),a ∼π(·|s ) r(s, s ) + γQ(s , a ) , and r(s, s ) = log µ E (s,s ) µ R (s,s ) . Proof. The first term in the RHS of the above equation can be reduced to the following: E (s,a,s )∼µ π (s,a,s ) [r(s, s ) − (B π Q − Q)(s, a)] =E (s,a)∼µ π (s,a) E s ∼P (·|s,a) r(s, s ) − (B π Q − Q)(s, a) =E (s,a)∼µ π (s,a) E s ∼P (·|s,a) [r(s, s )] + Q(s, a) − E s ∼P (·|s,a) [B π Q(s, a)] =E (s,a)∼µ π (s,a) E s ∼P (·|s,a) X X X X [r(s, s )] + Q(s, a) − E s ∼P (·|s,a),a ∼π(·|s ) [ X X X X r(s, s ) + γQ(s , a )] =E (s,a)∼µ π (s,a) Q(s, a) − γE s ∼P (·|s,a),a ∼π(·|s ) [Q(s , a )]
∞ t=0 γ
t=0t+1 E s∼µ π t ,a∼π(·|s),s ∼P (·|s,a),a ∼π(·|s ) [Q(s , a )]]=(1 − γ) ∞ t=0 γ t E s∼µ π t ,a∼π(s) [Q(s, a)] − (1 − γ) ∞ t=0 γ t+1 E s∼µ π t+1 ,a∼π(·|s) [Q(s, a)]] =(1 − γ)E s∼p0,a0∼π(·|s0) [Q(s 0 , a 0 )]. Therefore: J opolo (π, Q) = (1 − γ)E s∼p0,a0∼π(·|s0) [Q(s 0 , a 0 )] + E (s,a)∼µ R [f * ((B π Q − Q)(s, a))].
E
f [µ π (s, s )||µ E (s, µ π (s,s ) [−x(s, s )] + E µ E (s,s ) [f * (x(s, s ))],
Sec 3.2 to learn a value function v(s, s ): v(s, s ) := −x(s, s ) + γE a ∼π(.|s ),s ∼P (.|s ,a ) [v(s , s )] = −x(s, s ) + B π v(s, s ).
E
µ π (s,s ) [−x(s, s )] + E µ E (s,s ) [f * (x(s, s )0 , s 1 )] + E µ E (s,s ) [f * ((B π v − v)(s, s ))] term 2 .
Table 2 :
2Evaluated performance of off-policy approaches. Results are averaged over 50 trajectories.
Table 5 :
5Expert Policy π E .
s |s)||µπ(s |s)] − D KL [µ E (s |s)||µ π (s |s)]=E µ E (s,s ) log µ π (s |s)] − log µπ(s |s)=E µ E (s,s ) log E a∼π(·|s) [P (s |s, a)] − E µ E (s,s ) log E a∼π(·|s) [P (s |s, a)] =E µ E (s,s ) log E a∼π(·|s) [P (s |s, a)] − E µ E (s,s ) log E a∼µ R (·|s,s ) [P (s |s, a)] =E µ E (s,s ) log E a∼π(·|s) [P (s |s, a)] − E µ E (s,s ) log E a∼µ R (·|s,s ) [1] =E µ E (s,s ) log E a∼π(·|s) [P (s |s, a)] ≤E µ E (s,s ) log E a∼π(·|s) [1]=0.Remark 2.In a deterministic MDP, assuming the support of µ E (s, s ) is covered by µ R (s, s),s.t. µ E (s, s ) > 0 =⇒ µ R (s, s ) > 0,then regulating policy using µ R (·|s, s ) can minimize D KL [µ E (s |s)||µ π (s |s)]: ∃π : S → A, s.t. ∀(s, s ) ∼ µ E (s, s ),π(·|s) ∝ µ R (·|s, s ) =⇒π = arg min π D KL [µ E (s |s)||µ π (s |s)]. Proof. Based on Lemma 3, we have that: ∀π : S → A, D KL [µ E (s |s)||µ π (s |s)] ≥ D KL [µ E (s |s)||µπ(s |s)]. Therefore,π = arg min π D KL [µ E (s |s)||µ π (s |s)]. 9.5.3 Estimating the Inverse Action Distribution Theorem 5. max P I :S×S→A −D KL [µ R (a|s, s )||P I (a|s, s )] ≡ max P I :S×S→A E (s,a,s )∼µ R (s,a,s ) [log P I (a|s, s )]. . − D KL [µ R (a|s, s )||P I (a|s, s )] S×S×A µ R (s, s )µ R (a|s, s ) log µ R (a|s, s ) − log P I (a|s, s ) dadsds = H[µ R (a|s, s )]fixed w.r.t. P I S×S×A µ R (s, s )µ R (a|s, s ) log P I (a|s, s )dadsds = H[µ R (a|s, s )]fixed w.r.t. P I +E µ R (s,a,s ) [log P I (a|s, s )]. Note that we use H[µ R (a|s, s )] to denote the conditional entropy of µ R (a|s, s ), with H[µ R (a|s, s )] = E µ R (s,a,s ) [− log µ R (a|s, s )].=E µ E (s,s ) log
µ E (s |s)
µπ(s |s)
− log
µ E (s |s)
µ π (s |s)
Corollary 2
Proof= −
S×S×A
µ R (s, s )µ R (a|s, s ) log
µ R (a|s, s )
P I (a|s, s )
dadsds
= −
+
Value Shared Parameters for Off-Policy Approaches Q, π update frequency / gradient steps 10 3 /10 3 D update frequency / gradient steps 500/10 Shared Parameters for On-Policy ApproachesBuffer size
10 7
Batch size
100
Learning rate
3e −4
Discount factor γ
0.99
Network architecture
MLP [400, 300]
Batch size
2048
mini-Batch size
256
Learning rate
3e −4
Discount factor γ
0.99
Network architecture
MLP [400, 300]
BCO
P I pre-train gradient steps
10 4
P I update frequency / gradient steps
10 3 /100
DAC
Number of extra absorbing states
1
OPOLO
P I update frequency / gradient steps
500/50
P I regularization coefficient λ
0.1
A reduction of imitation learning and structured prediction to no-regret online learning. Stéphane Ross, Geoffrey Gordon, Drew Bagnell, Proceedings of the fourteenth international conference on artificial intelligence and statistics. the fourteenth international conference on artificial intelligence and statisticsStéphane Ross, Geoffrey Gordon, and Drew Bagnell. A reduction of imitation learning and structured prediction to no-regret online learning. In Proceedings of the fourteenth international conference on artificial intelligence and statistics, pages 627-635, 2011.
Generative adversarial imitation learning. Jonathan Ho, Stefano Ermon, Advances in neural information processing systems. Jonathan Ho and Stefano Ermon. Generative adversarial imitation learning. In Advances in neural information processing systems, pages 4565-4573, 2016.
Imitation learning via off-policy distribution matching. Ilya Kostrikov, Ofir Nachum, Jonathan Tompson, International Conference on Learning Representations (ICLR). Ilya Kostrikov, Ofir Nachum, and Jonathan Tompson. Imitation learning via off-policy distribu- tion matching. International Conference on Learning Representations (ICLR), 2019.
Discriminator-actor-critic: Addressing sample inefficiency and reward bias in adversarial imitation learning. Ilya Kostrikov, Krishna Kumar, Debidatta Agrawal, Sergey Dwibedi, Jonathan Levine, Tompson, ICLRIlya Kostrikov, Kumar Krishna Agrawal, Debidatta Dwibedi, Sergey Levine, and Jonathan Tompson. Discriminator-actor-critic: Addressing sample inefficiency and reward bias in adversarial imitation learning. ICLR, 2019.
Variational discriminator bottleneck: Improving imitation learning, inverse rl, and gans by constraining information flow. Angjoo Xue Bin Peng, Sam Kanazawa, Pieter Toyer, Sergey Abbeel, Levine, ICLRXue Bin Peng, Angjoo Kanazawa, Sam Toyer, Pieter Abbeel, and Sergey Levine. Variational discriminator bottleneck: Improving imitation learning, inverse rl, and gans by constraining information flow. ICLR, 2019.
Mastering the game of go without human knowledge. David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, Nature. 5507676David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. Nature, 550(7676):354-359, 2017.
Human-level control through deep reinforcement learning. Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, G Marc, Alex Bellemare, Martin Graves, Andreas K Riedmiller, Georg Fidjeland, Ostrovski, Nature. 5187540Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Andrei A Rusu, Joel Veness, Marc G Bellemare, Alex Graves, Martin Riedmiller, Andreas K Fidjeland, Georg Ostrovski, et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529-533, 2015.
Apprenticeship learning via inverse reinforcement learning. Pieter Abbeel, Y Andrew, Ng, Proceedings of the twenty-first international conference on Machine learning. the twenty-first international conference on Machine learningACMPieter Abbeel and Andrew Y Ng. Apprenticeship learning via inverse reinforcement learning. In Proceedings of the twenty-first international conference on Machine learning, page 1. ACM, 2004.
Generative adversarial imitation from observation. Faraz Torabi, Garrett Warnell, Peter Stone, ICML Workshop on Imitation, Intent, and Interaction (I3). Faraz Torabi, Garrett Warnell, and Peter Stone. Generative adversarial imitation from observa- tion. ICML Workshop on Imitation, Intent, and Interaction (I3), 2019.
Imitation learning from observations by minimizing inverse dynamics disagreement. Chao Yang, Xiaojian Ma, Wenbing Huang, Fuchun Sun, Huaping Liu, Junzhou Huang, Chuang Gan, Advances in Neural Information Processing Systems. Chao Yang, Xiaojian Ma, Wenbing Huang, Fuchun Sun, Huaping Liu, Junzhou Huang, and Chuang Gan. Imitation learning from observations by minimizing inverse dynamics disagree- ment. In Advances in Neural Information Processing Systems, pages 239-249, 2019.
Recent advances in imitation learning from observation. IJICAI. Faraz Torabi, Garrett Warnell, Peter Stone, Faraz Torabi, Garrett Warnell, and Peter Stone. Recent advances in imitation learning from observation. IJICAI, 2019.
Playing hard exploration games by watching youtube. Yusuf Aytar, Tobias Pfaff, David Budden, Thomas Paine, Ziyu Wang, Nando De Freitas, Advances in Neural Information Processing Systems. Yusuf Aytar, Tobias Pfaff, David Budden, Thomas Paine, Ziyu Wang, and Nando de Freitas. Playing hard exploration games by watching youtube. In Advances in Neural Information Processing Systems, pages 2930-2941, 2018.
Learning robust rewards with adversarial inverse reinforcement learning. Justin Fu, Katie Luo, Sergey Levine, Justin Fu, Katie Luo, and Sergey Levine. Learning robust rewards with adversarial inverse reinforcement learning. ICLR, 2017.
Deep reinforcement learning framework for autonomous driving. E L Ahmad, Mohammed Sallab, Etienne Abdou, Senthil Perot, Yogamani, Electronic Imaging. 19Ahmad EL Sallab, Mohammed Abdou, Etienne Perot, and Senthil Yogamani. Deep rein- forcement learning framework for autonomous driving. Electronic Imaging, 2017(19):70-76, 2017.
High speed obstacle avoidance using monocular vision and reinforcement learning. Jeff Michels, Ashutosh Saxena, Andrew Y Ng, Proceedings of the 22nd international conference on Machine learning. the 22nd international conference on Machine learningJeff Michels, Ashutosh Saxena, and Andrew Y Ng. High speed obstacle avoidance using monoc- ular vision and reinforcement learning. In Proceedings of the 22nd international conference on Machine learning, pages 593-600, 2005.
Sample efficient imitation learning for continuous control. Fumihiro Sasaki, Tetsuya Yohira, Atsuo Kawaguchi, ICLRFumihiro Sasaki, Tetsuya Yohira, and Atsuo Kawaguchi. Sample efficient imitation learning for continuous control. ICLR, 2019.
Behavioral cloning from observation. Faraz Torabi, Garrett Warnell, Peter Stone, Proceedings of the 27th International Joint Conference on Artificial Intelligence (IJCAI 2018). the 27th International Joint Conference on Artificial Intelligence (IJCAI 2018)Faraz Torabi, Garrett Warnell, and Peter Stone. Behavioral cloning from observation. Proceedings of the 27th International Joint Conference on Artificial Intelligence (IJCAI 2018), 2018.
Reinforcement learning: An introduction. S Richard, Andrew G Sutton, Barto, MIT pressRichard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018.
Huang Xiao, Michael Herman, Joerg Wagner, Sebastian Ziesche, Jalal Etesami, Thai Hong Linh, arXiv:1906.08113Wasserstein adversarial imitation learning. arXiv preprintHuang Xiao, Michael Herman, Joerg Wagner, Sebastian Ziesche, Jalal Etesami, and Thai Hong Linh. Wasserstein adversarial imitation learning. arXiv preprint arXiv:1906.08113, 2019.
f-gan: Training generative neural samplers using variational divergence minimization. Sebastian Nowozin, Botond Cseke, Ryota Tomioka, Advances in neural information processing systems. Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-gan: Training generative neural sam- plers using variational divergence minimization. In Advances in neural information processing systems, pages 271-279, 2016.
State alignment-based imitation learning. ICLR. Fangchen Liu, Zhan Ling, Tongzhou Mu, Hao Su, Fangchen Liu, Zhan Ling, Tongzhou Mu, and Hao Su. State alignment-based imitation learning. ICLR, 2019.
Adversarial imitation learning from state-only demonstrations. Faraz Torabi, Garrett Warnell, Peter Stone, Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems. the 18th International Conference on Autonomous Agents and MultiAgent SystemsInternational Foundation for Autonomous Agents and Multiagent SystemsFaraz Torabi, Garrett Warnell, and Peter Stone. Adversarial imitation learning from state-only demonstrations. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, pages 2229-2231. International Foundation for Autonomous Agents and Multiagent Systems, 2019.
Provably efficient imitation learning from observation alone. Wen Sun, Anirudh Vemula, Byron Boots, J Andrew Bagnell, arXiv:1905.10948arXiv preprintWen Sun, Anirudh Vemula, Byron Boots, and J Andrew Bagnell. Provably efficient imitation learning from observation alone. arXiv preprint arXiv:1905.10948, 2019.
Extrapolating beyond suboptimal demonstrations via inverse reinforcement learning from observations. Wonjoon Daniel S Brown, Prabhat Goo, Scott Nagarajan, Niekum, arXiv:1904.06387arXiv preprintDaniel S Brown, Wonjoon Goo, Prabhat Nagarajan, and Scott Niekum. Extrapolating beyond suboptimal demonstrations via inverse reinforcement learning from observations. arXiv preprint arXiv:1904.06387, 2019.
Generative adversarial nets. Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio, Advances in neural information processing systems. Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pages 2672-2680, 2014.
Soft actor-critic: Offpolicy maximum entropy deep reinforcement learning with a stochastic actor. Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, Sergey Levine, arXiv:1801.01290arXiv preprintTuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. Soft actor-critic: Off- policy maximum entropy deep reinforcement learning with a stochastic actor. arXiv preprint arXiv:1801.01290, 2018.
Addressing function approximation error in actor-critic methods. Scott Fujimoto, Herke Van Hoof, David Meger, arXiv:1802.09477arXiv preprintScott Fujimoto, Herke Van Hoof, and David Meger. Addressing function approximation error in actor-critic methods. arXiv preprint arXiv:1802.09477, 2018.
P Timothy, Jonathan J Lillicrap, Alexander Hunt, Nicolas Pritzel, Tom Heess, Yuval Erez, David Tassa, Daan Silver, Wierstra, arXiv:1509.02971Continuous control with deep reinforcement learning. arXiv preprintTimothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. arXiv preprint arXiv:1509.02971, 2015.
Asymptotic evaluation of certain markov process expectations for large time, i. D Monroe, Donsker, Sr Srinivasa Varadhan, Communications on Pure and Applied Mathematics. 281Monroe D Donsker and SR Srinivasa Varadhan. Asymptotic evaluation of certain markov process expectations for large time, i. Communications on Pure and Applied Mathematics, 28(1):1-47, 1975.
Dualdice: Behavior-agnostic estimation of discounted stationary distribution corrections. Ofir Nachum, Yinlam Chow, Bo Dai, Lihong Li, Advances in Neural Information Processing Systems. Ofir Nachum, Yinlam Chow, Bo Dai, and Lihong Li. Dualdice: Behavior-agnostic estimation of discounted stationary distribution corrections. In Advances in Neural Information Processing Systems, pages 2315-2325, 2019.
Algaedice: Policy gradient from arbitrary experience. Ofir Nachum, Bo Dai, Ilya Kostrikov, Yinlam Chow, Lihong Li, Dale Schuurmans, arXiv:1912.02074arXiv preprintOfir Nachum, Bo Dai, Ilya Kostrikov, Yinlam Chow, Lihong Li, and Dale Schuurmans. Al- gaedice: Policy gradient from arbitrary experience. arXiv preprint arXiv:1912.02074, 2019.
Estimating divergence functionals and the likelihood ratio by convex risk minimization. Xuanlong Nguyen, J Martin, Michael I Jordan Wainwright, IEEE Transactions on Information Theory. 5611XuanLong Nguyen, Martin J Wainwright, and Michael I Jordan. Estimating divergence func- tionals and the likelihood ratio by convex risk minimization. IEEE Transactions on Information Theory, 56(11):5847-5861, 2010.
Gendice: Generalized offline estimation of stationary values. Ruiyi Zhang, Bo Dai, Lihong Li, Dale Schuurmans, International Conference on Learning Representations (ICLR. 2020Ruiyi Zhang, Bo Dai, Lihong Li, and Dale Schuurmans. Gendice: Generalized offline estimation of stationary values. International Conference on Learning Representations (ICLR), 2020.
Understanding the relation between maximum-entropy inverse reinforcement learning and behaviour cloning. Seyed Seyed Kamyar, Shane Ghasemipour, Richard Gu, Zemel, ICLR. Seyed Kamyar Seyed Ghasemipour, Shane Gu, and Richard Zemel. Understanding the relation between maximum-entropy inverse reinforcement learning and behaviour cloning. ICLR 2019 Workshop, 2019.
Efficient training of artificial neural networks for autonomous navigation. A Dean, Pomerleau, Neural computation. 31Dean A Pomerleau. Efficient training of artificial neural networks for autonomous navigation. Neural computation, 3(1):88-97, 1991.
A game-theoretic approach to apprenticeship learning. Umar Syed, Robert E Schapire, Advances in neural information processing systems. Umar Syed and Robert E Schapire. A game-theoretic approach to apprenticeship learning. In Advances in neural information processing systems, pages 1449-1456, 2008.
Maximum entropy inverse reinforcement learning. D Brian, Andrew L Ziebart, Andrew Maas, Anind K Bagnell, Dey, Aaai. Chicago, IL, USA8Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, and Anind K Dey. Maximum entropy inverse reinforcement learning. In Aaai, volume 8, pages 1433-1438. Chicago, IL, USA, 2008.
. Martin Arjovsky, Soumith Chintala, Léon Bottou, arXiv:1701.07875Wasserstein gan. arXiv preprintMartin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein gan. arXiv preprint arXiv:1701.07875, 2017.
D Ashley, Himanshu Edwards, Yannick Sahni, Charles L Schroecker, Isbell, Imitating latent policies from observation. ICML. Ashley D Edwards, Himanshu Sahni, Yannick Schroecker, and Charles L Isbell. Imitating latent policies from observation. ICML, 2019.
State-only imitation with transition dynamics mismatch. ICLR. Tanmay Gangwani, Jian Peng, Tanmay Gangwani and Jian Peng. State-only imitation with transition dynamics mismatch. ICLR, 2020.
Imitation from observation: Learning to imitate behaviors from raw video via context translation. Yuxuan Liu, Abhishek Gupta, Pieter Abbeel, Sergey Levine, 2018 IEEE International Conference on Robotics and Automation (ICRA). IEEEYuXuan Liu, Abhishek Gupta, Pieter Abbeel, and Sergey Levine. Imitation from observation: Learning to imitate behaviors from raw video via context translation. In 2018 IEEE International Conference on Robotics and Automation (ICRA), pages 1118-1125. IEEE, 2018.
Third-person imitation learning. C Bradly, Pieter Stadie, Ilya Abbeel, Sutskever, Bradly C Stadie, Pieter Abbeel, and Ilya Sutskever. Third-person imitation learning. ICLR, 2017.
Reinforcement learning in robotics: A survey. Jens Kober, Andrew Bagnell, Jan Peters, The International Journal of Robotics Research. 3211Jens Kober, J Andrew Bagnell, and Jan Peters. Reinforcement learning in robotics: A survey. The International Journal of Robotics Research, 32(11):1238-1274, 2013.
Deep direct reinforcement learning for financial signal representation and trading. Yue Deng, Feng Bao, Youyong Kong, Zhiquan Ren, Qionghai Dai, IEEE transactions on neural networks and learning systems. 28Yue Deng, Feng Bao, Youyong Kong, Zhiquan Ren, and Qionghai Dai. Deep direct rein- forcement learning for financial signal representation and trading. IEEE transactions on neural networks and learning systems, 28(3):653-664, 2016.
| [] |
[
"Case Tool: Fast Interconnections with New 3-Disjoint Paths MIN Simulation Module",
"Case Tool: Fast Interconnections with New 3-Disjoint Paths MIN Simulation Module",
"Case Tool: Fast Interconnections with New 3-Disjoint Paths MIN Simulation Module",
"Case Tool: Fast Interconnections with New 3-Disjoint Paths MIN Simulation Module"
] | [
"Ravi Rastogi \nDepartment of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia\n",
"Amit Singh \nDepartment of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia\n",
"Nikhil Singhal \nDepartment of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia\n",
"Nitin \nCollege of Information Science and Technology\nThe Peter Kiewit Institute\nUniversity of Nebraska at Omaha\nOmaha-68182-0116, Nebraska\n",
"Durg Singh Chauhan \nUttarakhand Technical University\nPost Office Chandanwadi, Prem Nagar, Sudohwala, Dehradun-248007UttarakhandIndia\n",
"Ravi Rastogi \nDepartment of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia\n",
"Amit Singh \nDepartment of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia\n",
"Nikhil Singhal \nDepartment of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia\n",
"Nitin \nCollege of Information Science and Technology\nThe Peter Kiewit Institute\nUniversity of Nebraska at Omaha\nOmaha-68182-0116, Nebraska\n",
"Durg Singh Chauhan \nUttarakhand Technical University\nPost Office Chandanwadi, Prem Nagar, Sudohwala, Dehradun-248007UttarakhandIndia\n"
] | [
"Department of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia",
"Department of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia",
"Department of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia",
"College of Information Science and Technology\nThe Peter Kiewit Institute\nUniversity of Nebraska at Omaha\nOmaha-68182-0116, Nebraska",
"Uttarakhand Technical University\nPost Office Chandanwadi, Prem Nagar, Sudohwala, Dehradun-248007UttarakhandIndia",
"Department of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia",
"Department of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia",
"Department of CSE and IT\nDepartment of Computer Science\nJaypee University of Information Technology\nWaknaghat, Solan-173234Himachal PradeshIndia",
"College of Information Science and Technology\nThe Peter Kiewit Institute\nUniversity of Nebraska at Omaha\nOmaha-68182-0116, Nebraska",
"Uttarakhand Technical University\nPost Office Chandanwadi, Prem Nagar, Sudohwala, Dehradun-248007UttarakhandIndia"
] | [
"International Journal of Computer Applications",
"International Journal of Computer Applications"
] | Multi-stage interconnection networks (MIN) can be designed to achieve fault tolerance and collision solving by providing a set of disjoint paths. In this paper, we are discussing the new simulator added to the tool designed for developing faulttolerant MINs. The designed tool is one of its own kind and will help the user in developing 2 and 3-disjoint path networks. The java technology has been used to design the tool and have been tested on different software platform. | 10.5120/2365-3110 | [
"https://arxiv.org/pdf/1202.0616v1.pdf"
] | 8,009,065 | 1202.0616 | 38dacbbaa8b8037b208412add15d821461132c44 |
Case Tool: Fast Interconnections with New 3-Disjoint Paths MIN Simulation Module
April 2011
Ravi Rastogi
Department of CSE and IT
Department of Computer Science
Jaypee University of Information Technology
Waknaghat, Solan-173234Himachal PradeshIndia
Amit Singh
Department of CSE and IT
Department of Computer Science
Jaypee University of Information Technology
Waknaghat, Solan-173234Himachal PradeshIndia
Nikhil Singhal
Department of CSE and IT
Department of Computer Science
Jaypee University of Information Technology
Waknaghat, Solan-173234Himachal PradeshIndia
Nitin
College of Information Science and Technology
The Peter Kiewit Institute
University of Nebraska at Omaha
Omaha-68182-0116, Nebraska
Durg Singh Chauhan
Uttarakhand Technical University
Post Office Chandanwadi, Prem Nagar, Sudohwala, Dehradun-248007UttarakhandIndia
Case Tool: Fast Interconnections with New 3-Disjoint Paths MIN Simulation Module
International Journal of Computer Applications
196April 2011United States of AmericaMulti-stage Interconnection NetworksFault-tolerance3- Disjoint Paths
Multi-stage interconnection networks (MIN) can be designed to achieve fault tolerance and collision solving by providing a set of disjoint paths. In this paper, we are discussing the new simulator added to the tool designed for developing faulttolerant MINs. The designed tool is one of its own kind and will help the user in developing 2 and 3-disjoint path networks. The java technology has been used to design the tool and have been tested on different software platform.
INTRODUCTION AND MOTIVATION
In a multiprocessor system, many processors and memory modules are tightly coupled together with an interconnection network. A properly designed interconnection network certainly improves the performance of such multiprocessor system. Multistage Interconnection Networks (MINs) [1-10] are highly suitable for communication among tightly coupled nodes. For ensuring high reliability in complex systems, fault tolerance is an important issue. The basic idea for fault tolerance is to provide multiple paths for a source-destination pair, so that alternate paths can be used in case of a fault in a path . However, to guarantee 1-fault tolerance, a network should have a pair of alternate paths for every source destination pair which are disjoint in nature [1-8, [24][25][26][27][28][29][30][31]. Now-a-days applications of MINs are widely used for on-Chip communication. In past number of techniques has been used to increase the reliability and fault-tolerance of MINs, a survey of the fault-tolerance attributes of these networks is found in [1-6]. The modest cost of unique paths MINs makes them attractive for large multiprocessors systems, but their lack of fault-tolerance, is a major drawback. To mitigate this problem, three hardware options are available [1-5, 20-23]:
1. Replicate the entire network, 2. Add extra stages, 3. And /or Add chaining links. 4. Rearranging of the connection patterns with the addition or deletion of hardware links. In addition to this, MINs can be designed to achieve fault tolerance and collision solving by providing a set of disjoint paths. Many researchers have done sufficient work on providing 1-fault tolerance to the MINs however; little attention has been paid to design the 3-Disjoint Paths Fault-tolerant MINs. We have been inspired by the work presented by the authors in [24][25][26][27][28][29][30][31].
A Multi-stage interconnection network is fully able to meet the reliability demands if it is at least one fault tolerant that is there is at least one alternative path to deal with faults or collisions. This alternative path should be disjoint in nature with the existing routing path followed so that there is no such implication that if a switch or a link fails in the existing routing path then the alternative path will also fail. Most design of Multi-stage interconnection networks do not generate at least two disjoint paths and hence are not always fault tolerant resulting in packet losses and eventual performance degradation. Hence, this approach of two disjoint paths will always guarantee a way out of the problem of faults or collisions in a network [32][33][34].
Whenever we want to design a interconnection network, we used to design them manually using the windows word and then hardwired them through the programming. At present, we do not have any tool through which we can develop the interconnection networks tool or this remains out of limelight therefore in this paper; we have discussed a tool designed for developing faulttolerant multi-stage interconnection networks. The designed tool is one of its own kind and will help the user in developing 2 and 3-disjoint path networks.
The rest of the paper is as follows: Section 2 discusses the testbed and experimental setup, new modules added to the existing case tool [32][33][34] and algorithm supported by the screen shots and the pseudocode followed by the conclusion and references.
CASE TOOL: FAST INTERCONNECTIONS 2.1 Testbed and Experimental Setup
CASE stands for "Computer Assisted Software Engineering. A CASE tool is a software tool that helps software designers and developers specify, generate and maintain some or all of the software components of an application. Many popular CASE tools provide functions to allow developers to draw database schemas and to generate the corresponding code in a data description language (DDL). Other CASE tools support the analysis and design phases of software development, for example by allowing the software developer to draw different types of UML diagrams [35].
We have designed both the networks using the Fast Interconnections tool and the architectural design of the software is already published in [33][34]. We have used Eclipse, is a multi-language software development environment comprising an integrated development environment (IDE) and an extensible plug-in system. It is written mostly in Java and can be used to develop applications in Java and, by means of various plug-ins, other programming languages. The IDE is often called Eclipse JDT for Java (i.e. JDK 1.6) and IDE is running on top of the IBM System x, running with Novell's SUSE Linux Enterprise Server 11. We have used advanced java features to build our system. The most important part of the tool is designing of the components, which are used to design disjoint paths MINs. We have design them in paint and stored them in component library. We have provided the access of this component within the tool using ComponentChooser class.
New Module added to the Case Tool
Added a new 3-Disjoint Paths Multi-stage Interconnection
Network Simulator, 2. Design a circuit, enter a custom path, faulty component numbers, and click on simulate button, 3. Simulation will start and the path will turn green/red, depending on the packet drop with the faulty component marked by red cross every time a packet is dropped.
Algorithm
Algorithmic
Step 1: Get path in A.
Algorithmic
Step 2: Extract individual wire numbers in path1[].
Algorithmic
Step 3: Get faulty_components in B.
Case Tool: Screen Shots
Code for the New Module
CONCLUSION AND FUTURE WORK
In this paper, we have discussed a the newly added module to the existing tool called as Fast Interconnections, which have been designed to develop the 2 and 3-disjoint path multi-stage interconnection network. We have provided the algorithm of the new simulator supported by the screen shots and pseudocode.
The current of the newly added module are as followssimulation run time and the amount of packets dropped is currently fixed, faulty components are to be input before simulation starts. Further work-we will maintain a database for dropped packets, and allow user to dynamically drop packets from anywhere in the circuit during simulation.
REFERENCES
[1] T.Y. Feng, A survey of interconnection networks, IEEE Computer 14, pp. [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]1981.
[2] G.B. Adams III, D.P. Agrawal and H.J. Siegel, A survey and comparison of fault-tolerant multi-stage interconnection networks, IEEE Computer 20, pp. [14][15][16][17][18][19][20][21][22][23][24][25][26][27]1987.
Fig 1 :Fig 2 :
12The front end of the Case Tool with the Welcome Message from the Fast Interconnections Group. The front window with different widths of the wires, MIN Components and Color Chooser Applet.Fig 3: A case tool with various components and size of the application window can be fixed in terms of horizontal and vertical distance.
Fig 4 :Fig 5 :
45Shows that the elements have been aligned. Shows that the components drawn using the draw method. We have changed the draw method which we have presented last time because of the addition of the new module.
Fig 6 :
6Highlighting the path in the MIN.Fig 7: MIN with one faulty Component and dropping of Packet.
cmp[line[j][2]].top_bot(line[j][1])==1) wtm2=2*cmp[line[j][2]].getW(); else wtm2=(int) (1.5 *cmp[line[j][2]].getW()); if(p1.x<=p3.x) { g.drawLine(p1.x,p1.y,p2.x+wtm1,p2.y); g.drawLine(p2.x+wtm1,p2.y,p3.x+wtm2,p3.y); g.drawLine(p3.x+wtm2,p3.y,p4.x,p4.y); } else { g.drawLine(p1.x,p1.y,p2.x-wtm1,p2.y); g.drawLine(p2.x-wtm1,p2.y,p3.x-wtm2,p3.y); g.drawLine(p3.x-wtm2,p3.y,p4.x,p4.y);@Action
public Task simulate() throws Exception
{
sim_paint();
return new SimulateTask(getApplication());
}
public void load_sim(File file) throws Exception
{
this.path = file.getPath();;
System.out.println(this.path);
ois = new ObjectInputStream(new
FileInputStream(this.path));
for(int i = 0;i < 100;i++)
{
cmp[i] = (cmp1)ois.readObject();
lc[i] = (Color)ois.readObject();
thick[i] = (Integer)ois.readObject();
line2[i] = (Integer)ois.readObject();
}
for(int i = 0;i < 100;i++)
{
for(int j = 0;j < 4;j++)
{
line[i][j] = (Integer)ois.readObject();
}
}
no_cmp = (Integer)ois.readObject();
no_line = (Integer)ois.readObject();
ois.close();
redraw();
}
void sim_paint()
{
int pkt=0;
Point p1,p2,p3,p4;
Graphics2D g=(Graphics2D)
canvas1.getGraphics();
g.setStroke(new BasicStroke());
g.setColor(Color.BLACK);
int x=0;
String a=jTextField1.getText();
int[] path1=new int[a.length()];
for(int i=0;i<a.length();i++)
{
path1[i]=Character.getNumericValue(a.charAt(i));
}
String b=jTextField2.getText();
int[] cmp1=new int[b.length()];
for(int i=0;i<b.length();i++)
{
cmp1[i]=Character.getNumericValue(b.charAt(i));
}
int flag1=0;
for (int i=0;i<a.length();i++)
{
if(path1[i]>no_line)
{
flag1=1;
break;
}
}
int flag2=0;
for (int i=0;i<b.length();i++)
{
if(cmp1[i]>no_cmp)
{
flag2=1;
break;
}
}
int time=5000000;
if(flag1==1 || flag2==1)
{
time=0;
if(flag1==1)
JOptionPane.showMessageDialog(this.canvas1,
"Invalid Path. Please check the input.");
else if(flag2==1)
JOptionPane.showMessageDialog(this.canvas1,
"Invalid Component number. Please check the
input.");
}
for(int i=0;i<time;i++)
{
int j;
if(i%(time/125)<24000)
{
g.setColor(Color.RED);
for(j=0;j<b.length();j++)
{
p3=cmp[cmp1[j]].centre;
pkt++;
g.drawLine(p3.x-40, p3.y-40, p3.x+40,p3.y+40);
g.drawLine(p3.x-40, p3.y+40, p3.x+40,p3.y-40);
}
}
else
{
g.setColor(g.getBackground());
for(j=0;j<b.length();j++)
{
p3=cmp[cmp1[j]].centre;
g.drawLine(p3.x-40, p3.y-40, p3.x+40,p3.y+40);
g.drawLine(p3.x-40, p3.y+40, p3.x+40,p3.y-40);
}
g.setColor(Color.GREEN);
}
for(j=0;j<a.length();j++)
{
Stroke d= new BasicStroke(3);
g.setStroke(d);
p1=cmp[line[path1[j]][0]].getPoint(line[path1[j]
][1]);
p2=cmp[line[path1[j]][2]].getPoint(line[path1[j]
][3]);
g.drawLine(p1.x,p1.y,p2.x,p2.y);
}
}
}
//the paint function has been modified a little
too, //new code is :
void redraw()
{
Point p1,p2,p3,p4;
Graphics2D g=(Graphics2D)
canvas1.getGraphics();
canvas1.paint(g);
for(int j=0;j<no_cmp;j++)
{
cmp[j].draw(g);
g.drawString(Integer.toString(j),cmp[j].
centre.x,cmp[j].centre.y);
}
g.setStroke(new BasicStroke());
g.setColor(Color.BLACK);
for(int j=0;j<no_line;j++)
{
Stroke d= new BasicStroke(thick[j]);
g.setStroke(d);
g.setColor(lc[j]);
if(line2[j]!=1)
{
p1=cmp[line[j][0]].getPoint(line[j][1]);
p2=cmp[line[j][2]].getPoint(line[j][3]);
g.drawLine(p1.x,p1.y,p2.x,p2.y);
g.drawString(Integer.toString(j),p1.x ,(p1.y-2)
);
}
else
{
p1=cmp[line[j][0]].getPoint(line[j][1]);
p2=cmp[line[j][0]].getPoint(line[j][1]);
p3=cmp[line[j][2]].getPoint(line[j][3]);
p4=cmp[line[j][2]].getPoint(line[j][3]);
int wtm1,wtm2;
if(cmp[line[j][0]].top_bot(line[j][1])==1)
wtm1=2*cmp[line[j][0]].getW();
else
wtm1=(int) (1.5 * cmp[line[j][0]].getW());
if(}
}
}
}
Scalable Switching Fabrics for Internet Routers, White paper, Avici Systems Incorporation. W J , W.J. D`ally, Scalable Switching Fabrics for Internet Routers, White paper, Avici Systems Incorporation, 2001.
Special issue of interconnection networks. L N Bhuyan, IEEE Computer. 206L.N. Bhuyan, Special issue of interconnection networks, IEEE Computer, Vol. 20 (6), June 1987.
Interconnection Network for Large Scale Parallel Processing: Theory and Case Studies. H J Siegel, ISBN 0-07-057561-4McGraw HillH.J. Siegel, Interconnection Network for Large Scale Parallel Processing: Theory and Case Studies, McGraw Hill, ISBN 0-07-057561-4, 1990.
Advanced Computer Architecture: Parallelism. K Hwang, ISBN 0-07-053070-XScalability, Programmability, Tata McGraw-Hill, IndiaK. Hwang, Advanced Computer Architecture: Parallelism, Scalability, Programmability, Tata McGraw-Hill, India, ISBN 0-07-053070-X, 2000.
Interconnection Networks: An Engineering Approach. J Duato, S Yalamanchili, L M Ni, 1-55860-852-4Morgan KaufmannJ. Duato, S. Yalamanchili and L.M. Ni, Interconnection Networks: An Engineering Approach, Morgan Kaufmann, ISBN 1-55860-852-4, 2003.
Principles and Practices of Interconnection Networks. W J Dally, B Towles, 978-0-12-200751-4Morgan KaufmannSan FranciscoW.J. Dally and B. Towles, Principles and Practices of Interconnection Networks, Morgan Kaufmann, San Francisco, ISBN 978-0-12-200751-4, 2004.
Arbitrary Rotation of Raster Images with SIMD Machine Architectures. H R Arabnia, M A Oliver, International Journal of Eurographics Association (Computer Graphics Forum). 61H.R. Arabnia and M.A. Oliver, Arbitrary Rotation of Raster Images with SIMD Machine Architectures, International Journal of Eurographics Association (Computer Graphics Forum), 6(1), pp. 3-12, 1987.
A Reconfigurable Architecture For Image Processing And Computer Vision. S M Bhandarkar, H R Arabnia, J W Smith, International Journal of Pattern Recognition And Artificial Intelligence. 92S.M. Bhandarkar, H.R. Arabnia and J.W. Smith, A Reconfigurable Architecture For Image Processing And Computer Vision, International Journal of Pattern Recognition And Artificial Intelligence, 9(2), pp. 201-229, 1995.
The Hough Transform on a Reconfigurable Multi-Ring Network. S M Bhandarkar, H R Arabnia, Journal of Parallel and Distributed Computing. 241S.M. Bhandarkar and H.R. Arabnia, The Hough Transform on a Reconfigurable Multi-Ring Network, Journal of Parallel and Distributed Computing, 24(1), pp. 107-114, 1995.
Parallel Edge-Region-Based Segmentation Algorithm Targeted at Reconfigurable Multi-Ring Network. M A Wani, H R Arabnia, The Journal of Supercomputing. 251M.A. Wani and H.R. Arabnia, Parallel Edge-Region-Based Segmentation Algorithm Targeted at Reconfigurable Multi- Ring Network, The Journal of Supercomputing, 25(1), pp. 43-63, 2003.
A New Theory of Deadlock-free Adaptive Routing in Wormhole Networks. J Duato, IEEE Transactions on Parallel and Distributed Systems. 412J. Duato, A New Theory of Deadlock-free Adaptive Routing in Wormhole Networks, IEEE Transactions on Parallel and Distributed Systems, 4(12), pp. 1320- 1331,1993.
A Necessary and Sufficient Condition for Dead lock-free Adaptive Routing in Wormhole Networks. J Duato, IEEE Transactions on Parallel and Distributed Systems. 610J. Duato, A Necessary and Sufficient Condition for Dead lock-free Adaptive Routing in Wormhole Networks, IEEE Transactions on Parallel and Distributed Systems, 6(10), pp. 1055-1067,1995.
Deadlock-Free Message Routing in Multiprocessor Interconnection Networks. W J Dally, C L Seitz, IEEE Transactions on Computers, C. 365W.J. Dally and C.L. Seitz, Deadlock-Free Message Routing in Multiprocessor Interconnection Networks, IEEE Transactions on Computers, C-36(5), pp. 547-553, 1987.
Deadlock-Free Adaptive Routing in Multi computer Networks Using Virtual Channels. W J Dally, H Aoki, IEEE Transactions on Parallel Distributed Systems. 44W.J. Dally and H. Aoki, Deadlock-Free Adaptive Routing in Multi computer Networks Using Virtual Channels, IEEE Transactions on Parallel Distributed Systems, 4(4), 1993.
Deadlock-Free Adaptive Routing Algorithms for the 3DTorus: Limitations and Solutions. J Duato, Proceedings of Parallel Architectures and Languages Europe 93. Parallel Architectures and Languages Europe 93J. Duato, Deadlock-Free Adaptive Routing Algorithms for the 3DTorus: Limitations and Solutions, In Proceedings of Parallel Architectures and Languages Europe 93, 1993.
Efficient Algorithms to Solve Dynamic MINs Stability Problems using Stable Matching with Complete TIES. Nitin , A Subramanian, Journal of Discrete Algorithms. 63Nitin and A. Subramanian, Efficient Algorithms to Solve Dynamic MINs Stability Problems using Stable Matching with Complete TIES, Journal of Discrete Algorithms, 6(3), pp. 353-380, 2008.
Adaptive Channel Queue Routing on k-ary n-cubes. A Singh, W J Dally, A K Gupta, B Towels, Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures. the sixteenth annual ACM symposium on Parallelism in algorithms and architecturesA. Singh, W.J. Dally, A.K. Gupta, B. Towels, Adaptive Channel Queue Routing on k-ary n-cubes, Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures, 2004.
Designing a Faulttolerant Fully-chained Combining Switches Multi-stage Interconnection Network with Disjoint Paths. S Nitin, N Garhwal, Srivastava, DOI10.1007/s11227-009-0336-zThe Journal of Supercomputing. Nitin, S. Garhwal and N. Srivastava, Designing a Fault- tolerant Fully-chained Combining Switches Multi-stage Interconnection Network with Disjoint Paths, The Journal of Supercomputing, DOI 10.1007/s11227-009-0336-z, pp. 1-32, 2009.
Comparative Analysis of Traffic Patterns on k-ary n-tree using Adaptive Algorithms based on Burton Normal Form. Nitin , D S Chauhan, 10.1007/s11227-010-0454-7Journal of Supercomputing. Nitin and D.S. Chauhan, Comparative Analysis of Traffic Patterns on k-ary n-tree using Adaptive Algorithms based on Burton Normal Form, Journal of Supercomputing, DOI: 10.1007/s11227-010-0454-7, pp. 1-20, 2010.
On MTTF analysis of a Fault-tolerant Hybrid MINs. V K Nitin, P K Sehgal, Bansal, 1991-8755WSEAS Transactions on Computer Research. 22Nitin, V.K. Sehgal and P.K. Bansal, On MTTF analysis of a Fault-tolerant Hybrid MINs, WSEAS Transactions on Computer Research, ISSN 1991-8755, 2(2), pp. 130-138, 2007.
Component Level Reliability analysis of Faulttolerant Hybrid MINs. Nitin , 1109-2750WSEAS Transactions on Computers. 59Nitin, Component Level Reliability analysis of Fault- tolerant Hybrid MINs, WSEAS Transactions on Computers, ISSN 1109-2750, 5(9), pp. 1851-1859, 2006.
Designing a disjoint path interconnection network with collision solving and fault tolerance. C W Chen, C P Chung, The Journal of Supercomputing. 341C.W. Chen and C.P. Chung, Designing a disjoint path interconnection network with collision solving and fault tolerance, The Journal of Supercomputing, 34(1), pp. 63- 80, 2005.
Design schemes of dynamic rerouting networks with destination tag routing for tolerating faults and preventing collisions. C W Chen, The Journal of Supercomputing. 383C.W. Chen, Design schemes of dynamic rerouting networks with destination tag routing for tolerating faults and preventing collisions, The Journal of Supercomputing, 38(3), pp. 307-326, 2006.
Fortes, Destination tag routing techniques based on a state model for the IADM network. H J Siegal, D R Jose, A B , IEEE Transaction on Computers. 413H.J. Siegal, D.R. Jose and A.B. Fortes, Destination tag routing techniques based on a state model for the IADM network, IEEE Transaction on Computers, 41(3), pp. 274- 285, 1992.
Design of dynamic rerouting networks with destination tag routing for tolerating faults and preventing collisions. B Smith, Springer ScienceB. Smith, Design of dynamic rerouting networks with destination tag routing for tolerating faults and preventing collisions, Springer Science, 2006.
The gamma network. D S Parker, C S Raghavendra, IEEE Transactions on Computers. 33D.S. Parker and C.S. Raghavendra, The gamma network, IEEE Transactions on Computers, 33, pp. 367-373, 1984.
CGIN: A fault tolerant modified gamma interconnection network. P J Chuang, IEEE Transactions on Parallel and Distributed Systems. 712P.J. Chuang, CGIN: A fault tolerant modified gamma interconnection network, IEEE Transactions on Parallel and Distributed Systems, 7(12), pp. 1301-1306, 1996.
Faulttolerant gamma interconnection networks by chaining. C W Chen, N P Lu, T F Chen, C P Chung, IEE Proceedings on Computers and Digital Techniques. 147C.W. Chen, N.P. Lu, T.F. Chen, and C.P. Chung, Fault- tolerant gamma interconnection networks by chaining, In IEE Proceedings on Computers and Digital Techniques, 147(2), pp. 75-80, 2000.
3-Disjoint gamma interconnection networks. C W Chen, N P Lu, C P Chung, The Journal of Systems and Software. 66C.W. Chen, N.P. Lu, and C.P. Chung, 3-Disjoint gamma interconnection networks, The Journal of Systems and Software, 66, pp .129-134, 2003.
3-Disjoint Paths Faulttolerant Omega Multi-stage Interconnection Network with Reachable Sets and Coloring Scheme. R Rastogi, Nitin , D S Chauhan, Proceedings of the 13th IEEE International conference on Computer Modeling and Simulation. the 13th IEEE International conference on Computer Modeling and SimulationEmmanuel College, Cambridge, UKIEEE UKSimR. Rastogi, Nitin and D.S. Chauhan, 3-Disjoint Paths Fault- tolerant Omega Multi-stage Interconnection Network with Reachable Sets and Coloring Scheme, Proceedings of the 13th IEEE International conference on Computer Modeling and Simulation (IEEE UKSim), Emmanuel College, Cambridge, UK, March 30-April 1, 2011.
Fast Interconnections: A Case Tool for Developing Fault-tolerant Multi-stage Interconnection Networks. R Rastogi, Nitin , D S Chauhan, International Journal of Advancements in Computing Technology. 25R. Rastogi, Nitin and D.S. Chauhan, Fast Interconnections: A Case Tool for Developing Fault-tolerant Multi-stage Interconnection Networks, International Journal of Advancements in Computing Technology, ISSN: 2005- 8039, 2(5), December 2010, pp. 13-24.
On a Fast Interconnections. R Rastogi, Nitin , International Journal of Computer Science and Network Security. 108R. Rastogi and Nitin, On a Fast Interconnections, International Journal of Computer Science and Network Security, ISSN: 1738-7906, 10(8), August 2010, pp. 74-79.
| [] |
[
"EQUATIONS, INEQUATIONS AND INEQUALITIES CHARACTERIZING THE CONFIGURATIONS OF TWO REAL PROJECTIVE CONICS",
"EQUATIONS, INEQUATIONS AND INEQUALITIES CHARACTERIZING THE CONFIGURATIONS OF TWO REAL PROJECTIVE CONICS"
] | [
"Emmanuel Briand "
] | [] | [
"Applicable Algebra in Engineering"
] | Couples of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well-adapted to the study of the relative position of two conics defined by equations depending on parameters.MSC2000: 13A50 (invariant theory), 13J30 (real algebra). | 10.1007/s00200-006-0023-8 | [
"https://arxiv.org/pdf/math/0505628v2.pdf"
] | 13,496,725 | math/0505628 | 5a92a12695e21789467f1edc0089e045f8714f5d |
EQUATIONS, INEQUATIONS AND INEQUALITIES CHARACTERIZING THE CONFIGURATIONS OF TWO REAL PROJECTIVE CONICS
13 Feb 2006
Emmanuel Briand
EQUATIONS, INEQUATIONS AND INEQUALITIES CHARACTERIZING THE CONFIGURATIONS OF TWO REAL PROJECTIVE CONICS
Applicable Algebra in Engineering
13 Feb 2006
Couples of proper, non-empty real projective conics can be classified modulo rigid isotopy and ambient isotopy. We characterize the classes by equations, inequations and inequalities in the coefficients of the quadratic forms defining the conics. The results are well-adapted to the study of the relative position of two conics defined by equations depending on parameters.MSC2000: 13A50 (invariant theory), 13J30 (real algebra).
Introduction
Couples of proper real projective conics, admitting real points, can be classified modulo ambient isotopy. The goal of this paper is to provide equations, inequations and inequalities characterizing each class. This is particularly well-suited for the following problem: given two conics whose equations depend on parameters, for which values of the parameters are these conics in a given ambient isotopy class ?
Such problems are of interest in geometric modeling. They are considered for instance in the articles [9,20] (and [21] for the similar problem for ellipsoids). We consider this paper as the systematization of their main ideas. Specially, in [9], an algorithm was proposed to determine the configuration of a pair of ellipses, by means of calculations of Sturm-Habicht sequences. Our approach is different: when there, computations were performed for each particular case, we perform the computations once for all in the most general case. The formulas obtained behave well under specialization.
Instead of working with ambient isotopy, we consider another equivalence relation, rigid isotopy 1 , corresponding to real deformation of the equations of the conics that doesn't change the nature of the (complex) singularities (definition 1, following the ideas of [12]). Figures 1 and 2 provide a drawing for a representative of each class. Rigid isotopy happens to be an equivalence relation just slightly finer than ambient Date: November 2005. 1 We follow the terminology used in real algebraic geometry in similar situations.
IN IS
IaN( * ) IaS IbN isotopy. So we get the classification under ambient isotopy directly from the one under rigid isotopy. The classification of pairs of real projective conics under rigid isotopy was first obtained by Gudkov and Polotovskiy [13,14,15] in their work on quartic real projective curves. Nevertheless, we start (section 2) with re-establishing this classification. We emphasize the following key ingredient: that any rigid isotopy decomposes into a path in one orbit of the space of pencils of conics under projective transformations, and a rigid isotopy stabilizing some pencil of conic (Lemma 2). As a consequence, each rigid isotopy class is determined by an orbit of pencils of conics and the position of the two conics with respect to the degenerate conics in the pencil they generate. This has two direct applications.
First, because there are finitely many orbits of pencils of conics under projective transformations, we get easily a finite set of couples of conics meeting at least once each rigid isotopy class.
Second, it indicates clearly how to derive the equations, inequations, inequalities characterizing the classes, which is done in section 3. The determination of the position of the conics with respect to the degenerate conics in a pencil essentially reduces to problems of location of roots of univariate polynomials. They can be treated using standard tools from real algebra, namely Descartes' law of signs and subresultant sequences. This contributes both to the characterization of the orbits of pencils of conics, and the characterization of the rigid isotopy class associated to each orbit of pencils of conics. Classical invariant theory is also used for the first task.
Last, section 4 provides some examples of computations using the previous results.
Generalities and notations. The real projective space of dimension k will be denoted with RP k ; in particular, RP 2 denotes the projective plane. The space of real ternary quadratic forms will be denoted with S 2 R 3 * . We will consider P(S 2 R 3 * ), the associated projective space (see [6] for the definitions of the notions of projective geometry needed here). The term conic will be used with two meanings:
• an algebraic meaning: an element of P(S 2 R 3 * ). The algebraic conic associated to the quadratic form f will be denoted with [f ]. • a geometric meaning: the zero locus, in RP 2 , of a non-zero quadratic form f . It will be denoted with [f = 0]. A (geometric or algebraic) conic is said proper if it comes from a nondegenerate quadratic form; degenerate if it comes from a degenerate quadratic form. Note that, with this definition, the empty set is a proper (geometric) conic. Algebraic and geometric proper non-empty conics are in bijection, and can be identified.
We define the discriminant of the quadratic form f to be Disc(f ) = det(Matrix(f )).
Any proper non-empty conic cuts out the real projective plane into two connected components. They are topologically non-equivalent: one is homeomorphic to a Möbius strip, the other to an open disk. The former is the outside of the conic, the latter is its inside. Let f 0 = x 2 + y 2 − z 2 . The inside of [f 0 = 0] is the solution set of the inequation f 0 < 0, or, equivalently, the set of points where f 0 has the sign of Disc(f 0 ). These signs change together under linear transformations. Now any proper, non-empty conic is obtained from [f 0 = 0] by means of a transformation of P GL (3, R). Thus the inside of [f = 0] is the set of points where f takes the sign of Disc(f ).
The tangential quadratic form associated to the quadratic form f on R 3 is the quadratic formf on R 3 * whose matrix is the matrix of the cofactors of the matrix of f . The tangential conic associated to [
f ] (resp. [f = 0]) is [f ] (resp. [f = 0]).
A pencil of quadratic forms is a plane (through the origin) in S 2 R 3 * ; the associated (projective) pencil of conics is the corresponding line in P(S 2 R 3 * ). It is said to be non-degenerate if it contains proper conics 2 . The common points of all conics of a given pencil are called the base points of the pencil. They are also the common points of any two distinct conics of the pencil. A non-degenerate pencil of conics has always four common points in the complex projective space, when counted with multiplicities.
Note that we will distinguish between (ordered) couples of conics ((C 1 , C 2 ) distinct from (C 2 , C 1 ), except when C 1 = C 2 ) and (unordered)
pairs of conics ({C 1 , C 2 } = {C 2 , C 1 }).
The characteristic form of the couple (f, g) of real ternary quadratic forms is the binary cubic in (t, u):
Φ(f, g; t, u) := Disc(tf + ug).
Its coefficients will be denoted as follows:
Φ(f, g; t, u) = Φ 30 t 3 + Φ 21 t 2 u + Φ 12 tu 2 + Φ 30 u 3 .
We will also consider the de-homogenized polynomial obtained from Φ by setting u = 1. It will be denoted with φ(f, g; t), or φ(t) when there is no ambiguity about f, g. So:
φ(t) := Disc(tf + g)
Note that Φ 30 = Disc(f ) and Φ 03 = Disc(g).
An isotopy of a manifold M is a continuous mapping θ : I × M → M, where I is an interval containing 0, such that for each t ∈ I, the mapping x → θ(t, x) is an homeomorphism of M onto itself, and x → θ(0, x) is the identity of M.
Two subsets N 1 , N 2 of M are ambient isotopic if there is an isotopy of M such that, at some instant t ∈ I, θ(t,
N 1 ) = N 2 .
This definition is immediately generalized to couples of subsets: (N 1 , N ′ 1 ) and (N 2 , N ′ 2 ) are ambient isotopic if there is an isotopy of M such that, at some instant t, θ(t, N 1 ) = N 2 and θ(t, N ′ 1 ) = N ′ 2 .
Classification
Rigid isotopy.
To classify the couples of conics up to ambient isotopy, we introduce a slightly finer equivalence relation, rigid isotopy, corresponding to a continuous path in the space of couples of distinct proper conics, that doesn't change the nature of the complex singularities of the union of the conics. Before stating formally the definition (definition 1 below), we clarify this point. The complex singularities of the union of the conics correspond to the (real and imaginary) intersections of the conics. For a given multiplicity, there is only one analytic type of intersection point of two conics. Thus the nature of the singularities for the union of two distinct proper conics is determined by the numbers of real and imaginary intersections of each multiplicity. This is narrowly connected to the projective classification of pencils of conics, that can be found in [7,17]. The connection is the following theorem.
Theorem 1. ( [7,17]) Two non-degenerate pencils of conics are equivalent modulo P GL(3, R) if and only if they have the same numbers of real and imaginary base points of each multiplicity.
The space of couples of distinct real conics is an algebraic fiber bundle over the variety of pencils, which is a grassmannian of the RP 1 's in a RP 5 . The fibers are isomorphic to the space of couples of distinct points in RP 1 . The sets of couples of distinct conics with given numbers of real and imaginary intersections of each multiplicity are, after Theorem 1, exactly the inverse images of the orbits of the variety of pencils under P GL(R 3 ), and are thus also smooth real algebraic submanifolds.
We can now state the following definition. We will now show that rigidly isotopic implies ambient isotopic. We first show it for some special rigid isotopies. Proof. Let B be the set of the base points of the pencil of [f ] and [g]. A stratification of RP 2 × I is given by:
S 1 = B × I S 2 = ([g = 0] × I) \ S 1 S 3 = {(p; t) | (f + tkg)(p) = 0} \ S 1 S 4 = RP 2 × I \ (S 1 ∪ S 2 ∪ S 3 )
One checks this stratification is Whitney. The projection from RP 2 × I to I is a proper stratified submersion. The lemma now follows, by direct application of Thom's isotopy lemma, as it is stated in [11].
Lemma 2. Consider two couples of distinct proper non-empty conics. If they are rigidly isotopic, then they can also be connected by a rigid isotopy α t (s t ) where
• s t is a sequence of slidings along one given pencil.
• α t is a path in P GL(3, R) with α 0 = id
Proof. Let (C 0 , D 0 ) and (C 1 , D 1 ) be the couples of conics, and t → (C t , D t ), t ∈ [0; 1] be the rigid isotopy that connects them. It projects to a path in one P GL(3, R)-orbit of the variety of pencils. This path lifts to a path α t of P GL(3, R) with α 0 = id (indeed, the group is a principal fiber bundle over each orbit; specially, it is a locally trivial fiber bundle: [3], ch. I, 4). The mapping t → α −1 t (C t , D t ) is a rigid isotopy drawn inside one pencil of conics. Such an isotopy is easy to describe: the pencil is a space RP 1 with a finite set Γ of degenerate conics. Let E = RP 1 \ Γ. A rigid isotopy inside the pencil is exactly a path in E × E \ Diag(E × E). There exists a finite sequence s t of horizontal and vertical paths, i.e. of slidings, having also origin (C 0 , D 0 ) and extremity α −1 1 (C 1 , D 1 ). Consider now α t (s t ). This is a rigid isotopy connecting (C 0 , D 0 ) to (C 1 , D 1 ).
Theorem 2. Two couples of distinct proper non-empty conics that are rigidly isotopic are also ambient isotopic.
Proof. Let (C 0 , D 0 ) and (C 1 , D 1 ) be rigidly isotopic. Consider a path t ∈ [0, 1] → α t (s t ) connecting them, as in Lemma 2. After Lemma 1, s t lifts to an ambient isotopy β t with β 0 = id. Then α t • β t is an ambient isotopy carrying (C 0 , D 0 ) to (C 1 , D 1 ).
2.2.
Orbits of pencils of conics. After [7,17], there are nine orbits of non-degenerate pencils of conics under the action of P GL(3, R). We follow Levy's nomenclature [17] for them. It is presented in the following table, where the second and third lines display the multiplicities of the real and imaginary base points. For instance, 211 stands for one We will also use the representatives of the orbits provided by Levy [17]. Each representative is given by a pair of generators of the corresponding pencil of quadratic forms. They are presented in Table 1. We provide, in figures 5 and 6, graphical representations of characteristic features of the pencils in each orbit. This has two goals: finding how to discriminate between the different orbits of pencils, and determining the possible rigid isotopy classes corresponding to each orbit of pencils. Each pencil is displayed as a circle, as it is topologically. In addition, the following information is represented:
Orbit f 0 g 0 I x 2 − y 2 x 2 − z 2 Ia x 2 + y 2 + z 2 xz Ib x 2 + y 2 − z 2 xz II yz x(y − z) IIa y 2 + z 2 xz III xz y 2 IIIa x 2 + y 2 z 2 IV xz − y 2 xy V xz − y 2 x 2
• the degenerate conics of the pencil. They are given by roots of the characteristic form, so the multiplicity of this root is also indicated, following the encoding shown in Figure 3. • In the case where the conics of one arc are nested, we indicate, by means of an arrow, which are the inner ones. 2.3. Rigid isotopy classification for pairs. We first classify pairs of proper conics, that is: couples of distinct proper conics, under rigid isotopy and permutation of the two conics. Later, for each pair class, we will check whether it is also a couple class (that is: the exchange of the two conics corresponds to a rigid isotopy) or it splits into two couple classes. Any pencil of conics is cut into arcs by its degenerate conics. Two proper conics are either on a same arc, or on distinct arcs on the pencil they generate.
Lemma 3. If a pencil of conics has (at least) two arcs of non-empty conics, then there are two equivalence classes for pairs of conics generating it. They correspond to the following situations:
• the conics are on a same arc.
• the conics are on distinct arcs. If the pencil has only one arc with proper non-empty conics, there is only one class.
Proof. The orbit of pencils is assumed to be fixed. Because of Lemma 1, to get (at least) one representative for each class, it is enough:
• to choose arbitrarily one conic on each arc and consider all the possible pairs of these conics. • to choose arbitrarily two conics on each arc and consider these pairs for each arc. But one observes that for a pencil in one of the orbits Ia, II, IIa, III, there is a projective automorphism that leaves it globally invariant and exchanges its two arcs (the two arcs bearing non-empty conics for orbit Ia). Again, this is proved by considering only Levy's representatives: the reflection x ↔ −x is suitable. Similarly, a pencil in orbit I is left globally invariant by some projective automorphism that permutes cyclically the three arcs. For Levy's representative, one can take the cyclic permutation of coordinates:
x → y → z → x.
Pencils in the four other orbits have only one arc with non-empty proper conics.
We have shown it is enough:
• to choose arbitrarily one arc with non-empty conics and two conics on this arc. • to choose arbitrarily two arcs and one conic on each arc. This gives nine representatives for the pairs of conics on a same arc, denoted with IN, . . . , VN (N like neighbors) and five representatives for pairs of conics on distinct arcs, denoted with IS, IaS, IIS, IIaS, IIIS (S like separated ).
Now it remains to check that for orbits I, Ia, II IIa and III, the Srepresentative and the N-representative are not equivalent. We use that a rigid isotopy conserves the topological type of (RP 2 , Table 2. They correspond to the graphical representations displayed in Figures 1 and 2. 2.4. Rigid isotopy classification for couples. We now derive from our classification for pairs of conics the classification for couples of conics.
Lemma 4. For each of the following representatives: IN, IS, IaS, IbN, IIS, IIaS, IIIS, IVN, there is a rigid isotopy that swaps the two conics. As a consequence, each of these classes for pairs is also a class for couples. Let C be the conic [f = 0] (resp. D the conic [g = 0]), I (resp. J) its inside andĪ (resp.J) the topological closure of this inside.
class f g IN 3x 2 − 2y 2 − z 2 3x 2 − y 2 − 2z 2 IS 3x 2 − 2y 2 − z 2 x 2 − 2y 2 + z 2 IaN( * ) x 2 + y 2 + z 2 + 3xz x 2 + y 2 + z 2 + 4xz IaS x 2 + y 2 + z 2 + 3xz x 2 + y 2 + z 2 − 3xz IbN x 2 + y 2 − z 2 + xz x 2 + y 2 − z 2 − xz IIN( * ) yz + xy − xz yz + 2xy − 2xz IIS yz + xy − xz yz − xy + xz IIaN( * ) y 2 + z 2 + xz y 2 + z 2 + 2xz IIaS y 2 + z 2 + xz y 2 + z 2 − xz IIIN( * ) xz+ 2 xz + 2y 2 IIIS xz + y 2 xz − y 2 IIIaN( * ) x 2 + y 2 − z 2 x 2 + y 2 − 2z 2 IVN xz − y 2 + xy xz − y 2 − 2xy VN( * ) xz − y 2 − x 2 xz − y 2 + x 2
Then one checks that the numbers of connected components of the four following sets are suitable for separating the ambient isotopy classes:
C ∩ D, RP 2 \ (C ∪ D), I \J, J \Ī
Remark : It is legitimate curiosity to compare these isotopy classes of couples of conics with the isotopy classes of projective quartic curves presented in [16], the union of two conics being a quartic. One then observes that IN corresponds to 17p, IS to 16p, IaS to 22p, IIS to 34p, IIaS to 44p, IIIS to 38p, IaN ∪ IIIaN to 21p, IIN to 36p and IIaN ∪ VN to 43p. Finally both IbN ∪ IVN and IIIN correspond to 18p. ⊡
Characterizing the isotopy classes by equations, inequations and inequalities
3.1. Preliminaries.
The invariants and covariants of two ternary quadratic forms.
Invariants and covariants (see [18] for a modern reference about classical invariant theory) are the convenient objects to discriminate, by means of equations and inequalities, between the different orbits for couples of complex conics under the group P GL(3, C).
Invariants and covariants of a couple of quadratic ternary forms have been calculated by the classics, and can be found in Glenn's book [10] or Casey's treatise [5]. Proposition 1. The algebra of invariants of a couple of ternary quadratic forms is freely generated by the coefficients of the characteristic form 4 .
The invariants alone are not sufficient to discriminate between the complex orbits. One has to consider the covariants. Some remarkable covariants of a couple of ternary quadratic forms are:
• The apolar covariant of the tangential quadratic formsf and g. We will denote it with F . This is a quadratic form that depends quadratically on f , as on g. Proposition 2. The algebra of covariants of a couple of ternary quadratic forms (f, g) is generated by the invariants, the ground forms f and g, the apolar covariant F and the autopolar triangle covariant G.
The covariant F will not be needed in this paper, but G will be used. We now explain how to derive a formula for it. Consider a generic couple of forms f, g. Let t 1 , t 2 , t 3 be the three roots of Disc(tf + g). Each of the t i f +g has rank two. Their respective associated tangential quadratic forms have all rank one: they are the squares of three linear forms p 1 , p 2 , p 3 of the dual space, and the associated points [p 1 ], [p 2 ], [p 3 ] of P(R 3 ) are exactly the vertices of the autopolar triangle. The sides of the triangle are obtained as the zero loci of the product of determinants: det(p 1 , p 2 , p) det(p 1 , p 3 , p) det(p 2 , p 3 , p).
Working in coordinates, with p i = p i1 X + p i2 Y + p i3 Z, where X, Y, Z are coordinates on R 3 * dual to x, y, z, one expands this and replace the products p ij p ik by the corresponding term given by the equality
p 2 i = f t i + g.
This product is antisymmetric in t 1 , t 2 , t 3 , and thus can be divided by the Vandermonde determinant (t 1 − t 2 )(t 1 − t 3 )(t 2 − t 3 ). The quotient happens to be free of t i 's: it is, up to a rational number in factor, the covariant G.
One finds that the formula for this covariant can be displayed shortly. Denote:
tf + g =ft 2 + Ω(f, g)t +g and
f = ã ijk X i Y j Z k , Ω(f, g) = ω ijk X i Y j Z k ,g = b ijk X i Y j Z k .
Consider the matrix of their coefficients:
M = ãU) + deg(V ) − 1 down to 0 of t deg(V )−1 U, t deg(V )−2 U, . . . , U, t deg(U )−1 V, t deg(U )−2 V, . . . , V.
A few classical formulas about resultants will be needed.
Res(U, V ) = c deg(V ) V (ρ)
where the product is carried over the complex roots ρ of U, counted with multiplicities 5 . This is Lemma 4.27 in [1], where a proof is provided. Only the following particular consequence will be needed in the sequel: It is obtained by applying Descartes' law of signs to U(t) and U(−t).
Lemma 11. Let U(t) = u 3 t 3 + u 2 t 2 + u 1 t + u 0 of degree 3.
Subresultant sequences.
Here we briefly introduce another tool: subresultant sequences. More details about them can be found in the book [1].
Let U(t), V (t) be two univariate polynomials. One wants to know on how many 6 of the (real) roots of V the polynomial U is positive, negative, and zero. The Sturm query of U for V is defined as the number of roots of V making U > 0, minus the number of roots of V making U < 0. This information is easily accessible once one knows the signs of the deg(V ) + 1 signed subresultants principal coefficients of V and W , where W is the remainder in the euclidean division of U · V ′ by V .
We give the formulas for these signed subresultant principal coefficients, and the procedure for getting the Sturm query from their signs, only for the particular case needed: when V has degree 3. Write
V = v 3 t 3 + v 2 t 2 + v 1 t + v 0 W = w 2 t 2 + w 1 t + w 0 .
Then
sr 3 (V, W ) = v 3 , sr 2 (V, W ) = w 2 , sr 1 (V, W ) = v 3 v 2 v 1 0 w 2 w 1 w 2 w 1 w 0 , sr 0 (V, W ) = v 3 v 2 v 1 v 0 0 0 v 3 v 2 v 1 v 0 0 0 w 2 w 1 w 0 0 w 2 w 1 w 0 0 w 2 w 1 w 0 0 0 . 6
When dealing with subresultant sequences, the multiplicities of the roots are not taken into account, e.-g. a double root will be counted as one root.
Note that sr 0 (V, W ) = − Res(V, W ), the opposite of the resultant of V and W . The Sturm query is obtained from the sequence of the signs of sr 3 , sr 2 , sr 1 , sr 0 the following way 7 :
(1) If there is a pair of consecutive zeros, remove it and change the signs that were following to their opposites. (2) From the resulting sequences of consecutive non-zero terms, compute the difference: number of sign permanences (identical consecutive terms, ++ or −−) minus number of sign exchanges (opposite consecutive terms, +− or −+). This gives the Sturm query 8 .
3.2.
Discriminating between the orbits of pencils. In this section, we give the equations and inequations characterizing the couples (f, g) of non-degenerate quadratic forms generating a pencil of each of the orbits. We first use invariants and covariants whose vanishing depends only of the generated pencil, that is those C that, besides the good behavior with respect to the action of SL(3, C):
C(f • θ, g • θ;
x, y, z; t, u) = C(f, g; θ(x, y, z); t, u) ∀θ ∈ SL(3, C)
are covariant with respect to combinations of f and g:
C(θ(f, g); x, y, z; θ(t, u)) = C(f, g; x, y, z; t, u) ∀θ ∈ SL(2, C).
Such objects are called combinants.
Obviously, the characteristic form Φ and its covariants are combinants. Remember that the algebra of the covariants of a binary cubic form Φ(t, u) is generated by the ground form Φ, its discriminant 9 , and its Hessian determinant, which are
Disc(Φ) = Res(φ, φ ′ ) 27Φ 30 , H(t, u) = ∂ 2 Φ dt 2 ∂ 2 Φ dt du ∂ 2 Φ dt du ∂ 2 Φ du 2
(the division is a simplification in the definition of the discriminant, that is: there remains no Φ 30 at the denominator). The covariant G is also a combinant.
The vanishing or non-vanishing of each of the combinants are properties of the orbits of pencils of conics. The sign of Disc(Φ) is also invariant on each orbit of pencils of conics (because Disc(Φ) has even degree in f as well as in g). Thus we just evaluate the combinants on Levy's representatives, and we get the following result: 7 for this specific case with 4 terms in the sign sequence. 8 So for instance, the sign sequence +0 − 0 has no sign permanence, nor sign change (because there are no consecutive non-zero terms). For the sign sequence +00−, the Sturm query is computed as for ++: one permanence, no change, this gives 1. 9 The discriminant of the characteristic form, Disc(Φ), is called the Tact invariant by the classics, because it vanishes exactly when the two conics are tangent [5].
Proposition 3. Let f, g be two non-proportional non-degenerate ternary quadratic forms.
• If Disc(Φ) < 0 then f, g generate a pencil in orbit I or Ia.
• If Disc(Φ) > 0 then f, g generate a pencil in orbit Ib.
• If Disc(Φ) = 0 then f, g generate a pencil in one of the six other orbits. The following table indicates how the vanishings of H and G discriminate further between the orbits of pencils (under the hypothesis that the discriminant vanishes):
H = 0 H = 0 G = 0 II, IIa IV G = 0 III, IIIa V .
Remark : The fact that the coefficients of G are linear combinations of maximal minors of the matrix M defined in 3.1.1 suggests that the vanishing of G is equivalent to: M takes rank two. This is true. To see this, consider the image of the (complex) pencil generated by [f ] and [g] by the quadratic mapping "tangential quadratic form" from P(S 2 C 3 * ) to P(S 2 C 3 ). It is an irreducible conic, thus either a proper conic or a line. One checks on Levy's representative that it is a line exactly when G = 0 (see also [2], 16.5.6.2). Finally, remark that the rows of M are the coordinates of generators of the linear span of this conic. ⊡ It remains now to discriminate between I and Ia, between II and IIa and between III and IIIa.
det(v · I − Matrix(tf + g)) that expands into v 3 − µ(t)v 2 + ψ(t)v − φ(t).
A degenerate conic of the pencil corresponds to a parameter t that annihilates φ, and is
• a pair of lines when Matrix(tf + g) has one eigenvalue positive, one negative, and one zero. Then ψ(t) < 0. • an isolated point when the matrix has an eigenvalue zero and the two other both positive or both negative. Then ψ(t) > 0.
• a single line when the matrix has two eigenvalues zero, and one non-zero. Then ψ(t) = 0. Thus the discriminations can be performed by a Sturm query of ψ for φ. In order not to introduce denominators, we consider the euclidean division of Φ 30 ψφ ′ by φ, instead of the division of ψφ ′ by φ suggested by 3.1.4. Set
P = Remainder(Φ 30 · Ψ · φ ′ , φ) = p 2 t 2 + p 1 t + p 0 and
A i = sr i (φ, P ) for i between 0 and 3. The consideration of the sign permanences and sign exchanges in Φ 30 = A 3 , A 2 , A 1 , A 0 gives the Sturm query of Φ 30 ψ for φ. The Sturm query of ψ for φ is the same as the Sturm query of Φ 2 30 ψ for φ. Using that sr i (φ, Φ 30 P ) = Φ 4−i 30 sr i (φ, P ), we get that this Sturm query is obtained by considering the sign permanences and sign exchanges in
Φ 30 , Φ 30 A 2 , A 1 , Φ 30 A 0 ; or, simpler, those in 1, A 2 , Φ 30 A 1 , A 0 . The polynomial A 1 is Φ 30 Φ 21 Φ 12 0 p 2 p 1 p 2 p 1 p 0 .
And
A 0 = − Res(φ, P ). This simplifies. Applying Lemma 9, one gets Res(Φ 30 ψφ ′ , φ) = Φ 2 30 Res(φ, P ). And, on the other hand, from Lemma 8 and the definition of Disc(Φ):
Res(Φ 30 ψφ ′ , φ) = Φ 4 30 · Res(ψ, φ) · Disc(Φ). Thus A 0 = −Φ 2 30 Res(ψ, φ) Disc(Φ). 3.2.1.
Discriminating between I and Ia. Suppose f and g generate a pencil in orbit I or Ia. The Sturm query of ψ for φ is −3 for orbit I and 1 for orbit Ia.
The assumption that f, g generate a pencil in orbit I or Ia gives more information:
Lemma 12. If f, g generate a pencil in orbit I or Ia, then A 0 < 0.
Proof. We had established that A 0 = −Φ 2 30 Res(ψ, φ) Disc(Φ). For orbit I or Ia, one has Disc(Φ) < 0. Moreover, Res(ψ, φ) < 0 because, from lemmas 6 and 7,
Res(ψ, φ) = Res(φ, ψ) = Φ 2 30 ψ(ρ),
where the product is carried over the three roots ρ of φ. They make either ψ three times negative (orbit I), either one time negative and two times positive (orbit Ia). In both cases, the product is negative.
There is only one sign sequence giving Sturm query −3 and beginning with + and finishing with −, that is + − +−. There are several sign sequences giving Sturm query 1, beginning with +, finishing with −:
+ + + − + + − − + + 0 − + − − − +0 − − + 00 − .
We deduce from this the criterion stated in the following proposition.
Proposition 4. Let f, g be non-degenerate quadratic forms generating a pencil in orbit I or Ia.
• if it is orbit I then p 2 < 0 and Φ 30 A 1 > 0.
• if it is orbit Ia, then p 2 > 0, or Φ 30 A 1 < 0, or p 2 = A 1 = 0.
Discriminating between II and
IIa. Suppose f and g generate a pencil in orbit II or IIa. Note first that A 0 = 0. The Sturm query of ψ for φ is −2 for orbit II and 0 for orbit IIa.
There is only one sign sequence with beginning with +, finishing with 0 that gives Sturm query −2, this is + − +0. Those giving Sturm query 0 are
+ + −0 + − − 0 + 0 + 0 + 0 − 0 + 000.
Proposition 5. Let f, g be non-degenerate quadratic forms generating a pencil in orbit II or IIa.
• if it is in orbit II then p 2 < 0 and Φ 30 A 1 > 0.
• if it is in orbit IIa, then p 2 = 0 or Φ 30 A 1 < 0.
Discriminating between III and
IIIa. Suppose f and g generate a pencil in orbit III or IIIa. Once again, A 0 = 0. The Sturm query of ψ for φ is −1 for orbit III, 1 for orbit IIIa. The only sign sequence (beginning with +, terminating with 0) giving Sturm query −1 is + − 00. There is also only one giving Sturm query 1, that is + + 00.
Proposition 6. Let f, g be non-degenerate quadratic forms generating a pencil in orbit III or IIIa.
• if it is in orbit III, then p 2 < 0.
• if it is in orbit IIIa, then p 2 > 0.
3.3.
Characterizing the rigid isotopy classes for pairs inside each pencil. Given f, g two non-proportional non-degenerate quadratic forms, we suppose we know the orbit of the pencil they generate.
Φ 30 Φ 12 > 0 ∧ Φ 03 Φ 21 > 0 3.4.
Which is inside ? Suppose the pair of conics is in one of the classes : IaN, IIN, IIaN, IIIN, IIIaN, VN. Which conic lies inside the other ? Otherwise stated, for any given class of pairs inside, we want to characterize the corresponding classes of couples.
The antisymmetric invariant solves the problem for pair classes IIN, IIaN, IIIN, IIIaN. The antisymmetric invariant is
A = Φ 30 Φ 3 12 − Φ 03 Φ 3 21 .
First it is homogeneous of even degree, 6, in f , as well as in g. So its sign depends only on the algebraic conics, not on the quadratic forms defining them. Consider again Table 1. Set
(1) f = f 0 + t 1 g 0 , g = f 0 + t 2 g 0 .
From figures 5 and 6, for the cases Ia,II IIa, III, IIIa, the inner conic is the one nearer from f 0 , that is the one whose parameter (t 1 or t 2 ) has smaller absolute value. For case V, it is the one with whose parameter is smaller. Evaluate the antisymmetric invariant on (f, g). For II, IIa, III, IIIa, we get each time a positive rational number times
(t 1 t 2 (t 1 − t 2 )) 2 (t 2 1 − t 2 2 ).
This proves the following proposition. For VN, the evaluation of the antisymmetric invariant gives zero, and for IaN it gives the expression
(t 2 1 − t 2 2 )(t 1 − t 2 ) 2 (t 1 + t 2 ) 2 − (t 1 t 2 − 3) 2
whose sign is not clear. We need other methods to solve the question in these two cases.
The antisymmetric covariant solves the problem for class VN.
Instead of considering the antisymmetric invariant, we can consider the following antisymmetric covariant 11 :
B(f, g) = Φ 12 f − Φ 21 g.
We consider its value on f , g generating a pencil in orbit V. It is enough to look at Levy's representative. Consider f , g as in (1) for Levy's representative of orbit V. Then
B(xz − y 2 + t 1 x 2 , xz − y 2 + t 2 x 2 ) = t 1 − t 2 4 x 2 .
Thus B(f, g) is a semi-definite quadratic form, negative when t 1 < t 2 (that is [f = 0] lies inside [g = 0]) and positive in the opposite case.
For the purpose of calculation, we use that one decides if a semidefinite quadratic form is negative or positive merely by considering the sign of the trace of its matrix. Define T = tr(B(f, g)).
Proposition 9. If f, g generate a pencil in orbit V, then the conic [f = 0] lies in inside the conic [g = 0] if and only if T < 0.
Case
IaN. This case is more difficult than the previous ones.
Suppose (f, g) is in class IaN. After Figure 5, φ(t) has three roots of the same sign, two making ψ > 0 (conics of the pencil degenerating into isolated points) and one making ψ < 0 (conic degenerating into a double line). Denote them with t 1 , t 2 , t 3 , such that |t 1 | < |t 2 | < |t 3 |. Denote also with ν their common sign (note it is obtained as the sign of −Φ 30 Φ 03 ). The sign of Φ 30 φ ′′ (t 1 ) is −ν and the sign of Φ 30 φ ′′ (t 3 ) is ν (because Φ 30 φ ′′ is linear, with leading coefficient 6 Φ 2 30 , positive, so it is increasing; its root lies between t 1 and t 3 ).
The sign of Φ 30 φ ′′ (t 2 ) is unknown, denote it with ε. After Figure 5, [f = 0] (resp. [g = 0]) is inside the other iff ψ(t 1 ) < 0 (resp. ψ(t 3 ) < 0). Thus we have the following table of signs:
t 1 t 2 t 3 Φ 30 φ ′′ −ν ε ν ψ [f = 0] inside − + + [g = 0] inside + + − Φ 30 φ ′′ ψ [f = 0] inside ν ε ν [g = 0] inside −ν ε −ν
One sees that a Sturm query of Φ 30 φ ′′ ψ for φ will give 3 or 1 in one case, −3 or −1 in the other, allowing to obtain the relative position of the conics. Precisely, the Sturm queries corresponding to the situations [f = 0] inside vs. [g = 0] inside are given by the following table:
ν = + ν = − ε = + 3 vs. − 1 −1 vs. 3 ε = − 1 vs. − 3 −3 vs. 1 ε = 0 2 vs. − 2 −2 vs. 2 Let Q = 1 2 Remainder(Φ 30 φ ′′ φ ′ ψ, φ) = q 2 t 2 + q 1 t + q 0 .
Note that Q can be defined in a simpler way from the already introduced polynomial P = Remainder(Φ 30 φ ′ ψ, φ), that is: Q = 1 2 Remainder(φ ′′ P, φ). Define B i = sr i (φ, Q) for i between 0 and 3. Then
B 3 = Φ 30 B 2 = q 2 B 1 = Φ 30 Φ 21 Φ 12 0 q 2 q 1 q 2 q 1 q 0 . Finally B 0 = − Res(φ, Q)
. This last polynomial simplifies. Using Lemma 9, one gets:
Res(P φ ′′ , φ) = −8Φ 30 Res(φ, Q) = 8Φ 30 B 0 .
On the other hand, from Lemma 8, Res(P φ ′′ , φ) = Res(P, φ) Res(φ ′′ , φ).
From Lemma 6, Res(P, φ) = Res(φ, P ), and this is −A 0 , which was proved to be equal to:
Φ 2 30
Res(ψ, φ) Disc(Φ). Gathering this information, we get that:
B 0 = 1 8 Φ 30 Res(ψ, φ) Res(φ ′′ , φ) Disc(Φ).
It is convenient to remark here that Φ 30 divides Res(φ ′′ , φ). We will define
R := Res(φ ′′ , φ) 8Φ 30 Thus B 0 = Φ 2 30
Res(ψ, φ)R Disc(Φ). From lemmas 6 and 7, it comes that Res(ψ, φ) = Res(φ, ψ) < 0 and Res(φ ′′ , φ) = − Res(φ, φ ′′ ) has the sign ε. Last, Disc(Φ) < 0. Thus B 0 has the sign of εΦ 30 .
The sign sequences s 1 , s 2 , s 3 , s 4 giving 3 or −3 are characterized by s 1 s 3 > 0 with s 2 s 4 > 0.
The sign sequences giving 2 are + + +0 and − − −0, those giving −2 are + − +0 and − + −0. The first are characterized with respect to the second by s 1 s 2 > 0.
If εν > 0, then [f = 0] is inside iff Φ 30 B 1 > 0 with q 2 εΦ 30 > 0. If εν < 0, then this characterizes [g = 0] inside. If ε = 0, [f = 0] is inside iff νΦ 30 q 2 > 0.
Using that ε is obtained as the sign of Res(φ ′′ , φ):
Proposition• when Φ 03 R < 0, it is Φ 30 B 1 > 0 and Φ 03 q 2 < 0; • when Φ 03 R > 0, it is Φ 30 B 1 ≤ 0 or Φ 03 q 2 ≤ 0; • when R = 0, it is Φ 03 q 2 < 0.
3.5. Recapitulation. Here we display the explicit definitions of the polynomials appearing in the description of the rigid isotopy classes. Note that all these formulas are short: the complicated polynomials express simply in terms of the less complicated ones. We also display the explicit description of the rigid isotopy classes.
3.5.1. Formulas. We will denote the two forms as follows:
f (x, y, z) = a 200 x 2 + a 020 y 2 + a 002 z 2 + a 110 xy + a 101 xz + a 011 yz g(x, y, z) = b 200 x 2 + b 020 y 2 + b 002 z 2 + b 110 xy + b 101 xz + b 011 yz.
We will denote similarly the coefficients off ,g, Ω withã ijk ,b ijk , ω ijk respectively. One has: The (de-homogenized) characteristic form is
aφ(t) = Φ 30 t 3 + Φ 21 t 2 + Φ 12 t + Φ 03 = Disc(tf + g).
Note that: Φ 30 = a 200ã200 + a 110ã110 + a 101ã101 , and
Φ 21 = b 200ã200 + b 002ã002 + b 020ã020 + b 110ã110 + b 101ã101 + b 011ã011 .
There are similar formulas for Φ 03 and Φ 12 , by exchanging a and b.
The discriminant of the characteristic form can be obtained as
Disc(Φ) = 1 81 3Φ 30 2Φ 21 Φ 12 0 0 3Φ 30 2Φ 21 Φ 12 Φ 21 2Φ 12 3Φ 03 0 0 Φ 21 2Φ 12 3Φ 03
, and its Hessian determinant as
H = H 20 t 2 + H 11 tu + H 02 u 2 (2) = 4 3Φ 30 Φ 21 Φ 21 Φ 12 t 2 + 4 3Φ 30 Φ 12 Φ 21 3Φ 03 t u + 4 Φ 21 Φ 12 Φ 12 3Φ 03 u 2 .(3)
The autopolar triangle covariant is: whose columns have been labeled 1, 2, 3,1,2,3.
G = −[1 2 3]x 3 − [
Denote the coefficients of ψ as follows:
ψ(t) = Ψ 20 t 2 + 2 Ψ 11 t + Ψ 02 ,
(beware the coefficient of t is 2Ψ 11 ) then Ψ 20 =ã 200 +ã 020 +ã 002 , Ψ 02 is the corresponding expression with b instead of a, and Ψ 11 = 1 2 (ω 200 + ω 020 + ω 002 ) .
There is also µ = µ 10 t + µ 01 . Then µ 10 = a 200 + a 020 + a 002 and µ 01 is defined by the corresponding formula with b instead of a. The polynomial P for the Sturm query of ψ for φ is
P = Remainder(Φ 30 φ ′ ψ, φ) = p 2 t 2 + p 1 t + p 0 with p 2 = 3Φ 2 30 Ψ 02 − 2Φ 21 Φ 30 Ψ 11 − 2Φ 12 Φ 30 Ψ 20 + Φ 2 21 Ψ 20 , p 1 = 2Φ 21 Φ 30 Ψ 02 − 4Φ 12 Φ 30 Ψ 11 + Φ 12 Φ 21 Ψ 20 − 3Φ 03 Φ 30 Ψ 20 , p 0 = Φ 12 Φ 30 Ψ 02 − 6Φ 03 Φ 30 Ψ 11 + Φ 03 Φ 21 Ψ 20 .
The subresultant A 1 is 3 21 . and the trace of the antisymmetric covariant is
A 1 = Φ 30 Φ 21 Φ 12 0 p 2 p 1 p 2 p 1 p 0 .
The antisymmetric invariant is
A = Φ 30 Φ 3 12 − Φ 03 ΦT = Φ 12 µ 10 − Φ 21 µ 01 .
The polynomial Q for the Sturm query of Φ 30 φ ′′ ψ for φ is:
Q = Remainder(P φ ′′ , φ) = P φ ′′ − 6p 2 φ = q 2 t 2 + q 1 t + q 0 .
Its coefficients are
q 2 = 3p 1 Φ 30 − 2p 2 Φ 21 , q 1 = 3p 0 Φ 30 + p 1 Φ 21 − 3p 2 Φ 12 , q 0 = p 0 Φ 21 − 3p 2 Φ 03 .
The subresultant B 1 is
B 1 = Φ 30 Φ 21 Φ 12 0 q 2 q 1 q 2 q 1 q 0 .
The last quantity to consider is
R = 27Φ 2 30 Φ 03 + 2Φ 3 21 − 6Φ 30 Φ 21 Φ 12 .
Each of these expressions is homogeneous in the coefficients of f and as well in the coefficients of g. The following table gives their bi-degree.
I : Disc(Φ) < 0 ∧ p 2 < 0 ∧ Φ 30 A 1 > 0 Ia : Disc(Φ) < 0 ∧ [p 2 > 0 ∨ Φ 30 A 1 < 0 ∨ [A 1 = 0 ∧ p 2 = 0]] Ib : Disc(Φ) > 0 II : Disc(Φ) = 0 ∧ H = 0 ∧ G = 0 ∧ p 2 < 0 ∧ Φ 30 A 1 > 0 IIa : Disc(Φ) = 0 ∧ H = 0 ∧ G = 0 ∧ [p 2 = 0 ∨ Φ 30 A 1 < 0] III : Disc(Φ) = 0 ∧ H = 0 ∧ G = 0 ∧ p 2 < 0 IIIa : Disc(Φ) = 0 ∧ H = 0 ∧ G = 0 ∧ p 2 > 0 IV : H = 0 ∧ G = 0 V : H = 0 ∧ G = 0.
Second step: decide the class of pairs. There is only one rigid isotopy class for pairs (class N) corresponding to each of the orbits of pencils Ib, IIIa, IV, V. There are two classes (N or S) corresponding to I Ia, II, IIa, III. The criterion for being in the class N is:
Φ 30 Φ 12 > 0 ∧ Φ 03 Φ 21 > 0.
Third step (nested cases): decide which of the conics is inside the other. The classes of pairs splitting into two classes of couples are: IaN, IIaN, IIIN, IIIaN, VN.
The criteria for [f = 0] lies inside [g = 0] are the following:
• IIN, IIaN, IIIN, IIIaN: A < 0.
• VN: T < 0.
• IaN:
sign of Φ 03 R − + 0 criterion Φ 30 B 1 > 0 Φ 03 B 1 ≤ 0 for [f = 0] ∧ ∨ Φ 03 q 2 < 0 inside Φ 03 q 2 < 0 Φ 03 q 2 ≤ 0
Examples and applications
We consider examples and applications for our work. In all of them, we specialize the above general formulas to pairs of quadratic forms depending on parameters. We obtain a complicated description of the partition of the parameters space into the subsets corresponding to the isotopy classes. We then use Christopher Brown's program SLFQ of simplification of large quantifier-free formulas [4] to get simpler descriptions.
4.1. Two ellipsoids. We consider two ellipsoids given by the equations (example 2 in [21]):
x 2 + y 2 + z 2 − 25 = 0, (x − 6) 2 9 + y 2 4 + z 2 16 − 1 = 0.
We consider then as equations in x, y of two affine conics depending on a parameter z. This corresponds to using a sweeping plane to explore the two ellipsoids. We homogenize the equations in x, y with t, thus considering:
f = x 2 + y 2 + t 2 (z 2 − 25), g = (x−6t) 2 9 + y 2 4 + t 2 z 2 16 − 1 .
The quantity Disc(Φ) is here h = 49z 4 + 2516z 2 − 229376. One checks easily that h has two single real roots z 0 , −z 0 with 0 < z 0 < 4. When the two conics are proper and non-empty, that is when −4 < z < 4, one finds, using our formulas, that the following classes can occur:
• IaS when h > 0, that is −4 < z < −z 0 or z 0 < z < 4.
• IIaS when h = 0, that is z = ±z 0 .
• IbN when h < 0, that is −z 0 < z < z 0 . The ellipsoids go each through the other.
4.2.
A paraboloid and an ellipsoid. Our equations, inequations, inequalities can tell the relative position of any two conics, not only ellipses, because of the choice of working in the projective plane.
Thus we can apply also the method of the previous example to any kind of quadric. In the following example, one considers a paraboloid and an ellipsoid: 4x 2 − 4xy + 2y 2 − 4xz + 14x − 6y + 2z 2 − 10z + 12 = 0, 3x 2 − 4xy + 2y 2 − 4xz + 16x + 2yz − 12y + 2z 2 − 16z + 39 = 0
As before, we consider z as a parameter and homogenize the equations in x, y with t, thus considering: f = 4x 2 − 4xy + 2y 2 + t(−4xz + 14x − 6y) + t 2 (2z 2 − 10z + 12), g = 3x 2 − 4xy + 2y 2 + t(−4xz + 16x + 2yz − 12y) + t 2 (2z 2 − 16z + 39)
We specialize our equations, inequations and inequalities and run SLFQ. Let z 0 = −1/4 and z 1 < z 2 be the two roots of z 2 − 12z + 34. One finds that z 0 < z 1 , and [f = 0] is proper non-empty when z > z 0 , [g = 0] is proper non-empty when z 1 < z < z 2 . When both are proper and non-empty, one finds that the isotopy class is always IaN. Thus the ellipsoid is inside the paraboloid.
4.3.
Uhlig's canonical forms. In [19], Uhlig presented representatives for the orbits under GL(n, R) of couples of quadratic forms generating a non-degenerate pencil.
For conics (n = 3), it follows from Uhlig's presentation that any couple of conics can be transformed, by means of P SL(3, R), into one with associated couple of matrices among:
1 1 1 , λ 1 λ 2 λ 3 (U 11 ); 1 1 −1 , λ 1 λ 2 −λ 3 (U 12 ); 1 1 1 , λ 1 λ 1 1 λ 2 (U 21 ); 1 1 −1 , λ 1 λ 1 1 −λ 2 (U 22 ); 1 1 1 , b a a −b λ (U 31 ); 1 1 −1 , b a a −b −λ (U 32 ); 1 1 1 , λ λ 1 λ 1 (U 4 ).
To which configuration corresponds each of these normal forms ?
We find simple description for the subsets of each parameters space corresponding to the isotopy classes.
As an illustration, we show the result for U 21 .
• g is degenerate when λ 1 = 0 or λ 2 = 0.
Final remarks
For clarity of the exposition, we have not considered the case when one conic, or both conics, are degenerated; but it is easy to list the corresponding isotopy classes and describe them with equations, inequations and inequalities. Remark that the polynomials involved in the description of the classes, specially invariants and covariants, have often very compact expressions in function of the smaller ones. Thus they can be evaluated with substantial saving of arithmetic operations, as was pointed out in [8].
The following, more ambitious, step is the classification of couples of quadrics drawn in RP 3 . Hopefully some of the methods developed in the present paper will be useful in this task, on which we wish to return in another paper.
It follows from our study that the rigid isotopy classes for couples of conics are characterized nearly totally by the behavior of the signature function on the pencil generated by the quadratic forms. For the nongeneric classes, this provides a precise answer to a question formulated in [20]. We plan also to develop this point in a forthcoming paper with B. Mourrain.
Finally, the reader will find some implementations and complements on the subject on the author's web page devoted to the paper: http://emmanuel.jean.briand.free.fr/publications/twoconics Ackowledgements. The research and the redaction of this paper have been possible thanks to the successive supports of the European projects GAIA II (Intersection algorithms for geometry based IT-applications using approximate algebraic methods); AIM@SHAPE; and of the European RT Network Real Algebraic and Analytic Geometry (contract No. HPRN-CT-2001-00271).
The author wants to thank specially Laureano González-Vega for introducing him the subject; Bernard Mourrain for fruitful discussions; Marie-Françoise Coste-Roy and Ioannis Emiris for their interest; Mercedes Rosas for her careful reading; and the University of Cantabria and INRIA Sophia Antipolis for their welcoming environment.
The author also wants to thank the anonymous referees for their useful comments, and for pointing out the work of Gudkov and Polotovskiy.
Finally, the author is much indebted to the people and institutions that provide free access to their work on the world wide web: Christopher Brown for his software SLFQ [4], the project Gutenberg, and the Bibliothèque Nationale de France for its project Gallica.
Figure 1 .Figure 2 .
12The rigid isotopy classes for generic pairs of conics. The rigid isotopy classes for non-generic pairs of proper conics.
Definition 1 .
1Two couples of distinct proper conics are rigidly isotopic if they are connected by a path in the space of couples of distinct proper conics, along which the numbers of real and imaginary intersections of each multiplicity don't change.
Definition 2 .
2Let f, g be two non-degenerate non-proportional quadratic forms. We define a sliding for [f ] and [g] as a path of the formt → ([f + tkg], [g]) (or t → ([f ], [g + tkf ])) for t in a closed interval containing 0; no t with f + tkg (resp. g + tkf ) degenerate; and k some real number.Let α be an homeomorphism of RP 2 . For a couple ([f ], [g]) of nonempty proper conics, we write α([f ], [g]) for the couple of algebraic conics corresponding to (α([f = 0]), α([g = 0])).
Lemma 1 .
1Any sliding, for a couple of non-empty conics, lifts to an ambient isotopy.
a sliding: t → ([f + tkg], [g]), t ∈ I with [f = 0] and [g = 0] non-empty, this means that there exists a family of homeomorphisms β t of RP 2 , with β 0 = id and β t ([f ], [g]) = ([f + tkg], [g]).
Figure 3 .−
3Degenerate conics corresponding to multiple roots of the discriminant, in the representations of the pencils. base point of multiplicity 2 and two base points of multiplicity 1. 1111 11 − 11 − 22 − −
• the nature of the proper conics (empty or non-empty) and of the degenerate conics (pair of lines, line or isolated point). The nature of the proper conics is constant on each arc between two degenerate conics.
Figure 4 .Figure 5 .
45Nature of the conics, in the representations of the pencils. Pencils of conics up to projective equivalence (beginning).
Figure 6 .
6Pencils of conics up to projective equivalence (end). These features are conserved under projective equivalence. Thus the representations are established by considering Levy's representative.
), after Theorem 2. To distinguish between IN and IS, one can count the number of connected components of the complement of [f = 0] ∪ [g = 0]: there are 6 in the first case and 5 in the second, the topological types are different, so are the rigid isotopy classes. For the other four orbits of pencils, one conic lies in the inside of the other (at least at the neighborhood of the double point for II) for the N-representative, while there is no such inclusion for the S-representative.
Corollary 1 .
1There are 14 equivalence classes for pairs of proper nonempty conics under rigid isotopy and exchange. Representatives for them are given in
•
The autopolar triangle covariant G, a cubic form that is also cubic in f , as in g, and that always factorizes as a product of three linear forms. When [f = 0] and [g = 0] have four distinct intersections, they are equations of the sides of the unique autopolar triangle associated to them (see[2],14.5.4 and 16.4.10).
Lemma 6 .
6One has Res(U, V ) = (−1) deg(U ) deg(V ) Res(V, U). Lemma 7. Let c be the leading coefficient of U. Then
Lemma 8 .
8Let U(t), V (t), W (t) be three univariate polynomials. Then Res(U, V W ) = Res(U, V ) Res(U, W ). And last: Lemma 9. Let U(t), V (t) be two univariate polynomials, and W the remainder in the euclidean division of U by V . Let c be the leading coefficient of V . Then Res(U, V ) = (−1) deg(U ) deg(V ) c deg(U )−deg(W ) Res(V, W ).
5 e.-g. a double real root should be here counted as two roots.3.1.3. Descartes' law of signs.Let U(t) be an univariate polynomial. Then Descartes' law of signs give some insight about its number N (U) of positive real roots, counted with multiplicities.Consider the sequence of the signs (+'s and −'s) of the (non-zero) coefficients of U and denote with V(U) the number of changes in consecutive terms. The following lemma is Descartes' Law of signs. It can be found as Theorem 2.34 in[1].Lemma 10. One has V(U) ≥ N (U), and V(U) − N (U) is even.
Suppose U has all its roots real and non-zero. Then they have all the same sign if and only if: u 3 u 1 > 0 and u 2 u 0 > 0.
For this we use that the numbers of degenerate conics of each type (pair of lines, isolated point or double line) in a pencil characterize its orbit, as shown in the table below, established by considering figures 5
Proposition 8 .
8Suppose ([f ], [g]) is a couple of distinct proper nonempty conics, such that {[f ], [g]} is in class IIN, IIaN, IIIN or IIIaN. Then [f = 0] lies inside 10 of [g = 0] if and only if A(f, g) < 0.
= a 200 a 110 /2 a 110 /2 a 020 ,ã 110 = −2 a 002 a 011 /2 a 101 /2 a 110 /2 . Similarly theb ijk 's are defined from the b ijk 's, and ω 200 = a 020 b 002 + a 002 b 020 − a 011 b 011 /2, ω 020 = a 002 b 200 + a 200 b 002 − a 101 b 101 /2, ω 002 = a 020 b 200 + a 200 b 020 − a 110 b 110 /2, ω 011 = a 200 b 011 + a 011 b 200 − a 110 b 101 /2 − a 101 b 110 /2, ω 101 = a 020 b 101 + a 101 b 020 − a 011 b 110 /2 − a 110 b 011 /2, ω 110 = a 002 b 110 + a 110 b 002 − a 011 b 110 /2 − a 110 b 011 /2.
ω
200 ω 020 ω 002 ω 011 ω 101 ω 110 b 200b020b002b011b101b110
. 2 .
2The decision procedure. First step: decide the orbit of pencils. Here are the descriptions of the sets of couples of distinct proper conics generating a pencil in a given orbit.
• the conics are in class VN when λ 1 = λ 2 = 0. In this case,[g = 0] lies inside [f = 0].• in the other cases, the conics are in class IIN, IIS, IIaN or IIaS as shown inFigure 7.
Figure 7 .
7isotopy classes for representatives U 21 .
Table 1 .
1Levy's representatives for each orbit of pen-
cils. Each representative is the pencil generated by [f 0 ]
and [g 0 ].
Triple
0 0 0
1 1 1
00 00 00
11 11 11
00 00 00
11 11 11
00 00 00
11 11 11
Simple
Double
Table 2 .
2The rigid isotopy classes.Proof. For IN, IS, IaS, IIS, IIaS, IIIS, it is enough to exhibit projective automorphisms that stabilize the corresponding Levy's representative and swap two arcs of non-empty proper conics. It was already done in the proof of Lemma 3, except for IN and IS. For them, the reflection y ↔ z is convenient.For IbN and IVN, it is enough to exhibit projective automorphisms that stabilize the corresponding Levy's representative and reverse the pencil's orientation. For IbN, the reflection x → −x is convenient; for IVN, one may use the transformation x → −x, z → −z.Lemma 5.•Each of the classes of pairs IaN, IIaN, IIIN, IIIaN, VN splits into two classes for couples, corresponding to one conic lying inside the other (except for the base points). • The class of pairs IIN also splits into two classes for couples, corresponding to one conic lying inside the other in a neighborhood of the double point (except the double point itself ). Proof. The property that one conic lies inside the second is conserved under ambient homeomorphism, and thus under rigid isotopy. The same holds for inclusion at the neighborhood of a double intersection point. Thus it is enough to consider the representatives of the given pair classes and check the inclusion to show the theorem. The computations are trivial, hence we omit them. Theorem 3. There are 20 classes of couples under rigid isotopy. A set of representatives is given by Table 2, where the reader should add the couple obtained by swapping f and g for each of the lines marked with ( * ). Corollary 2. The ambient isotopy classes for couples of conics are the following unions of rigid isotopy classes: • classes where the two conics can be swapped: IN, IS, IaS, IbN ∪ IVN, IIS, IIaS and IIIS. • pair classes splitting into two classes for couples, one with [f = 0] inside [g = 0], one with [g = 0] inside [f = 0]: IaN ∪ IIIaN, IIN, IIaN ∪ VN and IIIN. Proof. (sketch) One shows that IbN and IVN are ambient isotopic by building explicitly a homeomorphism 3 of RP 2 sending a representative of the first to a representative of the second. Details on how to do it are tedious, we skip them. Idem (with [f = 0] inside [g = 0]) for IaN and IIIaN, and for IIaN and VN. Next, one shows the displayed rigid isotopy classes or unions of rigid isotopy classes are not equivalent modulo ambient isotopy. This is done by considering topological invariants of the triples (RP 2 , [f = 0], [g = 0]) which take different values on the 15 representatives.
Contrary, for instance, to the pencil of the zero loci of the f (x, y, z) = λxy+µxz.
This is enough. Indeed, the set of the homeomorphisms of RP 2 is connected, so any homeomorphism is the extremity of some ambient isotopy.
The analogue assertion is still true for a couple of quadratic forms in n variables, for any n.
only at the neighborhood of the double intersection point for class IIN.
This is a quadratic form, and actually the antisymmetric invariant of the previous paragraph is its discriminant.
Algorithms in real algebraic geometry. Saugata Basu, Richard Pollack, Marie-Françoise Roy, Algorithms and Computation in Mathematics. 10Springer-VerlagSaugata Basu, Richard Pollack, and Marie-Françoise Roy. Algorithms in real algebraic geometry, volume 10 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, 2003.
. Marcel Berger, Géométrie, Nathan, 2nd editionMarcel Berger. Géométrie. Nathan, 2nd edition, 1990.
. T Bröcker, T Tom Dieck, Representations of compact Lie groups. Springer-Verlagfirst editionT. Bröcker and T. tom Dieck. Representations of compact Lie groups, vol- ume 98 of GTM. Springer-Verlag, first edition, 1985.
SLFQ (Simplifying Large Formulas with QEPCAD B). Christopher Brown, Christopher Brown. SLFQ (Simplifying Large Formulas with QEPCAD B). http://www.cs.usna.edu/~qepcad/SLFQ/Home.html.
A treatise of the analytical geometry of the point, line, circle and conic sections. John Casey, Dublin University Presssecond edition, 1893. Digitalized by GallicaJohn Casey. A treatise of the analytical geometry of the point, line, circle and conic sections. Dublin University Press, second edition, 1893. Digitalized by Gallica, http://gallica.bnf.fr.
Undergraduate Texts in Mathematics. David Cox, John Little, Donal O' Shea, Springer-VerlagNew YorkIdeals, varieties, and algorithms. second editionDavid Cox, John Little, and Donal O'Shea. Ideals, varieties, and algorithms. Undergraduate Texts in Mathematics. Springer-Verlag, New York, second edi- tion, 1997.
Quadratic transformations RP 2 → RP 2. Alexander Degtyarev, Topology of real algebraic varieties and related topics. Providence, RIAmer. Math. Soc173Alexander Degtyarev. Quadratic transformations RP 2 → RP 2 . In Topology of real algebraic varieties and related topics, volume 173 of Amer. Math. Soc. Transl. Ser. 2, pages 61-71. Amer. Math. Soc., Providence, RI, 1996.
Algebraic methods and arithmetic filtering for exact predicates on circle arcs. Olivier Devillers, Alexandra Fronville, Bernard Mourrain, Monique Teillaud, 16th ACM Symposium on Computational Geometry. Hong Kong22Olivier Devillers, Alexandra Fronville, Bernard Mourrain, and Monique Teil- laud. Algebraic methods and arithmetic filtering for exact predicates on circle arcs. Comput. Geom., 22(1-3):119-142, 2002. 16th ACM Symposium on Com- putational Geometry (Hong Kong, 2000).
A new approach for characterizing the relative position of two ellipses depending on one or several parameters. Fernando Etayo, Laureano González-Vega, Natalia Del Rio, Computer Aided Geometric Design. In PressFernando Etayo, Laureano González-Vega, and Natalia del Rio. A new ap- proach for characterizing the relative position of two ellipses depending on one or several parameters. Computer Aided Geometric Design, 2005. In Press.
A treatise on the theory of invariants. Oliver E Glenn, Available as EText 9933 of project Gutenberg. Ginn and CompanyOliver E. Glenn. A treatise on the theory of invariants. Ginn and Company, 1915. Available as EText 9933 of project Gutenberg, at http://www.gutenberg.net.
Stratified Morse Theory. Mark Goresky, Robert Macpherson, Ergebnisse der Mathematik und ihrer Grenzgebiete. 14Springer-VerlagMark Goresky and Robert MacPherson. Stratified Morse Theory, volume 14 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1988.
Topology and geometry -Rokhlin seminar. D A Gudkov, Lecture Notes in Math. O. Ya. Viro1346Springer-VerlagPlane real projective plane curvesD. A. Gudkov. Plane real projective plane curves. In O. Ya. Viro, editor, Topology and geometry -Rokhlin seminar, Lecture Notes in Math. 1346, pages 341-347. Springer-Verlag, Heidelberg, 1988.
Stratification of the space of 4 th order curves. the joining of stratums. D A Gudkov, G M Polotovskiy, Engl. transl. in Russian Math. Surveys. 424152Uspekhi Mat. NaukD. A. Gudkov and G. M. Polotovskiy. Stratification of the space of 4 th order curves. the joining of stratums. Uspekhi Mat. Nauk, 42(4):152, 1987. Engl. transl. in Russian Math. Surveys 42 (1987), no. 4.
Stratification of the space of plane algebraic 4 th order curves with respect to algebraic-geometric types. I. Deposited in VINITI. D A Gudkov, G M Polotovskiy, RussianD. A. Gudkov and G. M. Polotovskiy. Stratification of the space of plane alge- braic 4 th order curves with respect to algebraic-geometric types. I. Deposited in VINITI, No 5600-B87, 1987. 1-55 (Russian).
Stratification of the space of plane algebraic 4 th order curves with respect to algebraic-geometric types. D A Gudkov, G M Polotovskiy, II. Deposited in VINITI. RussianD. A. Gudkov and G. M. Polotovskiy. Stratification of the space of plane alge- braic 4 th order curves with respect to algebraic-geometric types. II. Deposited in VINITI, No 6331-B90, 1990. 1-33 (Russian).
The isotopy classification of affine quartic curves. Anatoly B Korchagin, David A Weinberg, Rocky Mountain J. Math. 321Anatoly B. Korchagin and David A. Weinberg. The isotopy classification of affine quartic curves. Rocky Mountain J. Math., 32(1):255-347, 2002.
Projective and related geometries. The Macmillan Co. Harry Levy, New YorkHarry Levy. Projective and related geometries. The Macmillan Co., New York, 1964.
Classical invariant theory. J Peter, Olver, Cambridge University PressCambridgePeter J. Olver. Classical invariant theory. Cambridge University Press, Cam- bridge, 1999.
A canonical form for a pair of real symmetric matrices that generate a nonsingular pencil. Frank Uhlig, Linear Algebra and Appl. 143Frank Uhlig. A canonical form for a pair of real symmetric matrices that gen- erate a nonsingular pencil. Linear Algebra and Appl., 14(3):189-209, 1976.
Interference analysis of conics and quadrics. Wenping Wang, Rimvydas Krasauskas, Topics in algebraic geometry and geometric modeling. Providence, RIAmer. Math. Soc334Wenping Wang and Rimvydas Krasauskas. Interference analysis of conics and quadrics. In Topics in algebraic geometry and geometric modeling, volume 334 of Contemp. Math., pages 25-36. Amer. Math. Soc., Providence, RI, 2003.
An algebraic condition for the separation of two ellipsoids. Wenping Wang, Jiaye Wang, Myung-Soo Kim, Comput. Aided Geom. Design. 186Wenping Wang, Jiaye Wang, and Myung-Soo Kim. An algebraic condition for the separation of two ellipsoids. Comput. Aided Geom. Design, 18(6):531-539, 2001.
E-mail address: emmanuel.briand@gmail. Emmanuel Briand, Dpto Universidad De Cantabria, Matemáticas, Avda Estadística Y Computación, Los Castros S/N, 39005Santander, SpainEmmanuel Briand, Universidad de Cantabria, Dpto. Matemáticas, estadística y computación, Avda. Los Castros S/N, 39005 Santander, Spain. E-mail address: [email protected] URL: http://emmanuel.jean.briand.free.fr/
| [] |
[
"Lattices from tight equiangular frames",
"Lattices from tight equiangular frames"
] | [
"Albrecht Böttcher ",
"Lenny Fukshansky ",
"Stephan Ramon Garcia ",
"Hiren Maharaj ",
"Deanna Needell "
] | [] | [] | We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k, n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k + 1 and that there are infinitely many k such that a lattice emerges for n = 2k. We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher's recent observation that this lattice is perfect. | null | [
"https://arxiv.org/pdf/1607.05350v1.pdf"
] | 14,358,064 | 1607.05350 | 4377dd50fdd649be2b02670088dacf2b4dff5dcb |
Lattices from tight equiangular frames
18 Jul 2016
Albrecht Böttcher
Lenny Fukshansky
Stephan Ramon Garcia
Hiren Maharaj
Deanna Needell
Lattices from tight equiangular frames
18 Jul 2016arXiv:1607.05350v1 [math.FA]AMS classification Primary: 15B35Secondary: 05B3011H0642C1552C07 Keywords LatticeEquiangular linesTight frameConference matrix
We consider the set of all linear combinations with integer coefficients of the vectors of a unit tight equiangular (k, n) frame and are interested in the question whether this set is a lattice, that is, a discrete additive subgroup of the k-dimensional Euclidean space. We show that this is not the case if the cosine of the angle of the frame is irrational. We also prove that the set is a lattice for n = k + 1 and that there are infinitely many k such that a lattice emerges for n = 2k. We dispose of all cases in dimensions k at most 9. In particular, we show that a (7,28) frame generates a strongly eutactic lattice and give an alternative proof of Roland Bacher's recent observation that this lattice is perfect.
Introduction
Let 2 ≤ k < n and let G be a real k × n matrix. Denote the columns of G by f 1 , . . . , f n . These columns or G itself are called a unit tight equiangular (k, n) frame if GG ′ = γI with γ = n/k (tightness) and G ′ G = I + (1/α)C with α = k(n − 1)/(n − k) and a matrix C whose diagonal entries are zero and the other entries of which are ±1 (property of being equiangular unit vectors). Define Λ(G) = span Z {f 1 , . . . , f n }. Our investigation is motivated by the following question.
When is Λ(G) a lattice, that is, a discrete additive subgroup of R k ? In case it is a lattice, what are its geometric properties?
After having posed the question in its most concise form, some comments are in order. By R k we understand the column-wise written Euclidean R k with the usual scalar product (·, ·). The condition G ′ G = I +(1/α)C with C as above means that f j = 1 for all j and that |(f i , f j )| = 1/α for i = j. In other words, the vectors f j are all unit vectors and each pair of them makes the angle ϕ or π−φ such that | cos ϕ| = | cos(π−φ)| = 1/α. The equality GG ′ = γI is equivalent to the requirement that G ′ x 2 = γ x 2 for all x in R k , which in turn is the same as saying that n j=1 (f j , x) 2 = γ x 2 for all x ∈ R k . It is well known since [17,18] that the two equalities G ′ G = I + (1/α)C with C as above and GG ′ = γI necessarily imply that γ = n/k and α = k(n − 1)/(n − k).
Tight equiangular frames (TEFs) possess many properties similar to orthonormal bases, yet may also be highly overcomplete, making them very attractive in many applications. For this reason there has been a recent surge of work addressing the construction and analysis of these frames. They appear in many practical applications, such as error correcting codes [9,17], wireless communications [16,17], security [11], and sparse approximation [6,14,19,20].
In sparse approximation for example, the incoherence (small 1/α, the absolute value of pairwise inner products of vectors) of TEFs allows them to be used as sensing operators. Viewing a TEF as a matrix whose columns consist of the frame vectors, samples of a signal are acquired via inner products between the signal and the rows of this (typically highly underdetermined) matrix. Under the assumption that the signal vector is sparse (has a small number of nonzero coordinates), the signal can be accurately reconstructed from this compressed representation. However, in many applications there is more known about the signal than it simply being sparse. For example, in error correcting codes [5] and communications applications like MIMO [13] and cognitive radio [1], the signal vectors may come from some lattice. However, there has been very little rigorous mathematical developments on the intersection between arbitrary lattice-valued signals and sparse approximation (see e.g. [7] and references therein).
In this work, we attempt to take the first step toward a rigorous analysis of properties of tight equiangular frames and associated lattices. We are especially interested in the following questions. When does the integer span of a TEF form a lattice? Does this lattice have a basis of minimal vectors? Is the generating frame contained among the minimal vectors of this lattice? We also study further geometric properties of the resulting lattices, such as eutaxy and perfection. Our hope is that this investigation will contribute not only to the understanding of TEFs in general, but also to their explicit use in applications with lattice-valued signals. For example, if the integer span of a TEF is a lattice, then the image of that TEF viewed as a sensing matrix restricted to integer-valued signals forms a discrete set. In some sense this is analogous to the well-known Johnson-Lindenstrauss lemma [3], and may be used to provide reconstruction guarantees for TEF sampled signals. More concretely, if the lattice constructed from the TEF is such that its minimal vectors are the frame vectors themselves, this guarantees a minimum separation between sample vectors in its image. These types of properties are essential for sparse reconstruction and can be leveraged to design new sampling mechanisms and reconstruction guarantees. On the other hand, it is also useful to know when such properties are impossible. We leave a detailed analysis and link to applications as future work, and focus here on the mathematical underpinnings to the questions raised above.
Main results
Let L be a lattice in R k , and let V = span R L be the subspace of R k that it spans. Then the rank of L, denoted by rk(L), is defined to be the dimension of V . We say that L has full rank if V = R k . The minimal distance of a lattice L ⊂ R k is defined as d(L) = min{ x : x ∈ L \ {0}}. The set of minimal vectors, S(L), is the set of all x ∈ L with x = d(L). The lattice L is called well-rounded if R k = span R S(L), and we say that it is generated by its minimal vectors if L = span Z S(L). It is known that the second condition is strictly stronger than the first when rk(L) ≥ 5. An even stronger condition (at least when rk(L) ≥ 10) is that S(L) contains a basis for L, i.e., there exist R-linearly independent vectors f 1 , . . . , f rk(L) ∈ S(L) such that span Z {f 1 , . . . , f rk(L) } = L; if this is the case, we say that L has a basis of minimal vectors.
A finite subset {q 1 , . . . , q m } of the unit sphere Σ k−1 in R k is called a spherical t-design for a positive integer t if for every real polynomial p of degree ≤ t in k variables,
Σ k−1 p(x) dσ(x) = 1 m k i=1 p(q i ),
where dσ denotes the unit normalized surface measure on the sphere Σ k−1 . A full rank lattice in R k is called strongly eutactic if its set of minimal vectors (normalized to lie on Σ k−1 ) forms a spherical 2-design. We finally define the notion of a perfect lattice.
Recall that we write vectors x in R k as column vectors. A full rank lattice L in R k is called perfect if the set of symmetric k × k matrices {xx ′ : x ∈ S(L)} spans all real symmetric k × k matrices as an R-vector space.
Two lattices L and M in R k are called similar if L = aUM for some a ∈ R and some orthogonal k × k matrix U. Conditions such as well-roundedness, generation by minimal vectors, existence of bases of minimal vectors, strong eutaxy, and perfection are preserved on similarity classes of lattices. Furthermore, there are only finitely many strongly eutactic and only finitely many perfect similarity classes of lattices in R k for each k ≥ 1.
Given a full rank lattice L ⊂ R k , it is possible to associate a sphere packing to it by taking spheres of radius d(L)/2 centered at every point of L. It is clear that no two such spheres will intersect in their interiors. Such sphere packings are usually called lattice packings. One convenient way of thinking of a lattice packing is as follows. The Voronoi cell of L is defined to be
V(L) := {x ∈ R k : x ≤ x − y ∀ y ∈ L}.
Then R k is tiled with translates of the Voronoi cell by points of the lattice, and spheres in the packing associated to L are precisely the spheres inscribed in these translated Voronoi cells. A compact measurable subset of R k is called a fundamental domain for a lattice L if it is a complete set of coset representatives in the quotient group R k /L. All fundamental domains of the same lattice have the same volume, and the Voronoi cell of a lattice is an important example of a fundamental domain.
A central problem of lattice theory is to find a lattice in each dimension k ≥ 1 that maximizes the density of the associated lattice packing. There is an easy formula for the packing density of a lattice. A lattice L in R k can be written as L = BZ k , where B is a basis matrix of L, i.e., the columns of B form a basis for L. The determinant of L is then defined to be det L := det(B ′ B), which is an invariant of the lattice, since any two basis matrices of L are related by a integer linear transformation with determinant ±1. The significance of the determinant is given by the fact that it is equal to the volumes of the fundamental domains. It is then easy to observe that the density of the lattice packing associated to L is the volume of one sphere divided by the volume of the translated Voronoi cell that it is inscribed into, that is,
δ(L) := ω k d(L) k 2 k det L ,(1)
where ω k is the volume of the unit ball in R k . In fact, this packing density function δ is defined on similarity classes of lattices in a given dimension, and a great deal of attention in lattice theory is devoted to studying its properties. There is a natural quotient metric topology on the space of all full rank lattices in R k , given by identifying this space with GL k (R)/ GL k (Z): indeed, every A ∈ GL k (R) is a basis matrix of some lattice, and A, B ∈ GL k (R) are basis matrices for the same lattice if and only if A = UB for some U ∈ GL k (Z). A lattice is called extreme if it is a local maximum of the packing density function in its dimension: this is a particularly important class of lattices that are actively studied. A classical result of Voronoi states that perfect strongly eutactic lattices are extreme (see, for instance, Theorem 4 of [15]); on the other hand, if a lattice is strongly eutactic, but not perfect, then it is a local minimum of the packing density function (see Theorem 9.4.1 of [10]). A good source for further information about lattice theory is Martinet's book [10].
We now return to our construction Λ(G) from unit equiangular frames and describe our results. It is well known that unit tight equiangular (k, k + 1) frames exist for all k ≥ 2. According to [18], except for the (k, k + 1)-case, the only unit tight equiangular (k, n) frames with k ≤ 9 are (3, 6), (5, 10), (6, 16), (7,14), (7, 28), (9, 18)
frames. Our first result says the following.
Proposition 2.1 If Λ(G)
is a lattice, then α must be a rational number.
Thus, since α = 1/ √ 5 for the (3, 6) frame, α = 1/ √ 13 for the (7,14) frame, and α = 1/ √ 17 for the (9, 18) frame, these three frames do not generate lattices. We will show that there are unit tight equiangular (5, 10), (6,16), and (7, 28) frames which generate lattices. Moreover, we will prove the following results. Theorem 2.2 (a) For every k ≥ 2, there are unit tight equiangular (k, k + 1) frames G such that Λ(G) is a full rank lattice. The lattice Λ(G) has a basis of minimal vectors, it is non-perfect and strongly eutactic, and hence it is a local minimum of the packing density function in dimension k.
(b) There are infinitely many k for which there exist unit tight equiangular (k, 2k) frames G such that Λ(G) is a full rank lattice.
(c) There is a unit tight equiangular (7, 28) frame G for which Λ(G) has a basis of minimal vectors, is a perfect strongly eutactic lattice, and hence extreme.
Remark 2.3
We explicitly construct the lattices of Theorem 2.2. We show that those of parts (a) and (c) and those with k ≤ 13 of part (b) have the property that the set of minimal vectors consists precisely of ± the generating frame vectors. The well known result of Gerzon (see, for instance, Theorem C of [18]) asserts that for a (k, n) tight equiangular frame necessarily n ≤ k(k + 1)/2. On the other hand, k(k + 1)/2 is the minimal number of (± pairs of) minimal vectors necessary (but not sufficient) for a lattice in R k to be perfect. Since only very few tight equiangular frames achieve equality in Gerzon's bound, it is likely quite rare for perfect lattices to be generated by tight equiangular frames. Perfection is a necessary condition for extremality, and hence it is unreasonable to expect to obtain extreme lattices often in this way. The only such example we have discovered is the lattice from the (7, 28) frame in part (c) of our Theorem 2.2, perfection of which has also previously been discussed in [2].
The strong eutaxy of our lattice constructions in Theorem 2.2(a),(c) is established directly with the use of the following result.
Proposition 2.4 Suppose that Λ(G) is a lattice and S(Λ(G)) = {±f 1 , . . . , ±f n }. Then Λ(G) is strongly eutactic. Proof. A spanning set {g 1 , . . . , g m } for R k is called a Parseval frame if x 2 = m j=1 (g j , x) 2 for all x ∈ R k . Further, {g 1 , . . . , g m } is a spherical 2-design if and only if k/m g 1 , . . . , k/m g m
is a Parseval frame and m i=1 g i = 0 (see [9] for details, especially Proposition 1.2). Now let G = (f 1 . . . f n ) be a unit tight equiangular (k, n) frame, and assume that Λ(G) is a lattice such that S(Λ(G)) = {±f 1 , . . . , ±f n }. We then have
x 2 = k 2n n j=1 (x, f j ) 2 + (x, −f j ) 2 = n j=1 x, k 2n f j 2 + x, − k 2n f j 2 ,
for every x ∈ R k . Hence ± k/2n f 1 , . . . , ± k/2n f n is a Parseval frame, and therefore S(Λ(G)) is a spherical 2-design.
A summary of a part of our results is given in Table 1.
Rationality of the cosine of the frame
Suppose G is a unit tight (k, n) frame. Then GG ′ = γI and hence G has rank k. Let G 0 be the k × k matrix formed by arbitrarily chosen k linearly independent columns Table 1: Summary of a part of our results.
(k, n) cosine 1 α Volume of a S(Λ) = {±f 1 , . . . , ±f n }?
fundamental domain Basis of minimal vectors?
(k + 1, k) 1 k 1 √ k + 1 1 + 1 k k/2
Yes, Yes of G and denote by G 1 the k × (n − k) matrix constituted by the remaining columns. We may without loss of generality assume that G = (G 0 G 1 ). We emphasize that G 0 is invertible. Recall that Λ(G) is called a full-rank lattice if span R {f 1 , . . . , f n } is all of R k . Note that in the following proposition we do not require equiangularity.
(3, 6) 1 √ 5 = 0.Proposition 3.1 Let G = (G 0 G 1 ) be a unit tight (k, n) frame.
Then the following are equivalent.
(i) Λ(G) is a lattice. (ii) Λ(G) is a full rank lattice. (iii) There exist β ∈ Z \ {0} and X ∈ Z k×(n−k) such that G −1 0 G 1 = (1/β)X. If (iii) holds with β = 1, then G 0 is a basis matrix for Λ(G). Proof. Since G 0 is invertible, we have span R {f 1 , .
. . , f n } = R k , which proves the equivalence of (i) and (ii). Suppose (ii) holds. Then G 0 = BX 0 and G 1 = BX 1 with an invertible k × k matrix B and integer matrices X 0 , X 1 . The matrix X 0 is invertible, so B = G 0 X −1 0 and hence
G 1 = G 0 X −1 0 X 1 = G 0 1 det X 0 X 2 X 1 = 1 β G 0 X with β = det X 0 and X = X 2 X 1 . This proves (iii). Conversely, suppose (iii) is true. It is clear that Λ(G) = span Z {f 1 , . . . , f n } is an additive subgroup of R k . Put B = (1/β)G 0 . Then B is invertible, G 0 = BX 0 with X 0 = βI and G 1 = BX 1 with X 1 = X. It follows that Λ(G) is a subset of L B := {BZ : Z ∈ Z k×1 }.
As the latter set is discrete, so must be Λ(G). This proves (i). Finally, if β = 1, then B = G 0 , which implies that L B ⊂ Λ(G) and hence L B = Λ(G). Consequently, B is a basis matrix for Λ(G).
Proposition 3.2 Let G = (G 0 G 1 ) be a unit tight equiangular (k, n) frame. If Λ(G)
is a lattice, then α must be a rational number.
Proof. By Proposition 3.1, we may assume that Λ(G) is a full rank lattice. So G = BZ with an invertible matrix B and a matrix Z ∈ Z k×n . Multiplying the equality γI = GG ′ = BZZ ′ B ′ from the right by (B ′ ) −1 and then from the left by B ′ , we obtain γI = B ′ BZZ ′ and thus,
(I + (1/α)C)Z ′ = G ′ GZ ′ = Z ′ B ′ BZZ ′ = γZ ′ , which implies that CZ ′ = α(γ − 1)Z ′ .
If α is irrational, the last equality yields Z = 0, and this gives G = 0, a contradiction.
The previous proposition implies in particular that a unit tight equiangular (3, 6) frame does not induce a lattice. The reader might enjoy to see the reason for this failure also from the following perspective. Consider the tight unit equiangular (3,6) frame G that is induced by the 6 upper vertices of a regular icosahedron. As shown in [18], with p = (1 + √ 5)/2, this frame is given by the columns of the matrix
G = 1 1 + p 2 0 0 1 −1 p p 1 −1 p p 0 0 p p 0 0 1 −1 .
We have c = 1/ √ 5. By Dirichlet's approximation theorem, there are integers x n , y n such that y n → ∞ and
x n y n + p ≤ 1 y 2 n .
In particular, x n + py n → 0 as n → ∞ The linear combination of the columns of G with the coefficients x n + y n , y n − x n , y n , y n , x n , −x n , equals
1 1 + p 2 0 2(x n + y n p) 2(y n p + x n ) ,
which tends to zero as n → ∞. Consequently, Λ(G) is not a discrete subgroup of R 3 and thus it is not a lattice.
Unit tight equiangular (k, 2k) frames
We first consider the case n = 2k. Then γ = 2 and α = √ n − 1. We furthermore suppose that n = p r + 1 with an odd prime number p and a natural number r. If r is odd and p = 4ℓ + 3, then k is even, which implies that unit tight equiangular (k, n) frames do not exist (Theorem 17 of [18]). If r is odd and p = 4ℓ + 1, then unit tight equiangular (k, n) frames G exist, but Λ(G) is not a lattice because α is irrational. We are so left with the case where r is even.
Theorem 4.1 Let k ≥ 2 and n = 2k. If n = p 2m + 1 with an odd prime number p and a natural number m, then there exists a unit tight equiangular (k, n) frame G such that Λ(G) is a full rank lattice.
Comments. This theorem proves Theorem 2.2(b) and will be a consequence of the following Theorem 4.2. Before turning to the proof of Theorem 4.2, which is a combination of ideas of Goethals and Seidel [8] and Strohmer and Heath [17], some comments seem to be in order. Following [17], we start with a symmetric n × n conference matrix C, that is, with a symmetric matrix C that has zeros on the main diagonal and ±1 elsewhere and that satisfies C 2 = (n − 1)I. Under the hypothesis of Theorem 4.1, such matrices were first constructed by Paley [12]. Goethals and Seidel [8] showed that one can always obtain such matrices in the form
C = A D D −A(3)
where A and D are symmetric k × k circulant matrices. Let a and b be any rational numbers such that a 2 + b 2 = α 2 (= n − 1 = p 2m ). Theorem 3.4 of [8] says that, under certain conditions, one can in turn represent the matrix (3) as
A D D −A = I −N N I −1 aI bI bI −aI I −N N I (4)
with a symmetric circulant matrix N all entries of which are rational numbers. The conditions ensuring the representation (4) are that D + bI or A + aI are invertible. We have
N = (A + aI) −1 (bI − D) or N = (D + bI) −1 (A − aI)(5)
if A + aI or D + bI is invertible, respectively. (Note that all occurring blocks are symmetric circulant matrices and in particular commuting matrices.) As there are infinitely many different decompositions of p 2m into the sum of two squares of rationals, for example,
p 2m = t 2 − s 2 t 2 + s 2 p m 2 + 2ts t 2 + s 2 p m 2
with integers s and t, we can, for given A and D, always find rational a and b such that a 2 + b 2 = α 2 and both D + bI and A + aI are invertible.
Let, for example n = 10. A matrix (3) with symmetric circulant matrices A and D is completely given by its first line, which is of the form
0, ε 1 , ε 2 , ε 2 , ε 1 , ε 3 , ε 4 , ε 5 , ε 5 , ε 4 with ε j ∈ {−1,
The corresponding matrix A is always singular. We see that in all cases we may take a = 3 and b = 0 (3 2 + 0 2 = 9) because D is invertible. In the cases (6) and (8) we could also take a = 0 and b = 3 (0 2 + 3 2 = 9) since D + 3I is invertible. In fact, we will prove the following theorem. As shown above, the hypothesis of this theorem can always be satisfied, so that this theorem implies Theorem 4.1.
is a unit tight equiangular (k, n) frame G such that Λ(G) is a full rank lattice.
Proof of Theorem 4.2. The requirement a = −p m assures that α + a = 0. Let
W = W 11 W 12 W 21 W 22 = 1 2α(α + a) (I + N 2 ) −1/2 U 11 U 12 U 21 U 22 with U 11 U 12 U 21 U 22 = (α + a)I + bN bI − (α + a)N bI − (α + a)N −αI − (α + a)N .
Using (4) one can show by straightforward computation that
C U 11 U 21 = α U 11 U 21 , C U 12 U 22 = −α U 12 U 22 ,
which implies that
C U 11 U 12 U 21 U 22 = U 11 U 12 U 21 U 22 αI 0 0 −αI and thus C(I + N 2 ) 1/2 W = (I + N 2 ) 1/2 W αI 0 0 −αI .(11)
We have W 2 = I. Indeed,
U 2 11 + U 12 U 21 = U 21 U 12 + U 2 22 = [(α + a) 2 + b 2 ](I + N 2 ) = 2α(α + a)(I + N 2 ),
whence W 2 11 +W 12 W 21 = W 21 W 12 +W 2 22 = I, and similarly one gets that the off-diagonal blocks of W 2 are zero. From (11) we therefore get
C = (I + N 2 ) 1/2 W αI 0 0 −αI W (I + N 2 ) −1/2 I 0 0 I = W αI 0 0 −αI W,
or equivalently,
I + 1 α C = W 2I 0 0 0 W = 2 W 2 11 W 11 W 12 W 21 W 11 W 21 W 12 .(12)
The matrix G given by (10) is just G = √ 2(W 11 W 21 ). We claim that G is a unit tight equiangular (k, n) frame. First, since W 11 and W 21 are symmetric, we have
G ′ G = 2 W 11 W 21 W 11 W 21 = 2 W 2 11 W 11 W 21 W 21 W 11 W 2 21 ,(13)
and since W 21 = W 12 , the right-hand sides of (12) and (13) coincide. This proves that G is unit and equiangular. Secondly,
GG ′ = 2 W 11 W 21 W 11 W 21 = 2(W 2 11 + W 2 21 ) = 2I,
which shows that G is tight with γ = 2 = n/k. The equality GG ′ = 2I implies that the rank of G is k. Thus, G = √ 2(W 11 W 12 ) has k linearly independent columns. We permute the columns of G so that these k linearly independent columns become the first k columns. The resulting matrix, which is anew denoted by G, is a unit tight equiangular (k, n) frame of the form G = (G 0 G 1 ) with an invertible matrix G 0 . Furthermore, we have G 0 = (I + N 2 ) −1/2 R and G 1 = (I + N 2 ) −1/2 S with matrices R and S whose entries are rational numbers. We therefore obtain that G −1 0 G 1 = R −1 S is a matrix with rational entries, and hence, by Proposition 3.1, the set Λ(G) is a full rank lattice. is a unit tight equiangular (k, n) frame G and the set Λ(G) is a full rank lattice. If N ∈ Z k×k , then G may be written as
G = B + I −N with B + := I + (1/α)A,(14)
and B + is a basis matrix for Λ(G), while if N −1 ∈ Z k×k , then G may be written in the form
G = B − S −N −1 I with B − := I − (1/α)A,(15)
where S := D|D| −1 and |D| is the positive definite square root of D ′ D, and this time B − S is a basis matrix for Λ(G), Furthermore,
det B ± = det I ± 1 α A = 1 α k/2 det(αI ± A).
Remark. Recall that the determinant (= volume of a fundamental domain) of a lattice is defined as the square root of det(B ′ B) where B is any basis matrix. Thus, if N is an integer matrix, then the determinant of the lattice is simply det(B ′ + B + ) = det B + , while if N −1 has integer entries, the determinant of the lattice Λ(G) is
det(S ′ B ′ − B − S) = det(SB ′ − B − S) = det(B ′ − B − S 2 ) = det B − because S = S ′ and S 2 = I.
Proof. Since D is invertible, we may use Theorem 4.2 with a = α and b = 0 (α 2 + 0 2 = α 2 ) and with N = D −1 (A − αI). In this special case, formula (10) becomes
G = √ 2(I + N 2 ) −1/2 I −N ,(16)
and since N has rational entries, Proposition 3.1 implies that Λ(G) is a full rank lattice. Proposition 3.1 also shows that √ 2(I +N 2 ) −1/2 is a basis matrix for the lattice provided N ∈ Z k×k . Writing (16) as
G = − √ 2(I + N 2 ) −1/2 N −N −1 I
and permuting (−N −1 I) to (I − N −1 ) we can deduce from Proposition 3.1 that the matrix − √ 2(I + N 2 ) −1/2 N is a basis matrix provided N −1 ∈ Z k×k . It remains to show that these two basis matrices are just the matrices B ± .
As the square of the matrix (3) is α 2 I, we have A 2 + D 2 = α 2 I. Since 0 is not in the spectrum of D, the equality A 2 + D 2 = α 2 I implies that the spectrum (= set of eigenvalues) of A is contained in the open interval (−α, α). Hence αI ± A are positive definite. Moreover, we get D 2 = α 2 I − A 2 = (αI − A)(αI + A), and since all involved matrices are circulants and therefore commute, we obtain
I + N 2 = I + D −2 (A − αI) 2 = I + D −2 (αI − A)(αI − A) = I + (αI − A) −1 (αI + A) = (αI + A) −1 [αI + A + αI − A] = 2α(αI + A) −1 = 2(I + (1/α)A) −1 .
Consequently, √ 2(I + N 2 ) −1/2 = (I + (1/α)A) 1/2 = B + , which proves (14). The matrix |D| is again a circulant matrix and we have D = S|D| with a circulant matrix S satisfying S 2 = I. From the equality D 2 = (αI − A)(αI + A) we obtain that |D| = (αI − A) 1/2 (αI + A) 1/2 . Thus,
− √ 2(I + N 2 ) −1/2 N = (I + (1/α)A) 1/2 D −1 (αI − A) = 1 √ α (αI + A) 1/2 (αI − A) −1/2 (αI + A) −1/2 (αI − A)S = 1 √ α (αI − A) 1/2 S = (I − (1/α)A) 1/2 S = B − S.
This proves (15). The determinant formulas are obvious.
Two lattices from (5,10) frames. Let (A, D) be one of the four pairs given by (6) to (9). Thus, k = 5, n = 10, α = 3. In either case, D is invertible with det D = ±48 With the Fourier matrix
B + x 2 = UEU ′ x 2 = EU ′ x 2 > 1 4 U ′ x 2 = 1 4 x 2 , and hence B + x 2 > 1 if x 2 ≥ 4. So consider the x ∈ Z 5 \ {0} with x 2 ≤ 3.F 5 = (1/ √ 5)(ω (j−1)(k−1) ) 5 j,k=1 , ω = e 2πi/5 , this is B 1 = F * 5 EF 5 with E = diag 1, 1 − 1 3 √ 5, 1 + 1 3 √ 5, 1 + 1 3 √ 5, 1 − 1 3 √ 5 = diag 1, √ 5 − 1 √ 6 , √ 5 + 1 √ 6 , √ 5 + 1 √ 6 , √ 5 − 1 √ 6 .
It follows in particular that the numerical values shown above are 0.2000 = 1/5, 0.9303 = 1/5 + 2 2/15, −0.1651 = 1/5 − 2/15.
Three lattices from (13,26) frames. Let now k = 13, n = 26, α = 5. Let A and D be symmetric 13 × 13 circulant matrices whose first rows are (0, ε 1 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 , ε 6 , ε 5 , ε 4 , ε 3 , ε 2 , ε 1 ) and (ε 7 , ε 8 , ε 9 , ε 10 , ε 11 , ε 12 , ε 13 , ε 13 , ε 12 , ε 11 , ε 10 , ε 9 , ε we have N ∈ Z 13×13 . We denote the A and B + = I + (1/5)A corresponding to these cases by A 1 , . . . , A 6 and B 1 , . . . , B 6 . Corollary 4.3 implies that B j is a basis matrix for the jth lattice. We have det(5I + A j ) = 2 560 000 = 2 12 · 5 4 and hence det B j = 2 6 /5 9/2 ≈ 0.0458 for 1 ≤ j ≤ 6. In the other 6 cases, for instance if the first rows of A and D equal We know from Corollary 4.3 that S j B j is a basis matrix for the jth lattice. It turns out that det(5I − A j ) = 2 560 000 = 2 12 · 5 4 and hence again det B j = 2 6 /5 9/2 ≈ 0.0458 for 7 ≤ j ≤ 12.
Actually, For 1 ≤ j ≤ 6, the smallest eigenvalue of B j is about 0.3736, whence
B 1 = B 2 , B 3 = B 4 , B 5 = B 6 , S 7 B 7 = −S 8 B 8 , S 9 B 9 = −S 10 B 10 , S 11 B 11 = −S 12 B 12 , B 1 = U 1 S 11 B 11 , B 3 = U 2 S 9 B 9 , B 5 = U 3 S 7 B 7 with orthogonal matrices U 1 , U 2 , U 3 . The relation "X ∼ Y ifB j x 2 > 0.37 2 x 2 > 0.13 x 2 .
Thus, Bx 2 > 1 for x 2 ≥ 7. In the last 6 cases, the smallest eigenvalue of B j is about 0.4991 and so we have
S j B j c 2 = B j x 2 > 0.49 2 x 2 > 0.24 x 2 ,
which is greater than 1 for x 2 ≥ 4. We took all j ∈ {1, . . . , 12} and x ∈ Z 13 with x 2 ≤ 6 and checked wether B j x 2 < 1.1. For each j, we obtained exactly 52 vectors x ∈ Z 13 \ {0} such that B j x 2 < 1.1. The columns B j x are ± the 26 columns f 1 , . . . , f 26 of G. Consequently, in all cases the minimal distance of Λ(G) is 1, Λ(G) has a basis of minimal vectors, and S(Λ(G)) = {±f 1 , . . . , ±f 26 }.
Ten lattices from (25,50) frames. We finally take k = 25, n = 50, α = 7. We consider the 25 × 25 circulant matrices A and D whose first rows are (0, ε 1 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 , ε 7 , ε 8 , ε 9 , ε 10 , ε 11 , ε 12 , ε 12 , ε 11 , ε 10 , ε 9 , ε 8 , ε 7 , ε 6 , ε 5 , ε 4 , ε 3 , ε 2 , ε 1 ) and (ε 25 , ε 13 , ε 14 , ε 15 , ε 16 , ε 17 , ε 18 , ε 19 , ε 20 , ε 21 , ε 22 , ε 23 , ε 24 , ε 24 , ε 23 , ε 22 , ε 21 , ε 20 , ε 19 , ε 18 , ε 17 , ε 16 Fix k ≥ 2 and consider the set F of the k + 1 normalized columns of height k + 1 formed by the permutations of −k, 1, . . . , 1 (k ones),
f 1 = 1 √ k 2 + k −k 1 . . . 1 , f 2 = 1 √ k 2 + k 1 −k . . . 1 , . . . , f k+1 = 1 √ k 2 + k 1 1 . . . −k .
These k + 1 vectors are in the orthogonal complement of (1, . . . , 1) ′ ∈ R k+1 and may therefore be thought of as vectors in R k . Let
Λ(F ) = span Z {f 1 , . . . , f k+1 } ⊂ R k .
The following theorem in conjunction with Proposition 2.4 proves Theorem 2.2(a).
B = 1 √ k 2 + k −k 1 . . . 1 1 −k . . . 1 . . . . . . . . . 1 1 . . . −k 1 1 . . . 1 (k+1)×k ,(17)
is a basis matrix for Λ(F ), we have
det(B ′ B) = 1 k + 1 1 + 1 k k ,
the lattice Λ(F ) has a basis of minimal vectors, and S(Λ(F )) = {±f 1 , . . . , ±f k+1 }.
Proof. It is well known that F is a tight unit equiangular (k, k + 1) frame. We include the proof for the reader's convenience. First, the columns of the matrix B are easily seen to be linearly independent, which shows that span R {f 1 , . . . , f k } = R k . Secondly, it is clear that f j = 1 for all j. Thirdly, we have (f i , f j ) = (−k − 1)/(k 2 + k) = −1/k for i = j. And finally, if x = (x 1 , . . . , x k+1 ) and x 1 + · · · + x k+1 = 0, then
(f j , x) = 1 √ k 2 + k −kx j + i =j x i = 1 √ k 2 + k (−kx j − x j ) and hence k+1 j=1 (f j , x) 2 = 1 k 2 + k k+1 j=1 (−(k + 1)x j ) 2 = k + 1 k x 2 ,
that is, the frame is tight with γ = (k + 1)/k.
Since f 1 + · · · + f k = −f k+1 , we have Λ(F ) = span Z {f 1 , . . . , f k }. This shows that Λ(F ) is {BX : X ∈ Z k }. Consequently, Λ(F ) is a full rank lattice with the matrix B given by (17) as a basis matrix. The product B ′ B is
B ′ B = 1 k 2 + k a b . . . b b a . . . b . . . . . . . . . . . . b b . . . a k×k(18)
with a = k 2 + k and b = −k − 1. The determinant of a matrix of the form (18) is
known to be (a − b) k−1 (a + (k − 1)b). Thus, det B ′ B = 1 (k 2 + k) k (k 2 + k + k + 1) k−1 (k 2 + k − (k − 1)(k + 1)) = (k + 1) k−1 k k .
We are left with determining S(Λ(F )). Straightforward computation shows that the inequality Bx 2 ≥ 1 is equivalent to the inequality
(k + 1)(x 2 1 + · · · + x 2 k ) ≥ k + (x 1 + · · · + x k ) 2 ,(19)
and that equality holds in both inequalities only simultaneously. We first show (19) for integers (x 1 , . . . , x k ) ∈ Z k \ {0} by induction on k. For k = 1, inequality (19) is trivial. Suppose it is true for k − 1:
k(x 2 1 + · · · + x 2 k−1 ) ≥ k − 1 + (x 1 + · · · + x k−1 ) 2 .
If x 2 1 + · · · + x 2 k−1 ≥ 1, we may add x 2 1 + · · · + x 2 k−1 on the left and 1 on the right to get
(k + 1)(x 2 1 + · · · + x 2 k−1 ) ≥ k + (x 1 + · · · + x k−1 ) 2 .
This proves (19) in the case where one of the integers x 1 , . . . , x k is zero and one of them is nonzero. We are so left with the case where x j = 0 for all j. Then x 2 1 + · · · + x 2 k ≥ k and hence
k + (x 1 + · · · + x k ) 2 ≤ k + (|x 1 | + · · · + |x k |) 2 (20) ≤ k + k(x 2 1 + · · · + x 2 k ) ≤ x 2 1 + · · · + x 2 k + k(x 2 1 + · · · + x 2 k ) = (k + 1)(x 2 1 + · · · + x 2 k ),(21)
which completes the proof of (19). At this point we have shown that {f 1 , . . . , f k } is a basis of minimal vectors.
To identify all of S(Λ(F )), we have to check when equality in (19) holds. Suppose first that x j = 0 for all j. In that case we have (20) to ( Bx are −f k+1 and f k+1 . Suppose finally that one of the x j is zero, say x k = 0. From (19) with k replaced by k − 1 we know that
k(x 2 1 + · · · + x 2 k−1 ) ≥ k − 1 + (x 1 + · · · + x k−1 ) 2 .
If x 2 1 + · · · + x 2 k−1 > 1, we may add this inequality to the previous one to obtain that
(k + 1)(x 2 1 + · · · + x 2 k−1 ) ≥ k + (x 1 + · · · + x k−1 ) 2 .
Consequently, for x 2 1 +· · ·+x 2 k−1 > 1 equality in (19) does not hold. If x 2 1 +· · ·+x 2 k−1 = 1, then x j = ±1 for some j and x i = 0 for all i = j. In that case equality in (19) holds and the vector Bx is ±f j . In summary, we have proved that the set S(Λ(F )) of all minimal vectors is just {±f 1 , . . . , ±f k+1 }. 6 The remaining frames in dimensions at most 9
Recall that (2) lists the unit tight equiangular frames in dimensions k ≤ 9 different from the (k, k + 1) frames. By Proposition 2.1, the (3, 6), (7,14), and (9, 18) frames do not yield lattices, and the lattices resulting from the (5, 10) case were discussed in Section 4. We are left with the (6, 16) and (7, 28) cases.
A lattice from a (6,16) frame. In [18] we see the unit tight equiangular (6,16) Here GG ′ = (16/6)I and G ′ G = I +(1/3)C with a 16×16 matrix C whose diagonal entries are zero and the other entries of which are ±1. The six columns f 1 , f 2 , f 3 , f 4 , f 5 , f 9 of the matrix G are linearly independent and each of the remaining 10 columns is a linear combination with integer coefficients of these six columns. Consequently, by Proposition 3.1 with β = 1, these six columns form a basis matrix,
B = 1 √ 6 + + + + + + + + + + + − + + + + − + + + − − + + + − + − + + + − − + − − .
We have det(B ′ B) = 2 6 /3 6 .
With B ′ B = U ′ EU, we get Bx 2 = (EUx, Ux) ≥ 0.48 x 2 , and this is at least 6 if x 2 ≥ 13. So consider the x ∈ Z 6 \ {0} with x 2 ≤ 13. Such x contain only 0, ±1, ±2, ±3, and using Matlab we checked that Bx 2 < 6.1 for exactly 32 nonzero x of these 7 6 − 1 = 117 648 possible x. The 32 columns Bx are just ± the columns of √ 6G. Thus, Λ(F ) has a basis of minimal vectors and S(Λ(G)) = {±f 1 , . . . , ±f 16 }.
A perfect lattice from a (7,28) frame. It is well known that the 8 2 = 28 vectors resulting from the columns (−3, −3, 1, 1, 1, 1, 1, 1) ′ by permuting the entries form a tight equiangular (7, 28) frame. To be precise, let F be the set of the vectors
f 1 = 1 √ 24 −3 −3 1 1 1 1 1 1 , . . . , f 28 = 1 √ 24 1 1 1 1 1 1 −3 −3 .
These are unit vectors in R 8 . They are all orthogonal to the vector (1, 1, 1, 1, 1, 1, 1, 1) ′ , and after identifying the orthogonal complement of this vector with R 7 , we may think of f 1 , . . . , f 28 as unit vectors in R 7 . We consider the set
Λ(F ) = span Z {f 1 , . . . , f 28 } ⊂ R 7 .
The columns of the 8 × 7 matrix
B = 1 √ 24 1 1 1 1
are formed by 7 of the above 28 vectors. We denote these 7 vectors by f 1 , . . . , f 7 . For the reader's convenience, we show that {f 1 , . . . , f 28 } is a tight unit equiangular (7,28) frame. The rank of the matrix B is 7, and hence span R {f 1 , . . . , f 28 } = R 7 . Clearly, f j = 1 for all j. We have |(f i , f j )| = 8/24 = 1/3 for i = j (equiangularity). Finally, let x = (x 1 , . . . , x 8 ) ∈ R 7 . Then x 1 + · · · + x 8 = 0 and hence j x 2 j + j =k
x j x k = 0, which implies that 2
j =k
x j x k = −2 x 2 .
We have
28 ℓ=1 (f ℓ , x) 2 = 1 24 j<k −3x j − 3x k + m =j,k x m 2 = 1 24 j<k (−3x j − 3x k − x j − x k ) 2 = 2 3 j<k (x j + x k ) 2 = 1 3 j =k (x j + x k ) 2 = 1 3 j =k (x 2 j + 2x j x k + x 2 k ) = 1 3 (14 x 2 − 2 x 2 ) = 4 x 2 .
This proves the tightness with γ = 4 (which, as is should be, is just n/k = 28/7). We now prove that the minimal norm of Λ(F ) is 1. Let
B = √ 24B = −3 −3 −3 −3 −3 −3 1 −3 1 1 1 1 1 1 1 −3 1 1 1 1 −3 1 1 −3 1 1 1 1 1 1 1 −3 1 1 1 1 1 1 1 −3 1 −3 1 1 1 1 1 −3 1 1 1 1 1 1 1 1
. Take x ∈ Z 7 and consider y = Bx ∈ Z 8 . We are interested in the x for which y 2 ≤ 24. With s := x 1 + · · · + x 7 , we have
y 1 = −3s + 4x 7 , y 3 = s − 4x 2 − 4x 7 , y 6 = s − 4x 5 − 4x 7 , y 8 = s, y 2 = s − 4x 1 , y 4 = s − 4x 3 , y 5 = s − 4x 4 , y 7 = s − 4x 6 .
It suffices to search for all x ∈ Z 7 with s ≥ 0 and y 2 1 + · · · + y 2 8 ≤ 24. This is impossible for y 8 = s > 5. So we may assume that 0 ≤ s ≤ 4.
Suppose first that s = 4. We then must have y 2 1 + · · · + y 2 7 ≤ 9. Since y 1 is an even number, it cannot be ±3. Consequently, −2 ≤ −3s + 4x 7 = −12 + 4x 7 ≤ 2, which gives x 7 = 3. Analogously, as y 3 is even, we get −2 ≤ s − 4x 2 − 4x 7 = −8 − 4x 2 ≤ 2, which yields x 2 = −2. In the same way we obtain x 5 = −2. Finally, the even number s − 4x 1 = 4 − 4x 1 is at least −2, which implies that x 1 ≤ 1. Equally, x 3 , x 4 , x 6 ≤ 1. It follows that s = x 1 + · · · + x 7 ≤ 1 + 1 + 1 + 1 − 2 − 2 + 3 = 3 < 4 = s, which is a contradiction.
Thus, we may restrict our search to 0 ≤ s ≤ 3 and y 2 1 + · · · + y 2 7 ≤ 24. The inequality −4 ≤ −3s + 4x 7 ≤ 4 gives −4 ≤ 3s − 4 ≤ 4x 7 ≤ 3s + 4 ≤ 13, whence −1 ≤ x 7 ≤ 3. These are 5 possibilities. From −4 ≤ s − 4x j ≤ 4 we obtain that −1 ≤ x j ≤ 1 for j = 1, 3, 4, 6, which is 3 4 possibilities, and the inequality −4 ≤ s − 4x j − 4x 7 ≤ 4 delivers −16 ≤ s − 4 + 4x 7 ≤ 4y j ≤ 4 + s − 4x 7 ≤ 11 and hence −4 ≤ x j ≤ 2 for j = 2, 5, leaving us with 7 2 possibilities. In summary, we have to check 5 · 3 4 · 7 2 = 19 845 possibilities. Matlab does this with integer arithmetics within a second. The result is that 0 ≤ s ≤ 3 and y 2 1 + · · · + y 2 7 ≤ 24 happens in exactly 50 cases. One of these cases is y = 0, and in the remaining 49 cases y is ± one of the 2 · 28 = 56 vectors √ 24f j . (Recall that, by symmetry, we restricted ourselves to s ≥ 0. For −3 ≤ s ≤ 3 and y 2 1 + · · · + y 2 7 ≤ 24 to happen we would obtain exactly 57 cases: the case y = 0 and the 56 vectors y given by ± √ 24f j .) This proves that the minimal distance of Λ(G) is 1, that S(Λ(G)) = {±f 1 , . . . , ±f 28 }, and that Λ(G) has a basis of minimal vectors. From Proposition 2.4 we deduce that the lattice Λ(G) is strongly eutactic.
We finally show that this (7, 28) frame generates a perfect lattice. We have shown that the 28 lattice vectors f 1 , . . . , f 28 are minimal vectors. These vectors are given by their coordinates in the ambient R 8 . We use a special 7 × 8 matrix A to transform these vectors isometrically into R 7 . The jth row of A is 1 j 2 + j (1, . . . , 1, −j, 0, . . . , 0)
with j ones and 7 − j zeros. We have A = EA 0 with E = diag(1/ j 2 + j) 7 j=1 and with (1, . . . , 1, −j, 0, . . . , 0) being the jth row of A 0 . We then get the 28 minimal vectors Af j = EA 0 f j (j = 1, . . . , 28) in R 7 . These give us 28 symmetric 7 × 7 matrices C j = E(A 0 f j )(A 0 f j ) ′ E. The lattice Λ(F ) is perfect if the real span of these 28 matrices is the space of all 7 × 7 symmetric matrices. Each symmetric 28 × 28 matrix may be written as ET E with a symmetric matrix T , and hence we are left with showing that each symmetric 28×28 symmetric matrix T is a real linear combination of the matrices (A 0 f j )(A 0 f j ) ′ . For k = 1, . . . , 7, let ([C j ] k,k , [C j ] k+1,k , . . . , [C j ] 7,k ) ′ be the column formed by the entries of the kth column of C j that are on or below the main diagonal. Stack these columns to a column D j of height 7 + 6 + · · · + 1 = 28. The lattice is perfect if and only if the real span of D 1 , . . . , D 28 is all of R 28 , which happens if and only if the 28 × 28 matrix D constituted by the 28 columns D 1 , . . . , D 28 is invertible. Tables 2 and 3 show the matrix D.
1} =: {−, +}. These are 2 5 = 32 matrices. Exactly four of them satisfy C 2 = 9I. Their first lines and the eigenvalues of D are 0, −, +, +, −, −, +, +, +, +,
Theorem 4. 2
2Let k ≥ 2 and n = 2k. Suppose n = p 2m + 1 with an odd prime number p and a natural number m, let a and b be rational numbers such that a 2 + b 2 = p 2m and a = −p m . Let A and D be symmetric k × k circulant matrices such that (3) is a conference matrix, and assume A + aI or D + bI is invertible. Define N by(5) and put α = √ n − 1 = p m . Then G = 1 α(α + a) (I + N 2 ) −1/2 (α + a)I + bN bI − (α + a)N
Corollary 4. 3
3Let k ≥ 2 and n = 2k. Suppose n = p 2m + 1 with an odd prime number p and a natural number m, let A and D be symmetric k × k circulant matrices such that the matrix (3) is a conference matrix. Put α = √ n − 1 = p m . If the matrix D is invertible, then I ±(1/α)A are positive definite matrices and, with the invertible matrix N := D −1 (A − αI), G := √ 2(I + N 2 ) −1/2 I −N
and we have det(3I + A) = 48. (The eigenvalues of A are − circulant matrices N = D −1 (A − 3I) have the first rows (+, 0, −, −, 0), (−, 0, +, +, 0), (+, −, 0, 0, −), (−, +, 0, 0, +). Thus, N ∈ Z 5×5 , and so by Corollary 4.3, Λ(G) is a lattice, B + = I + (1/3)A is a basis matrix, and det B + = 3 −5/2 √ 48 = 2 2 /3 2 = 0.4444 . . .. (Incidentally, the matrices N −1 also have integer entries.) The eigenvalues of I + (1/3)A are
Such x contain only 0 ,
0+1, −1, and using Matlab we checked that B + x 2 < 1.1 for exactly 20 nonzero x of these 3 5 − 1 = 242 possible x. The 20 columns B + x are just ± the columns of G. Thus, the minimal distance of Λ(G) is 1, Λ(G) has a basis of minimal vectors, and S(Λ(G)) = {±f 1 , . . . , ±f 10 }. Note that if we denote the basis matrices for the lattices corresponding to (6) and (9) by B 1 , . . . , B 4 , then actually B 1 = B 2 and B 3 = B 4 . However, B 1 B −1 3 is not a scalar multiple of an orthogonal matrix. To "see" a concrete matrix B + , note that in the case where the matrices A, D are specified by(6), we obtain that B 1 = B + = I + (
8 ) with ε k ∈ {−1, +1} =: {−, +}, respectively. There are 2 13 = 8 192 such matrices. For exactly 12 of them the matrix (3) satisfies C 2 = 25I. The determinant of D is always det D = ±768 000 = ±2 12 · 3 · 5 4 . Thus, by Corollary 4.3, Λ(G) is a full rank lattice. In exactly 6 cases, for example if the first rows of A and D are (0, −, −, −, +, −, +, +, −, +, −, −, −) and (−, −, +, +, +, −, +, +, −, +, +, +, −),
( 0 ,
0−, +, +, +, −, +, +, −, +, +, +, −) and (−, +, −, −, +, +, +, +, +, +, −, −, +), we get that N −1 ∈ Z 13×13 . Let A 7 , . . . , A 12 , S 7 , . . . , S 12 , and B 7 , . . . , B 12 be the corresponding A, S = D|D| −1 , B − = I − (1/5)A.
and only if XY −1 is a nonzero scalar multiple of an orthogonal matrix" is an equivalence relation on every family of invertible k × k matrices. The equivalence classes of this relation on {B 1 , . . . , S 12 B 12 } are {B 1 = B 2 , S 11 B 11 , S 12 B 12 }, {B 3 = B 4 , S 9 B 9 , S 10 B 10 }, {B 5 = B 6 , S 7 B 7 , S 8 B 8 }. The first rows of (A 1 , D 1 ), (A 3 , D 3 ), (A 5 , D 5 ) are (0, −, −, −, +, −, +, +, −, +, −, −, −, −, −, +, +, +, −, +, +, −, +, +, +, −), (0, −, +, +, −, −, −, −, −, −, +, +, −, +, −, −, −, +, −, +, +, −, +, −, −, −), (0, +, −, −, −, +, −, −, +, −, −, −, +, +, −, +, +, −, −, −, −, −, −, +, +, −).
, ε 15 , ε 14 , ε 13 ), with ε k ∈ {−1, 1} =: {−, +}, respectively. These are 2 25 = 33 554 432 matrices. In exactly 20 cases the matrix C given by (3) satisfies C 2 = 49 I. One such case is where the first rows of A and D are (0, −, −, −, +, −, +, +, −, +, +, +, −, −, +, +, +, −, +, +, −, +, −, −, −) and (−, −, +, +, +, +, +, −, +, −, +, +, −, −, +, +, −, +, −, +, +, +, +, +, −), respectively. We have | det D| = det(7I + A) = det(7I − A) = 260 119 840 2 = 2 22 · 3 2 · 5 2 · 7 2 · 11 4 , N ∈ Z 25×25 and N −1 ∈ Z 25×25 in each of the 20 cases. Thus, by Corollary 4.3, we obtain 20 lattices Λ(G j ) with B j = I + (1/7)A j as a basis matrix and det B j = 2 11 · 3 · 5 · 7 1 ≤ j ≤ 20. In fact B j = B j+10 for 1 ≤ j ≤ 10, and the equivalence classes of the set {B 1 , . . . , B 10 } under the equivalence relation "B i ∼ B j if and only if B i B −1 j is a nonzero scalar multiple of an orthogonal matrix" are the ten singletons {B 1 }, . . . , {B 10 }. The smallest eigenvalue of B j is about 0.1415 for all j.
Theorem 5. 1
1The vectors f 1 , . . . , f k+1 form a unit tight equiangular (k, k + 1) frame and Λ(F ) is a full rank lattice. The matrix B constituted by f 1 , . . . , f k ,
21). Equality in (21) holds if and only if |x j | = 1 for all j, and equality in (20) is valid if and only if all the x j have the same sign. Thus, we get the two vectors x = (1, . . . , 1) ′ and x = (−1, . . . , −1) ′ . The corresponding products
+ − − − − + + + + − − − − + + − − + + − − + + − − + + − − + − + − + − + − + − + − + − + − + − − + − + + − − + + − + −
+ = UEU ′ with an orthogonal matrix U and the diagonal matrix E of the eigenvalues of B + . Consequently,1 −
1
3
√
5, 1 −
1
3
√
5, 1, 1 +
1
3
√
5, 1 +
1
3
√
5,
and hence the smallest eigenvalue of B + is 1 − (1/3)
√
5 = 0.50462... > 1/2. We have
B
Straightforward inspection shows that each of the vectors f 8 , . . . , f 28 is a linear combination with integer coefficients of the vectors f 1 , . . . , f 7 . Consequently, Λ(F ) is a full rank lattice in R 7 , {f 1 , . . . , f 7 } is a basis of Λ(F ), and B is a basis matrix. We haveB ′ B =
1
24
24 8
8
8 8
8 −8
8 24
8
8 8
8
8
8 8 24
8 8
8 −8
8 8
8 24 8
8 −8
8 8
8
8 24
8
8
8 8
8
8 8 24 −8
−8 8 −8 −8 8 −8 24
,
and straightforward computation gives
det B ′ B =
2 27
24 7 =
2 6
3 7 .
Table 2 :
2The first 14 columns of the matrix D.
Table 3 :
3The last 14 columns of the matrix D.
Unit tight equiangular (k, k + 1) frames Sometimes it is advantageous to represent a unit tight equiangular (k, n) frame by coordinates different from those in R k . This is in particular the case for (k + 1, k) frames.
−3 −3 −3 −3 −3 −3 1 −3 1 1 1 1 1 1 1 −3 1 1 1 1 −3 1 1 −3 1 1 1 1 1 1 1 −3 1 1 1 1 1 1 1 −3 1 −3 1 1 1 1 1 −3 1 1 1 1
Acknowledgement. Fukshansky acknowledges support of the NSA grant H98230-1510051. Garcia acknowledges support of the NSF grant DMS-1265973. Needell acknowledges support of the Alfred P. Sloan Fellowship and NSF Career grant number 1348721.The matrix D can be constructed with integer arithmetics. The determinant det D may be computed by the Gaussian algorithm and thus with integer arithmetics, too.In the intermediate steps, one may factor out powers of 2. For example, in the original matrix D we may draw out 16 from the first line, 4 from the second, 8 from the third, and so on. It results thatand we may start the Gaussian algorithm with det D. The final result isAs this is nonzero, we conclude that D is invertible and thus that Λ(F ) is perfect. At this point the proof of Theorem 2.2(c) is complete.The perfection of this lattice was also established by Bacher in[2](see Section 7, especially 7.1). However, Bacher's approach is different from ours: he obtains the lattice in question as the kernel of a certain linear map, establishes its perfection, and then remarks that its set of minimal vectors comprises an equiangular system. We, on the other hand, construct the lattice from the equiangular frame and show its perfection directly from this construction. Hence our argument here complements Bacher's, going in the opposite direction.Since Λ(F ) is perfect and strongly eutactic, the packing density of this lattice is a local maximum. As we know the minimal distance and the determinant of this lattice, the packing density can be easily computed using(1). It turns out to be 21.57 %. This is better than the packing density of the root lattice A 7 , which is 14.76 %. In[4], we studied lattices in R k that are generated by Abelian groups of the order k + 1. There the packing density of the lattices generated by Abelian groups of order 8 was shown to 20.88 %. Thus, Λ(F ) is also better than this. We nevertheless do not reach the best packing density for a 7-dimensional lattice, which is 29.53 % and is achieved for the well known lattice E 7 .
Spectrum sensing for cognitive radio: State-of-the-art and recent advances. E Axell, G Leus, E G Larsson, H V Poor, IEEE Signal Proc. Mag. 293E. Axell, G. Leus, E. G. Larsson, and H. V. Poor, Spectrum sensing for cogni- tive radio: State-of-the-art and recent advances. IEEE Signal Proc. Mag. 29(3) (2012), 101-116.
Constructions of some perfect integral lattices with minimum 4. R Bacher, J. Théor. Nombres Bordeaux. 273R. Bacher, Constructions of some perfect integral lattices with minimum 4. J. Théor. Nombres Bordeaux 27 (2015), no. 3, 655-687.
A simple proof of the restricted isometry property for random matrices. R Baraniuk, M Davenport, R Devore, M Wakin, Constr. Approx. 28R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, A simple proof of the restricted isometry property for random matrices. Constr. Approx. 28 (2008), 253- 263.
On lattices generated by finite Abelian groups. A Böttcher, L Fukshansky, S R Garcia, H Maharaj, SIAM J. Discrete Math. 29A. Böttcher, L. Fukshansky, S. R. Garcia, and H. Maharaj, On lattices generated by finite Abelian groups. SIAM J. Discrete Math. 29 (2015), 382-404.
Error correction via linear programming. E Candes, M Rudelson, T Tao, R Vershynin, IEEE Symp. Found. Comput. Sci. (FOCS'05). IEEEE. Candes, M. Rudelson, T. Tao, and R. Vershynin, Error correction via linear programming. In: IEEE Symp. Found. Comput. Sci. (FOCS'05), IEEE, 2005, 668- 681.
Steiner equiangular tight frames. M Fickus, D G Mixon, J C Tremain, Linear Algebra Appl. 436M. Fickus, D. G. Mixon, and J. C. Tremain, Steiner equiangular tight frames. Linear Algebra Appl. 436 (2012), 1014-1027.
Promp: A sparse recovery approach to lattice-valued signals. A Flinth, G Kutyniok, PreprintA. Flinth and G. Kutyniok, Promp: A sparse recovery approach to lattice-valued signals. Preprint 2016.
Orthogonal matrices with zero diagonal. J M Goethals, J J Seidel, Canad. J. Math. 19J. M. Goethals and J. J. Seidel, Orthogonal matrices with zero diagonal. Canad. J. Math. 19 (1967), 1001-1010.
Optimal frames for erasures. R B Holmes, V Paulsen, Linear Algebra Appl. 377R. B. Holmes and V. Paulsen, Optimal frames for erasures. Linear Algebra Appl. 377 (2004), 31-51.
Perfect Lattices in Euclidean Spaces. J Martinet, Springer-VerlagJ. Martinet, Perfect Lattices in Euclidean Spaces. Springer-Verlag, 2003.
Equiangular tight frame fingerprinting codes. D G Mixon, C Quinn, N Kiyavash, M Fickus, IEEE Int. Conf. Acoustics, Speech and Sig. Proc. (ICASSP). IEEED. G. Mixon, C. Quinn, N. Kiyavash, and M. Fickus, Equiangular tight frame fin- gerprinting codes. In: IEEE Int. Conf. Acoustics, Speech and Sig. Proc. (ICASSP), IEEE, 2011, 1856-1859.
On orthogonal matrices. R E A C Paley, J. Math. Phys. 12R. E. A. C. Paley, On orthogonal matrices. J. Math. Phys. 12 (1933), 311-320.
Spatial compressive sensing for mimo radar. M Rossi, A M Haimovich, Y C Eldar, IEEE Trans. Signal Process. 622M. Rossi, A. M. Haimovich, and Y. C. Eldar, Spatial compressive sensing for mimo radar. IEEE Trans. Signal Process., 62(2) (2014), 419-430.
Designing incoherent frames through convex techniques for optimized compressed sensing. C Rusu, N González-Prelcic, IEEE Trans. Signal Process. 649C. Rusu and N. González-Prelcic, Designing incoherent frames through convex techniques for optimized compressed sensing. IEEE Trans. Signal Process. 64(9) (2016), 2334-2344.
Perfect, strongly eutactic lattices are periodic extreme. A Schürmann, Adv. Math. 2255A. Schürmann, Perfect, strongly eutactic lattices are periodic extreme. Adv. Math. 225 (2010), no. 5, 2546-2564.
Optimal OFDM design for time-frequency dispersive channels. T Strohmer, S Beaver, IEEE Trans. Commun. 517T. Strohmer and S. Beaver, Optimal OFDM design for time-frequency dispersive channels. IEEE Trans. Commun., 51(7) (2003), 1111-1122.
Grassmannian frames with applications to coding and communication. T Strohmer, R W HeathJr, Appl. Comput. Harmon. Anal. 14T. Strohmer and R. W. Heath Jr., Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14 (2003), 257-275.
On the existence of equiangular tight frames. M A Sustik, J A Tropp, I S Dhillon, R W HeathJr, Linear Algebra Appl. 426M. A. Sustik, J. A. Tropp, I. S. Dhillon, and R. W. Heath Jr., On the existence of equiangular tight frames. Linear Algebra Appl. 426 (2007), 619-635.
Greed is good: Algorithmic results for sparse approximation. J A Tropp, IEEE Trans. Inform. Theory. 5010J. A. Tropp, Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inform. Theory, 50(10) (2004), 2231-2242.
Use of tight frames for optimized compressed sensing. E Tsiligianni, L P Kondi, A K Katsaggelos, Proc. Signal Process. Conf. (EUSIPCO). Signal ess. Conf. (EUSIPCO)IEEEE. Tsiligianni, L. P. Kondi, and A. K. Katsaggelos, Use of tight frames for op- timized compressed sensing. In: Proc. Signal Process. Conf. (EUSIPCO), IEEE, 2012, 1439-1443,
. A Böttcher, T U Fakultät Für Mathematik, Chemnitz, [email protected], GermanyA. Böttcher, Fakultät für Mathematik, TU Chemnitz, 09107 Chemnitz, Germany [email protected]
. L Fukshansky, Department of Mathematics. Claremont McKenna CollegeL. Fukshansky, Department of Mathematics, Claremont McKenna College,
. Columbia Ave, S R Claremont, Garcia, Department of Mathematics. [email protected] Ave, Claremont, CA 91711, USA [email protected] S. R. Garcia, Department of Mathematics, Pomona College, 610 N. College Ave, Claremont, CA 91711, USA [email protected]
H Maharaj, Department of Mathematics. Pomona College, 610 N. College Ave, Claremont, CA 91711, USA hirenmaharaj@gmailH. Maharaj, Department of Mathematics, Pomona College, 610 N. College Ave, Claremont, CA 91711, USA [email protected]
. D , Department of Mathematics, Claremont McKenna CollegeD. Needell, Department of Mathematics, Claremont McKenna College,
. Columbia Ave, Claremont, CA 91711, USA dneedell@cmcColumbia Ave, Claremont, CA 91711, USA [email protected]
| [] |
[
"Statistical study of electron density turbulence and ion-cyclotron waves in the inner heliosphere: Solar Orbiter observations",
"Statistical study of electron density turbulence and ion-cyclotron waves in the inner heliosphere: Solar Orbiter observations"
] | [
"F Carbone \nNational Research Council -Institute of Atmospheric Pollution Research\nC/o University of Calabria\n87036RendeItaly\n",
"L Sorriso-Valvo \nSwedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden\n\nCNR\nIstituto per la Scienza e Tecnologia dei Plasmi\nvia Amendola 122/D70126BariItaly\n",
"Yu V Khotyaintsev \nSwedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden\n",
"K Steinvall \nSwedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden\n",
"A Vecchio \nLESIA\nObservatoire de Paris\nUniversité PSL\nCNRS\nSorbonne\n\nUniversité, Univ. Paris Diderot\nSorbonne Paris Cité\n5 place Jules Janssen92195MeudonFrance\n\nAstrophysics and Particle Physics\nResearch Institute for Mathematics\nRadboud University\nNijmegenThe Netherlands\n",
"D Telloni \nNational Institute for Astrophysics -Astrophysical Observatory of Torino\nItaly\n",
"E Yordanova \nSwedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden\n",
"D B Graham \nSwedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden\n",
"N J T Edberg \nSwedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden\n",
"A I Eriksson \nSwedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden\n",
"E P G Johansson \nSwedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden\n",
"C L Vásconez \nDepartamento de Física\nEscuela Politécnica Nacional\nLadrón de Guevara E11-253170525QuitoEcuador\n",
"M Maksimovic \nAstrophysics and Particle Physics\nResearch Institute for Mathematics\nRadboud University\nNijmegenThe Netherlands\n",
"R Bruno \nInstitute for Space Astrophysics and Planetology (IAPS)\nNational Institute for Astrophysics (INAF)\nVia Fosso del Cavaliere, 10000133RomeItaly\n",
"R D'amicis \nInstitute for Space Astrophysics and Planetology (IAPS)\nNational Institute for Astrophysics (INAF)\nVia Fosso del Cavaliere, 10000133RomeItaly\n",
"S D Bale \nSpace Sciences Laboratory\nUniversity of California\nBerkeleyCAUSA\n\nPhysics Department\nUniversity of California\nBerkeleyCAUSA\n",
"T Chust \nLPP\nCNRS\nEcole Polytechnique\nSorbonne Université\nObservatoire de Paris\nUniversité Paris-Saclay\nPalaiseau, ParisFrance\n",
"V Krasnoselskikh \nLPC2E\nCNRS\n3A avenue de la Recherche Scientifique\nOrléansFrance\n",
"M Kretzschmar \nLPC2E\nCNRS\n3A avenue de la Recherche Scientifique\nOrléansFrance\n\nUniversité d'Orléans\nOrléansFrance\n",
"E Lorfèvre \nCNES\n18 Avenue Edouard Belin31400ToulouseFrance\n",
"D Plettemeier \nTechnische Universität Dresden\nWürzburger Str. 35D-01187DresdenGermany\n",
"J Souček \nInstitute of Atmospheric Physics\nCzech Academy of Sciences\nPragueCzechia\n",
"M Steller \nSpace Research Institute\nAustrian Academy of Sciences\nGrazAustria\n",
"Š Štverák \nInstitute of Atmospheric Physics\nCzech Academy of Sciences\nPragueCzechia\n\nAstronomical Institute of the Czech Academy of Sciences\nPragueCzechia\n",
"P Trávníček \nSpace Sciences Laboratory\nUniversity of California\nBerkeleyCAUSA\n\nInstitute of Atmospheric Physics\nCzech Academy of Sciences\nPragueCzechia\n",
"A Vaivads \nLPC2E\nCNRS\n3A avenue de la Recherche Scientifique\nOrléansFrance\n\nDepartment of Space and Plasma Physics\nSchool of Electrical Engineering and Computer Science\nRoyal Institute of Technology\nStockholmSweden\n",
"T S Horbury \nSpace and Atmospheric Physics\nThe Blackett Laboratory\nImperial College of London\nSW7 2AZLondonUK\n",
"H O'brien \nSpace and Atmospheric Physics\nThe Blackett Laboratory\nImperial College of London\nSW7 2AZLondonUK\n",
"V Angelini \nSpace and Atmospheric Physics\nThe Blackett Laboratory\nImperial College of London\nSW7 2AZLondonUK\n",
"V Evans \nSpace and Atmospheric Physics\nThe Blackett Laboratory\nImperial College of London\nSW7 2AZLondonUK\n"
] | [
"National Research Council -Institute of Atmospheric Pollution Research\nC/o University of Calabria\n87036RendeItaly",
"Swedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden",
"CNR\nIstituto per la Scienza e Tecnologia dei Plasmi\nvia Amendola 122/D70126BariItaly",
"Swedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden",
"Swedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden",
"LESIA\nObservatoire de Paris\nUniversité PSL\nCNRS\nSorbonne",
"Université, Univ. Paris Diderot\nSorbonne Paris Cité\n5 place Jules Janssen92195MeudonFrance",
"Astrophysics and Particle Physics\nResearch Institute for Mathematics\nRadboud University\nNijmegenThe Netherlands",
"National Institute for Astrophysics -Astrophysical Observatory of Torino\nItaly",
"Swedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden",
"Swedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden",
"Swedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden",
"Swedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden",
"Swedish Institute of Space Physics (IRF)\nÅngström Laboratory\nLägerhyddsvägen 175121UppsalaSweden",
"Departamento de Física\nEscuela Politécnica Nacional\nLadrón de Guevara E11-253170525QuitoEcuador",
"Astrophysics and Particle Physics\nResearch Institute for Mathematics\nRadboud University\nNijmegenThe Netherlands",
"Institute for Space Astrophysics and Planetology (IAPS)\nNational Institute for Astrophysics (INAF)\nVia Fosso del Cavaliere, 10000133RomeItaly",
"Institute for Space Astrophysics and Planetology (IAPS)\nNational Institute for Astrophysics (INAF)\nVia Fosso del Cavaliere, 10000133RomeItaly",
"Space Sciences Laboratory\nUniversity of California\nBerkeleyCAUSA",
"Physics Department\nUniversity of California\nBerkeleyCAUSA",
"LPP\nCNRS\nEcole Polytechnique\nSorbonne Université\nObservatoire de Paris\nUniversité Paris-Saclay\nPalaiseau, ParisFrance",
"LPC2E\nCNRS\n3A avenue de la Recherche Scientifique\nOrléansFrance",
"LPC2E\nCNRS\n3A avenue de la Recherche Scientifique\nOrléansFrance",
"Université d'Orléans\nOrléansFrance",
"CNES\n18 Avenue Edouard Belin31400ToulouseFrance",
"Technische Universität Dresden\nWürzburger Str. 35D-01187DresdenGermany",
"Institute of Atmospheric Physics\nCzech Academy of Sciences\nPragueCzechia",
"Space Research Institute\nAustrian Academy of Sciences\nGrazAustria",
"Institute of Atmospheric Physics\nCzech Academy of Sciences\nPragueCzechia",
"Astronomical Institute of the Czech Academy of Sciences\nPragueCzechia",
"Space Sciences Laboratory\nUniversity of California\nBerkeleyCAUSA",
"Institute of Atmospheric Physics\nCzech Academy of Sciences\nPragueCzechia",
"LPC2E\nCNRS\n3A avenue de la Recherche Scientifique\nOrléansFrance",
"Department of Space and Plasma Physics\nSchool of Electrical Engineering and Computer Science\nRoyal Institute of Technology\nStockholmSweden",
"Space and Atmospheric Physics\nThe Blackett Laboratory\nImperial College of London\nSW7 2AZLondonUK",
"Space and Atmospheric Physics\nThe Blackett Laboratory\nImperial College of London\nSW7 2AZLondonUK",
"Space and Atmospheric Physics\nThe Blackett Laboratory\nImperial College of London\nSW7 2AZLondonUK",
"Space and Atmospheric Physics\nThe Blackett Laboratory\nImperial College of London\nSW7 2AZLondonUK"
] | [] | Context. The recently released spacecraft potential measured by the RPW instrument on-board Solar Orbiter has been used to estimate the solar wind electron density in the inner heliosphere. Aims. Solar-wind electron density measured during June 2020 has been analysed to obtain a thorough characterization of the turbulence and intermittency properties of the fluctuations. Magnetic field data have been used to describe the presence of ion-scale waves.Methods. Selected intervals have been extracted to study and quantify the properties of turbulence. The Empirical Mode Decomposition has been used to obtain the generalized marginal Hilbert spectrum, equivalent to the structure functions analysis, and additionally reducing issues typical of non-stationary, short time series. The presence of waves was quantitatively determined introducing a parameter describing the time-dependent, frequency-filtered wave power. Results. A well defined inertial range with power-law scaling has been found almost everywhere. However, the Kolmogorov scaling and the typical intermittency effects are only present in part of the samples. Other intervals have shallower spectra and more irregular intermittency, not described by models of turbulence. These are observed predominantly during intervals of enhanced ion frequency wave activity. Comparisons with compressible magnetic field intermittency (from the MAG instrument) and with an estimate of the solar wind velocity (using electric and magnetic field) are also provided to give general context and help determine the cause for the anomalous fluctuations. | 10.1051/0004-6361/202140931 | [
"https://arxiv.org/pdf/2105.07790v1.pdf"
] | 234,742,664 | 2105.07790 | 540b1f75b4b5759b07c61878efda7a69026b8534 |
Statistical study of electron density turbulence and ion-cyclotron waves in the inner heliosphere: Solar Orbiter observations
May 18, 2021 May 18, 2021
F Carbone
National Research Council -Institute of Atmospheric Pollution Research
C/o University of Calabria
87036RendeItaly
L Sorriso-Valvo
Swedish Institute of Space Physics (IRF)
Ångström Laboratory
Lägerhyddsvägen 175121UppsalaSweden
CNR
Istituto per la Scienza e Tecnologia dei Plasmi
via Amendola 122/D70126BariItaly
Yu V Khotyaintsev
Swedish Institute of Space Physics (IRF)
Ångström Laboratory
Lägerhyddsvägen 175121UppsalaSweden
K Steinvall
Swedish Institute of Space Physics (IRF)
Ångström Laboratory
Lägerhyddsvägen 175121UppsalaSweden
A Vecchio
LESIA
Observatoire de Paris
Université PSL
CNRS
Sorbonne
Université, Univ. Paris Diderot
Sorbonne Paris Cité
5 place Jules Janssen92195MeudonFrance
Astrophysics and Particle Physics
Research Institute for Mathematics
Radboud University
NijmegenThe Netherlands
D Telloni
National Institute for Astrophysics -Astrophysical Observatory of Torino
Italy
E Yordanova
Swedish Institute of Space Physics (IRF)
Ångström Laboratory
Lägerhyddsvägen 175121UppsalaSweden
D B Graham
Swedish Institute of Space Physics (IRF)
Ångström Laboratory
Lägerhyddsvägen 175121UppsalaSweden
N J T Edberg
Swedish Institute of Space Physics (IRF)
Ångström Laboratory
Lägerhyddsvägen 175121UppsalaSweden
A I Eriksson
Swedish Institute of Space Physics (IRF)
Ångström Laboratory
Lägerhyddsvägen 175121UppsalaSweden
E P G Johansson
Swedish Institute of Space Physics (IRF)
Ångström Laboratory
Lägerhyddsvägen 175121UppsalaSweden
C L Vásconez
Departamento de Física
Escuela Politécnica Nacional
Ladrón de Guevara E11-253170525QuitoEcuador
M Maksimovic
Astrophysics and Particle Physics
Research Institute for Mathematics
Radboud University
NijmegenThe Netherlands
R Bruno
Institute for Space Astrophysics and Planetology (IAPS)
National Institute for Astrophysics (INAF)
Via Fosso del Cavaliere, 10000133RomeItaly
R D'amicis
Institute for Space Astrophysics and Planetology (IAPS)
National Institute for Astrophysics (INAF)
Via Fosso del Cavaliere, 10000133RomeItaly
S D Bale
Space Sciences Laboratory
University of California
BerkeleyCAUSA
Physics Department
University of California
BerkeleyCAUSA
T Chust
LPP
CNRS
Ecole Polytechnique
Sorbonne Université
Observatoire de Paris
Université Paris-Saclay
Palaiseau, ParisFrance
V Krasnoselskikh
LPC2E
CNRS
3A avenue de la Recherche Scientifique
OrléansFrance
M Kretzschmar
LPC2E
CNRS
3A avenue de la Recherche Scientifique
OrléansFrance
Université d'Orléans
OrléansFrance
E Lorfèvre
CNES
18 Avenue Edouard Belin31400ToulouseFrance
D Plettemeier
Technische Universität Dresden
Würzburger Str. 35D-01187DresdenGermany
J Souček
Institute of Atmospheric Physics
Czech Academy of Sciences
PragueCzechia
M Steller
Space Research Institute
Austrian Academy of Sciences
GrazAustria
Š Štverák
Institute of Atmospheric Physics
Czech Academy of Sciences
PragueCzechia
Astronomical Institute of the Czech Academy of Sciences
PragueCzechia
P Trávníček
Space Sciences Laboratory
University of California
BerkeleyCAUSA
Institute of Atmospheric Physics
Czech Academy of Sciences
PragueCzechia
A Vaivads
LPC2E
CNRS
3A avenue de la Recherche Scientifique
OrléansFrance
Department of Space and Plasma Physics
School of Electrical Engineering and Computer Science
Royal Institute of Technology
StockholmSweden
T S Horbury
Space and Atmospheric Physics
The Blackett Laboratory
Imperial College of London
SW7 2AZLondonUK
H O'brien
Space and Atmospheric Physics
The Blackett Laboratory
Imperial College of London
SW7 2AZLondonUK
V Angelini
Space and Atmospheric Physics
The Blackett Laboratory
Imperial College of London
SW7 2AZLondonUK
V Evans
Space and Atmospheric Physics
The Blackett Laboratory
Imperial College of London
SW7 2AZLondonUK
Statistical study of electron density turbulence and ion-cyclotron waves in the inner heliosphere: Solar Orbiter observations
May 18, 2021 May 18, 2021Astronomy & Astrophysics manuscript no. CARBONE ©ESO 2021
Context. The recently released spacecraft potential measured by the RPW instrument on-board Solar Orbiter has been used to estimate the solar wind electron density in the inner heliosphere. Aims. Solar-wind electron density measured during June 2020 has been analysed to obtain a thorough characterization of the turbulence and intermittency properties of the fluctuations. Magnetic field data have been used to describe the presence of ion-scale waves.Methods. Selected intervals have been extracted to study and quantify the properties of turbulence. The Empirical Mode Decomposition has been used to obtain the generalized marginal Hilbert spectrum, equivalent to the structure functions analysis, and additionally reducing issues typical of non-stationary, short time series. The presence of waves was quantitatively determined introducing a parameter describing the time-dependent, frequency-filtered wave power. Results. A well defined inertial range with power-law scaling has been found almost everywhere. However, the Kolmogorov scaling and the typical intermittency effects are only present in part of the samples. Other intervals have shallower spectra and more irregular intermittency, not described by models of turbulence. These are observed predominantly during intervals of enhanced ion frequency wave activity. Comparisons with compressible magnetic field intermittency (from the MAG instrument) and with an estimate of the solar wind velocity (using electric and magnetic field) are also provided to give general context and help determine the cause for the anomalous fluctuations.
Introduction
The turbulent nature of solar wind fluctuations has been investigated for more than half a century (see, e.g. Bruno & Carbone 2016). Advances are constantly achieved thanks to the increasingly accurate measurements of several dedicated space mission, lar wind flow and its relationship with the small-scale processes still need to be described in full (Tu & Marsch 1995;Bruno & Carbone 2013;Matthaeus & Velli 2011;Chen 2016). Magnetic field fluctuations have been characterized with great detail at magnetohydrodynamic and kinetic scales, for example through spectral and high-order moments analysis (Tu & Marsch 1995;Bruno & Carbone 2013). The anisotropic nature of magnetic turbulence has also been addressed, and is still being debated, due to the limited access to three-dimensional measurements in space (see, e.g. Horbury et al. 2008;Sorriso-Valvo, L. et al. 2010;Horbury et al. 2012;Yordanova et al. 2015;Verdini et al. 2018;Oughton & Matthaeus 2020). Velocity fluctuations have been studied thoroughly (see, e.g. Sorriso-Valvo et al. 1999;Bruno & Carbone 2013), although the kinetic scales still remain quite unexplored for instrumental limitations, most notably in sampling time resolution. Both velocity and magnetic field show highly variable turbulence properties, with well developed spectra, strong intermittency (Sorriso-Valvo et al. 1999), anisotropy, and linear third-order moments scaling (Sorriso-Valvo et al. 2007;Carbone et al. 2011). The level of Alfvénic fluctuations (mostly but not exclusively found in fast streams, see e.g. D' Amicis et al. 2011;Bruno, R. et al. 2019) are believed to be associated with the state of the turbulence. In particular, solar wind samples containing more Alfvénic fluctuations are typically associated with less developed turbulence, as inferred from both shallower spectra and reduced intermittency (see Bruno & Carbone 2013, and references therein). This is consistent with the expectation that uncorrelated Alfvénic fluctuations contribute to reduce the nonlinear cascade by sweeping away the interacting structures (Dobrowolny et al. 1980), as also confirmed by the observed anticorrelation between the turbulent energy cascade rate and the cross-helicity (Smith et al. 2009;Marino et al. 2011b,a).
Conversely, density fluctuations have been only partially explored, due in part to the unavailability of high-frequency time series, and in part to their supposedly secondary relevance in the nearly incompressible solar-wind dynamics. In recent years, studies have shown that proton density is also turbulent and intermittent (Hnat et al. 2003(Hnat et al. , 2005, with the turbulence characteristics often depending of the Alfvénic nature of each specific solar wind interval. In particular, it is understood that in the nearly incompressible Alfvénic solar wind the turbulence of density fluctuations is strongly similar to that of magnetic field magnitude, both behaving like scalar quantities passively advected by velocity and magnetic components turbulence (Goldreich & Sridhar 1995;Chen et al. 2012). Conversely, in the more compressive non-Alfvénic solar wind (more typically associated with slow streams) they are actively contributing to the nonlinear transfer of energy (Schekochihin et al. 2009;Boldyrev et al. 2013). The radial evolution of proton density turbulence was examined, providing evidence of complex, unexpected behaviours (Bruno et al. 2014). Modeling of the radial evolution was attempted based on the parametric instability, expected to generate increasingly compressive fluctuations as the solar wind expands away from the sun. However, despite the analysis provided some insight, the lack of statistical description still prevents the validation of models that could provide a prediction for the radial evolution of the density fluctuations. Finally, sub-ion scale density turbulence has been performed using high-resolution proton density measurements from Spektr-R (Chen et al. 2014;Riazantseva et al. 2019) and electron density from MMS (Roberts et al. 2020), but results could not be fully conclusive about the nature of the multifractal and intermittency properties of the fluctuations (Sorriso-Valvo et al. 2017;Carbone et al. 2018). Therefore, a deeper anal-ysis of solar wind density fluctuations is necessary in order to constrain modelings and to understand the role of density fluctuations in the magnetohydrodynamic turbulent cascade. Note that, although the above studies may refer either to proton or electron density, depending on the instrument used for the measurements, at the scales of interest for turbulence the two can be safely considered equal for the quasi neutrality condition of solar wind plasma.
Recent development in electromagnetic wave payload has allowed to obtain density fluctuations from the measurement of the spacecraft potential (Pedersen 1995;Roberts et al. 2017). This gave access to higher-frequency, accurate density measurements. In particular, the Solar Orbiter spacecraft (Müller, D. et al. 2020) was launched in February 2020, equipped with both remote and in situ instrumentation aiming at investigating the sun and the solar wind from and within the inner heliospehre ). The Radio and Plasma Waves (RPW) instrument (Maksimovic, M. et al. 2020) measures the spacecraft potential with unprecedented accuracy, allowing the estimation of the high-cadence solar wind electron density . Together with the enhanced data quality provided by Solar Orbiter instruments, novel data analysis techniques are emerging that improve the delicate measurement of turbulence parameters in solar wind data. Among these, the Empirical Mode Decomposition (EMD, Huang et al. 2008) has been successfully used to mitigate short-sample and large-scale structures effects (Carbone et al. 2018;Alberti et al. 2019). This paper aims at providing the first description of the properties of turbulence as obtained applying EMD-based analysis techniques to the Solar Orbiter RPW density measurements taken during the month of June 2020 .
The results are discussed in relation to solar wind parameters and magnetic field turbulence, studied applying the same EMD-based techniques to the MAG instrument (Horbury, T. S. et al. 2020). Moreover, after introducing a novel parameter to describe the presence of ion-scale waves, the relationship between the properties of turbulence and such waves has been analyzed.
Section 2 provides a description of the data used for the analysis. In Section 3 the techniques used for the analysis of turbulent fluctuations and the corresponding results are described. In Section 4 the wave parameter is introduced and the presence of ion-scale waves is discussed. Section 5 gives a detailed discussion on the existing correlations between solar wind, turbulence and wave parameters and their implications. Finally, conclusions are summarized in Section 6.
Description of data
In order to study the properties of turbulence of solar wind density fluctuations, we make use of Solar Orbiter measurements taken from 7 to 29 June 2020, when the spacecraft was orbiting the Sun between 0.52 AU and 0.55 AU. During that time interval, the Solar Wind Analyser (SWA) plasma instruments (Owen, C. J. et al. 2020) were not operational, so that direct measurements of proton and electron moments are not available. Here, we use the 16 Hz electron density n e , accurately estimated from the RPW probe-to-spacecraft potential measurements (Maksimovic, M. et al. 2020). The equilibrium floating potential of the spacecraft is reached when the current due to photo-electrons emitted from the spacecraft is balanced by the plasma current collected by the spacecraft. The equilibrium is reached instantaneously (0.1-1 ms) on the time scales of interest for the turbulence studies. From the current balance we can find that the electron density has an exponential relation to the spacecraft po-tential. Then making an exponential fit of the spacecraft potential to the electron density obtained from the high-frequency measurement of plasma quasi-thermal noise one can find the relation for approximate conversion of the probe-to-spacecraft potential to electron density. Details on the density estimation technique are given in Khotyaintsev, Y. et al. (2021).
Localized estimates of the solar wind speed from the deHoffmann-Teller (HT) analysis of electromagnetic fluctuations are used to provide general context, most notably discriminating between fast and slow solar wind streams. HT analysis is used to find the velocity of the frame in which the electric field is zero. In the solar wind, where current sheets and MHD turbulence are ubiquitous, the HT velocity is in general a good estimate of the solar wind velocity. In order to get reasonable coverage, estimates of the solar wind speed V sw were obtained every ten minutes by applying the HT analysis on one-hour running intervals of E and B data. Details about the HT analysis can be found in Steinvall et al. (2021).
Finally, magnetic field vector B measurements from the magnetometer (MAG) (Horbury, T. S. et al. 2020) are studied for comparison.
Due to the presence of large gaps in the electron density data, and to avoid strong violation of homogeneity and stationarity, a number of intervals of variable length were extracted from the electron density time series n e (t). All intervals were chosen as relatively stationary, covering at least 1 hour data, and could only include a few data gaps shorter than 5 seconds. The remaining missing points have been interpolated linearly. Such criteria resulted in the selection of 36 sub-intervals, whose list and macroscopic details (starting and ending time, interval duration, mean distance from the Sun, estimated mean solar wind bulk speed, and the angle between the ambient magnetic field and the radial direction averaged over each interval) are given in Table 1. Given the estimated solar wind speed, the Taylor hypothesis (Taylor 1938) is considered valid throughout the whole data set, and allows to interpret the time series as equivalent to a space ensemble, therefore enabling the standard tools for turbulence analysis. Figure 1 shows an overview of the above parameters for the whole month of June 2020. The overall solar wind conditions are variable, with alternation of fast and slow streams, as well as complex coronal mass ejection structures (as studied in detail by Telloni et al. 2021). However, the selected sub-intervals, whose duration is between one and six hours, are typically embedded in homogeneous solar wind conditions.
Analysis of solar wind electron density turbulence
The properties of turbulent fields are usually studied through standard statistical analysis techniques. Among others, these may include (Dudok de Wit et al. 2013): the autocorrelation function, used to determine specific scales of the data; the power spectral density E( f ), providing information on the self-similar energy redistribution among scales; and the structure functions (SF) S q ( t ) ≡ [n e (t + t ) − n e (t)] q ∼ ζ(q) t , with t representing the time scale of the field increments, whose anomalous scaling exponents ζ(q) are able to quantitatively describe the effects of intermittency, namely the inhomogeneous nature of the turbulent cascade (Frisch 1995). Related to the structure functions, the kurtosis of the distribution function of the fluctuations K( t ) = S 4 ( t )/S 2 2 ( t ) is often used to quantify the intermittency effects. However, such techniques may be sensitive to the data sample characteristics, resulting in undesired effects not attributed to the turbulent energy cascade, but rather due, for example, to finite sample size, limited stationarity, or presence of superposed structures larger than the typical turbulence scales (for example non-turbulence related current sheets or velocity shears). Such issues are often occurring in ecliptic solar wind intervals, where instrumental performance and the intrinsic wind variability may prevent ideal experimental conditions for the study of turbulence (Carbone et al. 2018). The magnification shown in the top two panels Figure 2 shows an example of selected intervals (specifically, sub-intervals 26 and 27, both on June 20). The presence of large-scale modulation of the density profile, sporadic sharp gradients similar to ramp-cliff structures, and the general lack of strict stationarity appear evident (Matthaeus & Goldstein 1982;Perri & Balogh 2010).
In order to mitigate the effects of such large-scale features, we make use of the Hilbert-Huang transform to obtain more precise estimators of the generalized high-order spectra.
Empirical Mode Decomposition
The technique used here is based on the Empirical Mode Decomposition (EMD) (Huang et al. 1998;Jánosi & Müller 2005;Carbone et al. 2016b). This is a self-consistent, data-driven projection of a time series (as in this case the solar wind electron density n e (t)) on a finite number n of empirical basis functions φ j (t), called intrinsic mode functions (IMFs), so that n e (t) = n j=1 φ j (t) + r n (t) .
(1)
The additive residual function r n (t) describes the mean trend. Each IMF can be characterized by the instantaneous timescale τ j (t). The decomposition is based on a recursive procedure, consisting of two main stages (Huang et al. 1998): (i) the local extrema of n e (t) are interpolated through cubic spline to provide superior and inferior envelops of the time series; and (ii) the average between the two envelops is subtracted from the original data. The resulting field is accepted as an IMF if it satisfies the following specific criteria: the number of local extrema and zero crossings does not differ by more than one, and the average between the IMF superior and inferior envelopes is zero at all times. Otherwise, the procedure is repeated on the remaining IMF until the criteria are met according to the so called 3thresholds stoppage criterion (with the following standard choice of parameters: δ = 0.05, ξ 1 = 0.05, and ξ 2 = 10ξ 1 , see Rilling et al. 2003). Some examples of IMFs and the associated residual, as obtained trough the above EMD decomposition of the solar wind electron density n e , are shown in the left panels of Figure 3 for sample 1. For clarity, only odd IMFs have been plotted. Although the instantaneous frequency of the modes is variable, the figure highlights that each mode j has a characteristic narrow frequency band, so that a mean period τ j and an associated variance Var(φ j ) can be properly defined. Indeed, EMD acts intrinsically as a dyadic filter bank (Wu & Huang 2004;Huang & Shen 2005), each IMF effectively capturing a narrow frequency band. However, the general features of the various IMFs depend on the specific process under analysis. For a turbulent field (Huang et al. 2008;Carbone et al. 2016aCarbone et al. , 2018Carbone et al. , 2020a or for a multifractal process Sorriso-Valvo et al. 2017), the characteristic mean period grows exponentially as τ j = αγ j , where · represents an ensemble average (in this case, average over time). The basis γ can be evaluated empirically from the IMFs. For an exact dyadic decomposition γ = 2. Additionally, the variance of the IMF scales as a power of the mean timescale, Var(φ j ) ∼ τ j 2H . The scaling exponent H is the Hurst number, a parameter describing the persistence or anti-persistence of the fluctuations of the process under analysis (Nava et al. 2016;Carbone et al. 2019).
The scaling of the mean period for sample 1 is shown in the central panel of Figure 3. The value γ = 1.88 ± 0.11 obtained through a least-square exponential fit is compatible with the expected value for a dyadic decomposition, showing that EMD is correctly decomposing the data. Moreover, the Hurst number H = 0.348 ± 0.04 is obtained from the regression of the IMFs variance versus the average period, as shown in the right panel of Figure 3. In this example, H is compatible with the standard value for classical fluid or magnetic turbulence H = 0.37 (Benzi et al. 1993;Arneodo et al. 1996;Bruno & Carbone 2016), and indicates persistence of the fluctuations typical of intermittency.
Using the above procedure, the Hurst number has been evaluated for all 36 intervals. The results, listed in Table A.1, show some variability that will be discussed and compared to other parameters in Section 5.
Arbitrary Order Hilbert Spectral analysis
The Hilbert Spectral Analysis (HSA) is an extension of the basic EMD designed to characterize scale-invariant properties directly in the amplitude-frequency space (Huang et al. 2008). It provides the equivalent of the power spectral density and high order moments of a field fluctuations (the structure functions), therefore representing a viable alternative to those standard tools. After decomposing the field under study in its IMFs, the Hilbert trans-form of each mode is computed as:
φ j (t) = 1 π P +∞ −∞ φ j (t ) t − t(2)
where P is the Cauchy principle value. The Hilbert representation allows to extract a time-dependent instantaneous frequency f j (t) ≡ τ −1 j (t) and a time-dependent amplitude modulation A j (t), by constructing the so called analytical signal (Cohen 1995).
Φ j (t) = φ j (t) + iφ j (t) ≡ A j (t)e iθ j (t)Here A j (t) = |Φ j (t)| ≡ φ 2 j (t) + φ j 2 (t) and θ j (t) = arctan[φ j (t)/φ j (t)]
are the instantaneous amplitude modulation and the instantaneous phase oscillation, respectively (the instantaneous frequency being defined as f j (t) = 2π −1 dθ j (t)/dt) (Long et al. 1995;Cohen 1995;Flandrin 1999). After rewriting the original signal in terms of A j and θ j , n e = Real j A j (t) exp i f i (t)dt , the energy as a function of the instantaneous frequency f and time, can be defined as
h( f ) = ∞ 0 H( f, t)dt, or the marginal integration of the Hilbert spectrum H( f, t) ≡ A 2 ( f, t) (
being H( f, t) a representation of the original signal at the local level) (Huang et al. 1998(Huang et al. , 2009). An equivalent definition of H( f, t) can be obtained from the joint probability density function of the instantaneous frequency and amplitude P( f j , A j ) (Long et al. 1995), extracted from the IMFs. In this case, the Hilbert marginal spectrum is the second statistical moment of such distribution, analog to the Fourier spectral energy density: The bottom plot (red line) represents the residual r n (t). For better readability, only odd IMFs have been plotted. Central panel: average timescale τ j of each IMF of sample 1 as a function of the mode j. Error bars represent the 95% confidence bounds. The dashed line is a least square fit obtained from the relation τ j = αγ j . Right panel: IMF variance Var(φ j ) as a function of the average period τ j for sample 1, with the associated 95% confidence bounds. The dashed line represents the relation Var(φ j ) ∼ τ j 2H , being H the Hurst number (see the list for all intervals in Table A.1).
h( f ) ≡ L 2 ( f ) = ∞ 0 P( f, A)A 2 dA .(3)
Article number, page 5 of 17 A&A proofs: manuscript no. CARBONE Table 1. List of 36 intervals selected for this work. For each interval, the list includes initial and final time, duration ∆W, estimated deHoffmann-Teller solar wind speed V sw , average angle between the magnetic field and the radial direction θ vb , and the identified group (see Section 3.3 The above definition can be then generalized to any arbitrary moment q ≥ 0, representing the analogous of the standard structure functions of the fluctuations:
L q ( f ) = ∞ 0 P( f, A)A q dA .(4)
Equations (3-4) are used here to estimate the spectral and intermittency properties of the electron density turbulence of the 36 sub-intervals of this study. The left panel of Figure 4 shows one example of the equivalent spectrum L 2 ( f ) (blue circles), obtained through the HSA described above, for sample 1. For comparison, the classical power spectral density E( f ) (red line), evaluated through the Welch's method (Welch 1967), and the second-order structure function S 2 ( −1 t ) are also shown. Both spectra clearly display power-law scaling The HSA equivalent spectrum L 2 ( f ) displays a slightly better power-law scaling than the traditional Fourier spectral density, which has some weak amplitude modulation. Thanks to the local nature of the EMD analysis, the sources of such modulation can be removed, allowing to obtain a more precise determination of the spectral scaling exponents. This corresponds to isolating the properties of the turbulent cascade from the possible effects of the instrumental noise, and of the larger-scale energy inhomogeneity (Huang et al. 2010;Carbone et al. 2018;. Note that the power-law scaling range can vary for the various samples, but always includes at least one decade of scales. The two spectral estimators are not always nicely superposing, with the Fourier spectrum occasionally presenting stronger modulations (not shown). This suggests the possible presence of large-scale modulations, which may affect the Fourier spectra but are well controlled by the HSA. In order to achieve robust estimate, the scaling exponent of the q-th Hilbert spectrum L q ( f ) was evaluated via the residual resampling (bootstrapping) procedure (Bradley & Robert 1994;Carbone et al. 2020b). First, a least square fit is performed on each L q ( f ), then the residuals are randomly resampled and added to the fit, generating a new data-set (replica). The replica is then fitted again, and the procedure is repeated for a number of times, in this case N boot = 10 4 (Boos & Stefanski 2010;Wilcox 2010). Such large number of replica is necessary for correctly evaluating confidence or prediction interval, wile in general simple statistical tests require a smaller number (N boot ∈ [50, 100]) (Dale L. et al. 2012). The probability distribution of the exponents β q (or the scaling exponents β q − 1) obtained from the N boot least square fits is finally used to estimate the 50-th percentile (median of the distribution), used as the best estimate of the exponents, and the statistical error (95% confidence interval) (Efron et al. 2015;Wilcox 2010). Figure 5 shows two examples of the distribution P(β 2 ) estimated for samples 1 and 9 using the bootstrapping technique. The red dotted vertical line represents the median, and the black dashed lines represents the statistical error. In the examples shown in the figure, the median of the distribution provides the scaling exponent β 2 = 1.72 ± 0.14 for sample 1 (the error indicating the 95% confidence interval), in excellent agreement with the classical scaling for the fully developed hydrodynamic turbulence β 2 ≈ 1.7 (Benzi et al. 1993), and a shallower β 2 = 1.51 ± 0.10 for sample 9, consistent with the Iroshnikov-Kraichnan spectrum for Alfvénic turbulence (Iroshnikov 1964;Kraichnan 1965). Values of β 2 were obtained for all 36 samples. A discussion about these values, collected in Table A.1, will be provided in next Subsection and in Section 5.
E( f ), L 2 ( f ) ∼ f −β 2 , in the frequency range f ∈ [3 × 10 −3 , 10 −1 ],
Scaling exponents and intermittency analysis
In the framework of the standard Kolmogorov turbulence, a direct link exists between the Fourier spectral exponent E ∼ f −β and the scaling exponent of the second-order structure function, S 2 ∼ ζ(2) t , so that β − 1 = ζ(2). This relationship can be extended to any moment order q of the generalized Hilbert spectra L q ∼ f −β q , yielding the generalized scaling exponents ξ(q) ≡ β q − 1 (Huang et al. 2010;Carbone et al. 2016a). These are the analogous of the scaling exponents ζ(q) obtained using the standard structure functions (Frisch 1995;Benzi et al. 1993;Arneodo et al. 1996), and can be used to retrieve quantitative information on the properties of turbulence. Additionally, the structure function scaling exponents are linked to the Hurst number via the relation ζ(q) = qH (in absence of intermittency corrections). This allows an alternative estimate of the Hurst number using, for example, the first-order exponent H = β 1 − 1 ≡ ξ(1).
The central and right panels of Figure 4 show two examples of L q ( f ) (for intervals 1 and 9), obtained from Equation 4 up to the 5th order, for the electron density n e in samples 1 and 9. All curves presents good power-law scaling for all orders, approximately in the inertial range of frequencies (shaded areas). The associated generalized Hilbert spectra scaling exponents β q , and hence the equivalent structure-function scaling exponents ξ(q), were obtained through the bootstrapping procedure described above and were used to determine the intermittency properties of the electron density. In order to check the quality of the procedure, the first-order exponents were initially used to obtain the alternative estimate of the Hurst number. For the example of sample 1, the value H = 0.32 ± 0.06 was obtained, in good agreement with the value obtained through the regression of the IMF variance versus the average timescale, illustrated in Figure 3. This was consistently observed for all the 36 samples. The power-law exponents of L q ( f ) are visibly different in the two intervals shown in the central and right paneThe power-law exponents of L q ( f ) are visibly different in the two intervals shown in the central and right panels of Figure 4. ls of Figure 4. For q = 2, this can be also more quantitatively noticed by comparing the distributions and median values shown in Figure 5 for the same intervals.
The scaling exponents ξ(q) are shown in the top-and bottomleft panels of Figure 6 for most of the samples, separated in two groups as will be described in the following. The scaling exponents for the magnetic field magnitude |B| (central panels) and radial component B r (right panels) are shown for comparison. For all cases included in the figure, the curvature of the exponents with respect to the linear prediction of the nonintermittent Kolmogorov phenomenology is evident. This universal behaviour of turbulence is due to the effects of the intermittency, or anomalous dissipation (Kolmogorov 1962;Schmitt et al. 1994;Schmitt 2003;Bruno & Carbone 2016;Carbone et al. 2019), and are related to the multifractal nature of the turbulent cascade (Meneveau & Sreenivasan 1991;Davis et al. 1994;Sorriso-Valvo et al. 2017). In all panels, the scaling exponents from a classical measure of fluid intermittent turbulence are also shown for comparison Benzi et al. (1993). In order to describe their intermittency behaviour, the 36 samples were then separated in three groups, according to the behaviour of the generalized scaling exponents and of the Hurst number, with respect to the standard fluid turbulence reference. Note that, for each sample, the same behaviour is consistently observed for density and magnetic field.
The first group (group 1, including 21 intervals) displays the typical statistical features of fully developed turbulence (top panels of Figure 6). For these samples, the scaling exponents are consistent with the reference values from fluid turbulence, and are well described (not shown) by models of intermittent turbulence (e.g. the p-model by Meneveau & Sreenivasan 1991, not shown). In particular, the equivalent spectral exponent β 2 > 1.55 is always compatible with the Kolmogorov scaling (β 2 = 5/3). Furthermore, for samples in this group H ∈ [0.30, 0.39], compatible with the classical value obtained for ordinary fluid turbulence H = 0.37. From these observations, we conclude that the samples in group 1 are characterized by a standard turbulence, with the expected power-law spectra, presence of small-scale intermittent structures and anti-persistent fluctuations. (Welch 1967) E( f ) (red line), and the secondorder structure function S 2 ( t ) (green squares, plotted as a function of the inverse timescale t ), for sample 1. Power-scaling is present in the same frequency range for all methods. The dashed line represents the classical Kolmogorov scaling L 2 ( f ) ∼ f −5/3 , while the dotted line represent the expected scaling for the second-order structure function S 2 ( t ) ∼ ζ(2) t (ζ(2) = β 2 −1). Central panel: generalized Hilbert spectra L q ( f ) for q ∈ [1, 5], obtained for sample 1. The curves have been vertically shifted for clarity. The shaded area represents the frequency range of the bootstrapping least-square fit. Right panel: same as in the central panel, for sample 9. The power behavior is still present, but the power-law exponents are considerably different.
In the second group (group 2, including 11 samples), the scaling exponents are characterized by more extreme deviation from the linear prediction, and by much smaller values (bottom panels of Figure 6). These exponents also deviate considerably from the fluid reference, and their order dependence cannot be described by standard models of turbulence (not shown). The equivalent spectral exponents β 2 < 1.55 are shallower than the Kolmogorov scaling, and in some cases are compatible with the Iroshnikov-Kraichnan scaling. The Hurst number is also consistently smaller than for standard turbulence, H ∈ [0.16, 0.26]. These observations suggest that, unlike in group 1, the density and magnetic fluctuations in these samples may not be generated by a standard turbulent cascade. Some other processes might coexist, modifying the statistics. Note that the EMD-based analysis constrains the effects of finite-size sample, poor stationarity and large-scale structures effects. Therefore, it can be claimed that the observed features might be related to the presence of smallor inertial-scale fluctuations that are not generated only by a turbulent cascade.
The third group (group 3, not shown) includes 4 samples that do not show clear power-law scaling of the generalized Hilbert spectra for all orders, so that not all the scaling exponents are available. These intervals, associated with small Hurst number and spectral exponent, are therefore not representative of turbulence.
The multifractal nature of the fluctuations can be quantitatively described fitting the scaling exponents ξ(q) to a log-normal model (Schmitt 2003;Medina et al. 2015):
ξ(q) = qH − µ 2 q 2 − q .(5)
The model is able to describe standard intermittent turbulence when the curvature parameter µ ≈ 0.02. For other multifractal processes, not generated by a nonlinear turbulent cascade, different values can be obtained. One example of log-normal model fit of the equivalent scaling exponents ξ(q) is shown in Figure 7 for group 1 (sample 10, red circles), giving µ = 0.019 ± 0.004. The model was also fitted to the exponents obtained ensemble-averaging all samples of group 1 (stars), providing µ = 0.023 ± 0.002. Alternatively, the average parameter computed using the results of the fit of all samples of group 1 is µ 1 = 0.028 ± 0.010 (the error representing the standard deviation). All the above values are in good agreement with those observed for standard turbulence. Similar values were obtained for the magnetic field magnitude |B| (e.g., µ 1 = 0.027 ± 0.010). The exponents for group 2 were also fitted to relation 5. In that case, the resulting parameters were generally smaller, with average µ 2 = 0.013 ± 0.01. The parameters from all 36 intervals are plotted in one of the panels of Figure 9 and will be discussed in Section 5. While there is a considerable spread in both groups, the parameters for group 2 appear generally smaller, confirming that the fluctuations have peculiar, strongly multifractal structures that do not simply originate from a nonlinear turbulent cascade.
Finally, in order to include also one example of standard data analysis technique for intermittent turbulence, we have estimated the kurtosis of the fluctuation distribution, namely the ratio between the fourth-order and the squared second-order structure functions K( t ) = S 4 /S 2 2 ∼ −κ t (Frisch 1995;Dudok de Wit et al. 2013). The kurtosis provides information on the shape of the distribution of the scale-dependent fluctuations. At large scales (comparable with the system correlation scale) the Gaussian value K = 3 is typically observed. As the scale decreases, the inhomogeneous turbulent cascade generates intermittent structures, so that the distribution deviates from Gaussian, corresponding to increasing K. The scaling properties of turbulence result in the power-law scaling of K in the inertial range. The scaling exponent κ is a good measure of the efficiency of the cascade, namely how rapidly the small-scale structures are generated. This depends on the nature of the nonlinear interactions and can be used as a quantitative measure of intermittency (Castaing et al. 1990;Carbone & Sorriso-Valvo 2014). The kurtosis was estimated for all intervals, and a power-law fit was performed whenever a long enough scaling range was observed. An example of K with power-law fit is shown in Figure 8. The resulting fitting parameters κ are collected in Table A
Observation of ion cyclotron waves
Observations of solar wind data often reveals the presence of wave activity near the end of the MHD inertial cascade range, and close to the kinetic plasma range. These are typically identified as kinetic Alfvén waves (KAW), or ion-cyclotron waves (ICW), among other modes (see e.g. Bale et al. 2005;Kiyani et al. 2012).
In order to explore the relationship between the observation of ion-scale waves and the properties of the inertial range turbulence, we have introduced a quantity that enables the identification of waves and quantitatively assesses their presence in the time series. The technique for the identification of the waves is described in detail in Khotyaintsev, Y. et al. (2021), where it has been used to determine that the observed fluctuations are most likely ion cyclotron waves. A brief description of the technique is given in the following. The first step is to rotate the magnetic field into the field-aligned coordinates using B low-pass filtered at 0.01 Hz as the background magnetic field. The powerspectrum of the resulting transverse component, δB ⊥ , is shown in the third panels from the top of Figure 2. Subsequently, the coherence between the two perpendicular magnetic field components is computed (fourth panel). This will have large values if circularly-polarized ion-scale waves are present. If the coherence is larger than an arbitrary threshold (0.65) in a frequency range near or below the proton gyro-frequency, the phase angle between the two perpendicular magnetic components is also computed (fifth panel), allowing to determine the fluctuations handedness; as we are interested in circularly-polarized waves, we exclude the phases outside the intervals +90 • ± 45 • (righthanded waves) and −90 • ± 45 • (left-handed waves).
Using the above indicators, it is therefore possible to unambiguously identify regions with circularly-polarized wave activity both in time and frequency. The perpendicular magnetic power is finally integrated in the identified wave intervals and frequency band (with lower and higher frequencies f 1 and f 2 respectively to be identified according to the above criteria), namely within the wave patches clearly visible in the phase angle plot. This procedure provides the time series of one frequencyintegrated local parameter Q w = f 1 f 2 δB 2 ⊥ d f , defined for each data point in the time series, indicating the total power associated to the wave-like fluctuations. Note that the parameter is not computed outside of the wave patches. The bottom panels of Figure 2 shows two examples of wave parameter Q w for samples 26 and 27. Interval 26 (left panel) with highly irregular and intermittent behaviour, capturing the corresponding wave patches observed in the scalograms. In the adjacent interval 27 (right panel), waves are nearly absent, and accordingly the wave parameter values are negligible.
Finally, using the time series of frequency-integrated wave power, two similar global parameters can be computed to quantitatively assess the occurrence of waves within each interval. The first one is simply obtained as the time-integrated power Q w = Q w dt, the integration being intended over each interval. The second one is the average over the interval Q w =Q w /∆W, taking into account the density of waves within each interval. The obtained values are listed in Table A.1 for all intervals. In some occasions, when no waves were identified, the parameters were set to 0. The wave parametersQ w and Q w will be used in Section 5 to determine possible correlations with the turbulence parameters.
Discussion
Once the turbulent properties of the fluctuations and the presence of ion-scale waves have been quantitatively assessed, it is possible to investigate correlations between the two phenomena. This may help understanding the dynamical processes of solar wind plasmas, and in particular the cross-scale coupling between fluid and sub-ion processes. Using the results of the analysis for the 36 samples, correlation coefficients have been computed between pairs of parameters of solar wind (V sw and θ vb ), turbulence (Hurst number, spectral exponent, kurtosis and intermittency), and waves (the two estimators presented in Section 4). Both linear (Pearson) and nonlinear (Spearman) coefficients have been computed. For each pair of parameters, the largest of the two has been considered. The complete list of coefficients for all pairs is presented in Table A for velocity fluctuations in the inertial range of hydrodynamic turbulence (red squares) are shown for reference (Benzi et al. 1993). The dashed line represents the theoretical expectation ζ(q) = q/3 (Kolmogorov 1941). Top-center and top-right panels: equivalent scaling exponents for the magnetic field radial component B r and magnitude |B|, for group 1. Bottom panels: same as the top panels, for the intervals of group 2.
related to different scaling exponents of the same field, or the two wave parameters. Others display known correlations, associated with the nature of the solar wind intervals. However, despite the high variability of the parameters and the experimental conditions, some non-trivially related pairs show moderate, non-negligible correlation. These are highlighted in bold in Table A.2. The most interesting correlation was found between the wave indicators (Q w and Q w ) and the intermittency parameters (µ and κ). For example, C( Q w , µ) = −0.5 indicates that intervals with substantial presence of ion-scale waves are likely to show reduced intermittency. This observation clearly highlights the link between the characteristics of the fluid-scale turbulent cascade to the excitation of waves at ion scales. A more visual description is provided by the scatter plots of pairs of parameters listed in Tables 1 and A.1, shown in Figure 9 for some pairs of parameters from Table A.2. In all panels, the samples are color-coded according to their group as determined in Section 3.1 (group 1: blue circles; group 2: red squares; group 3: green triangles). Whenever relevant, vertical or horizontal lines indicate typical value of the parameters for standard fluid turbulence. The top-left panel, plotting the wind speed and the density Hurst number, highlights the clear separation between group 1 (mostly large Hurst number) and groups 2 and 3 (smaller Hurst number). Additionally, it clearly shows that while intervals of group 1 belong to both fast and slow wind, nearly all intervals of groups 2 and 3 (with one single exception) belong to faster solar wind. It is worth noting that the flow speed does not necessarily act as an ordering parameter for Alfvénicity, though slow wind is generally less Alfvénic than fast wind (see for example D' Amicis et al. 2021, and references therein). Specifically, fast wind can exhibit different levels of Alfvénicity. In this respect, the top-left panel suggests that the lack of Alfvénicity in the slow wind assures a more developed turbulence, while possible enhancements in Alfvénic nature of the fluctuations in some (though not all) fast wind samples may prevent plasma from fully developing into a turbulent state. It turns out that in the samples studied here the fast wind can include both standard and reduced turbulence intervals (e.g. with shallower spectra), depending on the corresponding level of Alfvénic fluctuations. We remind that while solar wind turbulence is likely strongly driven by Alfvénic fluctuations, these need to include both counterpropagating modes in order to effectively generate nonlinear interactions. On the other hand, if the fluctuations are unbalanced, with one mode prevailing over the other (typically resulting in a definite sign large cross-helicity), then the sweeping effect strongly reduces the nonlinear interactions, resulting in weaker turbulence and shallower spectra (Dobrowolny et al. 1980).
The top-center panel of Figure 9 shows that the angle between the magnetic field and the radial direction (approximately corresponding to the velocity vector and in turn to the sampling direction, at such distances) is also relevant to the turbulence. 10 -1 10 0 10 1 10 2 10 3 10 0 10 1 Fig. 8. Scale-dependent kurtosis K( t ) for samples 26 (group 1) and 27 (group 2). Power-law fits K( t ) ∼ −κ t give the indicated exponents κ. The Gaussian value K = 3 is also shown (horizontal dotted line).
In particular, for group 1 intervals the Kolmogorov spectrum (dashed line) is observed at all angles. On the contrary, groups 2 and 3, characterized by a shallower spectrum, only include intervals with nearly radial field. This suggests that during intervals belonging to groups 2 and 3, Solar Orbiter sampled parallel fluctuations (namely the slab component of turbulence), which are generally less evolved and likely more Alfvénic with respect to 2D turbulence. Interestingly, this result is in contrast with critical balance theory (Goldreich & Sridhar 1995;, which predicts a steeper spectrum (with a scaling close to −2) for parallel fluctuations (an interesting discussion on the validity and relevance of critical balance in solar wind turbulence is provided in Oughton & Matthaeus 2020).
The top-right panel of Figure 9 shows that strong intermittency (large kurtosis) is mostly observed in intervals with quasiperpendicular field. Additionally, it highlights the good correlation existing between the angle and the intermittency exponent κ, demonstrating that θ vb is a good ordering parameter for intermittency. As mentioned above, in the studied intervals the solar wind plasma is likely to be more Alfvénic at quasiparallel angles, where the turbulence is only poorly developed. The stochastic nature of the Alfvénic fluctuations tends to reduce the intermittency, which is indeed lower at larger angles. On the other hand, at quasi-perpendicular angles, where the turbulence is more fully developed (possibly in association with reduced Alfvénic fluctuations), the mitigating effect of Alfvénicity is lower and the coherent structures advected by the wind tend to emerge, resulting in the observed increasing intermittency.
The bottom-left panel of Figure 9 shows the strong overall correlation between the spectral exponents of electron density and magnetic field magnitude. For the intervals of group 1, spectral exponents of both fields are mostly consistent with the standard Kolmogorov value. On the contrary, the evident linear correlation for the more variable exponents of groups 2 and 3 strongly suggests the Alfvénic nature of the fluctuations, with well correlated compressive magnetic magnitude and plasma density fluctuations. It is indeed worth reminding that density behaves as a passive scalar (it reproduces the magnetic field magnitude characteristics) only in the Alfvénic solar wind, where the contribution of compressive fluctuations is negligible. In this perspective, for intervals in groups 2 and 3 the plasma density can be considered as a proxy of the magnetic field for the turbulent properties (spectral scaling, intermittency, etc.).
In the bottom-center panel of Figure 9, the correlation between the angle and the wave density parameter is shown. In this case, no clear separation between the three groups is observed. However, it is evident that for intervals with perpendicular field the wave density is always small (note that 5 intervals of group 1 for which Q w = 0 have been artificially represented on the logarithmic vertical axis by the open blue circles at 0.1). The four intervals of group 3 are also characterized by large presence of waves. This is in good agreement with the expectations. Indeed, the presence of KAWs (at quasi-perpendicular angles) and ICWs (at quasi-parallel angles) strictly depends on the presence of Alfvénic fluctuations at fluid scales. Larger Alfvénicity is associated with enhanced presence of waves at ion scales, as first shown by Bruno & Telloni (2015) (see Fig. 2 therein) on a single case study, and then corroborated on a statistical data set by (see Fig. 3c therein).
Finally, the bottom-right panel of Figure 9 shows the correlation between the intermittency scaling exponent κ and the total wave power Q w . For group 1 intervals with no waves, the same representation as in the bottom-center panel has been adopted. The observed correlation is a very interesting result. Indeed, despite the scattered plot, a general trend is evident: stronger intermittency intervals have less wave activity. This is in striking agreement with a scenario in which for higher the Alfvénic fluctuations (which implies a lower intermittency), the presence of waves at ion scales (and, in turn, the related measured energy) is larger. This has been very recently validated by the statistical work by .
The overall conclusion gained from the examination of Figure 9 is that 2D fluctuations (fluctuations sampled at quasiperpendicular angles with the magnetic field) are always characterized by strong Kolmogorov turbulence, strong intermittency and absence of wave activity. A a less Alfvénic content is also suggested. As discussed above, all these fluid and kinetic characteristics are strictly related to each other. On the other hand, slab fluctuations (fluctuations sampled at quasi-radial directions) are associated with less developed turbulence (smaller spectral exponent and intermittency parameters) and stronger ion-scale wave activity. A higher Alfvénic content (which acts to make less efficient the nonlinear interactions) can be inferred for these intervals.
In this respect, the speed of the solar wind flows does not seem to be an order parameter, Alfvénicity being a more suitable one.
Conclusions
We have presented the first analysis of turbulence and intermittency of the solar wind electron density measured by the RPW instrument on-board the Solar Orbiter spacecraft. 36 intervals were selected during the month of June 2020, when Solar Orbiter was located in the inner heliosphere, approximately at 0.5 AU from the Sun. The study was performed using standard analysis techniques as well as the Empirical Mode Decomposition. It was found that the intervals could be separated in three groups, according to their agreement with standard turbulence parameters. The distinction was attributed to the different level of Alfvénic fluctuations, which in the absence of plasma measurements was inferred from the turbulence characteristics. Using the magnetic field measurements from the MAG instrument, the presence of ion-scale waves has been detected through coherence analysis. These waves have been identified as being mostly ion cyclotron waves. A new parameter has been introduced to determine quantitatively the energy associated to waves in a given frequency range. Making use of such parameter, two estimators were introduced to assess the overall wave activity within each interval. Comparing these wave parameters with the turbulence indicators, it has been found that intervals with enhanced presence of waves are also characterized by anomalous turbulence and weaker intermittency of the solar wind density. While the study of statistical correlations cannot determine the causality relation between the phenomena, it definitely provides constrains and, in this specific case, can help understanding the cross-scale connection between the fluid scales and the ion-scales. The possible role of Alfvénic fluctuations in driving both the observed ion cyclotron waves and the reduced turbulence and intermittency was highlighted.
The preliminary results described in this paper demonstrate the unprecedented high quality of the Solar Orbiter RPW electron density data estimated using the probe-to-spacecraft potential. The excellent performances of the EMD-based analysis allowed the accurate determination of turbulence parameters. The results described here represent the most detailed description of turbulence of solar wind density fluctuations so far. While these results are generally in line with previous observations at 1 AU, future studies of Solar Orbiter measurements will finally allow us to determine the radial evolution of the properties of density turbulence.
Furthermore, the novel parameter providing quantitative assessment of the presence of waves helped identify an important relationship between fluid-scale turbulence and ion-scale phenomena in the solar wind plasma. The future study of measure-ments including magnetic field and velocity fluctuations, and more extended statistical analysis could provide a deeper understanding of such relationship.
Acknowledgements. Solar Orbiter is a space mission of international collaboration between ESA and NASA, operated by ESA. We thank the entire Solar Orbiter team and instrument PIs for data access and support. Solar Orbiter data are available at http://soar.esac.esa.int/soar. The RPW instrument has been designed and funded by CNES, CNRS, the Paris Observatory, The Swedish National Space Agency, ESA-PRODEX and all the participating institutes. Swedish co-authors are supported by the Swedish Research Council, grant 2016-05507, and Swedish National Space Agency grant 20/136. CNES and CDPP are acknowledged for the support to the French co-authors. Solar Orbiter magnetometer operations are funded by the UK Space Agency (grant ST/T001062/1). TSH is supported by STFC grant ST/S000364/1. FC acknowledges the contribution received from EU-H2020 program ERA-PLANET through the project "Integrated Global Observing Systems for Persistent Pollutants" (iGOSP) (Grant Agreement: 689443), funded under H2020-SC5-15-2015 "Strengthening the European Research Area in the domain of Earth Observation", from FET Proactive project "Towards new frontiers for distributed environmental monitoring based on an ecosystem of plant seed-like soft robots" (I-Seed), funded under Horizon 2020 research and innovation programme (Grant agreement: 101017940), and from EU-H2020 project "EuroGEO Showcases: Applications Powered by Europe" (e-shape) (Grant Agreement: 820852), funded under H2020-SC5-2018-2 "Strengthening the benefits for Europe of the Global Earth Observation System of Systems (GEOSS) -establishing EuroGEO". LSV was funded by the Swedish Contingency Agency grant 2016-2102 and by SNSA grant 86/20. CLV was partially supported by EPN project PIM-19-01. Table 1 and A.1). Blue, red and green points indicate intervals of group 1, 2 and 3 respectively. Blue open circles in the bottom central and left panels represent intervals with wave zero parameters, for clarity, their value is set to 0.1 in order to be represented in the logarithmic vertical axis. Vertical and horizontal lines indicate standard fluid turbulence reference vales. The correlation coefficient is indicated ( Correlation coefficients between pairs of parameters, including the solar wind speed V sw , the angle between the mean magnetic field and the radial direction θ vb , and the turbulence and wave parameters listed in
Fig. 1 .
1From top to bottom: solar wind electron density n e (RPW), deHoffmann-Teller solar wind velocity estimate V sw (RPW) and interplanetary magnetic field components B i and magnitude |B| (MAG), measured by Solar Orbiter during the whole month of June 2020.
Fig. 2 .Fig. 3 .
23Two examples of adjacent intervals during day 20 June 2020. Left: interval 26; right panel: interval 27. From top to bottom: interplanetary magnetic field components B i and magnitude |B| (MAG); electron density n e (RPW); magnetic filed spectrogram; perpendicular magnetic field components coherence; perpendicular magnetic field components phase angle; wave parameter Q w (t) (see Section 4). Left panel: an example of IMFs φ j (t) for sample 1 (black lines).
compatible with the typical inertial range of time-scales. Similarly, the second-order structure function also show a very clear power law scaling. In all three cases, the power-law scaling of L 2 ( f ) is compatible with the standard Kol-
Fig. 4 .
4Left panel: second order Hilbert spectrum L 2 ( f ) (blue circles), the classical Fourier PSD
Fig. 5 .
5Probability distribution function P(β 2 ) of the scaling exponents β 2 , constructed via bootstrap resampling, for sample 1 (upper panel), and sample 9 (lower panel). In both panels, the vertical dotted line represents the median of the distribution (50th percentile), while the vertical dashed bars indicate the 95% confidence interval (enclosed between 2.5th and 97.5th percentile).
. 2 .Fig. 6 .
26As expected, some of the parameters are trivially correlated with each other, such for example those Top-left panel: HSA equivalent scaling exponents ξ(q) for group 1 (blue circles) and their average (black crosses). The exponents ζ(q)
Fig. 7 .
7Least square fit of the scaling exponents ξ(q) via the log-normal cascade model (5) (lines), for sample 10 (red circles) and for the average of all exponents (black stars). The fitting parameter µ is in agreement with that of classical hydrodynamic turbulence µ ≈ 0.02.
Fig. 9 .
9Scatter plots of pairs of solar wind, turbulence and wave parameters (see
) .
)Sample Start time End time ∆W [hour] V sw [km/s] θ vb Group07-June-2020
1
05:22:13 06:28:53
1.11
228
80
1
2
09:26:39 12:02:13
2.59
353.84
95
1
3
12:35:33 13:48:47
1.22
353.84
78
1
4
14:00:00 16:39:26
2.66
650
86
1
5
16:42:33 19:57:46
3.25
430±62
88
1
6
20:05:53 23:47:59
3.70
354±12
128
1
08-June-2020
7
00:00:00 01:51:06
1.85
430±25
130
1
8
01:56:39 03:52:23
1.93
398.07
117
2
9
04:00:00 07:08:53
3.15
415.76
127
2
10
07:31:06 10:28:53
2.96
500
129
2
11
10:39:59 11:53:11
1.22
415.76
123
1
12
13:35:00 15:52:59
2.30
440±12
131
2
09-June-2020
13
00:00:00 03:57:59
3.97
614±43
133
3
14
04:10:33 10:01:59
5.86
514±50
157
1
15
10:59:59 15:58:59
4.98
555±27
148
1
16
18:02:46 21:00:33
2.96
485±16
143
3
17
21:01:40 23:52:46
2.85
535±47
145
1
10-June-2020
18
04:05:17 07:36:23
3.52
522±25
155
2
19
07:47:30 12:58:37
5.19
443±35
136
2
20
19:00:50 22:09:44
3.15
425±25
149
2
11-June-2020
21
00:00:00 02:57:46
2.96
541±55
154
2
22
04:05:17 06:57:30
2.87
529±27
159
1
23
09:38:37 12:36:24
2.96
397±37
138
3
24
17:00:50 21:27:30
4.44
407±42
118
1
14-June-2020
25
00:00:00 01:01:06
1.02
415.76
149
2
20-June-2020
26
00:55:33 03:08:53
2.22
575 (e)
172
1
27
03:19:59 06:55:33
3.59
525 (e)
165
2
22-June-2020
28
01:58:09 04:33:42
2.59
542±75
118
1
29
16:08:47 18:54:38
2.76
517±59
120
1
24-June-2020
30
04:26:39 06:17:46
1.85
379±16
176
2
31
14:30:33 16:06:39
1.60
300
169
3
32
16:47:46 17:57:46
1.17
403
147
1
27-June-2020
33
05:33:19 07:57:46
2.41
283.07
79
1
34
08:19:59 11:06:39
2.78
303±48
61
1
35
19:02:01 22:55:21
3.89
386±28
118
1
29-June-2020
36
16:20:33 18:00:33
1.67
278
162
2
Table A.2). 22-June-2020 28 0.31±0.09 0.36±0.06 0.32±0.07 1.61±0.11 1.70±0.09 1.70±0.09 29±0.08 0.30±0.08 0.36±0.04 1.57±0.11 1.70±0.13 1.64±0.09 June-2020 30 0.23±0.14 0.25±0.05 0.27±0.04 1.51±0.13 1.54±0.09 1.54±0.08 0.002 31±0.07 0.25±0.07 0.31±0.09 1.72±0.10 1.60±0.09 1.68±0.10 35±0.06 0.40±0.05 0.34±0.08 1.65±0.06 1.67±0.08 1.64±0.19 35±0.14 0.34±0.04 0.35±0.03 1.69±0.19 1.69±0.05 1.72±0.0560.1
858
0.19
-
29 0.10 −5
0.07
0.21
-
24-9.7
0.15
-
31
-
0.17±0.15 0.21±0.09
-
1.31±0.20 1.46±0.14 0.037
216
0.10
-
32
-
0.25±0.06 0.23±0.17
-
1.56±0.09 1.50±0.20
0.29
1217 0.08 0.037
27-June-2020
33 0.10 −4
0.45
0.55 0.034
34 0.10 −4
1.06
0.39 0.039
35 0.10 −4
2.23
0.35 0.019
29-June-2020
36 0.24±0.12
-
0.27±0.10 1.48±0.18
-
1.50±0.21 0.156
938
-
-
Table A .2.
A
Table A.1. For each pair, the maximum between the linear (Pearson) and nonlinear (Spearman) coefficients is given. For parameters which are not trivially related, non-negligible correlation values are highlighted in bold.Parameter V sw
θ vb
H
H |B|
H r
β 2
β |B|
2
β r
2Q W
Q W
κ
µ
V sw
1
0.28 -0.13 -0.13 0.11 -0.14 -0.09 0.18
0.15
-0.24 -0.20 -0.28
θ vb
1
-0.43 -0.54 -0.18 -0.50 -0.51 -0.22 0.59
0.65
-0.55 -0.38
H
1
0.79
27
0.94
0.73
0.29 -0.19 -0.31
0.55
0.45
H |B|
1
0.27
0.82
0.91
0.24 -0.45 -0.43
0.55
0.53
H r
1
0.16
0.41
0.90 -0.10
0.16
0.29 -0.16
β 2
1
0.80
0.19 -0.45 -0.48 -0.63 0.61
β |B|
2
1
0.39 -0.43 -0.34
0.51
0.36
β r
2
1
-0.07 -0.11 -0.34 -0.14
Q W
1
0.92
-0.50 -0.31
Q W
1
-0.45 -0.50
κ
1
0.42
µ
1
Appendix A: TablesArticle number, page 15 of 17 A&A proofs: manuscript no. CARBONETable A.1. List of turbulence and wave parameters obtained from the analysis of the intervals (# indicates the interval number): the Hurst numbers H, H |B| H r (for electron density, magnetic field magnitude and magnetic radial component respectively); the Hilbert spectral exponents β 2 , β |B| 2 and β r 2 ; the total and averaged wave parametersQ W and Q W ; the kurtosis scaling exponent κ; and the intermittency parameter µ for the electron density fluctuations. Absent values (-) indicate lack of power-law scaling. For the wave parameters, null values (0) indicate no wave power above the selected threshold.
. T Alberti, G Consolini, V Carbone, Entropy. 21Alberti, T., Consolini, G., Carbone, V., et al. 2019, Entropy, 21
. A Arneodo, C Baudet, F Belin, Europhysics Letters. 34EPLArneodo, A., Baudet, C., Belin, F., et al. 1996, EPL (Europhysics Letters), 34
. S D Bale, P J Kellogg, F S Mozer, T S Horbury, H Reme, Phys. Rev. Lett. 94215002Bale, S. D., Kellogg, P. J., Mozer, F. S., Horbury, T. S., & Reme, H. 2005, Phys. Rev. Lett., 94, 215002
. R Benzi, S Ciliberto, R Tripiccione, Phys. Rev. E. 4829Benzi, R., Ciliberto, S., Tripiccione, R., et al. 1993, Phys. Rev. E, 48, R29
. S Boldyrev, H Konstantinos, Q Xia, J C Perez, The Astrophysical Journal. 77741Boldyrev, S., Konstantinos, H., Xia, Q., & Perez, J. C. 2013, The Astrophysical Journal, 777, 41
. D Boos, L Stefanski, Significance. 7186Boos, D. & Stefanski, L. 2010, Significance, 7, 186
E Bradley, J T Robert, An Introduction to the Bootstrap (Chapman & Hall CRC Monographs on Statistics & Applied Probability). Taylor & Francis Group1st edn.Bradley, E. & Robert, J. T. 1994, An Introduction to the Bootstrap (Chapman & Hall CRC Monographs on Statistics & Applied Probability), 1st edn. (CRC Press, Taylor & Francis Group)
. R Bruno, V Carbone, Living Reviews in Solar Physics. 102Bruno, R. & Carbone, V. 2013, Living Reviews in Solar Physics, 10, 2
Turbulence in the Solar Wind. R Bruno, V Carbone, Lecture Notes in Physics. 928Springer International Publishing1st edn.Bruno, R. & Carbone, V. 2016, Turbulence in the Solar Wind, 1st edn., Lecture Notes in Physics 928 (Springer International Publishing)
. R Bruno, D Telloni, ApJ. 81117Bruno, R. & Telloni, D. 2015, ApJ, 811, L17
. R Bruno, D Telloni, L Primavera, The Astrophysical Journal. 78653Bruno, R., Telloni, D., Primavera, L., et al. 2014, The Astrophysical Journal, 786, 53
. R Bruno, D Telloni, L Sorriso-Valvo, A&A. 62796Bruno, R., Telloni, D., Sorriso-Valvo, L., et al. 2019, A&A, 627, A96
. F Carbone, T Alberti, L Sorriso-Valvo, Atmosphere. 11Carbone, F., Alberti, T., Sorriso-Valvo, L., et al. 2020a, Atmosphere, 11
. F Carbone, C N Gencarelli, I M Hedgecock, Phys. Rev. E. 9463101Carbone, F., Gencarelli, C. N., & Hedgecock, I. M. 2016a, Phys. Rev. E, 94, 063101
. F Carbone, M S Landis, C N Gencarelli, Geophysical Research Letters. 43Carbone, F., Landis, M. S., Gencarelli, C. N., et al. 2016b, Geophysical Research Letters, 43, 7751, 2016GL069252
. F Carbone, L Sorriso-Valvo, The European Physical Journal E. 3761Carbone, F. & Sorriso-Valvo, L. 2014, The European Physical Journal E, 37, 61
. F Carbone, L Sorriso-Valvo, T Alberti, The Astrophysical Journal. 85927Carbone, F., Sorriso-Valvo, L., Alberti, T., et al. 2018, The Astrophysical Journal, 859, 27
. F Carbone, L Sorriso-Valvo, C Versace, G Strangi, R Bartolino, Phys. Rev. Lett. 106114502Carbone, F., Sorriso-Valvo, L., Versace, C., Strangi, G., & Bartolino, R. 2011, Phys. Rev. Lett., 106, 114502
. F Carbone, D Telloni, A G Bruno, Atmosphere. 10Carbone, F., Telloni, D., Bruno, A. G., et al. 2019, Atmosphere, 10
. F Carbone, D Telloni, L Sorriso-Valvo, UniverseCarbone, F., Telloni, D., Sorriso-Valvo, L., et al. 2020b, Universe, 6
. F Carbone, H Yoshida, S Suzuki, Europhysics Letters). 8946004EPLCarbone, F., Yoshida, H., Suzuki, S., et al. 2010, EPL (Europhysics Letters), 89, 46004
. B Castaing, Y Gagne, E Hopfinger, Physica D: Nonlinear Phenomena. 46177Castaing, B., Gagne, Y., & Hopfinger, E. 1990, Physica D: Nonlinear Phenom- ena, 46, 177
. C H K Chen, Journal of Plasma Physics. 82535820602Chen, C. H. K. 2016, Journal of Plasma Physics, 82, 535820602
. C H K Chen, C S Salem, J W Bonnell, F S Mozer, S D Bale, Phys. Rev. Lett. 10935001Chen, C. H. K., Salem, C. S., Bonnell, J. W., Mozer, F. S., & Bale, S. D. 2012, Phys. Rev. Lett., 109, 035001
. C H K Chen, L Sorriso-Valvo, J Šafránková, Z Němeček, The Astrophysical Journal Letters. 7898Chen, C. H. K., Sorriso-Valvo, L., Šafránková, J., & Němeček, Z. 2014, The Astrophysical Journal Letters, 789, L8
Time-frequency analysis. L Cohen, Prentice Hall PTR Englewood Cliffs, N.JCohen, L. 1995, Time-frequency analysis (Prentice Hall PTR Englewood Cliffs, N.J, 1995)
. L Dale, G William, L Ron, T , MIS Quarterly. 36981Dale L., G., William, L., & Ron, T. 2012, MIS Quarterly, 36, 981
. R D'amicis, R Bruno, B Bavassano, Journal of Atmospheric and Solar-Terrestrial Physics. 73653D'Amicis, R., Bruno, R., & Bavassano, B. 2011, Journal of Atmospheric and Solar-Terrestrial Physics, 73, 653
. R D'amicis, D Perrone, R Bruno, M Velli, e2020JA028996Journal of Geophysical Research: Space Physics. D'Amicis, R., Perrone, D., Bruno, R., & Velli, M. 2021, Journal of Geo- physical Research: Space Physics, n/a, e2020JA028996, e2020JA028996 2020JA028996
. A Davis, A Marshak, W Wiscombe, R Cahalan, Journal of Geophysical Research: Atmospheres. 998055Davis, A., Marshak, A., Wiscombe, W., & Cahalan, R. 1994, Journal of Geo- physical Research: Atmospheres, 99, 8055
. M Dobrowolny, A Mangeney, P Veltri, Phys. Rev. Lett. 45144Dobrowolny, M., Mangeney, A., & Veltri, P. 1980, Phys. Rev. Lett., 45, 144
. T Dudok De Wit, O Alexandrova, I Furno, L Sorriso-Valvo, G Zimbardo, Space Science Reviews. 178665Dudok de Wit, T., Alexandrova, O., Furno, I., Sorriso-Valvo, L., & Zimbardo, G. 2013, Space Science Reviews, 178, 665
B Efron, D Rogosa, R Tibshirani, International Encyclopedia of the Social & Behavioral Sciences. J. D. WrightOxfordElsevierSecond Edition. second edition edn.Efron, B., Rogosa, D., & Tibshirani, R. 2015, in International Encyclopedia of the Social & Behavioral Sciences (Second Edition), second edition edn., ed. J. D. Wright (Oxford: Elsevier), 492 -495
P Flandrin, Wavelet Analysis and Its Applications. Academic Press10Time-Frequency/Time-Scale AnalysisFlandrin, P. 1999, Time-Frequency/Time-Scale Analysis, 1st edn., Wavelet Anal- ysis and Its Applications 10 (Academic Press)
. P Flandrin, P Goncalves, International Journal of Wavelets, Multiresolution and Information Processing. 1477Flandrin, P. & Goncalves, P. 2004, International Journal of Wavelets, Multireso- lution and Information Processing, 1, 477
. P Flandrin, G Rilling, P Goncalves, IEEE Signal Processing Letters. 11112Flandrin, P., Rilling, G., & Goncalves, P. 2004, IEEE Signal Processing Letters, 11, 112
U Frisch, Turbulence: the legacy of A. N. Kolmogorov. Cambridge UKCambridge Univ. PressFrisch, U., ed. 1995, Turbulence: the legacy of A. N. Kolmogorov (Cambridge Univ. Press, Cambridge UK)
. P Goldreich, S Sridhar, The Astrophysical Journal. 438763Goldreich, P. & Sridhar, S. 1995, The Astrophysical Journal, 438, 763
. B Hnat, S C Chapman, G Rowlands, Phys. Rev. E. 6756404Hnat, B., Chapman, S. C., & Rowlands, G. 2003, Phys. Rev. E, 67, 056404
. B Hnat, S C Chapman, G Rowlands, Phys. Rev. Lett. 94204502Hnat, B., Chapman, S. C., & Rowlands, G. 2005, Phys. Rev. Lett., 94, 204502
. T Horbury, R Wicks, C Chen, Space. Sci. Rev. 172325Horbury, T., Wicks, R., & Chen, C. 2012, Space. Sci. Rev., 172, 325
. T S Horbury, M Forman, S Oughton, Phys. Rev. Lett. 101175005Horbury, T. S., Forman, M., & Oughton, S. 2008, Phys. Rev. Lett., 101, 175005
. T S Horbury, H O´brien, I Carrasco Blazquez, A&A. 6429Horbury, T. S., O´Brien, H., Carrasco Blazquez, I., et al. 2020, A&A, 642, A9
The Hilbert-Huang Transform and Its Applications. Huang, N. E. & Shen, S. S. P.SingaporeWorld ScientificHuang, N. E. & Shen, S. S. P., eds. 2005, The Hilbert-Huang Transform and Its Applications (World Scientific, Singapore)
. N E Huang, Z Shen, S R Long, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 454903Huang, N. E., Shen, Z., Long, S. R., et al. 1998, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 454, 903
. Y Huang, F G Schmitt, Z Lu, Y Liu, Journal of Hydrology. 373103Huang, Y., Schmitt, F. G., Lu, Z., & Liu, Y. 2009, Journal of Hydrology, 373, 103
. Y X Huang, F G Schmitt, Z M Lu, Phys. Rev. E. 8226319Huang, Y. X., Schmitt, F. G., Lu, Z. M., et al. 2010, Phys. Rev. E, 82, 026319
. Y X Huang, F G Schmitt, Z M Lu, Y L Liu, Europhysics Letters). 8440010EPLHuang, Y. X., Schmitt, F. G., Lu, Z. M., & Liu, Y. L. 2008, EPL (Europhysics Letters), 84, 40010
. P S Iroshnikov, Soviet Astronomy. 7566Iroshnikov, P. S. 1964, Soviet Astronomy, 7, 566
. I M Jánosi, R Müller, Phys. Rev. E. 7156126Jánosi, I. M. & Müller, R. 2005, Phys. Rev. E, 71, 056126
. Y Khotyaintsev, D B Graham, A Vaivads, A&A. this volume. submittedKhotyaintsev, Y., Graham, D. B., Vaivads, A., et al. 2021, A&A, this volume, submitted
. K H Kiyani, S C Chapman, F Sahraoui, The Astrophysical Journal. 76310Kiyani, K. H., Chapman, S. C., Sahraoui, F., et al. 2012, The Astrophysical Jour- nal, 763, 10
. A N Kolmogorov, C. R. Acad. Sci. U.R.S.S. 36301Kolmogorov, A. N. 1941, C. R. Acad. Sci. U.R.S.S, 36, 301
. A N Kolmogorov, Journal of Fluid Mechanics. 13Kolmogorov, A. N. 1962, Journal of Fluid Mechanics, 13, 82-85
R H Kraichnan, The Physics of Fluids. 81385Kraichnan, R. H. 1965, The Physics of Fluids, 8, 1385
. S R Long, N Huang, C Tung, IEEE Geoscience Remote Sensing Soc. Lett. 36Long, S. R., Huang, N., Tung, C., et al. 1995, IEEE Geoscience Remote Sensing Soc. Lett, 3, 6
. M Maksimovic, S D Bale, T Chust, A&A. 64212Maksimovic, M., Bale, S. D., Chust, T., et al. 2020, A&A, 642, A12
R Marino, L Sorriso-Valvo, V Carbone, cross-Scale Coupling in Plasmas. 59Marino, R., Sorriso-Valvo, L., Carbone, V., et al. 2011a, Planetary and Space Science, 59, 592, cross-Scale Coupling in Plasmas
. R Marino, L Sorriso-Valvo, R D'amicis, The Astrophysical Journal. 75041Marino, R., Sorriso-Valvo, L., D'Amicis, R., et al. 2011b, The Astrophysical Journal, 750, 41
. W H Matthaeus, M L Goldstein, Journal of Geophysical Research: Space Physics. 8710347Matthaeus, W. H. & Goldstein, M. L. 1982, Journal of Geophysical Research: Space Physics, 87, 10347
. W H Matthaeus, M Velli, Space Science Reviews. 160145Matthaeus, W. H. & Velli, M. 2011, Space Science Reviews, 160, 145
Energy Procedia, 76, 193 , european Geosciences Union General Assembly 2015 -Division Energy, Resources and Environment. O D Medina, F G Schmitt, R Calif, Medina, O. D., Schmitt, F. G., & Calif, R. 2015, Energy Procedia, 76, 193 , european Geosciences Union General Assembly 2015 -Division Energy, Re- sources and Environment, EGU 2015
. C Meneveau, K R Sreenivasan, Journal of Fluid Mechanics. 224Meneveau, C. & Sreenivasan, K. R. 1991, Journal of Fluid Mechanics, 224, 429-484
. D Müller, St, O C Cyr, I Zouganelis, A&A. 6421Müller, D., St. Cyr, O. C., Zouganelis, I., et al. 2020, A&A, 642, A1
. N Nava, T D Matteo, T Aste, Physica A: Statistical Mechanics and its Applications. 447434Nava, N., Matteo, T. D., & Aste, T. 2016, Physica A: Statistical Mechanics and its Applications, 447, 434
. S Oughton, W H Matthaeus, The Astrophysical Journal. 89737Oughton, S. & Matthaeus, W. H. 2020, The Astrophysical Journal, 897, 37
. C J Owen, R Bruno, S Livi, A&A. 64216Owen, C. J., Bruno, R., Livi, S., et al. 2020, A&A, 642, A16
. A Pedersen, Annales Geophysicae. 13118Pedersen, A. 1995, Annales Geophysicae, 13, 118
. S Perri, A Balogh, M O ; -Riazantseva, L S Rakhmanova, G N Zastenker, Y I Yermolaev, I G Lodkina, The Astrophysical Journal. 714127Geomagnetism and AeronomyPerri, S. & Balogh, A. 2010, The Astrophysical Journal, 714, 937- Riazantseva, M. O., Rakhmanova, L. S., Zastenker, G. N., Yermolaev, Y. I., & Lodkina, I. G. 2019, Geomagnetism and Aeronomy, 59, 127
G Rilling, P Flandrin, NSIP-03IEEE-EURASIP WORKSHOP ON NONLINEAR SIGNAL AND IMAGE PROCESSING. Rilling, G., Flandrin, P., & et al. 2003, in IEEE-EURASIP WORKSHOP ON NONLINEAR SIGNAL AND IMAGE PROCESSING NSIP-03
. O W Roberts, Y Narita, X Li, C P Escoubet, H Laakso, Journal of Geophysical Research: Space Physics. 1226940Roberts, O. W., Narita, Y., Li, X., Escoubet, C. P., & Laakso, H. 2017, Journal of Geophysical Research: Space Physics, 122, 6940
. O W Roberts, J Thwaites, L Sorriso-Valvo, R Nakamura, Z Vörös, Frontiers in Physics. 8464Roberts, O. W., Thwaites, J., Sorriso-Valvo, L., Nakamura, R., & Vörös, Z. 2020, Frontiers in Physics, 8, 464
. A A Schekochihin, S C Cowley, W Dorland, The Astrophysical Journal Supplement Series. 182310Schekochihin, A. A., Cowley, S. C., Dorland, W., et al. 2009, The Astrophysical Journal Supplement Series, 182, 310
Estimation Of Universal Multifractal Indices For Atmospheric Turbulent Velocity Fields. F Schmitt, D Schertzer, S Lovejoy, B & Y, World Scientific. Schmitt, F., Schertzer, D., Lovejoy, S., & Y., B. 1994, Estimation Of Universal Multifractal Indices For Atmospheric Turbulent Velocity Fields (World Sci- entific), 274-281
. F G Schmitt, The European Physical Journal B -Condensed Matter and Complex Systems. 3485Schmitt, F. G. 2003, The European Physical Journal B -Condensed Matter and Complex Systems, 34, 85
. C W Smith, J E Stawarz, B J Vasquez, M A Forman, B T Macbride, Phys. Rev. Lett. 103Smith, C. W., Stawarz, J. E., Vasquez, B. J., Forman, M. A., & MacBride, B. T. 2009, Phys. Rev. Lett., 103, 201101
. L Sorriso-Valvo, F Carbone, E Leonardis, Advances in Space Research. 591642Sorriso-Valvo, L., Carbone, F., Leonardis, E., et al. 2017, Advances in Space Research, 59, 1642
. L Sorriso-Valvo, V Carbone, P Veltri, G Consolini, R Bruno, Geophysical Research Letters. 261801Sorriso-Valvo, L., Carbone, V., Veltri, P., Consolini, G., & Bruno, R. 1999, Geo- physical Research Letters, 26, 1801
. L Sorriso-Valvo, R Marino, V Carbone, Phys. Rev. Lett. 99115001Sorriso-Valvo, L., Marino, R., Carbone, V., et al. 2007, Phys. Rev. Lett., 99, 115001
. L Sorriso-Valvo, E Yordanova, V Carbone, EPL. 9059001Sorriso-Valvo, L., Yordanova, E., & Carbone, V. 2010, EPL, 90, 59001
K Steinvall, Y Khotyaintsev, G Cozzani, A&A, this volume. Steinvall, K., Khotyaintsev, Y., Cozzani, G., et al. 2021, A&A, this volume, sub- mitted
. G I Taylor, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 164476Taylor, G. I. 1938, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 164, 476
. D Telloni, F Carbone, R Bruno, The Astrophysical Journal. 887160Telloni, D., Carbone, F., Bruno, R., et al. 2019, The Astrophysical Journal, 887, 160
. D Telloni, F Carbone, R Bruno, ApJ. 8855Telloni, D., Carbone, F., Bruno, R., et al. 2019, ApJ, 885, L5
A&A, this volume. D Telloni, C Scolini, C Möstl, Space Sci Rev. 731submitted Tu, C. & Marsch, E.Telloni, D., Scolini, C., Möstl, C., et al. 2021, A&A, this volume, submitted Tu, C. & Marsch, E. 1995, Space Sci Rev, 73, 1
. A Verdini, R Grappin, O Alexandrova, S Lion, The Astrophysical Journal. 85385Verdini, A., Grappin, R., Alexandrova, O., & Lion, S. 2018, The Astrophysical Journal, 853, 85
. N M Viall, J E Borovsky, Journal of Geophysical Research: Space Physics. 125Viall, N. M. & Borovsky, J. E. 2020, Journal of Geophysical Research: Space Physics, 125, e2018JA026005, e2018JA026005 2018JA026005
. P Welch, IEEE Transactions on Audio and Electroacoustics. 1570Welch, P. 1967, IEEE Transactions on Audio and Electroacoustics, 15, 70
R R Wilcox, Fundamentals of Modern Statistical Methods: Substantially Improving Power and Accuracy. New YorkSpringer-Verlag2nd edn.Wilcox, R. R. 2010, Fundamentals of Modern Statistical Methods: Substantially Improving Power and Accuracy, 2nd edn. (Springer-Verlag New York)
. Z Wu, N E Huang, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences. 4601597Wu, Z. & Huang, N. E. 2004, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 460, 1597
E Yordanova, S Perri, L Sorriso-Valvo, V Carbone, EPL (Europhysics Letters). 110Yordanova, E., Perri, S., Sorriso-Valvo, L., & Carbone, V. 2015, EPL (Euro- physics Letters), 110, 19001
. I Zouganelis, A De Groof, A P Walsh, A&A. 6423Zouganelis, I., De Groof, A., Walsh, A. P., et al. 2020, A&A, 642, A3
| [] |
[] | [] | [] | [] | In this article, we propose two-stage planning models for Electricity-Gas Coupled Integrated Energy System (EGC-IES), in which traditional thermal power plants (TTPPs) are considered to be retrofitted into carbon capture power plants (CCPPs), with power to gas (PtG) coupling CCPPs to gas system. The sizing and siting of carbon capture, utilization and storage (CCUS)/PtG facilities, as well as the operation cost of TTPPs/CCPPs/gas sources/PtG, are all considered in the proposed model, including penalty on carbon emissions and revenue of CCUS. With changing policy on climate change and carbon emission regulation, the uncertainties of carbon price and carbon tax are also analyzed and considered in the proposed planning model. The stochastic planning, and robust planning methods are introduced to verify mutually through economic and carbon indices. The proposed methods' effectiveness in reducing carbon emissions, increasing CCUSs' profit from EGC-IES are demonstrated through various cases and discussions.Index Terms-Carbon capture, utilization and storage, Electricity-Gas Coupled Integrated Energy System, Carbon tax and price uncertainty, Two-stage stochastic planning, Two-stage robust planning. | null | [
"https://arxiv.org/pdf/2107.09127v2.pdf"
] | 236,134,059 | 2107.09127 | f5358887cdbfead4c33e597719fd698afb8b1d07 |
In this article, we propose two-stage planning models for Electricity-Gas Coupled Integrated Energy System (EGC-IES), in which traditional thermal power plants (TTPPs) are considered to be retrofitted into carbon capture power plants (CCPPs), with power to gas (PtG) coupling CCPPs to gas system. The sizing and siting of carbon capture, utilization and storage (CCUS)/PtG facilities, as well as the operation cost of TTPPs/CCPPs/gas sources/PtG, are all considered in the proposed model, including penalty on carbon emissions and revenue of CCUS. With changing policy on climate change and carbon emission regulation, the uncertainties of carbon price and carbon tax are also analyzed and considered in the proposed planning model. The stochastic planning, and robust planning methods are introduced to verify mutually through economic and carbon indices. The proposed methods' effectiveness in reducing carbon emissions, increasing CCUSs' profit from EGC-IES are demonstrated through various cases and discussions.Index Terms-Carbon capture, utilization and storage, Electricity-Gas Coupled Integrated Energy System, Carbon tax and price uncertainty, Two-stage stochastic planning, Two-stage robust planning.
Matrixes:
A Gas node-gas source incidence matrix B Gas node-gas pipeline incidence matrix C Bus-generator incidence matrix
A. Background and Motivation
HE emission of greenhouse gas which gives priority to carbon dioxide (CO 2 ), is a primary driver of global climate change and one of the most pressing challenges nowadays [1]. In response to global climate change, the Paris Agreement was adopted by 196 parties at the 21st Conference of the United Nations Framework Convention on Climate Change in Paris on 12 Dec., 2015 and entered into force on 4 Nov., 2016. Its longterm goal is to keep the rise in global average temperature to well below 2°C above pre-industrial levels, and to pursue efforts to limit the increase to 1.5°C [2].
It has already sparked various low-carbon solutions since the Paris Agreement entered into force, more and more countries, regions and cities all over the world are establishing zerocarbon targets. Carbon capture, utilization and storage (CCUS) is the key technology that reduces and removes CO 2 after emission, which is a critical part of zero-carbon goals [3]. According to Special Report on CCUS [4] released by International Energy Agency (IEA), the next decade will be critical to the zero-carbon emission goal via retrofitting existing power and industrial facilities. Thus carbon capture power plant (CCPP), which is altered from traditional thermal power plant (TTPP), its carbon reduction performance in Gas Network, Electricity Network and even Electricity-Gas Coupled Integrated Energy System (EGC-IES) are worth investigating urgently.
Encouragingly, as more ambitious climate pledges are taken, many programs and strategies are factoring in the role and potential for carbon pricing and carbon markets [5]. How to translate commitment into reality in ensuring we can confine global warming to below 2°C, the careful planning for EGC-IES considering the global changing and divergent carbon price policy will be critical.
B. Literature Review
Several scholars have carried out related researches in developing planning models considering CCUS. In CCUS and its applications, research [6] introduced a scalable infrastructure model to determine where and how much CO 2 to capture and store, where to build and connect pipelines. A mixed integer linear programming (MILP) model was developed [7] to describe a general modeling approach for optimal planning of energy systems subject to carbon and land footprint constraints. Several literatures addressed the retro fitment [8], expansion ([9], [11]), transition [10] pipeline networks ( [12], [13]) planning problems considering CCUS.
Later in 2016, literature [14] presented a planning framework considering CO 2 supply, production, transportation, and emission costs . literature [15] developed a two-step linear optimization to make a trade-off between the cost of network modification and CO 2 emissions considering the network revamp. In recent years, CO 2 utilization has attracted more and more attention from scholars to industry. Review [16] and [17] summarized the existing and under-development technologies for CO 2 utilization in the world. The exact applications in gas field [18], oil field [19], and chemical products [20] were gradually studied in depth. CCUS is often planned with particular actual cases to observe performance. For example, a multi-stage mixed integer programming (MIP) model for CCUS planning was developed and tested in Beijing-Tianjin-Hebei, China [21], the United States [22] and the United Kingdom [23].
Specializing on CCUS's application in retrofitting existing T facilities, early research [24] presented linear models of the most common components in the CCPP. A post-combustion and solvent/sorbent separation technology based CCPP model is established in [25], the authors investigated the performance of CCPP in the carbon emission market [26] in 2012. The TTPP was equipped with nuclear units, renewable energy units, and different fossil fuel-fired units in [27] to form MINLP and solve using particle swarm optimization algorithm. Table I categories these researches by topics and mathematical method for CCUS planning in detail. MILP [6]- [8], [12], [14], [15], [24], [
C. Problem Identification and Main Contributions
Due to the long time horizons involved in CCUS planning, it is necessary to include uncertainties. Several literatures have studied relevant researches on uncertainties such as load and renewable generation uncertainties [28], generation and demand side uncertainties [29], user behavior uncertainty [30], etc. However, as the significant contents of low-carbon development, the volatilities of carbon price and carbon tax are essential sources of uncertainties that cannot be ignored.
Besides, gas-fired power generation units are considered as the coupling point of gas and electricity systems [31] for a long time. The development of power to gas (PtG) technology could be a new coupling point [33] in the future. However, related research in EGC-IES is still very few.
In this paper, two-stage planning methods for EGC-IES with CCUS considering carbon price uncertainty are proposed. Main contributions of the paper are therefore three-fold:
(1) A CCPP planning model retrofitted from TTPP, with PtG in EGC-IES for CO 2 utilization is proposed, in which the methane produced by PtG is transported to the gas network to relieve the gas supply pressure.
(2) The carbon penalty and carbon revenue are introduced in the objective function considering carbon tax and carbon price uncertainties globally.
(3) Three planning models aim at finding out a balance point where the external revenue and reduced penalty to cover the investment cost. The effectiveness of the proposed model is also verified.
The remainder of this paper is organized as follows: Section II would outline the CCUS business model under carbon market and carbon policy. Model formulation is displayed in Section III where the detailed objective function and constraints are described. Section IV discusses the proposed four models. Case studies and conclusions are given in Section V and VI.
II. PRELIMINARY OF CCUS AND ITS BUSINESS MODEL
CCUS is a crucial emissions reduction technology that can be applied across the energy system [3]. Fig. 1 illustrates the significant profits method of CCUS currently. Its business model can be summarized as follows [4]: CCUS involve CO 2 capture from fuel combustion or industrial processes, uncaptured CO 2 is subject to the mandatory penalty in the form of carbon tax; captured CO 2 is transported via vehicles or pipeline or permanently stored deep underground, CO 2 could be utilized as a resource to create valuable products and services such as CO 2 enhanced oil recovery, CO 2 enhanced coal bed methane, chemical and biological products; in an emission trading system, captured CO 2 can also be traded, as predicted in The New York Times that carbon will be the world's biggest commodity market [32]. Nowadays, there are 22 CCUS facilities worldwide with the capacity to capture more than 40 Mt CO 2 every year. Report [4] pointed out three aspects that summarize the growth trends in CCUS projects over the following decades: 1) Retrofitting of existing power and industrial facilities that significantly reduce emissions.
2) The scale-up of low-carbon hydrogen production with CCUS.
3) The rapid adoption of CCUS technologies and applications that are not yet widely used.
According to the technical development level, the current focus of CCUS is on retrofitting fossil fuel-based power and industrial plants.
Due to the vast differences in the economic and technological development levels and resource endowments around the world, this article uses the technical parameters published by IEA and related literatures to build a general planning model for TTPP retrofitting into CCPP, other than focus on one or several specific CCUS technologies; at the same time, based on the consideration of carbon policies are influenced by politics, culture and other factors, carbon price and carbon tax are regarded as uncertainties, the business models like carbon penalty and carbon revenue are adopted to form the planningoperation two-stage model.
III.MODEL FORMULATION
In Section III, the objective function, gas network model, electricity network model, CCPP and PtG coupling model, and constraints on facility investment and siting are elaborated separately.
A. Objective function
The objective function is to minimize the total cost (1)
B. Gas network model
A typical gas network model comprises natural gas transmission pipelines, natural gas sources and gas loads. The Weymouth equation [34] is adopted in (6) to describe natural gas transmission flow in this model, in which the gas flow is expressed as a quadratic equation of nodal gas pressure, , a b denote the input end and output node respectively, in accordance with the specified direction. Formula (7a) introduces the variable I to replace nodal pressure to avoid the nonconvexity.
2 2 2 , ,, , , , 1,gas gas GP p t p t p a t b t f f W p t T (6) 2 ,
, ,
gas gas GP p t p t p a t b t f f W I I p t T (7a) 2 2 , , , 1, GN m m m t I m t T (7b), , , 1,
Piecewise linearization are listed as constraints (8a)-(10), adopted from [35] with continuous variable and binary variable : the former indicating the proportion occupied in a specific segment and the latter indicating the selected status of the segment (the value of changing from 1 to 0 means the segment is selected). ,
, ,, , , , 1, , 1, 1GP p t k p t k p t T k seg (8a) , , 0 1, , 1, , 1, GP p t k p t T k seg (8b) , , 1 , , , , 1, , 1, 1 GP p t k p t k p t T k seg (8c) ,,1 , , , 1 , 1 , , 1, , 1,f F F F p t T k seg (9) 2 , ,,1 ,1 , , , 1 , 1 , , 1 , 1, , 1,W I I F F F F F F p t T k seg (10)
Natural gas source production is limited by output constraints (11a) and ramp constraints (11b). Eq. (12) is the power balance constraint in gas network model among gas source, gas flow, and gas load, where on right-hand side , m t L is the gas load at node m to be balanced, , mi A is the element of gas node-gas
source incidence matrix A , , 1 m i A if gas source i is connected to gas node m , ,GS i i i t P P P i t T (11a) ,, 1,GS i i i t i t P P P P i t T (11b) , ,, 1 , , 2,gas m i i t m p p t m t GN GS GP P f L m i p t T A B (12) min GS GEN PtG inv ope ope ope CC CS pen tra C C C C obj C C C C , , , , , , 1, (1 ) , ,(1 )1 L L dr dr PtG siting dr t (2b) , ,, 1,GS GS ope i i t i t C r P i t T (3a) , ,, 1,GEN GEN ope j j t j t C r P j t T (3b) , ,, 1,PtG PtG ope q q t q t C r P q t T (3c) , ,, 1,CC CC CC CCPP j t j t C r Q j t T (4a) , ,, 1,CS CS CS CCPP j t j t C r Q j t T (4b) , ,, , 1,tax EMI CC CCPP pen j t j t j t C r Q Q j t T (5a) ,, 1,pr CS CCPP tra j t j t C r Q j t T ， (5b)
C. Electricity network model A typical electricity power system comprises electricity transmission lines, generators, and electricity loads. The DC power flow model in eq. (13a)-(13b) can be adopted to estimate steady state for the distribution system, (14) is constraint for power flow on transmission line.
, , , / , , 1, ele TL l t a t b t l f X l t T (13a) , ,, 1,EB n t n t T (13b) , ,, 1,ele ele ele TL l l l t F f F l t T (14)
Generators' output constraints (15) and ramp constraints (16) of
GEN j j j t j t j t u P P u P j t T (15) ,, , 1,GEN j j j t j t P P P P j t T (16) , , 1 , , , on j t GEN on j tt j t j tt t T v u j t T T (17a) , , 1 1 , , , off j t GEN off j tt j t j tt t T w u j t T T (17b) ,, 1 , , 2,GEN j t j t j t j t u u v w j t T (18) ,, 1 , , , , 2,GEN j t j t v w j t T , 1, , 1,
Eq. (20) is the power balance constraint in electricity network among generator, power flow, and load, similar to eq. (12).
, ,, , , , , 1,ele n j j n l l t n t EB GEN TL P f L n j l t T C D(20)
D. CCPP and PtG coupling model A coupling model combined TTPP, CC, CS and PtG technology to retrofit into CCPP is illustrated below, where exhaust emission from TTPP is flowed into CCUS, then captured CO2 is provided to Sabatier reactor to react with H 2 electrolyzed to produce CH 4 . The external characteristics are adopted in planning problem instead of the internal chemical reaction process [36], Fig. 2 shows the flow chart retrofitting TTPP to CCPP with CCUS.
,, , , , , 1,CCPP PtG CC GEN j t j t j t j t P P P P j t T (21) , , , , 1, emi GEN j t j t Q emi P j t T (22) , , / , , 1, CC CC CC GEN j t j t Q P W j t T (23)
Eq. (21) denotes the TTPP output power supply for the external grid, PtG, and carbon capture device. In (22), considering that the CO 2 emissions of power plants changed with loads, the CO 2 emission volume accounts for a fixed proportion emi of output power based on statistical data from U.S. DOE [39]. The relationship between the captured CO 2
CH CH GN CCPP j t m j t m V V m j t T , , , 1,
Natural gas generated by PtG can be delivered to the gas network to relieve gas supply pressure. , m j s is the binary variable indicating the investment statue of transmission pipeline between gas node m and PtG j , which is equal to 1 if they are connected, otherwise it is 0. j PtG y is integer variable denoting the number of PtG module to be invested, with each module assumed to be 1MW. Constraint (28) means that every pipeline can only be invested after PtG was chosen to be expanded on CCPP due to investment logic, where M is a very large number used in the Big-M method. Similarly in (29), the transmission volume of natural gas between gas node m and PtG j at any time is bound by decision variable , m j s . (30) implicates the coupling relationship between natural gas produced by PtG and transmitted in pipeline. Fig. 3 illustrates the sketch on facility sizing and siting in EGC-IES.
CH G gas m i i t m p p t m j t m t j GN GS GP CCPP P f V L m i p j t T A B (31) , ,, , , , , , , , 1,CCPP ele n j j t n l l t n t EB CCPP TL P f L n j l t T C D, , , , , , 1,
After retrofitting TTPP to CCPP with CCUS and PtG, the original power balance functions in (12) and (20) are converted into (31) and (32) by adding an external gas source from PtG and replacing generator as CCPP.
IV.METHODOLOGY
In Section IV, three models are formulated based on previous constraints and objective function, and the effectiveness of proposed models are verified through case studies in Section V.
A. Without CCUS planning
Without CCUS planning is the original circumstance, in which it does not exist additional investment in CCUS equipment. The CO 2 generated by TTPP is directly emitted into the atmosphere, and the utilities have to pay the carbon penalty. Correspondingly, there is no carbon revenue due to the absence of CCUS.
The objective function in (28) consists of operation costs of gas sources, generators, and carbon penalty due to CO 2 emission. It could be put into a compact form:
, ,, 2 minC P C P C P i j t T s t (33)
B. Deterministic planning with CCUS
Deterministic planning with CCUS considers investment cost, operation cost, carbon capture/storage cost and carbon penalty/revenue in a specific value of carbon price and carbon tax. Different planning results can be obtained since the carbon price and carbon tax fluctuations along with policy changes. It could be put into a compact form:
, ,
,
c y c s C P C P C P C P C P C P C P s t b a i j q , 1, t T (34)
C. Two-stage stochastic planning with CCUS
There are over 61 carbon pricing initiatives in place or scheduled for implementation until 2020 [4], meanwhile, the carbon price level of the implemented carbon pricing mechanism varies greatly around the world. Therefore, it is essential to consider the carbon price and carbon tax uncertainty based on the implemented carbon pricing mechanism globally. In the two-stage stochastic planning model, the first-stage decisions include the PtG capacity and its siting, while the second-stage considers the operation cost, e. g. carbon capture & storage cost, carbon penalty & trading in all scenarios.
, ,
,
min ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) (, , , , , ( ) ( )C P s C P s C P s C P s s t b P 3 19 ,32 , , , 1,21 GS CCPP PtG a i j q t T (35)
D. Two-stage robust planning with CCUS Unlike the method using probabilistic weights of uncertainties in stochastic planning, two-stage robust planning further considers the robustness of uncertainty set in the second stage. Box uncertainty set is adopted to illustrate the uncertainty of policy considering carbon price and carbon tax, the boundaries of uncertainty set are obtained from recent Prices in implemented carbon pricing initiatives in [5].
,, , , , , , , , ( ) ( , . . 6 11 ,13 19 , 32 , , , , 21 , 1,
pr pr
GS CCPP PtG GN EN r r s t b a i j q m n t T (36)
V. CASE STUDIES An updated IEEE 24-bus electric system with the Belgian 20node natural gas system are employed to verify the effectiveness of the proposed models. A sketch map indicating sources of key parameters of PtGs and their siting planning is shown in Appendix Table. All the datasets are available in [40]. The investment cost of PtG is set as 3 M$/MW, the siting costs of PtG to gas system are set to different values according to distance. The MILP is modeled by MATLAB 2019b with YALMIP toolbox and solved by Gurobi 9.1 on a laptop with Intel® Core™ i7-6700U 3.40 GHz processor and 8GB RAM.
A. Analysis of planning results
Four cases are developed to observe corresponding planning and operation results based on the proposed models as follows: Table II summarizes the problem identifications of the proposed four cases. The detailed planning results are shown in Table III.
Overall, the investment cost increased from Case 2 to Case 4. The reasons can be traced back to the second stage, the second stage of Case 2 is calculated in a specific scenario ( ta x r =50 $/ton, pr r =40 $/ton); while the second stage of Case 3 is the weighted average based on 25 equal-probabilistic scenarios extracted from the current carbon price and carbon tax range globally, with more scenarios simulated, the planning result is likely to get closer to the average value ( ta x r =60 $/ton, pr r =40 $/ton); as for Case 4, the planning result reflects the optimal decisions in the worst scenario of the current global carbon policy ( ta x r =120 $/ton, pr r =1 $/ton), where the extreme situation probably won't happen in practice. The siting planning of CCUS obtained the same results, gas node #17 is selected as the connection node of gas network and electricity network due to joint optimization based on sitting cost and load conditions of gas network node. More detailed analysis of investment costs can be found in Section V.B.
As for the operation cost, the gradual rise of generators' and PtGs' indicate their outputs increased along with the carbon capture process and PtGs' increasing investment. The operation costs of gas sources decreased slightly while the generators generate more power (174.60 in Case 1 →334.62 in Case 4), revealing that gas sources' supply pressure is eased with the coupling PtGs producing gas from CCUS in EGC-IES.
Carbon-related indices are also our key concerns. Case 1 (without CCUS planning) accounted for maximum carbon emission volume and carbon penalty except for Case 4 (robust planning with the highest carbon tax), the carbon emission volume increases from Case 2 to Case 4 since outputs of generators to supply for CCUS is also increased to avoid more penalty due to higher carbon tax, while the carbon capture and storage volume of Case 2 and Case 3 varied little. In Case 4, with the highest carbon tax and lowest carbon price, we obtained the worst circumstance with maximum carbon penalty (1588.40M$) and minimum carbon revenue (15.04M$) of the EGC-IES.
In summary, for the global average carbon price and carbon tax level, the business model of installing CC-CS-PtG CCUS for TTPP and transmitting produced gas to the gas network is economically feasible. However, for those countries or regions with high carbon taxes and low carbon prices, the business model of retrofitting such kind of CCUS for policy arbitrage is still not applicable. The next subsection analyzed the approximate range of carbon prices and carbon taxes that have economic advantages in terms of the total cost.
B. Sensitivity analysis of carbon price and carbon tax
The investment decisions in different carbon prices and carbon taxes are simulated in Fig. 4 changed in steps of $10. It can be concluded from the figure that the investment tends to rise as carbon price decreases and carbon tax increases. It is reasonable under the current business model: the optimal choice is to sell the captured CO 2 for income when the carbon price is rather high; with relatively low carbon price, it is not costeffective to sell the captured CO 2 and it would be better to produce methane to reduce gas supply pressure. Furthermore, it can be observed from Fig. 4 that when carbon price is higher than carbon tax (lower right corner of Fig. 4), no more CCUS investment is preferred; on the contrary, when carbon price is lower than carbon tax, CCUS investment is adopted, and the investment increases with difference increases. The results depict that CCUS retrofitting is an investment strategy sensitive to the carbon emission policy, i.e., carbon tax and carbon price. 5 is to further analyze the trend of total cost along with carbon price and carbon tax compared to circumstance without CCUS planning, in which surface in pink color represents the total cost without CCUS. Generally speaking, the total cost presents a downward tendency along with the carbon price increases and carbon tax decreases. It can be seen from the figure that while the value of carbon price is larger than 40 $/ton, the business model of CCUS retrofitting with CC, CS and PtG for TTPP is approachable globally. This retrofit planning has achieved economically better for some high-carbon tax areas while the carbon price equals 30 $/ton.
It is worth mentioning that, despite carbon prices increasing in many jurisdictions all over the world, the carbon prices in most parts of the world are still below 40$/ton at present. Report [5] has pointed out that the carbon price level of 50$/ton-100$/ton by 2030 is required to cost-effectively reduce emissions in line with the temperature goals in the Paris Agreement, this is also consistent with the intersecting line shown in Fig. 5. Facing the policy foundation of a gradual increase in carbon prices, the retrofit planning could have great potential in the future.
C. Analysis of daily carbon indices
The effectiveness of the proposed models can be verified further by comparing daily carbon emission, capture, storage, and utilization volume in Appendix Fig. Case 1 in blue color can be regarded as a benchmark without CCUS planning, it results in vast amounts of carbon emissions and simultaneously wastes policy dividends. Case 2 and Case 3 can be compared and analyzed together, in which more sampled scenarios can lead closer to the global average level in Case 3. Case 4 in purple line reflects the "worst case" in robust optimization: compared with Case 2 and Case 3, the robust planning result is more inclined to carbon emission and averse to carbon capture/storage due to the extremely low carbon price, even when carbon tax is also high; at the same time, the planning result adopted maximum carbon emissions volume and minimum carbon capture/storage volume in the worst circumstance. Moreover, captured carbon is used for storage or utilization, they made a tradeoff with each other. Because of the minimum operating power constraint with the planned capacity of PtG (Constraints (27)), the carbon utilization volume in Case 4 still achieved the highest among all cases.
Carbon-related volume is also consistent with the load conditions. During the morning and evening load valley moment (1:00-6:00, 22:00-24:00), CCUS is sufficient to achieve zero carbon emissions; the changing trend of carbon capture and carbon storage curves is basically in accordance with load conditions for Case 2 and Case 3. During non-valley periods (7:00-22:00), abnormal changes of carbon capture and storage happened due to the worst carbon price in Case 4.
On the whole, it can be observed that the effectiveness of the proposed models (deterministic planning, two-stage stochastic planning & two-stage robust planning) in reducing carbon emissions, increasing carbon capture and utilization by making use of carbon policy dependent on carbon tax and prices.
VI.CONCLUSIONS
This paper presents a gas-electricity coupled integrated energy system planning model with CCUS, in which carbon price and tax uncertainties are included. With proper carbon tax and price range, the proposed model effectively reduces carbon emissions, increases carbon capture and usage, and makes use of policy to profit. In the proposed model, the capacity and siting of PtG are included in the objective function to optimize the first-stage investment cost, and the economic indices on operation cost of generator, gas source and PtG, carbon capture and storage cost, as well as carbon penalty and revenue are optimized by IES operation strategy in the second stage.
Moreover, carbon price and carbon tax were considered in the model as the key sources of the uncertainties under the changing carbon policy. Stochastic and robust planning methods are introduced to verify mutually through economic indices and carbon indices. The case studies demonstrate the effectiveness of the proposed gas-electricity coupled models by PtG and illustrated the benefits of CUSS installation. Future work could further consider gas turbine and power to gas to form bi-direction energy flows and renewable energy and load uncertainties in EGC-IES.
F
Lowest boundary point of gas flow in segment k on pipeline p , Maximum/minimum outputs power of gas source i , Mm 3 /h i P Maximum ramp up power of gas source i , Mm 3 /h , m t L Gas load of gas node m in hour t , Mm 3 Student Member, IEEE, Xinwei Shen, Senior Member, IEEE, Qinglai Guo, Senior Member, IEEE, Hongbin Sun,
Gas flow on pipeline p in hour t , Nodal pressure of gas node m in hour t , bar , , p t k Continuous variable of segment k on pipeline p in hour t indicating the proportion occupied in a specific segment , , p t k Binary variable of segment k on pipeline p in hour t indicating the selected status of Power flow on transmission line l in hour t , MW , n t Phase angle of electric node n in hour t
Fig. 1
1CCUS business model flow chart
of gas source i , generator j and PtG q , power of gas source i , generator j and PtG q in hour t, Q are CO 2 volume indicating emission, capture and storage of CCPP j in hour t, respectively.
F
represents the lowest boundary point of gas flow in segment k on pipeline p .
are similar to those of natural gas sources, in which u indicating the start-stop status, the value of u keeps 1 if the generator is in operation, otherwise being 0. The unit commitment constraints in formulas (17a)-(19) are employed to accurately describe generator output characteristics. v and w are binary variables to reflect startup and shutdown actions, the value of v is 1 if the generator started up from the prior moment, and the value of w is 1 if the generator shut down from the prior moment, otherwise they remain 0.
Fig. 3
3Sketch on facility investment and siting of EGC-
Deterministic planning results with CCUS( ta x r =50 $/ton, pr r =40 $/ton);3) Case 3Two-stage stochastic planning results with CCUS in 25 scenarios (the value of ta x r and pr r are selected as five-segment points on average within their range to form 5*5 equalprobabilistic scenarios, the range of ta x
Fig. 4
4Investment cost in different carbon tax and carbon price Fig.
Fig. 5
5Total cost in different carbon tax and carbon price
APPENDIX Appendix Fig. Daily carbon emission, capture, storage, and utilization volume (a). carbon emission volume. (b). carbon capture volume. (c). carbon storage volume. (d). carbon utilization volume. Appendix
Set of gas pipelinesAngXuan and Xinwei Shen are with Tsinghua-Berkeley Shenzhen Institute, Tsinghua Shenzhen International Graduate School, Tsinghua University. Qinglai Guo and Hongbin Sun are with Dept. of Electrical Engineering, Tsinghua University. This work is supported by the National Natural Science Foundation of China (No. 52007123) (Corresponding author: Xinwei Shen, email: [email protected]; Hongbin Sun, email: [email protected]).NOMENCLATURE
Abbreviations:
CCUS
carbon capture, utilization and storage
CC
carbon capture
CS
carbon storage
IEA
International Energy Agency
TTPP
traditional thermal power plant
CCPP
carbon capture power plant
EGC-IES
Electricity-Gas Coupled Integrated Energy
System
PtG
Power to gas
Sets:
GN
Set of gas nodes
GP
Table I
ISummary of Planning Model Considering CCUSCategory
Literature
Carbon capture & storage
[6]-[11]
Carbon transportation
(pipeline planning)
[12]-[15]
Topics
Carbon utilization
[16]-[23]
Decision tree method
[13]
1 seg
gas
gas
gas
gas
p t
p
p t k
p k
p k
k
GP
1 p
a t
b t
seg
gas
gas
gas
gas
gas
gas
p
p
p t k
p k
p k
p k
p k
k
GP
volume by carbon capture device, i. e.(unit: MWh/ton) is the energy consumption of capturing per ton CO 2 by carbon capture device adopted from[37]. MWh) is CO 2 consumed volume of PtG to generate unit work CH 4 adopted from[38]. Constraints (26) denote CH 4 production character of PtG, volume by PtG in CCPP j during hour t . Constraints(27) limits the output power of PtG in CCPP j during hour t withCC
Q , and unit work by
carbon capture device operation power
CC
P is approximated by
linear correlation with C C
W
in (23), C C
W
,
,
, ,
,
1,
CC
CS
CU
CCPP
j t
j t
j t
Q
Q
Q
j
t
T
(24)
Fig. 2 Framework of CCPP and PtG coupling in EGC-IES
The captured CO 2 is either used for PtG to produce methane
(
CU
Q ) as separated form or stored to sell to carbon market (
CS
Q
) in compressed form, which can be modeled by (24).
2
,
, ,
,
1,
CO
CU
PtG PtG
CCPP
j t
j t
Q
P
j
t
T
(25)
4
4
,
,
/
,
,
1,
CH
CH
PtG PtG
CCPP
j t
j t
V
P
H
j
t
T
(26)
,
,
,
1,
j
j
PtG
PtG
PtG
PtG
PtG
CCPP
j t
y P
P
y P
j
t
T
(27)
The total CO 2 volume consumed by PtG during time period
t can be calculated as (25), where
PtG
is the conversion
efficiency of PtG from electricity to CH 4 ,
2
C O
(unit:
ton/4
CH
H
is the calorific value of
methane, usually takes 36MJ/m 3 ,
4
,
CH
j t
V
signifies produced CH 4
boundaries
PtG
P
and
PtG
P
multiplied by planned module
j
PtG
y .
E. Constraints on Facility Investment and Siting
,
0
,
j
PtG
m j
m GN
s
y M j CCPP
(28)
4
, ,
,
0
,
,
,
1,
CH
GN
CCPP
m j t
m j
V
s M m
j
t
T
(29)
4
4
,
, ,
Table II
IISummary of Compared Cases
Case
With
CCUS
Deterministic
planning
Stochastic
planning
Robust
planning
Case 1
×
×
×
×
Case 2
√
√
×
×
Case 3
√
×
√
×
Case 4
√
×
×
√
Table III .
IIIPLANNING RESULTS Contains CC, CS and PtG deviceCategory
Indices
Case 1
Case 2
Case 3
Case 4
CCUS *
-
36.61
72.58
428.69
CCUS sittings
-
50.74
50.74
50.74
Annualized
Investment Cost
(M$)
Total
-
87.35
123.32
479.43
Gas Sources
4809.61 4761.19 4760.28 4751.27
Generators
174.60
288.44
292.56
334.62
PtGs
-
0.90
1.79
10.58
Capture
-
723.68
705.26
470.37
Storage
-
120.34
117.00
75.22
Penalty
967.89
110.55
186.31
1588.40
Revenue
-
962.74
936.05
15.04
Operational Cost
(M$)
Total
5952.10 5042.37 5127.17 7215.44
Total Cost (M$)
5952.10 5129.72 5250.49 7694.87
Emission
19.36
2.21
3.10
13.23
Capture
-
24.12
23.51
15.68
Storage
-
24.06
23.40
15.04
Carbon-related
Volume (Mt)
Utilization
-
0.06
0.11
0.64
*
Table .
.Source of Key ParametersCCPP energy consumption of capturing per ton CO 2Parameter
Value
Source
CO 2 emission factor of TTPP emi
1005 g/KWh
Website [39]
Carbon capture cost
≤30 USD/ton
Carbon storage cost
≤10 USD/ton
Website [4]
Carbon tax
1~120 USD/ton
Carbon prize
1~80 USD/ton
Report [5]
C
W
0.269 MWh/ ton
CO 2 consumed volumn of PtG per unit output
2
C O
0.2 ton /MWh
Conversion efficiency of PtG PtG
0.6
Literature [37]-[38]
Climate change 2013: the physical science basis: Working Group I contribution to the Fifth assessment report of the Intergovernmental Panel on Climate Change. Cambridge university pressClimate change 2013: the physical science basis: Working Group I contribution to the Fifth assessment report of the Intergovernmental Panel on Climate Change[M]. Cambridge university press, 2014.
P Agreement, Paris, Report of the Conference of the Parties to the United Nations Framework Convention on Climate Change. Paris21st Session. RetrivedAgreement P. Paris agreement[C]//Report of the Conference of the Parties to the United Nations Framework Convention on Climate Change (21st Session, 2015: Paris). Retrived December. 2015, 4: 2017.
The Global Status of CCUS 2020: Vital to Achieve Net Zero. Brad Page, Page, Brad, et al. "The Global Status of CCUS 2020: Vital to Achieve Net Zero." (2020).
CCUS in Clean Energy Transitions. ParisIEAIEA (2020), CCUS in Clean Energy Transitions, IEA, Paris https://www.iea.org/reports/ccus-in-clean-energy-transitions.
State and trends of carbon pricing 2020. World BankWorld Bank. State and trends of carbon pricing 2020[J]. 2020.
A scalable infrastructure model for carbon capture and storage: SimCCUS. R S Middleton, J M Bielicki, Energy policy. 37Middleton R S, Bielicki J M. A scalable infrastructure model for carbon capture and storage: SimCCUS[J]. Energy policy, 2009, 37(3): 1052- 1060.
Optimal energy planning models with carbon footprint constraints. Ł M Pękala, R R Tan, D C Y Foo, J]. Applied Energy. 876Pękala Ł M, Tan R R, Foo D C Y, et al. Optimal energy planning models with carbon footprint constraints[J]. Applied Energy, 2010, 87(6): 1903- 1910.
Crisp and fuzzy integer programming models for optimal carbon sequestration retrofit in the power sector. R R Tan, D K S Ng, D C Y Foo, Tan R R, Ng D K S, Foo D C Y, et al. Crisp and fuzzy integer programming models for optimal carbon sequestration retrofit in the power sector[J].
Considering carbon capture and storage in electricity generation expansion planning. H Saboori, R Hemmati, Saboori H, Hemmati R. Considering carbon capture and storage in electricity generation expansion planning[J].
. IEEE Transactions on Sustainable Energy. 74IEEE Transactions on Sustainable Energy, 2016, 7(4): 1371-1378.
Low-Carbon Electricity Network Transition Considering Retirement of Aging Coal Generators. W Shen, J Qiu, K Meng, X Chen, Z Y Dong, 10.1109/TPWRS.2020.2995753IEEE Transactions on Power Systems. 35W. Shen, J. Qiu, K. Meng, X. Chen and Z. Y. Dong, "Low-Carbon Electricity Network Transition Considering Retirement of Aging Coal Generators," in IEEE Transactions on Power Systems, vol. 35, no. 6, pp. 4193-4205, Nov. 2020, doi: 10.1109/TPWRS.2020.2995753.
Power System Capacity Expansion Under Higher Penetration of Renewables Considering Flexibility Constraints and Low Carbon Policies. X Chen, J Lv, M B Mcelroy, X Han, C P Nielsen, J Wen, 10.1109/TPWRS.2018.2827003IEEE Transactions on Power Systems. 33X. Chen, J. Lv, M. B. McElroy, X. Han, C. P. Nielsen and J. Wen, "Power System Capacity Expansion Under Higher Penetration of Renewables Considering Flexibility Constraints and Low Carbon Policies," in IEEE Transactions on Power Systems, vol. 33, no. 6, pp. 6240-6253, Nov. 2018, doi: 10.1109/TPWRS.2018.2827003.
Pipeline networks for CCUS by static programming in the Chinese mainland. S U N Liang, C Wenying, Journal of Tsinghua University (Science and Technology. 556Liang S U N, Wenying C. Pipeline networks for CCUS by static programming in the Chinese mainland[J]. Journal of Tsinghua University (Science and Technology), 2015, 55(6): 678-683.
Effects of pipeline distance, injectivity and capacity on CO2 pipeline and storage site selection. Z Wang, G A F Weihs, P R Neal, Wang Z, Weihs G A F, Neal P R, et al. Effects of pipeline distance, injectivity and capacity on CO2 pipeline and storage site selection[J].
. International Journal of Greenhouse Gas Control. 51International Journal of Greenhouse Gas Control, 2016, 51: 95-105.
Optimization and Simulation of Collaborative Networks for Sustainable Production and Transportation. G Liotta, T Kaihara, G Stecca, 10.1109/TII.2014.2369351IEEE Transactions on Industrial Informatics. 12G. Liotta, T. Kaihara and G. Stecca, "Optimization and Simulation of Collaborative Networks for Sustainable Production and Transportation," in IEEE Transactions on Industrial Informatics, vol. 12, no. 1, pp. 417- 424, Feb. 2016, doi: 10.1109/TII.2014.2369351.
Optimal revamp of multi-region carbon capture and storage (CCUS) systems by two-step linear optimization. J F D Tapia, R R Tan, J]. Energy Systems. 62Tapia J F D, Tan R R. Optimal revamp of multi-region carbon capture and storage (CCUS) systems by two-step linear optimization[J]. Energy Systems, 2015, 6(2): 269-289.
A review: CO2 utilization. C H Huang, C S Tan, Aerosol and Air Quality Research. 14Huang C H, Tan C S. A review: CO2 utilization[J]. Aerosol and Air Quality Research, 2014, 14(2): 480-499.
CO2 utilization: Developments in conversion processes. E Alper, O Orhan, 3Alper E, Orhan O Y. CO2 utilization: Developments in conversion processes[J]. Petroleum, 2017, 3(1): 109-126.
Process integration approaches to optimal planning of unconventional gas field development. D C Y Foo, R E H Ooi, R R Tan, Chemical Engineering Science. 150Foo D C Y, Ooi R E H, Tan R R, et al. Process integration approaches to optimal planning of unconventional gas field development[J]. Chemical Engineering Science, 2016, 150: 85-93.
Process systems engineering perspective on the planning and development of oil fields. M S Tavallali, I A Karimi, D Baxendale, Tavallali M S, Karimi I A, Baxendale D. Process systems engineering perspective on the planning and development of oil fields[J]. 2016.
Selecting emerging CO2 utilization products for short-to mid-term deployment. R Chauvy, N Meunier, D Thomas, J]. Applied energy. 236Chauvy R, Meunier N, Thomas D, et al. Selecting emerging CO2 utilization products for short-to mid-term deployment[J]. Applied energy, 2019, 236: 662-680.
Development and application of a multi-stage CCUS source-sink matching model. L Sun, W Chen, J]. Applied Energy. 185Sun L, Chen W. Development and application of a multi-stage CCUS source-sink matching model[J]. Applied Energy, 2017, 185: 1424-1432.
Nationwide, regional, and statewide CO2 capture, utilization, and sequestration supply chain network optimization. M M F Hasan, F Boukouvala, E L First, Hasan M M F, Boukouvala F, First E L, et al. Nationwide, regional, and statewide CO2 capture, utilization, and sequestration supply chain network optimization[J].
. Industrial & Engineering Chemistry Research. 5318Industrial & Engineering Chemistry Research, 2014, 53(18): 7489-7506.
Optimization of CCUS supply chains in the UK: A strategic role for emissions reduction. G Leonzio, D Bogle, P U Foscolo, 155Chemical Engineering Research and DesignLeonzio G, Bogle D, Foscolo P U, et al. Optimization of CCUS supply chains in the UK: A strategic role for emissions reduction[J]. Chemical Engineering Research and Design, 2020, 155: 211-228.
Linear Models for Optimization of Infrastructure for CO2 Capture and Storage. B H Bakken, I Von Streng Velken, 10.1109/TEC.2008.921474IEEE Transactions on Energy Conversion. 23B. H. Bakken and I. von Streng Velken, "Linear Models for Optimization of Infrastructure for CO2 Capture and Storage," in IEEE Transactions on Energy Conversion, vol. 23, no. 3, pp. 824-833, Sept. 2008, doi: 10.1109/TEC.2008.921474.
Modeling Flexible Operation Mechanism of CO2 Capture Power Plant and Its Effects on Power-System Operation. Q Chen, C Kang, Q Xia, 10.1109/TEC.2010.2051948IEEE Transactions on Energy Conversion. 25Q. Chen, C. Kang and Q. Xia, "Modeling Flexible Operation Mechanism of CO2 Capture Power Plant and Its Effects on Power-System Operation," in IEEE Transactions on Energy Conversion, vol. 25, no. 3, pp. 853-861, Sept. 2010, doi: 10.1109/TEC.2010.2051948.
Optimal Flexible Operation of a CO2 Capture Power Plant in a Combined Energy and Carbon Emission Market. Q Chen, C Kang, Q Xia, D S Kirschen, 10.1109/TPWRS.2012.2185856IEEE Transactions on Power Systems. 27Q. Chen, C. Kang, Q. Xia and D. S. Kirschen, "Optimal Flexible Operation of a CO2 Capture Power Plant in a Combined Energy and Carbon Emission Market," in IEEE Transactions on Power Systems, vol. 27, no. 3, pp. 1602-1609, Aug. 2012, doi: 10.1109/TPWRS.2012.2185856.
Considering Carbon Capture and Storage in Electricity Generation Expansion Planning. H Saboori, R Hemmati, 10.1109/TSTE.2016.2547911IEEE Transactions on Sustainable Energy. 74H. Saboori and R. Hemmati, "Considering Carbon Capture and Storage in Electricity Generation Expansion Planning," in IEEE Transactions on Sustainable Energy, vol. 7, no. 4, pp. 1371-1378, Oct. 2016, doi: 10.1109/TSTE.2016.2547911.
Optimal Stochastic Operation of Integrated Low-Carbon Electric Power, Natural Gas, and Heat Delivery System. Y Li, 10.1109/TSTE.2017.2728098IEEE Transactions on Sustainable Energy. 9Y. Li et al., "Optimal Stochastic Operation of Integrated Low-Carbon Electric Power, Natural Gas, and Heat Delivery System," in IEEE Transactions on Sustainable Energy, vol. 9, no. 1, pp. 273-283, Jan. 2018, doi: 10.1109/TSTE.2017.2728098.
Interactive Energy-Saving Dispatch Considering Generation and Demand Side Uncertainties: A Chinese Study. J Liu, J Li, 10.1109/TSG.2016.2623947IEEE Transactions on Smart Grid. 9J. Liu and J. Li, "Interactive Energy-Saving Dispatch Considering Generation and Demand Side Uncertainties: A Chinese Study," in IEEE Transactions on Smart Grid, vol. 9, no. 4, pp. 2943-2953, July 2018, doi: 10.1109/TSG.2016.2623947.
Robust Scheduling of Smart Appliances in Active Apartments With User Behavior Uncertainty. K Paridari, A Parisio, H Sandberg, K H Johansson, 10.1109/TASE.2015.2497300IEEE Transactions on Automation Science and Engineering. 131K. Paridari, A. Parisio, H. Sandberg and K. H. Johansson, "Robust Scheduling of Smart Appliances in Active Apartments With User Behavior Uncertainty," in IEEE Transactions on Automation Science and Engineering, vol. 13, no. 1, pp. 247-259, Jan. 2016, doi: 10.1109/TASE.2015.2497300.
Low carbon oriented expansion planning of integrated gas and power systems. J Qiu, Z Y Dong, J H Zhao, K Meng, Y Zheng, D J Hill, IEEE Trans. Power Syst. 302J. Qiu, Z. Y. Dong, J. H. Zhao, K. Meng, Y. Zheng and D. J. Hill, "Low carbon oriented expansion planning of integrated gas and power systems", IEEE Trans. Power Syst., vol. 30, no. 2, pp. 1035-1046, 2014.
Carbon trading: Where greed is green. James Kanter, The New York Times. Kanter, James (2007-06-20). "Carbon trading: Where greed is green". The New York Times. Retrieved 2010-01-27
Game-theoretic planning for integrated energy system with independent participants considering ancillary services of power-togas stations. X Zhang, 176EnergyX. Zhang et al., "Game-theoretic planning for integrated energy system with independent participants considering ancillary services of power-to- gas stations", Energy, vol. 176, pp. 249-264, 2019.
Petroleum production engineering. B Guo, Guo B. Petroleum production engineering, a computer-assisted approach[M].
Optimal power and gas flow with a limited number of control actions. Sheng Chen, IEEE Transactions on Smart Grid. 9Chen, Sheng, et al. "Optimal power and gas flow with a limited number of control actions." IEEE Transactions on Smart Grid 9.5 (2017): 5371- 5380.
Environment-friendly and economical scheduling optimization for integrated energy system considering power-to-gas technology and carbon capture power plant. X Zhang, Y Zhang, Journal of Cleaner Production. 276123348Zhang X, Zhang Y. Environment-friendly and economical scheduling optimization for integrated energy system considering power-to-gas technology and carbon capture power plant[J]. Journal of Cleaner Production, 2020, 276: 123348.
Optimization Scheduling of Virtual Power Plant with Carbon Capture and Waste Incineration Considering Power-to-gas Coordination. Sun Huijuan, Liu Yun, Peng Chunhua, Jinhui, 10.13335/j.1000-3673.pst.2020.1720Power System Technology. in ChineseSUN Huijuan, LIU Yun, PENG Chunhua, MENG Jinhui, et al. Optimization Scheduling of Virtual Power Plant with Carbon Capture and Waste Incineration Considering Power-to-gas Coordination[J]. Power System Technology, doi:10.13335/j.1000-3673.pst.2020.1720. (in Chinese).
Multi-objective optimal day-ahead dispatch of integrated energy system considering power-to-gas operation cost. Wang Dong Shuai, Chengfu, Liang Jun, 42Automation of Electric Power Systems，. in ChineseDONG Shuai, WANG Chengfu, LIANG Jun, et al. Multi-objective optimal day-ahead dispatch of integrated energy system considering power-to-gas operation cost[J]. Automation of Electric Power Systems， 2018, 42(11):8-15, 121(in Chinese).
Two-stage Planning for Electricity-Gas Coupled Integrated Energy System with CCUS Considering Carbon Price Uncertainty. figshare. Ang Xuan, 10.6084/m9.figshare.14906655.v1Xuan, Ang (2021): Two-stage Planning for Electricity-Gas Coupled Integrated Energy System with CCUS Considering Carbon Price Uncertainty. figshare. Dataset. https://doi.org/10.6084/m9.figshare.14906655.v1
| [] |
[
"Pair correlation and twin primes revisited",
"Pair correlation and twin primes revisited"
] | [
"Jonathan P Keating [email protected] ",
"Brian Conrey \nAmerican Institute of Mathematics\n600 East Brokaw Road95112San JoseCAUSA\n\nSchool of Mathematics\nUniversity of Bristol\nBS8 1TWBristolUK\n",
"Jonathan P Keating \nSchool of Mathematics\nUniversity of Bristol\nBS8 1TWBristolUK\n"
] | [
"American Institute of Mathematics\n600 East Brokaw Road95112San JoseCAUSA",
"School of Mathematics\nUniversity of Bristol\nBS8 1TWBristolUK",
"School of Mathematics\nUniversity of Bristol\nBS8 1TWBristolUK"
] | [] | We establish a connection between the conjectural two-over-two ratios formula for the Riemann zetafunction and a conjecture concerning correlations of a certain arithmetic function. Specifically, we prove that the ratios conjecture and the arithmetic correlations conjecture imply the same result. This casts a new light on the underpinnings of the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe. | 10.1098/rspa.2016.0548 | null | 17,856,647 | 1604.06124 | efe3209feb824343e73740e980a22c221c7ff0c4 |
Pair correlation and twin primes revisited
Jonathan P Keating [email protected]
Brian Conrey
American Institute of Mathematics
600 East Brokaw Road95112San JoseCAUSA
School of Mathematics
University of Bristol
BS8 1TWBristolUK
Jonathan P Keating
School of Mathematics
University of Bristol
BS8 1TWBristolUK
Pair correlation and twin primes revisited
10.1098/rspa.2016.0548Received: 9 July 2016 Accepted: 2 September 2016Research Cite this article: Conrey B, Keating JP. 2016 Pair correlation and twin primes revisited. Proc. R. Soc. A 472: 20160548. Author for correspondence:Subject Areas: number theoryprime numbersmathematical physics Keywords: pair correlationrandom matrix theorytwin primes
We establish a connection between the conjectural two-over-two ratios formula for the Riemann zetafunction and a conjecture concerning correlations of a certain arithmetic function. Specifically, we prove that the ratios conjecture and the arithmetic correlations conjecture imply the same result. This casts a new light on the underpinnings of the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe.
Introduction
Montgomery in his famous pair correlation paper [1] used heuristics based on the Hardy-Littlewood conjecture concerning the distribution of prime pairs [2] to conclude that pairs of zeros of the Riemann zetafunction have the same scaled statistics, in the limit in which their height up the critical tends to infinity, as pairs of eigenvalues of large random Hermitian matrices (or of unitary matrices with Haar measure). Montgomery did not give the details of the calculation involving twin primes in his paper, but that calculation has been repeated with variations several times in the literature (e.g. [3][4][5][6][7]). Goldston & Montgomery [8] proved rigorously that the pair correlation conjecture is equivalent to an asymptotic formula for the variance of the number of primes in short intervals, and Montgomery & Soundararajan [9] proved that this variance formula follows from the Hardy-Littlewood prime-pair conjecture, under certain assumptions.
In a slightly different vein, Bogomolny & Keating [10,11] and later Conrey & Snaith [12] developed methods to give more precise estimates for the pair correlation (and higher correlations) of Riemann zeros. Bogomolny and Keating gave four different heuristic methods to accomplish this, while Conrey and Snaith used a uniform version of what is known as the ratios conjecture from which assumption they could rigorously derive this precise form of pair correlation. All of these methods lead to the same formulae.
In this paper, we reconsider this circle of ideas from yet another perspective, namely that of deriving a form of the ratios conjecture from consideration of correlations between the values of a certain arithmetic function. This provides a new perspective on the underpinnings of the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe [13][14][15]. This is similar to how, in a recent series of papers [16][17][18][19] we have shown that moment conjectures previously developed using random matrix theory [14,20] may be recovered from correlations of divisor sums.
The twin prime conjectures are easily stated in terms of the von Mangoldt function Λ(n) which is the generating function for −ζ /ζ (e.g. [21]):
− ζ ζ (s) = ∞ n=1 Λ(n) n s or equivalently Λ(n) = log p if n = p k for some prime p 0 otherwise.
In the Conrey-Snaith approach, zeros of ζ (s) are detected as poles of (ζ /ζ )(s) which in turn is realized via
ζ ζ (s) = d dα ζ (s + α) ζ (s + γ ) α=0 γ =0 .
Passing to coefficients, we write
I α,γ (s) = ∞ n=1 I α,γ (n) n s = ζ (s + α) ζ (s + γ ) ; explicitly I α,γ (n) = de=n μ(e) d α e γ .
Note that I α,γ (n) = n r I α+r,γ +r (n) for any r. Also we have
Λ(n) = − d dα I α,γ (n) α=0 γ =0
.
Here we will investigate the averages
R α,β,γ ,δ (T) := ∞ 0 ψ t T ζ (s + α)ζ (1 − s + β) ζ (s + γ )ζ (1 − s + δ) dt,
where s = 1 2 + it and ψ(z) is holomorphic in a strip around the real axis and decreases rapidly on the real axis. Not surprisingly, R is related to averages of the (analytic continuation of the) rspa.royalsocietypublishing.org Proc. R. Soc.
Rankin-Selberg convolution
B α,β,γ ,δ (s) := ∞ n=1 I α,γ (n)I β,δ (n) n s .
In fact, the simplest case of the ratios conjecture asserts that
R α,β,γ ,δ (T) = ∞ 0 ψ t T B α,βγ ,δ (1) + t 2π −α−β B −β,−α,γ ,δ (1) dt + O(T 1−η ) (1.1)
for some η > 0. It is also not surprising that R is connected to weighted averages over n and h of
I α,γ (n)I β,δ (n + h).
It is this connection that we are elucidating. Using the δ-method, it transpires that these weighted averages may be expressed in terms of
C α,β,γ ,δ (s) := 1 (2π i) 2 |w−1|= |z−1|= χ (w + z − s − 1) ∞ q=1 ∞ h=1 r q (h) h s+2−w−z × ∞ m=1 I α,γ (m)e(m/q) m w ∞ n=1 I γ ,δ (n)e(n/q) n z dw dz,
where r q (h) denotes Ramanujan's sum and where χ (s) is the factor from the functional equation
ζ (s) = χ (s)ζ (1 − s)
; also here and elsewhere is chosen to be larger than the absolute values of the shift parameters α, β, γ , δ but smaller than 1 2 . The result that ties this all together is the following identity.
Theorem 1.1. Assuming the generalized Riemann hypothesis
C α,β,γ ,δ (s) = B −β,−α,γ ,δ (s + 1).
In a recent series of papers [16][17][18][19], we have outlined a method that involves convolutions of coefficient correlations and leads to conclusions for averages of truncations of products of shifted zeta-functions implied by the recipe of [14]. In this paper, we strike out in a new direction, using similar ideas to evaluate averages of truncations of products of ratios of shifted zeta-functions. In particular, the approach of Bogolmony & Keating [6,7] on convolutions of shifted coefficient sums guide the calculations and we are led, as in the previous series, to formulate a kind of multidimensional Hardy-Littlewood circle method. This first paper, as indicated above, may be viewed in a more classical context.
It turns out to be convenient to study an average of the ratios conjecture. To this end, let
I α,γ (s; X) = n≤X I α,γ (n)n −s .
We are interested in the average over t of I α,γĪβ,δ in the case that X = T λ for some λ > 1. (When λ < 1 this average is dominated by diagonal terms.) We give two different treatments of the average of 'truncated' ratios:
M α,β,γ ,δ (T; X) := ∞ 0 ψ t T I α,γ (s, X)I β,δ (1 − s, X) dt,
(where again s = 1/2 + it) which lead to the same answer. The first is by the ratios conjecture and the second is by consideration of the correlations of the coefficients.
In each case, we prove the following theorem. correlations of values of I α,γ (n) (conjecture 5.1, §5), we have for some η > 0 and some λ > 1,
M α,β,γ ,δ (T; X) = ∞ 0 ψ t T 1 2π i s=2 B α,β,γ ,δ (s + 1) + t 2π −α−β−s B −β,−α,γ ,δ (s + 1) X s s ds dt + O(T 1−η ).
This shows that the ratios conjecture follows not only from the 'recipe' of [14,15], but also relates to correlations of values of I α,γ (n).
Approach via the ratios conjecture
We have
I α,γ (s, X) = 1 2π i (2) I α,γ (s + w) X w w dw;
there is a similar expression for I β,δ (s, X). Inserting these expressions and rearranging the integrations, we have
M α,β,γ ,δ (T; X) = 1 (2π i) 2 w=2 z=2 X w+z wz R α+w,β+z,γ +w,δ+z (T) dw dz.
We observe from expression (1.1) for the ratios conjecture that the integrand R α+w,β+z,γ +w,δ+z is, to leading order in T, expected to be a function of z + w. We therefore make the change of variable s = z + w; now the integration in the s variable is on the vertical line s = 4. We retain z as our other variable and integrate over it. This turns out to be the integral
1 2π i z=2 dz z(s − z) = 1 s
as is seen by moving the path of integration to the left to z = −∞. Thus, we have that M α,β,γ ,δ (T; X) is given to leading order by 1 2π i s=4 X s s R α+s,β,γ +s,δ (T) ds.
We move the path of integration to s = , avoiding crossing any poles, insert the ratios conjecture (1.1) (cf. the uniform version as laid out in [12]), and observe that B α+s,β,γ +s,δ (1) = B α,β,γ ,δ (s + 1).
In this way, we have that the uniform ratios conjecture implies the conclusion of theorem 1.2.
Approach via coefficient correlations
We follow the methodology developed by Goldston
(a) Diagonal
The diagonal term is
Tψ(0) m≤X I α,γ (m)I β,δ (m) m .
By Perron's formula, the sum here is We replace the arithmetic terms by their average and express this as
1 2π i (2) B α,2T X T 1≤h≤X/T I α,γ (m)I β,δ (m + h) m∼u uψ Th 2π u du.
We compute the average heuristically via the delta-method [22]:
I α,γ (m)I β,δ (m + h) m∼u ∼ ∞ q=1 r q (h) I α,γ (m)e m q m∼u I β,δ (m)e m q m∼u ,
where r q (h) is the Ramanujan sum, a formula for which is r q (h) = d|h d|q dμ(q/d); note that to actually prove this formula would be as difficult as proving the Twin Prime conjecture. We formalize this as a precise conjecture in §5. It is this conjecture that we refer to in theorem 1.2. Now
I α,γ (m)e m q m∼u = 1 2π i |w−1|= ∞ m=1 I α,γ (m)e m q m −w u w−1 dw.
Thus, the off-diagonal contribution is We make the change of variables v = Th/2π u. The inequality u ≤ X then implies that Th/2πv ≤ X or h ≤ 2πvX/T. The above can be re-expressed as
2T ∞ 0 1≤h≤2πvX/T 1 (2π i) 2 |w−1|= |z−1|= ∞ q=1 r q (h)ψ(v) Th 2πv w+z−2 × ∞ m 1 =1 I α,γ (m 1 )e(m 1 /q) m w 1 ∞ m 2 =1 I β,δ (m 2 )e(m 2 /q) m z 2 dw dz dv v .
Using Perron's formula to capture, the sum over h gives
2T ∞ 0 1 (2π i) 3 s=2 |w−1|= |z−1|= ∞ q=1 ∞ h=1 r q (h) h sψ (v) Th 2πv w+z−2 2πvX T s × ∞ m 1 =1 I α,γ (m 1 )e(m 1 /q) m w 1 ∞ m 2 =1 I β,δ (m 2 )e(m 2 /q) m z 2 ds s dw dz dv v . Now 2 ∞ 0ψ (v)v A dv v = χ (1 − A) ∞ 0 ψ(t)t −A dt.T ∞ 0 ψ(t) 1 (2π i) 3 s=2 |w−1|= |z−1|= ∞ q=1 ∞ h=1 r q (h) h s+2−w−z Tt 2π w+z−2 2π X tT s χ (w + z − s − 1) × ∞ m 1 =1 I α,γ (m 1 )e(m 1 /q) m w 1 ∞ m 2 =1 I β,δ (m 2 )e(m 2 /q) m z 2 ds s dw dz dt.
Hence, by theorem 1.1, this is
∞ 0 ψ t T 1 2π i s=2 t 2π −α−β−s B −β,−α,γ ,δ (s + 1) X s s ds dt.
Thus, adding the diagonal and off-diagonal terms we obtain that the conjecture for the correlations of values of I α,γ (n) also implies the conclusion of theorem 1.2.
Proof of theorem 1.1
First of all, we have
∞ h=1 r q (h) h A = ∞ h=1 g|q g|h gμ(q/g) h A = g|q g 1−A μ q g ζ (A) = q 1−A Φ(1 − A, q)ζ (A), where Φ(x, q) = p|q 1 − 1 p x .
Using this and the functional equation for ζ , we have to evaluate
1 (2π i) 2 |w−1|= |z−1|= ∞ q=1 q w+z−s−1 Φ(w + z − s − 1, q) × ζ (w + z − s − 1) ∞ m 1 =1 I α,γ (m 1 )e(m 1 /q) m w 1 ∞ m 2 =1 I β,δ (m 2 )e(m 2 /q) m z 2 dw dz.
We can identify the polar structure of the Dirichlet series here by passing to characters via the formula
e m q = d|m d|q 1 φ(q/d) χ mod (q/d) τ (χ )χ m d .
Assuming GRH, the only poles near w = 1 arise from the principal characters
χ (0) q/d . Using τ (χ (0) q/d ) = μ q d ,
we have that the poles of ∞ m=1 I α,γ (m)e(m/q)m −w are the same as the poles of for r = qe/d. We use a lemma from [23] which asserts that if
d|q μ(q/d) φ(q/d) ∞ m=1 I α,γ (md)χ (0) q/d (m)m −w d −w = q −w d|q μ(d) φ(d) d w ∞ m=1 I α,γ (mq/d)χq −w d|q μ(d)d w φ(d) e|d μ(e)e −w ∞ m=1 I α,γ (meq/d) m w .A(w) = B(w)C(w), where A(w) = ∞ m=1 (a(m)/m w ), B(w) = ∞ m=1 (b(m)/m w ) and C(w) = ∞ m=1 (c(m)/m w ) then ∞ m=1 a(mr) m w = r=r 1 r 2 ∞ m=1 b(mr 1 ) m w ∞ m=1(m,r 1 )=1 c(mr 2 ) m w .
We apply this identity with a(m)
= I α,γ (m), with b(m) = m −α and with c(m) = μ(m)m −γ . Then ∞ m=1 b(mr 1 ) m w = r −α 1 ζ (w + α) and (m,r 1 )=1 c(mr 2 ) m w = (m,r 1 )=1 μ(mr 2 ) m w+γ r γ 2 = μ(r 2 ) r γ 2 (m,r)=1 μ(m)m −w−γ = μ(r 2 )r −γ 2 Φ(w + γ , r)ζ (w + γ ) . Now r=r 1 r 2 μ(r 2 )r −α 1 r −γ 2 = r −α r=r 1 r 2 μ(r 2 )r α−γ 2 = r −α Φ(γ − α, r).
Thus,
∞ m=1 I α,γ (mr) m w = ζ (w + α)r −α Φ(γ − α, r) Φ(w + γ , r)ζ (w + γ ) .
In particular, we see that the only pole near to w = 1 is at w = 1 − α with residue
r −α Φ(γ − α, r) Φ(1 + γ − α, r)ζ (1 + γ − α)
.
Inserting this with r = qe/d into the above, we now have that
Res w=1−α ∞ m=1 I α,γ (m)e(m/q) m w = q α−1 d|q μ(d)d 1−α φ(d) e|d μ(e)e α−1 (qe/d) −α Φ(γ − α, qe/d) Φ(1 + γ − α, qe/d)ζ (1 + γ − α) = F α,γ (q) qζ (1 + γ − α) , where F α,γ (q) = q α d|q μ(d)d 1−α φ(d) e|d μ(e)e α−1 (qe/d) −α Φ(γ − α, qe/d) Φ(1 + γ − α, qe/d)
is a multiplicative function of q. At a prime p, we have With w = 1 − α and z = 1 − β, we see that our sum is
F α,γ (p) = p α p −α Φ(γ − α, p) Φ(1 + γ − α, p) − p 1−α p − 1 1 − p α−1 p −α Φ(γ − α, p) Φ(1 + γ − α, p) = Φ(γ − α, p) Φ(1 + γ − α, p) 1 + 1 p − 1 − p p − 1 = p (p − 1) Φ(γ − α, p) Φ(1 + γ − α, p) − 1 = p (p − 1) (1 − p α−γ ) (1 − p −1+α−γ ) − 1 = p (p − 1) (−p α−γ + p −1+α−γ ) (1 − p −1+α−γ ) = −p α−γ (1 − p −1+α−γ ) = −p α−γ + O 1 p .ζ (1 − α − β − s) ζ (1 − α + γ )ζ (1 − β + δ) ∞ q=1 q −1−α−β−s Φ(1 − α − β − s, q)F α,γ (q)F β,δ (q). Because of F α,γ (p) = −p α−γ + O(1/p), we have ∞ q=1 q −1−α−β−s Φ(1 − α − β − s, q)F α,γ (q)F β,δ (q) = ζ (1 + γ + δ + s)B α,β,γ ,δ (s),
where B is an Euler product that is absolutely convergent for s near 0. We claim that B α,β,γ ,δ (s) = A −β,−α−s,γ +s,δ . This is easily seen to be equivalent to showing that
B α,β,γ ,δ (0) = A −β,−α,γ ,δ .
To prove this, we first note that for j ≥ 2 we have
F α,γ (p j ) = p jα p −jα Φ(γ − α, p) Φ(1 + γ − α, p) − p 1−α p − 1 p −(j−1)α Φ(γ − α, p) Φ(1 + γ − α, p) − p α−1 p −αj Φ(γ − α, p) Φ(1 + γ − α, p) = Φ(γ − α, p) Φ(1 + γ − α, p) 1 − p (p − 1) + p α−1 = Φ(γ − α, p) Φ(1 + γ − α, p) − 1 (p − 1) + 1 (p − 1) = 0. Now the sum of the series ∞ j=0 p (−1−α−β)j Φ(1 − α − β, p j )F α,γ (p j )F β,δ (p j ) is just 1 + p −1−α−β Φ(1 − α − β, p)F α,γ (p)F β,δ (p) = 1 + (1 − 1/p 1−α−β ) p 1+α+β p α−γ (1 − p −1+α−γ ) p β−δ (1 − p −1+β−δ ) = 1 + (1 − 1/p 1−α−β ) p 1+γ +δ (1 − p −1+α−γ )(1 − p −1+β−δ ) = 1 − 1 p 1+γ +δ −1 B (p) α,β,γ ,δ (0), where B (p) α,β,γ ,δ (0) = 1 − 1 p 1+γ +δ 1 + (1 − 1/p 1−α−β ) p 1+γ +δ (1 − p −1+α−γ )(1 − p −1+β−δ )
.
The identity will be proven provided we can show that
1 + (1 − 1/p 1−α−β ) p 1+γ +δ (1 − p −1+α−γ )(1 − p −1+β−δ ) = (1 − 1/p 1−α+γ − 1/p 1−β+δ + 1/p 1+γ +δ ) (1 − 1/p 1−β+δ )(1 − 1/p 1−α+γ ) .
This is equivalent to showing that
1 + XCD(1 − X/AB) (1 − XC/A)(1 − XD/B) = (1 − XC/A − XD/B + XCD) (1 − XD/B)(1 − XC/A) ,
where X = 1/p; A = p −α ; B = p −β ; C = p −γ ; D = p −δ . This reduces to
1 − XC A 1 − XD B + XCD 1 − X AB = 1 − XC A − XD B + XCD or (A − XC)(B − XD) + XCD(AB − X) = AB − XC − XD + XABCD,
which is easily checked.
Conjecture 1
We can use the results of the previous two sections to formulate the conjecture that is part of the input for theorem 1.2. We expect I α,γ (n)I β,δ (n + h) for n near u to behave on average like The integrals over w and z are
F α,γ (q)u −α qζ (1 + γ − α) F β,δ (q)u −β qζ (1 + δ − β) ,
respectively. Thus, I α,γ (n)I β,δ (n + h) behaves like
n −α−β ζ (1 + γ − α)ζ (1 + δ − β) ∞ q=1 r q (h)F α,γ (q)F β,δ (q) q 2 .
In particular, we expect that
∞ n=1 I α,γ (n)I β,δ (n + h) n s − ζ (s + α + β) ζ (1 + γ − α)ζ (1 + δ − β) ∞ q=1 r q (h)F α,γ (q)F β,δ (q) q 2
is analytic in σ > σ 0 for some σ 0 < 1.
This leads us to the following conjecture.
m(x, h) = 1 ζ (1 + γ − α)ζ (1 + δ − β) ∞ q=1 r q (h)F α,γ (q)F β,δ (q) q 2 x 1−α−β 1 − α − β .
Conclusion
In subsequent papers, we will extend this process to averages of truncated ratios with any number of factors in the numerator and denominator.
2016 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited. rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
d
(m) m w and the principal parts are the same. We replace χ (0) d (m) by e|d e|m μ(e). Thus, we have
rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conjecture 5. 1 .
1There are numbers φ < 1 and ψ > 0 such thatn≤x I α,γ (n)I β,δ (n + h) = m(x, h) + O(x φ ) uniformly for h x ψ where
Theorem 1.2. Let α, β, γ , δ be complex numbers smaller than 1/4 in absolute value. Then, assuming either a uniform version of the ratios conjecture or a uniform version of a conjectured formula for rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .β,γ ,δ (s + 1)
X s
s
ds.
rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160548
(b) Off-diagonal
For the off-diagonal terms, we need to analyse
2T
T≤m≤X 1≤h≤X/T
I α,γ (m)I β,δ (m + h)
mψ
Th
2π m
.
rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Incorporating this formula leads us to
rspa.royalsocietypublishing.org Proc. R. Soc. A 472: 20160548 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements. J.P.K. is grateful to the American Institute of Mathematics for hospitality during a visit, where this work started.First of all, we note that the Rankin-Selberg Dirichlet series has an Euler productwhere A α,β,γ ,δ (s) = p 1 − 1/p s+γ +δ 1 − 1/p s+β+γ − 1/p s+α+δ + 1/p s+γ +δNow it is an easy exercise to calculate that M α,β,γ ,δ (T; X)A −β,γ +δ−α,−δ,δ (1) dtfor some η > 0.
The pair correlation of zeros of the zeta function. H L Montgomery, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Analytic number theory. Proc. Symp. St. Louis, MO, USA; Providence, RIAmerican Mathematical SocietyXXIVSt. Louis UniversityMontgomery HL. 1973 The pair correlation of zeros of the zeta function. In Analytic number theory. Proc. Symp. Pure Math., vol. XXIV, St. Louis University, St. Louis, MO, USA, 1972, pp. 181- 193. Providence, RI: American Mathematical Society. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes. G H Hardy, J E Littlewood, 10.1007/BF02422942Acta Math. 412. Hardy GH, Littlewood JE. 1918 Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes. Acta Math. 41, 119-196. (doi:10.1007/BF02422942)
Über Die Montgomery'she Paarvermutung, Diplomarbeit. J Bolanz, 131Bolanz. J. 1987 Über Die Montgomery'she Paarvermutung, Diplomarbeit, 131 pages.
Quantum chaology and the Riemann zeta-function. J P Keating, ; G Casati, Guarneri, Smilansky, Quantum chaos. Amsterdam, The NetherlandsNorth-HollandKeating JP. 1993 Quantum chaology and the Riemann zeta-function. In Quantum chaos (eds G Casati, I Guarneri, U Smilansky), pp. 145-185. Amsterdam, The Netherlands: North-Holland.
Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series. D A Goldston, S M Gonek, Acta Arith. 84Goldston DA, Gonek SM. 1998 Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series. Acta Arith. 84, 155-192.
Random matrix theory and the Riemann zeros I: three-and four-point correlations. E B Bogomolny, J P Keating, 10.1088/0951-7715/8/6/013Nonlinearity. 8Bogomolny EB, Keating JP. 1995 Random matrix theory and the Riemann zeros I: three-and four-point correlations. Nonlinearity 8, 1115-1131. (doi:10.1088/0951-7715/8/6/013)
Random matrix theory and the Riemann zeros II: n-point correlations. E B Bogomolny, J P Keating, 10.1088/0951-7715/9/4/006Nonlinearity. 9Bogomolny EB, Keating JP. 1996 Random matrix theory and the Riemann zeros II: n-point correlations. Nonlinearity 9, 911-935. (doi:10.1088/0951-7715/9/4/006)
Pair correlation of zeros and primes in short intervals. D A Goldston, H L Montgomery, Analytic number theory and Diophantine problems. Stillwater, OK; Boston, MABirkhäuser Boston70Goldston DA, Montgomery HL. 1987 Pair correlation of zeros and primes in short intervals. In Analytic number theory and Diophantine problems (Stillwater, OK, 1984), pp. 183-203. Progress in Mathematics, vol. 70. Boston, MA: Birkhäuser Boston.
Primes in short intervals. H L Montgomery, K Soundararajan, 10.1007/s00220-004-1222-4Commun. Math. Phys. 252Montgomery HL, Soundararajan K. 2004 Primes in short intervals. Commun. Math. Phys. 252, 589-617. (doi:10.1007/s00220-004-1222-4)
Gutzwiller's trace formula and spectral statistics: beyond the diagonal approximation. E B Bogomolny, J P Keating, 10.1103/PhysRevLett.77.1472Phys. Rev. Lett. 77Bogomolny EB, Keating JP. 1996 Gutzwiller's trace formula and spectral statistics: beyond the diagonal approximation. Phys. Rev. Lett. 77, 1472-1475. (doi:10.1103/PhysRevLett.77.1472)
A method for calculating spectral statistics based on randommatrix universality with an application to the three-point correlations of the Riemann zeros. E B Bogomolny, J P Keating, 10.1088/1751-8113/46/30/305203J. Phys. A. 46305203Bogomolny EB, Keating JP. 2013 A method for calculating spectral statistics based on random- matrix universality with an application to the three-point correlations of the Riemann zeros. J. Phys. A 46, 305203. (doi:10.1088/1751-8113/46/30/305203)
Applications of the L-functions ratios conjectures. J B Conrey, N C Snaith, 10.1112/plms/pdl021Proc. Lond. Math. Soc. 943Conrey JB, Snaith NC. 2007 Applications of the L-functions ratios conjectures. Proc. Lond. Math. Soc. (3) 94, 594-646. (doi:10.1112/plms/pdl021)
Autocorrelation of random matrix polynomials. J B Conrey, D W Farmer, J P Keating, M O Rubinstein, N C Snaith, 10.1007/s00220-003-0852-2Commun. Math. Phys. 237Conrey JB, Farmer DW, Keating JP, Rubinstein MO, Snaith NC. 2003 Autocorrelation of random matrix polynomials. Commun. Math. Phys. 237, 365-395. (doi:10.1007/s00220- 003-0852-2)
Integral moments of L-functions. J B Conrey, D W Farmer, J P Keating, M O Rubinstein, N C Snaith, 10.1112/S0024611504015175Proc. Lond. Math. Soc. 91Conrey JB, Farmer DW, Keating JP, Rubinstein MO, Snaith NC. 2005 Integral moments of L-functions. Proc. Lond. Math. Soc. 91, 33-104. (doi:10.1112/S0024611504015175)
Autocorrelation of ratios of L-functions. J B Conrey, D W Farmer, M R Zirnbauer, 10.4310/CNTP.2008.v2.n3.a4Commun. Number Theory Phys. 2Conrey JB, Farmer DW, Zirnbauer MR. 2008 Autocorrelation of ratios of L-functions. Commun. Number Theory Phys. 2, 593-636. (doi:10.4310/CNTP.2008.v2.n3.a4)
Moments of zeta and correlations of divisor-sums: I. Phil. J B Conrey, J P Keating, 10.1098/rsta.2014.0313Trans. R. Soc. A. 373Conrey JB, Keating JP. 2015 Moments of zeta and correlations of divisor-sums: I. Phil. Trans. R. Soc. A 373, 20140313. (doi:10.1098/rsta.2014.0313)
Moments of zeta and correlations of divisor-sums: II. J B Conrey, J P Keating, Advances in the Theory of Numbers, Proc. of the 13th Conf. of the Canadian Number Theory Association, Fields Institute Communications. Alaca, S Alaca, KS Williams; Berlin, GermanySpringerConrey JB, Keating JP. 2015 Moments of zeta and correlations of divisor-sums: II. In Advances in the Theory of Numbers, Proc. of the 13th Conf. of the Canadian Number Theory Association, Fields Institute Communications (eds A Alaca, S Alaca, KS Williams), pp. 75-85. Berlin, Germany: Springer. (http://arxiv.org/abs/1506.06843)
Moments of zeta and correlations of divisor-sums: III. J B Conrey, J P Keating, 10.1016/j.indag.2015.04.005Indagationes Mathematicae. 26Conrey JB, Keating JP. 2015 Moments of zeta and correlations of divisor-sums: III. Indagationes Mathematicae 26, 736-747. (doi:10.1016/j.indag.2015.04.005)
J B Conrey, J P Keating, Moments of zeta and correlations of divisor-sums: IV. Conrey JB, Keating JP. 2016 Moments of zeta and correlations of divisor-sums: IV. (http://arxiv.org/abs/1603.06893)
Random matrix theory and ζ ( 1 2 + it). J P Keating, N C Snaith, 10.1007/s002200000261Commun. Math. Phys. 214Keating JP, Snaith NC. 2000 Random matrix theory and ζ ( 1 2 + it). Commun. Math. Phys. 214, 57-89. (doi:10.1007/s002200000261)
Are there infinitely many twin primes?. D A Goldston, PreprintGoldston DA. Are there infinitely many twin primes? Preprint.
A quadratic divisor problem. W Duke, J B Friedlander, H Iwaniec, 10.1007/BF01231758Invent. Math. 115Duke W, Friedlander JB, Iwaniec H. 1994 A quadratic divisor problem. Invent. Math. 115, 209-217. (doi:10.1007/BF01231758)
Simple zeros of the Riemann zeta-function. J B Conrey, A Ghosh, S M Gonek, 10.1112/S0024611598000306Proc. Lond. Math. Soc. 76Conrey JB, Ghosh A, Gonek SM. 1998 Simple zeros of the Riemann zeta-function. Proc. Lond. Math. Soc. 76, 497-522. (doi:10.1112/S0024611598000306)
| [] |
[
"Novel magnetic state in d 4 Mott insulators",
"Novel magnetic state in d 4 Mott insulators"
] | [
"O Nganba Meetei \nDepartment of Physics\nThe Ohio State University\n43210ColumbusOhioUSA\n",
"William S Cole \nDepartment of Physics\nThe Ohio State University\n43210ColumbusOhioUSA\n",
"Mohit Randeria \nDepartment of Physics\nThe Ohio State University\n43210ColumbusOhioUSA\n",
"Nandini Trivedi \nDepartment of Physics\nThe Ohio State University\n43210ColumbusOhioUSA\n"
] | [
"Department of Physics\nThe Ohio State University\n43210ColumbusOhioUSA",
"Department of Physics\nThe Ohio State University\n43210ColumbusOhioUSA",
"Department of Physics\nThe Ohio State University\n43210ColumbusOhioUSA",
"Department of Physics\nThe Ohio State University\n43210ColumbusOhioUSA"
] | [] | We show that the interplay of strong Hubbard interaction U and spin-orbit coupling λ in systems with the d 4 electronic configuration leads to several unusual magnetic phases. While in the atomic limit the system is in a non-magnetic J = 0 singlet state, we find that the competition between superexchange and atomic spin-orbit coupling dramatically changes the local moment, which challenges the conventional wisdom that local moments are well-defined in a Mott insulator. Most notably, we find that in the Mott limit at strong U there is a phase transition from a non-magnetic insulator of uncoupled J = 0 singlets to an orbitally entangled ferromagnetic insulator. We identify candidate materials and present predictions for Resonant X-ray Scattering (RXS) signatures for the unusual magnetism in d 4 Mott insulators and contrast them with the well-studied d 5 case. arXiv:1311.2823v1 [cond-mat.str-el] | 10.1103/physrevb.91.054412 | [
"https://arxiv.org/pdf/1311.2823v2.pdf"
] | 119,259,889 | 1311.2823 | 856c4a25e1ac83ad6109c84ef974e19b879e9cae |
Novel magnetic state in d 4 Mott insulators
(Dated: November 13, 2013)
O Nganba Meetei
Department of Physics
The Ohio State University
43210ColumbusOhioUSA
William S Cole
Department of Physics
The Ohio State University
43210ColumbusOhioUSA
Mohit Randeria
Department of Physics
The Ohio State University
43210ColumbusOhioUSA
Nandini Trivedi
Department of Physics
The Ohio State University
43210ColumbusOhioUSA
Novel magnetic state in d 4 Mott insulators
(Dated: November 13, 2013)
We show that the interplay of strong Hubbard interaction U and spin-orbit coupling λ in systems with the d 4 electronic configuration leads to several unusual magnetic phases. While in the atomic limit the system is in a non-magnetic J = 0 singlet state, we find that the competition between superexchange and atomic spin-orbit coupling dramatically changes the local moment, which challenges the conventional wisdom that local moments are well-defined in a Mott insulator. Most notably, we find that in the Mott limit at strong U there is a phase transition from a non-magnetic insulator of uncoupled J = 0 singlets to an orbitally entangled ferromagnetic insulator. We identify candidate materials and present predictions for Resonant X-ray Scattering (RXS) signatures for the unusual magnetism in d 4 Mott insulators and contrast them with the well-studied d 5 case. arXiv:1311.2823v1 [cond-mat.str-el]
Introduction: The interplay of strong interactions and spin-orbit coupling (SOC) has recently taken center stage in condensed matter research. The discovery of topological band insulators [1] has fueled tremendous interest in materials with strong spin-orbit coupling, while strong correlations have produced exciting phenomena, ranging from high T c superconductivity [2] to colossal magnetoresistance [3]. These two features naturally combine in the 5d transition metal materials, which hold the potential of hosting new phases of matter with entangled spin, orbital and charge degrees of freedom. Already there are many predictions for exotic topological matter, for example the topological Mott insulators [4] and Weyl semi-metals [5]. Recent experiments demonstrating that Sr 2 IrO 4 is an unusual Mott insulator with a half filled J = 1/2 band resulting from strong SOC [6] have prompted the search for Weyl semi-metals in iridium pyrochlores [7].
We emphasize that most of the focus in this field to date has been on iridium based materials with a d 5 electronic configuration. In the limit of strong spin orbit coupling the t 2g orbitals are split into J = 1/2 and J = 3/2 multiplets. For the d 5 configuration, all of the J = 3/2 states are filled and the only low-energy degree of freedom is a single hole in the J = 1/2 manifold which is sufficient to understand most phenomena. This picture is often given in terms of non-interacting atomic levels, but it also holds in the strongly interacting limit for the d 5 configuration.
The physics is dramatically different for other fillings. Mott insulators with d 1 and d 2 configuration have been shown to exhibit exotic magnetic phases [8,9] in the presence of large SOC. In the d 3 case, SOC is quenched in a cubic environment [10] and the problem reduces to a conventional spin-only model. This leaves only the d 4 case, which has been largely ignored because it is expected that large SOC and strong interactions give rise to a non-magnetic state in the atomic limit [9], though see [11].
We show that contrary to the above expectation, the d 4 configuration has a rich magnetic phase diagram as a function of SOC and Hubbard U . In particular, at large U there is a quantum phase transition from the expected non-magnetic insulator of local J = 0 singlets to a ferromagnetic insulator with a nonzero local moment as SOC is reduced. This moment arises from superexchangeinduced mixing, at a single site, of magnetic states largely from the higher energy triplet with the singlet. Our result provides an interesting counterexample to the commonly held notion that Mott insulators have well defined local moments that cannot be affected by small perturbations (compared to the interaction scale U ). We then present predictions for Resonant X-ray Scattering (RXS), which has proven to be an indispensable tool for studying local magnetic states. Considering resonant enhancement coming from virtual excitations to the J = 1/2 intermediate states in the d-orbitals, we find that the scattering cross-section at the L 2 edge is about an order of magnitude larger than at the L 3 edge. This is an indication of weight in the local state coming from the J = 1 triplet. In contrast, experiments on the d 5 Sr 2 IrO 4 find suppressed signal at the L 2 edge.
The paper is organized as follows: we start by describing the multi-orbital Hubbard model relevant for oxides with d 4 configuration. We then diagonalize it exactly for a two-site system with strong SOC and large U , which helps us to build intuition about the problem. In the Mott limit, the two-site calculation is expected to provide a reliable description of local properties and the nature of the magnetic interactions between neighboring sites. It also provides the basis for calculating RXS cross-sections which can be directly used to verify the unusual ferromagnetic Mott insulator. Following the exploration of the local physics, we propose an effective magnetic model on a lattice that leads to a Ginzburg-Landau description of the magnetic phase transition. We conclude by identifying some specific candidate materials among the ruthenates and osmates with a d 4 electron count where this phase transition could be observed by using strain to modify the effective interaction strength and SOC.
Hamiltonian: We consider a three-orbital Hubbard model with SOC that captures the essence of 4d 4 or 5d 4 materials with an octahedral cage of oxygen surround-ing each transition metal ion (as occurs in, e.g., the perovskites, double perovkites and pyrochlores). The crystal field splitting between the t 2g orbitals and the e g orbitals is the largest energy scale and for d 4 filling, only the t 2g levels are relevant. The Hamiltonian has the following form:
H = H hop + i (H i,U + H i,SOC )(1)
where
H hop = ij αβ σσ t ασ,βσ ij c † iασ c jβσ + h.c.(2)H i,SOC = λ αβ σσ s · l ασ,βσ c † iασ c iβσ(3)H i,U = U α n iα↑ n iα↓ + (U − 3J H ) α>β,σ n iασ n iβσ +(U − 2J H ) α>β,σ n iασ n iβσ −J H α>β c † iα↑ c † iβ↓ c iβ↑ c iα↓ +c † iα↑ c † iα↓ c iβ↓ c iβ↑ + h.c.(4)
Here c † iασ (c iασ ) creates(annihilates) an electron at site i in orbital α with spin σ. t ασ,βσ ij is the hopping matrix element from the state βσ at site j to ασ at site j. For simplicity we will consider only nearest neighbor hopping, and take it to be diagonal in both spin and orbital space (t ασ,βσ ij → t ij δ αβ δ σσ ). U , J H and λ are intra-orbital interaction strength, Hund's coupling and SOC strength respectively. s·l ασ,βσ are the matrix elements of atomic SOC in the t 2g basis. Note that the t 2g orbitals have an effective orbital momentum L = −1. Unless mentioned otherwise, all energy scales are measured in units of t.
Two-site results: We solve the multi-orbital Hubbard model described in Eq. 1 for a two-site system using exact diagonalization. As a function of λ and U the two-site system shows three different magnetic states as shown in Fig. 1(a): (i) a non-magnetic state (J = 0) in the large λ limit, (ii) a ferromagnet with J = 2 for small λ and moderate U and (iii) a different ferromagnet with J = 1 at large values of U and small λ. The magnetic phases can be understood easily in some limiting cases. The J = 0 state at large λ and small U corresponds to a band insulator with a completely filled J = 3/2 manifold. The post-perovskite material NaIrO 3 and perovskites BaOsO 3 and CaOsO 3 are believed to be in such a state [12,13]. With increasing U, this band insulator smoothly crosses over into a J = 0 Mott insulator, consistent with recent Gutzwiller and Dynamical Mean-Field Theory calculations [13]. We will discuss the nonmagnetic Mott insulator in greater detail below. In the limit of small λ and moderate U , the J = 2 ferromagnet is essentially the Stoner ferromagnet seen in SrRuO 3 [14]. The most interesting phase is the J = 1 ferromagnet observed at large U and small λ, which has been overlooked completely because the naive atomic limit analysis predicts a non-magnetic insulator. However, its existence can be understood as follows. For large U , each site has a well defined |L i |=1 and |S i |=1 as shown in Fig. 1(b). In Fig. 2 we sketch out why the superexchange interaction is ferromagnetic, and therefore stabilizes the total S = 2 state for the two-site problem. A thorough analysis is presented in the Supplementary material [15]. This sketch also helps explain why the total orbital momentum is not maximized. In fact, a total L = 1 state is realized, and including SOC in the L − S coupling scheme yields the observed J = 1 state. In sharp contrast, an atomic limit analysis will give J i = 0 singlet states at each site in accordance with Hund's third rule with the caveat that L i =-1 for t 2g orbitals. Having established that a J = 1
(b) (a) d 4 -d 4 virtual d 3 -d 5 d 4 -d 4 virtual d 3 -d 5
FIG. 2: Schematic rationalization for ferromagnetic superexchange. For the antiferromagnetic configuration (a), the intermediate state has a higher energy, due to Hund's coupling on the d 3 site, than the corresponding state in the ferromagnetic case (b). To make use of these virtual processes, the "down" electrons in (b) must also occupy different orbitals.
ferromagnetic state must exist for small λ, we next turn to an analysis of the competition between the ferromagnetic superexchange interaction and SOC and show that it leads to a phase transition from an unusual ferromag- netic Mott insulator to a non-magnetic insulator with increasing λ as shown in Fig. 1(b). More importantly, the local J moment changes from J i ≈ 1 to J i ≈ 0. This challenges the generally accepted notion that the local moment is a robust quantity in Mott insulators. A generalization of the two-site result to the thermodynamic limit will be discussed below. RXS scattering cross-section: We now make predictions for RXS cross-sections, which can be used to identify the ferromagnetic insulator. For Mott insulators, RXS matrix elements are usually calculated in the free ion approximation [6,16]. However, to include non-local effects, which we will show is crucial for understanding the d 4 ferromagnetic Mott insulator, we need to generalize the expression for RXS amplitude as follows [15]
∆f (ω) ∝ Tr ρ n (e .D) † |ψ n ψ n |e.D E n − E G − ω − iΓ n(5)
where ρ is the reduced density matrix at the scattering site, and the trace is over atomic states in the d 4 configuration. e(e ) is the incoming(outgoing) polarization, D is the dipole operator, |ψ n is an excited state (in the d 5 configuration) with energy E n , and E G is the ground state energy. Γ n is the inverse life time of the excited state |ψ n . The L 2 and L 3 edges corresponds to excitations from 2p 1/2 and 2p 3/2 levels to the intermediate d 5 states respectively. The d 5 states are also split by SOC into J = 1/2 and J = 3/2 multiplets, with a single hole occupying the J = 1/2 levels being lower in energy. Because of this we consider only the resonant enhancement from these J = 1/2 states. The magnetic scattering cross-section, as a function of scattering angle, is calculated from the two-site model in the ferromagnetic state. The results are shown in Fig. 3(a). If ρ had contributions only from the J = 0 state, which would be the case in the atomic limit, there could not be any magnetic scattering. The non-zero signal results from significant contribution to ρ from the higher energy local J = 1 states. This has significant consequences for the RXS cross-section: the L 2 edge is enhanced by about an order of magnitude more than the L 3 edge. This is in sharp contrast from the d 5 case, relevant for iridates, also shown in Fig. 3(b). The L 2 edge is suppressed in iridates.
In Fig. 4(a) and (b) we show a comparison between two-site and atomic σ − σ scattering cross-sections for the L 2 and L 3 edges respectively. In the d 4 configuration, the two-site result deviates significantly from the atomic result for the ferromagnetic phase. For the nonmagnetic phase, the atomic and two-site results agree. For comparison, we have also shown the atomic and twosite calculations for the d 5 configuration in Fig. 4, where the atomic limit calculation is adequate. The picture that emerges from our analysis is that the non-magnetic phase is well described by the atomic picture with J = 0 singlets at each site while the ferromagnetic phase implies significant occupation of the local J = 1 states. In the next section, we use this insight to construct an effective spin-orbital model and produce a Ginzburg-Landau theory for the phase transition.
Magnetic Hamiltonian and Ginzburg-Landau theory: The simplest spin-orbital Hamiltonian that we can write down which includes SOC and superexchange mediated by only the lowest-lying virtual state is (see Supplementary material for details [15])
H = − J F M 2 ij S i · S j P(L i + L j = 1) + λ 2 i L i · S i (6)
where S i = 1 and L i = 1 are local S and L moments at site i. From the Goodenough-Kanamori analysis shown in Fig. 2, it is easy to see that the superexchange scale J F M ∼ O(t 2 /U ). Each bond is projected by P(L i + L j = 1) on to the total L = 1 space, while the factor of 1/2 in the SOC term comes from rewriting the SOC Hamiltonian in the L − S coupling scheme relevant for the d 4 configuration. While the combination of spinorbit interaction and ferromagnetic exchange are sufficient to produce a phase transition, the projection operator is necessary to capture the orbital entanglement in the ferromagnet. We have checked explicitly against the exact diagonalization results that this model accurately describes the magnetic phase transition at the two-site level shown in Fig. 1(a) [15]. As described earlier, the non-magnetic phase can be understood quite simply as a local singlet at each site. The ferromagnetic phase, on the other hand, can be described as a Bose-Einstein condensation (BEC) of magnetic excitations to the J = 1 state, induced by the ferromagnetic superexchange interaction. To this end we define the operators s † i which creates a singlet at site i while T † i,(0,±) creates a triplet carrying J z = 0, ±1 at site i [17,18]. By calculating the matrix elements of S and L operators in the singlet-triplet space, we get
S α i = − 2 3 T † iα s i + s † i T iα − i 2 αβγ T † iβ T iγ L α i = 2 3 T † iα s i + s † i T iα − i 2 αβγ T † iβ T iγ(7)
where α, β, γ = x, y or z,
T † iz = T † i0 , T † ix = −(T † i1 − T † i−1 )/ √ 2 and T † iy = i(T † i1 +T † i−1 )/ √ 2.
Upon substituting these operators in the Hamiltonian, we obtain an effective action from the saddle point approximation [19]. Close to the magnetic phase transition, we assume s i ≈ 1 and T iα = φ iα 1, and thereby obtain the Ginzburg-Landau functional, up to second order in φ iα , as
L = λ 2 iα [φ * iα φ iα ] 1 0 0 1 φ iα φ * iα −ηJ F M ij ,α [φ * iα φ iα ] 1 a a 1 φ jα φ * jα(8)
where η and a are parameters of O(1) which depend on details of the model. This can be solved easily by a Bogoliubov transformation which gives a gap function ∆ k = (λ/2 − ηJ F M f (k)) 2 − (aηJ F M f (k)) 2 where f (k) = δ cos(k.δ) and δ is nearest-neighbor position. For ferromagnetic superexchange, it closes at k = 0 when λ c /J F M = 2zη(1 + |a|) or λ c ∼ zO(t 2 /U ) where z is the coordination number, thereby generating a condensate of triplets in the ground state and consequently a phase transition into a ferromagnet. Materials: We propose candidate materials from the double perovskite family which can be tuned across the magnetic transition by chemical substitution and/or pressure. They have the general formula A 2 BB O 6 where A is an alkaline earth element while B and B are two different transition metal ions, ordered in a 3D chequerboard pattern. If we choose the B sites to have completely filled shells, the bandwidth is suppressed, giving rise to a Mott insulator. When combined with a 4d or 5d element on the B site, we have the ideal model system where large SOC competes with J F M . Of particular interest to us is La 2 ZnRuO 6 which is an insulator with Ru in d 4 configuration. Two different samples grown by two different groups have shown very different magnetic states. One group has found a ferromagnetic state with T C ≈ 165K [21], while the other found a non-magnetic state [22]. We believe La 2 ZnRuO 6 is very close to the phase boundary and small differences in the lattice parameter may be the origin of this discrepancy. An RXS study under pressure will be an ideal experiment to observe the phase transition. Another closely related material is La 2 MgRuO 6 [21], which is also a promising candidate.
Comments: During the final stages of our work we came across Ref. [11] which also identifies a superexchange-driven magnetic phase transition in a d 4 Mott insulator. However, there are significant differences in our results. We obtain a phase transition from a nonmagnetic insulator to an entangled ferromagnetic insulator, consistent with the Goodenough-Kanamori analysis, whereas Ref. [11] predicts an anti-ferromagnetic phase and further identifies Ca 2 RuO 4 as a candidate material. Experimentally Ca 2 RuO 4 is known to show weak ferromagnetism [20] whose explanation would require additional DzyaloshinskiiMoriya type interactions. Our proposal, on the other hand, naturally explains the weak ferromagnetism as arising from superexchange that is itself ferromagnetic.
Conclusion: We have provided evidence for a d 4 ferromagnetic Mott insulator, despite the fact that each site in the atomic limit has no moment. RXS scattering cross sections are a direct probe of this unusual entangled magnetic state and we predict, in sharp contrast to the d 5 iridates, that the L 3 edge magnetic scattering is suppressed by an order of magnitude compared to the L 2 edge.
While we have focussed on the unusual ferromagnetic state at large U and λ, the phase diagram in Fig. 1(a) is considerably richer, allowing for a broader exploration of magnetic and metal-insulator phase transitions.
Acknowledgements: We thank Patrick Woodward In this supplement we provide details on the following: (1) Derivation and validation of the minimal spin-orbital Hamiltonian used to describe the magnetic phase transition from a non-magnetic phase with local J = 0 singlets to a novel ferromagnetic phase in d 4 Mott insulators. (2) A general theory for calculating Resonant X-ray Scattering amplitude including non-local effects.
MAGNETIC HAMILTONIAN
As discussed in the main text, the sign of the superexchange is ferromagnetic and it can be easily understood from a Goodenough-Kanamori analysis in the λ = 0 limit. Here, we present a detailed derivation and validation of the spin-orbital Hamiltonian in Eq. 6 of the main text.
In the atomic limit with no SOC, the d 4 Mott insulator has L i = 1 and S i = 1 at each site. For two sites, the ground state with d 4 −d 4 configuration is a direct product state
|Ψ GS = |S 1 = 1 ⊗ |L 1 = 1 ⊗ |S 2 = 1 ⊗ |L 2 = 1 (1)
which can give rise to total L = 2, 1 or 0 and total S = 2, 1 or 0, all of which are degenerate with energy E G . From second order perturbation theory, the magnetic exchange term that captures the correction to the atomic ground state energy has the form
H = H hop n |ψ n ψ n | E G − E n H hop(2)
where
H hop = −t α,σ (c † 1ασ c 2ασ + h.c.)(3)
is the kinetic energy and |ψ n is the intermediate exited atomic state with d 3 − d 5 configuration and energy E n . Let us now examine the excited states. The ground state for d 3 has L i = 0 and S i = 3/2 while d 5 has L i = 1 and S i = 1/2 in its ground state. So, the lowest lying excited state, which we will call |ψ 1 , can have total L = 1 and total S = 2 or 1. If |ψ 1 contributes to the exchange interaction in Eq. 2, it will be the dominant term. To see if |ψ 1 contributes to the magnetic exchange energy, we need to examine how H hop connects |Ψ GS to the excited state. The form of H hop in Eq. 3 is invariant under rotations in both spin and orbital space. It, therefore, commutes with totalL 2 and totalŜ 2 operators and only connects states with the same total L and where
J i = | ψ 1 |H hop |Ψ GS (L = 1, S = i) | 2 |E G − E 1 |(5)
Next, we are interested in describing correctly only the ground state and the first excited state. We will show later higher energy states are not important for understanding the magnetic phase transition shown in Fig. 1(b) of the main text. We can, then, replace the spin projection operators in Eq. 4 by the Heisenberg form as follows
H ≈ −J F M S 1 · S 2 P(L 1 + L 2 = 1)(6)
where J F M = (J 2 − J 1 )/2. From the Goodenough-Kanamori analysis and also from exact diagonalization, we know that J 2 > J 1 and, therefore, J F M > 0. P is the same as |L = 1 L = 1| which projects the total L of the arXiv:1311.2823v1 [cond-mat.str-el] 8 Nov 2013 two sites to L = 1, and it has the form
P(L 1 + L 2 = 1) = (1 − L 1 .L 2 )(2 + L 1 .L 2 ) 2 (7)
We can add to Eq. 6 the spin-orbit coupling term and generalize to a lattice in order to obtain the desired spinorbital Hamiltoniañ
H = −J F M ij S i · S j P(L i + L j = 1) + λ 2 i L i · S i (8)
We note that in our analysis, the intermediate oxygen orbitals are not taken into account explicitly. Including them, however, does not change the ferromagnetic nature of superexchange. We now present numerical results to show that the minimal spin-orbital Hamiltonian in Eq. 8 describes the magnetic phase transition as shown in Fig. 1(b) of the main text. In Fig. 1(a) we have reproduced the exact diagonalization results for two site Hubbard model (See Eq. 1 of main text) and in Fig. 1(b) we present exact diagonalization results for two sites of the magnetic Hamiltonian in Eq. 8. By comparing the two plots, it is clear that our minimal spin-orbital Hamiltonian captures the competition between superexchange and SOC. Not only does it accurately describe the phase transition from a non-magnetic (J = 0) state insulator to a ferromagnetic (J = 1) state, it also captures the non-trivial changes in the local moment (J i ≈ 0 in the non-magnetic state and J i ≈ 1 in the ferromagnet).
RESONANT X-RAY SCATTERING
In this section, we describe the general theory of resonant x-ray scattering (RXS) that we have used to calculate the results shown in Figs. 3 and 4 of the main text. The starting point is the scattering amplitude. Within second order perturbation theory and the dipole approximation, the resonant scattering amplitude has the following form [1]
∆f (ω) ∝ n Ψ GS |(e .D) † |ψ n ψ n |e.D|Ψ GS E n − E G −hω − iΓ n(9)
where |Ψ GS is the ground state with energy E G and |ψ n is an excited state with energy E n . Γ n corresponds to the inverse lifetime of the particular excited state |ψ n and e(e ) is the polarization of the incoming(outgoing) X-ray photon.
It is convenient to write the dipole operator D in second quantized form to facilitate calculation of the matrix elements in the numerator of Eq. (9). For the L 2(3) edge, absorbing a photon promotes a core 2p electron to the valence d shell e.D ≈ e ·r = αβσ e · d α |r|p β d † ασ p βσ + h.c.
where the d α |r|p β ≡ R αβ are easily determined by symmetry and tabulated
R ∝ |p x |p y |p z d yz | 0ẑŷ d zx |ẑ 0x d xy |ŷx 0 (11)
The proportionality constant depends on fine details of the atomic states, but symmetry dictates that it should be the same for all combinations of p and d orbitals.
Hence it is an overall constant which we hereafter ignore. Free ion approximation: A common practice in calculating RXS matrix elements is to approximate the scattering site as a free ion [1,2]. This usually gives a good description for Mott insulators where the scattering amplitude is primarily determined by local properties. The effect of the lattice comes only through the geometrical structure factor. Within the free ion approximation, the ground state in Eq. 9 is replaced by the atomic ground state and the excited states are replaced by atomic excited states.
Non-local effects: When interaction between different sites have significant effect on local properties, as in the case of the d 4 ferromagnetic Mott insulator, the free ion approximation breaks down. Substituting the ground state in Eq. 9 by the atomic ground state is no longer a good approximation. However, it turns out the excited states can still be substituted by the atomic exited states because the core hole generates an additional binding energy for the excited electron [1,3].
To include the non-local effects correctly, we need to write the ground state as a direct product of states defined only on the scattering site |ψ n and states defined on the rest of the lattice |φ n |Ψ GS = pq a pq |ψ p |φ q (12) Substituting this into Eq. 9, the matrix element in the numerator becomes Ψ GS |(e .D) † |ψ n ψ n |e.D|Ψ GS = pqrs a * rs a pq ψ r |(e .D) † |ψ n ψ n |e.D|ψ p φ s |φ q = pr ρ pr ψ r |(e .D) † |ψ n ψ n |e.D|ψ p = Tr ρ(e .D) † |ψ n ψ n |e.D
where ρ pr = q a * rq a pq is the reduced density matrix at the scattering site. Finally, we get the desired expression for the resonant scattering amplitude with non-local effects included correctly ∆f (ω) ∝ Tr ρ n (e .D) † |ψ n ψ n |e.D E n − E G −hω − iΓ n (14)
FIG. 1 :
1(a) Magnetic phase diagram of the two-site d 4 system in the U −λ plane. It consists of three phases: a non-magnetic (J = 0) phase and two different ferromagnetic phases (J = 2 and J = 1). (b) A magnetic phase transition occurs as a function of λ in the Mott limit (U/t = 40 and JH = 0.2U ). The total J-moment changes from J = 1 to J = 0 at λc/t ≈ 0.185. The local J-moment also changes from Ji ≈ 1 to Ji ≈ 0 at the phase transition. The local Si and Li moments, on the other hand, do not change across this phase transition.
FIG. 3 :
3Panel (a) shows the magnetic (σ − π) RXS cross section for the d 4 J = 1 ferromagnet as a function of scattering angle. Results for both the L2 and L3 edges are plotted. For comparison, the magnetic RXS scattering cross-section for the d 5 Mott insulator with half-filled J = 1/2 band is shown in panel (b). The L2 edge for the d 5 insulator is suppressed in comparison to the L3 edge; in sharp contrast, the L3 edge is suppressed in the d 4 ferromagnet.
FIG. 4 :
4Comparison of the RXS cross-section between atomic limit and two-site calculations. Panels (a) and (b) show the nonmagnetic (σ − σ) scattering cross-sections as a function of λ at the L2 edge and L3 edge respectively. Results for both d 4 and d 5 configurations are shown. In the d 5 case, the atomic calculation indeed gives a good description of the Mott insulator. However, for the d 4 configuration, the ferromagnetic superexchange interaction between neighbors strongly alters the local state.
FIG. 1 :
1Comparison of exact diagonalization results of the multi-orbital Hubbard model (a) with the spin-orbital Hamiltonian (b). The two plots are practically indistinguishable, which proves that the minimal spin-orbital Hamiltonian in Eq. 8 captures the competition between superexchange interaction and SOC. It accurately describes the phase transition from a non-magnetic state (J = 0) to an orbitally entangled ferromagnetic state (J = 1) as a function of λ. total S. |ψ 1 has L = 1, and consequently the energy gain from the exchange term is maximized if |Ψ GS is in the L = 1 state. Similarly, only the S = 2 and S = 1 components of |Ψ GS gains energy via virtual excitations to |ψ 1 . If we consider only the dominant contribution from |ψ 1 to the magnetic exchange energy, the magnetic Hamiltonian in Eq. 2 can be written in terms of spin and orbital projection operators as H ≈ −J 2 |S = 2 |L = 1 L = 1| S = 2| −J 1 |S = 1 |L = 1 L = 1| S = 1| (4)
. M Z Hasan, C L Kane, Rev. Mod. Phys. 823045M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
. P A Lee, N Nagaosa, X.-G Wen, Rev. Mod. Phys. 7817P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006).
. M B Salamon, M Jaime, Rev. Mod. Phys. 73583M. B. Salamon and M. Jaime, Rev. Mod. Phys. 73, 583 (2001).
. D Pesin, L Balents, Nat Phys. 6376D. Pesin and L. Balents, Nat Phys 6, 376 (2010).
. X Wan, A M Turner, A Vishwanath, S Y Savrasov, Phys. Rev. B. 83205101X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011).
. B J Kim, H Ohsumi, T Komesu, S Sakai, T Morita, H Takagi, T Arima, Science. 3231329B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi, and T. Arima, Science 323, 1329 (2009).
. K Ueda, J Fujioka, Y Takahashi, T Suzuki, S Ishiwata, Y Taguchi, Y Tokura, Phys. Rev. Lett. 109136402K. Ueda, J. Fujioka, Y. Takahashi, T. Suzuki, S. Ishiwata, Y. Taguchi, and Y. Tokura, Phys. Rev. Lett. 109, 136402 (2012).
. G Chen, R Pereira, L Balents, Phys. Rev. B. 82174440G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82, 174440 (2010).
. G Chen, L Balents, Phys. Rev. B. 8494420G. Chen and L. Balents, Phys. Rev. B 84, 094420 (2011).
. O N Meetei, O Erten, M Randeria, N Trivedi, P Woodward, Phys. Rev. Lett. 11087203O. N. Meetei, O. Erten, M. Randeria, N. Trivedi, and P. Woodward, Phys. Rev. Lett. 110, 087203 (2013).
. G Khaliullin, arXiv e-print 1310.0767G. Khaliullin, arXiv e-print 1310.0767 (2013).
. L Du, X Sheng, H Weng, X Dai, Europhys. Lett. 10127003L. Du, X. Sheng, H. Weng, and X. Dai, Europhys. Lett. 101, 27003 (2013).
. L Du, L Huang, X Dai, Eur. Phys. J. B. 861L. Du, L. Huang, and X. Dai, Eur. Phys. J. B 86, 1 (2013).
. I I Mazin, D J Singh, Phys. Rev. B. 562556I. I. Mazin and D. J. Singh, Phys. Rev. B 56, 2556 (1997).
. J Fink, E Schierle, E Weschke, J Geck, Rep. Prog. Phys. 7656502J. Fink, E. Schierle, E. Weschke, and J. Geck, Rep. Prog. Phys. 76, 056502 (2013).
. S Sachdev, R N Bhatt, Phys. Rev. B. 419323S. Sachdev and R. N. Bhatt, Phys. Rev. B 41, 9323 (1990).
. T Giamarchi, C Regg, O Tchernyshyov, Nat Phys. 4198T. Giamarchi, C. Regg, and O. Tchernyshyov, Nat Phys 4, 198 (2008).
J W Negele, H Orland, Quantum Many-particle Systems. Westview PressJ. W. Negele and H. Orland, Quantum Many-particle Systems (Westview Press, 2008).
. S Nakatsuji, S Ikeda, Y Maeno, J. Phys. Soc. Jpn. 661868S. Nakatsuji, S.-i. Ikeda, and Y. Maeno, J. Phys. Soc. Jpn. 66, 1868 (1997).
. R I Dass, J.-Q Yan, J B Goodenough, Phys. Rev. B. 6994416R. I. Dass, J.-Q. Yan, and J. B. Goodenough, Phys. Rev. B 69, 094416 (2004).
. K Yoshii, N Ikeda, M Mizumaki, Phys. Stat. Sol.(a). 203K. Yoshii, N. Ikeda, and M. Mizumaki, Phys. Stat. Sol.(a) 203 (2006).
As shown in Fig. 4 of the main text, the influence of the lattice on the scattering site through the super-exchange interaction is crucial in understanding the ferromagnetic d 4 Mott insulator. As shown in Fig. 4 of the main text, the influence of the lattice on the scattering site through the super-exchange interaction is crucial in understanding the ferromagnetic d 4 Mott insulator.
. J Fink, E Schierle, E Weschke, J Geck, Rep. Prog. Phys. 7656502J. Fink, E. Schierle, E. Weschke, and J. Geck, Rep. Prog. Phys. 76, 056502 (2013).
. B J Kim, H Ohsumi, T Komesu, S Sakai, T Morita, H Takagi, T Arima, Science. 3231329B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi, and T. Arima, Science 323, 1329 (2009).
. J P Hannon, G T Trammell, M Blume, D Gibbs, Phys. Rev. Lett. 611245J. P. Hannon, G. T. Trammell, M. Blume, and D. Gibbs, Phys. Rev. Lett. 61, 1245 (1988).
| [] |
[
"Geometric phases in open tripod systems",
"Geometric phases in open tripod systems"
] | [
"Ditte Møller \nDepartment of Physics and Astronomy\nLundbeck Foundation Theoretical Center for Quantum System Research\nUniversity of Aarhus\nDK-8000Denmark\n",
"Lars Bojer Madsen \nDepartment of Physics and Astronomy\nLundbeck Foundation Theoretical Center for Quantum System Research\nUniversity of Aarhus\nDK-8000Denmark\n",
"Klaus Mølmer \nDepartment of Physics and Astronomy\nLundbeck Foundation Theoretical Center for Quantum System Research\nUniversity of Aarhus\nDK-8000Denmark\n"
] | [
"Department of Physics and Astronomy\nLundbeck Foundation Theoretical Center for Quantum System Research\nUniversity of Aarhus\nDK-8000Denmark",
"Department of Physics and Astronomy\nLundbeck Foundation Theoretical Center for Quantum System Research\nUniversity of Aarhus\nDK-8000Denmark",
"Department of Physics and Astronomy\nLundbeck Foundation Theoretical Center for Quantum System Research\nUniversity of Aarhus\nDK-8000Denmark"
] | [] | We first consider stimulated Raman adibatic passages (STIRAP) in a closed four-level tripod system. In this case, the adiabatic eigenstates of the system acquire real geometric phases. When the system is open and subject to decoherence they acquire complex geometric phases that we determine by a Monte Carlo wave function approach. We calculate the geometric phases and the state evolution in the closed as well as in the open system cases and describe the deviation between these in terms of the phases acquired. When the system is closed, the adiabatic evolution implements a Hadamard gate. The open system implements an imperfect gate and hence has a fidelity below unity. We express this fidelity in terms of the acquired geometric phases. | 10.1103/physreva.77.022306 | [
"https://arxiv.org/pdf/0710.0450v1.pdf"
] | 119,233,552 | 0710.0450 | 0f564fc1fef7886c0d940a2dbad73080438e44cd |
Geometric phases in open tripod systems
2 Oct 2007 (Dated: February 2, 2008)
Ditte Møller
Department of Physics and Astronomy
Lundbeck Foundation Theoretical Center for Quantum System Research
University of Aarhus
DK-8000Denmark
Lars Bojer Madsen
Department of Physics and Astronomy
Lundbeck Foundation Theoretical Center for Quantum System Research
University of Aarhus
DK-8000Denmark
Klaus Mølmer
Department of Physics and Astronomy
Lundbeck Foundation Theoretical Center for Quantum System Research
University of Aarhus
DK-8000Denmark
Geometric phases in open tripod systems
2 Oct 2007 (Dated: February 2, 2008)numbers: 0367Lx0365Vf0365Yz
We first consider stimulated Raman adibatic passages (STIRAP) in a closed four-level tripod system. In this case, the adiabatic eigenstates of the system acquire real geometric phases. When the system is open and subject to decoherence they acquire complex geometric phases that we determine by a Monte Carlo wave function approach. We calculate the geometric phases and the state evolution in the closed as well as in the open system cases and describe the deviation between these in terms of the phases acquired. When the system is closed, the adiabatic evolution implements a Hadamard gate. The open system implements an imperfect gate and hence has a fidelity below unity. We express this fidelity in terms of the acquired geometric phases.
I. INTRODUCTION
Each eigenstate of a non-degenerate quantum system that evolves adiabatically in time acquires a well-defined phase in addition to the usual dynamic phase. The former is called the geometric phase because it depends on the path traversed in Hilbert space [1]. With the current interest in quantum information and computation, the geometric phase has received new attention because it is expected to be robust against some sources of decoherence [2,3,4,5] and hence useful, e.g., for creation of quantum gates. To study its robustness under decoherence it is essential to generalize the concept of geometric phase to open quantum systems. Various proposals have been made [6,7] and they all point to the problem that phase information tends to be lost when the system is open and decoheres, e.g., due to spontaneous emission.
The full system including decoherence can still be described, for example, by the density matrix approach, which predicts the relative phases between the involved basis states, but the information about the phases acquired by each eigenstate is not available nor is the information about the geometric or dynamic nature of the phase. In this work we are interested in the phase dynamics of the states of the physical system when the system is subject to decoherence. To this end we consider the Monte Carlo wave function (MCWF) approach [8,9]. We follow each wave function trajectory and calculate the complex geometric phases that are acquired by the adiabatic eigenstates. The MCWF approach has the advantage that we gain information about the evolution of the single trajectories and the dynamic or geometric nature of the phases. Not surprisingly, on average the trajectories reproduce the density matrix result. We show how the evolution of the open system can be described by adiabatic eigenstates that acquire complex geometric phases and we discuss the deviation from the closed system case * Electronic address: [email protected] in terms of the phases acquired. We consider a tripod system with three laser fields applied and present a full calculation of the adiabatic evolution of the wave function of the system. The adiabatic evolution considered consist of a sequence of stimulated Raman adiabatic passages (STIRAP) [10] and implements the Hadamard gate. We quantify the effect of decoherence in terms of the fidelity of this gate and identify the role of the complex geometric phases.
The paper is organized as follows. In Sec. II we review the theory of geometric phases, we explain their appearance in STIRAP processes and present the tripod system. In Sec. III we present calculations for the closed system and in Sec. IV we derive the full solution for the evolution of the open system evolution and calculate the fidelity of the Hadamard gate. Sec. V concludes.
II. BACKGROUND
A. Geometric phase
The adiabatic theorem states that for a given set of instantaneous eigenstates, ψ n (t), and eigenenergies, E n (t), of a timedependent Hamiltonian, there is no population transfer between the eigenstates if these vary slowly compared with the energy difference between eigenstates [11]
∂ψ n ∂t ≪ |E n − E m | .(1)
In this adiabatic regime the eigenstates do not only acquire a dynamic phase, ϑ n = t f ti E n (t ′ )/ dt ′ , but also a geometric phase, γ n
Ψ(0) = n c n (0)ψ n (0) ad −→ Ψ(t) = n c n (0)e i(ϑn+γn) ψ n (t),(2)
where the geometric part of the phase can be calculated directly from the eigenstates [1],
γ n = i R f Ri ψ n (R)|∇R|ψ n (R) · dR.(3)
HereR are time-dependent parameters of the Hamiltonian, and the geometric phase becomes an integral in the space of these parameters (See Eq. (7) below for a specific example).
B. STIRAP in lambda system
STIRAP is an efficient, adiabatic process for population transfer in three-level systems. In the method two laser fields are applied to the atomic lambda system [see Fig. 1(a)] [10,12,13,14,15,16,17,18]. The instantaneous adiabatic dressed states for this system are two bright states and one dark state. The explicit form of the dark state (|D ) with zero energy eigenvalue, ω D = 0, reads
|D = Ω 2 |Ω 0 | 2 + |Ω 2 | 2 |0 − Ω 0 |Ω 0 | 2 + |Ω 2 | 2 |2 (4) = cos θ|0 − sin θe iϕ2 |2 ,
where Ω 0 and Ω 2 are Rabi frequencies (see Fig. 1), tan θ = |Ω 0 |/|Ω 2 | and ϕ 2 is the time-dependent phase difference between the two laser fields. The real amplitudes of the fields, A 0 (t) and A 2 (t) are time-dependent and the Rabi frequencies explicitly read
Ω 0 (t) = A 0 (t),(5)Ω 2 (t) = A 2 (t)e −iϕ2(t) .
The pulses are modeled by sin 2 -pulses with different amplitudes
A j (t) = A max,j sin 2 ( π(t−tsj ) 2τ ) if t sj < t < t sj + 2τ 0 otherwise ,(6)
where t sj is the instant of time when the pulse starts and τ the FWHM. With all population initially in the |0 -state and only Ω 2 applied the system is in this dark state. Now, adiabatically increasing Ω 0 and decreasing Ω 2 causes population transfer from |0 to |2 , while all population remains in |D of Eq. (4). This implies that the excited state, |e is never populated and hence no population is lost due to spontaneous emission.
STIRAP is well-suited for studying geometric phases, because no dynamic phase is acquired, ω D = 0. The geometric phase is calculated by Eq. (3) withR = (θ, ϕ 2 ), . 1: (a) Three-level lambda system with two laser fields applied with Rabi frequencies Ω0 and Ω2. (b) Pulse sequence consisting of two STIRAP processes separated by ∆T in time. The first set of pulses transfers population from |0 to |2 , while the second set transfers it back. We show the real amplitudes, A0 and A2 of the Rabi frequencies Ω0 and Ω2 as defined in Eq. (5). With this pulse sequence sin θ = Ω0/( p |Ω0| 2 + |Ω2| 2 ) is zero before the first pair of pulses arrive, one in between the to processes and zero after the second pair of pulses. The FWHM of each pulse is τ and the delay between pulses within one process is ∆t.
γ D = i R f Ri D|∇R|D · dR = − t f tiφ 2 sin 2 θdt. (7) t Dt t A 2 A 2 DT t a t b t a +DT t +DT b A 0 A 0 |2> |e> W 2 |0> W 0 (b) (a) FIG
To obtain a non-zero γ D , the phase difference between the two laser fields, ϕ 2 , should be controlled and have a timedependence with non-vanishingφ 2 . In order to be able to control the actual value of γ D we require that sin 2 θ = 0 before and after the pulse sequence, ensuring that only during the sequence a geometric phase is acquired. As stated in Eq. (4) sin 2 θ = |Ω 0 | 2 /(|Ω 0 | 2 + |Ω 2 | 2 ), which implies that during one STIRAP process sin 2 θ is increased from zero to one. Applying a second STIRAP process, where Ω 0 is decreased while Ω 2 is increased, decreases sin 2 θ from one to zero. This second process transfers the population from |2 back to |0 . The whole pulse sequence is shown in Fig. 1(b) and after this sequence the system ends up in
|D = e iγD |0 ,(8)
where γ D is calculated form Eq. (7) with t i = t a and t f = t b + ∆T as defined in Fig. 1(b).
C. STIRAP in tripod system
In the lambda system described above the geometric phase acquired by the dark state, |D is a collective phase on the two atomic states, |0 and |2 . We now turn to the tripod system shown in Fig. 2(a). This system has the advantage that two dark states are populated and these acquire different geometric phases leading to relative phases between the atomic states as well as changes in their population due to changes in the interference of the two dark states. In the tripod system we can therefore calculate the geometric phase and see how it affects the measurable populations and relative phases between the atomic states. The tripod system further constitute a system, where a universal set of quantum gates can be implemented (see, e.g., [19] and references therein). The level structure consists of three lower states (|0 , |1 and |2 ) coupled to an excited state, |e , with The first set of pulses transfers population from {|0 , |1 } to |2 , while the second set transfers it back. We show the real amplitudes, A0, A1 and A2 of the Rabi frequencies Ω0, Ω1 and Ω2 as defined in Eq. (9). The FWHM of each pulse is τ and the delay between pulses within one process is ∆t. three laser fields. These have Rabi frequencies Ω 0 , Ω 1 and Ω 2 , respectively, and for simplicity we assume that they are all on resonance. We restrict ourselves to the case where the relative phase between Ω 0 and Ω 1 , ϕ 01 , is time-independent while the phase of Ω 2 , ϕ 2 (t), is timedependent. The real amplitudes of all fields, A 0 (t), A 1 (t) and A 2 (t) are time-dependent and the Rabi frequencies explicitly read
t Dt t A 2 A 2 A 1 DT t a t b t a +DT t +DT b A 1 A 0 A 0 |1> |2> |e> W 1 W 2 |0> W 0 (b) (a)Ω 0 (t) = A 0 (t),(9)Ω 1 (t) = A 1 (t)e −iϕ01
,
Ω 2 (t) = A 2 (t)e −iϕ2(t) .
The pulses are as in the lambda-system case modeled by the sin 2 -pulses in Eq. (6). When we apply the pulse sequence shown in Fig. 2(b), the STIRAP process transfers part of the population from |0 and |1 to |2 and back. During this process geometric phases are acquired. Calculations of the evolution of the system is presented in Sec. III for a closed system and in Sec. IV for an open system.
III. CLOSED SYSTEM
In the rotating wave approximation, and with Rabi frequencies as defined in Eq. (9), we derive the Hamiltonian
H(t) = 2 0 0 A 0 (t) 0 0 0 A 1 (t)e iϕ01 0 A 0 (t) A 1 (t)e −iϕ01 0 A 2 (t)e −iϕ2(t) 0 0 A 2 (t)e iϕ2(t) 0 (10)
expressed in the {|0 , |1 , |e , |2 } basis. We parameterize the complex Rabi frequencies as
Ω 0 (t) = sin θ 01 A 0 (t) 2 + A 1 (t) 2 ,(11)Ω 1 (t) = cos θ 01 A 0 (t) 2 + A 1 (t) 2 e −iϕ01 ,(12)Ω 2 (t) = cos θ H (t) A 0 (t) 2 + A 1 (t) 2 + A 2 (t) 2 e −iϕ2(t) ,(13)
where the two angles are defined as
tan θ 01 = A 0 (t)/A 1 (t),(14)
tan
θ H (t) = A 2 0 (t) + A 2 1 (t)/A 2 (t).(15)
A diagonalization of Eq. (10) gives the four energy eigenvalues
ω ± = ± 1 2 A 2 0 + A 2 1 + A 2 2 , ω Di = 0 (i = 1, 2),(16)
and eigenvectors
|+ = 1 √ 2 sin θ H (t)(sin θ 01 |0 + cos θ 01 e iϕ01 |1 ) − |e + cos θ H (t)e iϕ2(t) |2 ,(17)|− = 1 √ 2 sin θ H (t)(sin θ 01 |0 + cos θ 01 e iϕ01 |1 ) + |e + cos θ H (t)e iϕ2(t) |2 , |D 1 = − cos θ H (t)(sin θ 01 |0 + cos θ 01 e iϕ01 |1 ) + sin θ H (t)e iϕ2(t) |2 , |D 2 = cos θ 01 |0 − sin θ 01 e iϕ01 |1 .
We assume that we start out in a superposition of the two dark eigenstates at time t i
|D(t i ) = C D1 (t i )|D 1 (t i ) + C D2 (t i )|D 2 (t i ) ,(18)
and that the evolution is adiabatic. Then the population stays within the space spanned by the two dark states and at later times the wave function is given by
|D(t) = C D1 (t)|D 1 (t) + C D2 (t)|D 2 (t) .(19)
In order to determine the time evolution of the coefficients {C D1 (t), C D2 (t)} we follow Refs. [12,20]. Inserting Eq. (19) into the time-dependent Schrödinger equation yields two coupled differential equations that can readily be solved and we find the very simple evolution
C D1 (t) = e iγD 1 C D1 (t i ), (20) C D2 (t) = C D2 (t i ).
Here the phase
γ D1 = − t tiφ 2 sin 2 θ H dt ′ ,(21)
acquired by |D 1 is purely geometric because the dark states do not acquire any dynamic phases, ω Di = 0. Since no population is transferred between the two dark states the geometric phase could also be calculated using Eq. (3). This approach was used in [19]. STIRAP (Sec. II B) ensures control of the geometric phases. The exact pulse sequence is shown in Fig. 2(b). The first set of pulses transfers population partially from |0 and |1 to |2 while the second transfers all population back to |0 and |1 . The amplitudes of Ω 0 and Ω 1 are such that θ 01 is kept constant. The amplitude of Ω 2 is A max,2 = A 2 max,0 + A 2 max,1 and the pulse of Ω 2 is delayed with respect to the pulses of Ω 0 and Ω 1 and hence sin θ H (t) is varied from 0 to 1 when the first set of pulses arrive, while the second set of pulses adiabatically turns the sin θ H (t) factor back to 0. After the whole pulse sequence the system ends up in the final state
|D(t f ) =C D1 (t i )e iγD 1 (t f ) |D 1 (t f ) + C D2 (t i )|D 2 (t f ) (22) =[− sin θ 01 C D1 (t i )e iγD 1 + cos θ 01 C D2 (t i )]|0 + [− cos θ 01 C D1 (t i )e iγD 1 + sin θ 01 C D2 (t i )]|1 ,
where we have used Eq. (17) in the second line. In the {|0 , |1 }-basis, an arbitrary initial state
|ψ i = a i |0 + b i |1 is transferred by the unitary matrix U to a final state |ψ f = U |ψ i = a f |0 + b f |1 with U = cos 2 θ 01 + e iγD 1 sin 2 θ 01 cos θ 01 sin θ 01 e −iϕ01 (e iγD 1 − 1) cos θ 01 sin θ 01 e iϕ01 (e iγD 1 − 1) sin 2 θ 01 + e iγD 1 cos 2 θ 01 .(23)
By carefully adjusting the amplitudes and phases of the laser fields, the values of θ 01 , ϕ 01 and γ D1 can be controlled and thus generate rotations in the {|0 , |1 }-basis. We note that U is the identity when no geometric phase is acquired, γ D1 = 0. As a special case θ 01 = π 8 , φ 01 = π and γ D1 = −π implement a Hadamard gate
U = 1 √ 2 1 1 1 −1 .(24)
The value θ 01 = π 8 is obtained by choosing
A max,0 = A max,1 ( √ 2 − 1), A max,2 = A 2 max,0 + A 2 max,1 and ϕ 2 = t/τ , ∆T /τ = π assures γ D1 = −π.
All simulations presented throughout this work will use these parameters and the initial state |ψ i = |0 which is transferred to the final state |ψ f = U |ψ i = (|0 + |1 )/ √ 2 in the closed system case.
IV. OPEN SYSTEM
We use STIRAP to transfer population among |0 , |1 and |2 , while keeping the population in the excited state |e negligible. Decoherence due to spontaneous emission will therefore have very little effect, while dephasing caused by, e.g., collisions or phase fluctuations of the laser fields, will influence the evolution. The evolution of the open system can be found by solving the Lindblad master equation [21],
ρ = − i [H, ρ] − 1 2 m (C † m C m ρ + ρC † m C m ) + m C m ρC † m ,(25)
where H is the Hamiltonian for the closed system and the decoherence is described by the Lindblad operators, C m . The Lindblad master equation results in an ensemble average of the evolution, but does not reveal a clear distinction between the geometric and dynamic phases. We wish to follow the evolution of the wave functions, the acquired geometric and dynamic phases and how these affect the relative phases between and the population in the atomic states. Towards this end we use the quantum jump approach, where the wave function is evolved stochastically [8,9]. For a small timestep ∆t the evolution of the wave function is described as either a jump to C m |ψ(t) or by a no-jump evolution with the non-Hermitian Hamiltonian
H = H + H ′ , where H ′ = −i /2 m C † m C m .
After a timestep with either the jump or the no-jump evolution the wave function is normalized. The probability for a jump to C m |ψ(t) in ∆t is P m (t) = ∆t ψ(t)|C † m C m |ψ(t) . For the method to be valid the total probability for a jump in ∆t has to be small, P = m P m ≪ 1. The method leads to many different traces, which on average reproduce the density matrix. For further details see, e.g., Refs. [8,9].
We model the dephasing by a single Lindblad operator,
C 0 = √ 2Γ 0 |0 0|, yielding the master equatioṅ ρ = − i [H, ρ] − 0 Γ 0 ρ 01 Γ 0 ρ 0e Γ 0 ρ 02 Γ 0 ρ 10 0 0 0 Γ 0 ρ e0 0 0 0 Γ 0 ρ 20 0 0 0 . (26)
Given C 0 we can calculate the no jump evolution with H (Sec. IV A) as well as the jump traces (Sec. IV B), where the system is projected onto the state C m |ψ(t j ) ∝ |0 at the instant of time t j .
A. Non-Hermitian no jump evolution
With the Lindblad operator
C 0 = √ 2Γ 0 |0 0|, H ′ = −i Γ 0 |0 0|−2iΓ0 0 A0(t) 0 0 0 A1(t)e iϕ 01 0 A0(t) A1(t)e −iϕ 01 0 A2(t)e −iϕ 2 (t) 0 0 A2(t)e iϕ 2 (t) 0 3 7 7 5 .(27)
In order to determine the time evolution of the system, we transform into the interaction picture with respect to
H ′ H I (t) = 2 0 0 A 0 (t)e Γ0(t−ti) 0 0 0 A 1 (t)e iϕ01 0 A 0 (t)e −Γ0(t−ti) A 1 (t)e −iϕ01 0 A 2 (t)e −iϕ2(t) 0 0 A 2 (t)e iϕ2(t) 0 .(28)
The subscript I indicates that the evolution is described in the interaction picture. This Hamiltonian is non-Hermitian due to the Γ 0 -exponents and in order to determine the geometric phases we follow the procedure of Ref. [22]. We diagonalize H I and find the eigenvalues
ω ± I = ± 1 2 A 2 0 + A 2 1 + A 2 2 , ω Di I = 0 (i = 1, 2),(29)
and the right (subscript r) and left (subscript l) eigenvectors
|+ r I = 1 √ 2 sin θ H (t)(sin θ 01 e Γ0(t−ti) |0 + cos θ 01 e iϕ01 |1 ) − |e + cos θ H (t)e iϕ2(t) |2 ,(30)|− r I = 1 √ 2 sin θ H (t)(sin θ 01 e Γ0(t−ti) |0 + cos θ 01 e iϕ01 |1 ) + |e + cos θ H (t)e iϕ2(t) |2 , |D 1r I = − cos θ H (t)(sin θ 01 e Γ0(t−ti) |0 + cos θ 01 e iϕ01 |1 ) + sin θ H (t)e iϕ2(t|ψ i I = C D1 (t i )|D 1r (t i ) I + C D2 (t i )|D 2r (t i ) I . (31)
The adiabatic STIRAP evolution ensures that the system remains within the subspace spanned by {|D 1r (t) I , |D 2r (t) I }. Inserting Eq. (31) into the timedependent Schrödinger equation gives a set of equations that can be solved numerically for C D1 and C D2 . Without loss of generality we write the solutions as
C D1 (t) =e −Γ0α(t) e iγ1(t) C D1 (t i ),(32)C D2 (t) =e −Γ0β(t) e iγ2(t) C D2 (t i ),
and expand the wave function as
|ψ(t) I =e −Γ0α(t) e iγ1(t) C D1 (t i )|D 1r (t) I (33) +e −Γ0β(t) e iγ2(t) C D2 (t i )|D 2r (t) I .
The two dark states each acquire a complex geometric phase composed by real (γ 1 and γ 2 ) and imaginary parts (Γ 0 α(t) and Γ 0 β(t)) parameterized by the dephasing rate, Γ 0 . As an example we choose an initial state |ψ i = |0 and apply the pulse sequence in Fig. 2(b) with parameters leading to θ 01 = π 8 , φ 01 = π and γ D1 = −π (see details in caption of Fig. 3). As discussed in Sec. III this evolution corresponds to the Hadamard gate, Eq. (24) for the closed system. In Fig. 3 we show the evolution of the phases as a function of time for different values of the dephasing rate Γ 0 . The real geometric phases γ 1 (light grey) and γ 2 (grey) are unaffected by the dephasing and their values are identical to the analytical result for the closed system [Eq. (21)], which are marked with crosses in Fig. 3. The exponents α and β are also almost unaffected by the dephasing rate. The results for Γ 0 τ = 10 −5 and Γ 0 τ = 10 −3 (solid) are identical while increasing the dephasing to Γ 0 τ = 10 −1 (dotted) gives only a small deviation. Experimentally dephasing rates can be kept below Γ 0 τ = 10 −3 and for these values of Γ 0 , α and β are unaffected by the dephasing rate and hence the imaginary part of the phases Γ 0 α and Γ 0 β scales linear with the dephasing rate. It should be noted that the value of α and β depend on the initial state while γ 1 and γ 2 are unaffected and equal to the values in the closed system case. Going back to the Schrödinger picture yields
|ψ(t) = 1 √ N e −Γ0(t−ti)|0 0| [e −Γ0α e iγ1 C D1 (t i )|D 1r I (34) +e −Γ0β e iγ2 C D2 (t i )|D 2r I ],
where the wave function was re-normalized (factor 1/ √ N ) because the non-Hermitian Hamiltonian does not preserve the norm. In the {|0 , |1 }-basis an initial state in the Schrödinger picture
|ψ i = a i |0 + b i |1 is trans- ferred to a final non-normalized state |ψ f non−norm = L|ψ i = a f |0 + b f |1 , with L = e iγ2
The normalization constant, N = |C D1 (t i )| 2 e −2Γ0α + |C D2 (t i )| 2 e −2Γ0β , depends on the initial state, and the final state reads |ψ f = 1
√ N |ψ f non−norm . While the phases γ 1 and γ 2 are robust with respect to dephasing, the appearance of the imaginary phases, affects the population of the states. Figure 4 shows the evolution of the population of the four states (|0 , |1 , |e , |2 ) with initial state |ψ i = |0 and the pulse sequence in Fig. 2(b) with parameters θ 01 = π/8, ϕ 01 = π and γ D1 = −π (see detailed parameters in the caption of Fig. 4). In the closed system these parameters lead to an implementation of the Hadamard gate and hence final populations P 0 = 1 2 and P 1 = 1 2 . When we introduce dephasing |ψ i = |0 leads to final population found from the no-jump evolu-
tion [Eq. (35)] P 0 = (cos 2 θ 01 e −Γ0β − sin 2 θ 01 e −Γ0α ) 2 sin 2 θ 01 e −2Γ0α − cos 2 θ 01 e −2Γ0β ,(36)P 1 = (sin θ 01 sin θ 01 (e −Γ0β + e −Γ0α )) 2 sin 2 θ 01 e −2Γ0α − cos 2 θ 01 e −2Γ0β .
The deviation from the closed system case (P 0 = (cos 2 θ 01 − sin 2 θ 01 ) 2 , P 1 = 4 sin 2 θ 01 cos 2 θ 01 ) is thus determined by the the two imaginary geometric phases, Γ 0 α and Γ 0 β. The values of α and β are found by numerically solving the time-dependent Schrödinger equation.
On the scale of Fig. 4 there is no deviation between the closed and the open system results for realistic dephasing rates. 6)) and all parameters are given in units of the pulse width, τ : ϕ2 = t/τ , Amax,0τ /2π = 300, Amax,1 =
Amax,0/( √ 2 − 1), Amax,2 = q A 2 max,0 + A 2 max,1
, ∆t/τ = 1, and ∆T /τ = π. These parameters lead to θ01 = π 8 , φ01 = π and γ1 = −π. Dephasing rates Γ0τ = 10 −5 and Γ0τ = 10 −3 (solid) yield identical results, while Γ0τ = 10 −1 (dotted) shows a small deviation at t/τ 3.5. Amax,1 = Amax,0/( √ 2 − 1), Amax,2 = q A 2 max,0 + A 2 max,1 , ∆t/τ = 1, and ∆T /τ = π. These parameters lead to θ01 = π 8 , φ01 = π and γ1 = −π.
B. Jump evolution
For the present choice of parameters the system jumps in a small part of the Monte Carlo traces to the |0state due to C 0 = √ 2Γ 0 |0 0|. If a jump occurs at t j we expand the wave function (|0 ) in the adiabatic basis of the instantaneous eigenstates |ψ(t j ) =|0
(37) = − cos θ H (t j ) sin θ 01 |D 1r (t j ) + cos θ 01 |D 2r (t j )
+ 1 √ 2 sin θ H (t j ) sin θ 01 (| + r (t j ) + | − r (t j ) ).
After the jump the evolution is described by the nojump non-Hermitian Hamiltonian [Eq. (28)]. In the adiabatic basis we can describe this evolution by calculating the geometric and dynamic phases acquired by the four states. The dark part of the wave function evolves as described for the no-jump evolution. The bright eigenstates (|+ r , |− r ) are separated energetically from each other and from the dark states, such that there is no diabatic population transfer among these. The bright states acquire a dynamic phase as well as a complex geometric phase that can be calculated directly Eq. (3). The complex geometric phase is the same for the two bright states
γ B =i R f Ri ± l |∇R|± r · dR (38) = 1 2 iΓ 0 sin θ 2 01 t tj sin 2 θ H dt ′ − 1 2 t tjφ 2 cos 2 θ H dt ′ ≡iΓ 0 δ + γ b
while the dynamic phase is different for the (|+ r , |− r )states
ϑ ± = ∓ 1 2 t tj A 2 0 + A 2 1 + A 2 2 dt ′ .
The wave function at later times in the interaction picture is
|ψ(t) = − cos θ H (t j ) sin θ 01 e −Γ0α e iγ1 |D 1r(39)
+ cos θ 01 e −Γ0β e iγ2 |D 2r
+ 1 √ 2 sin θ H (t j ) sin θ 01 e −Γ0δ e iγ b e iϑ+ |+ r + 1 √ 2 sin θ H (t j ) sin θ 01 e −Γ0δ e iγ b e iϑ− |− r .
Going back to the Schrödinger picture and calculating the final populations in the atomic states yields
P 0 = 1 N | cos 2 θ 01 e −Γ0β − sin 2 θ 01 cos θ H (t j )e −Γ0α e iγ1 )| 2 ,(40)P 1 = 1 N sin 2 θ 01 cos 2 θ 01 |e −Γ0β + cos θ H (t j )e −Γ0α e iγ1 | 2 , P e = 1 N sin 2 θ 01 sin 2 θ H (t j )e −2Γ0δ sin 2 ϑ − , P 2 = 1 N sin 2 θ 01 sin 2 θ H (t j )e −2Γ0δ cos 2 ϑ − ,
where the normalization constant is given as
N = sin 2 θ 01 cos 2 θ H (t j )e −
C. Fidelity of Hadamard gate
The calculated Monte Carlo traces can be used to determine the fidelity of the Hadamard gate. For a given initial state |ψ i we can determine the fidelity as the overlap between the target (closed-system) wave function |ψ 0 and the final Monte Carlo wave functions (no jump |ψ nj or one jump at t j |ψ j (t j ) ) weighed by the probability of each trace (no jump P nj or one jump P j (t j )). The contributions to the fidelity from traces with more than one jump (F i,mj ) is negligible for realistic dephasing rates. 6)) and all parameters are given in units of the pulse width, τ : ϕ2 = t/τ , Amax,0τ /2π = 300,
Amax,1 = Amax,0/( √ 2 − 1), Amax,2 = q A 2 max,0 + A 2 max,1 , ∆t/τ = 1, and ∆T /τ = π.
The fidelity, accordingly reads
F i =P nj | ψ 0 |ψ nj | 2 + P j (t j )| ψ 0 |ψ j (t j ) | 2 dt j + F i,mj(41)
=| ψ 0 |ψ nj,non−norm. | 2
+ | ψ 0 |ψ j,non−norm. (t j ) | 2 dt j + F i,mj ,
where we use the non-normalized Monte Carlo wave functions to avoid calculating the probability distributions directly. As a first approximation the fidelity can be calculated neglecting the jump traces
F i,nj =| ψ 0 |ψ nj,non−norm. | 2 = | ψ i U † 0 |Lψ i | 2 (42) =e −2Γ0αnj |C D1 (t i )| 4 + e −2Γ0βnj |C D2 (t i )| 4 + 2e −Γ0(αnj +βnj) |C D1 (t i )| 2 |C D2 (t i )| 2 × cos(γ 1,nj − γ 2,nj − γ D1 ).
The decrease in the fidelity is governed by the geometric phases acquired during the no-jump evolution. The non-normalized wave function for traces with one jump at the instant t j is found in three steps. The system evolves under the non-Hermitian Hamiltonian until t j , |ψ j (t j ) = L|ψ j (t i ) , where it acquires geometric phases −Γ 0 α + iγ 1 and −Γ 0 β + iγ 2 . At t j the system jumps and |ψ j (t j ) = √ 2Γ 0 |0 0|ψ j (t j ) . Finally the system evolves under the non-Hermitian Hamiltonian from t j to t f , |ψ j (t f ) = L|ψ j (t j ) , where it acquires geometric phases −Γ 0 α ′ + iγ ′ 1 and −Γ ′ 0 β + iγ ′ 2 . The fidelity from the jump traces will hence be proportional to
√ 2Γ 0 2 F i = F i,nj + 2Γ 0 ξ i,j (α, γ 1 , α ′ , γ ′ 1 , β, γ 2 , β ′ , γ ′ 2 )dt j ,(43)where ξ i,j (α, γ 1 , α ′ , γ ′ 1 , β, γ 2 , β ′ , γ ′ 2 )
is determined by the geometric phases in the jump traces with jump at t j . Traces with more than one jump will contribute with terms proportional to higher orders of Γ 0 and are therefore neglected. The fidelity determined from the Monte Carlo traces can be compared with the Uhlmann state fidelity calculated from the final closed system density matrix ρ 0 and the final open system density matrix ρ (Γ0) [24] F ρ,i = Tr ρ
1/2 0 ρ (Γ0) ρ 1/2 0 2 .(44)
Both F i and F ρ,i give the fidelity for a given initial state.
The average fidelity can be found by integrating over the surface of the Bloch sphere
F = 1 4π F i dΩ.(45)
This averaging procedure can be simplified to only averaging over the six axial pure initial states on the Bloch sphere, Λ ={|0 , |1 , 1
√ 2 (|0 + |1 ), 1 √ 2 (|0 − |1 ), 1 √ 2 (|0 + i|1 ), 1 √ 2 (|0 − i|1 )}[25] F = 1 6 |ψi ǫΛ F i .(46)
In Fig. 6 we show the average fidelity as a function of the dephasing rate by the Monte Carlo method when only nojump traces are taken into account (solid, black curve) and when traces with no or one jump are included (dotted, black curve). These are compared with the full master equation solution (dashed, grey curve). The fidelities are all calculated for the Hadamard gate implemented by the parameters θ 01 = π/8, ϕ 01 = π and γ D1 = −π as in all previous numerical results. The fidelity decreases as expected when the dephasing rate increases, but is still acceptable when the system is subject to realistic dephasing rates. The no-jump results gives a lower bound on the fidelity but it is necessary to include traces with one jump in order to get a satisfactory accuracy for realistic dephasing rates. Eq. (42) shows explicitly how the nojump fidelity depends on the geometric phases acquired during the no-jump evolution.
V. SUMMARY AND CONCLUSION
We presented a method to describe the adiabatic evolution of an open system using the quantum Monte Carlo Amax,2 = q A 2 max,0 + A 2 max,1 , ∆t/τ = 1, and ∆T /τ = π.
method and keeping track of all acquired phases. This method has the advantage that it reveals the evolution of the single quantum trajectories and discloses the geometric or dynamic nature of the acquired phases in the adiabatic basis. We considered a tripod system with three laser fields applied and calculated its instantaneous adiabatic eigenstates. The tripod system is subject to a double STIRAP process and during this the adiabatic eigenstates acquire complex geometric phases. In the closed system case the geometric phases are purely real (γ D1 ) and all population is at all times in the two dark (with zero energy eigenvalues) adiabatic eigenstates. The acquired geometric phases create a phase difference between the two dark states and hence generate a rotation in the atomic {|0 , |1 }-basis. With the right parameter choice this rotation implements the Hadamard gate, which we have used as an example in the numerical simulations.
When dephasing is present we used the quantum Monte Carlo method, where the system either follow a non-Hermitian no-jump evolution during the whole time sequence or at one or more instants of time jump to the |0 -state. Before, in between and after jumps the system follows the non-Hermitian no-jump evolution. During the non-Hermitian evolution the instantaneous adiabatic eigenstates acquire complex geometric phases. These deviate from the closed system case mainly because they contain non-negligible imaginary parts, which lead to a decay of the adiabatic eigenstates and hence influence the populations resulting in imperfect gate performance.
The specific Hadamard gate simulations show that the fidelity is still appreciable at realistic dephasing rates.
FIG. 2 :
2(a) Four-level tripod system with three laser fields applied with Rabi frequencies Ω0, Ω1, Ω2. (b) Pulse sequence consisting of two STIRAP processes separated by ∆T in time.
)
|2 , |D 2r I = cos θ 01 e Γ0(t−ti) |0 − sin θ 01 e iϕ01 |1 , θ H (t)(sin θ 01 e −Γ0(t−ti) 0| + cos θ 01 e −iϕ01 1|) − e| + cos θ H (t)e −iϕ2(t) 2| ,I − l | I = 1 √ 2 sin θ H (t)(sin θ 01 e −Γ0(t−ti) 0| + cos θ 01 e −iϕ01 1|) + e| + cos θ H (t)e −iϕ2(t) 2| , I D 1l | = − cos θ H (t)(sin θ 01 e −Γ0(t−ti) 0| + cos θ 01 e −iϕ01 1|) + sin θ H (t)e −iϕ2(t) 2|, I D 2l | = cos θ 01 e −Γ0(t−ti) 0| − sin θ 01 e −iϕ01 1|. The left and right eigenvectors fulfill the biorthonormal condition i l |j r I = δ i,j [23]. Initially (t = t i ) the eigenvectors of the open system [Eq. (30)] coincide with the eigenvectors of the closed system [Eq. (17)], and hence the initial state of the closed system is also the initial state of the open system
cos 2 θ 01 e −Γ0β + e iγ1 sin 2 θ 01 e −Γ0α cos θ 01 sin θ 01 e −iϕ01 (e iγ1 e −Γ0α − e iγ2 e −Γ0β ) cos θ 01 sin θ 01 e iϕ01 (e iγ1 e −Γ0α − e iγ2 e −Γ0β ) e iγ2 sin 2 θ 01 e −Γ0β + e iγ1 cos 2 θ 01 e −Γ0α .
FIG. 3 :
3Time evolution of γ1, γ2, α and β for different dephasing rates, Γ0. The calculations were made with sin 2 pulses (Eq. (
FIG. 4 :
4Evolution of the population in the |0 -state (P0, solid), the |1 -state (P1, dashed),the |e -state (Pe, dasheddotted) and the |2 -state (P2, dotted) with all population initially in the |0 -state. The population evolve to P0 = 1/2 and P1 = 1/2 when no dephasing is present. With dephasing (Γ0T0 = 10 −3 ) the master equation as well as the no jump evolution [Eq. (35)] show no deviation from the no dephasing case on the scale of this figure. The calculations were made with sin 2 pulses [Eq. (6)] and all parameters are given in units of the pulse width, τ : ϕ2 = t/τ , Amax,0τ /2π = 300,
2Γ0α + cos 2 θ 01 e −2Γ0β + sin 2 θ 01 sin 2 θ H (t j )e −2Γ0δ . Phases are only acquired from the latest jump time, t j . The final populations can in this way be calculated after each Monte Carlo trace and the deviations from the closed system case can be explained by the complex geometric phases acquired by the adiabatic states. When all complex geometric phases are carefully taken into account an average over many traces reproduces the numerical solution of the master equation [Eq. (26)]. This is shown in Fig. 5, where we enlarge the last part of the evolution of the populations shown in Fig. 4. On this scale deviations between the closed and the open system are visible. The closed system (dash-dot-dot curves) yields final populations P 0 = 1/2 and P 1 = 1/2 as expected for the initial state |ψ i = |0 and the Hadamard gate applied (parameters are specified in the caption of Fig. 5). The evolution of the open system is determined either by solving the master equation (solid curves) or by the Monte Carlo method. The final populations predicted by the no-jump evolution (dashed curves) deviates from the master equation (solid curves) on the order of 10 −3 . An average over 10000 Monte Carlo traces (dash-dot-dash) reduces the deviation to an order of 10 −4 , while averaging over 200000 traces (dotted curves) reduces it further to an order of 10 −5 .
FIG. 5 :
5Population in the |0 -state (P0, black), the |1state (P1, grey), when the system is not subject to dephasing (Pi,Γ 0 =0, dot-dot-dash) as well a when dephasing (Γ0τ = 10 −3 ) is present. The graph shows only the final part of the time evolution, where the differences can be distinguished. With dephasing the full curve (Pi,ρ) shows the solution of the master equation [Eq. (25)], the dashed curve (Pi,nj) the nojump trace, while the last two curves are averages over 10000 (Pi,j 10000 traces dash-dot-dash) and 200000 (Pi,j 200000 traces , dotted) jump and no-jump traces. The calculations were made with sin 2 pulses (Eq. (
FIG. 6 :
6Fidelity as a function of the dephasing rate, Γ0. The dashed grey curve shows the density matrix calculation, Fρ [Eq. (44)], the dotted black curve the Monte Carlo calculation including traces with no jump as well as one jump (F ) and solid black includes only no-jump traces (Fnj ) [Eq. 42]. The calculations were made with sin 2 pulses [Eq. (6)] and all parameters are given in units of the pulse width, τ : ϕ2 = t/τ , Amax,0τ /2π = 300, Amax,1 = Amax,0/( √ 2 − 1),
AcknowledgmentsThis work is supported by the Danish Research Agency (Grant. No. 2117-05-0081).
. M V Berry, Proc. R. Soc. London, Ser. A. 39245M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984).
. D Ellinas, J Pachos, Phys. Rev. A. 6422310D. Ellinas and J. Pachos, Phys. Rev. A 64, 022310 (2001).
. P Solinas, P Zanardi, N Zanghi, Phys. Rev. A. 7042316P. Solinas, P. Zanardi, and N. Zanghi, Phys. Rev. A 70, 042316 (2004).
. I Fuentes-Guridi, F Girelli, E Livine, Phys. Rev. Lett. 9420503I. Fuentes-Guridi, F. Girelli, and E. Livine, Phys. Rev. Lett. 94, 020503 (2005).
. S L Zhu, P Zanardi, Phys. Rev. A. 7220301S. L. Zhu and P. Zanardi, Phys. Rev. A 72, 020301 (2005).
. E Sjöqvist, Acta Phys. Hung. B. 26195E. Sjöqvist, Acta Phys. Hung. B 26, 195 (2006).
. S Dasgupta, D A Lidar, J. Phys. B. 40127S. Dasgupta and D. A. Lidar, J. Phys. B 40, S127 (2007).
. K Mølmer, Y Castin, J Dalibard, J. Opt. Soc. Am. B. 10524K. Mølmer, Y. Castin, and J. Dalibard, J. Opt. Soc. Am. B 10, 524 (1993).
. K Mølmer, Y Castin, Quantum Semiclass.Opt. 849K. Mølmer and Y. Castin, Quantum Semiclass.Opt. 8, 49 (1996).
. K Bergmann, H Theuer, B W Shore, Rev. Mod. Phys. 701003K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998).
A Messiah, Quantum Mechanics. North-Holland publishing company2A. Messiah, Quantum Mechanics, vol. 2 (North-Holland publishing company, 1961).
. R G Unanyan, B W Shore, K Bergmann, Phys. Rev. A. 592910R. G. Unanyan, B. W. Shore, and K. Bergmann, Phys. Rev. A 59, 2910 (1999).
. U Gaubatz, P Rudecki, S Schiemann, K Bergmann, J. Chem. Phys. 925363U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, J. Chem. Phys. 92, 5363 (1990).
. B Broers, H B Van Linden Van Den Heuvell, L D Noordam, Phys. Rev. Lett. 692062B. Broers, H. B. van Linden van den Heuvell, and L. D. Noordam, Phys. Rev. Lett. 69, 2062 (1992).
. L S Goldner, C Gerz, R J C Spreeuw, S L Rolston, C I Westbrook, W D Phillips, P Marte, P Zoller, Phys. Rev. Lett. 72997L. S. Goldner, C. Gerz, R. J. C. Spreeuw, S. L. Rolston, C. I. Westbrook, W. D. Phillips, P. Marte, and P. Zoller, Phys. Rev. Lett. 72, 997 (1994).
. J L Sørensen, D Møller, T Iversen, J B Thomsen, F Jensen, P Staanum, D Voigt, M Drewsen, New J. Phys. 8261J. L. Sørensen, D. Møller, T. Iversen, J. B. Thomsen, F. Jensen, P. Staanum, D. Voigt, and M. Drewsen, New J. Phys. 8, 261 (2006).
. T Cubel, B K Teo, V S Malinovsky, J R Guest, A Reinhard, B Knuffman, P R Berman, G Raithel, Phys. Rev. A. 7223405T. Cubel, B. K. Teo, V. S. Malinovsky, J. R. Guest, A. Reinhard, B. Knuffman, P. R. Berman, and G. Raithel, Phys. Rev. A 72, 023405 (2005).
. J Lawall, M Prentiss, Phys. Rev. Lett. 72993J. Lawall and M. Prentiss, Phys. Rev. Lett. 72, 993 (1994).
. D Møller, L B Madsen, K Mølmer, Phys. Rev. A. 7562302D. Møller, L. B. Madsen, and K. Mølmer, Phys. Rev. A 75, 062302 (2007).
. F Wilczek, A Zee, Phys. Rev. Lett. 522111F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984).
S Haroche, J.-M Raimond, Exploring the Quantum. OxfordOxford University Presschapter 4S. Haroche and J.-M. Raimond, Exploring the Quantum (Oxford University Press, Oxford, 2006), chapter 4.
. J C Garrison, E M Wright, Phys. Lett. A. 128177J. C. Garrison and E. M. Wright, Phys. Lett. A 128, 177 (1988).
F H M , Theory of multiphoton processes. New YorkPlenum PressF. H. M. Faisal, Theory of multiphoton processes (Plenum Press, New York, 1987), chapter 11.2.
. A Uhlmann, Rep. Math. Phys. 9273A. Uhlmann, Rep. Math. Phys. 9, 273 (1976).
. M D Bowdrey, D K L Oi, A J Short, K Banaszek, J A Jones, Phys. Lett. A. 294258M. D. Bowdrey, D. K. L. Oi, A. J. Short, K. Banaszek, and J. A. Jones, Phys. Lett. A 294, 258 (2002).
| [] |
[
"Constraints and degrees of freedom in Lorentz-violating field theories",
"Constraints and degrees of freedom in Lorentz-violating field theories"
] | [
"Michael D Seifert \nDept. of Physics, Astronomy, and Geophysics\nConnecticut College *\n\n"
] | [
"Dept. of Physics, Astronomy, and Geophysics\nConnecticut College *\n"
] | [] | Many current models which "violate Lorentz symmetry" do so via a vector or tensor field which takes on a vacuum expectation value, thereby spontaneously breaking the underlying Lorentz symmetry of the Lagrangian. To obtain a tensor field with this behavior, one can posit a smooth potential for this field, in which case it would be expected to lie near the minimum of its potential. Alternately, one can enforce a non-zero tensor value via a Lagrange multiplier. The present work explores the relationship between these two types of theories in the case of vector models. In particular, the naïve expectation that a Lagrange multiplier "kills off" one degree of freedom via its constraint does not necessarily hold for vector models that already contain primary constraints. It is shown that a Lagrange multiplier can only reduce the degrees of freedom of a model if the field-space function defining the vacuum manifold commutes with the primary constraints. | 10.1103/physrevd.99.045003 | [
"https://arxiv.org/pdf/1810.09512v2.pdf"
] | 119,391,376 | 1810.09512 | 3e0d5e2e191b9b85b7aed6bec876dc8766d2f8be |
Constraints and degrees of freedom in Lorentz-violating field theories
5 Nov 2018
Michael D Seifert
Dept. of Physics, Astronomy, and Geophysics
Connecticut College *
Constraints and degrees of freedom in Lorentz-violating field theories
5 Nov 2018(Dated: November 6, 2018)
Many current models which "violate Lorentz symmetry" do so via a vector or tensor field which takes on a vacuum expectation value, thereby spontaneously breaking the underlying Lorentz symmetry of the Lagrangian. To obtain a tensor field with this behavior, one can posit a smooth potential for this field, in which case it would be expected to lie near the minimum of its potential. Alternately, one can enforce a non-zero tensor value via a Lagrange multiplier. The present work explores the relationship between these two types of theories in the case of vector models. In particular, the naïve expectation that a Lagrange multiplier "kills off" one degree of freedom via its constraint does not necessarily hold for vector models that already contain primary constraints. It is shown that a Lagrange multiplier can only reduce the degrees of freedom of a model if the field-space function defining the vacuum manifold commutes with the primary constraints.
Many current models which "violate Lorentz symmetry" do so via a vector or tensor field which takes on a vacuum expectation value, thereby spontaneously breaking the underlying Lorentz symmetry of the Lagrangian. To obtain a tensor field with this behavior, one can posit a smooth potential for this field, in which case it would be expected to lie near the minimum of its potential. Alternately, one can enforce a non-zero tensor value via a Lagrange multiplier. The present work explores the relationship between these two types of theories in the case of vector models. In particular, the naïve expectation that a Lagrange multiplier "kills off" one degree of freedom via its constraint does not necessarily hold for vector models that already contain primary constraints. It is shown that a Lagrange multiplier can only reduce the degrees of freedom of a model if the field-space function defining the vacuum manifold commutes with the primary constraints.
I. INTRODUCTION
Many classical field theories are constructed in such a way that the "most natural" solutions to the equations of motion involve a non-zero field value. This paradigm, where an underlying symmetry of the Lagrangian or Hamiltonian is spontaneously broken by the solutions of the equations of motion, has proven to be both compelling and fruitful over the years. In the context of particle physics, the best-known example is the Higgs field [1][2][3][4]; in the context of condensed-matter physics, this paradigm underlies the modern theory of phase transitions, most notably the Ginzburg-Landau theory of superconductivity [5]. Many non-linear sigma models can also be thought of in this way, if one view the model's target manifold as being embedded in some higher-dimensional space in which the fields are forced to a non-zero value. Such models have been used in the study of both particle physics [6,7] and in ferromagnetism [8].
In more recent years, this paradigm has also been used to study possible observational signatures of Lorentz symmetry violation in the context of modified gravity theories. Such models include a new vector or tensor field, and have equations of motion that are satisfied when the metric is flat and the new tensor field is constant but non-zero. In this sense, Lorentz symmetry is spontaneously broken in these theories, as the tensor field takes on a vacuum expectation value that has non-trivial transformation properties under the Lorentz group. The vacuum state of such a model is often said to be "Lorentzviolating", though it would be more accurate to say that Lorentz symmetry is spontaneously broken in the model. Examples of such models can be found in [9][10][11][12][13][14], as well as a particularly early example in [15].
In general, all of these models (Lorentz-violating or otherwise) have the property that the fields, collectively denoted by ψ α , will satisfy an equation of the form * [email protected] f (ψ α ) = 0 in the "vacuum", appropriately defined. Here, f is a real-valued function of the fields; thus, if the fields ψ α are specified by N real numbers, the equation f (ψ α ) = 0 will generically specify an (N −1)-dimensional hypersurface in field space. I will therefore refer to this function as the vacuum manifold function.
Two broad classes of models in which some fields have non-trivial background values can readily be conceived of. In one class, the constraint f (ψ α ) = 0 is enforced exactly via the introduction of a Lagrange multiplier λ in the Lagrange density:
L LM = (∇ψ α )(∇ψ α ) + λf (ψ α ),(1)
where we have denoted the kinetic terms for the fields ψ α schematically. I will call such models Lagrange-multiplier (LM) models.
In the other class, the fields ψ α are assigned a potential energy that is minimized when the field is non-zero. In particular, if we define a potential V (ψ α ) ∝ f 2 (ψ α ), and write down a Lagrange density
L P = (∇ψ α )(∇ψ α ) − V (ψ α ),(2)
then the lowest-energy state of the theory would be expected to occur when f (ψ α ) = 0 and ∇ψ α = 0. I will call such models potential models. A natural question then arises: for a given collection of fields ψ α and a given kinetic term (∇ψ α )(∇ψ α ), what is the relationship between the models (1) and (2)? In particular, one might expect the following statement to be true:
Conjecture. In a potential model such as (2), the fields can take on any value in the N -dimensional field space. By contrast, in a Lagrange-multiplier model, the fields are constrained to an (N − 1)-dimensional subspace of field space. Thus, the number of degrees of freedom of a Lagrange-multiplier model (1) should be one fewer than the degrees of freedom of the corresponding potential model (2).
The main purpose of this article is to show that this naïve conjecture not is true in general; if ψ α includes a spacetime vector or tensor field, it may be false. In such models, the fields may need to satisfy certain constraints due to the structure of the kinetic terms; adding a new "constraint" to such theories, in the form of a Lagrange multiplier, does not automatically reduce the number of degrees of freedom of the theory.
To demonstrate this, I will analyse the degrees of freedom of two types of symmetry-breaking models. After a brief description of Dirac-Bergmann analysis in Section II, I will first analyse a multiplet of Lorentz scalar fields with an internal symmetry, followed by a vector field (Sections III and IV respectively.) For the vector fields, the analysis will depend on the structure of the kinetic term chosen, and so three distinct sub-cases will need to be treated. In each case, I will examine a model where the fields are assigned a potential energy, and one where the fields are directly constrained via a Lagrange multiplier.
The bulk of the explicit analysis in this work will be done in the context of a fixed, flat background spacetime; however, I will briefly discuss these models in the context of a dynamical curved spacetime in Section V. In that section, I will also discuss the implications of these results for the broader relationship between these two classes of models. More general tensor fields will be examined in a forthcoming work [16].
Throughout this work, I will use units in whichh = c = 1; the metric signature will be ( − + + + ). Roman indices a, b, c, . . . will be used to denote spacetime tensor indices; i, j, k, . . . will be used to denote spatial indices, where necessary. Greek indices α, β, γ, ... will be used exclusively to denote indices in field space. All expressions involving repeated indices (either tensor indices or field space indices) can be assumed to obey the Einstein summation convention.
II. DIRAC-BERGMANN ANALYSIS
Our primary tool for finding the number of degrees of each model will be Dirac-Bergmann constraint analysis [17]; my methods and nomenclature below will draw heavily from the later work of Isenberg & Nester [18]. I will briefly summarize the method here, and then illustrate it in more detail via the example theories described in Section III.
The method of Dirac-Bergmann analysis involves the construction of a Hamiltonian which generates the timeevolution of the system. In the process of this construction, one may need to introduce constraints among various variables, thereby reducing the number of degrees of freedom of the system. One may also discover that the evolution of certain field combinations is undetermined by the equations of motion (for example, gauge degrees of freedom). These combinations of fields, which we will collectively call "gauges", must be interpreted as unphysical, again reducing the number of physical degrees of freedom of the model. If, as is usual, we count a field degree of freedom as a pair of real-valued functions (e.g., a field value and its conjugate momentum) that can be freely specified on an initial data surface, then the number of degrees of freedom N dof can be inferred quite simply once the above analysis is complete:
N dof = 1 2 no. of fields + no. of momenta − no. of constraints − no. of gauges .(3)
In general, of course, the number of fields and the number of conjugate momenta will be the same. Moreover, in the particular field theories I will be considering in this work, I will not find any "gauges", so the last term in (3) will vanish. Thus, for my purposes, the above equation reduces to
N dof = no. of fields − 1 2 no. of constraints .(4)
III. SCALAR MULTIPLET FIELDS
The first case we will consider is a multiplet of N Lorentz scalars: ψ α = φ α , with α = 1, 2, 3, . . . , N . We wish to construct a model where these scalars "naturally" take on values in some (N − 1)-dimensional hypersurface, defined by f (φ α ) = 0 (with f a real-valued function.) The "potential model" for this field will be derived from the Lagrange density
L = − 1 2 ∂ a φ α ∂ a φ α − κf (φ α ) 2 (5) = 1 2 (φ α )(φ α ) − ( ∇φ α ) · ( ∇φ α ) − κf (φ α ) 2 ,
where κ is a proportionality constant; the LM model for this field will be
L = − 1 2 ∂ a φ α ∂ a φ α − λf (φ α )(6)= 1 2 (φ α )(φ α ) − ( ∇φ α ) · ( ∇φ α ) − λf (φ α ),
where λ is a Lagrange multiplier field. In both cases, we have chosen a time coordinate t and performed a 3+1 decomposition of the tensors; a dot over a quantity (e.g., φ α ) will denote its derivative with respect to this time coordinate, while spatial derivatives will be denoted with either ∇ or ∂ i depending on the expression.
A. Potential model
The degrees of freedom for the model (5) are particularly easy to count. The momenta conjugate to the fields are all well-defined:
π α = ∂L ∂φ α =φ α(7)
The Hamiltonian density is therefore
H 0 = π αφα − L = 1 2 π α π α + ( ∇φ α ) · ( ∇φ α ) + κf (φ α ) 2 .
(8) Nothing further is required here; we have no primary constraints on the initial data, and the Hamiltonian obtained by integrating H 0 over space will generate the field dynamics. We thus have the N degrees of freedom one would expect.
B. Lagrange-multiplier model
Counting the degrees of freedom for the model (6) requires a bit more effort. As the kinetic term of (6) is the same as that of (5), the momenta conjugate to the scalars φ α are again defined by (7). The difficulty arises due to the Lagrange multiplier λ. From the perspective of the model, it is just another field; but its associated momentum vanishes automatically:
̟ = ∂L ∂λ = 0 ≡ Φ.(9)
We thus have a primary constraint, Φ = 0, on this theory. The "base Hamiltonian density" H 0 = π αφα − L must then be modified to obtain the "augmented Hamiltonian density" H A by adding this primary constraint multiplied by an auxiliary Lagrange multiplier u λ : 1
H A = H 0 + u λ ̟ = 1 2 (π α ) 2 + ( ∇φ α ) 2 + λf (φ α ) + u λ ̟(10)
We now need to ensure that this constraint is preserved by the time-evolution of the system; in other words, we must have̟ = {̟,
H A } = 0, where H A ≡ H A d 3 x. 2
If this Poisson bracket does not vanish identically, this demand will yield a secondary constraint Ψ 1 = 0. The demand that this constraint be preserved may lead to new secondary constraints Ψ 2 = 0, Ψ 3 = 0, and so forth, which must themselves be conserved. We will refer to the stage at which a secondary constraint arises as its "order". In other words, if Ψ 1 ensures the preservation of a primary constraint, it is a "first-order secondary constraint"; if Ψ 2 ensures the preservation of Ψ 1 , it is a "second-order secondary constraint"; and so on. In this process, it may occur that the preservation of these constraints allows us to determine the auxiliary Lagrange multiplier u λ introduced above. The process is continued until all constraints are known to be automatically conserved or a contradiction is reached.
With this in mind, we derive the secondary constraints for this model. We first have
0 =̟ = {̟, H A } = − δH A δλ = f (φ α ).(11)
Thus, Ψ 1 = f (φ α ) = 0 is a secondary constraint. We repeat this procedure, obtaining another secondary constraint:
Ψ 1 = {Ψ 1 , H A } = ∂f (φ α ) ∂φ β δH A δπ β = (δ β f )π β ≡ Ψ 2 ,(12)
where we have defined δ β f = δf /δφ β . (Higher derivatives will be defined similarly.) Ψ 2 must also be conserved, which leads to a third secondary constraint:
Ψ 2 = {Ψ 2 , H A } = −δ α f −∇ 2 φ α + λδ α f + π α π β (δ αβ f ) ≡ Ψ 3 .(13)
Finally, demanding that Ψ 3 be conserved allows us to determine the auxiliary Lagrange multiplier u λ ; this is because Ψ 3 is itself dependent on λ:
Ψ 3 = {Ψ 3 , H A } = ∂Ψ 3 ∂φ γ δH A δπ γ − ∂Ψ 3 ∂π γ δH A δφ γ + ∂Ψ 3 ∂λ δH A δ̟ .(14)
When the dust settles, we obtaiṅ
Ψ 3 = π γ (δ αγ f ) 3∇ 2 φ α − 4λδ α f + ∇ 2 (δ α f ) + π α π β (δ αβγ f ) − (δ α f )(δ α f )u λ . (15)
So long as δ α f = 0 when f = 0, this allows us to determine the previously-unknown auxiliary Lagrange multiplier u λ . Thus, the process terminates here.
Having determined the Hamiltonian and its constraints, we can count the degrees of freedom. We have N + 1 fields, namely the multiplet φ α (α = 1, . . . , N ) and the Lagrange multiplier λ; one primary constraint Φ = ̟ = 0; and three secondary constraints, {Ψ 1 , Ψ 2 , Ψ 3 } = 0. The single auxiliary Lagrange multiplier is determined, which means that there are no unphysical gauge degrees of freedom. With N + 1 fields and four constraints, the number of degrees of freedom of the model (6) is therefore
N dof = (N + 1) − 1 2 (4) = N − 1.
Thus, the Lagrange-multiplier theory (6) has one less degree of freedom than the corresponding potential theory (5); in this case, the naïve conjecture outlined in the introduction holds true.
IV. VECTOR FIELDS
We now consider the case of a vector field A a which spontaneously breaks Lorentz symmetry. We will again restrict our attention to the case of flat spacetime. As the motivation behind these models is usually a spontaneous breaking of Lorentz symmetry, we want our Lagrange density to be a Lorentz scalar, without any prior geometry specified.
In any such Lagrange density, we can identify a set of "kinetic" terms L K that depend on the derivatives of A a . The most general kinetic term that we can write down which is quadratic in the field A a is 3
L k = c 1 ∂ a A b ∂ a A b + c 2 (∂ a A a ) 2 + c 3 ∂ a A b ∂ b A a . (16) However, since (∂ a A a ) 2 = ∂ a A b ∂ b A a + ∂ a A a ∂ b A b − A b ∂ b A a ,(17)
we can eliminate one of c 2 or c 3 via an integration by parts. We will therefore set c 2 = 0 in what follows. The familiar "Maxwell" kinetic term
L K = − 1 4 F ab F ab , with F ab = 2∂ [a A b] corresponds to c 1 = −c 3 = − 1 2 .
We can now perform the usual 3 + 1 decomposition of the Lagrange density, writing A 0 for the t-component of A a and A (or A i ) for its spatial components. The kinetic term (16) then becomes
L K = 1 2 c 13 Ȧ 0 2 − c 1 2 ∇A 0 2 − c 1 2˙ A 2 − c 3˙ A · ∇A 0 + c 1 2 (∂ i A j )(∂ i A j ) + c 3 2 (∂ i A j )(∂ j A i ),(18)
where c 13 ≡ c 1 + c 3 . The momenta conjugate to A 0 and A are then
Π 0 = ∂L K ∂Ȧ 0 = c 13Ȧ0(19)
and
Π = ∂L K ∂˙ A = −c 1˙ A − c 3 ∇A 0 .(20)
These equations can be inverted, to find the velocitieṡ A 0 and˙ A, so long as c 1 + c 3 = 0 and c 1 = 0. If either of these expressions vanishes, (19) and (20) will instead yield constraint equations; we will have to handle these cases separately. For the potential term, meanwhile, our desire for the Lagrangian to be a Lorentz scalar implies that the only possible form for the vacuum manifold function is one which sets the norm of A a to some constant b. For simplicity's sake, we will therefore choose f (A a ) to be of the following form:
f (A a ) = A a A a − b(21)
where b is a constant. Depending on the sign of b, the "vacuum" manifold will consist of timelike vectors (b < 0), spacelike vectors (b > 0), or null vectors (b = 0). 4 Our potential model will then be
L P = L K − κf (A a ) 2 = L K − κ(−A 2 0 + A 2 − b) 2 ,(22)
where κ is again a proportionality constant; the LM model will be
L LM = L K − λf (A a ) = L K − λ(−A 2 0 + A 2 − b). (23)
For compactness, I will denote the four-norm of A a as A 2 = −A 2 0 + A 2 ; the norm of the spatial part on its own will always be denoted by A 2 .
A. General case: c13 = 0, c1 = 0
Potential model
Performing a Legendre transform on L P (22) to obtain the Hamiltonian density, we obtain a base Hamiltonian density of
H B = 1 2c 13 Π 2 0 − 1 2c 1 Π 2 + c 2 1 − c 2 3 2c 1 ∇A 0 2 − c 3 c 1 Π · ∇A 0 − c 1 2 (∂ i A j )(∂ i A j ) − c 3 2 (∂ i A j )(∂ j A i ) + κ A 2 − b 2 .(24)
The resulting theory has four fields (A 0 and A) and no constraints; so the process terminates here, and the base Hamiltonian is the complete Hamiltonian for the model. Counting the degrees of freedom, we find that
N dof = 4 − 1 2 (0) = 4.(25)
Lagrange multiplier model
For the Lagrange multiplier model (23), we have a primary constraint associated with λ:
̟ = ∂L ∂λ = 0.(26)
We must therefore augment the Hamiltonian with an auxiliary Lagrange multiplier u λ to enforce this constraint:
H A = Π 0 A 0 + Π · A − L LM + u λ ̟ = 1 2c 13 Π 2 0 − 1 2c 1 Π 2 + c 2 1 − c 2 3 2c 1 ∇A 0 2 − c 3 c 1 Π · ∇A 0 − c 1 2 (∂ i A j )(∂ i A j ) − c 3 2 (∂ i A j )(∂ j A i ) + λ(A 2 − b) + u λ ̟(27)
We now find the secondary constraints required for the primary constraint to be conserved under time-evolution. Taking the Poisson brackets of each constraint with the Hamiltonian in turn, we obtain three secondary constraints:
̟ = {̟, H A } = A 2 0 − A 2 + b ≡ Ψ 1 ;(28)Ψ 1 = {Ψ 1 , H A } = 2 1 c 13 A 0 Π 0 + 1 c 1 A · Π + c 3 ∇A 0 ≡ Ψ 2 ; (29) anḋ Ψ 2 = {Ψ 2 , H A } = 2λ A 2 0 c 13 − A 2 c 1 + Ξ ≡ Ψ 3 ,(30)
where
Ξ ≡ 1 c 2 13 Π 0 2 − 1 c 2 1 Π + c 3 ∇A 0 2 + c 3 c 1 c 13 A · ∇Π 0 − A 0 ∇ · Π + c 1 − c 3 c 1 A 0 ∇ 2 A 0 − c 3 c 1 A · ∇ ∇ · A + A · ∇ 2 A. (31)
All three of the quantities Ψ 1 , Ψ 2 , and Ψ 3 must vanish for the model to be consistent. When we take the Poisson bracket of Ψ 3 with H A , we will obtaiṅ
Ψ 3 = {Ξ, H A } + 2λ A 2 0 c 13 − A 2 c 1 , H A = {Ξ, H A } + 2u λ A 2 0 c 13 − A 2 c 1 + 4λ 1 c 2 13 A 0 Π 0 + 1 c 2 1 A · Π + c 3 ∇A 0(32)
This means that for generic initial data, for which
A 2 0 c 13 = A 2 c 1 ,(33)
we can solve (32) for u λ . 5 Thus, the auxiliary Lagrange multiplier u λ is determined via the self-consistency of 5 The full expression for {Ξ, H A } is complicated and not terribly the theory. We have five fields (A 0 , A, and λ), and self-consistency generates four constraints (one primary, three secondary); and so the total number of degrees of freedom of this model is
N dof = 5 − 1 2 (4) = 3.(34)
Again, as expected from the conjecture, we have lost one degree of freedom to the Lagrange multiplier. A related analysis was performed by Garfinkle, Isenberg, and Martin-Garcia [19] in the case of Einsteinaether theory. In such models, the vector field is constrained to satisfy A a A a = b = −1, i.e., the vector field is unit and timelike. In [19], the time component A 0 of the vector field was explicitly eliminated from the Lagrangian after performing a 3 + 1 decomposition, leaving the components of A as the three dynamical fields. It was found in that case that the model did not contain any extra constraints on these three dynamical fields, if (assuming c 2 = c 4 = 0, as we have done here)
c 1 = 0, c 3 c 1 ≤ 0.(35)
Such models were called "safe" by the authors of [19]; such a model would be expected to contain the three degrees of freedom present in A. Models with c 1 = 0 were called "endangered", in that they contained additional constraints on the initial data; such models would contain fewer than three degrees of freedom. Finally, models with c 1 = 0 and c 3 /c 1 > 0 were called "conditionally endangered", since the constraint structure of the equations differed at various points in configuration space.
To connect this to the present work, we note that (33) is equivalent to
c 1 A 2 0 − c 1 A 2 − c 3 A 2 = 0,(36)
or, since c 1 = 0 and −A 2
0 + A 2 = b under the constraint, b = − c 3 c 1 A 2 .(37)
In the case where c 3 /c 1 ≤ 0 and b < 0, this is guaranteed to hold, and we therefore have no additional constraints and three degrees of freedom. However, if c 3 /c 1 > 0, there can be non-generic points in configuration space where the number of constraints changes.
B. Maxwell case: c13 = 0, c1 = 0
Potential model
We now consider a vector field A a with a "Maxwell" kinetic term, for which c 1 = −c 3 in (16). We again have illuminating, so we will not present it here. However, from the form of Ξ and H A , we can see that it will depend on A 0 , A, Π 0 , Π, and λ-but it will be independent of both ̟ and (more importantly) u λ . four independent fields, namely the four components of A a . In this case, the canonical momentum Π 0 defined in (19) vanishes automatically, giving us a constraint:
Π 0 = 0 ≡ Φ 1 .(38)
The other three canonical momenta Π defined in (20) have an invertible relationship with the corresponding field velocities:
Π = c 1 −˙ A + ∇A 0 .(39)
Thus, the base Hamiltonian density H B , given by
H B = Π 0Ȧ0 + Π ·˙ A − L(40)
must be augmented by an auxiliary Lagrange multiplier term enforcing the constraint Π 0 = 0. After simplification, this yields
H A = − 1 2c 1 Π 2 + Π · ∇A 0 − c 1 2 (∂ i A j )(∂ i A j ) − (∂ i A j )(∂ j A i ) + κ(A 2 − b) 2 + u 0 Π 0 . (41)
Once again, we must ensure that the primary constraint Φ = Π 0 = 0 is conserved by the equations of motion; this again produces a series of secondary constraints:
Π 0 = {Π 0 , H A } = ∇ · Π + 4κ(A 2 − b)A 0 ≡ Ψ 1 (42) Ψ 1 = {Ψ 1 , H A } = −8κ ∇ · ((A 2 − b) A) − (−3A 2 0 + A 2 − b)u 0 + A 0 A · 1 c 1 Π − ∇A 0 .(43)
So long as −3A 2 0 + A 2 − b = 0, the demand that the secondary constraint Ψ 1 be preserved by the evolution determines the auxiliary Lagrange multiplier u 0 uniquely. We therefore have four fields, two constraints (one primary, one secondary), and no undetermined Lagrange multipliers; counting the degree of freedom therefore yields
N dof = 4 − 1 2 (2) = 3.
Lagrange multiplier model
We now apply the same process to the vector model with a Lagrange multiplier, (23). With the addition of the Lagrange multiplier λ, we must also introduce a conjugate momentum ̟. As in the scalar LM model, this vanishes identically, yielding a second primary constraint:
̟ = ∂L ∂λ = 0 ≡ Φ 2(44)
Including this constraint with an auxiliary Lagrange multiplier u λ in the Hamiltonian density gives us the augmented Hamiltonian density for the model:
H A = − 1 2c 1 Π 2 + Π · ∇A 0 − c 1 2 (∂ i A j )(∂ i A j ) − (∂ i A j )(∂ j A i ) + λ(A 2 − b) + u 0 Π 0 + u λ ̟. (45)
We now derive the secondary constraints and see if their time-evolution fixes the auxiliary Lagrange multipliers u 0 and u λ :
Φ 1 = {Π 0 , H A } = ∇ · Π + 2λA 0 ≡ Ψ 1 (46) Φ 2 = {̟, H A } = −A 2 + b ≡ Ψ 2(47)Ψ 1 = {Ψ 1 , H A } = − ∇ · λ A + 2A 0 u λ + 2λu 0 (48) Ψ 2 = {Ψ 2 , H A } = 2A 0 u 0 − 2 A · Π + ∇A 0(49)
Assuming A 0 = 0, the requirement that both (48) and (49) vanish determines the auxiliary Lagrange multipliers u λ and u 0 . The degree of freedom counting is therefore five fields (four components of A a , plus λ); four constraints (two primary, two secondary); and no undetermined auxiliary Lagrange multipliers, for a result of
N dof = 5 − 1 2 (4) = 3.
This is a surprising result: the number of degrees of freedom of the theory when the vector field is "constrained" to a vacuum manifold determined by f (A a ) = 0 is exactly the same as when it is "allowed" to leave this vacuum manifold. In other words, the Lagrangemultiplier "constraint" does not actually reduce the degrees of freedom of the model.
We can again connect this model to the terminology of [19], as we did for the "general" LM case in Section IV A 2. In that work, for a model of a timelike vector field with c 1 = 0 and c 13 = 0 (i.e., c 3 /c 1 = −1), the number of constraints was found to be zero for all points in configuration space; such a model was therefore "safe", with three degrees of freedom at all points in field space. This is in agreement with our work here: so long as the vector A a is constrained to be timelike (b < 0), we will always have A 0 = 0, and the above analysis holds.
C. V -field case: c13 = 0, c1 = 0
Potential model
In this case, we have a Lagrange density with c 1 = 0 and c 3 = 0; such a field is called a "V -field" by Isenberg & Nester [18]. When we calculate the conjugate momenta in this case, (19) allows us to solve for the velocityȦ 0 = Π 0 /c 3 , but (20) becomes a set of three constraints:
Φ ≡ Π + c 3 ∇A 0 = 0.(50)
The augmented Hamiltonian density for the potential model (22) is then
H A = Π 0 A 0 + Π ·˙ A − L P + u · Π + c 3 ∇A 0 = 1 2c 3 Π 0 2 − c 3 2 (∂ i A j )(∂ j A i ) + κ(A 2 − b) 2 + u · Π + c 3 ∇A 0 ,(51)
where u is a vector of auxiliary Lagrange multipliers enforcing the primary constraints (50). Enforcing these primary constraints under time evolution then yields a set of three secondary constraints Ψ:
Φ = { Φ, H A } = ∇Π 0 − c 3 ∇ ∇ · A − 4κ(A 2 − b) A ≡ Ψ.
(52) The time-evolution of these secondary constraints, written out in terms of spatial components, is theṅ
Ψ i = {Ψ i , H A } = M ij u j − 4κ∂ i (A 2 − b)A 0 + 8κ c 3 A 0 A i Π 0 + ∇ · A , (53) where M ij ≡ 4κ δ ij A 2 − b + 2A i A j .(54)
We require thatΨ i = 0. This can be guaranteed in equation (53) via an appropriate choice of u j so long as the matrix M ij is invertible. For general field values, this inverse can be calculated to be
M −1 ij = 1 4κ(A 2 − b) δ ij − 2A i A j 1 + 2 A 2(55)
and so we can solve the equation˙ Ψ = 0 for u so long as A 2 − b = 0. 6 The generic theory therefore has four fields, six constraints (three primary, three secondary) and
N dof = 4 − 1 2 (6) = 1(56)
degree of freedom.
Lagrange multiplier model
As in Section IV B 2, the switch from a potential Vfield model to a Lagrange-multiplier V -field model does 6 It is notable that this set of field values is precisely the vacuum manifold. This property becomes more important in the context of tensor models involving potentials and Lagrange multipliers, and will be discussed more extensively in an upcoming work [16]. not actually "kill off" any degrees of freedom. The augmented Hamiltonian density now contains one more auxiliary Lagrange multiplier u λ , which (as before) enforces the constraint ̟ = 0:
H A = 1 2c 3 Π 0 2 − c 3 2 (∂ i A j )(∂ j A i ) + λ(A 2 − b) + u · Π + c 3 ∇A 0 + u λ ̟. (57)
We thus have four primary constraints, Φ = 0 (as defined in (50)) and ̟ = 0. Requiring that˙ Φ = 0 anḋ ̟ = 0 then yields four secondary constraints, which I will denote by Ψ and Ψ:
Φ = { Φ, H A } = ∇Π 0 − c 3 ∇ ∇ · A − 2λ A ≡ Ψ, (58) ̟ = {̟, H A } = A 2 − b ≡ Ψ.(59)
The time-evolution of these secondary constraints is theṅ
Ψ = { Ψ, H A } = 2 ∇(λA 0 ) − u λ A + λ u (60) Φ = {Ψ, H A } = A · u − 1 c 3 A 0 Π 0 .(61)
These equations determine all four auxiliary Lagrange multipliers u and u λ so long as A = 0 and λ = 0; in this case, we have
u λ = 1 A 2 A · ∇ (λA 0 ) + 1 c 3 λA 0 Π 0 (62) and u = 1 λ u λ A − ∇ (λA 0 ) .(63)
Thus, for generic initial data, we are done. We have five fields, four primary constraints, and four secondary constraints; and so the number of degrees of freedom is
N dof = 5 − 1 2 (8) = 1.(64)
As for the Maxwellian vector theory in Section IV B 2, the addition of a Lagrange multiplier to a V -field model does not reduce its degrees of freedom. This analysis is again in agreement with the work of Garfinkle, Isenberg, & Martin-Garcia [19]. For a model with c 1 = 0, they find that Einstein-aether theory contains additional initial data constraints on the three dynamical fields A, and is therefore "endangered". In the present work, we have confirmed this result: this model does indeed contain fewer than three degrees of freedom. 7
V. DISCUSSION
A. Generalization
We have found that a field theory model in flat spacetime may or may not "lose" a degree of freedom when a constraint is added to the system via a Lagrange multiplier. Specifically, scalar models (Section III) and general vector models (Section IV A) lose a degree of freedom when we replace a potential with a Lagrange multiplier; but Maxwell-type and V -type vector models (Sections IV B & IV C, respectively) retain the same number of degrees of freedom regardless of whether the field values are governed by a potential or by a Lagrange multiplier.
There is an obvious difference between these cases. In those models where there are no primary constraints in the potential model, a Lagrange multiplier eliminates a degree of freedom. In contrast, in the models where the potential model does contain primary constraints, the field theory retains the same number of degrees of freedom when a constraint is imposed via a Lagrange multiplier.
The reason for this difference can be traced to a particular feature of the models we have examined. In those models containing primary constraints, the conservation of the first-order secondary constraints leads to an equation that determines the auxiliary Lagrange multiplier u λ (Eqns. (48) and (60) for the Maxwell-like and Vfield models, respectively). In those models without primary constraints, u λ is only determined once we require that higher-order secondary constraints (specifically, the third-order secondary constraints in Eqs. (13) and (30)) be conserved.
To extend this to a general statement, we first note that the primary constraints for a potential model and its corresponding Lagrange multiplier model are simply related. If the primary constraints for the potential model are a set of M functions {Φ 1 , · · · , Φ M }, then the primary constraints for the corresponding Lagrange multiplier model will simply be {Φ 1 , · · · , Φ M , ̟}, where ̟ is the conjugate momentum to the Lagrange multiplier λ. Moreover, ̟ will commute with all of the primary constraints that derive from the potential model, since none of these constraints depend on λ.
The augmented Hamiltonian density will then be the base Hamiltonian density with terms added to impose the constraints:
H A = H 0 + u I Φ I + u λ ̟.(65)
(Here and in what follows, repeated capitalized Roman indices are summed from 1 to M .) The first-order secondary constraint required in order to maintain ̟ = 0 under time evolution will then be
Ψ λ = {̟, H A } = − δH A δλ = f (ψ α ),(66)
where ψ α here stands for the collection of fields in the model. In addition, there will be a set of first-order sec-ondary constraints Ψ I (I = 1, . . . , M ), each derived from the requirement thatΦ I = 0; these are given by
Ψ I = {Φ I , H A }.(67)
Now consider the time-evolution of the first-order secondary constraints. The time derivative ofΨ λ will be independent of u λ , though it will generally depend on the other auxiliary Lagrange multipliers u I :
Ψ λ ⊃ u J {Ψ λ , Φ J } + u λ {Ψ λ , ̟} = u J {f (ψ α ), Φ J }. (68)
Note that {f (ψ α ), ̟} = 0 since f (ψ α ) is independent of λ. Meanwhile, the expressionΨ I (for arbitrary I) will contain terms of the forṁ
Ψ I = {Ψ I , H A } ⊃ u J {Ψ I , Φ J } + u λ {Ψ I , ̟},(69)
Equations (68) and (69) together imply that if
{Ψ I , ̟} = 0,(70)
then the equations for conservation of the constraints (Ψ I = 0 andΨ λ = 0) do not contain u λ , leaving this auxiliary Lagrange multiplier undetermined at this stage. If this occurs, then we must proceed to find additional second-and higher-order secondary constraints. Since we have more than two additional constraints, but only one additional degree of freedom from λ itself, we conclude that in such cases, the Lagrange-multiplier model will have fewer degrees of freedom than the potential model. 8 This condition (70) can be greatly elucidated via use of the Jacobi identity. Specifically, we have
{{Φ I , H A }, ̟} + {{H A , ̟}, Φ I } + {{̟, Φ I }, H A } = 0
(71) for any primary constraint Φ I . Since ̟ commutes with the rest of these primary constraints, the last term automatically vanishes; and applying (66) and (67) yields the equation
{Ψ I , ̟} = −{f (ψ α ), Φ I }.(72)
Thus, the equation (69) will leave u λ undetermined, and the Lagrange multiplier will reduce the degrees of freedom of the model, so long as
{f (ψ α ), Φ I } = 0,(73)
i.e., the vacuum manifold function f (ψ α ) commutes with all the primary constraints.
B. Lagrange-multiplier models in dynamical spacetimes
The number of degrees of freedom of a field theory in flat spacetime is not always simply related to the number of degrees of freedom it possesses in a curved, dynamical spacetime. It is well-known that diffeomorphisminvariant field theories have primary constraints corresponding to the non-dynamical nature of the lapse and shift functions; when we pass to a dynamical spacetime, we both introduce new fields (the ten metric components) as well as new constraints. 9 Perhaps less well-known, but equally important, is that degrees of freedom which are unphysical (gauge or constraint) in flat spacetime can become "activated" in a minimally coupled curvedspacetime theory [18]. This occurs due to the fact that the covariant derivative of a tensor field (unlike that of a scalar) depends on the derivatives of the metric. The "minimally coupled" kinetic term for a tensor field therefore contains couplings between the metric derivatives and the tensor field derivatives, which can turn equations that were constraints or gauge degrees of freedom in flat spacetime into dynamical equations in curved spacetime, and vice versa.
In light of these facts, we might then ask how much of the above analysis would carry over to dynamical spacetimes. Given the critical role played by the constraints in this analysis, it is natural to ask whether a Lagrangemultiplier model in a dynamical curved spacetime would lose any degrees of freedom relative to the corresponding potential model in a dynamical curved spacetime.
The condition (73) sheds some light on this question. We know that if u λ remains undetermined when we require conservation of the first-order secondary constraints, then we will in general have to find higher-order secondary constraints, leading to a reduction of the degrees of freedom of the theory relative to the corresponding potential model. This will occur when the vacuum manifold function f (ψ α ) commutes with the primary constraints of the theory.
Any diffeomorphism-invariant theory, when decomposed into 3+1 form, will contain terms involving the lapse N and shift N a ; these are related to the spacetime metric g ab and the induced spatial metric h ab by
g ab = h ab − 1 N 2 (t a − N a )(t b − N b ),(74)
where t a is the vector field we have chosen to correspond to "time flow" in our decomposition. We can then write down the Einstein-Hilbert action in terms of this induced metric, the lapse, and the shift. As the lapse and shift can be arbitrarily specified, the are effectively "gauge quantities" corresponding to diffeomorphism invariance; thus, their time derivatives do not appear in the Lagrange density of the theory when it is decomposed. In the Dirac-Bergmann formalism, there are therefore primary constraints on the momenta conjugate to these quantities:
Π ≡ ∂L ∂Ṅ = 0, Π a ≡ ∂L ∂Ṅ a = 0.(75)
The question is then whether the vacuum manifold function f (ψ α ) commutes with these primary constraints. But this is easy enough to see, since
{f (ψ α ), Π} = δf δN , {f (ψ α ), Π a } = δf δN a(76)
Thus, the question of whether the Lagrange multiplier reduces the number of constraints is reduced to the question of whether the vacuum manifold function depends on the lapse and shift. In particular, for a collection of scalar fields in curved spacetime (the dynamical-spacetime analogue of Section III), the vacuum manifold function will be independent of the metric, and so there is no way for the lapse or shift functions to enter into it. We would therefore expect that a Lagrange-multiplier model containing N scalars would have fewer than N degrees of freedom attributable to the scalars. 10 However, for a function of a vector field A a , the norm of the vector field A a will depend on the lapse and shift functions:
A a A b g ab = A a h ab A b − ((t a − N a )A a ) 2 N 2 = A ⊥ a h ab A ⊥ b − A t − N a A ⊥ a 2 N 2 ,(77)
where A t = t a A a and A ⊥ a = h a b A b . Any function of the spacetime norm of A a will therefore depend on the lapse and shift, and so the vacuum manifold function will not commute with the primary constraints of the theory. Given the results stated above, it seems unlikely that the Lagrange multiplier would reduce the number of degrees of freedom of such a theory.
It is interesting to note that this coupling occurs even if the flat-spacetime theory does not contain any primary constraints, as in the general vector models described in Section IV A. Since the conservation of the first-order secondary constraints determines the auxiliary multiplier u λ in this case, rather than giving rise to further constraints, one would conclude that the number of degrees of freedom of a general vector theory in curved spacetime would not be reduced by the presence of a Lagrange multiplier, in contrast to the situation in flat spacetime. In fact, this is confirmed by known results. A model consisting of a vector field in a curved spacetime with a "generic" kinetic term (as in Section IV A) will contain two "metric" degrees of freedom and three "vector" degrees of freedom, regardless of whether the vector is forced to a non-zero expectation value by a Lagrange multiplier [22] or by a potential [18,23].
C. Potential models in the low-energy limit
In classical particle mechanics, it is common to think of a constrained system in relation to an unconstrained system with a potential energy. In the limit where the potential energy becomes infinitely strong, it can be shown that the dynamics of the unconstrained system reduce to those of a system constrained to lie only in the minimum of the potential [24]. It is therefore common, in the analysis of constrained systems, to simply include one or more Lagrange multipliers that enforce the constraints. In general, each Lagrange multiplier reduces the number of degrees of freedom of the system by one.
One might think that this general picture could be carried over to field theory. In particular, a set of fields in a potential could be thought of as possessing a certain number of massive modes (corresponding to oscillations in field-space directions in which the potential increases) and a certain number of massless modes (oscillations in field-space directions in which the potential is flat.) One could then construct a low-energy effective field theory in which the massive modes have "frozen out", reducing the number of degrees of freedom of the model. In this low-energy limit, one would expect the fields to always lie in their vacuum manifold, effectively being constrained there. Hence, one would think that the Lagrange-multiplier version of a potential theory would nicely correspond to the low-energy behavior of the corresponding potential theory.
The results of this work, however, show that the picture is not so simple. While this simple picture holds for scalar fields in flat spacetime, it seems quite unlikely that the low-energy limit of a Maxwell-type or V -type vector field in a potential would correspond to a model with the same kinetic term but containing a Lagrange multiplier. One would expect the low-energy limit to have fewer degrees of freedom than the full potential model; but in these cases, the Lagrange-multiplier models and the corresponding potential models have the same number of degrees of freedom. While the low-energy limit of some such models has been investigated [23,25,26], the Lagrange-multiplier models would necessarily have a different behavior.
In fact, this feature was noted in [23] in the context of a vector field with a "Maxwell" kinetic term. In Section IV.C of that work, it was noted that the Lagrangemultiplier model only corresponded to the low-energy ("infinite-mass") limit of the potential model if the Lagrange multiplier λ was set to zero by fiat. However, for a generic solution λ will not vanish; the vanishing of the secondary constraint in Eq. (46) requires that λ = − ∇ · Π/2A 0 . In other words, one must restrict the class of solutions under considerations-i.e., further reduce the number of degrees of freedom-to obtain the low-energy limit of a potential model from the corresponding Lagrange-multiplier model. This work shows that this lack of direct correspondence is a common feature of models in which tensor fields take on a vacuum expectation value.
Appendix: Poisson brackets and functionals
In calculating the time-evolution of a field quantity in Hamiltonian field theory, one would like to take the Poisson bracket of a field ψ α (x) with the Hamiltonian H to find the time-evolution of the field at x: ψ α (x) = {ψ α (x), H}.
(A.1)
However, one does have to be careful with this notation, as the Poisson bracket is only rigorously defined on realvalued field functionals, not on functions of space like ψ α . Specifically, we have
{G 1 , G 2 } ≡ d 3 z δG 1 δψ α (z) δG 2 δπ α (z) − δG 1 δπ α (z) δG 2 δψ α (z) ,
(A.2) where π α is the conjugate field momentum to ψ α (and a summation over α is implied), and the functional derivatives are implicitly defined via the relation and similarly for π α , where the ellipses stand for higherorder derivatives of ψ α (or π α ), and the partial derivatives of f (and their gradients) are evaluated at the point z. Here and throughout, I will use partial derivatives ∂ to denote the variation of a locally constructed field quantity with respect to one of its arguments, while the δ notation will be reserved for functional derivatives. Under this extension, the Poisson bracket of a local field quantity f (ψ α (x), ∇ψ α (x), . . . ) with the Hamiltonian H = Hd 3 x is "really" the Poisson bracket of the functional F x with H. Restricting attention to quantities that only depend on the fields ψ α and π α and their first derivatives, this Poisson bracket is d dt
[f (ψ α , ∇ψ α , π α , ∇π α )] = {F x , H}
= d 3 z ∂f ∂ψ α δ 3 (x − z) − ∇ a ∂f ∂(∇ a ψ α ) δ 3 (x − z) δH δπ α (z) − ∂f ∂π α δ 3 (x − z) − ∇ a ∂f ∂(∇ a π α ) δ 3 (x − z) δH δψ α (z) (A.6) = ∂f ∂ψ α δH δπ α (x) + ∂f ∂(∇ a ψ α ) ∇ a δH δπ α (x) − ∂f ∂π α δH δψ α (x) − ∂f ∂(∇ a π α ) ∇ a δH δψ α (x) , (A.7)
where all the field quantities are now evaluated at x. Note that this definition implies that that timeevolution "commutes" with spatial derivatives when we take the Poisson bracket, as one would expect. For example, suppose that f = ψ α and H = Hd 3 x, where H is locally constructed from the fields. Then we have This definition can be extended straightforwardly to quantities depending on higher derivatives of ψ α and π α , and the above-mentioned commutativity extends to such cases as well. It can also be extended to the Poisson brackets of two local field quantities f (x) and g(y) by defining functionals F x and G y and following the same procedure. In such cases, the resulting Poisson bracket will contain a factor of δ 3 (x−y). However, in the interests of clarity, we will elide these factors when we take the Poisson bracket of two such quantities; in other words, we will take it as understood that the first argument of such a Poisson bracket is evaluated at x, the second at y, and that the result is multiplied by δ 3 (x − y) or its derivatives.
ACKNOWLEDGMENTS
I would like to thank D. Garfinkle, T. Jacobson, and J. Tasson for helpful discussion and correspondence on this subject. I would also like to thank Perimeter Institute for their support and hospitality during the period over which the majority of this research was conducted.
the definition (A.2) of a Poisson bracket to a local field quantity f (ψ α (x), ∇ψ α (x), . . . ) constructed from field quantities at a fixed point x, one introduces the functional F x ≡ d 3 y f (ψ α (y), ∇ψ α (y), . . . )δ 3 (x − y) . (A.4) The functional derivatives in (A.2) then becomeδF x δψ α (z) = ∂f ∂ψ α δ 3 (x − z) − ∇ a ∂f ∂(∇ a ψ α ) δ 3 (x − z) + . . . (A.5)
x. Meanwhile, if f = ∇ a ψ α , we have {∇ a ψ α , H} = ∇ a δH δπ α (x) = ∇ a ∂H ∂π α = ∇ a ({ψ α , H}) . (A.9)This fact simplifies the calculation of the Poisson brackets considerably.
Here and throughout, we will need to distinguish between the "real" Lagrange multiplier that appears in the original Lagrangian and the "auxiliary" Lagrange multipliers that are used to construct a Hamiltonian for the model. In general, we will only have one real Lagrange multiplier at a time, which we will denote with λ; auxiliary Lagrange multipliers will be denoted with the symbol u, possibly with subscripts or diacritical marks.2 We are playing a bit fast and loose with notation here; in Hamiltonian field theory, the Poisson bracket is only rigorously defined for a functional with a single real value, not for a field which is a function of space. A more rigorous definition of what we mean by an expression like {̟, H A } is given in the Appendix.
This follows the notation of[10], with the coefficient c 4 from that reference set equal to zero.
As spontaneous symmetry breaking implies a non-zero field value, one would normally exclude the case b = 0 to ensure that Aa = 0 is not in the vacuum manifold. However, this is not necessary for the analysis which follows.
The cases A = 0 and λ = 0 were excluded from the above analysis. In this case, one would have to look at the time evolution of the quantities in (60) and (61), generate one or more secondorder secondary constraints, and attempt to solve these for the auxiliary Lagrange multipliers. In any event, this would generate a model with no more than one degree of freedom (if the resulting model was even consistent at such points in configuration space.)
It is also conceivable that u λ could remain undetermined even after the process of finding the constraints is completed. This would also reduce the number of degrees of freedom in the final counting.
See[20,21] for a detailed description of the Hamiltonian formulation of general relativity.
As there is no coupling between the kinetic terms of the scalar and the metric, it seems likely that there would also still be two degrees of freedom attributable to the metric itself.
Broken symmetry and the mass of gauge vector mesons. F Englert, R Brout, Phys. Rev. Lett. 139F. Englert and R. Brout. Broken symmetry and the mass of gauge vector mesons. Phys. Rev. Lett., 13(9):321-323, 1964.
Broken symmetries and the masses of gauge bosons. W Peter, Higgs, Phys. Rev. Lett. 1316Peter W. Higgs. Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett., 13(16):508-509, 1964.
Global conservation laws and massless particles. G S Guralnik, C R Hagen, T W B Kibble, Phys. Rev. Lett. 1320G. S. Guralnik, C. R. Hagen, and T. W B Kibble. Global conservation laws and massless particles. Phys. Rev. Lett., 13(20):585-587, 1964.
Spontaneous symmetry breakdown without massless bosons. W Peter, Higgs, Phys. Rev. 1454Peter W. Higgs. Spontaneous symmetry breakdown with- out massless bosons. Phys. Rev., 145(4):1156-1163, 1966.
On the Theory of Superconductivity. V L Ginzburg, L D Landau, Zhurnal Eksp. noi i Teor. Fiz. 201064V. L. Ginzburg and L. D. Landau. On the Theory of Su- perconductivity. Zhurnal Eksp. noi i Teor. Fiz., 20:1064, 1950.
The axial vector current in beta decay. M Gell-Mann, M Lévy, Nuovo Cim. 164M. Gell-Mann and M. Lévy. The axial vector current in beta decay. Nuovo Cim., 16(4):705-726, may 1960.
On the symmetries of strong and weak interactions. F Gürsey, Nuovo Cim. 162F. Gürsey. On the symmetries of strong and weak inter- actions. Nuovo Cim., 16(2):230-240, apr 1960.
Classical Heisenberg Model. G S Joyce, Phys. Rev. 1552G. S. Joyce. Classical Heisenberg Model. Phys. Rev., 155(2):478-491, mar 1967.
Spontaneous breaking of lorentz symmetry in string theory. V , Alan Kostelecký, Stuart Samuel, Phys. Rev. D. 392V. Alan Kostelecký and Stuart Samuel. Spontaneous breaking of lorentz symmetry in string theory. Phys. Rev. D, 39(2):683-685, 1989.
Gravity with a dynamical preferred frame. Ted Jacobson, David Mattingly, Phys. Rev. D -Part. Fields, Gravit. Cosmol. 6429Ted Jacobson and David Mattingly. Gravity with a dy- namical preferred frame. Phys. Rev. D -Part. Fields, Gravit. Cosmol., 64(2):9, 2001.
Relativistic gravitation theory for the modified Newtonian dynamics paradigm. Jacob Bekenstein, Phys. Rev. D. 708Jacob Bekenstein. Relativistic gravitation theory for the modified Newtonian dynamics paradigm. Phys. Rev. D, 70(8):1-28, oct 2004.
Gravity, Lorentz violation, and the standard model. V , Alan Kostelecký, Phys. Rev. D. 6910105009V. Alan Kostelecký. Gravity, Lorentz violation, and the standard model. Phys. Rev. D, 69(10):105009, may 2004.
Scalar-tensor-vector gravity theory. J W Moffat, J. Cosmol. Astropart. Phys. 033J. W. Moffat. Scalar-tensor-vector gravity theory. J. Cosmol. Astropart. Phys., 03(3), 2006.
Lorentz violation with an antisymmetric tensor. Brett Altschul, Quentin G Bailey, V Alan Kostelecký, Phys. Rev. D. 81665028Brett Altschul, Quentin G. Bailey, and V. Alan Kost- elecký. Lorentz violation with an antisymmetric tensor. Phys. Rev. D, 81(6):065028, mar 2010.
Quantum Electrodynamics in Nonlinear Gauge. Yoichiro Nambu, Suppl. Prog. Theor. Phys., Extra Numb. Yoichiro Nambu. Quantum Electrodynamics in Nonlin- ear Gauge. Suppl. Prog. Theor. Phys., Extra Numb:190- 195, 1968.
. D Michael, Seifert, In preparationMichael D Seifert. In preparation. 2018.
A M Paul, Dirac, Lectures on Quantum Mechanics. Paul A. M. Dirac. Lectures on Quantum Mechanics.
The effect of gravitational interaction on classical fields: A hamilton-dirac analysis. J Isenberg, Nester, Ann. Phys. (N. Y). 1071-2J Isenberg and J Nester. The effect of gravitational in- teraction on classical fields: A hamilton-dirac analysis. Ann. Phys. (N. Y)., 107(1-2):56-81, sep 1977.
Constraint equations in Einstein-aether theories and the weak gravitational field limit. David Garfinkle, James Isenberg, Jose M Martin-Garcia, Phys. Rev. D -Part. Fields, Gravit. Cosmol. 868David Garfinkle, James Isenberg, and Jose M. Martin- Garcia. Constraint equations in Einstein-aether theories and the weak gravitational field limit. Phys. Rev. D - Part. Fields, Gravit. Cosmol., 86(8):1-6, 2012.
M Robert, Wald, General Relativity. University of Chicago PressRobert M Wald. General Relativity. University of Chicago Press, 1984.
Eric Poisson Relativist's Toolkit, The Mathematics of Black-Hole Mechanics. Cambridge University PressEric Poisson. A Relativist's Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press, 2004.
Einstein-aether waves. T Jacobson, D Mattingly, Phys. Rev. D -Part. Fields, Gravit. Cosmol. 702T. Jacobson and D. Mattingly. Einstein-aether waves. Phys. Rev. D -Part. Fields, Gravit. Cosmol., 70(2):1-5, 2004.
Spontaneous Lorentz and diffeomorphism violation, massive modes, and gravity. Robert Bluhm, Shu-Hong Fung, V Alan Kostelecký, Phys. Rev. D. 77665020Robert Bluhm, Shu-Hong Fung, and V. Alan Kostelecký. Spontaneous Lorentz and diffeomorphism violation, mas- sive modes, and gravity. Phys. Rev. D, 77(6):065020, mar 2008.
Mathematical Methods of Classical Mechanics. V I Arnold, SpringerV. I. Arnold. Mathematical Methods of Classical Mechan- ics. Springer, 1989.
Vector models of gravitational Lorentz symmetry breaking. D Michael, Seifert, Phys. Rev. D. 7912124012Michael D Seifert. Vector models of gravitational Lorentz symmetry breaking. Phys. Rev. D, 79(12):124012, jun 2009.
Generalized bumblebee models and Lorentz-violating electrodynamics. Michael D Seifert, Phys. Rev. D. 81665010Michael D. Seifert. Generalized bumblebee models and Lorentz-violating electrodynamics. Phys. Rev. D, 81(6):065010, mar 2010.
| [] |
[
"Signed Complete Graphs on Six Vertices",
"Signed Complete Graphs on Six Vertices"
] | [
"Deepak \nDepartment of Mathematics\nDepartment of Mathematics Indian Institute of Technology Guwahati Guwahati\nIndian Institute of Technology Guwahati Guwahati\n781039, 781039India -, India\n",
"Bikash Bhattacharjya \nDepartment of Mathematics\nDepartment of Mathematics Indian Institute of Technology Guwahati Guwahati\nIndian Institute of Technology Guwahati Guwahati\n781039, 781039India -, India\n"
] | [
"Department of Mathematics\nDepartment of Mathematics Indian Institute of Technology Guwahati Guwahati\nIndian Institute of Technology Guwahati Guwahati\n781039, 781039India -, India",
"Department of Mathematics\nDepartment of Mathematics Indian Institute of Technology Guwahati Guwahati\nIndian Institute of Technology Guwahati Guwahati\n781039, 781039India -, India"
] | [] | A signed graph is a graph whose edges are labeled positive or negative. The sign of a cycle is the product of the signs of its edges. Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs. It is also known that, up to switching isomorphism, there are two signed K 3 's, three signed K 4 's, and seven signed K 5 's. In this paper, we prove that there are sixteen signed K 6 's upto switching ismomorphism. | null | [
"https://arxiv.org/pdf/1812.08383v1.pdf"
] | 119,177,348 | 1812.08383 | 7f7409f210b8805bb8876538757ed1720cee057d |
Signed Complete Graphs on Six Vertices
20 Dec 2018 October 12, 2021
Deepak
Department of Mathematics
Department of Mathematics Indian Institute of Technology Guwahati Guwahati
Indian Institute of Technology Guwahati Guwahati
781039, 781039India -, India
Bikash Bhattacharjya
Department of Mathematics
Department of Mathematics Indian Institute of Technology Guwahati Guwahati
Indian Institute of Technology Guwahati Guwahati
781039, 781039India -, India
Signed Complete Graphs on Six Vertices
20 Dec 2018 October 12, 2021arXiv:1812.08383v1 [math.CO]Signed graphSwitchingSwitching isomorphism
A signed graph is a graph whose edges are labeled positive or negative. The sign of a cycle is the product of the signs of its edges. Zaslavsky proved in 2012 that, up to switching isomorphism, there are six different signed Petersen graphs. It is also known that, up to switching isomorphism, there are two signed K 3 's, three signed K 4 's, and seven signed K 5 's. In this paper, we prove that there are sixteen signed K 6 's upto switching ismomorphism.
Introduction
Throughout the paper we only consider simple graphs. For all the graph-theoretic terms that have not been defined but are used in the paper, see Bondy [1]. Harary [4] was the first to introduced signed graph and balance. Harary [2] used signed graphs to model social stress in small groups of people in social psychology. Subsequently, signed graphs have turned out to be valuable. The fundamental property of signed graphs is balance. A cycle is positive or negative according as the product of the signs of its edges is positive or negative. A signed graph is balanced if all its cycles are positive. The second basic property of signed graphs is switching equivalence. Switching is a way of turning one signature of a graph into another, without changing the sign of its cycles. Many properties of signed graphs are unaltered by switching, the set of unbalanced cycles is a notable example. The author in [9] described the non-isomorphic signed Petersen graph. In [6], the author have studied the non-isomorphic signatures on Heawood graph. In this paper we determine the non-isomorphic signatures on K 6 .
Preliminaries
A signified graph is a graph G together with an assignment of + or − signs to its edges. If Σ is the set of negative edges of a graph G, then we denote the signified graph by (G, Σ). The set Σ is called the signature of (G, Σ). Signature Σ of graph G can also be viewed as a function from E(G) to {+1, −1}. A switching (resigning) of a signified graph at a vertex v is to change the sign of each edge incident to v.
We say (G, Σ 2 ) is switching equivalent or simply equivalent to (G, Σ 1 ) if it is obtained from (G, Σ 1 ) by a sequence of switchings. Equivalently, we say (G, Σ 2 ) is switching equivalent to (G,
Σ 1 ) if there exist a function f from V to {+1, −1} such that Σ 2 (e) = f (u)Σ 1 (e)f (v) for each edge e = uv.
Switching defines an equivalence relation on the set of all signified graphs over G (also on the set of signatures). Each such equivalence class is called a signed graph and is denoted by [G, Σ], where (G, Σ) is any member of the class.
We say that two signified graphs (G, Σ 1 ) and (H, Σ 2 ) are isomorphic, denoted Σ 1 ∼ = Σ 2 , if there exist a graph isomorphism ψ : V (G) → V (H) which preserves the edge signs. Signified graphs (G, Σ 1 ) and (H, Σ 2 ) are called switching isomorphic if Σ 1 is isomorphic to a switching of Σ 2 . That is, there exist a signified graph (H, Σ ′ 2 ) which is equivalent to (H, Σ 2 ) such that Σ 1 ∼ = Σ ′ 2 . We use the notation Σ 1 ∼ Σ 2 to mean that Σ 1 is switching isomorphic to Σ 2 .
Proposition 2.1. [5] If G has m edges, n vertices and c components, then there are 2 (m−n+c) distinct signed graphs on G.
One of the first theorems in the theory of signed graphs is that the set of unbalanced cycles uniquely determines the class of signed graphs to which a signified graph belongs. More precisely, we state the following theorem. Thus if the signed graphs [G, Σ 1 ] and [G, Σ 2 ] have different sets of negative cycles, then they cannot be switching isomorphic.
In a signed graph [G, Σ], a signature Σ ′ which is equivalent to Σ is said to be a minimal signature if the number of edges in Σ ′ is minimum among all equivalent signatures of Σ. We denote the number of edges in Σ ′ by |Σ ′ |. Note that automorphism group of K 6 is S 6 , and it is vertex-transitive as well as edge-transitive. Thus it is easy to see that for any two isomorphic subgraphs of K 6 , there exist an automorphism of K 6 which maps one subgraph onto the other subgraph.
The notation Σ(e 1 , e 2 , ..., e k ) denotes a signature Σ which contains the edges e 1 , e 2 , ..., e k of a graph G. For example, in the graph K 6 of Figure 1, Σ(u 1 u 2 , u 3 u 4 ) denotes a signature containing the edges u 1 u 2 and u 3 u 4 .
Further, we say that two signatures Σ 1 and Σ 2 of a graph G are automorphic if there exists an The distance between two edges e 1 and e 1 of a graph G, denoted by d G (e 1 , e 2 ), is the number of vertices of a shortest path connecting their end points. For example, in the complete graph K 6 of Figure 1, for edges e 1 = u 1 u 2 and e 2 = u 3 u 4 , we have d G (e 1 , e 2 ) = 2.
automorphism f of G such that uv ∈ Σ 1 if and only if f (u)f (v) ∈ Σ 2 . If
Throughout this paper, a negative edge in a signed graph is drawn as dashed line, and a positive edge is drawn as solid line. In the next section, we discuss signings on K 6 .
Signings on K 6
The complete graph K 6 is shown in Figure 1. For a signed graph [G, Σ], let the graph G Σ be such that
V (G Σ ) = V (G) and E(G Σ ) = Σ.of Σ. Then d G Σ ′ (v) ≤ ⌊ n−1 2 ⌋ for each vertex v ∈ V (G Σ ′ ).
Proof. Let, if possible, there exists a vertex u ∈ V (G Σ ) such that d GΣ (u) > n−1 2 . Resign at u to get an equivalent signature Σ 1 . It is clear that |Σ| > |Σ 1 |. We apply the same operation on Σ 1 , if G Σ1 has a vertex of degree greater than n−1 2 . Repeated application, if needed, of this process will ultimately give us an equivalent signatureΣ of minimum number of edges such that degree of every vertex ofΣ is atmost
⌊ n−1 2 ⌋.
It is clear that |Σ| = |Σ ′ |, and every vertex of Σ ′ have degree atmost ⌊ n−1 2 ⌋.
As a particular case of Theorem 4.1, we get the following corollary. Proposition 2.1 tells us that K 6 has 2 10 distinct signed graphs. But in some respect, only 16 of them are different. Basically, we want to find the different signatures on K 6 upto switching isomorphism.
Corollary 4.1.1 suggests that, it is enough to find the non-isomorphic signatures on K 6 of size upto six and vertex degree is atmost two. Further, K 6 is vertex as well as edge-transitive. We use these facts to find distinct automorphic type signatures of various sizes of K 6 in the following lemmas.
We denote a signature of size zero by Σ 0 . We know that K 6 is edge-transitive, so all signatures of size one are automorphic and we denote this automorphic type signature by Σ 1 (u 1 u 2 ).
Lemma 4.1. The number of distinct automorphic type signatures of K 6 of size two is 2.
Proof. For a signature of size two in K 6 , followings are the only possibilities.
(i) Edges of the signature form a path of length two. One of such signatures is Σ 2 (u 1 u 2 , u 2 u 3 ).
(ii) Edges of the signature are at distance two. One of such signatures is Σ 3 (u 1 u 2 , u 3 u 4 ).
It is easy to see that, any other signature of K 6 of size two is either automorphic to Σ 2 or Σ 3 , and that Σ 2 is not automorphic to Σ 3 . This proves the lemma. Proof. For a signature of size three in K 6 , followings are the only possibilities.
(i) Edges of the signature form a path of length three. One of such signatures is Σ 4 (u 1 u 2 , u 2 u 3 , u 3 u 4 ).
(ii) Two edges form a path and the third edge is at distance two from that path. One of such signatures is Σ 5 (u 1 u 2 , u 2 u 3 , u 4 u 5 ).
(iii) All three edges are pairwise at distance two. One of such signatures is Σ 6 (u 1 u 2 , u 6 u 3 , u 5 u 4 ).
(iv) The three edges of the signature form a cycle. One of such signature is Σ 7 (u 1 u 2 , u 2 u 3 , u 3 u 1 ).
It is clear that any other signature of size three of K 6 is automorphic to one of Σ 4 , Σ 5 , Σ 6 , or Σ 7 . Further, these four signatures are pairwise non-automorphic. This proves the lemma. Proof. For a signature of size three in K 6 , followings are the only possibilities.
(i) All four edges of the signature form a path. One of such signatures is Σ 8 (u 1 u 2 , u 2 u 3 , u 3 u 4 , u 4 u 5 ).
(ii) Three edges of the signature form a path and the remaining edge is at distance two from this path.
One of such signatures is Σ 9 (u 1 u 2 , u 2 u 3 , u 3 u 4 , u 5 u 6 ).
(iii) Two edges of the signature lie on a path and other two edges lie on an another path, disjoint from the first path. One of such signatures is Σ 10 (u 1 u 2 , u 6 u 1 , u 3 u 4 , u 4 u 5 ).
(iv) Three edges of the signature form a cycle and the remaining edge is at distance two from that cycle.
One of such signatures is Σ 11 (u 1 u 2 , u 2 u 3 , u 3 u 1 , u 5 u 6 ).
(v) All four edges of the signature form a cycle. One of such signatures is Σ 12 (u 1 u 2 , u 2 u 3 , u 3 u 6 , u 6 u 1 ).
It is easy to see that any other signature of size four in K 6 is automorphic to one of Σ 8 , Σ 9 , Σ 10 , Σ 11 or Σ 12 . Further, these five signatures are pairwise non-automorphic. This proves the lemma. Proof. It is easy to see that any subgraph of K 6 having five edges and having maximum degree two cannot have two disjoint paths of length 1 and 4 or 2 and 3. Therefore for a signature of size five in K 6 , followings are the only possibilities.
(i) All the edges of the signature form a path of length five. One of such signatures is Σ 13 (u 1 u 2 , u 2 u 3 , u 3 u 4 , u 4 u 5 , u 5 u 6 ).
(ii) Four edges of the signature form a cycle and the remaining edge is just a path of length one and disjoint from that cycle. One of such signatures is Σ 14 (u 1 u 2 , u 2 u 3 , u 3 u 4 , u 4 u 1 , u 5 u 6 ).
(iii) Three edges of the signature form a cycle and the remaining two edges form a path of length two.
One of such signatures is Σ 15 (u 1 u 2 , u 2 u 3 , u 3 u 1 , u 4 u 6 , u 5 u 6 ).
(iv) All edges of the signature form a cycle. One of such signatures is Σ 16 (u 1 u 2 , u 2 u 3 , u 3 u 4 , u 4 u 5 , u 5 u 1 ).
It is clear that any other signature of size five of K 6 is automorphic to one of Σ 13 , Σ 14 , Σ 15 or Σ 16 . Also, these four signatures are pairwise non-automorphic. This proves the lemma.
Lemma 4.5. The number of distinct automorphic type signatures of K 6 of size six is 2.
Proof. It is easy to see that a subgraph of K 6 having six edges and having maximum degree two is either a spanning cycle or union of two 3-cycles. One of such signatures whose edges form a spanning cycle is Σ 17 (u 1 u 2 , u 2 u 3 , u 3 u 4 , u 4 u 5 , u 5 u 6 , u 6 u 1 ). Again, one of such signatures whose edges form two disjoint 3-cycles is Σ 18 (u 1 u 2 , u 2 u 3 , u 3 u 1 , u 4 u 5 , u 5 u 6 , u 6 u 4 ). It is clear that any signature of size six of K 6
is automorphic to one of Σ 17 or Σ 18 . These two signatures are non-automorphic too. This proves the lemma.
The signatures obtained in the previous five lemmas along with Σ 0 and Σ 1 give us 19 distinct automorphic type signatures of K 6 , viz., Σ 0 , Σ 1 , . . . , Σ 18 . Notice that any two signatures belonging to any
one of {Σ 0 , Σ 1 }, {Σ 2 , Σ 3 }, {Σ 4 , Σ 5 , Σ 6 , Σ 7 }, {Σ 8 , Σ 9 , Σ 10 , Σ 11 , Σ 12 }, {Σ 13 , Σ 14 , Σ 15 , Σ 16 } or {Σ 17 , Σ 18 }
are not automorphic. However, two among these 19 signatures may be switching isomorphic to each other. We have the following observations.
• In Σ 6 , by resigning at {u 2 , u 3 , u 4 }, we get a signature which is automorphic to Σ 17 . Thus Σ 6 is switching isomorphic to Σ 17 , that is, Σ 6 ∼ Σ 17 .
• In Σ 10 , by resigning at {u 1 , u 3 , u 5 }, we get a signature which is auotomprphic to Σ 14 . Thus Σ 10 is switching isomorphic to Σ 14 , that is, Σ 10 ∼ Σ 14 .
• In Σ 13 , by resigning at {u 2 , u 4 , u 6 }, we get a signature which is automorphic to Σ 9 . Thus Σ 9 is switching isomorphic to Σ 13 , that is, Σ 9 ∼ Σ 13 .
Thus we are left with the signatures Σ 0 , Σ 1 , Σ 2 , Σ 3 , Σ 4 , Σ 5 , Σ 6 , Σ 7 , Σ 8 , Σ 9 , Σ 10 , Σ 11 , Σ 12 , Σ 15 , Σ 16 and Σ 18 , and their corresponding signified graphs are shown in Figure 2. Now we show that no two of these 16 signatures are switching isomorphic. Proof. Let the number of negative 3-cycles, number of negative 4-cycles and number of negative 5-cycles of a signified graph (K 6 , Σ) be denoted by |C − 3 |, |C − 4 |, and |C − 5 |, respectively. These numbers for the sixteen signed K 6 depicted in Figure 2 are given in Table 1.
From Table 1 and Theorem 2.2, it is easy to see that all the signatures depicted in Figure 2 are pairwise non-isomorphic. This concludes the proof of the theorem. Figure 2: The sixteen signed K 6 .
We described the different signatures of complete graph K 6 upto switching isomorphism. In [8], Zaslavsky introduced the concepts of signed graph colouring and signed chromatic number of signed graphs. Zaslavsky also shown that these parameters are invariant under switching. So, study of these invariants for the sixteen non-isomorphic signatures of K 6 would be interesting.
Theorem 2 . 2 .
22[7] Two signatures Σ 1 and Σ 2 are equivalent if and only if they have the same set of unbalanced cycles.
For
example, if [G, Σ] is balanced then Σ ′ = ∅ and so |Σ ′ | = 0. Notice that there may be two or more than two minimal signatures for a signed graph [G, Σ]. For example, in the signed graph [K 3 , Σ], where Σ = {12, 23, 31}, the equivalent signatures Σ 1 = {12} and Σ 2 = {23} are minimal signatures. This shows that minimal signature of a signed graph is not unique.
two signatures are automorphic then they are said to be automorphic type signatures. If two signatures Σ 1 and Σ 2 of a graph G are not automorphic to each other, then we say that they are distinct automorphic type signatures. For example, in signed graphs [K 6 , {u 1 u 2 }] and [K 6 , {u 3 u 5 }], the signatures {u 1 u 2 } and {u 3 u 5 } are automorphic type signatures.
Theorem 4 . 1 .
41Let [G, Σ] be a signed graph on n vertices and let Σ ′ be an equivalent minimal signature
Figure 1 :
1The complete graph K 6 .
Corollary 4.1. 1 .
1Let [K6 , Σ] be a signed graph and Σ ′ be an equivalent minimal signature of Σ. Then the size of Σ ′ is at most 6.
Lemma 4. 2 .
2The number of distinct automorphic type signatures of K 6 of size three is 4.
Lemma 4. 3 .
3The number of distinct automorphic type signatures of K 6 of size four is 5.
Lemma 4. 4 .
4The number of distinct automorphic type signatures of K 6 of size five is 4.
Theorem 4. 2 .
2There are exactly 16 different signatures on K 6 upto switching isomorphism.
Σ 0 Σ 1 Σ 2 Σ 3 Σ 4 Σ 5 Σ 6 Σ 7 Σ 8 Σ 9 Σ 10 Σ 11 Σ 12 Σ 15 Σ 16 Σ 18Table 1: Number of negative 3-cycles, 4-cycles and 5-cycles in different signed K 6 .|C −
3 |
0
4
6
8
8
10 12 10 10 12
12
14
8
16
10
20
|C −
4 |
0
12 18 20 24 22 24 18 26 24
20
18
24
12
30
0
|C −
5 |
0
24 24 32 40 36 24 36 36 32
40
36
48
48
36
72
Σ 0
Σ 1
Σ 2
Σ 3
Σ 4
Σ 5
Σ 6
Σ 7
Σ 8
Σ 9
Σ 10
Σ 11
Σ 12
Σ 15
Σ 16
Σ 18
J A Bondy, U S R Murty, Graph T heory. SpringerJ. A. Bondy, and U. S. R. Murty, Graph T heory, Graduate Text in Mathematics, Springer, 2007.
Structural balance: a generalization of Heider's theory. D Cartwright, Frank Harary, Psychol. Rev. 63277293D. Cartwright, and Frank Harary, Structural balance: a generalization of Heider's theory, Psychol. Rev. 63 (1956) 277293.
The groups of the generalized Petersen graphs. R Frucht, J E Graver, M E Watkins, Proc. Cambridge Philos. Soc. 70211218R. Frucht, J. E. Graver, and M. E. Watkins, The groups of the generalized Petersen graphs, Proc. Cambridge Philos. Soc., 70 (1971), pp. 211218.
On the notion of balance of a signed graph. F Harary, Michigan Math J. 2143146F. Harary, On the notion of balance of a signed graph. Michigan Math J 2 (1953-54), 143146.
Homomorphisms of Signed Graphs. R Naserasr, E Rollov, Sopena, 10.1002/jgt.21817J. Graph Theory. 79R. Naserasr, Rollov, and E. Sopena, Homomorphisms of Signed Graphs. J. Graph Theory, 79: 178- 212. doi:10.1002/jgt.21817.
Seven signings of the Heawood graph. V Sivaraman, The Ohio State UniversityPhd ThesisV. Sivaraman, Seven signings of the Heawood graph. Phd Thesis, The Ohio State University, 2012.
Signed graphs. T Zaslavsky, Discrete Appl Math. 414774T. Zaslavsky, Signed graphs. Discrete Appl Math 4(1) (1982), 4774.
Signed graph coloring. T Zaslavsky, Discrete Math. 392T. Zaslavsky, Signed graph coloring, Discrete Math. 39 (1982), no. 2, 215-228.
Six signed Petersen graphs, and their automorphisms. T Zaslavsky, Discrete Math. 3129T. Zaslavsky, Six signed Petersen graphs, and their automorphisms, Discrete Math. 312 (2012), no. 9, 1558-1583.
| [] |
[
"On Possible Study of Quark-Pomeron Coupling Structure at the COMPASS spectrometer",
"On Possible Study of Quark-Pomeron Coupling Structure at the COMPASS spectrometer"
] | [
"S V Goloskokov \nBogoliubov Laboratory of Theoretical Physics\nJoint Institute for Nuclear Research\n141980Dubna, Moscow regionRussia\n"
] | [
"Bogoliubov Laboratory of Theoretical Physics\nJoint Institute for Nuclear Research\n141980Dubna, Moscow regionRussia"
] | [] | We analyse the diffractive QQ production and final jet kinematics in polarized deep-inelastic lp scattering at √ s = 20GeV . We show that this reaction can be used in the new spectrometer of the COMPASS Collaboration at CERN to study the quarkpomeron coupling structure.1 | 10.1142/s0217732397000170 | [
"https://arxiv.org/pdf/hep-ph/9611350v1.pdf"
] | 11,921,016 | hep-ph/9611350 | 4a041196c4f0b222c8d305fcd9f7f2fb0831bd31 |
On Possible Study of Quark-Pomeron Coupling Structure at the COMPASS spectrometer
arXiv:hep-ph/9611350v1 18 Nov 1996
S V Goloskokov
Bogoliubov Laboratory of Theoretical Physics
Joint Institute for Nuclear Research
141980Dubna, Moscow regionRussia
On Possible Study of Quark-Pomeron Coupling Structure at the COMPASS spectrometer
arXiv:hep-ph/9611350v1 18 Nov 1996
We analyse the diffractive QQ production and final jet kinematics in polarized deep-inelastic lp scattering at √ s = 20GeV . We show that this reaction can be used in the new spectrometer of the COMPASS Collaboration at CERN to study the quarkpomeron coupling structure.1
The diffractive events with a large rapidity gap in deep inelastic lepton-proton scattering e + p → e ′ + p ′ + X
have recently been investigated (see, e.g. [1,2]). These experiments have given an excellent tool to test the structure of the pomeron and its couplings. As a result, the study of the pomeron properties becomes again popular now. The diffractive lepton-proton reactions (1) is described usually in terms of the kinematic variables
Q 2 = −q 2 , t = r 2 , y = pq p l p , x = Q 2 2pq , x p = q(p − p ′ ) qp , β = x x p ,(2)
where p l , p ′ l and p, p ′ are the initial and final lepton and proton momenta, respectively, q = p l − p ′ l , r = p − p ′ are the virtual photon and pomeron momenta. The cross section of this reaction is related to the diffractive structure function
d 4 σ dxdQ 2 dx p dt = 4πα 2 xQ 4 [1 − y + y 2 2 ]F D(4) 2 (x, Q 2 , x p , t),(3)
which is determined by the pomeron contribution and usually represented at small x p in the factorized form F
D(4) 2 (x, Q 2 , x p , t) = f (x p , t)F P 2 (β, Q 2 , t).(4)
Here f (x p , t) is the pomeron flux factor and F P 2 (β, Q 2 , t) is the pomeron structure function. The function f (x p , t) at small x p behaves as [3] f (x p , t) ∝ 1
x 2α P (t)−1 p ,(5)
where α P (t) is the pomeron trajectory
α P (t) = α P (0) + α ′ t, α ′ = 0.25(GeV ) −2 .(6)
The future polarized diffractive experiments at DESY, CERN and Brookhaven [4] might give the possibility to study the spin structure of the pomeron. One of the places to perform such an experiment is the future detector of the COMPASS collaboration at CERN [5] which will use the polarized muon beam and fixed polarized hadron target. The important feature of COMPASS is the possibility to detect the hadron component of the process within an angle of about 200-250 Mrad.
The question on the value of the spin-flip component of the pomeron should be very important for the diffractive scattering of polarized particles. In the nonperturbative twogluon exchange model [6] and the BFKL model [7] the pomeron couplings have a simple matrix structure (the standard coupling in what follows):
V µ hhI P = β hhI P γ µ .(7)
In this case, the spin-flip effects are suppressed as a power of s.
It was shown in [8,9] that in addition to the standard pomeron vertex (7) determined by the diagrams where gluons interact with one quark in the hadron [6], the large-distance gluon-loop effects (see Fig.1) should complicate the structures of the pomeron coupling. Really, if we consider the gluon loop correction of Fig.1a for the standard pomeron vertex (7) and the massless quark, we obtain in addition to the γ µ term, new structures
γ α (/ k + / r)γ µ / kγ α ≃ −2[2(/ k + / r 2 )k µ + iǫ µαβρ k α r β γ ρ γ 5 ],(8)
where k is a quark momentum, r is a momentum transfer. The perturbative calculations [8] of both graphs, Fig.1, give the following form for this vertex:
V µ qqI P (k, r) = γ µ u 0 + 2M Q k µ u 1 + 2k µ / ku 2 + iu 3 ǫ µαβρ k α r β γ ρ γ 5 + iM Q u 4 σ µα r α ,(9)
where M Q is the quark mass. We shall call the form (9) the spin-dependent pomeron coupling. It has been shown [10] that the functions u 1 (r) − u 4 (r) can reach 20 − 30% of the standard pomeron term ∼ γ µ for |r 2 | ≃ few GeV 2 . Moreover, they result in the spin-flip effect at the quark-pomeron vertex in contrast with the term γ µ . So, the loop diagrams lead to a complicated spin structure of the pomeron couplings. The phenomenological vertex V µ qqI P with the γ µ and u 1 terms was proposed in [11]. The modification of the standard pomeron vertex (7) might be obtained from the instanton contribution [12]. The test of the spin properties of the pomeron coupling can be done in future polarized experiments. At small x p , the contribution where all the energy of the pomeron goes into the QQ production [13,14] might be very important. The role of these contributions in the spin asymmetries of diffractive two-jet production has been studied in [9,15]. It has been found that the A ll asymmetry in the light quark production in deep inelastic lp scattering, Fig.2, is dependent on the pomeron coupling structure. This asymmetry for cross sections integrated over the transverse momentum of jet could reach 10 −20% [15]. The dependence of polarized cross sections and double-spin longitudinal asymmetry on the transverse momentum of a produced jet k 2 ⊥ and their sensitivity to the quark-pomeron coupling structure have been studied in [16].
In this paper, we analyze the effects of the quark-pomeron coupling in the polarized diffractive e + p → e ′ + p ′ + QQ reaction at the energy √ s = 20GeV . We estimate the cross section, the longitudinal double-spin asymmetry A ll and the kinematics of the final jet to show that these events can be studied at future spectrometer of the COMPASS Collaboration [5].
The diffractive light QQ production in lepton-proton reaction is determined by the diagram of Fig. 2. The spin-average cross section can be written in the form [16]
σ(t) = d 5 σ( → ⇐ ) dxdydx p dtdk 2 ⊥ + d 5 σ( → ⇒ ) dxdydx p dtdk 2 ⊥ = 3(1 − y + y 2 /2)β 4 0 F (t) 2 [9 i e 2 i ]α 2 128x 2α P (t) p yQ 2 π 3 N(β, k 2 ⊥ , x p , t) 1 − 4k 2 ⊥ β/Q 2 (k 2 ⊥ + M 2 Q ) 2 .(10)
Here σ( → ⇒ ) and σ( → ⇐ ) are the cross sections with parallel and antiparallel longitudinal polarization of the leptons and protons, β 0 is the quark-pomeron coupling, F (t) is the pomeronproton form factor, e i are the quark charges. The leading x p dependence is extracted in the coefficient of Eq.(10) which is determined by the pomeron flux factor (5). The trace over the quark loop -N may be decomposed as follows
N(β, k 2 ⊥ , t) = N s (β, k 2 ⊥ , t) + δN(β, k 2 ⊥ , t).(11)
Here N s is the contribution of the standard pomeron vertex (7) and δN contains the contribution of the u 1 (r) − u 4 (r) terms from (9). For N s in the case of light quarks in the loop and x p = 0 we find
N s (β, k 2 ⊥ , t) = 32[2(1 − β)k 2 ⊥ − β|t|]|t|.(12)
The form of δN is more complicated. We have found it in the β → 0 limit. For the massless quarks only the u 3 terms contribute to δN:
δN(k 2 ⊥ , t) = 32k 2 ⊥ |t|[(k 4 ⊥ + 4k 2 ⊥ |t| + |t| 2 )u 3 − 4k 2 ⊥ − 2|t|]u 3 .(13)
Note that δN is positive because u 3 ≤ 0. Higher twist terms of an order of M 2 Q /Q 2 and |t|/Q 2 have been dropped in (12,13).
The difference of the cross section for the supercritical pomeron can be written in the form
∆σ(t) = d 5 σ( → ⇐ ) dxdydx p dtdk 2 ⊥ − d 5 σ( → ⇒ ) dxdydx p dtdk 2 ⊥ = 3(2 − y)β 4 0 F (t) 2 [9 i e 2 i ]α 2 128x 2α P (t)−1 p Q 2 π 3 A(β, k 2 ⊥ , x p , t) 1 − 4k 2 ⊥ β/Q 2 (k 2 ⊥ + M 2 Q ) 2 .(14)
The notation here is similar to that used in Eqs. (10). The function A is determined by the trace over the quark loop. It can be written in the x p → 0 limit as follows:
A(β, k 2 ⊥ , t) = A s (β, k 2 ⊥ , t) + δA(β, k 2 ⊥ , t).(15)
Here A s is the contribution of the standard pomeron vertex (7) and δA is determined by the u 1 (r) − u 4 (r) terms from (9). The function A s for the light quarks looks like
A s (β, k 2 ⊥ , t) = 16(2(1 − β)k 2 ⊥ − |t|β)|t|.(16)
We have calculated δA in the β → 0 limit. For the massless quarks we have
δA(β, k 2 ⊥ , t) = −16(3k 2 ⊥ + 2|t|)k 2 ⊥ |t|u 3 .(17)
The leading twist terms have been calculated here as previously.
It can be seen that σ has a more singular behaviour than δσ as x p → 0. This is determined by the fact that the leading term in δσ is proportional to ǫ µναβ r β ... ∝ x p p. The same is true for the lepton part of the diagram of Fig.1. As a result, the additional term yx p appears in δσ.
We calculate the cross section integrated over momentum transfer because it is difficult to detect the recoil proton in COMPASS detector
σ[∆σ] = 0 tm dtσ(t)[∆σ(t)], |t m | = 7(GeV ) 2 .(18)
The exponential form of the proton form factor F (t) = e bt with b = 1.9(GeV ) −2 has been used.
As an example, we calculate the cross sections and asymmetry for β = 0.175, y = 0.7, x p = 0.1 and Q 2 = 5GeV 2 . The results for the cross section of the light quark production in diffractive deep inelastic scattering for the pomeron with the pomeron intercept α P (0) = 1.1 are shown in Fig. 3 for the standard and spin-dependent pomeron couplings. The shape of both the curves is very similar and for the spin-dependent pomeron coupling the cross section is almost twice that for the standard pomeron coupling.
The longitudinal double spin asymmetry is determined by the relation
A ll = ∆σ σ = σ( → ⇐ ) − σ( → ⇒ ) σ( → ⇒ ) + σ( → ⇐ ) ,(19)
The asymmetry of the diffractive light QQ production is shown in Fig. 4. It can be seen from the cross section (14,10) that the asymmetry for the standard quark-pomeron vertex is very simple in form
A ll = yx p (2 − y) 2 − 2y + y 2 .(20)
There is no any k ⊥ and β dependence here. For the spin-dependent pomeron coupling the asymmetry is more complicated because of different contributions to δA and δN proportional to k 2 ⊥ . In this case the A ll asymmetry is smaller than for the standard pomeron vertex. Thus, the A ll asymmetry can be used to test the quark-pomeron coupling structure.
Let us estimate now the kinematics of jet events. The jet momenta are:
j 1 = q − k, j 2 = r + k.(21)
The photon momentum can be written in the center-mass system in the form
q = y √ s, −Q 2 √ s , q ⊥ , | q ⊥ | = (1 − y)Q 2 .(22)
The transverse momentum r can be written as follows
r = −|t| √ s , x p √ s, r ⊥ , | r ⊥ | = (1 − x p )|t|.(23)
From the mass-shell conditions for jet momenta j 2 1 = j 2 2 = M 2 Q the quark momentum k has been found to be
k ≃ ( r ⊥ + k ⊥ ) 2 + M 2 Q √ sx p , − yQ 2 + ( q ⊥ − k ⊥ ) 2 + M 2 Q √ sy , k ⊥ .(24)
In (22-24) the light-cone variables have been used. The jet momenta and its angles in the rest system of the initial proton can be expressed in terms of (22-24)
P J1 ≃ yx p s − k 2 ⊥ − M 2 Q 2x p m , sin θ J1 2 ≃ m ( k ⊥ − q ⊥ ) 2 ys ; (25) P J2 ≃ m 2 + M 2 Q + k 2 ⊥ 2x p m , sin θ J2 2 ≃ mx p m 2 + M 2 Q + k 2 ⊥ .
Here m is the proton mass. The invariant mass of a produced system is
M 2 2Jet = x p ys.(26)
The momenta and jet angles for √ s = 20GeV, x p = 0.1, y = 0.7 and Q 2 = 5GeV 2 are shown in Figs 5,6 for the azimuth angle between the lepton scattering plane and k ⊥ is equal to 90 degree. It is seen that both jets can be detected by the COMPASS detector whose angular acceptance is about 200-250Mrad.
Thus, we have found that the structure of the quark-pomeron coupling should modify the spin average and spin-dependent cross section. The spin-dependent form of V qqP almost twice increases the cross section. However, the shape of the cross sections is very similar for the standard and spin-dependent pomeron vertices. The A ll asymmetry is more convenient to test the pomeron coupling structure. The asymmetry is free from normalization factors and is sensitive to the dynamics of pomeron interaction. We have found a well-defined prediction for A ll for the standard pomeron vertex. This conclusion is similar to the results of [9] where the single-spin asymmetry in the diffractive QQ production has been studied.
The predicted cross sections are not small for the experimental investigation of this reaction. Our analysis of jet kinematics shows that they might be detectable by the COMPASS spectrometer. There is no possibility to detect the final proton. However, the analysis of the diffractive events similar to that done in HERA experiments [1,2] can be performed in this case, too.
We can conclude that the study of the longitudinal double spin asymmetry and the cross section of the diffractive deep inelastic scattering at the new spectrometer of the COMPASS Collaboration at CERN can give important information about the complicated spin structure or the pomeron coupling.
Fig. 1
1Gluon-loop contribution to the quark-pomeron coupling. Broken line -the pomeron exchange.
Fig. 2 Diffractive
2QQ production in deep inelastic scatteringFig.1 k 2 ⊥ -dependence of cross-sections at √ s = 20(GeV ). Solid line -for the standard vertex; dot-dashed line -for the spin-dependent quark-pomeron vertex.
Fig.2 k 2 ⊥
2-dependence of A ll asymmetry at √ s = 20(GeV ). Solid line -for the standard vertex; dot-dashed line -for the spin-dependent quark-pomeron vertex.
Fig.5 k 2 ⊥
2-dependence of jet momenta. Solid and dot-dashed line -for jet1 and jet2 respectively.
Fig.6 k 2 ⊥
2-dependence of jet angles. Solid and dot-dashed line -for jet1 and jet2 respectively.
. M Derrick, ZEUS CollaborationZ.Phys. 68569ZEUS Collaboration, M.Derrick et al. Z.Phys. C68 (1995) 569.
. T Ahmed, H1 CollaborationPhys.Lett. 348681H1 Collaboration, T.Ahmed et al, Phys.Lett. B348 (1995) 681.
. A Donnachie, P V Landshoff, Nucl.Phys. 303634A.Donnachie, P.V.Landshoff, Nucl.Phys. B303 (1988) 634.
. G Bunce, Phys. World. 31G.Bunce et al., Phys. World 3 (1992) 1;
W.-D Nowak, AIP conference proceedings 343 on High Energy Spin Physics. K. J. Heller and S. L. Smith, Woodbury, NY412W.-D.Nowak, in AIP conference proceedings 343 on High Energy Spin Physics, eds. K. J. Heller and S. L. Smith, Woodbury, NY (1995), p.412;
A Schäfer, J Feltesse, Proc. of the Workshop "Future Physics at HERA. of the Workshop "Future Physics at HERAA.Schäfer and J.Feltesse, in Proc. of the Workshop "Future Physics at HERA", DESY May 29-31, 1996.
Proposal COMPASS, the. G Baum, COMPASS CollaborationCERN/SPSLC 96-16. G.Baum et al. Proposal COMPASS, the COMPASS Collaboration, CERN/SPSLC 96- 16, 1996.
. P V Landshoff, O Nachtmann, Z.Phys. 35405P.V.Landshoff, O. Nachtmann, Z.Phys. C35 (1987) 405.
. E A Kuraev, L N Lipatov, V S Fadin, Sov.Phys. JETP. 44443E.A.Kuraev, L.N.Lipatov, V.S.Fadin, Sov.Phys. JETP 44 (1976) 443;
. Y Y Balitsky, L N Lipatov, Sov.J.Nucl.Phys. 28822Y.Y.Balitsky,L.N.Lipatov, Sov.J.Nucl.Phys. 28 (1978) 822.
. S V Goloskokov, Phys. Lett. 315459S.V. Goloskokov, Phys. Lett. B315 (1993) 459.
. S V Goloskokov, Phys.Rev. 535995S.V. Goloskokov, Phys.Rev. D53 (1996) 5995.
. S V Goloskokov, O V Selyugin, Yad. Fiz. 57727S.V.Goloskokov, O.V. Selyugin, Yad. Fiz. 57 (1994) 727.
. J Klenner, A Schäfer, W Greiner, Z.Phys. 352203J.Klenner, A.Schäfer, W.Greiner, Z.Phys. A352 (1995) 203.
. A E Dorokhov, N I Kochelev, Yu A Zubov, Int.Journ.Mod.Phys. 8603A.E.Dorokhov, N.I.Kochelev, Yu.A.Zubov, Int.Journ.Mod.Phys. A8 (1993) 603;
. M Anselmino, S Forte, Phys.Rev.Lett. 71223M.Anselmino, S.Forte, Phys.Rev.Lett. 71 (1993) 223.
. A Donnachie, P V Landshoff, Phys.Lett. 285172A.Donnachie, P.V.Landshoff, Phys.Lett. B285 (1992) 172.
. J C Collins, L Frankfurt, M Strikman, Phys.Lett. 307161J.C.Collins, L.Frankfurt, M.Strikman, Phys.Lett. B307 (1993) 161.
S V Goloskokov, E -Prints, hep-ph 9506347. 9509238S.V.Goloskokov, E-prints: hep-ph 9506347; hep-ph 9509238;
S V Goloskokov, Proc. of the Workshop "Future Physics at HERA. of the Workshop "Future Physics at HERAS.V.Goloskokov, in Proc. of the Workshop "Future Physics at HERA", DESY May 29-31, 1996.
. S V Goloskokov, E , 9604359S.V. Goloskokov, E-print hep-ph 9604359;
E-print hep-ph 9610342. S V Goloskokov, Proc. of 12 Int. Symposium on High Energy Spin Physics. of 12 Int. Symposium on High Energy Spin PhysicsAmsterdamS.V. Goloskokov, E-print hep-ph 9610342, to be published in Proc. of 12 Int. Symposium on High Energy Spin Physics, Amsterdam, 10-14 September 1996.
| [] |
[
"NODAL GEOMETRY, HEAT DIFFUSION AND BROWNIAN MOTION",
"NODAL GEOMETRY, HEAT DIFFUSION AND BROWNIAN MOTION"
] | [
"Bogdan Georgiev ",
"Mayukh Mukherjee "
] | [] | [] | We use tools from n-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold M . On one hand we extend a theorem of Lieb (see [L]) and prove that any nodal domain Ω λ almost fully contains a ball of radius ∼ 1 √ λ | 10.2140/apde.2018.11.133 | [
"https://arxiv.org/pdf/1602.07110v4.pdf"
] | 55,931,461 | 1602.07110 | 60cdaa1ced1dd6f90772dbeb28468816de7398ca |
NODAL GEOMETRY, HEAT DIFFUSION AND BROWNIAN MOTION
Bogdan Georgiev
Mayukh Mukherjee
NODAL GEOMETRY, HEAT DIFFUSION AND BROWNIAN MOTION
We use tools from n-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold M . On one hand we extend a theorem of Lieb (see [L]) and prove that any nodal domain Ω λ almost fully contains a ball of radius ∼ 1 √ λ
Introduction
We consider a compact n-dimensional smooth Riemannian manifold M , and the Laplacian (or the Laplace-Beltrami operator) −∆ on M 1 . For an eigenvalue λ of −∆ and a corresponding eigenfunction ϕ λ , recall that a nodal domain Ω λ is a connected component of the complement of the nodal set N ϕ λ := {x ∈ M : ϕ λ (x) = 0}. In this paper, we are interested in the asymptotic geometry of a nodal domain Ω λ as λ → ∞.
In this note we address the following two questions. First, we start by discussing the problem of whether a nodal domain can be squeezed in a tubular neighbourhood around a certain subset Σ ⊆ M . A result of Steinerberger (see Theorem 2 of [St]) states that for some constant r 0 > 0 a nodal domain Ω λ cannot be contained in a r0 √ λ -tubular neighbourhood of hypersurface Σ, provided that Σ is sufficiently flat in the following sense: Σ must admit a unique metric projection in a wavelength (i.e. ∼ 1 √ λ ) tubular neighbourhood. The proof involves the study of a heat process associated to the nodal domain, where one also uses estimates for Brownian motion and the Feynman-Kac formula.
We relax the conditions imposed on Σ. Our first result is a direct extension of Theorem 2 of [St]. Before stating the result, we begin with the following definition: Definition 1.1 (Admissible Collections). For each fixed eigenvalue λ, we consider a natural number m λ ∈ N and a collection Σ λ := ∪ m λ i=1 Σ i λ , where Σ i λ is an embedded smooth submanifold (without boundary) of dimension k, (1 ≤ k ≤ n − 1).
We call Σ λ admissible up to a distance r if the following property is satisfied: for any x ∈ M with dist(x, Σ λ ) ≤ r there exists a unique index 1 ≤ i x (λ) ≤ m λ and a unique point y ∈ Σ ix(λ) λ realizing dist(x, Σ λ ) -that is, dist(x, y) = dist(x, Σ λ ).
We note that if Σ λ consists of one submanifold which is admissible up to distance r, then Definition 1.1 means that r is smaller than the normal injectivity radius of Σ λ . Moreover, if Σ λ consists of more submanifolds, then these submanifolds must be disjoint and the distance between every two of them must be greater than r.
Let us also remark that, Theorem 2 of [St] holds true when the hypersurface Σ is allowed to vary with respect to λ in a controlled way, which is made precise by Definition 1.1. With that clarification in place, Theorem 1.2 is an extension of Theorem 2 of [St].
Theorem 1.2. There is a constant r 0 depending only on (M, g) such that if a submanifold Σ λ ⊂ M is admissible up to distance 1 √ λ , then no nodal domain Ω λ can be contained in a r0 √ λ -tubular neighbourhood of Σ λ . Further, it turns out that we can select Σ λ to be a union of submanifolds of varying dimensions, having relaxed admissibility conditions.
Elaborating on this, we observe that getting entirely rid of the admissibility condition, as in Definition 1.1 allows situations where Σ i λ is dense in M , for example, M = T 2 and Σ 1 λ being a generic geodesic. By assuming Σ i λ is compact, we avoid such situations. Also, since we are considering unions of surfaces, the restriction of "unique projection" of nearby points, as in Definition 1.1, makes no sense any more, and one can see that the approach of the proof of Theorem 1.2 does not work.
First, for ease of presentation, we adopt the following notation.
Definition 1.3. Given a compact subset K of M , let ψ K (t, x) denote the probability that a particle undergoing a Brownian motion starting at the point x will reach K within time t.
We now introduce the following relaxed notion of admissibility.
Definition 1.4 (α-admissible Collections). Let 0 < α < 1 be a constant. For each fixed eigenvalue λ, we consider a natural number m λ ∈ N and a collection Σ λ :=
∪ m λ i=1 Σ i λ , where Σ i λ is a compact embedded smooth submanifold (without boundary) of dimension k i , (1 ≤ k i ≤ n − 1). Denote the respective tubular neighbourhoods by N ε (Σ i λ ) := {x ∈ M : dist (x, Σ i λ ) < ε}, and let N ε (Σ λ ) = ∪ m λ i=1 N ε (Σ i λ )
. We say that the collection Σ λ is α-admissible, if for each sufficiently small ε > 0 and each x ∈ N ε (Σ λ ) we have
(1) ψ ∂B(x,2 )\Nε(Σ λ ) (4ε 2 , x) ≥ αψ ∂B(x,2 ) (4ε 2 , x).
Intuitively, using the above implicit formulation via Brownian motion hitting probabilities, we wish to ensure that N ε (Σ λ ) does not occupy too large a proportion of each B(x, 2 ) for x ∈ N ε (Σ λ ) (cf. diagram on page 12 below).
In other words, we allow the family Σ λ to intersect, but the intersections should not be "too dense". To illustrate the idea, let us for simplicity assume that M = R n and let us suppose that each member Σ i λ of the collection Σ λ is a line passing through the origin. If the collection of these lines gets sufficiently close together or in other words "dense", then no matter how small ε > 0 we take, the tubular neighbourhood N ε (Σ λ ) will contain the ball B(0, 2ε). In particular, the left hand side of (1) is vanishing and so, there is no α > 0 for which the collection Σ λ is αadmissible. Clearly, in the above example, replacing the lines Σ i λ by linear subspaces of varying dimensions will deliver a similar example of a collection, which is not α-admissible.
Having this intuition in mind, we have the following result.
Theorem 1.5. Given an α-admissible collection Σ λ , there exists a constant C, independent of λ, such that N C √ λ (Σ λ ) cannot fully contain a nodal domain Ω λ .
Theorem 1.5 gives a strong indication as to the "thickness" or general shape of a nodal domain in many situations of practical interest. For example, in dimension 2, numerics show nodal domains to look like a tubular neighbourhood of a tree. We also note that our proof of Theorem 1.5 reveals a bit more information, but for aesthetic reasons, we prefer to state the theorem this way. Heuristically, the proof reveals that the nodal domain Ω λ is thicker at the points where the eigenfunction ϕ λ attains its maximum, or at points where ϕ λ (x) ≥ βmax y∈Ω λ |ϕ λ (y)|, for a fixed constant β > 0.
Second, we study the problem of how large a ball one may inscribe in a nodal domain Ω λ at a point where the eigenfunction achieves extremal values on Ω λ . We show Theorem 1.6. Let dim M ≥ 3, 0 > 0 be fixed and x 0 ∈ Ω λ be such that |ϕ λ (x 0 )| = max Ω λ |ϕ λ |. There exists r 0 = r 0 ( 0 ), such that
(2) Vol B(x 0 , r 0 λ −1/2 ) ∩ Ω λ Vol B(x 0 , r 0 λ −1/2 ) ≥ 1 − 0 .
A celebrated theorem of Lieb (see [L]) considers the case of a domain Ω ⊂ R n and states that there exists a point x 0 ∈ Ω, where a ball of radius
C √ λ1(Ω)
can almost be inscribed (in the sense of our Theorem 1.6). A further generalization was obtained in the paper [MS] (see, in particular, Theorem 1.1 and Subsection 5.1 of [MS]). However, the point x 0 was not specified. Physically, one expects that x 0 is close to the point where the first Dirichlet eigenfunction of Ω attains extremal values. This is in fact the essential statement of Theorem 1.6 above. Also, in this context, it is illuminating to compare the main Theorem from [CD].
We take the space to reiterate that the proof of Theorem 1.6 uses estimates from [GS] (see (31)), and a certain isocapacitary estimate (see (32)) that work only in dimensions n ≥ 3. As far as dimension n = 2 is concerned, it is known due to Mangoubi (Theorem 1.2 of [Man1], see also [H]) that any nodal domain has wavelength inradius; see further discussion on this at the beginning of Section 4.
As a corollary of Theorem 1.6, we derive the following:
Corollary 1.7. Let M be a closed manifold of dimension n ≥ 3, and Ω λ ⊆ M be a nodal domain upon which the corresponding eigenfunction ϕ λ is positive. Let x 0 be a point of maximum of ϕ λ on Ω λ . Then there exists a ball B x 0 , C λ α(n) ⊆ Ω λ with α(n) = 1 4 (n − 1) + 1 2n and a constant C = C(M, g). This recovers Theorem 1.5 of [Man2], with the additional information that the ball of radius C λ α(n) is centered around the max point of the eigenfunction ϕ λ (for more discussion on this, see Section 4). We also point out that using Theorem 1.6, the first author has established in [G] using results from [JM], the following inner radius bounds for real analytic manifolds:
Theorem 1.8 ( [G]). Let (M, g) be a real-analytic closed manifold of dimension at least 3. Let ϕ λ be a Laplacian eigenfunction and Ω λ be a nodal domain of ϕ λ . Then, there exist constants c 1 , c 2 depending only on (M, g) such that
c 1 λ ≤ inrad(Ω λ ) ≤ c 2 √ λ .
Moreover, if ϕ λ is positive (resp. negative) on Ω λ , then a ball of this radius can be inscribed within a wavelength distance to a point where ϕ λ achieves its maximum (resp. minimum) on Ω λ .
For another improvement of inner radius estimates in the smooth setting under certain conditional bounds on ϕ λ L ∞ (Ω λ ) , see Theorem 1.7 of [GM].
A few assorted remarks: as advertised, in Section 3 we address the problem of inscribing a nodal domain Ω λ in a tubular neighbourhood around Σ. In this context, an interesting subcase one might also consider is Σ having conical singularities: at its singular points Σ looks locally like R n−1−k × ∂C k for some k = 1, . . . , n − 1, where ∂C k denotes the boundary of a generalized cone, i.e. the cone generated by some open set D ⊆ S n−1 .
In this situation a useful tool is an explicit heat kernel formula for generalized cones C ⊆ R n . One denotes the associated Dirichlet eigenfunctions and eigenvalues of the generating set D by m j , l j respectively. Using polar coordinates x = ρθ, y = rη, one has that the heat kernel of P C (t, x, y) of the generalized cone C is given by
(3) P C (t, x, y) = e − ρ 2 +r 2 2t t(ρr) n 2 +1 ∞ j=1 I √ lj +( n 2 −1) 2 ( ρr t )m j (θ)m j (η),
where I ν (z) denotes the modified Bessel function of order ν. For more on the formula (3) we refer to [BS]. An even more general formula can be found in [C]. The expression for P C (t, x, y) provides means for estimating p t (x) from below as in Section 3. However, some features of the conical singularity (i.e. the eigenvalues and eigenfunctions l j , m j of the generating set D) enter explicitly in the estimate. Such considerations appear promising in discussing theorems of the following type, for example, and their higher dimensional analogues (see also [St]): Theorem 1.9 (Bers, Cheng). Let n = 2. If −∆u = λu, then any nodal set satisfies an interior cone condition with opening angle α λ −1/2 . 1.1. Basic heuristics. We outline the main idea behind Theorems 1.2, 1.5 and 1.6.
First, one considers a point x 0 ∈ Ω λ where the eigenfunction achieves a maximum on the nodal domain (w.l.o.g. we assume that the eigenfunction is positive on Ω λ ). One then considers the quantity p(t, x 0 ) -i.e. the probability that a Brownian motion started at x 0 escapes the nodal domain within time t.
The main strategy is to obtain two-sided bounds for p(t, x 0 ). On one hand, we have the Feynman-Kac formula (see Subsection 2.1) which provides a straightforward upper bound only in terms of t (see Equation (13) below).
On the other hand, depending on the context of the theorems above, we provide a lower bound for p(t, x 0 ) in terms of some geometric data. To this end, we take advantage of various tools some of which are: formulas for hitting probabilities of spheres and the parabolic scaling between the space and time variables; comparability of Brownian motions on manifolds with similar geometry (see Subsection 2.2); bounds for hitting probabilities in terms of 2-capacity (cf. [GS]), etc.
1.2. Outline of the paper. In Section 2, we recall tools from n-dimensional Brownian motion and the Feynman-Kac formulation of heat diffusion, and discuss the parabolic scaling technique we referred to above. We include some background material on stochastic analysis on Riemannian manifolds, some of which (to our knowledge) is not widely known, but is important to our investigation. We also believe such results to be of independent interest to the community. Of particular mention is Theorem 2.2, which roughly says that if the metric is perturbed slightly, hitting probabilities of compact sets by Brownian particles are also perturbed slightly. This allows us to apply Brownian motion formulae from R n to compact manifolds, on small distance and time scales.
In Section 3, we begin by proving Theorem 1.2. As mentioned before, we then take the generalization one step further, by considering intersecting surfaces of different dimensions. Our main result in this direction is Theorem 1.5, which gives a quantitative lower bound on how "thin" or "narrow" a nodal domain can be.
In Section 4, we take up the investigation of inradius estimates of Ω λ . As mentioned before, our main result in this direction is Theorem 1.6. We also establish Corollary 1.7.
1.3. Acknowledgements. It is a pleasure to thank Stefan Steinerberger for his detailed comments on a draft version of this paper, as well as Dan Mangoubi for his comments and remarks. The authors further thank Yuval Peres and Itai Benjamini for advice regarding Martin capacity, as well as Steve Zelditch for discussions on the Feynman-Kac formula and Brownian motion. Thanks are also due to the anonymous referee for a substantial improvement in the final presentation. Lastly, the authors would also like to thank Werner Ballmann, and gratefully acknowledge the Max Planck Institute for Mathematics, Bonn for providing ideal working conditions.
Preliminaries: heat equation, Feynman-Kac and Bessel processes
2.1. Feynman-Kac formula. We begin by stating a Feynman-Kac formula for open connected domains in compact manifolds for the heat equation with Dirichlet boundary conditions. Such formulas seem to be widely known in the community, but since we were unable to find out an explicit reference, we also indicate a line of proof.
Theorem 2.1. Let M be a compact Riemannian manifold. For any open connected
Ω ⊂ M , f ∈ L 2 (Ω), we have that (4) e t∆ f (x) = E x (f (ω(t))φ Ω (ω, t)), t > 0, x ∈ Ω,
where ω(t) denotes an element of the probability space of Brownian motions starting at x, E x is the expectation with regards to the measure on that probability space, and
φ Ω (ω, t) = 1, if ω([0, t]) ⊂ Ω 0, otherwise.
A proof of Theorem 2.1 can be constructed in three steps. First, one proves the corresponding statement when Ω = M . This can be found, for example, in [BP], Theorem 6.2. One can then combine this with a barrier potential method to prove a corresponding statement for domains Ω with Lipschitz boundary. Lastly, the extension to domains with no regularity requirements on the boundary is achieved by a standard limiting argument. For details on the last two steps, see [T], Chapter 11, Section 3.
2.2.
Euclidean comparability of hitting probabilities. Implicit in many of our calculations is the following heuristic: if the metric is perturbed slightly, hitting probabilities of compact sets by Brownian particles are also perturbed slightly, provided one is looking at small distances r and at small time scales t = O(r 2 ).
To describe the set up, let (M, g) be a compact Riemannian manifold and cover M by charts (U k , φ k ) such that in these charts g is bi-Lipschitz to the Euclidean metric. Consider an open ball B(p, r) ⊂ M , where r is considered small, and in particular, smaller than the injectivity radius of M . Let B(p, r) sit inside a chart (U, φ) and let φ(p) = q and φ(B(p, r)) = B(q, s) ⊂ R n . Let K be a compact set inside B(p, r) and let K := φ(K) ⊂ B(q, s). Now, let ψ M K (T, p) denote the probability that a Brownian motion on (M, g) started at p and killed at a fixed time T hits K within time T . ψ e K (t, q) is defined similarly for the standard Brownian motion in R n started at q and killed at the same fixed time T . Now, we fix the time T = cr 2 , where c is a constant. The following is the comparability result:
Theorem 2.2. There exists constants c 1 , c 2 , depending only on c and M such that T, q). The proof uses the concept of Martin capacity (see [BPP], Definition 2.1):
(5) c 1 ψ e K (T, q) ≤ ψ M K (T, p) ≤ c 2 ψ e K (
Definition 2.3. Let Λ be a set and B a σ-field of subsets of Λ. Given a measurable function F : Λ × Λ → [0, ∞] and a finite measure µ on (Λ, B), the F -energy of µ is
I F (µ) = Λ Λ F (x, y)dµ(x)dµ(y).
The capacity of Λ in the kernel F is
(6) Cap F (Λ) = inf µ I F (µ) −1 ,
where the infimum is over probability measures µ on (Λ, B), and by convention, ∞ −1 = 0. Now we quote the following general result, which is Theorem 2.2 in [BPP].
Theorem 2.4. Let {X n } be a transient Markov chain on the countable state space Y with initial state ρ and transition probabilities p(x, y). For any subset Λ of Y , we have
(7) 1 2 Cap M (Λ) ≤ P ρ [∃n ≥ 0 : X n ∈ Λ] ≤ Cap M (Λ),
where M is the Martin kernel M (x, y) = G(x,y) G(ρ,y) , and G(x, y) denotes the Green's function.
For the special case of Brownian motions, this reduces to (see Proposition 1.1 of [BPP] and Theorem 8.24 of [MP]):
1 2 Cap M (A) ≤ P ρ {B(t) ∈ A for some 0 < t ≤ T } ≤ Cap M (A).(8)
An inspection of the proofs reveals that they go through with basically no changes on a compact Riemannian manifold M , when the Brownian motion is killed at a fixed time T = cr 2 , and the Martin kernel M (x, y) is defined as G(x,y) G(ρ,y) , with G(x, y) being the "cut-off" Green's function defined as follows: if h M (t, x, y) is the heat kernel of M ,
G(x, y) := T 0 h M (t, x, y)dt.
Now, to state it formally, in our setting, we have Theorem 2.6.
(9) 1 2 Cap M (K) ≤ ψ M K (T, p) ≤ Cap M (K). Now, let h R n (t,
x, y) denote the heat kernel on R n . To prove Theorem 2.2, it suffices to show that for y ∈ K, and y = φ(y) ∈ K , we have constants C 1 , C 2 (depending on c and M ) such that
(10) C 1 T 0 h R n (t, q, y )dt ≤ T 0 h M (t, p, y)dt ≤ C 2 T 0 h R n (t, q, y )dt.
In other words, we need to demonstrate comparability of Green's functions "cut off" at time T = cr 2 . Recall that we have the following Gaussian two-sided heat kernel bounds on a compact manifold (see, for example, Theorem 5.3.4 of [Hs] for the lower bound and Theorem 4 of [CLY] for the upper bound, also (4.27) of [GS] where Γ(s, x) is the (upper) incomplete Gamma function. Since r is a small constant chosen independently of λ, we observe that C 1 , C 2 are constants in (10) depending only on c, c 1 , c 2 , c 3 , c 4 , c 5 , r and M , which finally proves (5).
Remark 2.7. Theorem 2.2 is implicit in [St], but it was not precisely stated or proved there. Since we are unable to find an explicit reference, here we have given a formal statement and indicated a proof. We believe that the statement of Theorem 2.2 will also be of independent interest for people interested in stochastic analysis on manifolds.
Brownian motion on a manifold and Euclidean Bessel processes.
Using the probabilistic formulation of the heat equation for the study of nodal geometry, we are largely inspired by the methods in [St]. Of course, such ideas have appeared in the literature before; for example, they are implicit in [GJ]. Here we extend some ideas of Steinerberger with the help of tools from n-dimensional Brownian motion.
Given an open subset V ⊂ M , consider the solution p t (x) to the following diffusion process:
(∂ t − ∆)p t (x) = 0, x ∈ V p t (x) = 1, x ∈ ∂V p 0 (x) = 0, x ∈ V.
By the Feynman-Kac formula (see Subsection 2.1), this diffusion process can be understood as the probability that a Brownian motion particle started in x will hit the boundary within time t. Now, fix an eigenfunction ϕ (corresponding to the eigenvalue λ) and a nodal domain Ω, so that ϕ > 0 on Ω without loss of generality. Calling ∆ the Dirichlet Laplacian on Ω and setting Φ(t, x) := e t∆ ϕ(x), we see that Φ solves
(∂ t − ∆)Φ(t, x) = 0, x ∈ Ω Φ(t, x) = 0, on {ϕ = 0} (11) Φ(0, x) = ϕ(x), x ∈ Ω.
Using the Feynman-Kac formula given by Theorem 2.1, we have,
(12) e t∆ f (x) = E x (f (ω(t))φ Ω (ω, t)), t > 0,
where ω(t) denotes an element of the probability space of Brownian motions starting at x, E x is the expectation with regards to the measure on that probability space, and
φ Ω (ω, t) = 1, if ω([0, t]) ⊂ Ω 0, otherwise.
Now, consider a nodal domain Ω corresponding to the eigenfunction ϕ, and consider the heat flow (11). Let x 0 ∈ Ω such that ϕ(x 0 ) = ϕ L ∞ (Ω) . We use the following upper bound derived in [St]:
Φ(t, x) = e −λt ϕ(x) = E x (ϕ(ω(t))φ Ω (ω, t)) (13) ≤ ϕ L ∞ (Ω) E x (φ Ω (ω, t)) = ϕ L ∞ (Ω) (1 − p t (x)).
Setting t = λ −1 and x = x 0 , we see that the probability of the Brownian motion starting at an extremal point x 0 leaving Ω within time λ −1 is ≤ 1 − e −1 . A rough interpretation is that maximal points x are situated deeply into the nodal domain. Using the notation introduced in the Introduction, the last derived upper estimate translates to ψ M \Ω (λ −1 , x) ≤ 1 − e −1 . Now, we consider an m-dimensional Brownian motion of a particle starting at the origin in R m , and calculate the probability of the particle hitting a sphere {x ∈ R m : x ≤ r} of radius r within time t. By a well known formula first derived in [Ke], we see that such a probability is given as follows:
(14) P( sup 0≤s≤t B(s) ≥ r) = 1 − 1 2 ν−1 Γ(ν + 1) ∞ k=1 j ν−1 ν,k J ν+1 (j ν,k ) e − j 2 ν,k t 2r 2 , ν > −1, where ν = m−2 2
is the "order" of the Bessel process, J ν is the Bessel function of the first kind of order ν, and 0 < j ν,1 < j ν,2 < ..... is the sequence of positive zeros of J ν . Choose x = x 0 , t = λ −1 , as before, and let r = C 1/2 λ −1/2 , where C is a constant to be chosen later, independently of λ. Plugging this in (14) then reads,
(15) P( sup 0≤s≤λ −1 B(s) ≥ Cλ −1/2 ) = 1 − 1 2 ν−1 Γ(ν + 1) ∞ k=1 j ν−1 ν,k J ν+1 (j ν,k ) e − j 2 ν,k 2C , ν > −1.
We need to make a few comments about the asymptotic behaviour of j ν,k here. For notational convenience, we write α k ∼ β k , as k → ∞ if we have α k /β k → 1 as k → ∞. [Wa], pp 506, gives the asymptotic expansion (16) j ν,k = (k + ν/2 + 1/4)π + o(1) as k → ∞, which tells us that j ν,k ∼ kπ. Also, from [Wa], pp 505, we have that
(17) J ν+1 (j ν,k ) ∼ (−1) k−1 √ 2 π 1 √ k .
These asymptotic estimates, in conjunction with (15), tell us that keeping ν bounded, and given a small η > 0, one can choose the constant C small enough (depending on η) such that
(18) P( sup 0≤s≤λ −1 B(s) ≥ Cλ −1/2 ) > 1 − η.
This estimate plays a role in Section 3. In this context, see also Proposition 5.1.4 of [Hs].
Admissibility conditions and intersecting surfaces
Proof of Theorem 1.2. If ϕ λ attains its maximum within Ω λ at x 0 , we already know from (13) that
(19) ψ M \Ω λ ( t 0 λ , x 0 ) ≤ 1 − e −t0 .
By the admissibility condition on Σ λ we know that x 0 has a unique metric projection on one and only one Σ ix 0 λ from the collection Σ λ . Now, suppose the result is not true. Choose R, t 0 small such that Theorem 2.2 applies. Choosing r 0 sufficiently smaller than R, we can find a λ such that Ω λ is contained in a r0 √ λ -tubular neighbourhood of Σ λ , denoted by N r0λ −1/2 (Σ λ ). From the remarks after Definition 1.1, it follows that Ω λ ⊆ N r0λ −1/2 (Σ ix 0 λ ). We start a Brownian motion at x 0 and, roughly speaking, we see that locally the particle has freedom to wander in n − k "bad directions", namely the directions normal to Σ ix 0 λ , before it hits ∂Ω λ . That means, we may consider a (n − k)dimensional Brownian motion B(t) starting at x 0 ; see the following diagram:
x Ω λ Brownian motion in n − k "bad directions"
S n−k−1 Σ λ
More formally, we choose a normal coordinate chart (U, φ) around x 0 adapted to Σ ix 0 λ , where the metric is comparable to the Euclidean metric. We have that
φ(Σ ix 0 λ ) = φ(U )∩{R k ×{0} n−k } and φ(N r0λ −1/2 (Σ ix 0 λ )) = φ(U )∩{R k ×[− r0 √ λ , r0 √ λ ] n−k }.
We take a geodesic ball B ⊂ U ⊂ M at x 0 of radius R √ λ . Using the hitting probability notation from Section 2 and monotonicity with respect to set inclusion we have
(20) ψ M \Ω λ t 0 λ , x 0 ≥ ψ B\Ω λ t 0 λ , x 0 ≥ ψ B\N r 0 λ −1/2 (Σ ix 0 λ ) t 0 λ , x 0 ,
and the comparability lemma implies that, if c = t0 R 2 , then there exists a constant C, depending on c and M , such that
(21) ψ B\N r 0 λ −1/2 (Σ ix 0 λ ) t 0 λ , x 0 ≥ Cψ e φ(B\N r 0 λ −1/2 (Σ ix 0 λ )) t 0 λ , φ(x 0 ) ,
where ψ e denotes the hitting probability in Euclidean space. We denote N e r0λ −1/2 := φ(N r0λ −1/2 (Σ ix 0 λ )). Let us consider the "solid cylinder"
S = B R √ λ × B r 0 √ λ
, a product of k and n − k dimensional Euclidean balls centered at φ(x 0 ). S is clearly the largest cylinder contained in N e r0λ −1/2 ∩ B. We denote S = B 1 × B 2 for convenience. By monotonicity,
(22) ψ e φ(B\N r 0 λ −1/2 (Σ ix 0 λ )) t 0 λ , φ(x 0 ) ≥ ψ e B1×∂B2 t 0 λ , φ(x 0 ) .
If B(t) = (B 1 (t), ..., B n (t)) is an n-dimensional Brownian motion, the components B i (t)'s are independent Brownian motions (see, for example, Chapter 2 of [MP]). Denoting by B k (t) and B n−k (t) the projections of B(t) onto the first k and last n − k components respectively, it follows that
ψ e B1×∂B2 t 0 λ , φ(x 0 ) ≥ P( sup 0≤s≤t0λ −1 B k (t) ≤ R √ λ ).P( sup 0≤s≤t0λ −1 B n−k (t) ≥ r 0 √ λ ) ≥ c k P( sup 0≤s≤t0λ −1 B n−k (t) ≥ r 0 √ λ ),
where c k is a constant depending on k and the ratio t 0 /R 2 , and can be calculated explicitly from (15). Using the estimate in Section 2, we may take r 0 ≤ R sufficiently small so that
(23) P( sup 0≤s≤t0λ −1 B n−k (t) ≥ r 0 √ λ ) > 1 − ε,
where ε is sufficiently small. Keeping c = t0 R 2 and (hence) C fixed, we take t 0 small enough and r 0 ≤ R appropriately, so that (23) contradicts (20) and the fact that
ψ M \Ω λ (t 0 λ −1 , x) ≤ 1 − e −t0 .
Remark 3.1. Note that the constant r 0 above is independent of Σ λ ; in other words, the same constant r 0 will work for Theorem 1.2 as long as the surface is admissible up to a wavelength distance. Indeed, this results from the fact that r 0 depends only on the diffusion process associated to the Brownian motion, and is an inherent property of the manifold itself. Now we address the generalizations of Theorem 1.2 for collections Σ λ which are more complicated, namely we assume Σ λ is a α-admissible collection in the sense of Definition 1.4.
Proof of Theorem 1.5. By assumption, we have an α-admissible collection Σ λ := ∪ m λ i=1 Σ i λ . Let us assume the contrary -if the statement is not true, we may select an arbitrarily small r 0 > 0 and find a corresponding inscribed nodal domain Ω λ ⊂ N r0λ −1/2 (Σ λ ).
As before, we choose a point x 0 ∈ Ω λ such that ϕ λ (x 0 ) = max x∈Ω λ |ϕ λ |. Monotonicity of the hitting probability function ψ K (., .) with respect to set inclusion in K, as well as the α-admissibility imply that
ψ M \Ω λ (t, x 0 ) ≥ ψ B(x0,2r0λ −1/2 )\Ω λ (t, x 0 ) (24) ≥ ψ B(x0,2r0λ −1/2 )\N r 0 λ −1/2 (Σ λ ) (t, x 0 ) = ψ ∂ B(x0,2r0λ −1/2 )\N r 0 λ −1/2 (Σ λ ) (t, x 0 ) ≥ ψ ∂B(x0,2r0λ −1/2 )\N r 0 λ −1/2 (Σ λ ) (t, x 0 ) ≥ αψ ∂B(x0,2r0λ −1/2 ) (t, x 0 ),
where we introduce the constant α > 0 coming from the α-admissibility condition. Moreover, following Definition 1.4 of α-admissibility, in (24) we also assume that the radius r0 √ λ is sufficiently small and that t := t0 λ with t 0 := 4r 2 0 .
N ε (Σ) Ω λ B(x 0 , r0 √ λ ) x 0 Σ 1 λ Σ 2 λ
The latter estimate (24) implies, in particular, that
(25) ψ M \Ω λ (t, x 0 ) ψ M \B(x0,2r0λ −1/2 ) (t, x 0 ) = ψ M \Ω λ (t, x 0 ) ψ ∂B(x0,2r0λ −1/2 ) (t, x 0 ) ≥ α.
We now observe that by setting t = t0 λ we still have the freedom to choose t 0 . We show that we can select t 0 such that (25) is violated. To this end we observe that the upper bound (19) along with (15) and Theorem 2.2 give:
ψ M \Ω λ ( t0 λ , x) ψ M \B(x0,2r0λ −1/2 ) ( t0 λ , x) 1 − e −t0 1 − 1 2 ν−1 Γ(ν+1) ∞ k=1 j ν−1 ν,k Jν+1(j ν,k ) e − j 2 ν,k t 0 2r 2 0 (26) = 1 − e −t0 1 − 1 2 ν−1 Γ(ν+1) ∞ k=1 j ν−1 ν,k Jν+1(j ν,k ) e −2j 2 ν,k = 1 − e −t0 C .
Now, we choose t 0 = 4r 2 0 small enough, so the last estimate yields a contradiction with (25). This proves the theorem.
Remark 3.2. We wish to comment that in the above proof, it is not essential to look at the nodal domain only around the maximum point x 0 . Given a predetermined positive constant β, choose a point y ∈ Ω λ such that ϕ λ (y) ≥ βϕ λ (x 0 ). Arguing similarly as in (13), we see that ψ M \Ω λ (t, y) ≤ 1 − βe −t0 . Following the computations in (26), we get a constant r 0 (depending on β) such that 1−βe −t 0 C < α, giving a contradiction. Also, it is clear that in Definitions 1.1 and 1.4, we do not actually need the submanifolds in the family Σ λ to be smooth, and the proofs of Theorems 1.2 and 1.5 work with submanifolds of much lower regularity (for example, C 1 submanifolds).
Large ball at a max point
In this section we discuss the asymptotic thickness of nodal domains around extremal points of eigenfunctions. More precisely, let us consider a fixed nodal domain Ω λ corresponding to the eigenfunction ϕ λ . Let x 0 ∈ Ω λ be such that
(27) ϕ λ (x 0 ) = max x∈Ω λ |ϕ λ |.
In the case dim M = 2, it was shown in Section 3 of [Man1] that at such maximal points x 0 one can fully inscribe a large ball of wavelength radius (i.e ∼ 1 √ λ ) into the nodal domain. In other words for Riemannian surfaces, one has that
(28) C 1 √ λ ≤ inrad (Ω λ ) ≤ C 2 √ λ ,
where C i are constants depending only on M . Note that the proof for this case, as carried out in [Man1] by following ideas in [NPS], makes use of essentially 2dimensional tools (conformal coordinates and quasi-conformality), which are not available in higher dimensions. To our knowledge, in higher dimensions the sharpest known bounds on the inner radius of a nodal domain appear in [Man2] (Theorem 1.5) and state that:
(29) C 1 λ α(n) ≤ inrad (Ω λ ) ≤ C 2 √ λ ,
where α(n) := 1 4 (n − 1) + 1 2n . A question of current investigation is whether the last lower bound on inrad (Ω λ ) in higher dimensions is optimal.
Here we exploit heat equation and Brownian motion techniques to show that at least, one can expect to "almost" inscribe a large ball having radius to the order of 1 √ λ , in all dimensions. Now we prove Theorem 1.6:
Proof. We denote t := t0 λ , and thus ψ M \Ω λ (t , x) ≤ 1 − e −t0 , where t 0 is a small constant to be chosen suitably later. Now, choosing t 0 small enough, and using monotonicity, we have,
(30) ψ B(x0,r0λ −1/2 )\Ω λ (t, x 0 ) < ψ M \Ω λ (t, x 0 ) < .
For convenience, let us denote E r0 := B(x 0 , r 0 λ −1/2 )\Ω λ -a relatively compact set. Observe that Theorem 2.2 applies to open balls and compact subsets contained in open balls. To adapt to the setting of Theorem 2.2, choose a number r 0 < r 0 such that B(x 0 , r 0 λ −1/2 ) satisfies
Vol B(x 0 , r 0 λ −1/2 ) \ B(x 0 , r 0 λ −1/2 ) Vol B(x 0 , r 0 λ −1/2 ) < .
Call E r 0 := E r0 ∩ B(x 0 , r 0 λ −1/2 ). Observe that proving that
Vol(E r 0 )
Vol(B(x0,r0λ −1/2 )) < will imply that Vol(Er 0 ) Vol(B(x0,r0λ −1/2 )) < 2 , which is what we want. Now, we would like to compare the volumes of the two sets E r 0 and B(x 0 , r 0 λ −1/2 ). Let r = r0 √ λ . Recall from [GS], Remark 4.1, the following inequality:
(31) c cap(E r 0 )r 2 Vol(B(x 0 , r 0 λ −1/2 )) e −C r 2 t ≤ ψ Er 0 (t , x 0 ) < ,
where cap(K) denotes the 2-capacity of the set K ⊂ M , and 0 < t < 2r 2 (see also Equation (3.20) of [GS]). Recall that the 2-capacity of a set K ⊂ M is defined as
cap(K) = inf η| K ≡1,η∈C ∞ (M ) M |∇η| 2 dM.
Formally, (31) holds on complete non-compact non-parabolic manifolds, which includes R n , n ≥ 3. For bringing in our comparability result Theorem 2.2, we fix the ratio t r 2 = 1 3 , say, and then choose t 0 small enough that (30) still works. Now (31) applies, albeit with a new constant c as determined by the ratio t/r 2 and Theorem 2.2. Now, to rewrite the capacity term in (31) in terms of volume, we bring in the following "isocapacitary inequality" (see [Maz], Section 2.2.3):
(32) cap(E r0 ) ≥ C Vol(E r0 ) n−2 n , n ≥ 3,
where C is a constant depending only on the dimension n. We note that the isocapacitary inequality (in combination with a suitable Poincare inequality) lies at the heart of the currently optimal inradius estimates, as derived by Mangoubi in [Man2]. Clearly, (31) and (32) together give (33) Vol(E r0 ) Vol(B(x 0 , r 0 λ −1/2 )) n−2 n cap(E r0 )r 2 Vol(B(x 0 , r 0 λ −1/2 )) ψ Er 0 (t, x) < .
The last inequalities contain constants depending only on M , so by taking even smaller we can arrange Vol(Er 0 ) Vol(B(x0,r0λ −1/2 )) < 0 for any initially given 0 .
Remark 4.1. We note that the heat equation method does not distinguish between a general domain and a nodal domain. This means that we cannot rule out the situation where B(x 0 , r0 √ λ ) \ Ω λ is a collection of "sharp spikes" entering into B(x 0 , r0 √ λ ). Indeed the probability of a Brownian particle hitting a spike, no matter how "thin" it is, or how far from x 0 it is, is always non-zero, a fact related to the infinite speed of propagation of heat diffusion. This is consistent with the heuristic discussed in [H] and [L]. Now we establish Corollary 1.7. First, we recall the following result, which gives a bound on the asymmetry between the volumes of positivity and negativity sets, as developed in [Man2]: . In other words, the relative volume of the error set E r0 decays as r 2n n−2 0 as r 0 → 0. This is slightly better than the scaling prescribed by Corollary 2 of [L].
Remark 4.4. There is a sizeable literature around optimizing the fundamental frequency of the complement of an obstacle inside a domain (for example, see [HKK] and references therein). As an explicit special case, consider a convex domain Ω ⊂ R n and a small ball B ⊆ Ω. The question is to find possible placements of translate x + B inside Ω such that λ 1 (Ω \ (x + B)) is maximized. For certain applications of Theorem 1.6 towards such questions, we refer to [GM1].
Theorem 2. 5 .
5Let {B(t) : 0 ≤ t ≤ T } be a transient Brownian motion in R n starting from the point ρ, and A ⊂ D be closed, where D is a bounded domain. Then,
,
): for all (t, p, y) ∈ (0, 1) × M × M , and positive constants c 1 , c 2 , c 3 , c 4 depending only on the geometry of M , where d denotes the distance function on M . Then, using the comparability of the distance function on M with the Euclidean distance function (which comes via metric comparability in local charts), for establishing (10), it suffices to observe that for any positive constant c 5 , we have that
Theorem 4.2.[Man2] Let B be a geodesic ball, so that 1 2 B ∩ {ϕ λ = 0} = ∅ with 1 2 B denoting the concentric ball of half radius. Then
.≤
Proof of Corollary 1.7. It suffices to combine the estimate (33) with (34).Let r := r0 √ λ be the radius of the largest inscribed ball in the nodal domain at x 0 . Noting that {ϕ λ < 0} ⊆ E r0 and combining Theorem 4.2 for B x0 (2r) with (33)Vol(E r0 ) Vol(B(x 0 , r 0 λ −1/2 )) Expanding the right hand side in Taylor series and rearranging finishes the proof.Remark 4.3. An inspection of the proof of Theorem 1.6 reveals that one can take
C Bär, F Pfäffle, Wiener measures on Riemannian manifolds and the Feynman-Kac formula. 40C. Bär and F. Pfäffle, Wiener measures on Riemannian manifolds and the Feynman-Kac formula, Mat. Contemp. 40 (2011), 37 -90.
Brownian motion in cones. R Bañuelos, R G Smits, Probability Theory and Related Fields. 108R. Bañuelos, R. G. Smits, Brownian motion in cones, Probability Theory and Related Fields, 108 (1997), no. 3, 299 -319.
Martin capacity for Markov chains. I Benjamini, R Pemantle, Y Peres, Ann. Probab. 233I. Benjamini, R. Pemantle and Y. Peres, Martin capacity for Markov chains, Ann. Probab., 23 (1995), no. 3, 1332 -1346.
Spectral geometry of singular Riemannian spaces. J Cheeger, J. Diff. Geom. 18J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Diff. Geom., 18 (1983), 575 -657.
Eigenfunctions and nodal sets. S.-Y Cheng, Comment. Math. Helv. 511S.-Y. Cheng, Eigenfunctions and nodal sets, Comment. Math. Helv., 51 (1) (1976), 43 -55.
On the upper estimate of the heat kernel of a complete Riemannian manifold. S.-Y Cheng, P Li, S.-T Yau, Amer. J. Math. 1035S.-Y. Cheng, P. Li and S.-T. Yau, On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math. 103 (1981), no. 5, 1021 -1063.
A lower bound for λ 1 on manifolds with boundary. C Croke, A Derdzinski, Comment. Math. Helv. 621C. Croke and A. Derdzinski, A lower bound for λ 1 on manifolds with boundary, Comment. Math. Helv. 62 (1987), no. 1, 106 -121.
B Georgiev, arXiv:1607.03816On the lower bound of the inner radius of nodal domains. B. Georgiev, On the lower bound of the inner radius of nodal domains, arXiv:1607.03816.
B Georgiev, M Mukherjee, arXiv:1608.05344Some remarks on nodal geometry in the smooth setting. B. Georgiev and M. Mukherjee, Some remarks on nodal geometry in the smooth setting, arXiv:1608.05344.
B Georgiev, M Mukherjee, arXiv:1706.02138On maximizing the fundamental frequency of the complement of an obstacle. B. Georgiev and M. Mukherjee, On maximizing the fundamental frequency of the comple- ment of an obstacle, arXiv:1706.02138.
The size of the first eigenvalue of a convex planar domain. D Grieser, D Jerison, J. Amer. Math. Soc. 111D. Grieser and D. Jerison, The size of the first eigenvalue of a convex planar domain, J. Amer. Math. Soc., 11 (1998), no. 1, 41 -72.
Hitting probabilities for Brownian motion on Riemannian manifolds. A Grigor'yan, L Saloff-Coste, J. Math. Pures Appl. 81A. Grigor'yan and L. Saloff-Coste, Hitting probabilities for Brownian motion on Riemannian manifolds, J. Math. Pures Appl., 81 (2002), 115 -142.
On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue. E Harrell, P Kröger, K Kurata, SIAM J. Math. Anal. 331E. Harrell, P. Kröger and K. Kurata, On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue, SIAM J. Math. Anal., 33 (2001), no. 1, 240 -259.
Some bounds for principal frequency. W Hayman, Applicable Anal. 73W. Hayman, Some bounds for principal frequency, Applicable Anal., 7 (1977/78), no. 3, 247 -254.
Stochastic analysis on manifolds. E Hsu, Graduate Studies in Mathematics. 38AMSProvidenceE. Hsu, Stochastic analysis on manifolds, Graduate Studies in Mathematics 38, AMS, Prov- idence, RI, 2002.
Tubular neighborhoods of nodal sets and Diophantine approximation. D Jakobson, D Mangoubi, Amer. J. Math. 1314D. Jakobson and D. Mangoubi, Tubular neighborhoods of nodal sets and Diophantine ap- proximation, Amer. J. Math., 131 (2009), no. 4, 1109 -1135.
Eigenvalue expansions for diffusion hitting times. J Kent, Z. Wahrsch. Verw. Gebiete. 523J. Kent, Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete, 52 (1980), no. 3, 309 -319.
On the lowest eigenvalue of the Laplacian for the intersection of two domains. E Lieb, Invent. Math. 743E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math., 74 (1983), no. 3, 441 -448.
On the inner radius of a Nodal domain. D Mangoubi, Canad. Math. Bull. 512D. Mangoubi, On the inner radius of a Nodal domain, Canad. Math. Bull. 51 (2008), no. 2, 249 -260.
Local asymmetry and the inner radius of nodal domains. D Mangoubi, Comm. PDE. 33D. Mangoubi, Local asymmetry and the inner radius of nodal domains, Comm. PDE, 33 (2008), 1611 -1621.
Sobolev spaces with applications to elliptic partial differential equations. V Maz'ya, Grundlehren der Mathematischen Wissenschaften. HeidelbergSpringer342V. Maz'ya, Sobolev spaces with applications to elliptic partial differential equations, Grundlehren der Mathematischen Wissenschaften, 342. Springer, Heidelberg, 2011.
Can one see the fundamental frequence of a drum?. V Maz'ya, M Shubin, Lett. Math. Phys. 74V. Maz'ya and M. Shubin, Can one see the fundamental frequence of a drum?, Lett. Math. Phys., 74 (2005), 135 -151.
Brownian motion. P Mörters, Y Peres, Oded Schramm and Wendelin WernerCambridge University PressP. Mörters and Y. Peres, Brownian motion, With an appendix by Oded Schramm and Wendelin Werner, Cambridge University Press, 2010.
Sign and area in nodal geometry of Laplace eigenfunctions. F Nazarov, L Polterovich, M Sodin, Amer. J. Math. 1274F. Nazarov, L. Polterovich and M. Sodin, Sign and area in nodal geometry of Laplace eigenfunctions, Amer. J. Math., 127 (2005), no. 4, 879 -910.
Lower bounds on nodal sets of eigenfunctions via the heat flow. S Steinerberger, Comm. PDE. 3912S. Steinerberger, Lower bounds on nodal sets of eigenfunctions via the heat flow, Comm. PDE, 39 (2014), no. 12, 2240 -2261.
M Taylor, Partial differential equations. II. Qualitative studies of linear equations, Applied Mathematical Sciences. New YorkSpringer-Verlag116M. Taylor, Partial differential equations. II. Qualitative studies of linear equations, Applied Mathematical Sciences, 116. Springer-Verlag, New York, 1996.
A treatise on the theory of Bessel functions. G Watson, Cambridge University Press2nd EditionG. Watson, A treatise on the theory of Bessel functions, 2nd Edition, Cambridge University Press, 1944.
E-mail address: [email protected] E-mail address: mukherjee@mpim-bonn. 7Bonn,, GermanyMax Planck Institute for Mathematicsmpg.deMax Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn,, Germany E-mail address: [email protected] E-mail address: [email protected]
| [] |
[
"Modulated cycles in an illustrative solar dynamo model with competing α-effects",
"Modulated cycles in an illustrative solar dynamo model with competing α-effects"
] | [
"L C Cole [email protected] \nSchool of Mathematics and Statistics\nNewcastle University\nNE1 7RUNewcastle Upon TyneUK\n",
"P J Bushby [email protected] \nSchool of Mathematics and Statistics\nNewcastle University\nNE1 7RUNewcastle Upon TyneUK\n"
] | [
"School of Mathematics and Statistics\nNewcastle University\nNE1 7RUNewcastle Upon TyneUK",
"School of Mathematics and Statistics\nNewcastle University\nNE1 7RUNewcastle Upon TyneUK"
] | [] | Context. The large-scale magnetic field in the Sun varies with a period of approximately 22 years, although the amplitude of the cycle is subject to long-term modulation with recurrent phases of significantly reduced magnetic activity. It is believed that a hydromagnetic dynamo is responsible for producing this large-scale field, although this dynamo process is not well understood. Aims. Within the framework of mean-field dynamo theory, our aim is to investigate how competing mechanisms for poloidal field regeneration (namely a time delayed Babcock-Leighton surface α-effect and an interface-type α-effect), can lead to the modulation of magnetic activity in a deep-seated solar dynamo model. Methods. We solve the standard αΩ dynamo equations in one spatial dimension, including source terms corresponding to both of the the competing α-effects in the evolution equation for the poloidal field. This system is solved using two different methods. In addition to solving the one-dimensional partial differential equations directly, using numerical techniques, we also use a local approximation to reduce the governing equations to a set of coupled ordinary differential equations (ODEs), which are studied using a combination of analytical and numerical methods. Results. In the ODE model, it is straightforward to find parameters such that a series of bifurcations can be identified as the time delay is increased, with the dynamo transitioning from periodic states to chaotic states via multiply periodic solutions. Similar transitions can be observed in the full model, with the chaotically modulated solutions exhibiting solar-like behaviour. Conclusions. Competing α-effects could explain the observed modulation in the solar cycle. | 10.1051/0004-6361/201323285 | [
"https://arxiv.org/pdf/1403.6604v1.pdf"
] | 55,935,997 | 1403.6604 | 09162b6d41193e4f833bd61843db00eb6ccf1551 |
Modulated cycles in an illustrative solar dynamo model with competing α-effects
26 Mar 2014 March 27, 2014 March 27, 2014
L C Cole [email protected]
School of Mathematics and Statistics
Newcastle University
NE1 7RUNewcastle Upon TyneUK
P J Bushby [email protected]
School of Mathematics and Statistics
Newcastle University
NE1 7RUNewcastle Upon TyneUK
Modulated cycles in an illustrative solar dynamo model with competing α-effects
26 Mar 2014 March 27, 2014 March 27, 2014Astronomy & Astrophysics manuscript no. AA c ESO 2014 Preprint online version:Dynamo -Magnetohydrodynamics (MHD) -Sun: activity -Sun: interior -Sun: magnetic fields
Context. The large-scale magnetic field in the Sun varies with a period of approximately 22 years, although the amplitude of the cycle is subject to long-term modulation with recurrent phases of significantly reduced magnetic activity. It is believed that a hydromagnetic dynamo is responsible for producing this large-scale field, although this dynamo process is not well understood. Aims. Within the framework of mean-field dynamo theory, our aim is to investigate how competing mechanisms for poloidal field regeneration (namely a time delayed Babcock-Leighton surface α-effect and an interface-type α-effect), can lead to the modulation of magnetic activity in a deep-seated solar dynamo model. Methods. We solve the standard αΩ dynamo equations in one spatial dimension, including source terms corresponding to both of the the competing α-effects in the evolution equation for the poloidal field. This system is solved using two different methods. In addition to solving the one-dimensional partial differential equations directly, using numerical techniques, we also use a local approximation to reduce the governing equations to a set of coupled ordinary differential equations (ODEs), which are studied using a combination of analytical and numerical methods. Results. In the ODE model, it is straightforward to find parameters such that a series of bifurcations can be identified as the time delay is increased, with the dynamo transitioning from periodic states to chaotic states via multiply periodic solutions. Similar transitions can be observed in the full model, with the chaotically modulated solutions exhibiting solar-like behaviour. Conclusions. Competing α-effects could explain the observed modulation in the solar cycle.
Introduction
At the solar photosphere, bipolar active regions are formed when loops of magnetic flux rise to the surface from the base of the convection zone due to the action of magnetic buoyancy (Parker 1955b). This implies that the properties of sunspot-bearing active regions can be used to deduce some of the features of the underlying large-scale magnetic field. It is well known (see, for example , Stix 2002;Charbonneau 2005;Jones et al. 2010) that zones of active region emergence follow a cyclic pattern with a period of approximately 11 years. At the beginning of each cycle, sunspots tend to be found at mid-latitudes, with zones of emergence drifting towards the equator as the cycle progresses. The underlying large-scale (predominantly azimuthal) magnetic field changes sign at the end of each cycle, giving a full magnetic period of approximately 22 years. However, the solar cycle is not strictly periodic. In particular, the peak amplitude (measured, for example, by the sunspot coverage) varies from one cycle to the next. Although this modulation does not usually disrupt the cycle, more extreme episodes of modulation have been recorded. For example, during a period known as the Maunder Minimum, very few sunspots were observed between approximately 1650 and 1720 (Eddy 1976;Ribes & Nesme-Ribes 1993). However, sunspot records are not the only indicators of modulation. Due to the fact that the Sun's strong magnetic field protects the Earth from cosmic rays, the abundance of certain isotopes in the Earth's atmosphere is known to be anti-correlated with the solar cycle. Therefore, by analysing Beryllium-10 deposits in ice cores (see, for example, Delaygue & Bard 2011) and Carbon-14 levels in tree rings (see, for example, Muscheler et al. 2007) it is possible to deduce the history of the solar cycle. Such studies have indicated that cyclic activity did persist throughout the Maunder Minimum, but at a significantly reduced level (Beer et al. 1998). Furthermore, it is clear that the Maunder Minimum is not exceptional -the solar cycle has often been interrupted by recurrent "Grand Minimum" phases of significantly reduced magnetic activity.
It is believed that the large-scale magnetic field in the solar interior is generated and maintained by a hydromagnetic dynamo. From a conceptual point of view, the large-scale field can usefully be decomposed into its toroidal (azimuthal) and poloidal (meridional) components -a working dynamo requires mechanisms that allow the poloidal field to be regenerated from toroidal field and vice versa. It is widely accepted that differential rotation (usually referred to as the Ω-effect in dynamo theory) is responsible for the generation of toroidal field from poloidal field. Surface observations indicate that equatorial regions rotate more rapidly than the poles, and helioseismological studies (see, for example, Schou et al. 1998) have shown that this rotation profile persists, approximately independently of radius, throughout most of the convection zone. At the base of the convection zone, a region of strong shear (the tachocline) couples the radiative zone, which rotates almost rigidly, to the differentially-rotating convective envelope. In most solar dynamo models, it is assumed that a significant fraction of the toroidal field is generated in the vicinity of the tachocline (where the Ω-effect should be very efficient due to the presence of strong differential rotation).
Although the Ω-effect is well understood, the reverse process that generates poloidal field from toroidal field is a topic of some debate. In classical interface dynamo models (see, for example , Parker 1993;Charbonneau & MacGregor 1996) the poloidal field is regenerated at the base of the convection zone by the action of cyclonic convection upon toroidal magnetic field lines (Parker 1955a). This process is usually referred to as the α-effect. Strong toroidal fields will tend to inhibit (or quench) the operation of the α-effect, so interface dynamo models are usually constructed in such a way that the α-effect is restricted to the region just above the base of the convection zone, whilst the Ω-effect operates just below the interface. The two layers are coupled by the effects of magnetic diffusion, as well as magnetic buoyancy and turbulent pumping (see, for example, Tobias et al. 2001). Even with strong α-quenching, it has been shown that an interface dynamo of this type can operate efficiently (Charbonneau & MacGregor 1996). In Babcock-Leighton dynamo models (Babcock 1961;Leighton 1964), the poloidal field is regenerated at the solar surface through the decay of active regions (which tend to emerge with a systematic tilt with respect to the azimuthal direction). This surface α-effect can only contribute to the dynamo if there is some mechanism that is capable of transporting the resultant poloidal field to the tachocline. This could be achieved by diffusion or by pumping, but meridional flows also could play an important role in this respect. A polewards meridional flow is observed at the solar surface (see, for example, Hathaway & Rightmire 2010) and, by mass conservation arguments, there must be a returning circulatory flow somewhere within the solar interior. A singlecell meridional circulation, with an equatorial flow at the base of the convection zone would couple the surface layers to the tachocline in an effective way, thus completing the dynamo loop.
A complete model of the solar dynamo must be able to explain the observed modulation as well as the 22-year magnetic cycle. It has been shown that it is possible to induce modulation by introducing stochastic effects into Babcock-Leighton models (Charbonneau & Dikpati 2000;Bushby & Tobias 2007), as well as into models of interface type (Ossendrijver 2000). However, fully deterministic models (with no random elements) can also produce modulated dynamo waves. Weiss et al. (1984) and Jones et al. (1985) considered a simple system in which the dynamo was modelled using a set of coupled ordinary differential equations, which included the nonlinear interactions between the magnetic field and the flow. They found that it was possible to generate quasiperiodic and chaotically-modulated solutions in addition to standard periodic dynamo waves. More recent studies have shown that the full mean-field equations also exhibit significant modulation when dynamical nonlinearities are included in the governing equations (see, for example, Tobias 1996;Brooke et al. 2002;Bushby 2006). An alternative approach was used by Yoshimura (1978) who demonstrated that modulation can arise if explicit time delays are built into the nonlinear terms in a simple system of model dynamo equations. A more sophisticated model was considered by Jouve et al. (2010) who investigated the effects of magnetic buoyancy-induced time delays in the context of a two-dimensional Babcock-Leighton dynamo. By introducing time delays into the surface α-effect term, they were able to demonstrate the existence of modulated cycles. They then went on to consider a simpler one-dimensional dynamo system in which the surface α-effect term was represented by the inclusion of a time-delayed toroidal field (with a parameterised time delay that was dependent upon the magnetic field strength). They were able to demonstrate the existence of a sequence of bifurcations from periodic to chaotically modulated solutions as the time delay parameter was increased.
The aim of this work is to investigate the competition between a deep-seated (interface) α-effect and a surface α-effect. Building on the approach described by Jouve et al. (2010), who did not include a deep-seated α-effect, the influence of the surface α-effect will be modelled using a time-delayed toroidal field. The use of a time delay is natural in this context: even if flux tubes rise rapidly to the surface, the time taken for the resultant poloidal field to be transported back to the tachocline will, in general, be non-negligible compared to the period of oscillation of the dynamo. Previous studies have investigated systems with competing α-effects (see, for example, Dikpati & Gilman 2001;Mason et al. 2002;Mann & Proctor 2009), but we believe that this is the first study to consider the effects of explicit time delays in a model of this type. The paper is structured as follows:
In Section 2, we describe the full model and an idealised system of equations that can derived from it (based upon a local analysis). This is followed in Section 3 by an analysis of the stability of the idealised model and then in Section 4 by the corresponding numerical results. In Section 5, we describe some numerical calculations which demonstrate the existence of modulated solutions in the full one-dimensional model. Finally, in Section 6, we present our conclusions and discuss the relevance of our results to the solar dynamo.
Model Setup
Following a similar approach to that adopted by Jouve et al. (2010), we consider a simple, illustrative model of the solar dynamo. This model is based upon the standard mean-field dynamo equation (see, e.g., Moffatt 1978),
∂B ∂t = ∇ × (αB + U × B) + η T ∇ 2 B,(1)
where U is the large-scale velocity field, α represents the standard mean-field α-effect, η T is the turbulent magnetic diffusivity (which we shall assume to be constant), whilst the mean magnetic field, B, satisfies ∇ · B = 0. Instead of solving this equation in spherical geometry, we consider the simpler problem of dynamo action in a flat Cartesian domain, with the axes oriented so that the y-axis would correspond to the azimuthal direction on a spherical surface. We can then look for dynamo solutions that depend only on a single spatial variable x (which can be regarded as being analogous to the co-latitude) and time t. The solenoidal constraint upon B can then be satisfied by writing the magnetic field in the following form:
B(x, t) = B(x, t)ŷ + ∇ × A(x, t)ŷ ,(2)
where B(x, t) is the toroidal field component, whilst A(x, t) corresponds to the poloidal potential. Our model is based on the assumption that the solar dynamo is operating primarily in the region around the base of the convection zone. For simplicity, we assume that α, which represents a deep-seated α-effect, is constant, i.e. α = α 0 . Furthermore, we adopt a fixed velocity profile of the form U = v 0x + Ω 0 zŷ, where v 0 and Ω 0 are both assumed to be constant in this illustrative model. This velocity field gives a constant meridional flow and a differential rotation profile that is independent of x. We also make the well known αΩ approximation, which assumes that differential rotation is the dominant mechanism for toroidal field regeneration in this region. Following Jouve et al. (2010), we also introduce a delayed toroidal field, Q(x, t), which lags behind the normal toroidal field with a time delay denoted by τ. However, unlike Jouve et al. (2010), who considered a time delay that was dependent upon the toroidal magnetic field strength, we assume τ to be constant throughout this study. The delayed toroidal field is coupled to the other equations via the inclusion of an additional poloidal source term, S Q(x, t), where S is a constant. This source term can be regarded as being the contribution to the local poloidal field from the non-local surface α-effect (which must, therefore, depend upon the strength of the toroidal field at earlier times). Finally, we introduce parameterised quenching nonlinearities into both of the α-effect terms in the poloidal field equation.
Having made these assumptions, we can now write down the three scalar partial differential equations for A(x, t), B(x, t) and Q(x, t):
∂A ∂t + v 0 ∂A ∂x = S Q 1 + λ|Q| 2 + α 0 B 1 + λ|B| 2 + η T ∂ 2 A ∂x 2 ,(3)∂B ∂t + v 0 ∂B ∂x = Ω 0 ∂A ∂x + η T ∂ 2 B ∂x 2 ,(4)∂Q ∂t = 1 τ (B − Q) ,(5)
where λ is a constant that determines the strength of the nonlinear quenching. In order to reduce the number of parameters that control the system, the variables can be rescaled as follows:
A = α 0 B 0 L 2 η T A ′ , B = B 0 B ′ , Q = B 0 Q ′ , t = L 2 η T t ′ , τ = L 2 η T τ ′ , x = Lx ′ , S = α 0 S ′ ,
where L is a characteristic length-scale and B 0 is a representative value of the magnetic field strength (which may be chosen so that the constant coefficient in the quenching terms equals unity in these scaled variables). On dropping the primes, we obtain the following set of partial differential equations (PDEs):
∂A ∂t + Re ∂A ∂x = S Q 1 + |Q| 2 + B 1 + |B| 2 + ∂ 2 A ∂x 2 ,(6)∂B ∂t + Re ∂B ∂x = D ∂A ∂x + ∂ 2 B ∂x 2 ,(7)∂Q ∂t = 1 τ (B − Q) .(8)
Thus the only parameters to control the system are the Reynolds number, Re = v 0 L/η T , which measures the strength of the meridional flow, and the dynamo number, D = α 0 Ω 0 L 3 /η 2 T , which indicates the strength of the dynamo sources relative to magnetic dissipation.
This system can be further simplified by carrying out a local analysis. Because we have the freedom to choose a convenient characteristic length-scale, local wavelike solutions can be assumed to have a unit wavenumber without any loss of generality. We therefore seek solutions of the form A =Ã(t)e ix , B =B(t)e ix and Q =Q(t)e ix , whereÃ(t),B(t) andQ(t) are complex functions of time only. Dropping the tildes, the governing equations for these quantities become:
dA dt + iReA = S Q 1 + |Q| 2 + B 1 + |B| 2 − A,(9)dB dt + iReB = iDA − B,(10)dQ dt = 1 τ (B − Q) .(11)
Following the methods used in Jones et al. (1985) and Jouve et al. (2010) it is possible to reduce the order of this system by using the following representation:
A = ρye iθ , B = ρe iθ , Q = ρze iθ ,
where ρ and θ are real quantities and y and z are complex numbers. Upon substituting these expressions into the governing equations (9) -(11), the following set of 5 real ODEs is obtained:
dρ dt = −Dρy 2 − ρ,(12)dy 1 dt = S z 1 1 + ρ 2 (z 2 1 + z 2 2 ) + α 1 + ρ 2 + 2y 1 y 2 D,(13)dy 2 dt = S z 2 1 + ρ 2 (z 2 1 + z 2 2 ) + Dy 2 2 − Dy 2 1 ,(14)dz 1 dt = 1 − z 1 τ + Dy 1 z 2 + Dy 2 z 1 − Rez 2 + z 1 ,(15)dz 2 dt = −z 2 τ − Dy 1 z 1 + Dy 2 z 2 + Rez 1 + z 2 ,(16)
where y 1 and y 2 represent the real and imaginary parts of y respectively and z 1 and z 2 represent the corresponding real and imaginary parts of z. This fifth-order system is a useful alternative representation of the local model.
Dynamo Transitions
In this section, we focus upon the local model that is described by Equations (9) -(11). To further our understanding of this system, we have carried out a series of calculations to determine the critical value of the dynamo number as the parameters S and τ are varied (at fixed Re). It is also possible to identify the value of τ that leads to quasi-periodic solutions analytically by studying the stability of the periodic solution.
Linear Theory
The critical dynamo numbers can be calculated by linearising the governing equations (9) -(11) and writing A, B and Q in the following form: A =Âe σt , B =Be σt and Q =Qe σt . The following characteristic equation is generated:
(σ + iRe + 1) 2 σ + 1 τ − iD σ + 1 τ − iS D τ = 0.(17)
Setting the real part of the growth rate to be zero and solving the characteristic equation for the imaginary part of σ will determine the critical value of the dynamo number, D c , at which the trivial (non-magnetic) solution loses stability to oscillatory dynamo waves. Letting σ = iω, we obtain the following:
− iω 3 − ω 2 τ − 2iω 2 Re − 2Reω τ − 2ω 2 + 2iω τ − iRe 2 ω − Re 2 τ − 2ωRe + 2iRe τ + iω + 1 τ + Dω − iD τ − iS D τ = 0.(18)
In the case of S = 0 the system corresponds to a standard αΩ dynamo. It is then straightforward to show that the critical dynamo number is 2, regardless of the magnitudes of τ or Re. For S 0, this equation can be solved numerically using a Newton-Raphson algorithm. As a specific example, Figure 1 shows how the critical dynamo number changes as S is varied between −50 and 50 with values of τ less than 1 and for fixed Re = 10. For these values of τ, D c < 2 for most values of S . In these regions of parameter space, it is easier to excite a dynamo than it would be in the corresponding αΩ system, so we can see that the non-local α-effect is enhancing the dynamo for these parameter values. However, for larger values of τ (where there is a significant time-lag between B and Q) there is a finite range of values of S in which D c > 2, with a local maximum in D c occurring somewhere in this range. In this case, the two competing α-effects appear to be impeding each other, thus making it more difficult to excite a dynamo. So it is clear that, even in linear theory, the interaction between competing α-effects is non-trivial. Additional calculations have been made to determine the critical dynamo number for different values of Re and a broader range of values for τ, but this case (which is of greatest relevance to the present study) is fairly representative in terms of the behaviour that is exhibited as S is varied.
The transition from periodic to quasi-periodic behaviour
Restoring the nonlinear terms to the governing equations, finite amplitude oscillations can be found when the dynamo number exceeds D c . The stability of the periodic solution can be analysed by expressing the magnetic fields in the following form: The Reynolds number varies between 0 and 50, with S varying between 0 and 70.
A ω (t) = A 0 e iωt ,B ω (t) = B 0 e iωt , Q ω (t) = Q 0 e iωt ,
where A 0 , B 0 and Q 0 are the complex wave amplitudes, ω is the frequency (with the ω subscript denoting the periodic state). The substitution of these expressions into the governing equations (9) -(11) produces the following simultaneous equations:
(iω + iRe + 1)A 0 = S Q 0 1 + |Q 0 | 2 + B 0 1 + |B 0 | 2 ,(19)(iω + iRe + 1)B 0 = iDA 0 ,(20)
iω
+ 1 τ Q 0 = 1 τ B 0 ,(21)
which can be solved using standard methods. Once the amplitude and frequency of the periodic solution have been determined, it is possible to perturb this solution to study its stability. Following the general method described by Jouve et al. (2010), this can be achieved by setting:
A= A ω 1 + α 1 e pt + α ⋆ 2 e p ⋆ t , B= B ω 1 + β 1 e pt + β ⋆ 2 e p ⋆ t , Q= Q ω 1 + γ 1 e pt + γ ⋆ 2 e p ⋆ t ,
where α 1 , α 2 , β 1 , β 2 , γ 1 and γ 2 are the coefficients of the perturbed fields, p is the complex growth rate of the perturbation and the symbol ⋆ represents the complex conjugate. Substituting these expressions into the governing equations (9) -(11) results in a system of 6 coupled equations that relates the coefficients of the perturbed fields to the growth rate p for a given set of parameters. After solving this system to find the growth rate of the perturbation, it is then possible to determine the stability of the periodic solutions. Tables 1 and 2 illustrate some of the results from this stability analysis. These tables show the parametric dependence of the critical value of τ for the transition from periodic to quasi-periodic solutions. The results in Table 1 correspond to D = 1000, with 0 ≤ Re ≤ 50 and 0 ≤ S ≤ 70. In Table 2, we have used the same values of Re, but D = −1000, whilst 0 ≥ S ≥ −70. An entry of (...) in either table indicates that no transition exists. Unsurprisingly, no modulation is found for S = 0. In this case the delayed toroidal field Q decouples from the system and we have a standard αΩ dynamo model. More unexpectedly, these results suggest that Re 0 is a necessary condition for modulation in this system. So the meridional flow seems to play a crucial role in driving the modulation, perhaps Here, D = 1000, S = 10 and τ = 0.1, whilst the Reynolds number is varied. All results are accurate to at least 1%. by introducing an additional (advective) time-scale into the problem. If it is simply the presence of an additional time-scale that is the key ingredient here, then it may still be possible to drive modulation in the absence of a flow if some other physical process (such as turbulent pumping) was included in the model. However, it is beyond the scope of this paper to investigate whether or not this is indeed the case. In the case of positive D, no modulation was found for negative S , whilst the same is true for positive values of S in the negative D case. Given the idealised nature of this local model, we should probably not read too much into this result, but (if nothing else) this again illustrates that competing α-effects interact in a rather non-trivial way in this system. Where modulation does occur, some trends can be identified. For example, for fixed Re in the D = 1000 case, the critical value of τ decreases with increasing S (whereas it increases with increasing |S | in the D = −1000 case). At fixed S , the critical value of τ tends to decrease with increasing values of Re, although this trend appears to reverse at low S and high Re in the D = 1000 case. We have no definitive physical explanation for this behaviour but can speculate that this is somehow related to the non-monotonicity that was observed in the D c calculations in the previous subsection.
Numerical simulations of the local model
In this section, we apply a numerical approach to the local model that is described by equations (9) -(11). Decomposing the system into its real and imaginary parts, we use a fourth-order Runge-Kutta scheme in Fortran to time-step the governing equations.
Validation of numerical calculations
To validate the code, it is possible to check that the results agree with the critical dynamo numbers that can be obtained from linear theory (see Section 3.1). Fixing the values of Re = 10, S = 20 and τ = 10, gives a prediction of D c = 2.499. This is consistent with the numerics: we find decaying oscillations for D = 2.4, whilst D = 2.6 gives a stable periodic solution. We can also compare the amplitude and frequency of the periodic solutions with the corresponding analytical predictions, using a Fourier transform to determine the frequency of oscillation in the numerical case. Table 3 shows the results of such a comparison, for variable Re, using D = 1000, S = 10 and τ = 0.1 (which includes the "reference case" below). All results are accurate to within 1% which clearly validates both the numerical scheme and the analytical calculations. Finally, fixing Re = 10, D = 1000 and S = 10, we find that the solution exhibits a transition from periodic to quasi-periodic dynamo waves at τ = 0.347, which is compares very favourably to the analytic value of τ = 0.348 (see Table 1).
Results
Initially, the parameters are chosen such that D = 1000, Re = 10, S = 10 and τ is varied (we will refer to this as the "reference case"). Figure 2 shows that a periodic solution can be found provided that τ is sufficiently small. Both B(t) and Q(t) oscillate with constant amplitude although Q(t) has a smaller amplitude of oscillation and, as expected, lags behind B(t). The effects of increasing the value of τ are shown in Figure 3. As τ is increased through the threshold value of τ = 0.347, the lag between B(t) and Q(t) increases to such an extent that we see a transition to a quasi-periodic state. Further increases in τ lead to further transitions, from multiply periodic to chaotically modulated states. Figure 4 shows the time evolution of the toroidal field energy B 2 for τ = 0.86, at which point the solution is chaotically modulated. It is clear that there are several phases of significantly reduced magnetic activity, and it is tempting to compare these to grand minima. The extent to which this behaviour is "solar-like" is a matter of some debate -this is, after all, a highly idealised model. Nevertheless, it is encouraging that this simple model, with competing α-effects, is capable of producing highly modulated dynamo waves when the time delay is large.
As indicated by the results in Table 1, the analysis of the stability of the periodic solution indicates that it is not possible to find a transition to a quasi-periodic solution for negative values of S , when D is positive. This tendency for the periodic state to be stable (for S ≤ 0) regardless of the value of τ has been confirmed numerically. However, for positive values of S it always appears to be possible to find a transition to quasi-periodic solutions, provided that the Reynolds number is non-zero, and these transitions are consistent with those predicted in Table 1. Once quasi-periodic solutions have been found it is usually possible to find chaotically-modulated states for sufficiently large values of the time delay. For negative values of the dynamo number, the results are again consistent with those predicted analytically. No modulation is found for positive S or for Re = 0. For negative S and positive Re, it is possible to find quasi-periodic and chaotically modulated solutions as the time delay is increased. One such solution is illustrated in Figure 5. As in Figure 4, it should again be noted that the chaotically modulated solution that is illustrated in the lower part of Figure 5 is characterised by phases of significantly reduced magnetic activity.
Solving the PDE system
Although the results from the local model are promising, it is important to verify that they are not crucially dependent upon the simplifying assumptions that have been made when deriving the model. In this section, we return to the original model of partial differential equations, as defined by Equations (3) -(5). In dimensionless units, we assume that 0 ≤ x ≤ π/2 (recalling that we interpret x as being analogous to the co-latitude on a spherical surface), imposing the boundary conditions that A = B = Q = 0 at x = 0 (the "North pole") and B = Q = ∂A/∂x = 0 at x = π/2 (the "Equator"). These boundary conditions correspond to the assumption that the global magnetic field has dipolar symmetry. Having neglected the effects of curvature, and having assumed constant α 0 , v 0 and Ω 0 , we should stress again that this should still be regarded as an illustrative model. Nevertheless, it contains the key physical ingredient of two competing α-effects with a surface α-effect contribution that depends upon a time-delayed toroidal field. In order to obtain dynamo waves that propagate towards the Equator, we focus primarily upon the D < 0 parameter regime (which would correspond to a negative deep-seated αeffect in the northern hemisphere). We solve the governing equations numerically, approximating derivatives using second-order finite differences. A 4th-order Runge-Kutta scheme is again used to time-step the governing equations.
Given that we are investigating the D < 0 regime, the local model suggests that we should be able to find modulation for negative values of S . However, in this region of parameter space there is an overwhelming tendency for steady modes to be preferred at onset (recall that wavelike solutions were assumed when the local model was derived). It is well known that steady and oscillatory modes can bifurcate from the trivial state at similar values of D in global αΩ dynamos (see, for exam- ple , Jennings & Weiss 1991), so this behaviour is not entirely unsurprising. However, it is almost certainly rather model specific -experimentation with the inclusion of different nonlinear quenching mechanisms suggests that it is possible to obtain oscillatory solutions in these parameter regimes. Furthermore, oscillatory solutions can be found for positive dynamo numbers and therefore, despite some differences, the results from the local model should not be discarded.
In fact, in the case of this global model, interesting solutions can be found for negative values of D and positive val-ues of S . This is illustrated by Figure 6 which shows solutions for D = −6000, S = 1 and Re = 10. A periodic solution can be found at τ = 0.01. This is characterised by an oscillatory magnetic field which propagates towards the Equator (note that these contour plots have been plotted as a function of latitude and time, for ease of comparison with observations). Increasing the time-delay leads to a transition to a quasi-periodic solution. Further increases in τ eventually lead to chaotically modulated oscillations (as illustrated in Figure 7). This solution is rather "solar-like" in many respects, with the dynamo confined to low latitudes, and with strong variations in the amplitudes of successive cycles. Furthermore, the modulation is characterised by periods of reduced magnetic activity. So although the modulation due to these competing α-effects was not in the expected parameter regime, it is clearly a robust feature of this system.
Conclusions
In this paper, we have investigated the properties of an illustrative mean-field dynamo model which includes two competing αeffects. The first of these is the standard deep-seated α-effect, the second is due to a surface α-effect (of Babcock-Leighton type). Following the approach described by Jouve et al. (2010), who did not consider competing α-effects, the contribution from the surface α-effect was modelled by assuming that it depends upon a time-delayed toroidal field (with a constant parameterised time delay τ). Two different approaches were applied to this model. Initially, a local approximation was made to reduce the governing equations to a system of coupled ordinary differential equations. A linearised version of these equations was used to determine the dependence of the critical dynamo number upon S (the magnitude of the surface α-effect) and τ. Generally, the larger the magnitude of S , the easier it becomes to excite the dynamo. However, there are some regions of parameter space in which the two competing α-effects appear to impede each other, thus inhibiting the dynamo. Moving beyond linear theory, it was found that there are significant regions of parameter space in which the periodic solution becomes unstable with increasing τ, leading to quasi-periodicity. This was verified numerically, where further increases in the time delay were shown to produce chaotically modulated states with phases of significantly reduced activity. The full PDE model was then investigated. Although modulation was found, this occurs in a different parameter regime to that predicted by the ODE model. This discrepancy could be model specific, although we expected to see some differences between the two models due to the fact that significant simplifications were made when deriving the set of coupled ODEs. Nevertheless, it was possible to find chaotically modulated solutions in the PDE model, and these solutions exhibit certain features that are (at least qualitatively) "solar-like".
There are many possible areas of future work. In particular, more could be done to explore the robustness of the PDE model to variations in the boundary conditions and the nonlinear quenching mechanisms. As has already been mentioned, preliminary calculations suggest that the adoption of different nonlinearities may make a very significant difference to the behaviour of the model. It may also be possible to improve the existing model by refining the way in which the time delay is implemented -the current approach is simple and effective, but is derived by truncating a Taylor series expansion at lowest order. Retaining higher order terms may make a difference to the behaviour of the model. Moving beyond the onedimensional Cartesian system, it would be natural to explore a two-dimensional version of this model in (axisymmetric) spherical geometry. This would open up the possibility of including a more realistic flow geometry (in both the meridional and azimuthal directions) as well as spatially dependent mean-field coefficients. Although this would still be within the framework of mean-field theory, a more realistic model would enable more detailed comparisons to be made between our results and the solar dynamo.
Fig. 1 :
1The critical dynamo number D c as a function of S , for values of τ less than 1 with Re = 10. Here, the solid line corresponds to τ = 0.1, the dotted line represents τ = 0.3 and the dash-dotted line shows τ = 0.5.
number varies between 0 and 50, with S varying between 0 and −70.
Fig. 2 :
2The reference case for τ = 0.2: This shows the toroidal field B (solid line) and the delayed toroidal field Q (dashed line) as a function of time (which is expressed in dimensionless units).
Fig. 4 :
4The reference solution for τ = 0.86. A plot of B 2 against time.
Fig. 5 :
5The effects of increasing τ for D = −1000, Re = 10, S = −10. The upper plot shows the time evolution of the toroidal field B (solid line) and the delayed toroidal field Q (dashed line) for a periodic solution at τ = 0.2. The lower plot shows the toroidal field energy, B 2 , as a function of time for τ = 1.08.
Fig. 3 :
3The reference case for τ = 0.35 (top), τ = 0.61 (middle) and τ = 0.86 (bottom). The plots on the left show the timedependence of the toroidal field, whilst the plots on the right show the phase portraits of the amplitudes of B(t) against Q(t) (as derived from the 5 th -order system).
Fig. 6 :Fig. 7 :
67Dynamo solutions from the full PDE system (D = −6000, S = 1 and Re = 10). Top: contours of toroidal field as a function of latitude and time (a latitude of 90 • corresponds to the pole, 0 • to the equator) for τ = 0.01. Middle: as above, but for τ = 0.05. Bottom: a plot of the energy in the toroidal field as a function of time for the quasi-periodic solution that is obtained for τ = 0.Chaotically-modulated solution from the full PDE system (D = −6000, S = 1, Re = 10 and τ = 0.09). Top: contours of toroidal field as a function of latitude and time. Bottom: a plot of the energy in the toroidal field as a function of time.
Table 1 :
1Critical values of τ for D = 1000.
Table 3 :
3Values of |B 0 |, |Q 0 | and ω from both analytical calculations and the numerical simulations.Re = 0
Re = 10
Re = 20
Analytic
Numerical
Analytic
Numerical
Analytic
Numerical
Calculation Simulation Calculation Simulation Calculation Simulation
|B 0 |
77.32
77.32
52.85
52.84
40.59
40.58
|Q 0 |
76.99
76.99
41.20
41.20
20.88
20.88
ω
0.92
0.93
8.03
8.04
16.67
16.68
Table 2 :
2Critical values of τ for D = −1000.
. H W Babcock, ApJ. 133572Babcock, H. W. 1961, ApJ, 133, 572
. J Beer, S Tobias, N Weiss, Sol. Phys. 181237Beer, J., Tobias, S., & Weiss, N. 1998, Sol. Phys., 181, 237
. J Brooke, D Moss, A Phillips, A&A. 3951013Brooke, J., Moss, D., & Phillips, A. 2002, A&A, 395, 1013
. P J Bushby, MNRAS. 371772Bushby, P. J. 2006, MNRAS, 371, 772
. P J Bushby, S M Tobias, ApJ. 6611289Bushby, P. J. & Tobias, S. M. 2007, ApJ, 661, 1289
. P Charbonneau, Living Reviews in Solar Physics. 22Charbonneau, P. 2005, Living Reviews in Solar Physics, 2, 2
. P Charbonneau, M Dikpati, ApJ. 5431027Charbonneau, P. & Dikpati, M. 2000, ApJ, 543, 1027
. P Charbonneau, K B Macgregor, ApJ. 47359Charbonneau, P. & MacGregor, K. B. 1996, ApJ, 473, L59
. G Delaygue, E Bard, Climate Dynamics. 362201Delaygue, G. & Bard, E. 2011, Climate Dynamics, 36, 2201
. M Dikpati, P A Gilman, ApJ. 559428Dikpati, M. & Gilman, P. A. 2001, ApJ, 559, 428
. J A Eddy, Science. 1921189Eddy, J. A. 1976, Science, 192, 1189
. D H Hathaway, L Rightmire, Science. 3271350Hathaway, D. H. & Rightmire, L. 2010, Science, 327, 1350
. R L Jennings, N O Weiss, MNRAS. 252249Jennings, R. L. & Weiss, N. O. 1991, MNRAS, 252, 249
. C A Jones, M J Thompson, S M Tobias, Space Sci. Rev. 152591Jones, C. A., Thompson, M. J., & Tobias, S. M. 2010, Space Sci. Rev., 152, 591
. C A Jones, N O Weiss, F Cattaneo, Physica D Nonlinear Phenomena. 14161Jones, C. A., Weiss, N. O., & Cattaneo, F. 1985, Physica D Nonlinear Phenomena, 14, 161
. L Jouve, M R E Proctor, G Lesur, A&A. 51968Jouve, L., Proctor, M. R. E., & Lesur, G. 2010, A&A, 519, A68
. R B Leighton, ApJ. 1401547Leighton, R. B. 1964, ApJ, 140, 1547
. P D Mann, M R E Proctor, MNRAS. 39999Mann, P. D. & Proctor, M. R. E. 2009, MNRAS, 399, L99
. J Mason, D W Hughes, S M Tobias, ApJ. 58089Mason, J., Hughes, D. W., & Tobias, S. M. 2002, ApJ, 580, L89
H K Moffatt, Magnetic field generation in electrically conducting fluids. CambridgeCambridge University PressMoffatt, H. K. 1978, Magnetic field generation in electrically conducting fluids (Cambridge: Cambridge University Press)
. R Muscheler, F Joos, J Beer, Quaternary Science Reviews. 2682Muscheler, R., Joos, F., Beer, J., et al. 2007, Quaternary Science Reviews, 26, 82
. M A J H Ossendrijver, A&A. 359364Ossendrijver, M. A. J. H. 2000, A&A, 359, 364
. E N Parker, ApJ. 122293Parker, E. N. 1955a, ApJ, 122, 293
. E N Parker, ApJ. 121491Parker, E. N. 1955b, ApJ, 121, 491
. E N Parker, ApJ. 408707Parker, E. N. 1993, ApJ, 408, 707
. J C Ribes, E Nesme-Ribes, A&A. 276549Ribes, J. C. & Nesme-Ribes, E. 1993, A&A, 276, 549
. J Schou, H M Antia, S Basu, ApJ. 505390Schou, J., Antia, H. M., Basu, S., et al. 1998, ApJ, 505, 390
M Stix, The sun: an introduction. BerlinSpringer-VerlagStix, M. 2002, The sun: an introduction (Berlin: Springer-Verlag)
. S M Tobias, A&A. 30721Tobias, S. M. 1996, A&A, 307, L21
. S M Tobias, N H Brummell, T L Clune, J Toomre, ApJ. 5491183Tobias, S. M., Brummell, N. H., Clune, T. L., & Toomre, J. 2001, ApJ, 549, 1183
. N O Weiss, F Cattaneo, C A Jones, Geophysical and Astrophysical Fluid Dynamics. 30305Weiss, N. O., Cattaneo, F., & Jones, C. A. 1984, Geophysical and Astrophysical Fluid Dynamics, 30, 305
. H Yoshimura, ApJ. 226706Yoshimura, H. 1978, ApJ, 226, 706
| [] |
[
"Compression of a confined semiflexible polymer under direct and oscillating fields",
"Compression of a confined semiflexible polymer under direct and oscillating fields"
] | [
"Keerthi Radhakrishnan \nDepartment of Physics\nIndian Institute Of Science Education and Research\n462 066Bhopal, Madhya PradeshIndia\n",
"Sunil P Singh \nDepartment of Physics\nIndian Institute Of Science Education and Research\n462 066Bhopal, Madhya PradeshIndia\n"
] | [
"Department of Physics\nIndian Institute Of Science Education and Research\n462 066Bhopal, Madhya PradeshIndia",
"Department of Physics\nIndian Institute Of Science Education and Research\n462 066Bhopal, Madhya PradeshIndia"
] | [] | The folding transition of biopolymers from the coil to compact structures has attracted wide research interest in the past and is well studied in polymer physics. Recent seminal works on DNA in confined devices have shown that these long biopolymers tend to collapse under an external field, contrary to the previously reported stretching. These long folded structures have a tendency to form knots that has profound implications in gene regulation and various other biological functions. These knots have been mechanically induced via optical tweezers, nano-channel confinement, etc., until recently, where uniform field driven compression lead to self entanglement of DNA. In this work, we capture the compression of a confined semiflexible polymer under direct and oscillating fields, using a coarse-grained computer simulation model in the presence of long-range hydrodynamics. Within this framework, we show that subjected to direct field, chains in stronger confinements exhibit substantial compaction, contrary to the one in moderate confinements or bulk, where such compaction is absent. Interestingly, an alternating field within an optimum frequency can effectuate this compression even in moderate or no confinement. Additionally, we show that the bending rigidity has a profound influence on the chain's folding favourability under direct and alternating fields. This field induced collapse is a quintessential hydrodynamic phenomenon, resulting in intertwined knotted structures, even for shorter chains, unlike DNA knotting experiments, where it happens exclusively for longer chains.arXiv:2204.05349v1 [cond-mat.soft] | null | [
"https://arxiv.org/pdf/2204.05349v1.pdf"
] | 248,118,639 | 2204.05349 | 3a5d004454046f3d6e9a0c6d896333d0ba8c36ea |
Compression of a confined semiflexible polymer under direct and oscillating fields
Keerthi Radhakrishnan
Department of Physics
Indian Institute Of Science Education and Research
462 066Bhopal, Madhya PradeshIndia
Sunil P Singh
Department of Physics
Indian Institute Of Science Education and Research
462 066Bhopal, Madhya PradeshIndia
Compression of a confined semiflexible polymer under direct and oscillating fields
(Dated: April 13, 2022)
The folding transition of biopolymers from the coil to compact structures has attracted wide research interest in the past and is well studied in polymer physics. Recent seminal works on DNA in confined devices have shown that these long biopolymers tend to collapse under an external field, contrary to the previously reported stretching. These long folded structures have a tendency to form knots that has profound implications in gene regulation and various other biological functions. These knots have been mechanically induced via optical tweezers, nano-channel confinement, etc., until recently, where uniform field driven compression lead to self entanglement of DNA. In this work, we capture the compression of a confined semiflexible polymer under direct and oscillating fields, using a coarse-grained computer simulation model in the presence of long-range hydrodynamics. Within this framework, we show that subjected to direct field, chains in stronger confinements exhibit substantial compaction, contrary to the one in moderate confinements or bulk, where such compaction is absent. Interestingly, an alternating field within an optimum frequency can effectuate this compression even in moderate or no confinement. Additionally, we show that the bending rigidity has a profound influence on the chain's folding favourability under direct and alternating fields. This field induced collapse is a quintessential hydrodynamic phenomenon, resulting in intertwined knotted structures, even for shorter chains, unlike DNA knotting experiments, where it happens exclusively for longer chains.arXiv:2204.05349v1 [cond-mat.soft]
INTRODUCTION
Nano-fluidic devices have been widely used for the characterization of macromolecules like separation [1,2], and sequencing of DNA [1][2][3][4][5][6][7], protein synthesis, transport of biopolymers, etc. [3][4][5][6][7][8][9][10][11][12] The fluid flow in such narrow capillaries profoundly influences the structural [1,2,[13][14][15][16][17] and dynamical behavior of soft-deformable macromolecules [1][2][3][4][5][6][7][18][19][20][21][22][23]. Few recent experiments have reported compression of confined long DNA strand into isotropic globules under uniform electric field [24][25][26], despite previously reported stretching of DNA seen under field [13,14,[27][28][29][30][31][32][33]. Similar shrinkage of a confined polyelectrolyte chain has been also captured in simulations [34] under AC field, with the exception that the shrinkage was manifested as backfoldings rather than an isotropic compression. These works substantiate the inevitability of hydrodynamic interactions in inducing such compression. Similar, hydrodynamic flow induced structural shrinkage is seen across diverse softmatter systems like U-shape bending of elastic rods under field [35], shear induced compaction [36,37], compaction of short chains under sedimenting fields [38,39], etc.
One direct repercussion of the chain compaction is the enhanced favourability of self knotting observed within these structures. The formation of these spontaneous knots along a long polymer chain like DNA [40][41][42][43] and other proteins [44,45] is a recurrent phenomenon in biological processes. The use of chain compaction either by spatial confinement, compaction against slit barriers [46], or molecular crowding [47] has been deemed extremely useful in effectuating these knottings. The use of direct or oscillatory fields in dielectrophoresis is another important protocol in obtaining a spectra of such self entangled structures.
In this article, we elucidate how a confined generic semiflexible chain exhibits large-scale compression under both direct and oscillating uniform fields. The course of investigation is mostly steered along deciphering the sensitiveness of this intriguing phenomenon to varying physical complexities like bending rigidity [48][49][50], degree of confinement [51][52][53], and chain length, along with plausible driving mechanism which remains elusive so far. While hydrodynamic flow field remains precursor to such a phenomenon, the above factors also acts as important ancillaries in dictating the folding favourability.
MODEL For this we model a semiflexible polymer along with solvent molecules confined in a cylindrical tube, periodic along its axis. The polymer is modelled as a bead-spring chain, which consists of N m monomers. The bond connectivity between adjacent monomers is ensured using a harmonic spring potential U s =
Nm−1 i=1 ks 2 (r i,i+1 − l 0 ) 2 , where l 0 , k s , r i,i+1 = |r i+1 − r i |,
and r i denotes the equilibrium bond length, the spring constant, magnitude of the bond vector, and position of the ith monomer, respectively. The semi-flexibility is introduced via a bending potential,
U b = Nm−2 i=1 k b 2 (r i,i+1 − r i+1,i+2 ) 2 ,
where k b is the bending rigidity that dictates the stiffness or persistence length of the chain.
Furthermore, excluded volume interactions among chain monomers are incorporated via standard repulsive shifted Lennard-Jones (LJ) potential. The solvent molecules are modelled using multi-particle collision dynamics [54,55], which is a particle based meso-scopic simulation technique that incorporates both hydrodynamic interactions and thermal fluctuations. The elaborate description of the method can be found in references cited herein [15,[54][55][56][57][58].
The interaction of monomers with wall is considered via a similar LJ potential described earlier, where a monomer feels a repulsive force within a distance R c ≤ 2 1/6 σ/2 from the wall. An external field G is imposed to all polymer monomers, while solvent molecules feels indirect drag via monomers. In case of an oscillating field, a square-wave periodic potential is applied to each monomer given as F x G = sign[sin(2πνt)]G. The force is directed along the channel axis, where ν (time period, τ ν = 1/ν) stands for the frequency.
The physical parameters are expressed in units of bond length l 0 , energy k B T , and time τ = m s l 2 0 /k B T , where m s is mass of a solvent particle i.e., taken to be unity. The MPC parameters are collision time τ s = 0.1τ , cell length a = l 0 and solvent density ρ s = 10m s /l 3 0 . Newton's equations of motion of the polymer is implemented using velocity-Verlet algorithm at fixed integration time step h m = 5 × 10 −3 τ . All results are for chain length N m = 200, if not mentioned otherwise. We use a certain mapping to obtain the experimental scales, equivalent to our coarse-grained model. This protocol has been previously found efficient in explaining many DNA based phenomenons [59,60].A DNA of width w ≈ 2nm, l p ≈ 50nm and contour L = 16µm is equivalent to a chain of bead diameter σ = 0.8, l p = 20, and N = 6400. Considering each base pair to be of mass 615 Da, a monomeric unit of length 2nm (12 bps) has a mass of 7380 Da which is equal to M = 10, a monomer's mass in simulations units. Hence, our simulation time unit t = m s l 2 0 /K B T = 1 translates as t = 4.3 × 10 −6 secs in real units. So, the chosen window of frequencies 0.0001 − 0.02 maps to 23Hz − 4600Hz for a chain length equivalent to N m = 200.
RESULTS
We present the structural response of a confined semi-flexible chain under an uniform field. For that, we follow a systemic approach of addressing the influence of direct fields followed by an investigation into oscillatory fields.
A. Chain collapse under direct field
In bulk, it is prestablished that a short flexible chain exhibits weak compression under direct field [36,61]. Figure 1 elucidates the structural response of a confined semiflexible polymer (N m = 200) under subjection to a constant DC field. This is parametrized in terms of average radius of gyration, given as R g =
Nm Nm i=1 (r i − R cm ) 2 ,
where R cm is the center-ofmass of the chain. The retrieved curve exhibits a nonmonotonic dependence over G, where within a moderate field strength chain exhibits a significant structural compression, followed by a stretching at higher field strengths [36]. This is qualitatively similar to the stretching response reported for the flexible chain in bulk, with the exception that there chain compaction was seen only for smaller chain lengths and weak field [38,39]. However, for a confined chain, not only the compaction becomes prominent for N m = 200, the favourability of the induced compression shifts toward higher fields with increasing persistence length. Also, the relative compaction gets more pronounced for higher bending rigidities such as, for l p = 25 it is roughly R g /R 0 g = 0.25, while for l p = 7, R g /R 0 g = 0.5, with R 0 g being the radius of gyration of the chain in equilibrium. This is particularly fascinating considering the long and semiflexible nature of a typical DNA molecule [62], which reportedly exhibits collapse under applied fields [24][25][26].
Effect of bending rigidity: In the equilibrium case, it is prestablished that a confined chain exhibits a monotonic stretching with increasing persistence length, where the chain with lower bending rigidity spanning the de Gennes regime/moderately confined de Gennes regime [63][64][65] (D/l p >> 1) takes a coil-like structure. However, with enhancing chain rigidity it transitions into Odijk's regime [16,66] (D/l p < 1), exhibiting stretching into linearly arranged array of the chain segments. The equilibrium curve is shown in Fig.2
-a .
However, the chain exhibits a counter intuitive structural response with bending rigidity under direct field, which is shown in Fig.2-b. The R g is normalized with the stretching seen for the case of a flexible chain R f lx g . Under DC, the monomers in the periphery experiences an enhanced hydrodynamic drag compared to the monomers inside which drifts faster under field in a random coil-like chain [38,39]. As a result of this spatial gradient in drift velocity of monomers, a re-circulation flow field builds up that pushes shear lagged particles from the surface into the middle in a circulatory fashion. As a result for a flexible chain, this recirculating hydrodynamic flow field results in a tadpole-like structure [38,39], made of a compact head followed by an extended tail as shown in Fig.2-c . However, with increasing l p , the size of this head grows, due to bending cost, leaving a folded structure constituting major fraction of the chain with shorter fluctuating tail. This manifests as the compression seen in intermediate l p regime, elucidated for l p = 13 in Fig.2-c. However, beyond a certain l p the chain once again stretches (see l p = 30 in Fig.2-c). At larger l p 's, the bending stiffness dominates and disfavours large curvatures, giving rise to elongated backfolded domains. In summary, the nonmonotonic response of the chain under DC field is due to the competition between hydrodynamic drag induced compaction under field [38,39], and the bending energy cost.
A similar non-monotonicity is also observed in the spontaneous knot formations seen in polymers [50,67]. This similarity can be drawn, considering the selfentangled and knotted structures seen in our case, which will be corroborated in later parts of the manuscript. A heuristic scaling obtained for the critical bending rigidity, beyond which the chain starts compressing indicates
l c p ∼ G β kind of dependence, where β ≈ 1.4.
Effect of confinement: Considering how the presence of confinement affects the conformational dynamics of a chain [16,64,65], its outright indispensable to look at the influence of the geometric constraint. Figure 3-a elucidates the field induced compression of a semi-flexible polymer (l p = 7), for a range of varying pore radii. A striking observation is that, while moderately confined polymer, D/2l p < 1.4, undergoes a substantial compression at lower field strengths followed by a stretching, chain under weak confinement/tending to bulk D/2l p > 1.4 exhibits no compression and undergoes a monotonic stretching. This is evident in R p ≤ 10 where a compressive dip in R g is seen for lower G values, while for R p > 10 only monotonic stretching is seen, devoid of any shrinkage. A heuristic scaling obtained in Fig. 3-a for the field G c beyond which stretching is effectuated, gives a G c ∼ R −3/2 p dependence. The deviation in R p scaling seen for R p = 30, we speculate is a repercussion of R p hitting a different regime, where the effect of confinement diminishes such that all R p > 20 tend to the bulk behaviour. This is elucidated in Fig.3-b, which shows the structural variation of a chain under varying confinement in equilibrium. It shows a typical non-monotonous pattern, as reported earlier [68]. The chain slightly swells back to its bulk value beyond R p > 20.
B. Chain under Oscillating-Field
The structural response of the chain under an oscillating field is shown in Fig. 4 for l p = 7 and R p = 10. At higher frequencies the chain remains nearly unperturbed retaining the equilibrium structure (R 0 g ). In the intermediate frequencies, the chain swells along the field direction, referred as the "stretch-state", until it reaches a critical frequency ν c below which the chain undergoes a significant collapse. This "compressed-state" essentially consists of a crumbled folded domain with a short fluctuating tail. This collapse is further followed by a stretching at lower frequencies approaching toward the DC like behavior. Unlike the stretch-state mentioned earlier, the structure of the chain is more like a "tadpole" with a leading bob-head and trailing long tail, that switches the direction at every field switch. This eventually merges with the DC limit, where for long chains a tadpolestructure is found with a long trailing end [36,38,39]. In summary, for a range of ν << ν c , ν < ν c , and ν ≥ ν c , the chain undergoes a three state transition involving tadpole-collapse-stretch states, respectively (see SI-movie-1). Additionally, we see similar behavior for various chain lengths (N m = 100, 150, 200, and 300), where the chain compression for longer chains happens at lower frequencies. This dependence over chain length A scaling of R g with ν/G is procured, where all the curves superimpose on each other, suggesting a linear dependence of ν c with G. The striking observation here is that for l p = 7, the chain stretches under DC field for G > 1 (see of Fig. 1). However, the oscillating field drives the system to a compressed state for the same field strengths. Similar AC induced compressions are seen for other bending rigidities l p = 16 and l p = 25 (see SI-movie-2), shown in Fig. 4-b, and c for a wide range of G beyond linear response regime, which is devoid of chain foldings in DC. Another, impressive feature of this AC driven compression is that same extent of collapse is attainable across all these field strengths. While, the maximum collapse possible varies from R g /R 0 g = 0.55, 0.25, 0.2 for persistence lengths l p = 10, 16, and 25, respectively. Further, the transition points for the collapse obtained in Fig. 4-a, b, c suggests that the retrieved critical frequency ν c /G is nearly independent of the persistence length of the chain.
R g /R g 0 R p =8 R p =10 R p =15 R p =20 R p =30
Effect of bending rigidity under AC field: Now we consider the explicit effect of bending rigidity on the chain's collapse mechanism. This is shown in Fig. 6 for different bending rigidities at G = 2 and pore radius R p = 10. More flexible chains undergo a stretch-collapsetadpole state transition, while going from higher to lower frequency in the respective window of ν ≥ ν c , ν < ν c , and ν << ν c . The stretching observed at higher frequencies, prior to the compression is suppressed with increasing bending stiffness. This again indicates toward the favoured chain stretching along the field direction, due to enhanced geometric constriction. For example, beyond l p > 13 stretching is completely absent, which corresponds to D/2l c p ≥ 1.0, that is close to the ratio obtained in the case of the pore radius variation. The discrepancy in the crossover regime here, we speculate is due to shorter chain lengths (N m = 200) used in the simulations. In summary, for a confining radius, there exists a critical persistence length D/2l c p ≈ 1.0, beyond which the chain eludes the pre-stretching tendency before collapse at high frequencies (ν > ν c ). Also, for higher persistence lengths, the folding favourability enhances dramatically and the chain exhibits collapse for a wide range of frequencies, even extending up to the DC limit.
Effect of confinement under AC field: The effect of confinement on the structural response of a chain for different pore radius is displayed in Fig. 7-a, at l p = 7 and G = 2.0. For a narrow pore R p ≤ 10, with decreasing frequency a three tier stretch-collapse-tadpole kind of transition is observed. While with increasing pore radii, the stretching prior to collapse (for ν ≥ ν c ) diminishes, as is evident in the case of R p = 15, 20, and 30, which is devoid of such stretching.
To gain insights into various stretching responses, we distinctly look into the chain's expanse in the longitudinal and transverse directions to the applied field. Figure 8 shows the radius of gyration R 2 g = 1 2N 2 i,j (r i − r j ) 2 (in black circles), its component along x-direction of confinement axis 2 (in red squares) and its component in the perpendicular y-z plane G yy + diamonds) for varying pore radii R p = 10, 15, 30, and bulk depicted in clockwise manner from top left, respectively. Interestingly for R p = 10, the G xx curve maps the overall R 2 g , suggesting that the stretching seen here is by virtue of the chain extension along the channel axis. With the transverse component contributing barely, this gives a highly anisotropic stretched state. However, with increasing pore radii, for R p = 15, the xcomponent departs away from the overall R g and the contribution from the transverse component enhances, decreasing the anisotropy along the field direction. Interestingly, with further increase in pore radius like for R p = 30 the structure again shows an increasing anisotropy, by preferentially aligning in the y-z plane. Further, tending to the bulk case the chain again exhibits prominent transverse alignment for higher frequencies, where (G yy + G zz )/G xx > 2. Hence, the stretching seen for R p = 10 and below in Fig. 7-a is a consequence of stretching along the field direction with transverse alignment of the chain constricted by the narrow wall, in contrast to the bulk where the chain preferably stretches and orients in the transverse direction. Also, it can be inferred that the chain undergoes a direct collapse, bypassing the stretched state at higher frequencies, approximately in the regime D/2l p > 1.4.
G xx = 1 2N 2 i,j (x i − x j )G zz = 1 2N 2 ( i,j (y i − y j ) 2 + i,j (z i − z j ) 2 ) (in blue
Additionally, a non-monotonic shift in the critical frequency ν c with pore radius is seen, where beyond R p = 20 the ν c decreases. This maybe stemming from similar nonmonotonicity in R g seen in equilibrium fig.3-b, which reflects in the associated timescales as well.
Further, structural response of the chain obtained for R p = 20, and R p = 30 for varying fields shown in Fig. 7-b,c, exhibits a strong coil to compressed-state transition at the critical frequency, before tending to the DC limit. See SI-movie-2 for compressed states. Here, we retrieve a scaling for the critical frequency with field strength as ν c ∼ G for R p = 20, similar to R p = 10 case, while for R p = 30 we get ν c ∼ G 3/5 . This difference is speculated to be a result of the chain hitting the bulk regime beyond R p = 20, see Fig.3-b. Previously, we have seen that in DC for narrow pores, strong compression is seen at lower fields in contrast to wider pores (like R p = 20 and 30), where no such folding is seen (see Fig. 3-a). Interestingly, under oscillating field not only narrow pores( R p = 10 in Fig. 4-a) even larger pores (R p = 20 and 30 in Fig. 7-b,c) exhibit remarkable compression of the chain for a range of field strengths, where chain remains stretched under DC field. Hence, AC driven collapse mechanism is accessible over a wider range of confining radii (including bulk) and field strengths beyond the linear response regime.
C. Transient dynamics under DC:
The dynamic pathway of the tadpolar formation under DC field involves an initial compaction followed by an extending tail [38,39]. We corroborate this by conducting 50 independent trials of a chain under DC, where majority of conformations undergo an initial compaction into a compressed-state prior to the stretching, while a few others undergo direct stretching, where a small head like structure grows over time to form an extended tadpolar structure. The transient dynamics obtained categorically by averaging over respective events (with and without initial collapse) are shown in Fig. 9-(a) and (b). Interestingly, a chain which started from the coil-state attains a collapsed-state, on average within the time window of t c = 1000−10000. This time-scale translates to frequency as ν = 1/t c = 0.001 − 0.0001, which coincides with the frequency window of chain collapse in AC (see Fig. 7 the allowed time window before every field switch is such that it captures the initial collapse process, partially forbidding the complete tail extension, the chain wouldn't relax to its actual DC field extension values (see Fig. 3-a). Further, it is prestablished that the chain compaction under DC, is a result of a recirculating flow field. Under such a scenario, a diffusive timescale of the chain can be given as (τ D = R 3 µ0ak B T ), where the chain traverses a distance of its own size in equilibrium. Similarly, there exists a drift timescale of the chain (τ v = R/v G ), which essentially signifies the time any part of the chain takes to traverse a distance of R under the field, enforcing proper recirculatory dynamics. The later is a precursor to obtain chain compaction under DC. Hence, in case of an oscillating field at G, if the time-period of the applied field follows 1/ν > τ v (G) (i.e. ν < 1/τ v (G)), the chain is allowed proper recirculating dynamics leading to a compact state in AC. This recirculation timescales as a function of G DC is presented in Fig. 10-a. Since in our case, τ v spans a window of τ v = 100 − 1000, roughly, translating as 1/τ v = 0.01 − 0.001, then 1/τ v provides a frequency window below which chain compaction can be obtained. However, at much lower frequencies (i.e larger allowed times) under field, the tail extension again gets prominent leading to the DC limit. As a result the compaction is seen only in a narrow window.
While, this roughly sets the upper frequency cutoff for the collapse to reinforce, the lower frequency cutoff for the oscillating field below which the chain reaches to the DC limit, is obtained from the relaxation of the max extension under the DC field. We estimate the relaxation of the maximum extension of chain defined as C X (t) = δX(t)δX(0) δX(0)δX(0) , where δX = X(t) − X(t) is the fluctuation in maximum extension, and X(t) and < X(t) > are the instantaneous and average max extension, respectively. The correlation shows exponential decay in short time limit followed by oscillatory behavior for large time. The behavior of the correlation can be parametrized by the following expression, C X (t) = a 0 exp(−t/τ ) cos( 2πt T ). The retrieved zero crossing time τ DC X at which C X (τ DC X ) = 0 values are shown in Fig. 10-b, for varying G with different pore radii. The obtained relaxation time τ DC X falls in the range of 10000 − 40000, which translates to 1/τ DC X = 0.0001 − 0.000025. This signifies that beyond τ DC X the chain attains proper conformational relaxation of its fluctuating head and tail ends. This results in stretched tadpolar states seen in the frequency window ν << ν c , corresponding to the DC limit.
D. Field induced knotting:
Now our goal is to corroborate the presence of knots driven by the AC field and how frequency modulation influences the knotting tendency and its complexity. For this purpose, we use a software package Kymoknot [69], which uses the arc closure algorithm for the analysis of knotted structures in linear chains [70,71]. Importantly, the average fragment l h that is part of a knot/knots exhibits a non-monotonic behavior with frequency for various pore radii, as depicted in Fig. 11-a. The l h shows the extent of knotted length feasible within a polymer under field. In the equilibrium or at higher frequency limit (like for R p = 20, ν ≥ 5 × 10 −3 ) we found l h = 0, which indicates that the structure is devoid of any such knots. While for the intermediate frequencies, the value of l h sharply grows even reaching beyond 0.5, suggesting an enhanced knotting favourability, where a large portion of the chain is topologically entangled. The attainability of the chain self-entanglement to such a wide extent is ensued from the oscillatory nature of the inducing field leading to crumbled structures with low structural expansion. Further, as we go to the DC limit (ν << 10 −4 , l h is shown with arrows), the fraction constituting the knot falls off dramatically, but exhibits a non-zero l h (see Fig. 11-a). In DC, the formation of a well taut and stretched tadpole constitutes a tightly knotted head (see Fig. 11-b-ii), resulting in smaller values of l h , contrary to the loosely formed knotted fragments observed at intermediate frequencies in AC (see Fig. 11-b-i). This can be understood by considering a piece of rope with a simple knot tied on it, such that if we pull the chain from its terminus, the knotted part will become tight and localized. The top row (i) and (ii) of Fig. 11-b, shows the native chain structures formed under AC field (knotted portion in orange), while the bottom row (iii) and (iv) shows the corresponding tight and localized knot formed after stretching the field favoured native structures at its extremities.
DISCUSSION
We have presented the phenomenon of chain collapse to self entangled structures orchestrated by direct and oscillatory field using a generic polymer model. Under any direct force, the generation of recirculating hydrodynamic flow fields paves way to tadpoles structures, but the imposition of spatial constriction profoundly influences the intra-chain intertwining, leading to field compressed structures even in longer chains. For larger bending rigidity, this collapse exhibits a remarkable non-monotonic response, where the compression is dictated by the competition between the bending cost and stretching force. Apart from the confinement and chain rigidity, nature of the jostling force also brings fascinatingly complex dynamics into picture [72][73][74] . Following this, we elucidate that a semi-flexible chain under an oscillating field exhibits remarkable compression in certain frequency window. The AC field orchestrated compression is attainable across a wide range of confinements (including bulk), bending rigidities and field strengths, where chain simply stretches under DC. The field switching under AC captures the initial recirculating flow induced compaction, forbidding proper tail extension, such that the chain essentially gets arrested in a collapsed state with fluctuating ends.
Within the cellular structure, self organisation and dynamical instabilities causes mechanical oscillations [75]. These periodic jostling forces might profoundly influence the conformations of the biopolymer, mediated via the cellular fluid. Further, dielectrophoresis (DEP) experiments [26,76,77] generally involves high fields of alternating nature, where this oscillatory field is leveraged for more controlled manipulation and efficient separation schemes for DNA and other polymers.
FIG
. 1. a. Radius of gyration Rg of a semi-flexible polymer under DC field (G) for various persistence length lp in pore radius Rp = 10.
FIG
. 2. a) Scaled radius of gyration Rg of a semi-flexible polymer chain with varying bending rigidity lp in confinement radius Rp = 10 for two different chain lengths. The inset shows variation in end-to-end distance Re. b) The Rg of a confined chain as a function of lp for a fixed G is shown in the main plot. The chain undergoes a collapse in DC beyond a critical lp. c. Snapshots of the chain under DC for various lp at G = 1 and Nm = 200.
FIG
. 3. a. The normalized radius of gyration of a chain at lp = 7, as a function of G for a range of confinements. Note that the x-axis is scaled with R 3/2 p . b. The equilibrium radius of gyration of a confined semiflexible chain as function of pore radius at lp = 7 for chain length Nm = 200.
FIG. 4 .
4The variation of Rg in response to varying field frequencies for different G and fixed Rp = 10. Different panels correspond to persistence length a) lp = 7, b) lp = 16, and c) lp = 25, respectively. The Rg for the respective DC values are denoted by a horizontal arrow. is shown in Fig.5. Further, a scaling of ν c ∼ N −3/4 m for critical frequency with chain length is retrieved by superimposing all the curves on to each other.
FIG. 5 .
5Scaled radius of gyration Rg of polymer chain in pore Rp = 10 in response to varying field frequencies at G = 2, for different chain length Nm. Horizontal lines denote the Rg for the same DC field.
radius of gyration Rg/R 0 g of the chain at Rp = 10 in response to varying frequency at G = 2, for different persistence length lp. The horizontal arrows denote the DC values of Rg/R 0 g (ν = 0).
FIG. 7 .
7(a) Scaled radius of gyration Rg of semi-flexible chain in confinement in response to varying field frequencies at G = 2, for different pore radius Rp. Horizontal lines denote the DC values. Here, (b) and (c) corresponds to the variation in Rg of polymer chain in confinement in response to frequencies for different field strengths G, for pore radii Rp = 20 and 30, respectively.
FIG. 8 .
8Transverse and longitudinal components of radius of gyrations Gxx and Gyy + Gzz for a few pore radius, including bulk. The values correspond to Nm = 200 and lp = 7 for all pores, while bulk is shown for Nm = 50, lp = 7). The stretching in lower Rp = 10 is effectuated by pore enhanced elongation along the field-axis. The stretching in bulk is attributed to the stretching and alignment of the chain along the transverse direction.
FIG. 9 .
9The average maximum extension of chain as a function time under DC field for pore radius Rp = 20 (Left panel) and Rp = 30 (Right panel) at G = 2.0. The solid line shows the average over all the confirmations. The red line shows the average over trajectories which undergoes straight stretching, while the blue is for those which undergoes an initial compaction followed by stretching.
FIG
-b and c). The transient dynamics of the chain under homogeneous DC field indicates that if for an oscillating field, . 10. a) Recirculation timescale τν = R0/vcm for different pores as a function varying field strength G in DC. b) The DC relaxation timescale obtained from the correlation of the max extension.
FIG. 11 .
11The fraction of chain length l h that constitutes a knot as a function ν at field strength G = 2.0, for different pore radii obtained using Kymoknot software[69]. The error bar depicts the standard deviation at each frequencies.b) Top row depicts few snapshots of the chain conformations obtained under field, with the knotted portion highlighted and, bottom row corresponds to the tight and localized knots obtained after stretching respective conformations at its ends.
Acknowledgments: This work received financial support from the DST SERB Grant No. CRG/2020/000661. High-performance computing facility provided at IISER Bhopal and Paramshivay NSM facility at IIT-BHU are also acknowledged.
. J Han, H G Craighead, Science. 2881026J. Han and H. G. Craighead, Science 288, 1026 (2000).
Polymer translocation (Taylor & Francis US. M Muthukumar, M. Muthukumar, Polymer translocation (Taylor & Fran- cis US, 2011).
. D Branton, D W Deamer, A Marziali, H Bayley, S A Benner, T Butler, M Di Ventra, S Garaj, A Hibbs, X Huang, Nature biotechnology. 261146D. Branton, D. W. Deamer, A. Marziali, H. Bayley, S. A. Benner, T. Butler, M. Di Ventra, S. Garaj, A. Hibbs, X. Huang, et al., Nature biotechnology 26, 1146 (2008).
. S Benner, R J Chen, N A Wilson, R Abu-Shumays, N Hurt, K R Lieberman, D W Deamer, W B Dunbar, M Akeson, Nature nanotechnology. 2718S. Benner, R. J. Chen, N. A. Wilson, R. Abu-Shumays, N. Hurt, K. R. Lieberman, D. W. Deamer, W. B. Dunbar, and M. Akeson, Nature nanotechnology 2, 718 (2007).
. P Chen, J Gu, E Brandin, Y.-R Kim, Q Wang, D Branton, Nano Letters. 42293P. Chen, J. Gu, E. Brandin, Y.-R. Kim, Q. Wang, and D. Branton, Nano Letters 4, 2293 (2004).
. I Vodyanoy, S Bezrukov, Biophysical journal. 6210I. Vodyanoy and S. Bezrukov, Biophysical journal 62, 10 (1992).
. J J Kasianowicz, E Brandin, D Branton, D W Deamer, Proceedings of the National Academy of Sciences. 9313770J. J. Kasianowicz, E. Brandin, D. Branton, and D. W. Deamer, Proceedings of the National Academy of Sci- ences 93, 13770 (1996).
. C Y Kong, M Muthukumar, Electrophoresis. 232697C. Y. Kong and M. Muthukumar, Electrophoresis 23, 2697 (2002).
. B M Venkatesan, R Bashir, Nature nanotechnology. 6615B. M. Venkatesan and R. Bashir, Nature nanotechnology 6, 615 (2011).
. I M Derrington, T Z Butler, M D Collins, E Manrao, M Pavlenok, M Niederweis, J H Gundlach, Proceedings of the National Academy of Sciences. 10716060I. M. Derrington, T. Z. Butler, M. D. Collins, E. Man- rao, M. Pavlenok, M. Niederweis, and J. H. Gundlach, Proceedings of the National Academy of Sciences 107, 16060 (2010).
. M B.-J. Jeon, Muthukumar, The Journal of chemical physics. 14015101B.-j. Jeon and M. Muthukumar, The Journal of chemical physics 140, 015101 (2014).
. C T A Wong, M Muthukumar, The Journal of chemical physics. 13345101C. T. A. Wong and M. Muthukumar, The Journal of chemical physics 133, 045101 (2010).
. M Jonsson, U Jacobsson, M Takahashi, B Nordén, Journal of the Chemical Society. 892791Faraday TransactionsM. Jonsson, U. Jacobsson, M. Takahashi, and B. Nordén, Journal of the Chemical Society, Faraday Transactions 89, 2791 (1993).
. R Netz, Physical review letters. 90128104R. Netz, Physical review letters 90, 128104 (2003).
. S P Singh, M Muthukumar, The Journal of chemical physics. 141S. P. Singh and M. Muthukumar, The Journal of chemical physics 141, 09B610 1 (2014).
. T Odijk, Physical Review E. 7760901T. Odijk, Physical Review E 77, 060901 (2008).
. A Muralidhar, D R Tree, K D Dorfman, Macromolecules. 478446A. Muralidhar, D. R. Tree, and K. D. Dorfman, Macro- molecules 47, 8446 (2014).
D Stein, F H J Heyden, W J A Koopmans, C Dekker, Proc. Natl. Acad. Sci. USA. Natl. Acad. Sci. USA10315853D. Stein, F. H. J. Heyden, W. J. A. Koopmans, and C. Dekker, Proc. Natl. Acad. Sci. USA 103, 15853 (2006).
. R M Jendrejack, D C Schwartz, J J De Pablo, M D Graham, J. of Chem. Phys. 1202513R. M. Jendrejack, D. C. Schwartz, J. J. de Pablo, and M. D. Graham, J. of Chem. Phys. 120, 2513 (2004).
. R M Jendrejack, E T Dimalanta, D C Schwartz, M D Graham, J J De Pablo, Physical review letters. 9138102R. M. Jendrejack, E. T. Dimalanta, D. C. Schwartz, M. D. Graham, and J. J. de Pablo, Physical review let- ters 91, 038102 (2003).
. R Chelakkot, R G Winkler, G Gompper, Europhysics Letters). 9114001EPLR. Chelakkot, R. G. Winkler, and G. Gompper, EPL (Europhysics Letters) 91, 14001 (2010).
. D Steinhauser, S Köster, T Pfohl, ACS Macro Letters. 1541D. Steinhauser, S. Köster, and T. Pfohl, ACS Macro Letters 1, 541 (2012).
. O A Hickey, C Holm, The Journal of chemical physics. 138194905O. A. Hickey and C. Holm, The Journal of chemical physics 138, 194905 (2013).
J Tang, N Du, P S Doyle, Proceedings of the National Academy of Sciences. the National Academy of SciencesJ. Tang, N. Du, and P. S. Doyle, Proceedings of the National Academy of Sciences (2011).
. C Zhou, R Riehn, Physical Review E. 9212714C. Zhou and R. Riehn, Physical Review E 92, 012714 (2015).
. C Zhou, W W Reisner, R J Staunton, A Ashan, R H Austin, R Riehn, Physical review letters. 106248103C. Zhou, W. W. Reisner, R. J. Staunton, A. Ashan, R. H. Austin, and R. Riehn, Physical review letters 106, 248103 (2011).
. M Ueda, H Oana, Y Baba, M Doi, K Yoshikawa, Biophysical chemistry. 71113M. Ueda, H. Oana, Y. Baba, M. Doi, and K. Yoshikawa, Biophysical chemistry 71, 113 (1998).
. M Ueda, Journal of biochemical and biophysical methods. 41153M. Ueda, Journal of biochemical and biophysical meth- ods 41, 153 (1999).
. N Kaji, M Ueda, Y Baba, Applied physics letters. 833413N. Kaji, M. Ueda, and Y. Baba, Applied physics letters 83, 3413 (2003).
. H Liu, Y Zhu, E Maginn, Macromolecules. 434805H. Liu, Y. Zhu, and E. Maginn, Macromolecules 43, 4805 (2010).
. P.-Y Hsiao, Y.-F Wei, H.-C Chang, Soft Matter. 71207P.-Y. Hsiao, Y.-F. Wei, and H.-C. Chang, Soft Matter 7, 1207 (2011).
. S Frank, R G Winkler, The Journal of chemical physics. 131234905S. Frank and R. G. Winkler, The Journal of chemical physics 131, 234905 (2009).
. R Netz, The Journal of Physical Chemistry B. 1078208R. Netz, The Journal of Physical Chemistry B 107, 8208 (2003).
. K Radhakrishnan, S P Singh, Macromolecules. 547998K. Radhakrishnan and S. P. Singh, Macromolecules 54, 7998 (2021).
. M C Lagomarsino, I Pagonabarraga, C Lowe, Physical review letters. 94148104M. C. Lagomarsino, I. Pagonabarraga, and C. Lowe, Physical review letters 94, 148104 (2005).
S P Singh, G Gompper, R G Winkler, The Journal of chemical physics. 14884901S. P. Singh, G. Gompper, and R. G. Winkler, The Jour- nal of chemical physics 148, 084901 (2018).
. C Sendner, R Netz, The European Physical Journal E. 3075C. Sendner and R. Netz, The European Physical Journal E 30, 75 (2009).
. X Schlagberger, R R Netz, Physical review letters. 98128301X. Schlagberger and R. R. Netz, Physical review letters 98, 128301 (2007).
. X Schlagberger, R R Netz, Macromolecules. 411861X. Schlagberger and R. R. Netz, Macromolecules 41, 1861 (2008).
. V V Rybenkov, N R Cozzarelli, A V Vologodskii, Proceedings of the National Academy of Sciences. 905307V. V. Rybenkov, N. R. Cozzarelli, and A. V. Vologod- skii, Proceedings of the National Academy of Sciences 90, 5307 (1993).
. X R Bao, H J Lee, S R Quake, Physical review letters. 91265506X. R. Bao, H. J. Lee, and S. R. Quake, Physical review letters 91, 265506 (2003).
. R W Deibler, S Rahmati, E L Zechiedrich, Genes & development. 15748R. W. Deibler, S. Rahmati, and E. L. Zechiedrich, Genes & development 15, 748 (2001).
. Y Arai, R Yasuda, K Akashi, Y Harada, H Miyata, K Kinosita, H Itoh, Nature. 399446Y. Arai, R. Yasuda, K.-i. Akashi, Y. Harada, H. Miyata, K. Kinosita, and H. Itoh, Nature 399, 446 (1999).
. W R Taylor, K Lin, Nature. 42125W. R. Taylor and K. Lin, Nature 421, 25 (2003).
. P Virnau, L A Mirny, M Kardar, PLoS computational biology. 2122P. Virnau, L. A. Mirny, and M. Kardar, PLoS computa- tional biology 2, e122 (2006).
. S Amin, A Khorshid, L Zeng, P Zimny, W Reisner, Nature communications. 91S. Amin, A. Khorshid, L. Zeng, P. Zimny, and W. Reis- ner, Nature communications 9, 1 (2018).
. G Adamo, C Micheletti, Macromolecules. 486337G. D'Adamo and C. Micheletti, Macromolecules 48, 6337 (2015).
. R Matthews, A A Louis, C N Likos, ACS macro letters. 11352R. Matthews, A. A. Louis, and C. N. Likos, ACS macro letters 1, 1352 (2012).
. E Orlandini, M Tesi, S Whittington, Journal of Physics A: Mathematical and General. 38795E. Orlandini, M. Tesi, and S. Whittington, Journal of Physics A: Mathematical and General 38, L795 (2005).
. H Zhu, F Tian, L Sun, S Wang, L Dai, Macromolecules. 541623H. Zhu, F. Tian, L. Sun, S. Wang, and L. Dai, Macro- molecules 54, 1623 (2021).
. C H Nakajima, T Sakaue, Soft Matter. 93140C. H. Nakajima and T. Sakaue, Soft Matter 9, 3140 (2013).
. C Micheletti, E Orlandini, Soft Matter. 810959C. Micheletti and E. Orlandini, Soft Matter 8, 10959 (2012).
. J G Reifenberger, K D Dorfman, H Cao, Analyst. 1404887J. G. Reifenberger, K. D. Dorfman, and H. Cao, Analyst 140, 4887 (2015).
. R , Adv. Chem. Phys. 14089R. Kapral, Adv. Chem. Phys. 140, 89 (2008).
. G Gompper, T Ihle, D M Kroll, R G Winkler, Adv. Polym. Sci. 2211G. Gompper, T. Ihle, D. M. Kroll, and R. G. Winkler, Adv. Polym. Sci. 221, 1 (2009).
. M Ripoll, K Mussawisade, R G Winkler, G Gompper, Europhys. Lett. 68106M. Ripoll, K. Mussawisade, R. G. Winkler, and G. Gompper, Europhys. Lett. 68, 106 (2004).
. A Malevanets, R , J. Chem. Phys. 1108605A. Malevanets and R. Kapral, J. Chem. Phys. 110, 8605 (1999).
. A Lamura, G Gompper, T Ihle, D Kroll, Euro. Phys. Lett. 56319A. Lamura, G. Gompper, T. Ihle, and D. Kroll, Euro. Phys. Lett. 56, 319 (2001).
. I Ali, D Marenduzzo, J Yeomans, Physical review letters. 96208102I. Ali, D. Marenduzzo, and J. Yeomans, Physical review letters 96, 208102 (2006).
. C R Locker, S C Harvey, Multiscale Modeling & Simulation. 51264C. R. Locker and S. C. Harvey, Multiscale Modeling & Simulation 5, 1264 (2006).
. R R Netz, J. Phy. Chem. B. 1078208R. R. Netz, J. Phy. Chem. B 107, 8208 (2003).
Annual review of biophysics and biophysical chemistry. P J Hagerman, 17265P. J. Hagerman, Annual review of biophysics and bio- physical chemistry 17, 265 (1988).
. M Daoud, P De Gennes, Journal de physique. 3885M. Daoud and P. De Gennes, Journal de physique 38, 85 (1977).
. Y Wang, D R Tree, K D Dorfman, Macromolecules. 446594Y. Wang, D. R. Tree, and K. D. Dorfman, Macro- molecules 44, 6594 (2011).
. W Reisner, J N Pedersen, R H Austin, Reports on Progress in Physics. 75106601W. Reisner, J. N. Pedersen, and R. H. Austin, Reports on Progress in Physics 75, 106601 (2012).
. T Odijk, Macromolecules. 161340T. Odijk, Macromolecules 16, 1340 (1983).
. L Coronel, E Orlandini, C Micheletti, Soft matter. 134260L. Coronel, E. Orlandini, and C. Micheletti, Soft matter 13, 4260 (2017).
. Z Benkovaé, P Cifra, Macromolecules. 452597Z. Benkovaé and P. Cifra, Macromolecules 45, 2597 (2012).
. L Tubiana, G Polles, E Orlandini, C Micheletti, The European Physical Journal E. 411L. Tubiana, G. Polles, E. Orlandini, and C. Micheletti, The European Physical Journal E 41, 1 (2018).
. L Tubiana, E Orlandini, C Micheletti, Physical review letters. 107188302L. Tubiana, E. Orlandini, and C. Micheletti, Physical review letters 107, 188302 (2011).
. L Tubiana, E Orlandini, C Micheletti, Progress of Theoretical Physics Supplement. 191192L. Tubiana, E. Orlandini, and C. Micheletti, Progress of Theoretical Physics Supplement 191, 192 (2011).
. A Belmonte, M J Shelley, S T Eldakar, C H Wiggins, Physical review letters. 87114301A. Belmonte, M. J. Shelley, S. T. Eldakar, and C. H. Wiggins, Physical review letters 87, 114301 (2001).
. B W Soh, I R Gengaro, A R Klotz, P S Doyle, Physical Review Research. 133194B. W. Soh, I. R. Gengaro, A. R. Klotz, and P. S. Doyle, Physical Review Research 1, 033194 (2019).
. E Ben-Naim, Z Daya, P Vorobieff, R E Ecke, Physical review letters. 861414E. Ben-Naim, Z. Daya, P. Vorobieff, and R. E. Ecke, Physical review letters 86, 1414 (2001).
. K Kruse, F Jülicher, Current opinion in cell biology. 1720K. Kruse and F. Jülicher, Current opinion in cell biology 17, 20 (2005).
. M Viefhues, R Eichhorn, Electrophoresis. 381483M. Viefhues and R. Eichhorn, Electrophoresis 38, 1483 (2017).
. C L Asbury, A H Diercks, G Van Den, Engh, Electrophoresis. 232658C. L. Asbury, A. H. Diercks, and G. Van Den Engh, Electrophoresis 23, 2658 (2002).
| [] |
[
"Extracting Persistent Clusters in Dynamic Data via Möbius inversion",
"Extracting Persistent Clusters in Dynamic Data via Möbius inversion"
] | [
"Woojin Kim [email protected] \nDepartment of Mathematics\nDuke University\n\n",
"Facundo Mémoli [email protected] \nDepartment of Mathematics and Department of Computer Science and Engineering\nThe Ohio State University\n\n"
] | [
"Department of Mathematics\nDuke University\n",
"Department of Mathematics and Department of Computer Science and Engineering\nThe Ohio State University\n"
] | [] | Identifying and representing clusters in time-varying network data is of particular importance when studying collective behaviors emerging in nature, in mobile device networks or in social networks. Based on combinatorial, categorical, and persistence theoretic viewpoints, we establish a stable functorial pipeline for the summarization of the evolution of clusters in a time-varying network.We first construct a complete summary of the evolution of clusters in a given time-varying network over a set of entities X of which takes the form of a formigram. This formigram can be understood as a certain Reeb graph R which is labeled by subsets of X . By applying Möbius inversion to the formigram in two different manners, we obtain two dual notions of diagram: the maximal group diagram and the persistence clustergram, both of which are in the form of an 'annotated' barcode. The maximal group diagram consists of time intervals annotated by their corresponding maximal groups -a notion due to Buchin et al., implying that we recognize the notion of maximal groups as a special instance of generalized persistence diagram by Patel. On the other hand, the persistence clustergram is mostly obtained by annotating the intervals in the zigzag barcode of the Reeb graph R with certain merging/disbanding events in the given time-varying network.We show that both diagrams are complete invariants of formigrams (or equivalently of trajectory grouping structure by Buchin et al.) and thus contain more information than the Reeb graph R. | null | [
"https://arxiv.org/pdf/1712.04064v5.pdf"
] | 246,863,349 | 1712.04064 | 0c97475cb193bed430bfb9a14698de27cf703162 |
Extracting Persistent Clusters in Dynamic Data via Möbius inversion
February 15, 2022
Woojin Kim [email protected]
Department of Mathematics
Duke University
Facundo Mémoli [email protected]
Department of Mathematics and Department of Computer Science and Engineering
The Ohio State University
Extracting Persistent Clusters in Dynamic Data via Möbius inversion
February 15, 2022
Identifying and representing clusters in time-varying network data is of particular importance when studying collective behaviors emerging in nature, in mobile device networks or in social networks. Based on combinatorial, categorical, and persistence theoretic viewpoints, we establish a stable functorial pipeline for the summarization of the evolution of clusters in a time-varying network.We first construct a complete summary of the evolution of clusters in a given time-varying network over a set of entities X of which takes the form of a formigram. This formigram can be understood as a certain Reeb graph R which is labeled by subsets of X . By applying Möbius inversion to the formigram in two different manners, we obtain two dual notions of diagram: the maximal group diagram and the persistence clustergram, both of which are in the form of an 'annotated' barcode. The maximal group diagram consists of time intervals annotated by their corresponding maximal groups -a notion due to Buchin et al., implying that we recognize the notion of maximal groups as a special instance of generalized persistence diagram by Patel. On the other hand, the persistence clustergram is mostly obtained by annotating the intervals in the zigzag barcode of the Reeb graph R with certain merging/disbanding events in the given time-varying network.We show that both diagrams are complete invariants of formigrams (or equivalently of trajectory grouping structure by Buchin et al.) and thus contain more information than the Reeb graph R.
Figure 1
: Summarization process of a dynamic graph. The evolution of connected components in a dynamic
Introduction
One of the most frequent tasks in data analysis consists of finding clusters in datasets. Since data in the real world are often spatiotemporal (i.e. data encode for both space and time), identifying and analyzing clusters in such data is of critical importance. Spatiotemporal data includes flocking, swarming, or herding behavior of animals [7,43,44,46,57,65,77,82], sensor networks [1,2,31,32,40], swarm robots [73], convoys [49], moving clusters [50], or mobile groups [47,83]. Furthermore, with the development of tracking and mobile devices with positioning capabilities, numerous object trajectory data are being collected, which makes the task of identifying and analyzing clusters in spatiotemporal datasets increasingly more important.
Buchin et al. [15] introduced a formal definition of groups in a given set X of trajectories and introduced a notion of trajectory grouping structure which encodes all groups in X represented as a Reeb graph along the timeline that is decorated by maximal groups.
The first goal of this work is to make clear identification of the trajectory grouping structure (and similar concepts that appeared in applications; e.g. [71,81])) as a mathematical object, which we call formigrams [52]. 1 A formigram is a constructible cosheaf over the real line [30,29] valued in the category of partitions; in plain language, a formigram is a 'zigzag' diagram of partitions. This category can be seen as either a lattice (when an underlying set is specified) or a category whose morphisms are monic (when an underlying set is not specified). This identification allows us to put formigrams into the context of the theories of persistence [17,37], lattices [8], and categories [14,58]. The second goal is to exploit this established identity of formigrams for metrizing the space of formigrams, summarizing formigrams, and smoothing formigrams. In particular, the metrization of the space of formigrams addresses a question formulated by Buchin et al. regarding the quantification of the difference between two collective motions via trajectory grouping structures [15,Section 6]. In order to achieve our second goal, we deploy recent ideas from zigzag or multiparameter persistence [9,11,18,34], constructible cosheaves [29,33], and generalized persistence diagrams [53,67].
Contributions.
In a nutshell, we establish a stable and (mostly) functorial summarization process illustrated in Figures 1 and 2. This entails appropriate identification of categories (Table 1), functors (Table 2), and metrics (Table 3), as well as appropriate usage of generalized rank and Möbius inversion (Table 4). More details follow:
• We adapt recent ideas from generalized persistence diagrams and Möbius inversion [72] in order to define the persistence clustergram of a formigram (cf. Figure 1 (B),(G)). The persistence clustergram is a finer invariant than the Reeb graph of a formigram and thus it is also finer than the zigzag barcode of the Reeb graph (cf. Figure 1 (E),(F)). In fact, the persistence clustergram is a complete invariant of formigrams. See Definition 6.27 and Remark 6.28.
• We show that the maximal groups [15] of a set X of trajectories can be obtained from the formigram θ X associated to X. More specifically, to obtain the maximal groups, we use the Möbius inversion of a suitable notion of rank invariant associated to the formigram. This leads us to the fact that the collection of all maximal groups (which we call the maximal group diagram) is another complete invariant of formigrams. This also implies that the collection of all maximal groups defined by Buchin et al. [15] is not just an invariant of collections of trajectories but it is also an invariant of formigrams. See Definition 6. 16 and Remark 6.20.
• By utilizing the join operation on the lattice of (sub)partitions, we metrize the space of formigrams with a distance function d F I (the interleaving distance between formigrams). d F I dovetails with celebrated metrics in topological data analysis, which makes the summarization pipeline that is illustrated in Figure 1 is entirely stable. See Proposition 4.13, Corollary 4.16, Theorems 4.17 and 6.35.
Several subsidiary contributions follow:
• We define dynamic graphs as constructible cosheaves over R valued in the lattice of all subgraphs of a given complete graph. By doing this, we are able to harness machinery from applied topology which allows us to both quantify the difference between dynamic graphs and induce the invariants of dynamic graphs illustrated in Figure 1. See Definition 3.1.
• We identify a sufficient condition on a dynamic metric space γ X [54] so that its dynamic Rips graph R 1 δ (γ X ), δ ≥ 0 (1-skeleton of Rips complexes), define a dynamic graph as described in the previous item. For example, our result implies that a finite collection of piecewise linear trajectories in Euclidean space induces a dynamic graph via the Rips graph functor. See Proposition 7.5.
• We show that the λ-slack interleaving distance between dynamic metric spaces [54] is a metric. Also we show that different choice of λ > 0 does not change the topology of the metric space of dynamic metric spaces. See Theorem 7. 12 and Proposition A. 15. • We observe that robust grouping structure by Buchin et al. [15] can be formulated as an intrinsic smoothing operation on formigrams. Smoothing operations on formigrams is entirely compatible with the Reeb graph smoothing operation. We also clarify the effect of smoothing operations on the zigzag barcode of an input formigram. See Remark C.1, Propositions C.2 and C.5.
• We clarify the computational complexity of every metric that is introduced in this paper. See Theorem 4.7, Remark 6.37 ((ii)), and Theorems 7. 15 and A.7 (Theorem A.7 appeared in the conference paper by the same authors [52]).
Other related work. In [55], a collection of dynamic graphs that naturally arise from an artificial life program, called Boids [69], were successfully classified by the bottleneck distance on their zigzag barcodes (cf. Figure 1 (F)). An implementation is available in [26].
In [79], van Goethem et al. presented algorithms and data structures that support the interactive analysis of the trajectory grouping structure. In [80,85], Van Kreveld et al. proposed an alternative to the aforementioned definition of groups of [15] together with experimental evaluation related to identifying groups from real or simulated pedestrian data.
Sinhuber and Ouellette carried out statistical analysis on time-varying connectivity graphs in order to characterize swarms of midges [76]. Munch established stability of time-varying persistence diagrams derived from dynamic point clouds in R d [64]. Kim and Mémoli proposed a method to encode spatiotemporal topology of dynamic metric spaces into multiparameter persistence modules [54]. While [54] provides useful tools for characterizing and classifying dynamic metric spaces according to their spatiotemporal homological features [27], no visualizable summary was provided which is one of the novel contributions of the present work.
We remark that, in this paper, only clustering features of dynamic graphs [1,31,40,45] are studied (i.e. the zeroth order homological features), whereas [54] considers arbitrary order homological features of dynamic metric spaces [76,78,86].
R. González-Díaz et al. devised the so-called spatiotemporal barcode as a visual tool to encode the lifespan of connected components on an image sequence over time [42]. Dey and Hou proposed efficient algorithms for computing zigzag persistent homology of dynamic graphs [34]. Kim
Preliminaries
In Sections 2.1 and 2.2 we introduce the posets, categories, and functors that are considered throughout the paper. In Section 2.3 we review the notion of interval decomposable persistence modules. In Section 2.4 we recall both the notion of constructible cosheaf over the real line and a suitable notion of interleaving.
Poset theory elements
Any nonempty set with a partial order determines a poset. Given a poset P, we call any subset Q of P with the partial order obtained by restricting that of P to Q a subposet of P. We introduce posets that are considered in this paper. First, the following posets will be used as domain posets of order-preserving maps or indexing posets of functors. 1. A poset P is said to be a join-semilattice (resp. meet-semilattice) if P contains the least upper bound p ∨ q (resp. greatest lower bound p ∧ q) of any pair of points p, q ∈ P. If P is both join-and meet-semilattice, then P is said to be a lattice. 2. The poset R n with order (a 1 , a 2 , · · · , a n ) ≤ (b 1 , b 2 , · · · , b n ) if and only if a i ≤ b i for all 1 ≤ i ≤ n. 3. Z + and R + will denote the sets of nonnegative integers and nonnegative real numbers, respectively. 4. Let P be a poset. By P op we mean the opposite poset of P, i.e. for any p, q ∈ P, p ≤ q in P op if and only if q ≤ p in P. 5. The poset R op × R where (a 1 , a 2 ) ≤ (b 1 , b 2 ) if and only if a 1 ≥ b 1 and a 2 ≤ b 2 for the usual order ≤ on R. 6. The poset Int consists of nonempty finite closed real intervals ordered by inclusion. Int will be identified with the subposet of R op × R consisting of (a, b) with a ≤ b in R via the bijection [a, b](∈ Int) ↔ (a, b)(∈ R op × R) (cf. Figure 4 (A)).
Lattices of subgraphs and subpartitions.
In the rest of this section, X will stand for some nonempty finite sets. We review lattice structures on the collection of subgraphs of the complete graph on X and on the collection of (sub)partitions of X . Table 2. Table 1.
Graph(X ) SubPart(X ) Part ωset set vec π 0 Def. 2.4 U X L Def. 2.9 (i) A Def. 2.9 (ii) U ω Def. 2.9 (iii) F Def. 2.6
Definition 2.1 (Lattice of subgraphs). By Graph(X ) we denote the lattice of subgraphs of the complete graph on the vertex set X ordered by inclusion. For any G, H ∈ Graph(X ), the union of G and H is the join G ∨ H , i.e. the graph whose vertex set (resp. edge set) is the union of the vertex sets (resp. edge sets) of G and H . The intersection of G and H is the meet G ∧ H .
Definition 2.2 (Lattice of subpartitions)
. We call any partition P of a subset X of X a subpartition of X . In this case we call X the underlying set of P , i.e. X = P . Each element of P is called a block. A partition of the empty set is defined as the empty set. By SubPart(X ), we denote the set of all subpartitions of X . By Part(X ), we denote the subcollection of SubPart(X ) consisting solely of partitions of the entire X .
For example, for X := {x 1 , x 2 , x 3 }, both {{x 1 }, {x 2 }} and {{x 1 , x 2 , x 3 }} belong to SubPart(X ). These subpartitions will henceforth be written simply as {x 1 |x 2 } and {x 1 x 2 x 3 } respectively.
The collection SubPart(X ) forms a lattice. Given P,Q ∈ SubPart(X ), by P ≤ Q we mean "P is finer than or equal to Q", i.e. for all B ∈ P , there exists C ∈ Q such that B ⊂ C . Given any P,Q ∈ SubPart(X ), the join P ∨ Q is the finest common coarsening of P and Q. The meet P ∧Q is the coarsest common refinement of P and Q.
Example 2.3. Let X := {x 1 , x 2 , x 3 , x 4 }. For P 1 = {x 1 |x 2 }, P 2 = {x 2 x 3 }, and P 3 = {x 1 x 4 } in SubPart(X ), we have: 2 i =1 P i = {x 2 }, 3 i =1 P i = , 2 i =1 P i = {x 1 |x 2 x 3 }, and 3 i =1 P i = {x 1 x 4 |x 2 x 3 }.
Remark 2.5 (Posets as categories).
(i) Any poset P will be regarded as a category: Objects are elements in P. For any p, q ∈ P, there exists a unique morphism p → q if and only if p ≤ q. Therefore, p ≤ q in P will denote the unique morphism p → q. In this perspective, any order-preserving map between posets is a functor. Any subposet Q of P is a full subcategory of P.
(ii) Since Graph(X ) is a poset, given any poset P and any functor F : P → Graph(X ) (i.e. order-preserving map), the limit and colimit of F are p∈P F p and p∈P F p , respectively. Similarly, since SubPart(X ) is a poset, given any poset P and any functor F : P → SubPart(X ), the limit and colimit of F are p∈P F p and p∈P F p , respectively.
The category set consists of finite sets with set maps. The category vec consists of finite-dimensional vector spaces over a fixed field F with linear maps. Definition 2.6. The free functor F : set → vec sends any set S to the vector space F (S) which consists of formal linear combinations i a i s i (a i ∈ F, s i ∈ S) of finite terms of elements in S over the field F. Also, given a set map f : S → T , F ( f ) is the linear map from F (S) to F (T ) obtained by linearly extending f .
For two categories C and D, the category C D stands for the category of functors from D to C with objects being functors C → D and arrows being natural transformations. For two functors F,G : C → D, we write F ∼ = G whenever F and G are naturally isomorphic, i.e. there exists a natural transformation τ :
F → G such that τ c : F (c) → G(c) is invertible for each c ∈ Ob(C).
We introduce the category of partitions (without specifying a set to partition) and the category of weighted sets. These categories appear in Figure 1 (B) and (C).
Definition 2.7.
In the category of partitions, denoted by Part, an object is a pair (X , P X ) of a finite set X and a partition P X of X . A morphism f : (X , P X ) → (Y , P Y ) is an injective map f : X → Y which preserves the equivalence relation on X induced by P X , i.e. if x and x belong to the same block in P X , then so do f (x) and f (x ) in P Y .
Note that f induces the map f * : P X → P Y that sends each block B to the unique block C containing the image f (B ). The morphism f is an isomorphism if both f and f * are bijective. For example, let X := {x 1 , x 2 , x 3 }. Whereas P X = {x 1 x 2 |x 3 } and Q X = {x 1 |x 2 x 3 } are non-isomorphic objects in SubPart(X ), the pairs (X , P X ) and (X ,Q X ) are isomorphic in Part.
Next we define the category ωset of weighted sets.
Definition 2.8. In ωset, an object is a pair (X , w X ) of a finite set X and a positive-integer-valued weight map
w X : X → N. A morphism f : (X , w X ) → (Y , w Y ) is a weight-observing set map f : X → Y , i.e. for all y ∈ Y , x∈ f −1 (y) w X (x) ≤ w Y (y).
Notice the following: (1) If there exists a morphism from (X , w X ) to (Y , w Y ), then the total weight of (Y , w Y ) is at least that of (X , w X ), i.e. x∈X w X (x) ≤ y∈Y w Y (y). (2) Two objects (X , w X ) and (Y , w Y ) are isomorphic if and only if there exists a weight-preserving bijection X → Y . (3) If there exists a pair of morphisms in the opposite directions between two objects in ωset, then the two object must be isomorphic.
We introduce the following three functors which are utilized for the summarization procedure depicted as the arrows (B) → (C) → (D) → (E) in Figure 1. See also Table 2. Let X be any nonempty finite set. Definition 2.9 (Three functors).
(i) The unlabeling functor U X L sends each P ∈ SubPart(X ) to the pair ( P, P ). For any pair P ≤ Q in SubPart(X ), the corresponding morphism U X L (P ≤ Q) : ( P, P ) → ( Q,Q) is the inclusion map P → Q.
(ii) The agglomerating functor A sends each (X , P X ) ∈ ob(Part) to the pair (P X , | − |) ∈ ob(ωset) where | − | : P X → N is the size function, i.e. any block in P X is sent to its size (i.e. cardinality). Since any morphism f : (X , P X ) → (Y , P Y ) in Part is an injective set map X → Y , the induced map f * : P X → P Y is a weightobserving map. (iii) The unweighting functor U ω simply forgets the weight function (i.e. the second entry) from each object in ωset.
Interval decomposable persistence modules
In this section we review the notion of interval decomposability of persistence modules. Let P be a poset. Any functor F : P → vec will be called a P-indexed module. This means that each p ∈ P is sent to a finite dimensional vector space F (p) and each p ≤ q in P is sent to a linear map F (p ≤ q) : F (p) → F (q). In particular, for any p ∈ P, p ≤ p is sent to the identity map on F (p). Also, for any p ≤ q ≤ r in P, we have
F (p ≤ r ) = F (q ≤ r ) • F (p ≤ q).
Definition 2.10 (Intervals [11]). Given a poset P, an interval J of P is any subset J ⊂ P such that (i) J is nonempty. (ii) If p, r ∈ J and p ≤ q ≤ r , then q ∈ J . (iii) (connectivity) For any p, q ∈ J , there is a sequence p = p 0 , p 1 , · · · , p l = q of elements of J with p i and
p i +1 comparable for 0 ≤ i ≤ l − 1.
For J an interval in P, the interval module I J : P → vec is the P-indexed module where
I J (p) = F if p ∈ J , 0 otherwise. I J (p ≤ q) = id F if p, q ∈ J , 0 otherwise.
The direct sums of P-indexed modules are defined pointwise at each index t ∈ P. We say that a P-indexed module F is decomposable if F is naturally isomorphic to G 1 G 2 for some non-trivial P-indexed modules G 1 and G 2 . Otherwise, we say that F is indecomposable. Any interval module is indecomposable [11,Proposition 2.2]. A P-indexed module F is interval decomposable if there exists a multiset barc(F ) of intervals in P such that
F ∼ = J ∈barc(F ) I J .
By Azumaya-Krull-Remak-Schmidt [3], the multiset barc(F ) is unique if F is interval decomposable. In this case, we call barc(F ) the barcode of F.
Constructible cosheaves over R, their interleavings, and their smoothing operations
In this section we review the notion of constructible cosheaves over R, their interleavings, and their smoothing operations. Constructible cosheaves over R are fully characterized by "zigzag diagrams over R" [29,33]. Definition 2.11 (Zigzag poset). Let ZZ be the subposet of R op × R given by ZZ := {(k, l ) : k ∈ Z, l ∈ {k, k − 1}} (cf. Figure 3). By Int(ZZ) we denote the poset of all finite intervals in ZZ ordered by inclusion. ([11]). Letting < denote the strict partial order on Z 2 (not on Z op × Z), every interval in ZZ falls into one of the four types below:
Notation 2.12
(b, d ) ZZ := {(i , j ) ∈ ZZ : (b, b) < (i , j ) < (d , d )} for b < d ∈ Z, [b, d ) ZZ := {(i , j ) ∈ ZZ : (b, b) ≤ (i , j ) < (d , d )} for b < d ∈ Z, (b, d ] ZZ := {(i , j ) ∈ ZZ : (b, b) < (i , j ) ≤ (d , d )} for b < d ∈ Z, [b, d ] ZZ := {(i , j ) ∈ ZZ : (b, b) ≤ (i , j ) ≤ (d , d )} for b ≤ d ∈ Z.
See Figure 3 for examples. We let 〈b, d 〉 ZZ denote any of the above sets without specifying its type. Also, by 〈b, d 〉 for b ∈ {−∞} ∪ R and d ∈ R ∪ {∞}, we denote any of the intervals Interleavings between Int-indexed functors [11]. For ε ≥ 0, and
(b, d ), [b, d ), (b, d ] or [b, d ] in R. (−2, −2) (−1, −1) (0, 0) (1, 1) (2, 2) (−1, −2) (0, −1) (1, 0) (2, 1) (−1, 1) ZZ (−2, −2) (−1, −1) (0, 0) (1, 1) (2, 2) (−1, −2) (0, −1) (1, 0) (2, 1) [−1, 1) ZZ (−2, −2) (−1, −1) (0, 0) (1, 1) (2, 2) (−1, −2) (0, −1) (1, 0) (2, 1) (−1, 1] ZZ (−2, −2) (−1, −1) (0, 0) (1, 1) (2, 2) (−1, −2) (0, −1) (1, 0) (2, 1) [−1, 1] ZZI = [b, d ] ∈ Int, let I ε := [b − ε, d + ε] ∈ Int.
Definition 2.13 (Interleaving distance). Let ε ≥ 0 and C be an arbitrary category. Two functors F,G : Int → C are said to be ε-interleaved if there exist collections of morphisms f = ( f I : F (I ) → G(I ε )) I ∈Int and g = (g J : G(J ) → F (J ε )) J ∈Int satisfying the following:
1. For all I , J ∈ Int, g I ε • f I = F (I ≤ I 2ε ) and f J ε • g J = G(J ≤ J 2ε ). 2. For all I ≤ J ∈ Int, G(I ε ≤ J ε ) • f I = f J • F (I ≤ J ) and F (I ε ≤ J ε ) • g I = g J • G(I ≤ J ).
In this case, we call ( f , g ) an ε-interleaving pair. For F,G ∈ Ob(C Int ), the interleaving distance between them is d I (F,G) := inf{ε ≥ 0 : F and G are ε-interleaved}.
If F and G are not ε-interleaved for any ε ≥ 0, then d I (F,G) = +∞ by definition. Constructible cosheaves over R [29,30,33]. We describe constructible cosheaves over R by adapting notation from [11]. In a nutshell, a constructible cosheaf over R is a cosheaf over R [13] which is fully determined by a certain "zigzag diagram" over R.
Given a strictly increasing function c : Z → R such that lim i →±∞ c(i ) = ±∞ and I :
= [b, d ] ∈ Int, we define ZZ[ι c ≤ I ] := {(i , j ) ∈ ZZ : (c(i ), c( j )) ≤ (b, d ) in R op × R}.
The following definition is depicted in Figure 4. Figure 4 (B)). For all I ≤ J in Int, the morphism F (I ) → F (J ) is specified by the initial property of the colimit F (I ). We call im(c) a set of critical points of F . 2 Constructible cosheaves over R are fully characterized by "zigzag diagrams over R" (cf. Figure 4 (C)). Definition 2.15 (Names of cosheaves). We call a constructible cosheaf F : Int → C as given in Table 3, depending on the target category C. Convention 2.16. In the rest of the paper, every functor Int → C will be assumed to be a constructible cosheaf and thus will be often simply called a cosheaf.
About cosheaves Int → C when C is a join-semilattice, vec, or set. Next we provide several remarks and review known results about constructible cosheaves Int → C when C is either a join-semilattice, vec, or set. Definition 2.17 (Cosheaf-inducing map). Let C be a join-semilattice (Item 1 in Section 2.1 and Remark 2.5 (i)). A map f : R → ob(C) is said to be cosheaf-inducing if the following hold: There exists a locally finite set C ⊂ R of 'critical points' such that (1) f is constant between any two consecutive points in C , and (2) for every c ∈ C , f (c) is locally maximal, i.e. there exists r c > 0 with f (c − ε) ≤ f (c) ≥ f (c + ε) for all ε ∈ [0, r c ). Remark 2.18. Let C be a join-semilattice. Then
(i) A constructible cosheaf F : Int → C induces the map f : R → ob(C) defined by t → F ([t , t ])
. Then, f is cosheaf-inducing. Note also that F is recovered from f via the formula F (I ) = { f (t ) : t ∈ I } for I ∈ Int. 3 (ii) Conversely, assume that a map g : R → ob(C) is cosheaf-inducing. Then, G : Int → C defined by I → {G(t ) : t ∈ I } is a constructible cosheaf.
For t ∈ R and ε ≥ 0, let [t ] ε := [t − ε, t + ε].
(iii) For any two constructible cosheaves F,G : Int → C, the interleaving distance between them is:
d I (F,G) = inf ε ≥ 0 : for all I ∈ Int, F (I ) ≤ G(I ε ) and G(I ) ≤ F (I ε ) . = inf ε ≥ 0 : for all t ∈ R, F ([t , t ]) ≤ G([t ] ε ) and G([t , t ]) ≤ F ([t , t ] ε ) .
.
The following proposition follows from the fact that any ZZ-indexed module is interval decomposable [12,39]. barc ZZ (F ) will be regarded as a multiset of real intervals by identifying R :y=x with R via the bijection (t , t ) ↔ t . Bauer, Ge and Wang [4] observed that the 0-th levelset barcode of a Reeb graph encodes all non-trivial persistent homology information of the Reeb graph. A recent theorem on stability of the 0-th levelset barcode of a Reeb graph is the following (see also [4,6,11,33,36] and cf. the arrow (E) → (F) in Figure 1). ([9, 11]). For any two Reeb graphs M , N : Int → set, we have that
Theorem 2.22
d B (barc ZZ (F • M ), barc ZZ (F • N )) ≤ 2 · d R I (M , N ),
where F is the free functor Set → Vec (Definition 2.6) and d B is the bottleneck distance (Definition A.1).
Smoothing of cosheaves.
The following definition is a straightforward adaptation of the notion of Reeb graph smoothing [33,Section 4.4].
Definition 2.23. For any ε ≥ 0, the ε-smoothing of a functor F : Int → C is the functor S ε F : Int → C defined by
I → F (I ε ) and (I ≤ J ) → F (I ε ≤ J ε ).
In Appendix C we study basic properties of this smoothing operation on formigrams.
Remark 2.24.
For ε 1 , ε 2 ≥ 0 and I ∈ Int, since I ε 1 +ε 2 = (I ε 1 ) ε 2 , we have S ε 1 +ε 2 F = S ε 1 S ε 2 F . This is a straightforward generalization of [33,Proposition 4.13]. ). If a given functor F : Int → C is constructible, then S ε F is also constructible. In particular, if C is a set of critical points of F , then one set of critical points of S ε F is
(C − ε) ∪ (C + ε) where C ± ε := {c ± ε ∈ R : c ∈ C }.
From dynamic graphs to formigrams, Reeb graphs, and zigzag barcodes
In this section we define objects that appear in Figure 1 (A)-(F). In Sections 3.1 and 3.2 we provide rigorous definitions of dynamic graphs and formigrams respectively. Formigrams are used to encode the evolution of connected components of dynamic graphs. In Section 3.3 we define several invariants of formigrams. Throughout this section X and Y are nonempty finite sets.
Dynamic graphs (DGs)
In this section we define the notion of dynamic graphs (cf. Figure 1 (A)) and also a suitable notion of isomorphism.
Definition 3.1 (Dynamic graphs). A dynamic graph (DG) over X is any cosheaf-inducing map G X : R → Graph(X ) (cf. Definition 2.17) which, in addition, also satisfies the condition that every x ∈ X admits a closed interval lifespan I x ⊂ R, i.e. x belongs to the vertex set of G X (t ) if and only if t ∈ I x . When all x ∈ X have the same lifespan, the dynamic graph G X is said to be saturated (e.g. the DG illustrated in Figure 1 (A)).
Remark 2.18 (ii) allows us to view any DG as a constructible cosheaf over R valued in the lattice Graph(X ). This will in turn allow us to quantify the difference between DGs using the interleaving distance (Remark 2.18 (iii)). 4 We now specify a suitable notion of isomorphism in the class of DGs. Definition 3.3 (Isomorphism for DGs). Two DGs G X and G Y over X and Y respectively are isomorphic if there exists a bijection ϕ : X → Y such that for all t ∈ R, the map ϕ serves as a graph isomorphism between G X (t ) and G Y (t ). Namely, for all t ∈ R, ϕ restricted to the vertex set of G X (t ) is a graph isomorphism from G X (t ) to G Y (t ).
Example 3.4.
Let G X and G Y be the two DGs in Example 3.2. Although the two graphs G X (t ) and G Y (t ) are isomorphic for each t ∈ R, it is not difficult to see that G X and G Y are not isomorphic as DGs. It is important to point out that d dynG I and also all the invariants of DGs which we consider in this paper (see Figures 1 and 2) are able to discriminate these two DGs.
Formigrams
We introduce a notion of formigram to encode the evolution of connected components in dynamic graphs.
Definition 3.5.
A formigram 5 over a finite set X is any cosheaf-inducing map θ X : R → SubPart(X ) (cf. Definition 2.17) which, in addition, also satisfies that every x ∈ X admits a closed interval lifespan I x ⊂ R, i.e. x belongs to θ X (t ) if and only if t ∈ I x . By the support of θ X , denoted by supp(θ X ), we mean the union x∈X I x of all lifespans of points in X . Remark 2.18 (ii) allows us to view any formigram as a constructible cosheaf over R valued in the lattice SubPart(X ). We will interchangeably write both θ X : R → SubPart(X ) and θ X : Int → SubPart(X ) as needed.
Recall the unlabeling functor U X L : SubPart(X ) → Part from Definition 2.9 (i). 4 In [76], this type of DGs was utilized for topological characterization of insect swarms. A sensor network [31,32] is another example of such DGs arising from viewing each sensor as a point in the dynamic metric space of sensors. 5 The name formigram is a combination of the words formicarium and diagram. Remark 3.6 (Essentially unique labeling). Given any unlabeled formigram θ : Int → Part (cf. Definition 2.15), one can always find a formigram θ X such that U X L •θ X . For example, in Figure 1 (C), by labeling the eight 'orbits' using any set X of eight elements, we obtain such a formigram θ X . Assuming that the colimit of θ is isomorphic to a pair (A, P A ) ∈ ob(Part), the size of A equals the size of the labeling set.
R 1 (γ X (t )) and R 1 (γ Y (t )) for t ∈ [−5, 5].
Remark 3.7.
Formigrams generalize the classical notion of dendrogram, a 1-parameter nested family of partitions [19,48]. Namely, any formigram θ X : R → SubPart(X ) is called a dendrogram if the following properties hold: Figure 6 for an illustrative example. Definition 3.8. Given a formigram θ X : R → SubPart(X ), we call θ X saturated if all x ∈ X have the same lifespan (e.g. any dendrogram or the formigram that is depicted in Figure 1 (B)). Definition 3.9. Two formigrams θ X and θ Y over X and Y , respectively, are isomorphic if there exists a bijection ϕ :
(1) every x ∈ X has the lifespan [0, ∞), (2) if t 1 ≤ t 2 , then θ X (t 1 ) ≤ θ X (t 2 ), (3) there exists T > 0 such that θ X (t ) = {X } for t ≥ T . SeeX → Y such that for all t ∈ R, θ X (t ) = ϕ * (θ Y (t )) := {ϕ −1 (B ) ⊂ X : B ∈ θ Y (t )}.
We remark that θ X and θ Y are isomorphic if and only if their unlabeled counterparts U X
L • θ X , U X L • θ Y : Int → Part are naturally isomorphic.
Functorial summarization of dynamic graphs and formigrams
We can summarize a dynamic graph at different levels by post-composing the functors from Table 2. First of all, a formigram serves as a summary of the evolution of connected components in a DG via the path components functor π 0 : Graph(X ) → SubPart(X ) (cf. Definition 2.4): Figure 1 (A) and (B)).
Definition 3.10. Given a DG G X : R → Graph(X ), the formigram of G X is defined as π 0 • G X : R → SubPart(X ) (cf.
Remark 3.11.
We remark that the lifespan of any x ∈ X in G X is inherited by the formigram π 0 •G X . In particular, the formigram of a saturated DG is itself saturated (Definitions 3.1 and 3.8). Definition 3.12 (Summaries of formigrams). Let θ X : Int → SubPart(X ) be a formigram (i) The unlabeled formigram of θ X is the cosheaf obtained by post-composing the unlabeling functor U X L : SubPart(X ) → Part to θ X . (ii) The (underlying) weighted Reeb graph of θ X , denoted by ω(θ X ), is the cosheaf obtained by post-composing the agglomeration functor A : Part → ωset to the unlabeled formigram of θ X . (iii) The (underlying) Reeb graph of θ X , denoted by Reeb(θ X ), is the cosheaf obtained by post-composing the unweighting functor U ω : ωset → set to the weighted Reeb graph of θ X .
The above three items can sometimes be difficult to visualize for example due to possible non-planarity (cf. Example 3.14 below). This motivates us to further summarize formigrams into more easily visualizable invariants. One such invariant is the (zigzag) barcode [18]:
(iv) The (zigzag) barcode of θ X is the zigzag barcode of the cosheaf Int → vec obtained by post-composing the free functor F : set → vec to the underlying Reeb graph of θ X . In other words, barc ZZ (θ X ) is the 0-th levelset barcode of the underlying Reeb graph of θ X .
We now describe how to visualize a formigram and its underlying weighted/unweighted Reeb graphs. The results of the visualization are topological graphs over the real line with or without labels; see Figure 7 for an example. Let us fix a formigram θ X : R → SubPart(X ) with a set C ⊂ R of critical points (cf. Remark 2.18 (ii)).
Step 1. for each c ∈ C , we specify the vertex set V c := θ X (c), which lie over c ∈ R. [10,15]. (B) depicts the Reeb graph of θ X (with labels).
= {x 1 , x 2 , x 3 }. θ X (t ) is {x 1 x 2 x 3 } for t ∈ [0, 2] ∪ [17, 20], it is {x 1 x 2 |x 3 } for t ∈ (2, 6], it is {x 1 |x 2 |x 3 } for t ∈ (6, 10) ∪ (15, 17), and it is {x 1 |x 2 x 3 } for t ∈
Step 2. for each pair of consecutive critical points c 1 < c 2 in C , we specify the edge set E c 1 ,c 2 := θ X (t ) for any t ∈ (c 1 , c 2 ), which lie over the interval (c 1 , c 2 ),
Step 3. for each pair of consecutive critical points c 1 < c 2 in C , we define left and right attaching maps l c 1 ,c 2 : E c 1 ,c 2 → V c 1 and r c 1 ,c 2 : E c 1 ,c 2 → V c 2 by sending each block B ∈ E c 1 ,c 2 to the blocks in V c 1 and V c 2 which contain B , respectively.
As a result, we obtain a topological graph along the real line such as the one depicted in Figure 7 (B). By replacing the labeling of the elements of vertex sets and edge sets by their cardinalities, we obtain the weighted Reeb graph of θ X . Alternatively, weights can be represented by the thickness of nodes and edges as in Figure 1 (D). By forgetting those weights, we obtain the Reeb graph of θ X .
The following example shows that two different non-isomorphic formigrams (Definition 3.9) can have the same underlying weighted/unweighted Reeb graph. Example 3.13. For the sets X := {x 1 , x 2 , x 3 } and Y := {y 1 , y 2 , y 3 }, we consider the following two formigrams θ X and θ Y over X and Y respectively:
θ X (t ) := {x 1 x 2 |x 3 }, t ∈ (−3, −1) ∪ (1, 3) {x 1 x 2 x 3 }, otherwise, θ Y (t ) := {y 1 y 2 |y 3 }, t ∈ (−3, −1) {y 1 |y 2 y 3 }, t ∈ (1, 3) {y 1 y 2 y 3 }, otherwise.
It is clear that θ X and θ Y are not isomorphic (cf. Definition 3.9) whereas their weighted Reeb graphs are both isomorphic to the weighted Reeb graph depicted in Figure 8 (A). This also implies that their unweighted Reeb graphs are isomorphic.
Reeb graphs of formigrams are not always planar, which can make the visualization of a formigram difficult [81]:
Example 3.14. Consider the formigram θ X over the set X = {x i } 3 i =1 ∪ {y i } 3 i =1 ∪ {z i } 3 i =1 given by: θ X (t ) := x 1 x 2 x 3 |y 1 y 2 y 3 |z 1 z 2 z 3 , t ∈ (−∞, 1]
x 1 |x 2 |x 3 |y 1 |y 2 |y 3 |z 1 |z 2 |x 3 , t ∈ (1, 2)
x 1 y 1 z 1 |x 2 y 2 z 2 |x 3 y 3 z 3 , t ∈ [2, ∞).
See Figure 8 (B) for the Reeb graph of θ X : this graph has the complete bipartite graph K 3,3 as a minor, which implies that it is not planar by Kuratowski's theorem [10].
Interleavings between dynamic graphs and between formigrams
Given any nonempty finite set X , recall that Graph(X ) and SubPart(X ) are lattices. Therefore, we readily have the interleaving distance between two DGs over the same underlying set X or between two formigrams over the same underlying set X (cf. Remark 2.18 (iii)). However, we often wish to quantify the difference between two given DGs (or between two formigrams) over possibly different underlying sets. For achieving this, we blend ideas related to the Gromov-Hausdorff distance (Definition A.2) with the interleaving distance. This type of idea has already appeared in the literature e.g. [20,54,61,70].
Tripods and their compositions. We recall the notion of tripod from [61] as a preliminary to blending the Gromov-Hausdorff distance with the interleaving distance.
Definition 4.1 (Tripods)
. Let X and Y be any two sets. A tripod R between X and Y is a pair of surjections from another set Z to X and Y , respectively. Namely, R can be expressed as a diagram R : X For any sets X , Y and W , consider any two tripods
ϕ X −−− − Z ϕ Y − −−− Y .R 1 : X ϕ X −−− − Z 1 ϕ Y − −−− Y and R 2 : Y ψ Y −−− − Z 2 ψ W − −−− W . Consider the set Z := (z 1 , z 2 ) ∈ Z 1 × Z 2 : ϕ Y (z 1 ) = ψ Y (z 2 )
and let π 1 : Z → Z 1 and π 2 : Z → Z 2 be the canonical projections to the first and the second coordinate, respectively. We define the composite tripod R 2 • R 1 as follows:
R 2 • R 1 : X ω X −−− − Z ω W − −−− W, where ω X := ϕ X • π 1 , ω W := ψ W • π 2 . (1) Z Z 1 Z 2 X Y W π 1 π 2 ϕ X ϕ Y ψ Y ψ W Notation 4.2. Given a tripod R : X ϕ X −−− − Z ϕ Y − −−− Y , for
x ∈ X and y ∈ Y , we write (x, y) ∈ R whenever there exists z ∈ Z such that ϕ X (z) = x and ϕ Y (z) = y.
Interleaving distance between dynamic graphs
In this section we introduce the interleaving distance between DGs. Let G X = (X , E X ) be any graph and let Z be any set. For any map ϕ : Z → X , the pullback G Z := ϕ * G X of G X via ϕ is the graph on the vertex set Z with the edge set
E Z = {z, z } : ϕ(z), ϕ(z ) ∈ E X . Let G X = (V X (·), E X (·)) be a DG over X . The pullback G Z := ϕ * G X of G X via ϕ is a DG over Z defined as follows: for all t ∈ R, G Z (t ) is the graph on the vertex set V Z (t ) = ϕ −1 (V X (t )) with the edge set E Z (t ) = {z, z } : ϕ(z), ϕ(z ) ∈ E X (t ) .
Let G X and G Y be two DGs. Given a tripod R :
X ϕ X −−− − Z ϕ Y − −−− Y , we write G X ≤ R G Y if for all t ∈ R ϕ * X G X (t ) ≤ ϕ * Y G Y (t ) in the poset Graph(Z ).
Remark 4.3.
Let G X , G Y , and G W be any three DGs. Let R 1 be any tripod between X and Y and let R 2 be any tripod between Y and W . If
G X ≤ R 1 G Y and G Y ≤ R 2 G W , then it is easy to check that G X ≤ R 2 •R 1 G W as well.
Recall that a DG G X can be viewed as a constructible cosheaf Int → Graph(X ) (Definition 3.1 and Remark 2.18 (ii)). This cosheaf sends each I ∈ Int to I G X := {G X (t ) : t ∈ I } = {G X (t ) : t ∈ I }. Therefore, we can utilize the interleaving distance between cosheaves (Definition 2.13) and tripods for quantifying the difference between two DGs over possibly different underlying sets.
Definition 4.4. Let G X : Int → Graph(X ) and G Y : Int → Graph(Y ) be two DGs. A tripod R : X ϕ X −−− − Z ϕ Y − −−− Y is called an ε-tripod between G X and G Y if d Graph(Z ) I (ϕ * X G X , ϕ * Y G Y ) ≤ ε. The interleaving distance between DGs G X and G Y is defined as d dynG I (G X , G Y ) = min{ε ≥ 0 : there exists an ε-tripod between G X and G Y }.
If there is no ε-tripod between G X and G Y for any ε ≥ 0, then we declare d We need the following lemma for proving Theorem 4.5.
dynG I (G X , G Y ) = +∞.
Lemma 4.6. Let ϕ : Z X be a surjective map. Then, for any DG G X and any I ∈ Int, ϕ * (
I G X ) = I ϕ * G X . Proof. Fix z, z ∈ Z (it is possible that z = z ). Note that I G X is the union ∪ t ∈I G X (t ). We have that {z, z } ∈ ϕ * ( I G X ) iff {ϕ(z), ϕ(z )} ∈ I G X iff ∃t ∈ I , {ϕ(z), ϕ(z )} ∈ G X (t ) iff ∃t ∈ I , {z, z } ∈ ϕ * G X (t ) iff {z, z } ∈ I ϕ * G X .
Proof of Theorem 4.5. Reflexivity and symmetry of d dynG I are clear and thus we only show the triangle inequality: Let X , Y and W be some finite sets and let G X , G Y and G W be DGs over the three sets respectively. We wish to prove that d
dynG I (G X , G W ) ≤ d dynG I (G X , G Y ) + d dynG I (G Y , G W ). Let 0 < ε 1 , ε 2 < ∞, and suppose that there are an ε 1 -tripod R 1 : X ϕ X −−− − Z 1 ϕ Y − −−− Y between G X and G Y and an ε 2 -tripod R 2 : Y ψ Y −−− − Z 2 ψ W − −−− W between G Y and G W . It suffices to prove that R 2 • R 1 is an (ε 1 + ε 2 )-tripod between G X and G W . Let I ∈ Int. Since R 1 is an ε 1 -tripod between G X and G Y , we have I ϕ * X G X ≤ I ε 1 ϕ * Y G Y . Since R 2 is an ε 2 -tripod between G Y and G W , we have I ε 1 ψ * Y G Y ≤ I ε 1 +ε 2 ψ * W G W . Therefore, by Lemma 4.6, I π * 1 ϕ * X G X ≤ I ε 1 π * 1 ϕ * Y G Y = I ε 1 π * 2 ψ * Y G Y ≤ I ε 1 +ε 2 π * 2 ψ * W G W .
By symmetry we also have I π * 2 ψ * W G W ≤ I ε 1 +ε 2 π * 1 ϕ * X G X . Since I ∈ Int is arbitrary, we have shown that R 2 • R 1 is an (ε 1 + ε 2 )-tripod between G X and G W , as desired.
Given a DG G X : R → Graph(X ), let crit(G X ) denote the set of points of discontinuity, i.e. critical points (cf. Definition 2.17). We prove Theorem 4.7 in Appendix A.2 by showing that certain NP-hard instances of the Gromov-Hausdorff distance between finite metric spaces can be reduced to the computation of the interleaving distance d dynG I between DGs.
Remark 4.8 (When is
d dynG I ≤ ε). In Definition 4.4, the condition d Graph(Z ) I (ϕ * X G X , ϕ * Y G Y ) ≤ ε with respect to the tripod R : X ϕ X −−− − Z ϕ Y − −−− Y is equivalent to the following: For (x, y), (x , y ) ∈ R, (i) if x ∈ V X (t ), then there exists s ∈ [t ] ε := [t − ε, t + ε] such that y ∈ V Y (s). (ii) if {x, x } ∈ E X (t ), then there exists s ∈ [t ] ε such that {y, y } ∈ E Y (s).
Furthermore, the two statements obtained by exchanging the roles of X and Y in the above two items also hold.
A sufficient condition for a pair of DGs to be ε-interleaved is described in the following example. Example 4.9. Fix ε ≥ 0. Let G X and G Y be two DGs such that for every x, x ∈ X (resp. every y, y ∈ Y ), for any time interval I of length ε, there exists t ∈ I such that the edge {x, x } (resp. {y, y }) is present at G X (t ) (resp. G Y (t )). Then, d dynG I (G X , G Y ) ≤ ε. Indeed, by invoking Remark 2.18 (iii), it can be checked that any tripod R between X and Y is an ε-tripod between G X and G Y .
Interleaving distance between formigrams
In this section we introduce the interleaving distance between formigrams. This metric quantifies the structural difference between two grouping/disbanding behaviors over time. We also show that this metric dovetails with other celebrated metrics, which makes the summarization pipeline that is illustrated in Figure 1 is entirely stable; see Remark 4.12, Proposition 4.13, Corollary 4.16, and Theorem 4.17.
Let X and Z be any two sets and let P X ∈ SubPart(X ). For any map ϕ : Z → X , the pullback P Z := ϕ * P X of P X via ϕ is the subpartition of Z defined as P Z = {ϕ −1 (B ) : B ∈ P X }. Let θ X be a formigram over X and assume that ϕ is surjective. Then the pullback of θ X via ϕ is the formigram
θ Z := ϕ * θ X over Z defined as θ Z (t ) = ϕ * θ X (t ) for all t ∈ R.
Definition 4.10 (Comparison of formigrams via pullbacks). Consider a tripod R :
X ϕ X −−− − Z ϕ Y − −−− Y . (i) Let P X ∈ SubPart(X ) and P Y ∈ SubPart(Y ). we write P X ≤ R P Y if ϕ * X P X ≤ ϕ * Y P Y in the poset SubPart(Z ). (ii) Let θ X , θ Y be any two formigrams. we write θ X ≤ R θ Y if for all t ∈ R, ϕ * X θ X (t ) ≤ ϕ * Y θ Y (t ) in SubPart(Z )
. Let us recall that a formigram θ X can be viewed as a constructible cosheaf Int → SubPart(X ) (Remark 2.18 (ii)); this cosheaf sends each I ∈ Int to I θ X := {θ X (t ) : t ∈ I }. We now utilize both the interleaving distance between cosheaves (Definition 2.13) and the notion of tripod for quantifying the degree of difference between two formigrams over possibly different underlying sets.
Definition 4.11. Let θ X and θ Y be two formigrams. A tripod R : X ϕ X −−− − Z ϕ Y − −−− Y is called an ε-tripod between θ X and θ Y if d SubPart(Z ) I (ϕ * X θ X , ϕ * Y θ Y ) ≤ ε. The interleaving distance between formigrams θ X and θ Y is defined as d F I (θ X , θ Y ) = min{ε ≥ 0 : there exists an ε-tripod between θ X and θ Y }. If there is no ε-tripod between θ X and θ Y for any ε ≥ 0, then we declare d F I (θ X , θ Y ) = +∞.(θ, θ ) := d F I (θ X , θ Y ) (2) where θ = U X L • θ X and θ = U Y L • θ Y .
The choices of the labeling sets X and Y do not affect the RHS of equation (2) and thus d F I (θ, θ ) is well-defined.
(ii) d F I between dendrograms agrees with (twice) the Gromov-Hausdorff distance between their canonically associated ultrametrics (cf. Proposition A.6). d F I is more discriminative than the interleaving distance for Reeb graphs (cf. Remark 2.21). Proposition 4.13. For any two formigrams θ X and θ Y , we have
d R I (Reeb(θ X ), Reeb(θ Y )) ≤ d F I (θ X , θ Y ).
In the appendix, we define a distance d ωR I between weighted Reeb graphs which mediates between d F I and d R I (cf. Theorem B.3 and Remark 6.38). Example 4.14. Since θ X and θ Y in Example 3.13 have the same (un)weighted Reeb graph, they are not discriminated by the interleaving distance between (un)weighted Reeb graphs. However, d F I (θ X , θ Y ) = 1. Indeed, any tripod R between X and Y fails to be an ε-tripod between θ X and θ Y for ε < 1 and the tripod y 1 ), (x 2 , y 2 ), (x 2 , y 3 )} ⊂ X × Y and let ϕ X and ϕ Y be the canonical projections to the first coordinate and the second coordinate, respectively. Let us recall from Definition 2.2 that the underlying set of P ∈ SubPart(X ) is defined as P . When x ∈ X belongs to P , we denote the block containing x by [x]. The following remark describes an if-and-only-if condition for a tripod to be an ε-tripod between formigrams.
R : X ϕ X −−− − Z ϕ Y − −−− Y which is defined as follows is a 1-tripod: Let Z = {(x 1 ,
Remark 4.15.
Let θ X , θ Y be any two formigrams and consider a tripod R :
X ϕ X −−− − Z ϕ Y − −−− Y . Then the condition d SubPart(Z ) I (ϕ * X θ X , ϕ * Y θ Y )
≤ ε holds if and only if the following hold: for any I ∈ Int, (i) If (x, y) ∈ R, x belongs to the underlying set of I θ X then y belongs to the underlying set of I ε θ Y .
(ii) If (x, y), (x , y ) ∈ R, whenever [x] = [x ] in I θ X , it holds that [y] = [y ] in I ε θ Y .
Also, the two statements obtained by exchanging the roles of X and Y in the above two items hold.
Proof of Proposition 4.13. Let R : X ϕ X −−− − Z ϕ Y − −−− Y bed B (barc ZZ (θ X ), barc ZZ (θ Y )) ≤ 2 · d F I (θ X , θ Y ).
Proof. The claim directly follows from Theorem 2.22 and Proposition 4.13.
Summarizing DGs into formigrams via the path components functor π 0 (Definition 3.10) is stable:
Theorem 4.17.
Let θ X and θ Y be the formigrams of two DGs G X and G Y respectively. Then,
d F I (θ X , θ Y ) ≤ d dynG I (G X , G Y ).
This inequality is tight (cf. Remark 6.37 (i)).
Proof. Let ε ≥ 0 and let R :
X ϕ X −−− − Z ϕ Y − −−− Y be an ε-tripod between G X and G Y .
We prove that R is also an ε-tripod between the formigrams θ X and θ Y . By symmetry, it suffices to show that I θ X ≤ R I ε θ Y . Let z ∈ Z and let x := ϕ X (z), and y := ϕ Y (z). Fix t ∈ R. Suppose that x is in the underlying set of I θ X . Since
I G X ≤ R I ε G Y , we have y ∈ ∪ s∈I ε V Y (s)
, which is the underlying set of I ε θ Y . Pick another z ∈ Z and let x := ϕ X (z ) and y := ϕ Y (z ). Assume that x, x belong to the same block of θ X (t ), meaning that there is a sequence x = x 0 , x 1 , . . . ,
x n = x in X such that {x i , x i +1 } ∈ E X (t ) for 0 ≤ i ≤ n − 1. For each 1 ≤ i ≤ n − 1, pick y i ⊂ Y such that (x i , y i ) ∈ R.
Since R is an ε-tripod between G X and G Y , we have {y i , y i +1 } ∈ s∈I ε E Y (s) (Remark 4.8). Then, y, y belong to the same connected component of the graph I ε G Y and in turn, by Lemma 4.6, the same block of I ε θ Y .
Categorical aspects of SubPart(X ), Part and set
In this section we establish a few categorical results that will be useful in later sections: We show that (1) the unlabeling functor preserves limits and colimits of connected diagrams (cf. Proposition 5.1) and that (2) the composition of the three functors in Definition 2.9 preserves colimits of connected diagrams (cf. Proposition 5.4). Lastly, we compute coimages in the category Part (cf. Proposition 5.8).
(Co)limit preserving properties of unlabeling and collapsing functors
Let X be a nonempty finite set. The unlabeling functor U X L : SubPart(X ) → Part (Definition 2.9 (i)) preserves limits and colimits of connected diagrams in SubPart(X ): Proposition 5.1. Let P be a connected poset. For any θ X :
P → SubPart(X ), we have U X L lim ← − − θ X ∼ = lim ← − − U X L (θ X ) and U X L lim − − → θ X ∼ = lim − − → U X L (θ X ).
The proof is rather straightforward and hence we omit it.
Definition 5.2. Let P,Q ∈ SubPart(X ) such that P ≤ Q. The canonical map P → Q is the unique map which sends each block B ∈ P to the unique block C ∈ Q such that B ⊂ C . In other words, the canonical map is the image of P ≤ Q via the three functors in Definition 2.9.
Definition 5.3. The collapsing functor C : SubPart(X ) → set is defined as the composition of the three functors in Definition 2.9, i.e. U ω • A • U X L . Namely, C sends each P ∈ SubPart(X ) to P ∈ ob(set) and each morphism P ≤ Q in SubPart(X ) to the canonical map P → Q.
The collapsing functor preserves the colimit of any diagram in SubPart(X ) indexed by a connected poset (Definition 2.10 (iii)). For example, let X := {x 1 , x 2 , x 3 , x 4 } and consider the diagram θ X :
{1 ≤ 2 ≥ 3 ≤ 4} → SubPart(X ) defined as {x 1 |x 2 |x 3 |x 4 } ≤ {x 1 x 2 |x 3 x 4 } ≥ {x 1 |x 2 |x 3 x 4 } ≤ {x 1 |x 2 x 3 x 4 }. Then, lim − − → θ X = {x 1 x 2 x 3 x 4 } ∈ SubPart(X ) and C lim − − → θ X = {{x 1 , x 2 , x 3 , x 4 }} ∈ ob(set), which is the colimit of the set diagram {{x 1 }, {x 2 }, {x 3 }, {x 4 }} → {{x 1 , x 2 }, {x 3 , x 4 }} ← {{x 1 }, {x 2 }, {x 3 , x 4 }} → {{x 1 }, {x 2 , x 3 , x 4 }}
where every map in the cocone is a canonical map. We omit the proof of the following proposition.
Proposition 5.4. Let P a connected poset. For any θ X : P → SubPart(X ), we have C lim
− − → θ X ∼ = lim − − → C (θ X ).
We remark that the collapsing functor C does not preserve limits (which is actually a consequence that the agglomeration functor A does not preserve limits). For example, let Y = {y 1 , y 2 } and consider the diagram
θ Y : {1 ≤ 2 ≥ 3} → SubPart
(Co)images in the category of partitions
The goal of this section is to compute coimages in Part (cf. Proposition 5.8).
Images and coimages in category theory [62]. A morphism f : a → b is said to be a monomorphism (mono in short) if f is left-cancellative: for any morphisms k 1 ,
k 2 : c → a, if f • k 1 = f • k 2 , then k 1 = k 2 .
Such f is written as f : a → b and a is called a subobject of b. On the other hand, a right-cancellative morphism g : a → b is said to be an epimorphism (epi in short), written as g : a b, and b is called a quotient object of a. The image of a morphism f : X → Y is defined as the smallest subobject of Y which f factors through. The object I will be sometimes denoted by im( f ).
The coimage of a morphism is the dual notion of the image of a morphism. In many categories (such as the category of sets, the category of groups, the category of rings, or the category of vector spaces, and etc.), the coimage is canonically isomorphic to the image, often called the first isomorphism theorem. However, in the category Part, those two are turned out to be different in general. We remark that the bijectivity of f : X → Y does not imply that f is an isomorphism in between (X , P X ) and (Y , P Y ).
Definition 5.6 (Coimages). Given a morphism
f : X → Y , a coimage of f (if it exists) is an epi c : X → C such that there is a morphism f c : C → Y with f = f c • c, for any epi z : X → Z for which there is a map f z : Z → Y with f = f z • z, there is a unique map u : Z → C such that c = u • z.
Proof. For the forward direction, we prove the contrapositive. Suppose that f is not surjective, i.e. there exists y ∈ Y to which no x ∈ X is mapped via f . Let Z = {z 1 , z 2 }. Let k 1 , k 2 : Y → Z be any pair of maps such that they differ only at y, i.e. k 1 = k 2 on Y \ {y} and k 1 (y) = k 2 (y). Then both k 1 and k 2 are morphisms from (Y , P Y ) to (Z , {z 1 z 2 }). Although
k 1 • f = k 2 • f , we do not have that k 1 = k 2 . Hence, f is not an epi.
We prove the backward direction. Consider any two morphisms k 1 ,
k 2 : (Y , P Y ) → (Z , P Z ) such that k 1 • = k 2 • f . Let y ∈ Y . Since f is surjective, there exists x ∈ X such that f (x) = y. Hence, k 1 • f (x) = k 2 • f (x) implies k 1 (y) = k 2 (y).
Since y was arbitrarily chosen in Y , we have that k 1 = k 2 .
Consider any morphism f : (X ,
P X ) → (Y , P Y ) in Part. Let f −1 (P Y ) := f −1 (B ) : B ∈ P Y
which is a partition of X . Notice that the identity map id X on X is a morphism from (X , P X ) to (X , f −1 (P Y )). Now we will see that, in the category Part, the image is not isomorphic to the coimage in general. (i) f itself is an image of f . (ii) The morphism id X : (X , P X ) → (X , f −1 (P Y )), the identity set map on X , is a coimage of f .
Proof. Item (i) directly follows from the fact that f is a mono. We prove Item (ii). Since id X : X → X is bijective, by Proposition 5.7, id X is an epi from (X , P X ) to (X , f −1 (P Y )). Note also that f is a morphism from (X , f −1 (P Y )) to (Y , P Y ) with the obvious identity f • id X = f . Assume that there exists a pair of an epi z : (X , P X ) (Z , P Z ) and a morphism f z : (Z , P Z ) → (Y , P Y ) such that f = f z • z. Since z is an epi, the set map z : X → Z must be bijective. The proof ends by observing that the inverse z −1 : Z → X is the unique morphism u from (Z , P Z ) to (X , f −1 (P Y )) such that id X = u • z.
Example 5.9. Consider i : ({x 1 , x 2 }, {x 1 |x 2 }) → ({x 1 , x 2 , x 3 }, {x 1 x 2 x 3 }) where i is the canonical inclusion {x 1 , x 2 } → {x 1 , x 2 , x 3 }. The coimage of i is the morphism id {x 1 ,x 2 } : ({x 1 , x 2 }, {x 1 |x 2 }) → ({x 1 , x 2 }, {x 1 x 2 }).
Summarizing formigrams via Möbius inversion
In this section we describe two novel summaries of formigrams: maximal group diagrams (cf. Figure 2 (C)) and persistence clustergrams (cf. Figure 1 (G)); Definitions 6.16 and 6.27.
In Section 6.1 we review the Möbius inversion formula for a function from a locally finite poset. In Section 6.2, we introduce a notion of silhouette 6 for a group-valued map from a poset; this notion allows us to have the summarization process that is illustrated as the arrow from (G) to (H) in Figure 1. In Section 6.3, we show that the celebrated notion of maximal groups by Buchin et al. [15] is an instance of the generalized persistence diagrams considered in [53,67]. Namely, we prove that maximal groups can be obtained by computing the Möbius inversion of a suitably defined rank function. In Section 6.4, we introduce persistence clustergrams, which can be regarded as being "dual" to maximal group diagrams. In Section 6.5 and Section 6.6, we consider the silhouette of a persistence clustergram and establish its stability. Convention 6.1. In Section 6 we only consider formigrams with finite support whose critical points are contained in Z (cf. Definitions 2.17 and 3.5). We will not distinguish between the real interval 〈a, b〉 and 〈a, b〉 ZZ ∈ Int(ZZ) for any a ≤ b in Z (cf. Notation 2.12).
Möbius inversion of a function on a poset
We review the Möbius inversion formula for a function f on a locally finite poset [72]. In a nutshell, Möbius inversion is a discrete analogue of the derivative of a real-valued map from calculus. For example, given a map f : Z → R (where Z is equipped with the canonical order), its Möbius inversion f : Z → R is defined as f (n) = f (n) − f (n − 1) for n ∈ Z, capturing the change of f at the point n relative to the value of f at the point n − 1. More generally, we can define Möbius inversion of a function on any locally finite poset.
Definition 6.2.
A poset P is said to be locally finite if for all p, q ∈ P with p ≤ q the set {r ∈ P : p ≤ r ≤ q} is finite.
Let P be a locally finite poset. The Möbius function µ P : P × P → Z of P is defined recursively as We are interested in carrying out Möbius inversion of abelian-group-valued maps defined on a given poset. Let G be an abelian group. For any m ∈ Z + and x ∈ G, we will interchangeably use m · x and x · m to denote x + · · · + x m terms . For a negative m ∈ Z, we will also use m · x or x · m to denote −((−m) · x). Theorem 6.4 (Möbius inversion formula). Let P be a locally finite poset and let G be an abelian group. Suppose that for a function f : P → G, there exists p ∈ P such that f (q) = 0 unless q ≥ p. Then, a function g : P → G satisfies f (q) = r ≤q g (r ) for all q ∈ P if and only if g (q) = r ≤q f (r ) · µ P (r, q) for all q ∈ P. We will often write g = f .
µ P (p, q) := 1, p = q, − p≤r <q µ P (p, r ), p < q,
A difference between this theorem and [72, Proposition 2 (p.344)] is that the codomain of the functions f and g is an abelian group, whereas the codomain is a field in [72]. Nevertheless, the verbatim proof in [72] works in our setting. In [67,53], Möbius inversion of abelian-group-valued functions is considered in order to obtain generalized persistence diagrams. The example below directly follows from Example 6.3, Theorem 6.4, and the following fact: For the opposite poset P op of P, the Möbius function µ P op is obtained by µ P op (p, q) = µ P (q, p) for all p, q ∈ P [72, p.345]. Note that (1) this formula is reminiscent of the original definition of persistence diagrams [28,38] and (2) this formula was utilized in [53] to define generalized persistence diagrams for functors indexed by ZZ.
Silhouettes and formal sums of clusters
We introduce the notion of the silhouette of a free-abelian-group valued map on a poset. Let A be a nonempty finite set. By ZA we denote the free abelian group generated by A. This means that any element in ZA is a formal sum of elements from A with integer coefficients.
Remark 6.7. The map |−| : ZA → Z that sends i =1 m i a i (m i ∈ Z, a i ∈ A) to i =1 m i is a group homomorphism.
In the rest of this section let P be a locally finite poset.
p ∈ P if f (p) = i =1 m i a i (m i ∈ Z, a i ∈ A), then f (p) := i =1 m i .
Given an f : P → ZA, supp( f ) is contained in supp( f ) but the converse is not always true. For example, if f (p) = a 1 − a 2 with a 1 = a 2 , then p ∈ supp( f ) but p ∉ supp( f ). Notation 6.9. We will often represent f : P → G by the set {(p, f (p)) : p ∈ supp( f )}. In particular, when G = Z and im( f ) ⊂ {0, 1}, f will be identified with supp( f ).
The silhouette, as an operation, commutes with Möbius inversion over any locally finite poset: Proposition 6.10. Given a map f : P → ZA, let f be the Möbius inversion of f , i.e. for all p ∈ P, f (p) = q≤p f (q) · µ P (q, p). Then, the Möbius inversion of f : P → Z equals f , i.e. f = f .
Proof. Let p ∈ P.
We have:
f (p) = q≤p f (q) · µ P (q, p) by Definition 6.8 (ii) = q≤p f (q) · µ P (q, p) by Remark 6.7 = f (p).
In the rest of the paper, the basis A of the free abelian group ZA will be pow(X ), the power set of some finite set X . Example 6.11. Let X := {x 1 , x 2 , x 3 } and assume that a given map f : Int(ZZ) → Zpow(X ) satisfies
f ((1, 2) ZZ ) = {x 1 } + {x 2 } + {x 3 }, f ([1, 2) ZZ ) = {x 1 } + {x 2 , x 3 }, f ((1, 2] ZZ ) = {x 1 , x 2 } + {x 3 }, f ([1, 2] ZZ ) = {x 1 , x 2 , x 3 }.(4)
Then, from Example 6.6, we have We define a natural inclusion ι : SubPart(X ) → Zpow(X ) as follows. For any P = {B i } i =1 ∈ SubPart(X ), let ι(P ) := i =1 1 · B i . Let us also define a natural left-inverse Π : Zpow(X ) → SubPart(X ) of ι as follows. Definition 6.12 (Left-inverse to ι : SubPart(X ) → Zpow(X )). Let g ∈ Zpow(X ) be any nonzero element that is given as
i =1 m i B i (m i ( = 0) ∈ Z) where B i = B j for i = j . Then, Π(g ) is defined as the finest common coarsening {{B i }} n i =1 . The trivial element 0 ∈ Zpow(X ) is sent to via Π. For example, if X := {x 1 , x 2 , x 3 }, Π sends {x 1 , x 2 } + {x 2 , x 3 } ∈ Zpow(X ) to {{x 1 , x 2 }} ∨ {{x 2 , x 3 }} = {{x 1 , x 2 , x 3 }} ∈ SubPart(X ).
Maximal group diagrams
Buchin et al. defined the concept of maximal groups 7 for a set X of trajectories in Euclidean space [15] (cf. Remark 6.18). We observe that maximal groups are totally determined by the formigram induced by the δconnectivity dynamic graph of the set of trajectories (cf. Example 3.2), which enables us to prove that maximal groups are obtained by Möbius inversion (cf. Theorem 6.17). A corollary is that the collection of all maximal groups of X contains the same information as the formigram θ induced by X (cf. Remark 6.18). Definition 6.13. Let θ X be a formigram over X . A subset G ⊂ X is called a group over an interval I G ⊂ R if for all t ∈ I , there exists B t ∈ θ X (t ) such that G ⊂ B t . A group H covers group G if G ⊂ H and I G ⊂ I H . Definition 6.14. Let θ X be a formigram and let G ⊂ X be a group over I G ⊂ R. Then, G is said to be a maximal group (over I G ) if there is no group H ⊂ X that covers G.
Let us recall the natural injection ι : SubPart(X ) → Zpow(X ) which sends {B i } i =1 ∈ SubPart(X ) to i =1 1 · B i , a formal sum of blocks of P . In what follows, {B i } i =1 ∈ SubPart(X ) will be identified with i =1 1 · B i . Definition 6.15. Let θ X be a formigram. The -rank function θ X : Int(ZZ) → SubPart(X ) is defined as I → I θ X := {θ X (i , j ) : (i , j ) ∈ I }. By Convention 6.1, when I = 〈a, b〉 ZZ , we have I θ X = t ∈〈a,b〉 θ X (t ).
For I ∈ Int(ZZ), note that I θ X is a collection of groups over I . In particular, G ∈ I θ X is a maximal group if there is no group G ⊂ X covering G over either I − or I + (cf. Notation 6.5). Let dgm (θ X ) be the Möbius inversion of θ X over Int(ZZ) op (cf. Example 6.6). This means that for all I ∈ Int(ZZ)
dgm (θ X )(I ) := I θ X − I + θ X − I − θ X + I ± θ X ,(5)
where these sums and subtractions should be interpreted as operations in Zpow(X ). Let us recall the map Π : Zpow(X ) → SubPart(X ) from Definition 6.12.
Definition 6.16. The maximal group diagram of θ X is defined as Π • dgm (θ X ) : Int(ZZ) → SubPart(X ). We will simply write Πdgm (θ X ) for Π • dgm (θ X ).
We adapt Notation 6.9 as follows. For I ∈ Int(ZZ), if Πdgm (θ X )(I ) contains a block G ⊂ X , then we write (I ,G) ∈ Πdgm (θ X ). The following theorem says that Πdgm (θ X ) encodes all the maximal groups of θ X . See Proof. Fix any G ∈ I θ X . Since I + θ X ≤ I θ X , either (a) G ∈ I + θ X or (b) there exists the subcollection {H j ∈ I + θ X : H j ⊂ G} of I + θ X which does not include G. Similarly, since I − θ X ≤ I θ X , there are the two analogous cases: (a') G ∈ I − θ X , or (b') there exists the subcollection {K j ∈ I θ X : K j ⊂ G} which does not include G. In combination, there are the four possible cases: (aa'), (ab'), (ba'), and (bb'). Among these, observe that G appears in the subpartition Πdgm (θ X )(I ) only in case (bb'), i.e. when G is a maximal group over I . In the other three cases, Πdgm (θ X )(I ) does not contain G nor any of its subsets. The claim follows. Remark 6.18. Let X be a set of trajectories in Euclidean space. In [15], a group of X is defined according to three different parameters. For any positive m ∈ Z and ε, δ ∈ R + , a set G ⊂ X is said to form an (m, ε, δ)-group during a time interval I if and only if (1) G contains at least m points, (2) the length of I is not less than ε and (3) for any two points x, x ∈ G and any time t ∈ I , there is a chain x = x 0 , x 1 , . . . , x n = x of points in X such that any two consecutive ones are at distance ≤ δ at time t . Note that (1) groups are totally determined by the formigram θ X induced by the δ-connectivity dynamic graph on X , and that (2) for any positive m ∈ Z and any ε ∈ R + the collection of (m, ε, δ)-groups is the subcollection of (1, 0, δ)-groups. Therefore, we restrict our attention to maximal (1, 0, δ)-groups which are exactly the maximal groups of θ X in Definition 6.14.
Example 6.19. Assume that the formigram θ X in Figure 10 (B) is obtained from some three trajectories x 1 , x 2 and x 3 in Euclidean space as described in Remark 6.18 (cf. Example 3.2 and Definition 3.10). Figure 11 is another representation of the maximal group diagram of θ X .
Consider any R-indexed module M (cf. Section 2.3). Then, for any t ∈ R, the dimension of M (t ) equals the total multiplicity of those intervals I in the barcode of M which contain t . We will establish an analogous result for θ X : R → SubPart(X ). Let us observe that for any nonempty B ∈ pow(X ), we have that {B } ∈ SubPart(X ). Figure 10 (B) is represented as an annotated standard persistence diagram [28] ([a, b] ∈ Int is identified with (a, b) ∈ R op × R). In the 3-tuple (m, ε, δ) of parameters for maximal groups (cf. Remark 6.18), m corresponds to the minimal size of maximal groups (depicted as the "mass" of points above), and ε corresponds to the minimal duration of maximal groups. By the monotonicity of maximal groups (cf. Remark 6.20), the masses of points near the diagonal line y = x tend to be relatively large. Remark 6.20 (Monotonicity of maximal groups and completeness). By Theorem 6.17 and the definition of maximal groups, Πdgm (θ X ) satisfies a monotonicity property: if B ⊂ B with (I , B ), (I , B ) ∈ Πdgm (θ X ), then I ⊃ I . This in turn implies that, for t ∈ R, θ X (t ) is equal to {{B } ∈ pow(X ) \ { } : there exists I t such that (I , {B }) ∈ Πdgm (θ X )}. Therefore, we can recover a formigram θ X from its maximal group diagram Πdgm (θ X ). Example 6.21. Consider the formigram depicted in Figure 10 (B). Note that θ X (5) = {x 1 |x 2 x 3 } and the blocks corresponding to the four intervals containing t = 5 are {x 1 }, {x 2 }, {x 3 } and {x 2 x 3 }. We have that
θ X (5) = {x 1 |x 2 x 3 } = {{x 1 }, {x 2 }, {x 3 }, {x 2 x 3 }}.
Let us characterize the silhouette of the maximal group diagram of a dendrogram (cf. Remark 3.7).
Example 6.22.
Let θ X be a dendrogram, and let G ⊂ X . If there exists t ∈ R such that G ∈ θ X (t ), then define b(G) := min{t ∈ R : G ⊂ θ X (t )}. Then, the silhouette of Πdgm (θ X ) amounts to the set
{[b(G), ∞) : there exists t ∈ R such that G ∈ θ X (t )} (cf. Notation 6.9).
For example, consider the dendrogram θ X over X := {x 1 , x 2 , x 3 } that is depicted in Figure 10
(A). Since t ∈R θ X (t ) = {{x 1 }, {x 2 }, {x 3 }, {x 1 , x 2 }, {x 1 , x 2 , x 3 }},
the silhouette of Πdgm (θ X ) consists of the five half-infinite intervals whose left endpoints are
b({x 1 }), b({x 2 }), b({x 3 }), b({x 1 , x 2 }), and b({x 1 , x 2 , x 3 }).
Remark 6.23 below motivates us to consider a different summary of a formigram, leading to the next section.
Remark 6.23.
(i) The maximal group diagram can drastically change under perturbations of an input formigram with respect to d F I (for example, see Figure 12). This is not unexpected since the distance d F I utilizes the join operation on subpartitions to quantify the difference between formigrams whereas the maximal group diagram is extracted from the -rank function (Definition 6.15).
(ii) The silhouette of the maximal group diagram almost always looks completely different from the barcode of θ X (Definition 3.12 (iv)); for example, observe that the silhouettes of Πdgm ∧ (θ X ) in Figure 10 (A),(B) are completely different from the silhouettes of dgm (θ X ) in Figure 10 (C),(D). The latter silhouettes are actually the zigzag barcodes of the respective formigrams, as it will turn out in the next section (cf. Theorem 6.30).
Persistence clustergrams
In this section we define the persistence clustergram of a formigram. When a given formigram θ X is saturated, its persistence clustergram can be regarded as an "annotated" zigzag barcode of θ X (cf. Figure 1 (C)). We begin by defining the -rank function of a formigram ( is a fusion of and , as will be clear below). The Möbius inversion of the -rank function of θ X will be the persistence clustergram of θ X . Let θ X be a formigram. For I ∈ Int, we define I θ X as the partition of the underlying set X (⊂ X ) of I θ X that is obtained by restricting the equivalence relation induced by I θ X to X . In other words,
I θ X = B ∩ X : B ∈ I θ X and B ∩ X = .(6)
Note that when θ X is saturated, I θ X = I θ X for any interval I ⊂ supp(θ X ).
x 2 } such that θ X (t ) = {x 1 |x 2 } for t ∈ (0, 2ε) and θ X (t ) = {x 1 x 2 } for t ∈ R \ (0, ε). (B)
The ε-smoothing of θ X (Definition 2.23). The bottleneck distance between the silhouettes of Πdgm (θ X ) and
Πdgm (S ε θ X ) is ∞, whereas d F I (θ X , S ε θ X ) = ε. Nature of F : P → C P C rk(F )(I ) Representation of rk(F )(I ) (i) Sublevelset persistence [28] R vec coim lim ← − − F | I → lim − − → F | I
An integer (dimension) (ii) Levelset persistence [18] Int (iii) Merge tree [63] R set An integer (cardinality) (iv) Reeb graph [33] Int (v) Unlabeled dendrogram [19] R Part An integer partition (vi) Unlabeled formigram Int Table 4: rk(F )(I ) denotes the rank of the functor F over an interval I ⊂ R up to isomorphism [53,Definition 3.5].
In each of rows (i)-(iv), the coimage of lim
← − − F | I → lim − − → F | I can be replaced by the image of lim ← − − F | I → lim − − → F | I ,
since they are isomorphic. However, it is not the case in row (v) nor in row (vi); see Proposition 5.8.
Definition 6.24.
Let θ X be a formigram. The -rank function θ X : Int(ZZ) → SubPart(X ) of θ X is defined by I → I θ X . Remark 6.25. In the category Part (cf. Definition 2.7), the pair ( ( I θ X ) , I θ X ) is the coimage of the morphism ( ( I θ X ) , I θ X ) → ( ( I θ X ), I θ X ) which is given by the inclusion ( I θ X ) → ( I θ X ) (cf. Proposition 5.8). In this respect, the -rank function is a rendition of the generalized rank invariant from [53]. Table 4 clarifies the analogy.
I θ X = {x 1 x 2 |x 3 x 4 }, and I θ X = {x 1 x 2 |x 3 }.
Note that whereas the -rank function classifies x 1 and x 3 into different clusters over I , the -rank function puts x 1 and x 3 into the same cluster over I . The -rank function θ X is completely depicted in Figure 9 (B).
= I θ X − I + θ X − I − θ X + I ± θ X .(7)
By the following remark, dgm is a complete invariant of formigrams.
Remark 6.28. By Theorem 6.4, for any I ∈ Int(ZZ), we have that I θ X = J ⊃I dgm (θ X )(J ). In particular, for any t ∈ R, θ X (t ) = [t ,t ] θ X = J t dgm (θ X )(J ). Thus θ X can be recovered from dgm (θ X ).
Whereas the H 0 barcode or the underlying merge tree is not a complete invariant of a dendrogram, the persistence clustergram of a dendrogram is a complete invariant. Example 6.29 (Persistence clustergram of dendrograms). Consider the dendrogram θ X over X := {x 1 , x 2 , x 3 } which is depicted as in Figure 10 (C). Then, dgm (θ X ) consists of the three elements (cf. Notation 6.9)
([0, 2), {x 1 |x 2 } − {x 1 x 2 }), ([0, 4), {x 1 x 2 |x 3 } − {x 1 x 2 x 3 }), ([0, ∞), {x 1 x 2 x 3 }),(8)θ X (1) = dgm (θ X )[0, 2) + dgm (θ X )[0, 4) + dgm (θ X )[0, ∞) = ({x 1 |x 2 } − {x 1 x 2 }) + ({x 1 x 2 |x 3 } − {x 1 x 2 x 3 }) + {x 1 x 2 x 3 } = {x 1 |x 2 |x 3 }.
Also, the silhouette {[0, 2), [0, 4), [0, ∞)} of dgm (θ X ) coincides with the zigzag barcode of θ X . This is not merely a coincidence, as we will see in the following theorem.
Theorem 6.30.
If θ X is a saturated formigram (Definition 3.8), then the barcode of θ X is equal to the silhouette dgm (θ X ) of dgm (θ X ) (cf. Figures 1 (F),(G),(H) and 10 (C),(D)).
We prove this theorem in Appendix A.4 by harnessing results from [53].
Corollary 6.31.
If θ X is a saturated formigram, the silhouette dgm (θ X ) : Int(ZZ) → Z is nonnegative.
Example 6.32. When a formigram is not saturated, the silhouette of its persistence clustergram is not necessarily the same as the barcode of θ X . See Figure 9 (C) for an illustrative example.
Since dgm is a complete invariant of formigrams, the distance d F I between formigrams can also be viewed as a distance between their persistence clustergrams. Since computing d F I is NP-hard, it is desirable to find many computable lower bounds for d F I . One such lower bound is the bottleneck distance between the barcodes of formigrams (cf. Corollary 4.16). One additional lower bound based on persistence clustergrams is proposed in the next section.
Persistent cluster counting functor
We define the persistent cluster counting functor of a formigram θ X as the silhouette of θ X . We will see later that persistent cluster counting functors of formigrams can be used for discriminating formigrams which cannot be discriminated by the underlying Reeb graphs of formigrams. Throughout this section, X will denote nonempty finite sets. Convention 6.1 still applies in this section.
For P ∈ SubPart(X ), |P | denotes the number of blocks in P . By viewing P as the formal sum B ∈P 1 · B ∈ Zpow(X ), this notation is consistent with the notation |−| which is defined in Remark 6.7. Definition 6.33. The persistent cluster counting functor of a formigram θ X is the map | θ X | : Int(ZZ) → Z + defined by I → | I θ X |.
For t ∈ R, | θ X | (t , t ) = |θ X (t )| , the number of the blocks in θ X (t ). The value | θ X | (I ) in Definition 6.33 is the number of independent groups over I (cf. Definition 6.13); Two groups G 1 and G 2 over an interval I are called independent if there is no pair of x 1 ∈ G 1 and x 2 ∈ G 2 such that x 1 and x 2 belong to the same block of θ X (t ) for some t ∈ I . The independence of G 1 and G 2 is stronger condition than G 1 ∩ G 2 = . If G 1 and G 2 are not independent, then they are called dependent. For the equivalence relation generated by dependence, the value | θ X | (I ) = I θ X is the number of equivalence classes of groups over I . Note that | θ X | is analogous to the rank invariant of R-indexed and ZZ-indexed modules [22,28,53].
We have that
θ X (I ) = J ⊃I dgm (θ X ) (J )(9)
by Theorem 6.4 and Proposition 6.10. In Appendix A.5 we also prove that | θ X | can be obtained from the unlabeled formigram of θ X (cf. Definition 3.12 (i)); see Proposition A.10. This establishes the arrow (C)→(H) in Figure 1, invoking that | θ X | and dgm (θ X ) can recover each other.
Stability of the persistent cluster counting functor
Convention 6.1 still applies in this section. For any two intervals I ⊂ J , the number of independent maximal groups over I is greater than equal to that of J , i.e. J θ X ≤ | I θ X |. This monotonicity allows us to quantify the difference between two persistent cluster counting functors via the so-called erosion distance. Definition 6.34 ([67, 68]). Let Y 1 , Y 2 : Int → Z + be any two order-reversing maps. The erosion distance between Y 1 and Y 2 is defined as
d E (Y 1 , Y 2 ) := inf ε ∈ [0, ∞) : for all I ∈ Int, Y i (I ) ≤ Y j (I ε ), for i , j ∈ {1, 2} .
Throughout this section, X and Y will denote nonempty finite sets. Persistent cluster counting functor enjoys stability: Theorem 6.35. For any two formigrams θ X and θ Y , we have
d E θ X , θ Y ≤ d F I (θ X , θ Y ).
We prove this theorem at the end of this section. Invoking Proposition 4.17, this theorem implies that DGs can be summarized into persistent cluster counting functors with a guarantee of stability. Example 6.36 below shows that persistent cluster counting functors can sometimes be more discriminative than the Reeb graph of formigrams (cf. Definition 3.12). Example 6.36. Consider the two DGs G X = (V X (·), E X (·)) and G Y = (V Y (·), E Y (·)) over X := {x 1 , x 2 } and Y := {y 1 } respectively given as follows (cf. Figure 13 (A)):
V X (t ) ∪ E X (t ) = {{x 1 }}, t ∈ [0, 1) ∪ (2, 3] {{x 1 }, {x 2 }, {x 1 , x 2 }}, t ∈ [1, 2] , otherwise, V Y (t ) ∪ E Y (t ) = {{y 1 }}, t ∈ [0, 3]
, otherwise.
Let θ X and θ Y be the formigrams of G X and G Y , respectively (cf. Definition 3.10 and Figure 13 (B)). Then | θ X | , | θ Y | : Int → Z + are described in Figure 13 (C). By Definition 6.34, we have d E (| θ X | , | θ Y |) = 1 (More details are provided below). Note that these two DGs cannot be discriminated by computing their Reeb graphs nor their zigzag barcodes.
Remark 6.37.
(i) The inequalities in Theorems 4.17 and 6.35 are tight. For the DGs G X , G Y and formigrams
θ X , θ Y in Example 6.36, we have that d dynG I (G X , G Y ) = d F I (θ X , θ Y ) = 1. Indeed, X π 1 −−− {(x 1 , y 1 ), (x 2 , y 1 )} π 2
−−− Y (where π 1 and π 2 are the canonical projections) is a 1-tripod between G X and G Y , and also between θ X and θ Y .
(ii) Assume that θ X , θ Y are two formigrams with the same n critical points. Once | θ X | and | θ Y | are computed, computing d E (| θ X | , | θ Y |) via ordinary binary search requires the (expected) cost O(n 2 log n), see [54,Section 5]. Figure 13: The summarization process of the two DGs G X and G Y from Example 6.36.
Details for Example 6.36. Let us compute the formigram θ X induced from G X (cf. Definition 3.10):
θ X (t ) = {{x 1 }}, t ∈ [0, 1) {{x 1 , x 2 }}, t ∈ [1, 2] {{x 2 }}, t ∈ (2, 3]
, t ∈ R \ [1,3] (cf. Figure 13 (B)).
Next we compute | θ X | : Int → Z + (cf. Figure 13 (C)). Note that [s,t ] Figure 13 (C). Notice that the cardinality of [s,t ]
θ X = {{x 1 , x 2 }}, 1 ≤ s ≤ t ≤ 2,: R → SubPart(Y ) and | θ Y | : Int → Z + as θ Y (t ) = {y 1 }, t ∈ [0, 3] , otherwise, and θ Y [s, t ] = 1, 0 ≤ s ≤ t ≤ 3 0, otherwise.
(see Figure 13 (B),(C)) Observe from Figure 13
(C) that d E (| θ X | , | θ Y |) = 1.
Remark 6.38. In general, we have the following "poset of distances":
d F I (θ X , θ Y ) d ωR I (ω(θ X ), ω(θ Y )) d R I (Reeb(θ X ), Reeb(θ Y )) d E (| θ X | , | θ Y |) d B (barc ZZ (θ X ), barc ZZ (θ Y ))
where the distance d ωR I between weighted Reeb graphs is defined in Appendix B. When θ X and θ Y are saturated formigrams, we have the following totally ordered hierarchy:
d F I (θ X , θ Y ) ≥ d ωR I (ω(θ X ), ω(θ Y )) ≥ d R I (Reeb(θ X ), Reeb(θ Y )) ≥ d B (barc ZZ (θ X ), barc ZZ (θ Y )) ≥ d E ( θ X , θ Y ).
The last inequality follows from Theorem 6.30 and the well-known inequality d B ≥ d E for (standard) persistence diagrams [59,67].
Proof of Theorem 6.35. If d F I (θ X , θ Y ) = +∞, there is nothing to prove. For some ε ∈ R + , assume that R :
X ϕ X −−− − Z ϕ Y − −−− Y
is an ε-tripod between θ X and θ X (cf. Definition 4.11). Fix I ∈ Int and we will only prove that | θ X | (I ε ) ≤ | θ Y | (I ). If I ε θ X is empty, then | θ X | (I ε ) = 0 by definition and hence there is nothing to prove. Suppose that I ε θ X is nonempty. We show that there are two set maps ψ 1 : I ε θ X → I θ Y and ψ 2 : I θ Y → I ε θ X which make the following diagram commutes.
I ε θ X I ε θ X I θ Y I θ Y . φ X (I ε ) ψ 1 φ Y (I ) ψ 2
Here the maps φ X (I ε ) and φ Y (I ) are the canonical maps in SubPart(X ) and SubPart(Y ), respectively (cf. Definition 5.2). Indeed, if we have such maps ψ 1 and ψ 2 , then we have:
θ X (I ε ) = im φ X (I ε ) ≤ im φ Y (I ) = θ Y (I ),
as desired. Let us construct two such maps ψ 1 and ψ 2 . First, define set maps f : X → Y and g : Y → X such that
{(x, f (x)) : x ∈ X } ∪ {(g (y), y) : y ∈ Y } ⊂ R,(10)
which is possible since ϕ X and ϕ Y are surjective (cf. Notation 4.2). We now construct the map ψ 1 : I ε θ X → I θ Y . Let B ∈ I ε θ X and pick any x ∈ B . By definition of I ε θ X , this implies that x is in the underlying set of θ X (t ) for all t ∈ I ε .
We claim that the lifespan
I f (x) of f (x) ∈ Y in the formigram θ Y contains the interval I =: [s, t ]
. Since I f (x) is an interval in R (cf. Definition 3.5), it suffices to prove that I f (x) contains some α ∈ (−∞, s] and β ∈ [t , ∞). Invoking that R is an ε-tripod between θ X and θ Y , we have that
θ X (s − ε) ≤ R [s−ε] ε θ Y = [s−2ε,s] θ Y .(11)
Also, since x is in the underlying set of θ X (s −ε) and (x, f (x)) ∈ R (from inclusion (10)), relation (11) implies that the element f (x) ∈ Y must be in the underlying set of [s−2ε,s] θ Y (cf. Remark 4.15 (i)). This implies that there exists α ∈ [s − 2ε, s] such that y is in the underlying set of θ Y (α). Hence, α belongs to I f (x) . Similarly, one can also check that there exists β ∈ [t , t + 2ε] such that β ∈ I f (x) . Therefore, we have that I ⊂ I f (x) . This inclusion implies that I θ Y contains a block C which in turn contains f (x). We define ψ 1 as the map sending B to C . Note that the definition of ψ 1 depends on the choice of a representative element x for each block B in I ε θ X .
Next, by invoking that R is an ε-tripod between θ X and θ Y , inclusion (10), and Remark 4.15 (ii), we define the map ψ 2 : I θ Y → I ε θ X as follows: for each C ∈ I θ Y , let ψ 2 (C ) be the unique block B ∈ I ε θ X containing the image of C via the map g : Y → X .
It remains to verify that φ X (I ε ) = ψ 2 • φ Y (I ) • ψ 1 . Pick any block B ∈ I ε θ X . Since φ X (I ε ) is the canonical map in SubPart(X ), B is sent to the unique block B ∈ I ε θ X containing the whole block B . Now, let x ∈ B be the representative element which was used for defining ψ 1 . Via ψ 1 , the block B is sent to the block C ∈ I θ Y containing f (x). Then, the map φ Y (I ) sends C to the unique block C ∈ I θ Y containing f (x) (and the whole block C ). From Remark 4.15 (ii), we have that I θ Y ≤ R I ε θ X . Thus, by inclusion (10), we conclude that ψ 2 sends C ( f (x)) to the block B ( x), completing the proof.
Analysis of dynamic metric spaces (DMSs)
Our work is motivated by the desire to construct a well-defined summarization tool of clustering behavior of time-varying metric data, which is modeled as dynamic metric spaces (DMSs). In Section 7.1, we define DMSs. In Section 7.2, we establish a sufficient condition for DMSs to be converted into DGs via the Rips graph functor (cf. Proposition 7.5). This enables us to produce summaries of DMSs such as those which are illustrated in Figure 1. 8 is a metric. 8 In [55,Section 5], DMSs generated by Boid [69] were successfully classified by the bottleneck distance on their zigzag barcodes (cf.
DMSs
In this section we introduce definitions pertaining to our model for dynamic metric spaces (DMSs). In particular, tameness (cf. Definition 7.4) is a crucial requirement on DMSs, which permits transforming DMSs into DGs via the Rips graph functor (cf. Proposition 7.13), thus subsequently into formigrams, persistence clustergrams, Reeb graphs, and barcodes. Definition 7.1 ([54]). A dynamic metric space is a pair γ X = (X , d X (·)) where X is a nonempty finite set and d X : R × X × X → R + satisfies:
(i) For every t ∈ R, γ X (t ) = (X , d X (t )) is a pseudo-metric space. (ii) For any x, x ∈ X with x = x the function d X (·)(x, x ) : R → R + is not identically zero. (iii) For fixed x, x ∈ X , d X (·)(x, x ) : R → R + is continuous.
We refer to t as the time parameter.
We remark that a DMS γ X is not just a continuous curve in the Gromov-Hausdorff space [25], but it also keeps track of the identities of points in X . Item (ii) is assumed to avoid redundancy; otherwise there could be two points which are forever identified.
Example 7.2. An example is given by n particles/animals moving continuously inside an environment Ω ⊂ R d where particles are allowed to coalesce. If the n trajectories are p 1 (t ), . . . , p n (t ) ∈ R d , then let P := {1, . . . , n} and define a DMS γ P := (P, d P (·)) as follows: for t ∈ R and i , j ∈ {1, . . . , n}, let d P (t )(i , j ) := p i (t ) − p j (t ) , where · denotes the Euclidean norm. Buchin et al. considered this setting [15].
We now introduce a notion of equality between two DMSs. Definition 7.3. Let γ X = (X , d X (·)) and γ Y = (Y , d Y (·)) be two DMSs. We say that γ X and γ Y are isomorphic if there exists a bijection ϕ : X → Y such that ϕ is an isometry between γ X (t ) and γ Y (t ) across all t ∈ R.
From DMSs to DGs
We introduce a notion of tame DMS which will ultimately ensure that the Rips graph functor induces DGs satisfying our definition (cf. Definition 3.1 and Example 3.2). A continuous function f : R → R is called tame, if for any c ∈ R and any finite interval I ⊂ R, the set f −1 (c)∩ I ⊂ R has only finitely many connected components (and is possibly empty). Elementary functions including polynomial functions (in particular, constant functions), piecewise linear functions with locally finitely many critical points are tame.
Definition 7.4. A DMS γ X = (X , d X (·)) is said to be tame if for any x, x ∈ X the function d X (·)(x, x ) : R → R + is tame.
For δ ≥ 0 and for any finite metric space (X , d X ), let R 1 δ (X , d X ) be the 1-skeleton of the δ-Rips complex of X , i.e. a simple graph over the vertex set X with the edge set
E X = {x, x } ⊂ X : d X (x, x ) ≤ δ and x = x .
Proposition 7.5. Let γ X be a tame DMS over X and let δ ≥ 0. Then, by defining
R 1 δ (γ X )(t ) := R 1 δ (γ X (t )) for t ∈ R, R 1 δ (γ X ) : R → Graph(X ) is a saturated DG over X .
The proof of Proposition 7.5 is given in Appendix A.6.1. Let γ X be a tame DMS over X . By Proposition 7.5 and Definition 3.10, one can obtain a formigram θ X := π 0 R 1 δ (γ X ) .
The λ-slack interleaving distance between DMSs
The main goal of this section is to introduce a [0, ∞)-parametrized family d
dynM I,λ λ∈[0,∞)
of extended metrics for DMSs. Each metric in this family is a hybrid between the Gromov-Hausdorff distance [16] and the interleaving distance [23,33]. We obtain a stability result with respect to the most stringent metric (the metric corresponding to λ = 0) in the family (cf. Theorem 7.14).
Definition 7.6. Let ε ≥ 0. Given any map d : X × X → R, by d + ε we denote the map from X × X to R defined by
(d + ε)(x, x ) = d (x, x ) + ε for all (x, x ) ∈ X × X .
Definition 7.7. Given any DMS γ X = (X , d X (·)) and any interval I ⊂ R, define the map I d X :
X × X → R + by ( I d X ) (x, x ) := min t ∈I d X (t )(x, x ) for all (x, x ) ∈ X × X .
Given any map d : X × X → R, let Z be any set and let ϕ : Z → X be any map. Then, we define the pullback
ϕ * d : Z × Z → R of d under ϕ by ϕ * d (z, z ) := d ϕ(z), ϕ(z )
for all (z, z ) ∈ Z × Z . Consider any two functions d 1 :
X × X → R and d 2 : Y × Y → R. Given a tripod R : X ϕ X −−− − Z ϕ Y − −−− Y between X and Y , by d 1 ≤ R d 2 we mean ϕ * X d 1 (z, z ) ≤ ϕ * Y d 2 (z, z ) for all (z, z ) ∈ Z × Z . Recall that for any t ∈ R, [t ] ε := [t − ε, t + ε].
Definition 7.8 (λ-distortion of a tripod). Fix λ ≥ 0. Let γ X = (X , d X (·)) and γ Y = (Y , d Y (·)) be any two DMSs. Let
R : X ϕ X −−− − Z ϕ Y − −−− Y be a tripod between X and Y such that for all t ⊂ R, [t ] ε d X ≤ R d Y (t ) + λε and [t ] ε d Y ≤ R d X (t ) + λε.(12)
We call any such R a (λ, ε)-tripod between γ X and γ Y . Define the λ-distortion dis dyn λ (R) of R to be the infimum of those ε ≥ 0 for which R is a (λ, ε)−tripod. (ii) (Temporal distortion) Fix τ ≥ 0. Let γ X = (X , d X (·)) be any DMS and define any continuous map α : R → R such that α − id R ∞ ≤ τ. Define the DMS γ X • α := (X , d X (α(·))), i.e. for t ∈ R, γ X • α(t ) = (X , d X (α(t ))).
Take the tripod R :
X id X −−− − X id X − −−− X .
Then, for any λ ≥ 0, it is easy to check that dis dyn λ (R) ≤ τ. Remark 7.10. In Definition 7.8, if R is a (λ, ε)-tripod, then R is also a (λ, ε )-tripod for any ε > ε: Fix any t ⊂ R.
If for some ε ≥ 0, [t ] ε d X ≤ R d Y (t ) + λε, then for any ε > ε, [t ] ε d X ≤ [t ] ε d X ≤ R d Y (t ) + λε < d Y (t ) + λε .
Now we introduce a family of metrics for DMSs. Definition 7.11 (The λ-slack interleaving distance between DMSs). For each λ ≥ 0, we define the λ-slack interleaving distance between any two DMSs γ X = (X , d X (·)) and γ Y = (Y , d Y (·)) as (γ X , γ Y ) ≤ ε for some ε ≥ 0, then we say that γ X and γ Y are ε-interleaved or simply interleaved. By definition, it is clear that for all λ > 0, d
dynM I,λ ≤ d dynM I .
For r > 0, we call any DMS γ X = (X , d X (·)) r −bounded if the distance between any pair of points in X does not exceed r across all t ∈ R. If γ X is r −bounded for some r > 0, then γ X is said to be simply bounded. In the rest of this section, we discuss several properties of d dynM I . Stability results. The following proposition provides a gateway for extending the stability results that are illustrated in Figure 1 to the setting of DMSs. Proposition 7.13 (Stability of summarizing DMSs into DGs). Let γ X = (X , d X (·)) and γ Y = (Y , d Y (·)) be any tame DMSs. Fix any δ ≥ 0. Consider saturated DGs G X :
= R 1 δ (γ X ), G Y := R 1 δ (γ Y )
, as in Proposition 7.5. Then,
d dynG I (G X , G Y ) ≤ d dynM I (γ X , γ Y ).
Proof. The proof follows from the fact that for any ε ≥ 0, any (0, ε)-tripod R between γ X and γ Y (Definition 7.11) is also an ε-tripod between G X and G Y (cf. Definition 4.4).
A priori Proposition 7.13 does not seem to be a satisfactory stability theorem in that the RHS can be infinity in many cases in comparison with the LHS. Nevertheless, this 'weak' stability seems to be the most we can expect for the mapping [DMSs] → [DGs] via the Rips graph functor because DMSs change continuously over time (cf. Definition 7.1 (iii)) whereas DGs are discontinuous in the sense that edges appear and disappear over time.
Proposition 7.13 together with Theorem 2.22, Proposition 4.13 and Theorem 4.17 directly imply:
Theorem 7.14. Let γ X = (X , d X (·)) and γ Y = (Y , d Y (·)) be any two tame DMSs. Fix any δ ≥ 0. Consider the saturated formigrams θ X := π 0 R 1 δ (γ X ) and θ Y := π 0 R 1 δ (γ Y ) . 9 Then,
d B (barc ZZ (θ X ), barc ZZ (θ Y )) ≤ 2 d dynM I (γ X , γ Y ).
We discuss the generalization of Theorem 7.14 to higher dimensional homology barcodes in Appendix E.
Computational complexity d dynM I
. A DMS γ X = (X , d X (·)) is said to be piecewise linear if for all x, x ∈ X , the function d X (·)(x, x) : R → R + is piecewise linear. We denote by S X the set of all breakpoints of all distance functions d X (·)(x, x ), x, x ∈ X . given by Theorem 7.14 is a realistic approach to comparing DMSs, especially when one is interested in a specific spatial scale.
Conclusion
We have established a stable mathematical framework for summarizing dynamic graphs, which often arise as discrete representations of spatiotemporal data. The evolution of connected components of a dynamic graph is fully encoded in a formigram, a constructible cosheaf over R valued in the lattice of subpartitions. 10 The lattice structure of subpartitions of a given set has been indispensable for obtaining:
(i) the formigram interleaving distance d F I which is more discriminative than the well-known Reeb graph interleaving distance. (ii) the maximal group diagram and persistence clustergram which are complete and visualizable invariants of formigrams (and thus more discriminative invariants than the Reeb graph of a formigram).
Note that we can directly extend the definitions of the maximal group diagram and the persistence clustergram to any cellular cosheaf over a topological space other than R [30]. Indeed, our two main ingredients, Möbius inversion and the lattice structure of subpartitions, do not depend on any specific property of R. Exploring this extension in relation to Reeb spaces [66] is left for future work. Extending the functorial pipeline in the left column of Figure 1 to the entire diagram is also left for the future [60]. Devising a bottleneck-type distance which can directly measure the difference between persistence clustergrams or between maximal group diagrams is of independent interest.
A Details and Proofs
A.1 Bottleneck distance
Recall that injective partial functions are referred to as matchings. We use σ : A B to denote a matching σ ⊂ A × B between sets A and B . The canonical projections of σ onto A and B are denoted by coim(σ) and im(σ), respectively.
Many equivalent expressions for the bottleneck distance have been given in the TDA literature. We adopt the following form from [5]: Recall Notation 2.12. Letting A be a multiset of intervals in R and ε ≥ 0,
A ε := {〈b, d 〉 ∈ A : b + ε < d } = {I ∈ A : [t , t + ε] ⊂ I for some t ∈ R} .
Note that A 0 = A . Definition A.1 ([5]). Let A and B be multisets of intervals in R. We define a δ-matching between A and B to be a matching σ : A B such that A 2δ ⊂ coim(σ), B 2δ ⊂ im(σ), and if σ〈b, d 〉 = 〈b , d 〉, then
〈b, d 〉 ⊂ 〈b − δ, d + δ〉, 〈b , d 〉 ⊂ 〈b − δ, d + δ〉.
with the convention +∞ + δ = +∞ and −∞ − δ = −∞. We define the bottleneck distance d B by We declare d B (A , B) = +∞ when there is no δ-matching between A and B for any δ ∈ [0, ∞).
A.2 Proof of Theorem 4.7
We recall the Gromov-Hausdorff distance between metric spaces. Let (X , d X ) and (Y , d Y ) be any two metric spaces and let R : 10 In the preprint of this work, we also considered formigrams derived from directed dynamic graphs [51].
X ϕ X −−− − Z ϕ Y − −−− Yd X ϕ X (z), ϕ X (z ) − d Y ϕ Y (z), ϕ Y (z ) .
Definition A.2 (Gromov-Hausdorff distance [16,Section 7.3]). Let (X , d X ) and (Y , d Y ) be any two compact metric spaces. Then,
d GH ((X , d X ), (Y , d Y )) = 1 2 inf
Proof. We first show that the LHS ≥ the RHS. Let ε ≥ 0 and let R :
X ϕ X −−− − Z ϕ Y − −−− Y
be any ε-tripod between the two dendrograms θ X and θ Y . Let (x, y), (x , y ) ∈ R and let t := u X (x, x ). This implies that x, x belong to the same block of the partition θ X (t ). Since θ X (t ) ≤ R [t ] ε θ Y = θ Y (t + ε), y and y must belong to the same block of θ Y (t + ε), and in turn this implies that u Y (y, y ) ≤ t + ε = u X (x, x ) + ε. By symmetry, we also have u Y (y, y ) ≤ u X (x, x )+ε and in turn u X (x, x ) − u Y (y, y ) ≤ ε. By Definition A.2, this implies that d GH ((X , u X ), (Y , u Y )) ≤ ε/2.
Next, we prove the opposite inequality. Let R :
X ϕ X −−− − Z ϕ Y − −−− Y
be a tripod between X and Y such that dis(R) = ε. it suffices to show that for all t ∈ R, θ X (t ) ≤ R θ Y (t + ε) and θ Y (t ) ≤ R θ X (t + ε). By symmetry, we only prove that θ X (t ) ≤ R θ Y (t + ε) for all t ∈ R. For t < 0, since θ X (t ) = , θ X (t ) ≤ R θ Y (t + ε) trivially holds. Now pick any t ≥ 0 and pick any (x, y), (x , y ) ∈ R. Assume that x, x belong to the same block of θ X (t ), implying that u X (x, x ) ≤ t . Since u X (x, x ) − u Y (y, y ) ≤ ε, we know u Y (y, y ) ≤ t +ε, and hence y, y belong to the same block of θ Y (t + ε). Therefore, θ X (t ) ≤ R θ Y (t + ε) for all t ∈ R. Theorem A.7 (Complexity of computing d F I ). Fix ρ ∈ (1, 6). It is not possible to obtain a ρ approximation to the distance d F I (θ X , θ Y ) between formigrams in time polynomially depending on |X |, |Y |, |crit(θ X )|, |crit(θ Y )| unless P = N P .
Proof. Pick any two dendrograms and invoke Proposition A.6 to reduce the problem to the computation of the Gromov-Hausdorff distance between the ultra-pseudometric spaces associated to the dendrograms. The rest of the proof follows along the same lines as that of Theorem 4.7.
A.4 Proof of Theorem 6.30
Theorem 6.30 will directly follow from Theorem A.9 below.
We explicitly represent the colimit of M : ZZ → set as follows. For k, l ∈ ZZ, assume that x ∈ M (k) and y ∈ M (l ). We write x ∼ y if k and l are comparable and one of x and y is mapped to the other via the internal map between M (k) and M (l ). The colimit of M is the pair (C , (i k ) k∈ZZ ) described as follows:
C := k∈ZZ M (k) ≈,(14)
where ≈ is the equivalence relation generated by the relations x k ∼ x l for x k ∈ M (k) and x l ∈ M (l ) with k, l being comparable. For the quotient map q : k∈ZZ M (k) → C , each i k is the composition M k → k∈ZZ M (k) q → C . Let I ∈ Int(ZZ). For any functor N : I → set, we can construct the limit and colimit of N in the same way; namely, in the above description, replace M and ZZ by N and I , respectively. In what follows, we use this explicit construction whenever considering colimits of (interval restrictions of ) ZZ-indexed set-diagrams. Proof of Theorem 6.30. By Proposition 6.10, for every I ∈ Int(ZZ), dgm (θ X ) (I ) = I θ X − I + θ X − I − θ X + I ± θ X .
Therefore, by Theorem A.9, it suffices to show that J θ X = full(Reeb(θ X )| J ) for all J ∈ Int(ZZ). If J ∈ Int(ZZ) is not a subset of supp(θ X ), then clearly 0 = J θ X = full(Reeb(θ X )| J ). Now assume that J ∈ Int(ZZ) is contained in supp(θ X ). Then, since θ X is saturated, J θ X = J θ X . Also, J θ X is equal to the number of full components of Reeb(θ X )| J , completing the proof.
B Distance between weighted Reeb graphs
In this section we introduce a distance between weighted Reeb graphs which mediates between d F I and d R I (cf. Definition 2.15 and Table 3). The Reeb graph of the formigram S ε θ X and its barcode. Small loops in Reeb(θ X ) disappear in Reeb(S ε θ X ). In the barcodes, bars with "[" on the left stand for half-closed intervals of the form [a, b). Open intervals in barc ZZ (θ X ) that are shorter than 2ε do not have corresponding intervals in barc ZZ (S ε θ X ). Also, disbanding and merging events in θ X which do not correspond to vertices on small loops in Reeb(θ X ) are replicated in S ε θ X : disbanding events in θ X are reflected in S ε θ X but with delay ε, whereas merging events in θ X are advanced by ε. For example, observe from the graphs Reeb(θ X ) and Reeb(S ε θ X ) that the disbanding event in θ X at t = t 0 is delayed to t = t 0 + ε in S ε θ X . Proof. Since F is constructible is defined via colimits over restrictions of a zigzag diagram over R (cf. Definition 2.14), for any J ∈ Int and for any x ∈ F (J ), there exist t ∈ J and y ∈ F ([t , t ]) such that F ([t , t ] ⊂ J )(y) = x. This property directly implies that the interval module I [b,a] BL cannot be a summand of F • F .
Proof of Proposition C.2. By Definitions 2.20 and 2.23, barc ZZ (S ε θ X ) is equal to the multiset
B ∩ R y=x+2ε : B ∈ barc(F • F ) .
where R y=x+2ε is the line y = x + 2ε identified with the real line via the bijection (r − ε, r + ε) ↔ r . The table (17) is directly obtained by Lemma C.3 and the block decomposability of F • C • θ X [11, Section 3].
The bijective correspondence of barcodes given in Proposition C.2 directly implies the following: Corollary C.4. Let θ X be any formigram over X . Then, for ε ≥ 0, d B (barc ZZ (S ε θ X ) , barc ZZ (θ X )) ≤ ε.
The smoothing operations defined for formigrams and Reeb graphs (cf. Definition 2.23) are compatible in the following sense: Figure 17: The interleaving condition. The thick blue curve and the thick red curve represent the graphs of ψ 0 (t ) = 1 + cos(t ) and ψ 1 (t ) = 1 + cos(t + π/4), respectively. Fixing ε ≥ 0, define a function S ε (ψ 0 ) : R → R by S ε (ψ 0 )(t ) := min s∈[t ] ε ψ 0 (s). The thin curves below the thick blue curve illustrate the graphs of S ε (ψ 0 ) for several different choices of ε. Note that for ε ≥ π/4 0.785, it holds that S ε (ψ 0 ) ≤ ψ 1 .
The following example generalizes the previous one. X ) where, for t ∈ R, ψ 0 (t ) = 1 + cos(ωt ), ψ 1 (t ) = 1 + cos(ω(t + τ)), for fixed ω > 0 and 0 < τ < 2π ω . Since in this case ψ 1 (t ) = ψ 0 (t + τ) for all t , one would expect that the interleaving distance between γ ψ 0 X and γ ψ 1 X is able to uncover the precise the value of τ. In this respect, we have:
E Higher dimensional persistent homology barcodes of dynamic metric spaces.
In this section we discuss extendibility of Theorem 7.14. The zigzag barcodes barc ZZ (θ X ) and barc ZZ (θ Y ) in Theorem 7.14 encodes the clustering behaviors of the given DMSs for a fixed scale δ ≥ 0. However, we do not need to restrict ourselves to clustering features of DMSs. Imagine that a flock of birds flies while keeping a circular arrangement from time t = 0 to t = 1. Regarding this flock as a DMS (trajectory data in R 3 ), we may want to have an interval containing [0, 1] in its 1-dimensional homology barcode. This idea can actually be implemented as follows. For a fixed δ ≥ 0, we substitute the Rips complex functor R δ for the Rips graph functor R 1 δ in Proposition 7.13. What we obtain is a dynamic simplicial complex or zigzag simplicial filtration, a generalization of Definition 3.1, induced from any tame DMS γ X . We then can apply the k-th homology functor to this zigzag simplicial filtration for each k ≥ 0 in order to obtain a vec-valued constructible cosheaf over R. This zigzag module will be a signature summarizing the time evolution of k-dimensional homological features of γ X . By virtue of Proposition 2.19 we eventually obtain the k-th homology barcode barc ZZ H k R δ (γ X ) of γ X with respect to the fixed scale δ ≥ 0; see also [35] for the computation of barc ZZ H k R δ (γ X ) for various δ. In particular, the 0-th homology barcode of the resulting zigzag module coincides with barc ZZ π 0 R 1 δ (γ X ) as defined in Theorem 7.14.
A natural question is then to ask whether our stability theorem (Theorem 7.14) can be extended to higher dimensional homology barcodes: Question E.1. For any pair of tame DMSs γ X = (X , d X (·)) and γ Y = (Y , d Y (·)), is it true that for any δ ≥ 0 and for any k ≥ 1,
d B barc ZZ H k R δ (γ X ) , barc ZZ H k R δ (γ Y ) ≤ 2 d dynM I γ X , γ Y ?
Interestingly, we found a family of counter-examples that indicates that stability, as expressed by Theorem 7.14, is a phenomenon which seems to be essentially tied to clustering (i.e. H 0 ) information.
Theorem E.2. For each integer k ≥ 1 there exist two different tame DMSs γ X k and γ Y k , and δ k ≥ 0 such that d dynM I γ X k , γ Y k < ∞ but such that the bottleneck distance between the barcodes of H k R δ k γ X k and H k R δ k γ Y k is unbounded.
Proof. Fix any k ≥ 1. We will illustrate DMSs γ X k and γ Y k as collections of trajectories of points in R k+1 , with the metric inherited from the Euclidean metric of R k+1 across all t ∈ R. For k = 1 or k = 2, see Figure 18.
Define γ X k to be the constant DMS consisting of 2(k + 1) points ±e i = (0, . . . , 0, ±1, 0, . . . , 0) ∈ R k+1 for i = 1, 2, . . . , k + 1. On the other hand, define γ Y k to be obtained from γ X k by substituting the still point +e 1 of γ X k by the oscillating point (1 + sin 2 (t ))e 1 = (1 + sin 2 (t ), 0, . . . , 0) for t ∈ R.
It is not difficult to check that d dynM I γ X k , γ Y k ≤ π/2. However, with the connectivity parameter δ = 2, their barcodes of the k-th zigzag persistent homology are barc ZZ (H k (R δ (X k ))) = {(−∞, ∞)} and barc ZZ (H k (R δ (Y k ))) = {[nπ, nπ] : n ∈ Z}, respectively. Therefore, d B (barc ZZ (H k (R δ (X k ))) , barc ZZ (H k (R δ (Y k )))) = +∞. (1 + sin 2 (t ))e 1 e 3 Figure 18: Pairs of DMSs (γ X i , γ X i ) for i = 1, 2 such that d dynM I γ X i , γ X i ≤ π/2. In contrast, for k = 1 (or k = 2), the bottleneck distance between their k-dimensional zigzag-persistence barcodes is infinite for δ ∈ [ 2, 2). DMS γ X 1 , described as the left-most figure, (γ X 2 , the third figure from the left) consists of four (eight) static points located at ±e 1 = (±1, 0, 0) and ±e 2 = (0, ±1, 0) (and ±e 3 = (0, 0, ±1)), respectively. On the other hand, DMS γ X 1 , illustrated at the second from the left (γ X 2 , at the right-most), contains a single oscillating point, denoted by a star shape, with trace (1+sin 2 (t ))e 1 for t ∈ R along with three (five) static points located at −e 1 , +e 2 and −e 2 , (and ±e 3 ), respectively. Then, the 1-dimensional (2-dimensional) zigzag-persistent homology barcode for γ X 1 (for γ X 2 ) consists of exactly one interval (−∞, ∞), indicating the presence of a loop (a void) for all time. However, the barcode of γ X 1 (γ X 2 ) consists of an infinite number of ephemeral intervals [nπ, nπ], n ∈ Z, indicating the onand-off presence of a loop (a void) that exists only at t = nπ for n ∈ Z in its configuration.
Figure 2 :
2The maximal groups introduced by Buchin et al.[15] can be obtained via the Möbius inversion of the -rank function of a formigram (cf. Definition 6.16). NB: In panel (C), commas are omitted in each set, e.g. {23} means the set {2, 3}.
Figure 3 :
3The points falling into the shaded regions comprise the intervals (−1, 1) ZZ , [−1, 1) ZZ , (−1, 1] ZZ and [−1, 1] ZZ of the poset ZZ, respectively in order.
45 Figure 4 :
454Illustration for Definition 2.14. (A) The shaded region stands for the poset Int. We have I ≤ J for the intervals I , J ∈ Int shown in the figure. The diagonal line is identified with the real line via the bijection (t , t ) ↔ t . (B) The canonical morphism F (I ) → F (J ) for I ≤ J in Int. (C) The zigzag diagram anchored over the real line completely determines the cosheaf F : Int → C defined by I → lim − − → M | ZZ[ι c ≤I ] .
3 :
3Objects in panels (A),(B),(C),(D),(E),(F) of Figure 1 and their corresponding metrics. Metrics in this table are totally ordered, i.e. in Figure 1, each process in (A)→(B)→(C)→(D)→(E)→(F) is stable. * Extra assumptions are necessary that are given in Definitions 3.1 and 3.5. **Visualized by zigzag barcode (Definition 2.20).
Definition 2. 14 (
14Constructible cosheaves and critical points). A functor F : Int → C is called a constructible cosheaf over R valued in C, if there exist a strictly increasing function c : Z → R such that lim i →±∞ c(i ) = ±∞ and a functor M : ZZ → C such that for all I ∈ Int, F (I ) = lim − − → M | ZZ[ι c ≤I ] (cf.
Proposition 2.19([11]). Any constructible cosheaf F : Int → vec is interval decomposable.Let R :y=x be the diagonal line y = x in R 2 . By Proposition 2.19, we have: Definition 2.20. Given any constructible cosheaf F : Int → vec, the zigzag barcode of F is defined as barc ZZ (F ) := barc(F ) ∩ R :y=x .
Remark 2 . 21 .
221In Definition 2.13, when C = set and M , N are constructible, we obtain the interleaving distance d R I between Reeb graphs M and N[11,33].
Proposition 2 .
225 ([33, Propositions 4.16 and 4.17]
Example 3. 2 .
2Let us consider the two dynamic point clouds depicted in Figure 5 (A) and (B). For δ = 1, their time-varying δ-Rips complexes are the DGs depicted in Figure 5 (A') and (B') (see Proposition 7.13 for a general statement).
Figure 5 :
5An illustration for Example 3.2. (A) and (B): Two dynamic point clouds γ X and γ Y each consisting of three points x 1 , x 2 , x 3 and y 1 , y 2 , y 3 , respectively. While the two points x 1 and x 3 (resp. y 1 and y 3 ) are fixed at vertical coordinate values −2 and 2, the other point x 2 (resp. y 2 ) moves according to α(t ) (resp. |α(t )|). (A') and (B') show the 1-Rips complexes
Figure 6 :
6An example of a dendrogram over the set X = {x 1 , x 2 , x 3 , x 4 }.
Figure 1 (
1B)-(F) illustrates the following definition. Recall the three functors in Definition 2.9.
Figure 7 :
7Visualization of the Reeb graph of a formigram θ X : R → SubPart(X ). (A) The formigram θ X is defined as follows over the interval [0, 20]: Let X :
Figure 8 :
8(A) The underlying weighted Reeb graph of the two formigrams from Example 3.13. (B) The underlying Reeb graph of the formigram from Example 3.14.
an extended pseudo metric on DGs.
Fix ρ ∈ (1, 6). Then, it is not possible to compute a ρ-approximation to d dynG I (G X , G Y ) between DGs in time polynomial in |X |, |Y |, |crit(G X )|, and |crit(G Y )|, unless P = N P .
an ε-tripod between θ X and θ Y . For I ∈ Int, let us define the set map f R,I :I θ X → I ε θ Y as f R,I ([x]) := [y] ∈ I ε θ Y where (x, y) ∈ R (this map is well-defined byRemark 4.15). Also, define g R,I : I θ Y → I ε θ X in a similar way. The collections f R := { f R,I } I ∈Int and g R := {g R,I } I ∈Int form an ε-interleaving pair between Reeb(θ X ) and Reeb(θ Y ), thus completing the proof.
Corollary 4. 16 .
16For any two formigrams θ X and θ Y , we have
Remark 4. 18 .
18By Remark 4.12 (ii), constant factor approximations to d F I cannot be obtained in polynomial time (Theorem A.7). The lower bound for d F I given by Proposition 4.13 can also lead to difficult computational problems (e.g. the graph-isomorphism problem [33, Section 5]). Hence, computing the lower bound for d F I given by Corollary 4.16 is a realistic approach to comparing formigrams. Another tractable lower bound for d F I is introduced in the next section; Theorem 6.35, which can sometimes be more discriminative than the lower bounds in Proposition 4.13 and Corollary 4.16.
(Y ) given by {y 1 } ≤ {y 1 y 2 } ≥ {y 2 }. Then, lim ← − − θ Y = which is the greatest lower bound of {y 1 }, {y 1 y 2 }, {y 2 } in SubPart(Y ). However, C (θ X ) is represented as the set diagram {•} → {•} ← {•} and thus lim ← − − C (θ X ) contains a single element.
Definition 5. 5 (
5Images). Given a morphism f : X → Y , an image of f (if it exists) is a mono m : I → Y such that there is a morphism f m : X → I such that f = m • f m , for any mono z : Z → Y and a morphism f z : X → Z with f = z • f z , there is a unique morphism u : I → Z such that m = z • u.
Images and coimages inPart. It is not difficult to check that every morphism f : (X , P X ) → (Y , P Y ) in Part is a mono. On the other hand, we have the following characterization of epis in Part. Proposition 5.7. A morphism f : (X , P X ) → (Y , P Y ) in Part is an epi if and only if f : X → Y is surjective (hence f is bijective).
Proposition 5. 8 .
8For any morphism f : (X , P X ) → (Y , P Y ) in Part,
. 3 .
3Let P := Int(ZZ) (Definition 2.11). Then, µ P (I , J ) I = J or I J with |J \ I | = 2, −1, I J with |J \ I | = 1, 0, otherwise.
Figure 9 :
9(A) A formigram θ X over the set X := {x 1 , x 2 , x 3 , x 4 } with support [0, 5] (i.e. union of the lifespans of all points) and the persistence clustergram dgm (θ X ). (B) An illustration of θ X : Int → SubPart(X ) for θ X in (A). The persistence clustergram dgm (θ X ) in (A) is the Möbius inversion of θ X . (C) A formigram θ Y over Y := {x 1 , x 2 , x 3 } and its persistence clustergram. Grey intervals carry silhouette values that are either negative or zero; −1 for {x 1 x 2 x 3 } − {x 1 } − {x 3 } and 0 for {x 1 |x 2 |x 3 } − {x 1 x 2 |x 3 } − {x 1 |x 2 x 3 } + {x 1 x 2 x 3 }. (D) An illustration of θ Y : Int → SubPart(X ) for θ Y in (C). The persistence clustergram dgm (θ Y ) in (C) is the Möbius inversion of this θ Y .
Notation 6. 5 (
5Neighborhood extension). Let I ∈ Int(ZZ).
(i) By I − we denote I ∪ {(i , j )} ∈ Int(ZZ) where (i , j ) ∈ ZZ is the "lower-left" neighborhood of I in ZZ. For example, for I = (−1, 1] ZZ (cf.Figure 3), we have I − = [−1, 1] ZZ . (ii) By I + we denote I ∪ {(i , j )} ∈ Int(ZZ) where (i , j ) ∈ ZZ is the "upper-right" neighborhood of I in ZZ. For example, for I = (−1, 1] ZZ , we have I + = (−1, 2) ZZ . (iii) By I ± we denote I − ∪ I + .
Example 6. 6 .
6Assume that for a function f : Int(ZZ) → G, there exists J 0 ∈ Int(ZZ) such that f (I ) = 0 unless J 0 ⊃ I . Let g : Int(ZZ) → G be such that f (I ) = J ⊃I g (J ) for all I ∈ Int(ZZ). Then, f (I ) = g (I ) = f (I ) − f (I − ) − f (I + ) + f (I ± ) for all I ∈ Int(ZZ).
Definition 6. 8 (
8Support and silhouette). Let G be an abelian group. Consider a map f : P → G.
(i) The support of f is defined as supp( f ) := {p ∈ P : f (p) = 0 ∈ G}.(ii) Assume that G = ZA. The silhouette f : P → Z of f is defined as follows. For
f
((1, 2) ZZ ) = {x 1 } + {x 2 } + {x 3 } − ({x 1 } + {x 2 , x 3 }) − ({x 1 , x 2 } + {x 3 }) + {x 1 , x 2 , x 3 }, and thus f ((1, 2) ZZ ) = 1 + 1 + 1 − (1 + 1) − (1 + 1) + 1 = 0. Next we consider f : Int(ZZ) → Z. By equations in (4) f ((1, 2) ZZ ) = 3, f ([1, 2) ZZ ) = f ((1, 2] ZZ ) = 2, f ([1, 2] ZZ ) = 1. Again from Example 6.6, we have f ((1, 2) ZZ ) = 3 − 2 − 2 + 1 = 0 which equals f ((1, 2) ZZ ).
Figures 10 (A) and (B) for examples; any pair (I ,G) ∈ Πdgm (θ X ) is depicted as the interval I annotated by G. Theorem 6.17. Let θ X be a formigram. A subset G ⊂ X is a maximal group of θ X on an interval I ⊂ R if and only if (I ,G) ∈ Πdgm (θ X ).
Figure 10 :
10(A) and (B) depict maximal group diagrams (cf. Definition 6.16) while (C) and (D) depict persistence clustergrams (cf. Definition 6.27).
Figure 11 :
11The maximal group diagram in
Figure 12 :
12(A) The formigram θ X over X := {x 1 ,
Example 6. 26 .
26Consider the formigram depicted in Figure 9 (A). For I := [0, 5], we have I θ X = {x 1 |x 2 |x 3 },
Definition 6. 27 .
27Let θ X be a formigram. The persistence clustergram dgm (θ X ) : Int(ZZ) → Zpow(X ) is defined as the Möbius inversion of θ X over the poset Int(ZZ) op . Namely, for all I ∈ Int(ZZ), dgm (θ X )(I ) :
as illustrated in the figure. Invoking Remark 6.28, let us compute θ X (1) = I 1 dgm (θ X )(I ). Since the silhouette of dgm (θ X ) is {[0, 2), [0, 4), [0, ∞)}, and 1 ∈ R belongs to all of [0, 2), [0, 4) and [0, ∞) we have:
, s ∈ [0, 1) and s ≤ t ≤ 2, Region (b) {{x 2 }}, t ∈ (2, 3] and 1 < s ≤ t , Region (c) , otherwise, Region (d), where regions (a),(b),(c), and (d) are marked in
Figure 1 (
1F)).
Example 7. 9 .
9For λ > 0, dis dyn λ (R) takes into account both spatial and temporal distortion of the tripod R between γ X and γ Y : (i) (Spatial distortion) Let a, b ≥ 0. For the two metric spaces (X , d X ,a ) = {x, x }, 0 a a 0 and (X , d X ,b ) = {x, x }, 0 b b 0 , consider the two constant DMSs γ X ,a ≡ (X , d X ,a ) and γ X ,b ≡ (X , d X ,b ). Take the tripod R : X id X −−− − X id X − −−− X . Then, for λ > 0, it is easy to check that dis dyn λ (R) = |a−b| λ .
X , γ Y ) := min R dis dyn λ (R)where the minimum ranges over all tripods between X and Y . For simplicity, when λ = 0, we write d
Fix ρ ∈ (1, 6) and let γ X and γ Y be piecewise linear DMSs. Then, it is not possible to compute a ρ-approximation to d dynM I (γ X , γ Y ) in time depending polynomially on |X |, |Y |, |S X |, and |S Y |, unless P = N P . Theorem 7.15 will be proved in Appendix D. More examples about d dynM I are provided in Appendix D as well. The theorem above indicates that computing the lower bound for d dynM I
d
B (A , B) := inf{δ ∈ [0, ∞) : ∃δ-matching between A and B}.
Definition A. 8 .
8Let I ∈ Int(ZZ) and let N : I → set by any functor. Let c ∈ lim − − → N . We define the support of c as supp(c) := {k ∈ I : ∃x k ∈ N k , i k (x k ) = c}. In particular, if supp(c) = I , we call c a full component of the functor N . Given M : ZZ → set and I ∈ Int(ZZ), we denote the number of full components of M | I by full(M | I ). Recall Notation 6.5. Theorem A.9 ([53, Corollary 4.10]). For any functor M : ZZ → set, the multiplicity of I in barc(F • M ) is full(M | I ) − full(M | I + ) − full(M | I − ) + full(M | I ± ).
Figure 14 :
14Two weighted Reeb graphs in Example B.4 (ii).
Figure 15 :
15An illustration for Proposition C.2. Top: The Reeb graph of a formigram θ X and its barcode. Bottom:
Figure 16 :
16An illustration of [b, a] ZZ for a < b in Z.
Example D. 3 (
3An interleaved pair of DMSs II). Fix the two-point metric space (X , d X ) = {x, x }
X
) = min τ, 2π ω − τ =: η(ω, τ).
, Mémoli, and Stefanou showed in[56] that
Categories that are considered throughout the paper. Those are connected by the functors inCategories
Objects
Morphisms
Graph(X )
Subgraphs of the complete graph on X
Inclusion
Def. 2.1
SubPart(X ) Subpartitions of X
Refinement
Def. 2.2
Part
Pairs (X , P X ) where X is a finite set
and P X is a partition of X
Block-preserving injective maps Def. 2.7
ωset
Pairs (X , w X ) where X is a finite set and
a weight function w X : X → N
Weight-observing maps
Def. 2.8
set
Finite sets
Set maps
vec
Fin. dim. vector spaces
Linear maps
Table 1:
Table 2 :
2Functors that connect the categories in
Table
d F I is is an extended pseudo metric on formigrams, which can be proved in a similar way to d For unlabeled formigrams θ, θ : Int → Part (cf. Definition 2.15), the interleaving distance d Part I (θ, θ ) (cf. Definition 2.13) can easily be infinite and thus might not be useful in practice; it is not difficult to check that d Part I (θ, θ ) is finite only if the colimits of θ and θ have the same cardinality. In this respect, by Remark 3.6, d F I can also be viewed as a practical metric between unlabeled formigrams by defining d F IdynG
I
.
Remark 4.12.
(i)
In Section 7.3, we define the λ-slack interleaving distance ddynM
I,λ
between DMSs. Although this
metric was utilized in [54], all properties of d
dynM
I,λ
mentioned therein are proved in this paper, including the
fact that d
dynM
I,λ
Theorem 7.12. For each λ ≥ 0, d dynM I,λ is an extended metric between DMSs modulo isomorphism. In particular, for λ > 0, d dynM I,λis a metric between bounded DMSs modulo isomorphism.The proof of Theorem 7.12 together with details pertaining to the following facts are deferred to Appendix A.6.2:(i) For λ > 0, d dynM I,λgeneralizes the Gromov-Hausdorff distance (cf. Remark A.14).(ii) The metrics d dynM I,λ , for different λ > 0, are bilipschitz-equivalent (cf. Proposition A.15). (iii) In Proposition D.1 we will elucidate a link between d dynM I and the Gromov-Hausdorff distance. This link will be useful for determining the computational complexity of d dynM I (cf. Theorem 7.15).
be a tripod between X and Y . Then, the distortion of R is defined asdis(R) := sup
z,z ∈Z
The name formigram is a combination of the words formicarium and diagram. A formicarium or ant farm is an enclosure for keeping ants under semi-natural conditions[84]. Visually, a formigram is reminiscent of a formicarium (cf.Figure 1 (B)).
The function c associated to F is not unique and thus we refer to im(c) as 'a' set of critical points and not as 'the' set of critical points.
F ([t , t ])is said to be the costalk of F at t in the literature (e.g.[13,30]).
Silhouettes in this paper have no relation with the persistence silhouettes in[24].
In this section groups do not stand for groups in abstract algebra. See Remark 6.18.
These are formigrams by Proposition 7.5 and Definition 3.10.
dis(R)where the infimum is taken over all tripods R between X and Y . In particular, any tripod R between X and Y is said to be an ε-tripod between (X , d X ) and (Y , d Y ) if dis(R) ≤ ε. Proposition A.3. Let (X , d X ) and (Y , d Y ) be any two finite metric spaces. Then, there exist two DGs G X = (V X (·), E X (·)) and G Y = (V Y (·), E Y (·)) corresponding to (X , d X ) and (Y , d Y ) respectively such thatProof. Let T be the diameter of (X , d X ). For t ∈ R, we define:We define G X by t → (V X (t ), E X (t )). Define G Y similarly. We show that d dynG ITo this end, suppose that for some ε ≥ 0, R : Xis any ε-tripod between G X and G Y (Definition 4.4). Then, by the construction of G X , G Y , it must hold that dX) can be similarly proved.Definition A.4. An ultrametric space is a metric space (X , d ) in which the following ultra-triangle inequality holds: for all x, y, z ∈ X , d (x, z) ≤ max d (x, y), d (y, z) .If (X , d ) were a pseudometric, then d is called an ultra-pseudometric.Proof of Theorem 4.7 Pick any two ultrametric spaces (X , u X ) and (Y , u Y ). Then, by Proposition A.3, there exist DGs G X = (V X (·), E X (·)) and G Y = (V Y (·), E Y (·)) such that the interleaving distance between G X and G Y is identical to twice the Gromov-Hausdorff distance ∆ := d GH ((X , u X ), (Y , u Y )) between (X , u X ) and (Y , u Y ). However, according to[75,Corollary 3.8], ∆ cannot be approximated within any factor less than 3 in polynomial time, unless P = N P . The author shows this by observing that any instance of the 3-partition problem can be reduced to an instance of the bottleneck ∞-Gromov-Hausdorff distance (∞-BGHD) problem between ultrametric spaces (cf.[75, p.865]). The proof follows.A.3 Details about Remark 4.18Remark A.5 (Interleaving between dendrograms). When θ X , θ Y are dendrograms over sets X and Y respec-Since both θ X and θ Y get coarser as t ∈ R increases, the interleaving condition in Definition 4.11 can be rewritten as follows: for all t ∈ R it holds thatLet X be a finite set and let θ X be a dendrogram over X (cf. Remark 3.7). Recall from[19]that this θ X induces a canonical ultra-pseudometric u X : X × X → R + on X (cf. Definition A.4) defined by u X (x, x ) := inf{ε ≥ 0 : x, x belong to the same block of θ X (ε)}(13)Proposition A.6. Given any two dendrograms θ X , θ Y over sets X , Y , respectively, let u X , u Y be the canonical ultra-pseudometrics on X and Y , respectively. Then, d F I (θ X , θ Y ) = 2 d GH ((X , u X ), (Y , u Y )).A.5 From unlabeled formigrams to persistent cluster counting functorsLet θ X be a formigram. We shall prove that the persistent counting functor | θ X | (cf. Definition 6.33) can be obtained from the unlabeled formigram of θ X (cf. Definition 3.12 (i)).Proposition A.10. Let θ be the unlabeled formigram of θ X . For any I ∈ Int(ZZ), consider the canonical limitto-colimit morphism ϕ I : lim ← − − θ| I → lim − − → θ| I in the category Part. Then, coim(ϕ I ) ∼ = ( t ∈I θ X (t ), I θ X ).Proof. By Proposition 5.1, lim ← − − θ| I ∼ = I θ X and lim − − → θ| I ∼ = I θ X , and the morphism ϕ I in Part is the inclusion t ∈I θ X (t ) → t ∈I θ X (t ). Now by Proposition 5.8 (ii) the desired isomorphism follows.Proposition A.10 implies that we can extract | θ X | from θ: Namely, | θ X | (I ) = | I θ X | equals the number of blocks in the second entry of coim(ϕ I ). Reciprocally, one may wonder whether | θ X | contains enough information to reconstruct θ. That is not true; there exists a pair of formigrams which have identical persistent cluster functor, whereas their underlying weighted/unweighted Reeb graphs are different. This implies that their unlabeled formigrams are also different.A.6 Details from Section 7A.6.1 Details from Section 7.2Proof of Proposition 7.5. Clearly, R 1 δ (γ X ) is a function R → Graph(X ). We show that R 1 δ (γ X ) is cosheaf-inducing (Definition 2.17). First we prove that locally R 1 δ (γ X ) admits only finitely many points of discontinuity (those points are called critical points). Let I ⊂ R be any nonempty finite interval. For i , j ∈ X := {1, . . . , n}, let f i , j := d X (·)(i , j ) : R → R + . Note that discontinuity points of R 1 δ (γ X ) can occur only at endpoints of connected components of the set f i , j −1 (δ) for some i , j ∈ X . Fix any i , j ∈ X . Then, by Definition 7.4, the set f i , j −1 (δ) ∩ I has only finitely many connected components and thus there are only finitely many endpoints arising from those components. Since the set X is finite, this implies that R 1 δ (γ X ) can have only finitely many points of discontinuity in I . Fix any point c ∈ R on which R 1 δ (γ X ) is discontinuous. Consider the following two subsets of X × X :The continuity of d X (·)(i , j ) for each (i , j ) ∈ X × X guarantees that there exists ε > 0 such thatand in turn for λ > 0 can be found in[54].between any two bounded DMSs is finite. More specifically, for any r -bounded DMSs γ X = (X , d X (·)) and γ Y = (Y , d Y (·)) for some r > 0, any tripod R between X and Y is a (λ, r λ )-tripod between γ X and γ Y . This implies thatDefinition A.12 (Equivalent tripods). Let X , Y be any two sets. For any two tripods R :Remark A.13. Let γ X = (X , d X (·)) and γ Y = (Y , d Y (·)) be any two DMSs. Suppose that R and S are equivalent tripods between X and Y (Definition A.12). Then, it is not difficult to check that for any λ, ε ≥ 0, R is a (λ, ε)tripod between γ X and γ Y if and only if S is a (λ, ε)-tripod between γ X and γ Y .Proof of Theorem 7.12. We prove the triangle inequality. Take any DMSs γ X , γ Y and γ W over X ,Y and W , respectively. For some ε, ε > 0, lettripod between γ X and γ Y and (λ, ε )-tripod between γ Y and γ W (Definition 7.8), respectively. Consider the set Z := (z 1 , z 2 ) ∈ Z 1 × Z 2 : ϕ Y (z 1 ) = ψ Y (z 2 ) and let π 1 : Z → Z 1 and π 2 : Z → Z 2 be the canonical projections to the first and the second coordinate, respectively. Define the tripod R 2 • R 1 between X and W as in equation (1). It is not difficult to check that R 2 • R 1 is a (λ, ε + ε )-tripod between γ X and γ W and thus we haveWe outline the proof of the fact that γ X and γ Y are isomorphic (Definition 7.3). Because there are only finitely many tripods between X and Y up to equivalence (Definition A.12),R becomes an (λ, ε)-tripod between γ X and γ Y for any ε > 0. In order to show that γ X and γ Y are isomorphic, one needs to prove that that R is in fact (λ, 0)-tripod. After that, invoke Definition 7.1, (ii) and (iii) to verify thatWe have the following bilipschitz-equivalence relation between the metrics d dynM I,λ for different λ > 0.Proof. Fix any two DMSs γ X and γ Y over X and Y . That d(γ X , γ Y ) follows from the observation that any (λ, ε)-tripod R between γ X and γ Y is also a (λ , ε)-tripod (Definition 7.8). We next proveFor some ε ≥ 0 let R be any (λ , ε)-tripod between γ X and γ Y . It suffices to show that R is also a (λ, λ λ ε)-tripod. Fix any t ∈ T. Then,By symmetry, we also haveThe proof is rather trivial and thus we omit it. In general, a realization of a weighted Reeb graph as an unlabeled formigram is not unique; see Example 3.13. Proposition B.1 allows us to define the following dissimilarity measure on weighted Reeb graphs. From equation(2), recall how to define d F I between unlabeled formigrams. For all weighted Reeb graphs F,G : Int → ωset, we define:Since a realization of a weighted Reeb graph as an unlabeled formigram is not necessarily unique, we have to possibly take into account multiple realizations of F and G to compute W (F,G). This leads to the fact that W does not necessarily satisfy the triangle inequality and thus we consider the maximal sub-dominant metric of W[21]as a metric on weighted Reeb graphs:Definition B.2 (Metric on weighted Reeb graphs). For any two weighted Reeb graphsProof. From the definition of d R I and Proposition 4.13, we have thatSince d ωR I is the greatest metric on weighted Reeb graphs among those upper bounded by W , the left inequality in(16)follows. The right inequality in(16)(ii) Let F and G be weighted Reeb graphs depicted inFigure 14. Their unweighted Reeb graphs A • F and A • G are clearly isomorphic and thus d ROn the other hand, d ωR I (F,G) = 1/2; this follows from the observation that both F and G are uniquely realized (up to natural isomorphism) by unlabeled formigrams θ and θ , which leads to d ωR I (F,G) = d F I (θ, θ ). Also, it is not difficult to check that d F I (θ, θ ) = 1/2.C Smoothing formigramsThe goal of this section is to establish a few basic properties of smoothing of formigrams. In particular, we reveal its effect on the zigzag barcodes of formigrams and its compatibility with smoothing of Reeb graphs in[33]; see Propositions C.2 and C.5.Recall that a formigram θ X will be regarded as either a cosheaf-inducing function R → SubPart(X ) or a constructible cosheaf Int → SubPart(X ) (cf. Definition 3.5, Remark 2.18 (i) and (ii)). By Definition 2.23 and Proposition 2.25, a smoothing operation on formigrams can be induced via the join operation on subpartitions. Namely, S ε θ X sends each I ∈ Int to I ε θ X := {θ X (t ) : t ∈ I ε }.Remark C.1 (Comparison with robust grouping structure). The use of the join operation is an important element that distinguishes our notion of smoothing from the robust grouping structure in[15]. In particular, given a dynamic metric space (DMS), the induced formigram of this DMS (which is obtained by combining Definition 3.10 and Proposition 7.5) can be smoothed out using the join operation. We emphasize that this smoothing operation is intrinsic in contrast to the robust grouping structure from[15]. Namely, our smoothing operation can be carried out without constructing any topological space in the spatiotemporal ambient space of the DMS as illustrated in [15,Figure 11]. One consequence of this 'intrinsicality' is that, when a dynamic graph is the input data (as opposed to a dynamic metric space), we can smooth out its induced formigram (cf. Definition 3.10), while[15]does not propose such a method. Since the coordinates of entities are not always available in applications (e.g. sensor networks[31,32], low-cost swarm robots[73], etc.), this intrinsicality is a desirable property.Given a formigram θ X ,Figure 15illustrates both the relationship between Reeb(θ X ) and Reeb(S ε θ X ) and the relationship between their zigzag barcodes. The following proposition precisely describes the relationship between barc ZZ (θ X ) and barc ZZ (S ε θ X ). For any r ∈ R, we define −∞ + r to be −∞.Proposition C.2.Let θ X be a formigram over X and let ε ≥ 0. Then, we have the following bijection between barc ZZ (θ X ) and barc ZZ (S ε θ X ) (cf.Figure 15):Recall the free functor F : set → vec (cf. Definition 2.6) and the fact that any constructible cosheaf Int → vec is interval decomposable (cf. Proposition 2.19).Figure 16).Proposition C.5.Let θ X be a formigram over X . Then, for any ε ≥ 0, Reeb(S ε θ X ) = S ε Reeb(θ X ).Proof. Let I ∈ Int. We have:Reeb(S ε θ X )(I ) = (C • (S ε θ X )) (I ) by Definitions 2.9 and 5.3by Definition 2.23Formigrams change in a continuous manner under ε-smoothing:Proposition C.6. For any ε ≥ 0 and any formigram θ X ,The following proposition is analogous to[33,Proposition 4.14]:Proposition C.7. For any ε ≥ 0, S ε is a contraction on formigrams, i.e. for any formigrams θ X and θ YProof. For δ ≥ 0, assume that R :We claim that R is also a δ-tripod between S ε θ X and S ε θ Y . First, we remark that ϕ * X S ε θ X = S ε ϕ * X θ X . Indeed, for any I ∈ Int, (ϕ * X S ε θ X )(I ) = ϕ * X (S ε θ Y (I )) = ϕ * X (θ X (I ε )) = (S ε ϕ * X θ X )(I ). Therefore,and by symmetry we have S δ ϕ * Y S ε θ Y ≥ ϕ * X S ε θ X , completing the proof.D About the 0-slack interleaving distance between DMSsWe clarify the computational complexity of d[74,75]we obtain the claim of Theorem 7.15.Given a ultrametric space (X , u X ), define a DMS D(X , u X ) := (X , d X (·)) where for all x, x ∈ X and for all t ∈ R, d X (t )(x, x ) := max(0, u X (x, x ) − t ). It is noteworthy that for any x, x ∈ X , d X (·)(x, x ) : R → R + is decreasing down to zero and that d X (0) = u X , a legitimate metric (i.e. not just pseudo-metric), satisfying the second item of Definition 7.1. Furthermore, note that D(X , u X ) is clearly piecewise linear and that the set of breakpoints isProposition D.1. For any two ultrametric spaces (X , u X ) and (Y , u Y ) we haveProof. Let D(X , u X ) = (X , d X (·)) and D(Y , u Y ) = (Y , d Y (·)). Observe that for any x, x ∈ X , any t ∈ R, and any ε ≥ 0, min s∈[t ] ε d X (s)(x, x ) = d X (t + ε)(x, x ) since d X is decreasing over time. Thus, for some ε ≥ 0, a tripod2) if and only if for all z, z ∈ Z and for all t ∈ R, d X (t +ε) ϕ X (z), ϕ X (z ) ≤ d Y (t ) ϕ Y (z), ϕ Y (z ) and d Y (t +ε) ϕ Y (z), ϕ Y (z ) ≤ d X (t ) ϕ X (z), ϕ X (z ) , if and only if for all z, z ∈ Z and for all t ∈ R, max 0, u X ϕ X (z), ϕ X (z ) − t − ε ≤ max 0, u Y ϕ Y (z), ϕ Y (z ) − t and max 0, u Y ϕ Y (z), ϕ Y (z ) − t − ε ≤ max 0, u X ϕ X (z), ϕ X (z ) − t if and only if for all z, z ∈ Z ,completing the proof.Proof of Theorem 7.15 Pick any two ultrametric spaces (X , u X ) and (Y , u Y ). Then, by Proposition D.1, the interleaving distance between D(X , u X ) and D(Y , u Y ) is identical to twice the Gromov-Hausdorff distance ∆ := d GH ((X , u X ), (Y , u Y )) between (X , u X ) and (Y , u Y ). The rest of the proof follows along the same lines as that of Theorem 4.7.Next we discuss a few computational examples of d dynM I. Let ψ : R → R + be any non identically zero continuous function. Then, for any finite metric space (X , d X ) we have the DMS γ ψ X = (X , d ψ X (·)) where for t ∈ R, dFigure 17).
Evasion paths in mobile sensor networks. H Adams, G Carlsson, The International Journal of Robotics Research. 341H. Adams and G. Carlsson. Evasion paths in mobile sensor networks. The International Journal of Robotics Research, 34(1):90-104, 2015.
Efficient evader detection in mobile sensor networks. H Adams, D Ghosh, C Mask, W Ott, K Williams, arXiv:2101.09813arXiv preprintH. Adams, D. Ghosh, C. Mask, W. Ott, and K. Williams. Efficient evader detection in mobile sensor net- works. arXiv preprint arXiv:2101.09813, 2021.
Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt's theorem. G Azumaya, Nagoya Mathematical Journal. 1G. Azumaya et al. Corrections and supplementaries to my paper concerning Krull-Remak-Schmidt's the- orem. Nagoya Mathematical Journal, 1:117-124, 1950.
Measuring distance between Reeb graphs. U Bauer, X Ge, Y Wang, Proceedings of the thirtieth annual symposium on Computational geometry. the thirtieth annual symposium on Computational geometryU. Bauer, X. Ge, and Y. Wang. Measuring distance between Reeb graphs. In Proceedings of the thirtieth annual symposium on Computational geometry, pages 464-473, 2014.
Induced matchings and the algebraic stability of persistence barcodes. U Bauer, M Lesnick, Journal of Computational Geometry. 62U. Bauer and M. Lesnick. Induced matchings and the algebraic stability of persistence barcodes. Journal of Computational Geometry, 6(2):162-191, 2015.
Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphs. U Bauer, E Munch, Y Wang, of Leibniz International Proceedings in Informatics (LIPIcs). L. Arge and J. PachGermany34Schloss Dagstuhl-Leibniz-Zentrum fuer InformatikU. Bauer, E. Munch, and Y. Wang. Strong Equivalence of the Interleaving and Functional Distortion Met- rics for Reeb Graphs. In L. Arge and J. Pach, editors, 31st International Symposium on Computational Geometry (SoCG 2015), volume 34 of Leibniz International Proceedings in Informatics (LIPIcs), pages 461- 475, Dagstuhl, Germany, 2015. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.
Reporting flock patterns. M Benkert, J Gudmundsson, F Hübner, T Wolle, Computational Geometry. 413M. Benkert, J. Gudmundsson, F. Hübner, and T. Wolle. Reporting flock patterns. Computational Geometry, 41(3):111-125, 2008.
Lattice theory. G Birkhoff, American Mathematical Society25New YorkG. Birkhoff. Lattice theory, volume 25. American Mathematical Society New York, 1948.
On the stability of interval decomposable persistence modules. H B Bjerkevik, Discrete & Computational Geometry. 661H. B. Bjerkevik. On the stability of interval decomposable persistence modules. Discrete & Computational Geometry, 66(1):92-121, 2021.
Graph theory (graduate texts in mathematics). J Bondy, U Murty, SpringerNew YorkJ. Bondy and U. Murty. Graph theory (graduate texts in mathematics). Springer New York, 2008.
Algebraic stability of zigzag persistence modules. Algebraic & geometric topology. M Botnan, M Lesnick, 18M. Botnan and M. Lesnick. Algebraic stability of zigzag persistence modules. Algebraic & geometric topol- ogy, 18(6):3133-3204, 2018.
Interval decomposition of infinite zigzag persistence modules. M B Botnan, Proceedings Of The American Mathematical Society. 1458M. B. Botnan. Interval decomposition of infinite zigzag persistence modules. Proceedings Of The American Mathematical Society, 145(8):3571-3577, 2017.
Sheaf theory. G E Bredon, Springer Science & Business Media170G. E. Bredon. Sheaf theory, volume 170. Springer Science & Business Media, 2012.
Categorification of persistent homology. P Bubenik, J A Scott, Discrete & Computational Geometry. 513P. Bubenik and J. A. Scott. Categorification of persistent homology. Discrete & Computational Geometry, 51(3):600-627, 2014.
. K Buchin, M Buchin, M J Van Kreveld, B Speckmann, F Staals, Trajectory grouping structure. JoCG. 61K. Buchin, M. Buchin, M. J. van Kreveld, B. Speckmann, and F. Staals. Trajectory grouping structure. JoCG, 6(1):75-98, 2015.
A Course in Metric Geometry. D Burago, Y Burago, S Ivanov, AMS Graduate Studies in Math. 33American Mathematical SocietyD. Burago, Y. Burago, and S. Ivanov. A Course in Metric Geometry, volume 33 of AMS Graduate Studies in Math. American Mathematical Society, 2001.
Topology and data. G Carlsson, Bulletin of the American Mathematical Society. 462G. Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255-308, 2009.
Zigzag persistence. G Carlsson, V. De Silva, Foundations of computational mathematics. 104G. Carlsson and V. De Silva. Zigzag persistence. Foundations of computational mathematics, 10(4):367- 405, 2010.
Characterization, stability and convergence of hierarchical clustering methods. G Carlsson, F Mémoli, Journal of Machine Learning Research. 11G. Carlsson and F. Mémoli. Characterization, stability and convergence of hierarchical clustering methods. Journal of Machine Learning Research, 11:1425-1470, 2010.
Multiparameter hierarchical clustering methods. G Carlsson, F Mémoli, Classification as a Tool for Research. SpringerG. Carlsson and F. Mémoli. Multiparameter hierarchical clustering methods. In Classification as a Tool for Research, pages 63-70. Springer, 2010.
Classifying clustering schemes. G Carlsson, F Mémoli, Foundations of Computational Mathematics. 132G. Carlsson and F. Mémoli. Classifying clustering schemes. Foundations of Computational Mathematics, 13(2):221-252, 2013.
The theory of multidimensional persistence. G Carlsson, A Zomorodian, Discrete & Computational Geometry. 421G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. Discrete & Computational Geometry, 42(1):71-93, 2009.
Proximity of persistence modules and their diagrams. F Chazal, D Cohen-Steiner, M Glisse, L J Guibas, S Oudot, Proc. 25th ACM Sympos. 25th ACM SymposF. Chazal, D. Cohen-Steiner, M. Glisse, L. J. Guibas, and S. Oudot. Proximity of persistence modules and their diagrams. In Proc. 25th ACM Sympos. on Comput. Geom., pages 237-246, 2009.
Stochastic convergence of persistence landscapes and silhouettes. F Chazal, B T Fasy, F Lecci, A Rinaldo, L Wasserman, Proceedings of the thirtieth annual symposium on Computational geometry. the thirtieth annual symposium on Computational geometryF. Chazal, B. T. Fasy, F. Lecci, A. Rinaldo, and L. Wasserman. Stochastic convergence of persistence land- scapes and silhouettes. In Proceedings of the thirtieth annual symposium on Computational geometry, pages 474-483, 2014.
Explicit geodesics in Gromov-Hausdorff space. S Chowdhury, F Mémoli, Electronic Research Announcements. 25S. Chowdhury and F. Mémoli. Explicit geodesics in Gromov-Hausdorff space. Electronic Research An- nouncements, 25:48-59, 2018.
Zigzag persistent homology and dynamic networks. N Clause, 2021N. Clause. Zigzag persistent homology and dynamic networks. https://github.com/ndag/DynGraphZZ, 2021.
Spatiotemporal persistent homology computation tool. N Clause, W Kim, 2020N. Clause and W. Kim. Spatiotemporal persistent homology computation tool. https://github.com/ ndag/PHoDMSs, 2020.
Stability of persistence diagrams. Discrete & Computational Geometry. D Cohen-Steiner, H Edelsbrunner, J Harer, 37D. Cohen-Steiner, H. Edelsbrunner, and J. Harer. Stability of persistence diagrams. Discrete & Computa- tional Geometry, 37(1):103-120, 2007.
Classification of constructible cosheaves. Theory and Applications of Categories. J Curry, A Patel, 35J. Curry and A. Patel. Classification of constructible cosheaves. Theory and Applications of Categories, 35(27):1012-1047, 2020.
J M Curry, Sheaves, cosheaves and applications. University of PennsylvaniaJ. M. Curry. Sheaves, cosheaves and applications. University of Pennsylvania, 2014.
Coordinate-free coverage in sensor networks with controlled boundaries via homology. V , De Silva, R Ghrist, The International Journal of Robotics Research. 2512V. De Silva and R. Ghrist. Coordinate-free coverage in sensor networks with controlled boundaries via homology. The International Journal of Robotics Research, 25(12):1205-1222, 2006.
Homological sensor networks. V. De Silva, R Ghrist, Notices of the American mathematical society. 541V. De Silva, R. Ghrist, et al. Homological sensor networks. Notices of the American mathematical society, 54(1), 2007.
Categorified Reeb graphs. V. De Silva, E Munch, A Patel, Discrete & Computational Geometry. 554V. De Silva, E. Munch, and A. Patel. Categorified Reeb graphs. Discrete & Computational Geometry, 55(4):854-906, 2016.
Computing zigzag persistence on graphs in near-linear time. T K Dey, T Hou, of Leibniz International Proceedings in Informatics (LIPIcs). 189:15. Schloss Dagstuhl -Leibniz-Zentrum für InformatikT. K. Dey and T. Hou. Computing zigzag persistence on graphs in near-linear time. In 37th International Symposium on Computational Geometry (SoCG 2021), volume 189 of Leibniz International Proceedings in Informatics (LIPIcs), pages 30:1-30:15. Schloss Dagstuhl -Leibniz-Zentrum für Informatik, 2021.
Updating zigzag persistence and maintaining representatives over changing filtrations. T K Dey, T Hou, arXiv:2112.02352arXiv preprintT. K. Dey and T. Hou. Updating zigzag persistence and maintaining representatives over changing filtra- tions. arXiv preprint arXiv:2112.02352, 2021.
The edit distance for Reeb graphs of surfaces. B , Di Fabio, C Landi, Discrete & Computational Geometry. 552B. Di Fabio and C. Landi. The edit distance for Reeb graphs of surfaces. Discrete & Computational Geom- etry, 55(2):423-461, 2016.
Persistent homology-a survey. H Edelsbrunner, J Harer, Contemporary mathematics. 453H. Edelsbrunner, J. Harer, et al. Persistent homology-a survey. Contemporary mathematics, 453:257-282, 2008.
Topological persistence and simplification. H Edelsbrunner, D Letscher, A Zomorodian, Discrete Comput. Geom. 28H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28:511-533, 2002.
Unzerlegbare darstellungen i. P Gabriel, Manuscripta mathematica. 61P. Gabriel. Unzerlegbare darstellungen i. Manuscripta mathematica, 6(1):71-103, 1972.
Applied topology in static and dynamic sensor networks. J Gamble, H Chintakunta, H Krim, 2012 International Conference on Signal Processing and Communications (SPCOM). IEEEJ. Gamble, H. Chintakunta, and H. Krim. Applied topology in static and dynamic sensor networks. In 2012 International Conference on Signal Processing and Communications (SPCOM), pages 1-5. IEEE, 2012.
R Ghrist, H Riess, arXiv:2007.04099Cellular sheaves of lattices and the Tarski Laplacian. arXiv preprintTo appear in Homology, Homotopy and ApplicationsR. Ghrist and H. Riess. Cellular sheaves of lattices and the Tarski Laplacian. arXiv preprint arXiv:2007.04099, To appear in Homology, Homotopy and Applications, 2022.
Spatiotemporal barcodes for image sequence analysis. R Gonzalez-Diaz, M.-J Jimenez, B Medrano, International Workshop on Combinatorial Image Analysis. SpringerR. Gonzalez-Diaz, M.-J. Jimenez, and B. Medrano. Spatiotemporal barcodes for image sequence analysis. In International Workshop on Combinatorial Image Analysis, pages 61-70. Springer, 2015.
Computing longest duration flocks in trajectory data. J Gudmundsson, M Van Kreveld, Proceedings of the 14th annual ACM international symposium on Advances in geographic information systems. the 14th annual ACM international symposium on Advances in geographic information systemsACMJ. Gudmundsson and M. van Kreveld. Computing longest duration flocks in trajectory data. In Proceedings of the 14th annual ACM international symposium on Advances in geographic information systems, pages 35-42. ACM, 2006.
Efficient detection of patterns in 2d trajectories of moving points. J Gudmundsson, M Van Kreveld, B Speckmann, Geoinformatica. 112J. Gudmundsson, M. van Kreveld, and B. Speckmann. Efficient detection of patterns in 2d trajectories of moving points. Geoinformatica, 11(2):195-215, 2007.
Visual detection of structural changes in time-varying graphs using persistent homology. M Hajij, B Wang, C Scheidegger, P Rosen, arXiv:1707.066833arxiv preprint. arXiv preprintM. Hajij, B. Wang, C. Scheidegger, and P. Rosen. Visual detection of structural changes in time-varying graphs using persistent homology. arxiv preprint. arXiv preprint arXiv:1707.06683, 3, 2017.
Modeling herds and their evolvements from trajectory data. Y Huang, C Chen, P Dong, International Conference on Geographic Information Science. SpringerY. Huang, C. Chen, and P. Dong. Modeling herds and their evolvements from trajectory data. In Interna- tional Conference on Geographic Information Science, pages 90-105. Springer, 2008.
Mining mobile group patterns: A trajectory-based approach. S.-Y Hwang, Y.-H Liu, J.-K Chiu, E.-P Lim, PAKDD. Springer3518S.-Y. Hwang, Y.-H. Liu, J.-K. Chiu, and E.-P. Lim. Mining mobile group patterns: A trajectory-based ap- proach. In PAKDD, volume 3518, pages 713-718. Springer, 2005.
Mathematical taxonomy. N Jardine, R Sibson, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons LtdN. Jardine and R. Sibson. Mathematical taxonomy. John Wiley & Sons Ltd., London, 1971. Wiley Series in Probability and Mathematical Statistics.
Discovery of convoys in trajectory databases. H Jeung, M L Yiu, X Zhou, C S Jensen, H T Shen, Proceedings of the VLDB Endowment. 11H. Jeung, M. L. Yiu, X. Zhou, C. S. Jensen, and H. T. Shen. Discovery of convoys in trajectory databases. Proceedings of the VLDB Endowment, 1(1):1068-1080, 2008.
On discovering moving clusters in spatio-temporal data. P Kalnis, N Mamoulis, S Bakiras, SSTD. Springer3633P. Kalnis, N. Mamoulis, and S. Bakiras. On discovering moving clusters in spatio-temporal data. In SSTD, volume 3633, pages 364-381. Springer, 2005.
Stable signatures for dynamic graphs and dynamic metric spaces via zigzag persistence. W Kim, F Memoli, arXiv:1712.04064v4arXiv preprintW. Kim and F. Memoli. Stable signatures for dynamic graphs and dynamic metric spaces via zigzag persis- tence. arXiv preprint arXiv:1712.04064v4, 2017.
Formigrams: Clustering summaries of dynamic data. W Kim, F Mémoli, Proceedings of 30th Canadian Conference on Computational Geometry (CCCG18). 30th Canadian Conference on Computational Geometry (CCCG18)W. Kim and F. Mémoli. Formigrams: Clustering summaries of dynamic data. In Proceedings of 30th Cana- dian Conference on Computational Geometry (CCCG18), 2018.
Generalized persistence diagrams for persistence modules over posets. W Kim, F Mémoli, Journal of Applied and Computational Topology. 54W. Kim and F. Mémoli. Generalized persistence diagrams for persistence modules over posets. Journal of Applied and Computational Topology, 5(4):533-581, 2021.
Spatiotemporal persistent homology for dynamic metric spaces. Discrete & Computational Geometry. W Kim, F Mémoli, 66W. Kim and F. Mémoli. Spatiotemporal persistent homology for dynamic metric spaces. Discrete & Com- putational Geometry, 66(3):831-875, 2021.
Analysis of dynamic graphs and dynamic metric spaces via zigzag persistence. W Kim, F Mémoli, Z Smith, Topological data analysis. SpringerW. Kim, F. Mémoli, and Z. Smith. Analysis of dynamic graphs and dynamic metric spaces via zigzag per- sistence. In Topological data analysis, pages 371-389. Springer, 2020.
Interleaving by parts for persistence in a poset. W Kim, F Mémoli, A Stefanou, arXiv:1912.04366arXiv preprintW. Kim, F. Mémoli, and A. Stefanou. Interleaving by parts for persistence in a poset. arXiv preprint arXiv:1912.04366, 2019.
Swarm: Mining relaxed temporal moving object clusters. Z Li, B Ding, J Han, R Kays, Proceedings of the VLDB Endowment. the VLDB Endowment3Z. Li, B. Ding, J. Han, and R. Kays. Swarm: Mining relaxed temporal moving object clusters. Proceedings of the VLDB Endowment, 3(1-2):723-734, 2010.
Categories for the working mathematician. S Mac Lane, Springer Science & Business Media5S. Mac Lane. Categories for the working mathematician, volume 5. Springer Science & Business Media, 2013.
Bottleneck stability for generalized persistence diagrams. A Mccleary, A Patel, Proceedings of the. theAmerican Mathematical Society148A. McCleary and A. Patel. Bottleneck stability for generalized persistence diagrams. Proceedings of the American Mathematical Society, 148(7):3149-3161, 2020.
Edit distance and persistence diagrams over lattices. A Mccleary, A Patel, arXiv:2010.07337arXiv preprintA. McCleary and A. Patel. Edit distance and persistence diagrams over lattices. arXiv preprint arXiv:2010.07337, 2020.
F Mémoli, arXiv:1704.03965A distance between filtered spaces via tripods. arXiv preprintF. Mémoli. A distance between filtered spaces via tripods. arXiv preprint arXiv:1704.03965, 2017.
Theory of categories. B Mitchell, Academic Press17B. Mitchell. Theory of categories, volume 17. Academic Press, 1965.
Interleaving distance between merge trees. Discrete and Computational Geometry. D Morozov, K Beketayev, G Weber, 49D. Morozov, K. Beketayev, and G. Weber. Interleaving distance between merge trees. Discrete and Compu- tational Geometry, 49:22-45, 2013.
Applications of persistent homology to time varying systems. E Munch, Duke UniversityPhD thesisE. Munch. Applications of persistent homology to time varying systems. PhD thesis, Duke University, 2013.
Animal groups in three dimensions: how species aggregate. J K Parrish, W M Hamner, Cambridge University PressJ. K. Parrish and W. M. Hamner. Animal groups in three dimensions: how species aggregate. Cambridge University Press, 1997.
Reeb spaces and the robustness of preimages. A Patel, Duke UniversityA. Patel. Reeb spaces and the robustness of preimages. Duke University, 2010.
Generalized persistence diagrams. A Patel, Journal of Applied and Computational Topology. 13A. Patel. Generalized persistence diagrams. Journal of Applied and Computational Topology, 1(3):397-419, 2018.
Erosion distance for generalized persistence modules. Homology, Homotopy and Applications. V Puuska, 22V. Puuska. Erosion distance for generalized persistence modules. Homology, Homotopy and Applications, 22(1):233-254, 2020.
Flocks, herds and schools: A distributed behavioral model. C W Reynolds, ACM SIGGRAPH computer graphics. 214C. W. Reynolds. Flocks, herds and schools: A distributed behavioral model. ACM SIGGRAPH computer graphics, 21(4):25-34, 1987.
Stable and consistent density-based clustering. A Rolle, L Scoccola, arXiv:2005.09048arXiv preprintA. Rolle and L. Scoccola. Stable and consistent density-based clustering. arXiv preprint arXiv:2005.09048, 2020.
Community discovery in dynamic networks: a survey. G Rossetti, R Cazabet, ACM Computing Surveys (CSUR). 512G. Rossetti and R. Cazabet. Community discovery in dynamic networks: a survey. ACM Computing Surveys (CSUR), 51(2):1-37, 2018.
On the foundations of combinatorial theory i. theory of Möbius functions. G.-C Rota, Probability theory and related fields. 2G.-C. Rota. On the foundations of combinatorial theory i. theory of Möbius functions. Probability theory and related fields, 2(4):340-368, 1964.
Kilobot: A low cost scalable robot system for collective behaviors. M Rubenstein, C Ahler, R Nagpal, 2012 IEEE international conference on robotics and automation. IEEEM. Rubenstein, C. Ahler, and R. Nagpal. Kilobot: A low cost scalable robot system for collective behaviors. In 2012 IEEE international conference on robotics and automation, pages 3293-3298. IEEE, 2012.
Shape matching and mesh segmentation. F Schmiedl, Technische Universität MünchenPhD thesisF. Schmiedl. Shape matching and mesh segmentation. PhD thesis, Technische Universität München, 2015.
Computational aspects of the Gromov-Hausdorff distance and its application in non-rigid shape matching. F Schmiedl, Discrete & Computational Geometry. 574F. Schmiedl. Computational aspects of the Gromov-Hausdorff distance and its application in non-rigid shape matching. Discrete & Computational Geometry, 57(4):854-880, 2017.
Phase coexistence in insect swarms. M Sinhuber, N T Ouellette, Physical review letters. 11917178003M. Sinhuber and N. T. Ouellette. Phase coexistence in insect swarms. Physical review letters, 119(17):178003, 2017.
Collective animal behavior. D J Sumpter, Princeton University PressD. J. Sumpter. Collective animal behavior. Princeton University Press, 2010.
Topological data analysis of biological aggregation models. C M Topaz, L Ziegelmeier, T Halverson, PloS one. 105126383C. M. Topaz, L. Ziegelmeier, and T. Halverson. Topological data analysis of biological aggregation models. PloS one, 10(5):e0126383, 2015.
Grouping Time-Varying Data for Interactive Exploration. A Van Goethem, M Van Kreveld, M Löffler, B Speckmann, F Staals, 32nd International Symposium on Computational Geometry (SoCG 2016). :16. Schloss Dagstuhl-Leibniz-Zentrum fuer InformatikA. van Goethem, M. van Kreveld, M. Löffler, B. Speckmann, and F. Staals. Grouping Time-Varying Data for Interactive Exploration. In 32nd International Symposium on Computational Geometry (SoCG 2016), vol- ume 51 of Leibniz International Proceedings in Informatics (LIPIcs), pages 61:1-61:16. Schloss Dagstuhl- Leibniz-Zentrum fuer Informatik, 2016.
A refined definition for groups of moving entities and its computation. M Van Kreveld, M Löffler, F Staals, L Wiratma, International Journal of Computational Geometry & Applications. 2802M. van Kreveld, M. Löffler, F. Staals, and L. Wiratma. A refined definition for groups of moving entities and its computation. International Journal of Computational Geometry & Applications, 28(02):181-196, 2018.
Visualizing the evolution of communities in dynamic graphs. C Vehlow, F Beck, P Auwärter, D Weiskopf, Computer Graphics Forum. 341C. Vehlow, F. Beck, P. Auwärter, and D. Weiskopf. Visualizing the evolution of communities in dynamic graphs. Computer Graphics Forum, 34(1):277-288, 2015.
On-line discovery of flock patterns in spatio-temporal data. M R Vieira, P Bakalov, V J Tsotras, Proceedings of the 17th ACM SIGSPATIAL international conference on advances in geographic information systems. the 17th ACM SIGSPATIAL international conference on advances in geographic information systemsACMM. R. Vieira, P. Bakalov, and V. J. Tsotras. On-line discovery of flock patterns in spatio-temporal data. In Proceedings of the 17th ACM SIGSPATIAL international conference on advances in geographic information systems, pages 286-295. ACM, 2009.
Efficient algorithms for mining maximal valid groups. The VLDB Journal-The International Journal on Very Large Data Bases. Y Wang, E.-P Lim, S.-Y Hwang, 17Y. Wang, E.-P. Lim, and S.-Y. Hwang. Efficient algorithms for mining maximal valid groups. The VLDB Journal-The International Journal on Very Large Data Bases, 17(3):515-535, 2008.
Wikipedia. Formicarium -Wikipedia, the free encyclopedia. 12Wikipedia. Formicarium -Wikipedia, the free encyclopedia, 2021. [Online; accessed 12-December- 2021].
An experimental evaluation of grouping definitions for moving entities. L Wiratma, M Van Kreveld, M Löffler, F Staals, Proceedings of the 27th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems. the 27th ACM SIGSPATIAL International Conference on Advances in Geographic Information SystemsL. Wiratma, M. van Kreveld, M. Löffler, and F. Staals. An experimental evaluation of grouping definitions for moving entities. In Proceedings of the 27th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, pages 89-98, 2019.
Capturing dynamics of time-varying data via topology. L Xian, H Adams, C M Topaz, L Ziegelmeier, Foundations of Data Science. 41L. Xian, H. Adams, C. M. Topaz, and L. Ziegelmeier. Capturing dynamics of time-varying data via topology. Foundations of Data Science, 4(1):1-36, 2022.
| [
"https://github.com/ndag/DynGraphZZ,"
] |
[
"Direct Observation of Interband Spin-Orbit Coupling in a Two-Dimensional Electron System",
"Direct Observation of Interband Spin-Orbit Coupling in a Two-Dimensional Electron System"
] | [
"Hendrik Bentmann \nExperimentelle Physik VII and Röntgen Research Center for Complex Materials (RCCM)\nUniversität Würzburg Am Hubland\nD-97074WürzburgGermany\n\nKarlsruhe Institute of Technology KIT\nGemeinschaftslabor für NanoanalytikD-76021KarlsruheGermany\n",
"Samir Abdelouahed \nTexas A&M University at Qatar\nP.O. Box 23874DohaQatar\n",
"Mattia Mulazzi \nExperimentelle Physik VII and Röntgen Research Center for Complex Materials (RCCM)\nUniversität Würzburg Am Hubland\nD-97074WürzburgGermany\n\nKarlsruhe Institute of Technology KIT\nGemeinschaftslabor für NanoanalytikD-76021KarlsruheGermany\n",
"Jürgen Henk \nInstitut für Physik -Theoretische Physik\nMartin-Luther-Universität Halle-Wittenberg\nD-06099Halle (Saale)Germany\n",
"Friedrich Reinert \nExperimentelle Physik VII and Röntgen Research Center for Complex Materials (RCCM)\nUniversität Würzburg Am Hubland\nD-97074WürzburgGermany\n\nKarlsruhe Institute of Technology KIT\nGemeinschaftslabor für NanoanalytikD-76021KarlsruheGermany\n"
] | [
"Experimentelle Physik VII and Röntgen Research Center for Complex Materials (RCCM)\nUniversität Würzburg Am Hubland\nD-97074WürzburgGermany",
"Karlsruhe Institute of Technology KIT\nGemeinschaftslabor für NanoanalytikD-76021KarlsruheGermany",
"Texas A&M University at Qatar\nP.O. Box 23874DohaQatar",
"Experimentelle Physik VII and Röntgen Research Center for Complex Materials (RCCM)\nUniversität Würzburg Am Hubland\nD-97074WürzburgGermany",
"Karlsruhe Institute of Technology KIT\nGemeinschaftslabor für NanoanalytikD-76021KarlsruheGermany",
"Institut für Physik -Theoretische Physik\nMartin-Luther-Universität Halle-Wittenberg\nD-06099Halle (Saale)Germany",
"Experimentelle Physik VII and Röntgen Research Center for Complex Materials (RCCM)\nUniversität Würzburg Am Hubland\nD-97074WürzburgGermany",
"Karlsruhe Institute of Technology KIT\nGemeinschaftslabor für NanoanalytikD-76021KarlsruheGermany"
] | [] | We report the direct observation of interband spin-orbit (SO) coupling in a two-dimensional (2D) surface electron system, in addition to the anticipated Rashba spin splitting. Using angle-resolved photoemission experiments and first-principles calculations on Bi/Ag/Au heterostructures we show that the effect strongly modifies the dispersion as well as the orbital and spin character of the 2D electronic states, thus giving rise to considerable deviations from the Rashba model. The strength of the interband SO coupling is tuned by the thickness of the thin film structures. | 10.1103/physrevlett.108.196801 | [
"https://arxiv.org/pdf/1203.6190v1.pdf"
] | 27,211,168 | 1203.6190 | 18dbbb801b28fdb991ccdc792f22c2b4cfcaec0b |
Direct Observation of Interband Spin-Orbit Coupling in a Two-Dimensional Electron System
28 Mar 2012
Hendrik Bentmann
Experimentelle Physik VII and Röntgen Research Center for Complex Materials (RCCM)
Universität Würzburg Am Hubland
D-97074WürzburgGermany
Karlsruhe Institute of Technology KIT
Gemeinschaftslabor für NanoanalytikD-76021KarlsruheGermany
Samir Abdelouahed
Texas A&M University at Qatar
P.O. Box 23874DohaQatar
Mattia Mulazzi
Experimentelle Physik VII and Röntgen Research Center for Complex Materials (RCCM)
Universität Würzburg Am Hubland
D-97074WürzburgGermany
Karlsruhe Institute of Technology KIT
Gemeinschaftslabor für NanoanalytikD-76021KarlsruheGermany
Jürgen Henk
Institut für Physik -Theoretische Physik
Martin-Luther-Universität Halle-Wittenberg
D-06099Halle (Saale)Germany
Friedrich Reinert
Experimentelle Physik VII and Röntgen Research Center for Complex Materials (RCCM)
Universität Würzburg Am Hubland
D-97074WürzburgGermany
Karlsruhe Institute of Technology KIT
Gemeinschaftslabor für NanoanalytikD-76021KarlsruheGermany
Direct Observation of Interband Spin-Orbit Coupling in a Two-Dimensional Electron System
28 Mar 2012(Dated: May 5, 2014)
We report the direct observation of interband spin-orbit (SO) coupling in a two-dimensional (2D) surface electron system, in addition to the anticipated Rashba spin splitting. Using angle-resolved photoemission experiments and first-principles calculations on Bi/Ag/Au heterostructures we show that the effect strongly modifies the dispersion as well as the orbital and spin character of the 2D electronic states, thus giving rise to considerable deviations from the Rashba model. The strength of the interband SO coupling is tuned by the thickness of the thin film structures.
The spin-orbit interaction plays a fundamental role in the rapidly developing field of spintronics as it allows for the electrostatic manipulation of the spin degrees of freedom in the conduction channels of nanoscale heterostructures [1,2]. Such applications are often based on the Rashba effect in a two-dimensional electron gas (2DEG) that prescribes a lifting of the spin degeneracy in the presence of structural inversion asymmetry and strong spin-orbit (SO) coupling [3]. Other contributions to the spin splitting arise from the Dresselhaus effect for constituent bulk crystal structures without a center of inversion [4]. Moreover, new effects due to the SO interaction have been discovered recently that give rise to topologically protected, spin-polarized states on the surfaces of a number of heavy-element semiconductors, referred to as topological insulators [5,6].
In the phenomenological treatment of the mentioned effects the spin-polarized electronic states are usually assumed as pure spin states. In a real system, on the other hand, the SO interaction couples spin and orbital angular momentum which will result in a mixing of orthogonal spinors in the singleparticle eigenstates [7]. Recent ab-initio calculations suggest that this can result in considerable reductions of the spin polarization of spin-split two-dimensional electronic states in the presence of strong SO interaction [8]. Even more profound effects of spin-mixing are known from the three-dimensional (3D), exchange-split band structures of ferromagnets where the SO coupling allows for hybridizations between spin-up and spin-down bands. This interband SO coupling lies at the origin of several magnetic phenomena, e.g. magneto crystalline anisotropy [9] or ultrafast demagnetization [10]. Note that spin-mixing is expected to gain increasingly in importance for materials with high atomic number and thus enhanced SO interaction. It is therefore of fundamental importance to explore whether interband SO coupling effects due to spin-mixing are present in heavy-element 2DEGs and how they modify the spin-split electronic structure. Indeed, recent theoretical reports predict interband SO coupling phenomena in 2DEGs formed in zinc blende quantum wells [11,12] and on high-Z metal surfaces [13] as a result of higher-order perturbation theory corrections in the SO interaction. Yet, to best of our knowledge, these effects have not been addressed in experiment so far.
In this Letter we report the direct observation of interband SO coupling in a 2DEG with large Rashba splitting that is formed in a BiAg 2 surface alloy grown on Ag quantum films supported by a Au(111) substrate [ Fig. 1(b)]. Using angle-resolved photoelectron spectroscopy (ARPES) with high energy-resolution we find avoided crossings in the spinsplit electronic structure that provide evidence for the hybridization of states with opposite spin due to SO coupling. These findings are in line with relativistic first-principles computations and model calculations. Further, we demonstrate that the strength of the interband SO effect varies upon changing the thickness of the Ag film whereas the Rashba coupling remains unmodified. The new interband coupling hence emerges to be tunable, independently from other SO effects, by nanostructural design capabilities as shown here for the layered Ag/Au heterostructure. Our investigations prove that the SO interaction can considerably influence the electronic states in 2D systems, in addition to established effects such as Rashba splitting. We hence expect the present findings to be highly relevant for spintronic applications based on strongly SO coupled compounds including the recently reported heavy-element semiconductors with large Rashba splitting [14,15] and topological insulators [16]. The BiAg 2 system and related isostructural alloys have been shown previously to feature an unusually large Rashba effect in their electronic structure resulting from the strong SO interaction of the Bi atoms [17,18,19,20]. The band structure of the surface alloys consists of two parabolic, Rashba-split states E ± 1,2 with negative effective mass [ Fig. 1(a)], whose spin-polarization has been verified by spin-resolved photoemission experiments [21,22,23].
The ARPES data were collected by a SCIENTA R4000 electron spectrometer employing a monochromatized He discharge lamp operating at an excitation energy of 21.22 eV (He I α ). The energy and angular resolution of the setup are ∆E = 3 meV and ∆θ = 0.3 • , respectively. We performed all measurements at base pressures lower than 2 · 10 −10 mbar and at temperatures of 20 K. The preparation of the Au(111) substrate by cycles of Ar sputtering and annealing resulted in a clean and well-ordered surface as verified by the spectral linewidth of the Au(111) surface state [24]. Ag was evaporated on the cooled substrate at ∼150 K. Subsequent mild annealing resulted in homogeneous films as confirmed by the characteristic shift in binding energy of the surface state for low Ag coverages [25]. The BiAg 2 surface alloy was obtained after evaporation of 1/3 ML Bi on the substrate film at elevated temperatures of 400 K. We observed no change in the binding energy of the Shockley state for Ag/Au(111) at this temperature providing evidence for no or neglegible Ag/Au intermixing at the interface [26]. Low energy electron diffraction measurements confirmed the ( √ 3x √ 3) reconstruction of the surface alloy.
BiAg 2 (a) (b) (c) Interband -0.6 -0.4 -0.2 0.0 E-E F [eV] -0.4 -0.2 0.0 0.2 0.4 Wave vector k x [1/Å] Rashba u E 2 + E 1 - E 2 - E 1 + E 1 ± E
First-principles calculations were carried out using the relativistic full-potential linearized augmented plane wave (FLAPW) method as implemented in the Fleur code [27]. Exchange and correlation were treated within the generalized gradient approximazion (GGA) [28]. We used a plane-wave cutoff of 8.6Å −1 while the charge density and potential cutoffs were 21.9Å −1 . A ten-layer slab (BiAg 2 /4ML Ag/5 ML Au) was used to simulate the surface electronic structure. The opposite slab surface was saturated by hydrogen atoms to suppress the formation of the Au(111) surface state. The vertical relaxation of the Bi atoms was 0.95Å. Fig. 1(c) shows the experimental electronic structure of a BiAg 2 surface alloy grown on a 4 ML Ag film on Au(111). We identify two pairs of states E ± 1,2 , both showing Rashba spin-splitting around theΓ point, and a backfolded Au bulk band u. The measured surface band structure thus complies with the scheme in Fig. 1(a) with a position of the Fermi energy as indicated. Deviations from the pure Rashba scenario [full line in Fig. 1(a)] are observed in the experimental data at the expected crossing points between the two branches E + 1 and E − 2 . Near these points the two states hybridize which leads to a gap opening and pronounced, kink-like changes in the dispersion [highlighted by the circle in (c)]. Note that these observations cannot be explained within the Rashba model which assigns pure opposite spin states to these two bands. A hybridization between them should be prohibited in this case. The findings are therefore distinctively different from the previously found hybridization phenomena between Rashba-split states and spin-degenerate quantum well states [22,29,30]. Our experimental results indicate an additional interband SO effect that couples orbitals with opposite spin and thereby induces the hybridization.
To gain a first understanding of the observed effect we adopt an effective 2D model Hamiltonian taking into account Rashba and interband contributions of the SO coupling but neglecting the orbital part of the wave functions for the moment, similar as it is done for the Rashba model in single band systems. The interband SO coupling term introduces a finite hybridization ∆ = E + 1 |H so |E − 2 between the two purely Rashba-split states. For the modified eigenvalues S ± we then
have S ± = 1 2 (E + 1 + E − 2 ) ± 1 4 (E + 1 − E − 2 ) 2 + ∆ 2 ,+ 1 − a 2 k |E − 2 and |S − = 1 − a 2 k |E + 1 − a k |E − 2 .
The coefficient a k can be expressed by |a k | 2 = (1+
(S+−E + 1 ) 2 ∆ 2 ) −1 .
Our experimental data allow us not only to determine the modified dispersions S ± but also to trace the k-dependence of the coefficient |a k | 2 [Fig. 2]. The latter is possible due to considerably differing photoionization cross sections for the two states |E + 1 (high cross section) and |E − 2 (low cross section). As a result, the branches S ± show drastic intensity changes close to the hybridization gap where their state character is strongly modified. We identify the k-dependent intensity evolution I ± of the branches S ± with the relative contributions of |E + 1 and |E − 2 to |S ± . These contributions are directly given by a k and we have the simple correspondence I + = I min + (I max − I min )|a k | 2 and accordingly for I − . Within this approximation we neglect photoemission matrix element effects. Note that a k and I ± depend on the hybridization strength ∆.
A comparison of the experimental data with the results of the model Hamiltonian is shown in Fig. 2. In (a) we display the measured band structure and the model dispersions close to the hybridization gap. The bands E + 1 and E − 2 [red (light) lines in Fig. 2(a)] are determined from the experimental data in (E, k) regions sufficiently far away from the hybridization gap. For E − 2 we use a linear dispersion which is an adequate approximation for the small (E, k) window which is of interest here. A close match between the experimental bands and S ± is obtained for ∆ = 31 meV. Fig. 2(b) shows the photoemission intensities I ± obtained from energy and momentum distribution curves as a function of the wave vector k x . The data for each branch were normalized to the respective maximal value of the individual data set. For increasing wave vector k x the intensity I + is enhanced whereas I − is reduced reflecting the change in state character of both branches near the hybridization gap. We confirmed that this behavior is not affected by the finite k-resolution of the ex-periment (∼0.01Å −1 ). The full lines in (b) correspond to the expected intensity change according to the model Hamiltonian using the same value for ∆ and the same dispersion relations as in (a). Again, we find a good agreement with the experimental data. We hence conclude that the observed dispersion modification and change in state character can be consistently described by a Rashba-type free electron model including an additional interband SO coupling term. Comparing the energy scales of the two SO effects in this system we find the interband contribution (∼30 meV) to be about one order of magnitude weaker than the Rashba contribution (∼300 meV).
Qualitative insight into the coupled orbital and spin parts of the electronic states |E + 1 and |E − 2 which goes beyond the discussed free electron model can be attained by inspecting the spatial symmetry and the contributing atomic orbitals in the system. Group-theoretical considerations show that both states belong to the same representation and that their wave function along k x can be written as |ψ = |sp z ↑ + |p x ↑ + |p y ↓ , where the spinors are quantized with respect to the k y axis [13]. It is important to note that the SO interaction induces a mixing of opposite spinors in the wave function and therefore facilitates a hybridization between the two bands which is indeed observed in our measurements. The hybridization mechanism is therefore comparable to the effect of SO coupling on the 3D spin-split band structures in ferromag- netic systems [10,31].
To further substantiate the experimental results on a quantitative theoretical level we examine the spin-polarized electronic structure by a first-principles calculation [ Fig. 3]. In accordance with experiment our calculation finds two Rashbasplit states and a hybridization gap between the two branches E + 1 and E − 2 . Similar to the measured results the strength of interband coupling is smaller than that of the Rashba coupling. Note that the spin-polarization of the branch E − 2 shows a sign change near the hybridization kink clearly reflecting the mixed and k-dependent spin character of the corresponding wave function. We further find the degree of spin-polarization to be reduced from 100 %, as predicted by the Rashba model, to maximal values of ∼66 %.
It is interesting to explore the possibility to modify the electronic structure of the BiAg 2 alloy by choosing a different thickness for the supporting Ag quantum well film. We have recently shown that the spatial localization of quantum well states in the system Ag/Au(111) changes considerably as a function of layer thickness, especially in the low coverage regime up to ∼20 ML [32]. As a result the charge density close to the surface, where the BiAg 2 alloy is located, varies strongly depending on the precise Ag film thickness. Hence, one may expect concomitant influences of these variations on the electronic states in the surface alloy. Indeed, we find such changes as is inferred from Fig. 4 which shows the electronic structure of BiAg 2 close to the hybridization gap for three different Ag layers. Whereas the general features in the electronic structure, such as binding energy and Rashba splitting, are very similar for all three films we find considerable changes in the size of the hybridization gap ∆ and therefore in the interband coupling strength. The largest gap is found for 2 ML (∆ = 42 meV) whereas at higher thicknesses it is reduced (∆ = 14 meV for the case of 16 ML). The strength of the interband SO coupling is therefore determined by details of the geometric and electronic substrate properties. This provides possibilities to tailor this new SO effect by a controlled variation of the corresponding parameters in custom designed nanostructures, as exemplified here by the electronically tunable quantum well system Ag/Au.
We have established the presence of interband SO coupling in 2DEGs with Rashba spin splitting by photoemission spectroscopy experiments on multilayered Bi/Ag/Au heterostructures. The coupling induces hybridization between bands of different spin polarizations and thereby causes considerable deviations from the Rashba model in the dispersion of the 2D electronic states as well as in their spin and orbital momentum character. Such effects will certainly influence the transport properties in 2DEGs, especially in compounds with very strong SO interactions, and may therefore be exploited to improve the performance of new spintronic functionalities [33,34].
HB acknowledges helpful discussions with Frank Forster and Kazuyuki Sakamoto. This work was supported by the Bundesministerium für Bildung und Forschung (Grants No. 05K10WW1/2 and No. 05KS1WMB/1) and the Deutsche Forschunsgsgemeinschaft within the Forschergruppe 1162 and the Sonderforschungsbereich 762.
FIG. 1 :
1(Color online) (a) Band structure of a two-band system with Rashba interaction (schematic). Two different cases with (dashed line) and without (full line) an additional interband spin-orbit coupling are shown. (b) Sketch of the studied multilayer sample: The BiAg 2 layer is grown on a Ag film of variable thickness supported by the Au(111) substrate. (c) Experimental band structure for BiAg 2 on 4 ML Ag/Au(111) along theΓK direction of the surface Brillouin zone. The dashed line indicates the edge of the projected bulk band gap of Au(111).
FIG. 2 :
2(Color online) Comparison of the experimental electronic structure close to the SO-induced hybridization gap with the solution of a model Hamiltonian (see text). A closeup of the hybridization gap taken from the dataset inFig. 1(c) is shown in (a). The ARPES spectrum is overlaid by the modelled dispersions S± (dark lines) for an interaction strength ∆ = 31 meV. The scale bar above the graph indicates the k-interval of panel (b). In (b) the datapoints mark the measured k-dependent intensity evolution I± of the two branches S±. The experimental results are compared with the calculated intensities (full lines) based on the model Hamiltonian. online) Spin-resolved surface electronic struture of BiAg 2 on 4 ML Ag/Au(111) as obtained by a first-principles calculation. The size of the markers scales with the degree of spin-polarization of the electronic states and the color refers to the spin-orientations up (red,light) and down (blue,dark) with the spinquantization axis along the y-direction. The hybridization gap ∆ between the branches E + 1 and E − 2 is indicated.
FIG. 4 :
4Modification of the SO-induced hybridization gap ∆ arising from changes in the thickness of the supporting Ag film. The three panels show second-derivative spectra of the experimental data for Ag layers of 2 ML, 4 ML and 16 ML.
where we omit the k-dependence for clarity. Close to the crossing point of E + 1 and E − 2 the SO coupling mixes the states |E +1
and |E −
2
resulting in new eigenstates |S + = a k |E +
1
. I Žutić, J Fabian, S. Das Sarma, Rev. Mod. Phys. 76323I.Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
. H C Koo, J H Kwon, J Eom, J Chang, S H Han, M Johnson, Science. 3251515H. C. Koo, J. H. Kwon, J. Eom, J. Chang, S. H. Han, and M. Johnson, Science 325, 1515 (2009).
. Y A Bychkov, E I Rashba, JETP Lett. 3978Y. A. Bychkov and E. I. Rashba, JETP Lett. 39, 78 (1984).
. G Dresselhaus, Phys. Rev. 100580G. Dresselhaus, Phys. Rev. 100, 580 (1955).
. L Fu, C L Kane, Phys. Rev. B. 7645302L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).
. D Hsieh, Y Xia, L Wray, D Qian, A Pal, J H Dil, J Osterwalder, F Meier, G Bihlmayer, C L Kane, Science. 323919D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Oster- walder, F. Meier, G. Bihlmayer, C. L. Kane, et al., Science 323, 919 (2009).
. J Henk, T Scheunemann, S V Halilov, R Feder, Journal of Physics: Condensed Matter. 847J. Henk, T. Scheunemann, S. V. Halilov, and R. Feder, Journal of Physics: Condensed Matter 8, 47 (1996).
. O V Yazyev, J E Moore, S G Louie, Phys. Rev. Lett. 105266806O. V. Yazyev, J. E. Moore, and S. G. Louie, Phys. Rev. Lett. 105, 266806 (2010).
. P Bruno, Phys. Rev. B. 39865P. Bruno, Phys. Rev. B 39, 865 (1989).
. M Pickel, A B Schmidt, F Giesen, J Braun, J Minár, H Ebert, M Donath, M Weinelt, Phys. Rev. Lett. 10166402M. Pickel, A. B. Schmidt, F. Giesen, J. Braun, J. Minár, H. Ebert, M. Donath, and M. Weinelt, Phys. Rev. Lett. 101, 066402 (2008).
. E Bernardes, J Schliemann, M Lee, J C Egues, D Loss, Phys. Rev. Lett. 9976603E. Bernardes, J. Schliemann, M. Lee, J. C. Egues, and D. Loss, Phys. Rev. Lett. 99, 076603 (2007).
. R S Calsaverini, E Bernardes, J C Egues, D Loss, Phys. Rev. B. 78155313R. S. Calsaverini, E. Bernardes, J. C. Egues, and D. Loss, Phys. Rev. B 78, 155313 (2008).
. H Mirhosseini, J Henk, A Ernst, S Ostanin, C.-T Chiang, P Yu, A Winkelmann, J Kirschner, Phys. Rev. B. 79245428H. Mirhosseini, J. Henk, A. Ernst, S. Ostanin, C.-T. Chiang, P. Yu, A. Winkelmann, and J. Kirschner, Phys. Rev. B 79, 245428 (2009).
. K Ishizaka, M S Bahramy, H Murakawa, M Sakano, T Shimojima, T Sonobe, K Koizumi, S Shin, H Miyahara, A Kimura, Nat Mater. 10521K. Ishizaka, M. S. Bahramy, H. Murakawa, M. Sakano, T. Shimojima, T. Sonobe, K. Koizumi, S. Shin, H. Miyahara, A. Kimura, et al., Nat Mater 10, 521 (2011).
. P D C King, R C Hatch, M Bianchi, R Ovsyannikov, C Lupulescu, G Landolt, B Slomski, J H Dil, D Guan, J L Mi, Phys. Rev. Lett. 10796802P. D. C. King, R. C. Hatch, M. Bianchi, R. Ovsyannikov, C. Lupulescu, G. Landolt, B. Slomski, J. H. Dil, D. Guan, J. L. Mi, et al., Phys. Rev. Lett. 107, 096802 (2011).
. Y Xia, D Qian, D Hsieh, L Wray, A Pal, H Lin, A Bansil, D Grauer, Y S Hor, R J Cava, Nat. Phys. 5398Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, et al., Nat. Phys. 5, 398 (2009).
. D Pacilé, C R Ast, M Papagno, C Silva, L Moreschini, M Falub, A P Seitsonen, M Grioni, Phys. Rev. B. 73245429D. Pacilé, C. R. Ast, M. Papagno, C. Da Silva, L. Moreschini, M. Falub, A. P. Seitsonen, and M. Grioni, Phys. Rev. B 73, 245429 (2006).
. C R Ast, J Henk, A Ernst, L Moreschini, M C Falub, D Pacilé, P Bruno, K Kern, M Grioni, Phys. Rev. Lett. 98186807C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub, D. Pacilé, P. Bruno, K. Kern, and M. Grioni, Phys. Rev. Lett. 98, 186807 (2007).
. L Moreschini, A Bendounan, H Bentmann, M Assig, K Kern, F Reinert, J Henk, C R Ast, M Grioni, Phys. Rev. B. 8035438L. Moreschini, A. Bendounan, H. Bentmann, M. Assig, K. Kern, F. Reinert, J. Henk, C. R. Ast, and M. Grioni, Phys. Rev. B 80, 035438 (2009).
. H Bentmann, F Forster, G Bihlmayer, E V Chulkov, L Moreschini, M Grioni, F Reinert, EPL. 8737003H. Bentmann, F. Forster, G. Bihlmayer, E. V. Chulkov, L. Moreschini, M. Grioni, and F. Reinert, EPL 87, 37003 (2009).
. F Meier, H Dil, J Lobo-Checa, L Patthey, J Osterwalder, Phys. Rev. B. 77165431F. Meier, H. Dil, J. Lobo-Checa, L. Patthey, and J. Osterwalder, Phys. Rev. B 77, 165431 (2008).
. K He, Y Takeichi, M Ogawa, T Okuda, P Moras, D Topwal, A Harasawa, T Hirahara, C Carbone, A Kakizaki, Phys. Rev. Lett. 104156805K. He, Y. Takeichi, M. Ogawa, T. Okuda, P. Moras, D. Topwal, A. Harasawa, T. Hirahara, C. Carbone, A. Kakizaki, et al., Phys. Rev. Lett. 104, 156805 (2010).
. H Bentmann, T Kuzumaki, G Bihlmayer, S Blügel, E V Chulkov, F Reinert, K Sakamoto, Phys. Rev. B. 84115426H. Bentmann, T. Kuzumaki, G. Bihlmayer, S. Blügel, E. V. Chulkov, F. Reinert, and K. Sakamoto, Phys. Rev. B 84, 115426 (2011).
. G Nicolay, F Reinert, S Hüfner, P Blaha, Phys. Rev. B. 6533407G. Nicolay, F. Reinert, S. Hüfner, and P. Blaha, Phys. Rev. B 65, 033407 (2001).
. D Popović, F Reinert, S Hüfner, V G Grigoryan, M Springborg, H Cercellier, Y Fagot-Revurat, B Kierren, D Malterre, Phys. Rev. B. 7245419D. Popović, F. Reinert, S. Hüfner, V. G. Grigoryan, M. Spring- borg, H. Cercellier, Y. Fagot-Revurat, B. Kierren, and D. Mal- terre, Phys. Rev. B 72, 045419 (2005).
. H Cercellier, C Didiot, Y Fagot-Revurat, B Kierren, L Moreau, D Malterre, F Reinert, Phys. Rev. B. 73195413H. Cercellier, C. Didiot, Y. Fagot-Revurat, B. Kierren, L. Moreau, D. Malterre, and F. Reinert, Phys. Rev. B 73, 195413 (2006).
. J P Perdew, K Burke, M Ernzerhof, Phys. Rev. Lett. 773865J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
. K He, T Hirahara, T Okuda, S Hasegawa, A Kakizaki, I Matsuda, Phys. Rev. Lett. 101107604K. He, T. Hirahara, T. Okuda, S. Hasegawa, A. Kakizaki, and I. Matsuda, Phys. Rev. Lett. 101, 107604 (2008).
. E Frantzeskakis, S Pons, H Mirhosseini, J Henk, C R Ast, M Grioni, Phys. Rev. Lett. 101196805E. Frantzeskakis, S. Pons, H. Mirhosseini, J. Henk, C. R. Ast, and M. Grioni, Phys. Rev. Lett. 101, 196805 (2008).
. A Rampe, G Güntherodt, D Hartmann, J Henk, T Scheunemann, R Feder, Phys. Rev. B. 5714370A. Rampe, G. Güntherodt, D. Hartmann, J. Henk, T. Scheune- mann, and R. Feder, Phys. Rev. B 57, 14370 (1998).
. F Forster, E Gergert, A Nuber, H Bentmann, L Huang, X G Gong, Z Zhang, F Reinert, Phys. Rev. B. 8475412F. Forster, E. Gergert, A. Nuber, H. Bentmann, L. Huang, X. G. Gong, Z. Zhang, and F. Reinert, Phys. Rev. B 84, 075412 (2011).
. J C Egues, G Burkard, D Loss, Applied Physics Letters. 822658J. C. Egues, G. Burkard, and D. Loss, Applied Physics Letters 82, 2658 (2003).
. J C Egues, G Burkard, D Loss, Phys. Rev. Lett. 89176401J. C. Egues, G. Burkard, and D. Loss, Phys. Rev. Lett. 89, 176401 (2002).
| [] |
[
"THE MAXIMUM DIMENSION OF A LIE NILPOTENT SUBALGEBRA OF M n (F ) OF INDEX m",
"THE MAXIMUM DIMENSION OF A LIE NILPOTENT SUBALGEBRA OF M n (F ) OF INDEX m",
"THE MAXIMUM DIMENSION OF A LIE NILPOTENT SUBALGEBRA OF M n (F ) OF INDEX m",
"THE MAXIMUM DIMENSION OF A LIE NILPOTENT SUBALGEBRA OF M n (F ) OF INDEX m"
] | [
"J Szigeti ",
"J Van Den Berg ",
"L Van Wyk ",
"M Ziembowski ",
"J Szigeti ",
"J Van Den Berg ",
"L Van Wyk ",
"M Ziembowski "
] | [] | [] | The main result of this paper is the following: if F is any field and R any F -subalgebra of the algebra M n (F ) of n × n matrices over F with Lie nilpotence index m, then. . , k m+1 nonnegative integers. This answers in the affirmative a conjecture by the first and third authors. The case m = 1 reduces to a classical theorem of Schur (1905), later generalized by Jacobson (1944) to all fields, which asserts that if F is an algebraically closed field of characteristic zero, and R is any commutative F -subalgebra of M n (F ), then dim F R n 2 4 + 1. Examples constructed from block upper triangular matrices show that the upper bound of M (m + 1, n) cannot be lowered for any choice of m and n. An explicit formula for M (m + 1, n) is also derived. | 10.1090/tran/7821 | [
"https://arxiv.org/pdf/1608.04562v1.pdf"
] | 119,156,536 | 1608.04562 | f77c4fdd76c84317b9f2a9cb445ae0eff93a0e22 |
THE MAXIMUM DIMENSION OF A LIE NILPOTENT SUBALGEBRA OF M n (F ) OF INDEX m
16 Aug 2016
J Szigeti
J Van Den Berg
L Van Wyk
M Ziembowski
THE MAXIMUM DIMENSION OF A LIE NILPOTENT SUBALGEBRA OF M n (F ) OF INDEX m
16 Aug 2016
The main result of this paper is the following: if F is any field and R any F -subalgebra of the algebra M n (F ) of n × n matrices over F with Lie nilpotence index m, then. . , k m+1 nonnegative integers. This answers in the affirmative a conjecture by the first and third authors. The case m = 1 reduces to a classical theorem of Schur (1905), later generalized by Jacobson (1944) to all fields, which asserts that if F is an algebraically closed field of characteristic zero, and R is any commutative F -subalgebra of M n (F ), then dim F R n 2 4 + 1. Examples constructed from block upper triangular matrices show that the upper bound of M (m + 1, n) cannot be lowered for any choice of m and n. An explicit formula for M (m + 1, n) is also derived.
Introduction
In 1905 Schur [15,Satz I,p. 67] proved that the dimension over the field of complex numbers C of any commutative subalgebra of M n (C) is at most n 2 4 + 1, where ⌊ ⌋ denotes the integer floor function. Some forty years later, Jacobson [ In a subsequent further improvement, Gustafson [7, Section 2, p. 558] showed that Schur's theorem in its most general form could be proved with much greater efficiency using module theoretic methods. We record here that Gustafson's elegant arguments are the inspiration for a key proposition in this paper.
There have also appeared in the literature a number of papers offering alternative proofs of Schur's theorem and its subsequent extensions. In this regard, we refer the reader to [18], [11] and [9].
In response to a question posed in [7, Section 5, Open problem (a), p. 562] Cowsik [2] has proved a version of Schur's theorem for artinian rings that are not algebras, in which the module length of a faithful module substitutes for the dimension of the F -space on which the matrices act.
The common approach to establishing Schur's upper bound has been to show that if F is a field and R a commutative F -subalgebra of M n (F ), then there exist positive integers k 1 and k 2 such that k 1 +k 2 = n and dim F R k 1 k 2 + 1.
An application of rudimentary Calculus then shows that max{k 1 k 2 + 1 : (k 1 , k 2 ) ∈ N × N and k 1 + k 2 = n} = n 2 4 + 1,
whence dim F R n 2 4 + 1.
The upper bound of n 2 4 + 1 is, moreover, easily seen to be optimal. Indeed, let F be any field and (k 1 , k 2 ) any pair of positive integers satisfying k 1 + k 2 = n. Define rectangular array B by B def = {(i, j) ∈ N × N : 1 i k 1 < j n}, and subset J of M n (F ) by
J def = (i,j)∈B b ij E (i,j) : b ij ∈ F ∀(i, j) ∈ B ,(1)
where E (i,j) denotes the matrix unit in M n (F ) associated with position (i, j). Observe that J comprises the set of all block upper triangular matrices that correspond with B; it has the following illuminating pictorial representation (the unshaded region in the picture below corresponds with zero entries):
J = k 1 k 2
Denote by F I n def = {aI n : a ∈ F } (I n is the n × n identity matrix) the set of all n × n scalar matrices over F , and define R def = F I n + J. (2) It is easily seen that R is a local F -subalgebra of M n (F ) with (Jacobson) radical J(R) = J such that J 2 = 0. This entails R is commutative. It is clear too, that dim F R = k 1 k 2 + 1.
The above simple construction shows that the upper bound n 2 4 + 1 = max{k 1 k 2 + 1 : (k 1 , k 2 ) ∈ N × N and k 1 + k 2 = n} cannot be lowered for any n 2, and is thus optimal, as claimed.
We construct now an F -subalgebra R of M n (F ) similar to the one constructed above, but whose radical J comprises m blocks rather than a single block. We require first a compact notation for the description of such rings. To this end, let k 1 , k 2 , . . . , k m+1 be a sequence of positive integers such that k 1 + k 2 + · · · + k m+1 = n. For each p ∈ {1, 2, . . . , m}, define rectangular array
B p def = {(i, j) ∈ N × N : 1 i k 1 < j n}, if p = 1,
{(i, j) ∈ N × N : k 1 + k 2 + · · · + k p−1 < i k 1 + k 2 + · · · + k p < j n}, if p > 1.
Put
B def = m p=1 B p .(3)
Define J as in (1) but with B defined as in (3) above. The following pictorial representation of J reveals a stack of m blocks · · · · · · · · · · · · · · · J = k 1
k 2 k m k m+1
We shall call the F -algebra R defined as in (2), the algebra of n × n matrices over F of type (k 1 , k 2 , . . . , k m+1 ). We see that R is again a local F -subalgebra of M n (F ) with radical J(R) = J such that J m+1 = 0 and dim F R = k 1 (n − k 1 ) + k 2 (n − k 1 − k 2 ) + · · · + k m (n − k 1 − k 2 − · · · − k m ) + 1.
= m j=1 k j n − j i=1 k i + 1.(4)
A routine inductive argument shows that the expression (less 1) appearing on the right-hand-side of (4), simplifies as
m j=1 k j n − j i=1 k i = 1 2 n 2 − m+1 i=1 k 2 i = m+1 i,j=1, i<j k i k j ,
so that (4) becomes
dim F R = 1 2 n 2 − m+1 i=1 k 2 i + 1 = m+1 i,j=1, i<j k i k j + 1.(5)
The algebra of n × n matrices over F of type (k 1 , k 2 , . . . , k m+1 ) is clearly not commutative (unless m = 1), but it does satisfy a weak form of commutativity called Lie nilpotence. To put this notion in context, we first recall some basic facts about Lie algebras.
Let g be a Lie algebra 1 and x 1 , x 2 , . . . , x m a finite sequence of elements in g. We define element [x 1 , x 2 , . . . , x m ] * of g recursively as follows
[x 1 ] * def = x 1 , and [x 1 , x 2 , . . . , x m ] * def = [[x 1 , x 2 , . . . , x m−1 ] * , x m ], for m > 1.
Recall that if h is any ideal of g, then the Lower Central Series
{h [m] } m∈N of h is defined by h [m] def = {[x 1 , x 2 , . . . , x m ] * : x i ∈ h for 1 i m}.
We say g is nilpotent if g [m] = 0 for some m ∈ N, m > 1, and more specifically, nilpotent of index m, if g [m+1] = 0.
Every ring R may be endowed with the structure of a Lie algebra (over the centre of R), by choosing as bracket the commutator defined by
∀r, s ∈ R, [r, s] def = rs − sr.
Following [17, p. 4785 Lie nilpotent rings have been shown to play an important role in the proofs of certain classical results about polynomial and trace identities in the F -algebra M n (F ) (see [5] and [6]). For fields F of characteristic zero, Kemer's [10] pioneering work on the T-ideals of associative algebras has revealed the importance of identities satisfied by n × n matrices over the Grassmann (exterior) algebra E = F {x i : i ∈ N} :
x i x j + x j x i = 0 whenever 1 i j generated by an infinite family {x i : i ∈ N} of anticommutative indeterminates. For n × n matrices over a Lie nilpotent ring of index m, a Cayley-Hamilton identity of degree n m (with left-or right-sided scalar coefficients) was found in [16]. Since the Grassmann algebra E is Lie nilpotent of index m = 2, the aforementioned Cayley-Hamilton identity for matrices in M n (E) is of degree n 2 . In [3], Domokos presents a slightly modified version of this identity in which the coefficients are invariant under the conjugation action of GL n (F ). This paper is an attempt to answer a conjecture posed in [17, p. 4785]. The statement of this conjecture is rendered less cumbersome if expressed in terms of a function M(ℓ, n) of positive integer arguments ℓ and n, defined as follows
M(ℓ, n) def = max 1 2 n 2 − ℓ i=1 k 2 i + 1 : k 1 , k 2 , . . . , k ℓ are nonnegative integers such that ℓ i=1 k i = n .(6)
Conjecture. Let F be any field, m and n positive integers, and R an F -subalgebra of M n (F ) with Lie nilpotence index m. Then
dim F R M(m + 1, n).(7)
We shall henceforth refer to the above as 'the Conjecture'. More specifically, if F is any fixed field, we shall say that 'the Conjecture holds in respect of F ', if (7) holds for all positive integers m and n, and Fsubalgebras R of M n (F ) with Lie nilpotence index m.
If R is any algebra over a field F , then a module V over R is precisely a representation of R via action on the underlying F -space structure on V . If the module is faithful, then this representation is faithful thus yielding an embedding of R into End F V , the F -algebra of Fspace endomorphisms on V . If V is also finite dimensional over F , say dim F V = n, then End F V is isomorphic to M n (F ) and so we have an F -algebra embedding of R into M n (F ). (We point out that such a finite dimensional V is certain to exist if R is finite dimensional, for V can always be chosen to be R itself.) Thus, seen through a representation theoretic lens, inequality (7) sheds light on a possible lower bound for the dimension of a faithful module over a given Lie nilpotent algebra.
In the same spirit, Domokos [4, Theorem 1, p. 156] derives a lower bound for the dimension of a faithful module over a finite dimension algebra satisfying the polynomial identity [x 1 , y 1 ][x 2 , y 2 ] . . . [x m , y m ] = 0, in terms of m.
Our initial task, which is easily accomplished, shall be to argue that the upper bound (7) is optimal for all choices of m and n.
Suppose first that m + 1 n. It is proven in Corollary 27(a) that for such m and n, M(m + 1, n) = 1 2 n 2 − m+1 i=1 k 2 i + 1 for some sequence of positive integers k 1 , k 2 , . . . , k m+1 satisfying m+1 i=1 k i = n. Let F be any field and R the algebra of n × n matrices over F of type (k 1 , k 2 , . . . , k m+1 ). As noted earlier, R has the form R = F I n + J with radical J satisfying J m+1 = 0. Since the set F I n of scalar matrices is central in R, it can be shown that the kth terms of the Lower Central Series for R (interpreted as a Lie algebra via the commutator) and J coincide, that is to say,
R [k] = J [k] , for k > 1. It is also evident that J [k] ⊆ J k for every k ∈ N. Thus R [m+1] = J [m+1] ⊆ J m+1 = 0, so R is Lie nilpotent of index m. It follows from (5) that dim F R = 1 2 n 2 − m+1 i=1 k 2 i + 1 = M(m + 1, n). Now suppose m + 1 > n.
No generality is lost if we suppose n > 1. It is proven in Corollary 27(b) that for such m and n, M(m + 1, n) = M(n, n) = 1 2 (n 2 − n) + 1, and this, by (5), is equal to dim F R where R is the algebra of n × n matrices over field F of type (k 1 , k 2 , . . . , k n ) with k 1 = k 2 = · · · = k n = 1. (The reader will see that in this instance, R is just the algebra of all upper triangular matrices over F with constant main diagonal.) As shown in the previous paragraph, such an algebra R is Lie nilpotent of index n − 1 and thus Lie nilpotent of index m, since m n − 1.
The theorem below collects together the conclusions drawn above. The main body of theory in this paper is developed in Sections 5 and 6 with module theoretic methods our primary tools. Sections 3 and 4 show that the Conjecture reduces to a consideration of local subalgebras of upper triangular matrix rings over an algebraically closed field. Section 7, which can be read independently of earlier sections, establishes important properties of the function M(ℓ, n) required in earlier theory. An explicit formula for M(ℓ, n) is also derived which is then shown to have a more simplified form for small values of ℓ. In Section 8 the algebra of n × n matrices of type (d 1 , d 2 , . . . , d ℓ ) is used to provide a pictorial representation of the objects introduced in earlier theory. The content of Section 9, which is titled Open questions, is self-evident.
Preliminaries
The symbol ⊆ denotes containment and ⊂ proper containment for sets.
If X is any set, then X n denotes the cartesian product of n copies of X.
N and N 0 will denote the sets of positive integers, and nonnegative integers, respectively.
All rings are associative and possess identity, and all modules are unital.
Let R be a ring and V a right R-module. We write W V to indicate that W is a submodule of V . If X is a nonempty subset of V and I is a right ideal of R, then Observe that (0 : I X) is always a right ideal of R.
Let F be a field. For each n ∈ N, M n (F ) [resp. U n (F )] [resp. U * n (F )] shall denote the F -algebra of all n × n matrices over F [resp. upper triangular n × n matrices over F ] [resp. upper triangular n × n matrices over F with constant main diagonal].
3. The passage to local algebras over an algebraically closed field
In this section we show that the Conjecture reduces to a consideration of local algebras over an algebraically closed field. Lemma 3. Let F be a subfield of field K and R an F -algebra. Let
r 1 ⊗ b 1 , r 2 ⊗ b 2 , . . . , r m ⊗ b m ∈ R ⊗ F K with r i ∈ R and b i ∈ K for i ∈ {1, 2, . . . , m}. Then [r 1 ⊗ b 1 , r 2 ⊗ b 2 , . . . , r m ⊗ b m ] * = [r 1 , r 2 , . . . , r m ] * ⊗ (b 1 b 2 . . . b m ).
Proof. We provide only a proof of the inductive step.
Putting r = [r 1 , r 2 , . . . , r m ] * and b = b 1 b 2 . . . b m we see that
[r 1 ⊗ b 1 , r 2 ⊗ b 2 , . . . , r m+1 ⊗ b m+1 ] * = [r ⊗ b, r m+1 ⊗ b m+1 ] [by the inductive hypothesis] = (r ⊗ b)(r m+1 ⊗ b m+1 ) − (r m+1 ⊗ b m+1 )(r ⊗ b) = (rr m+1 ) ⊗ (bb m+1 ) − (r m+1 r) ⊗ (b m+1 b) = (rr m+1 ) ⊗ (bb m+1 ) − (r m+1 r) ⊗ (bb m+1 ) [because K is a field so b m+1 b = bb m+1 ] = (rr m+1 − r m+1 r) ⊗ (bb m+1 ) = [r 1 , r 2 , . . . , r m+1 ] * ⊗ (b 1 b 2 . . . b m+1 ).
Proposition 4. Let F be a subfield of field K and R an F -subalgebra of M n (F ). Then: (c) Suppose R is Lie nilpotent of index m. Take
(a) dim F R = dim K (R ⊗ F K). (b) R ⊗ F K is isomorphic to a K-subalgebra of M n (K). (c) If R is Lie nilpotent of index m, then so is R ⊗ F K.x 1 , x 2 , . . . , x m+1 ∈ R ⊗ F K. Since the expression [x 1 , x 2 , . . . , x m+1 ] * is additive in each of its m + 1 arguments, [x 1 , x 2 , . . . , x m+1 ] * is expressible as a sum of elements of the form [r 1 ⊗ b 1 , r 2 ⊗ b 2 , . . . , r m+1 ⊗ b m+1 ] * where r i ∈ R and b i ∈ K for i ∈ {1, 2, . . . , m + 1}. By Lemma 3 [r 1 ⊗ b 1 , r 2 ⊗ b 2 , . . . , r m+1 ⊗ b m+1 ] * = [r 1 , r 2 , . . . , r m+1 ] * ⊗ (b 1 b 2 . . . b m+1 ) = 0 ⊗ (b 1 b 2 . . . b m+1 ) [because R is Lie nilpotent of index m] = 0. It follows that [x 1 , x 2 , . . . , x m+1 ] * = 0, so R ⊗ F K is Lie nilpotent of index m.Proof. (b)⇒(a) is obvious since C ⊆ C.
(a)⇒(b) Let m and n be positive integers, F ∈ C, and R an Fsubalgebra of M n (F ) with Lie nilpotence index m. We must show that dim F R M(m + 1, n).
Choose field extension K of F such that K ∈ C. By Proposition 4 ((b) and (c)), the K-algebra R ⊗ F K is Lie nilpotent of index m and is isomorphic to a K-subalgebra of M n (K). By part (a) of this theorem, dim K (R ⊗ F K) M(m + 1, n). Hence by Proposition 4(a),
dim F R = dim K (R ⊗ F K) M(m + 1, n), as required.
It follows from Theorem 5 that the Conjecture will hold for a given field F , if it can be shown to hold for the algebraic closure of F . We shall exploit this fact in the next section.
Proposition 6. Every idempotent in a ring satisfying the Engel condition is central.
Proof. If R is an arbitrary ring and e = e 2 ∈ R, then a routine calculation shows that for each a ∈ R,
Interchanging the roles of e and 1 − e in the above argument yields
ea(1 − e) = 0.(9)
Equations (8) and (9) imply ae − eae = 0 and ea − eae = 0 whence ea = ae. We conclude that e is central.
Proposition 7. Every right artinian ring satisfying the Engel condition is isomorphic to a finite direct product of local rings.
Proof. It is known (see [1, Theorem 27.6, p. 304] or [12, Theorem 5.9, p. 49]) that every right artinian ring R contains a complete set of primitive orthogonal idempotents {e 1 , e 2 , . . . , e k } such that R decomposes as
R R ∼ = e 1 R ⊕ e 2 R ⊕ · · · ⊕ e k R,
where each e i R has unique maximal proper submodule e i J(R). If R satisfies the Engel condition, then each idempotent e i is central by Proposition 6, so the above decomposition is a decomposition of (twosided) ideals with each e i R = e i Re i a local ring. Proof. Since rank e = r, F (n) e has dimension r as an F -space, so F (n) e ∼ = F (r) as F -spaces. Then
eM n (F )e ∼ = End F F (n) e ∼ = End F F (r) ∼ = M r (F ).
The following theorem tells us that for a given field F , the Conjecture will hold for all F -subalgebras of M n (F ), if it can be shown to hold for all local F -subalgebras of M n (F ).
dim F R M(m + 1, n). Proof. (a)⇒(b) is obvious. (b)⇒(a)
Let m and n be positive integers and R an F -subalgebra of M n (F ) with Lie nilpotence index m. Note that R satisfies the Engel condition of index m. Since R is a finite dimensional F -algebra, it is right (and left) artinian, and so by Proposition 7, R ∼ = R 1 × R 2 × · · · × R k where each R i is a local right artinian ring. This entails the existence of a complete set of central primitive orthogonal idempotents {e 1 , e 2 , . . . , e k } in R such that
R R ∼ = e 1 R ⊕ e 2 R ⊕ · · · ⊕ e k R (10) with e i R = e i Re i ∼ = R i for each i ∈ {1, 2, . . . , k}. For each i ∈ {1, 2, . . . , k} put r i def = rank e i .(11)
The equation 1 R = I n = e 1 + e 2 + · · · + e k induces the F -space decomposition (11)]. (12) Observe that each local ring e i R is an F -subalgebra of e i M n (F )e i , and that e i M n (F )e i ∼ = M r i (F ) for each i ∈ {1, 2, . . . , k}, by Lemma 8. It is clear too that each e i R must be Lie nilpotent of index m, since R has the same property and e i R ⊆ R.
F (n) = F (n) e 1 ⊕ F (n) e 2 ⊕ · · · ⊕ F (n) e k . Thus n = dim F F (n) = dim F (F (n) e 1 ) + dim F (F (n) e 2 ) + · · · + dim F (F (n) e k ) = r 1 + r 2 + · · · + r k [by
The aforementioned facts, together with (b), imply that dim F (e i R) M(m + 1, r i ) for each i ∈ {1, 2, . . . , k}. Then (12)].
dim F R = k i=1 dim F (e i R) [by (10)] k i=1 M(m + 1, r i ) M m + 1, k i=1 r i [by Proposition 28] = M(m + 1, n) [by
Simultaneous triangularization and the passage to upper triangular matrix rings
The main result of this section (Theorem 12) shows that for algebraically closed fields F , the Conjecture reduces to a consideration of F -subalgebras of U * n (F ), the algebra of upper triangular matrices over F with constant main diagonal.
Recall that an F -subalgebra R of M n (F ) is said to be simultaneously
upper triangularizable in M n (F ) if there exists an invertible U ∈ M n (F ) such that U −1 RU ⊆ U n (F ).
A key result is the following. Although implicit in [8, Theorem 1, p. 434] we shall provide a proof in the absence of a suitable reference.
Proposition 10. Let F be an algebraically closed field.
(a) If R is a finite dimensional local F -algebra, then R has F -space decomposition R = F · 1 R ⊕ J(R). (b) If R is a local F -subalgebra of M n (F ), then there exists an in- vertible U ∈ M n (F ) such that U −1 RU ⊆ U * n (F ). Thus, R is isomorphic to an F -subalgebra of U * n (F ). Proof. (a) Since R is local, it follows that R/J(R) is a division algebra that is finite dimensional over F . Since F is algebraically closed this implies R/J(R) ∼ = F . Inasmuch as F · 1 R ∩ J(R) = 0, the equation dim F (F · 1 R + J(R)) = 1 + dim F J(R) = dim F (R/J(R)) + dim F J(R) = dim F R entails R = F · 1 R ⊕ J(R).
(b) It is known (see [13,Theorem 1.4.6,p. 12]) that for an algebraically closed field F , a necessary and sufficient condition for an Fsubalgebra R of M n (F ) to be simultaneously upper triangularizable in M n (F ) is that R/J(R) is commutative, a condition that is clearly met in our case. Hence U −1 RU ⊆ U n (F ) for some invertible U ∈ M n (F ). Putting S = U −1 RU we note that since S is local, S = F I n ⊕ J(S) by (a). Since every element of J(S) is a nilpotent matrix in U n (F ), and a nilpotent upper triangular matrix is strictly upper triangular, we have Proof. (a) and (b) are equivalent by Theorem 9 without any restriction on the field F .
U −1 RU = S = F I n ⊕ J(S) ⊆ U * n (F ).
The equivalence of (b) and (c) is a consequence of Proposition 10(b) which tells us that up to isomorphism, the local F -subalgebras of M n (F ) are precisely the F -subalgebras of U * n (F ).
5.
Subalgebras of U * n (F ) The main body of theory is developed in this section. Throughout this section and unless otherwise stated, F shall denote a field and R an F -subalgebra of U * n (F ). Let V be a faithful right R-module. We define a sequence {R k } k∈N of F -subalgebras of R, a sequence {J k } k∈N where each J k is an ideal of R k , and a sequence {U k } k∈N of F -subspaces of V as follows
R 1 def = R, J 1 def = J(R 1 ), and U 1 def = any F -subspace complement of V J 1 in V . For k ∈ N, k 2, define R k def = F I n + (0 : R k−1 U k−1 ), J k def = J(R k ), and U k def = any F -subspace complement of V J 1 J 2 . . . J k in V J 1 J 2 . . . J k−1 .(14)
It follows from the definition of U k that
V J 1 J 2 . . . J k−1 = U k ⊕ V J 1 J 2 . . . J k(15)
as F -spaces.
For convenience we put J 0 = R. Since (0 : R k−1 U k−1 ) ⊆ R k−1 and since every F -subalgebra of U * n (F ) contains F I n , it is clear from the definition of R k in (14) that R k−1 ⊇ R k for every k ∈ N, k 2. We thus have
R 1 ⊇ R 2 ⊇ · · ·(16)
It is easily shown that if S and T are any F -subalgebras of U * n (F ), then S ⊆ T if and only if J(S) ⊆ J(T ). In the light of this observation, (16) implies that
J 1 ⊇ J 2 ⊇ · · ·(17)
Since J k ⊆ J 1 for all k ∈ N, and J 1 is nilpotent, we must have J 0 J 1 . . . J k = 0 for k sufficiently large. Define
ℓ def = min{k ∈ N : J 0 J 1 . . . J k = 0}.(18)
It follows from (17)
that J 0 J 1 . . . J k−1 ⊇ J 0 J 1 . . . J k for each k ∈ N.
We thus have the descending chain
R = J 0 ⊇ J 0 J 1 ⊇ · · · ⊇ J 0 J 1 . . . J ℓ−1 ⊇ J 0 J 1 . . . J ℓ = 0.
This, in turn, induces a descending chain
V = V J 0 ⊇ V J 0 J 1 ⊇ · · · ⊇ V J 0 J 1 . . . J ℓ−1 ⊇ 0. (19) Note that V J 0 J 1 . . . J ℓ−1 = 0 since J 0 J 1 . . . J ℓ−1 = 0 and V is a faithful right R-module.
Recall that if R is an arbitrary ring, then a submodule N of a right R-module M is said to be superfluous if (14), and positive integer ℓ defined as in (18). Let k ∈ {1, 2, . . . , ℓ}. Then:
(a) V J 0 J 1 . . . J k is a superfluous R k -submodule of V J 0 J 1 . . . J k−1 ; (b) U k R k = V J 0 J 1 . . . J k−1 = U k ⊕ · · · ⊕ U ℓ ; (c) J k+1 = (0 : R k U k ); (d) V J 0 J 1 . . . J k−1 is a faithful right R k -module.
Proof. (a) That V J 0 J 1 . . . J k is a right R k -module is a consequence of the fact that J 0 J 1 . . . J k is an F -subspace of R that is closed under right multiplication by elements from R k .
Since R k ⊆ R k−1 , every right R k−1 -module is canonically a right R k -module. In particular,
V J 0 J 1 . . . J k−1 is a right R k -module. It remains to show that V J 0 J 1 . . . J k is superfluous in V J 0 J 1 . . . J k−1 . Put U = V J 0 J 1 . . . J k−1 .
Since J k ⊆ J 1 and J 1 is nilpotent, J k must also be nilpotent. It follows from Lemma 13 that UJ k is a superfluous submodule of U, as required.
(b) Since U k R k ⊇ U k , it follows from (15) that
V J 0 J 1 . . . J k−1 = U k R k + V J 0 J 1 . . . J k where the right-hand-side of the above equation is a sum of R k -submodules of V J 0 J 1 . . . J k−1 . Since V J 0 J 1 . . . J k is a superfluous R k -submodule of V J 0 J 1 . . . J k−1 by (a), we must have U k R k = V J 0 J 1 . . . J k−1 .
To establish the equation V J 0 J 1 . . . J k−1 = U k ⊕ · · · ⊕ U ℓ , we note first that the U i constitute an independent family of F -subspaces of V . This is clear from the definition of the U i in (14). This means that the sum U k ⊕ · · · ⊕ U ℓ is indeed a direct sum of F -subspaces. It remains to establish equality.
Since, by (14), U ℓ is an F -subspace complement of V J 0 J 1 . . . J ℓ in V J 0 J 1 . . . J ℓ−1 , and since V J 0 J 1 . . . J ℓ = 0 by definition of ℓ, we must have
V J 0 J 1 . . . J ℓ−1 = U ℓ .
Repeated application of the formula for U k in (14) shows that
V J 0 J 1 . . . J ℓ−2 = U ℓ−1 ⊕ U ℓ ,
and, more generally, that
V J 0 J 1 . . . J k−1 = U k ⊕ · · · ⊕ U ℓ , as required.
(c) Since k − 1 < ℓ, it follows from (b) and the minimality of ℓ that U k R k = V J 0 J 1 . . . J k−1 = 0, whence U k = 0. This means that (0 : R k U k ) must be a proper right ideal of R k and so cannot contain any units of R k . Inasmuch as R k is an F -subalgebra of U * n (F ), (0 : R k U k ) must therefore comprise strictly upper triangular matrices. Since, by (14), R k+1 = F I n + (0 : R k U k ), we must have J k+1 = J(R k+1 ) = (0 : R k U k ).
(d) We use induction on k. Take k = 1. Then V J 0 J 1 . . . J k−1 = V J 0 = V , which is a faithful R 1 -module by hypothesis. This establishes the base case.
To establish the inductive step, take t ∈ R k with k 2 and suppose
(V J 0 J 1 . . . J k−1 )t = 0.(20)
Since V J 0 J 1 . . . J k−1 = 0, t cannot be a unit of R k , and since R k is local, we must have t ∈ J k . By (c), J k = (0 :
R k−1 U k−1 ), so U k−1 t = 0.(21)
We thus have (20) and (21)].
(V J 0 J 1 . . . J k−2 )t = (U k−1 + V J 0 J 1 . . . J k−1 )t [by (15)] = 0 [by
By the inductive hypothesis, (14), and positive integer ℓ defined as in (18). Then
V J 0 J 1 . . . J k−2 is a faithful right R k−1 - module. Since t ∈ J k ⊆ R k ⊆ R k−1 , the above equation entails t = 0. We conclude that V J 0 J 1 . . . J k−1 is a faithful R k -module.
Remark 15. (a) Taking k = 1 in Lemma 14(b) yields the F -subspace decomposition
V = U 1 ⊕ U 2 ⊕ · · · ⊕ U ℓ . (22) Substituting the equation V J 0 J 1 . . . J k−1 = U k ⊕· · ·⊕U ℓ of Lemma 14(b) into (22) yields V = U 1 ⊕ · · · ⊕ U k−1 ⊕ V J 0 J 1 . . . J k−1 . (23) (b) The faithfulness of V J 0 J 1 . . . J k−1 proved in Lemma 14(d) means that (V J 0 J 1 . . . J k−1 )J k = 0 if and only if J k = 0. Moreover, since V is faithful as a right R-module, we have that V J 0 J 1 . . . J k = 0 ⇔ J 0 J 1 . . . J k = 0. It follows that J 0 J 1 . . . J k = 0 ⇔ J k = 0.R = R 1 ⊃ R 2 ⊃ · · · ⊃ R ℓ = R ℓ+1 = · · ·
is a strictly descending chain of F -subalgebras of U * n (F ) that stabilizes at R ℓ . Moreover, J ℓ = 0, so that R ℓ = F I n .
Proof. Suppose R k = R k+1 for some k ℓ. Note that we cannot have U k = 0 since this would imply, by Lemma 14(b), that V J 0 J 1 . . . J k−1 = 0, which contradicts the fact that
V J 0 J 1 . . . J k−1 ⊇ V J 0 J 1 . . . J ℓ−1 = 0. Now 0 = U k J k+1 [because J k+1 = (0 : R k U k ) by Lemma 14(c)] = U k J k [because J k = J(R k ) = J(R k+1 ) = J k+1 by hypothesis] = (U k R k )J k [because J k is an ideal of R k ] = (V J 0 J 1 . . . J k−1 )J k [by Lemma 14(b)].
Since k ℓ it follows from the minimality of ℓ that k = ℓ. We have thus proven that R k ⊃ R k+1 for k ∈ {1, 2, . . . , ℓ − 1}.
In Remark 15(b) we noted that J ℓ = 0. Since R ℓ ⊆ U * n (F ), this entails R ℓ = F I n . However, since every F -subalgebra of U * n (F ) contains F I n , the descending chain of F -subalgebras must stabilize at R ℓ .
Let the sequences {R k } k∈N , {J k } k∈N and {U k } k∈N be defined as in (14), and positive integer ℓ defined as in (18). For each k ∈ {1, 2, . . . , ℓ} define
d k def = dim F U k .(24)
A key step in the proof of Theorem 17(c) below is inspired by [ (14), positive integer ℓ defined as in (18), and {d k : 1 k ℓ} defined as in (24). Then: We next derive the recursive formula
(a) dim F (U k J k ) = dim F V − k i=1 d i for each k ∈ {1, 2, . . . , ℓ}. (b) dim F V = ℓ i=1 d i . (c) dim F R M(ℓ, dim F V ).
Proof. (a) Inasmuch as
V = U 1 ⊕ · · · ⊕ U k ⊕ V J 0 J 1 . . . J k [by (23)] = U 1 ⊕ · · · ⊕ U k ⊕ (V J 0 J 1 . . . J k−1 )J k = U 1 ⊕ · · · ⊕ U k ⊕ (U k R k )J k [by Lemma 14(b)] = U 1 ⊕ · · · ⊕ U k ⊕ U k J k [because J k is an ideal of R k ], we have dim F V = d 1 + · · · + d k + dim F (U k J k ),dim F J k d k dim F V − k i=1 d i + dim F J k+1 (1 k ℓ).(25)
To this end, take k ∈ {1, 2, . . . , ℓ}, X ∈ J k and let ρ X : U k → U k J k be the right multiplication by X map. Observe that ρ X is an F -linear map and thus a member of Hom F (U k , U k J k ).
Define the map Θ : J k → Hom F (U k , U k J k ) by Θ(X) = ρ X . It is also easily seen that Θ is an F -linear map. Note that
Ker Θ = {X ∈ J k : ρ X = 0} = {X ∈ J k : U k X = 0} = (0 : J k U k ) = J k ∩ (0 : R k U k ) = J k ∩ J k+1 [by Lemma 14(c)] = J k+1 [because J k ⊇ J k+1 ].(26)
We thus have
dim F J k = rank Θ + nullity Θ dim F (Hom F (U k , U k J k )) + dim F J k+1 [by (26)] = dim F U k · dim F (U k J k ) + dim F J k+1 = d k dim F V − k i=1 d i + dim F J k+1 [by (a)],
which is (25).
Letting k take on the values from 1 to ℓ − 1 in (25), we see that
dim F J 1 ℓ−1 j=1 d j dim F V − j i=1 d i + dim F J ℓ = ℓ−1 j=1 d j dim F V − j i=1 d i [because J ℓ = 0] = 1 2 (dim F V ) 2 − ℓ i=1 d 2 i [because dim F V = ℓ i=1 d i by (b)] M(ℓ, dim F V ) − 1 [by the definition of M(ℓ, dim F V ) noting that dim F V = ℓ i=1 d i ].(27)
Since R has F -space decomposition R = F I n ⊕ J, we have
dim F R = 1 + dim F J = 1 + dim F J 1 [because J = J 1 ] 1 + M(ℓ, dim F V ) − 1 [by (27)] = M(ℓ, dim F V ).
In Proposition 29 it is shown that M(ℓ, n) is an increasing function in both arguments. This means, with reference to Theorem 17(c), that the smaller the value of ℓ, the lower the upper bound M(ℓ, dim F V ) for dim F R.
We shall show presently that if the F -subalgebra R of U * n (F ) has radical J satisfying J m = 0 for some m ∈ N, then the value of ℓ cannot exceed m, and so
dim F R M(m, dim F V ).
In the next section we shall strengthen the above by proving that if R has Lie nilpotence index m (this is the case if J m+1 = 0), then the value of ℓ cannot exceed m + 1, from which we may deduce
dim F R M(m + 1, dim F V ).
Since the d i are positive in Theorem 17(b), we must have ℓ dim F V . A combination of Theorem 17(c), the fact that M(ℓ, n) is increasing in its first argument (Proposition 29), and the formula for M(n, n) derived in Corollary 27(a), yields:
Corollary 18. If R is an F -subalgebra of U * n (F ) and V any faithful right R-module, then dim F R M(dim F V, dim F V ) = 1 2 (dim F V ) 2 − dim F V + 1.
Remark 19. If V = F n = n times F × F × · · · × F is interpreted as a 1 × n matrix over F , then it has the canonical structure of a faithful right module with respect to any F -subalgebra of the matrix algebra M n (F ). For such a module V , we have
dim F V = n.
This allows us to replace dim F V with n in each of the results in this, and subsequent, sections. In particular, taking dim F V = n in the previous corollary yields the upper bound dim F R 1 2 (n 2 − n) + 1, an observation that has little value, since the expression 1 2 (n 2 − n) + 1 coincides with the dimension of the overlying F -algebra U * n (F ). Proposition 20. Let the sequences {R k } k∈N , {J k } k∈N and {U k } k∈N be defined as in (14), and positive integer ℓ defined as in (18). If J m = 0 for some m ∈ N, then ℓ m.
Proof. Inasmuch as J 0 J 1 . . . J m ⊆ J m = 0, it follows from the definition of ℓ in (18) that ℓ m.
Corollary 21. If R is an F -subalgebra of U * n (F ) satisfying J m = 0, and V is any faithful right R-module, then dim F R M(m, dim F V ).
Proof. It follows from Theorem 17(c) and Proposition 20 that there exists a positive integer ℓ m such that dim F R M(ℓ, dim F V ). By Proposition 29,
M(ℓ, dim F V ) M(m, dim F V ), whence dim F R M(m, dim F V ).
6. Lie nilpotent subalgebras of U * n (F ): the main theorem A routine inductive argument establishes the following. where c σ ∈ {−1, 0, 1} for all σ ∈ S m , and {σ ∈ S m : c σ = 0 and σ(1) = 1} is a singleton comprising the identity permutation.
Proposition 23. Let the sequences {R k } k∈N , {J k } k∈N and {U k } k∈N be defined as in (14), and positive integer ℓ defined as in (18). If R is Lie nilpotent of index m, then ℓ m + 1.
Proof. By Lemma 14(c), we have J 2 = (0 : R U 1 ). Pick arbitrary r ∈ R and b k ∈ J k for each k ∈ {2, . . . , m + 1}. Since U 1 J 2 = 0 and J 2 ⊇ J 3 ⊇ · · · ⊇ J m+1 , we have U 1 b k = 0 for all k ∈ {2, . . . , m + 1}. Thus, using Lemma 22, we see that
U 1 [r, b 2 , . . . , b m+1 ] * = U 1 rb 2 . . . b m+1 . But R is Lie nilpotent of index m, so [r, b 2 , . . . , b m+1 ] * = 0 whence U 1 rb 2 . . . b m+1 = 0. Since r is arbitrary, we get 0 = (U 1 R)b 2 . . . b m+1 = V b 2 . . . b m+1 [because U 1 R = V by Lemma 14(b)],
from which we infer b 2 . . . b m+1 = 0 since V is faithful. It follows that J 2 . . . J m+1 = 0, so ℓ m + 1 by definition of ℓ.
Theorem 24. For all positive integers m and n, and fields F , if R is any F -subalgebra of U * n (F ) with Lie nilpotence index m, then
dim F R M(m + 1, n).
Proof. Let m and n be arbitrary positive integers, and F an arbitrary field. Let R be an F -subalgebra of U * n (F ) with Lie nilpotence index m. If sequences {R k } k∈N , {J k } k∈N and {U k } k∈N are defined as in (14), and positive integer ℓ defined as in (18), then it follows from Theorem 17(c) that
dim F R M(ℓ, dim F V ).
Choose V to be F n , so that dim F V = n (see Remark 19). By Proposition 23, ℓ m + 1. Since M(ℓ, n) is increasing in its first argument by Proposition 29, we have dim F R M(ℓ, dim F V ) = M(ℓ, n) M(m + 1, n).
Remark 25. Let R be any F -subalgebra of U * n (F ) satisfying the polynomial identity
f (x 1 , x 2 , . . . , x m ) = σ∈Sm c σ x σ(1) x σ(2) . . . x σ(m) = 0
where c σ ∈ F for all σ ∈ S m , and {σ ∈ S m : c σ = 0 and σ(1) = 1} is a singleton comprising the identity permutation.
Arguments similar to those used earlier in this section show that
dim F R M(m + 1, n).
We are finally in a position to complete the proof of the Conjecture.
Proof of Conjecture. Let F be any field with algebraic closure K. Taking the field F of Theorems 12 and 24 to be K, we see that the latter is just Statement (c) of the former. It thus follows from Theorem 12 ((c)⇒(a)) that the Conjecture holds in respect of field K.
Taking the class of fields C in Theorem 5 to be the singleton C = {K} and noting that F is a subfield of K, we conclude that the Conjecture holds in respect of field F . Since F was chosen arbitrarily, the proof is complete.
The function M(ℓ, n)
The purposes of this section are twofold. First, to establish a number of important properties of the function M(ℓ, n) that are required in earlier theory, and second to obtain an explicit description of M(ℓ, n); without such a description, the important results of this paper remain somewhat opaque. This task will involve the solution of an integervariable optimization problem. Our methods, however, are first principled and require no background knowledge of integer optimization techniques.
We shall make use of the following notation: if k = (k 1 , k 2 , . . . , k ℓ ) ∈ N ℓ 0 , then: ⊲ supp k def = {i ∈ {1, 2, . . . , ℓ} : k i > 0}; and
⊲ |k| def = ℓ i=1 k 2 i 1/2 so that |k| 2 = ℓ i=1 k 2 i .
Proposition 26. Let ℓ and n be positive integers. Then the following statements are equivalent for k = (k 1 , k 2 , . . . , k ℓ ) ∈ N ℓ 0 such that
ℓ i=1 k i = n: (a) M(ℓ, n) = 1 2 (n 2 − |k| 2 ) + 1; (b) |k i − k j | 1 for all i, j ∈ {1, 2, . . . , ℓ}.
Proof. (a)⇒(b) Suppose (a) holds but |k p − k q | 2 for some p, q ∈ {1, 2, . . . , ℓ}. Without loss of generality, we may suppose that k p k q + 2. Define
k ′ = (k ′ 1 , k ′ 2 , . . . , k ′ ℓ ) ∈ N ℓ 0 by: k ′ i def = k i , if i / ∈ {p, q} k p − 1, if i = p k q + 1, if i = q. Note that ℓ i=1 k ′ i = ℓ i=1 k i = n. Then: 1 2 n 2 − |k ′ | 2 + 1 − M(ℓ, n) = 1 2 n 2 − |k ′ | 2 + 1 − 1 2 n 2 − |k| 2 + 1 = 1 2 ℓ i=1 k 2 i − (k ′ i ) 2 = 1 2 k 2 p + k 2 q − (k ′ p ) 2 − (k ′ q ) 2 = 1 2 k 2 p + k 2 q − (k p − 1) 2 − (k q + 1) 2 = 1 2 (2k p − 2k q − 2) = k p − k q − 1 > 0 [because k p k q + 2].
This implies that 1 2 (n 2 − |k ′ | 2 ) + 1 > M(ℓ, n), a contradiction.
(b)⇒(a) Suppose k = (k 1 , k 2 , . . . , k ℓ ) ∈ N ℓ 0 is such that ℓ i=1 k i = n and |k i − k j | 1 for all i, j ∈ {1, 2, .
. . , ℓ}. Inasmuch as each k i is nonnegative this implies the existence of some r ∈ N such that
k i ∈ {r − 1, r} ∀i ∈ {1, 2, . . . , ℓ}. (28) Now suppose M(ℓ, n) = 1 2 (n 2 − |k ′ | 2 ) + 1 with k ′ = (k ′ 1 , k ′ 2 , . . . , k ′ ℓ ) ∈ N ℓ 0 such that ℓ i=1 k ′ i = n.
It follows from implication (a)⇒(b) that |k ′ i − k ′ j | 1 for all i, j ∈ {1, 2, . . . , ℓ}, so there must exist some s ∈ N such that
k ′ i ∈ {s − 1, s} ∀i ∈ {1, 2, . . . , ℓ}.(29)
If r < s, then it follows from (28) and (29) that
k i r s − 1 k ′ i ∀i ∈ {1, 2, . . . , ℓ}. Since ℓ i=1 k i = ℓ i=1 k ′ i , the above inequalities can only be satisfied if k i = k ′ i for all i ∈ {1, 2, .
. . , ℓ}, whence k = k ′ . A similar argument shows that k = k ′ whenever r > s. Thus if r = s, then k = k ′ , whence 1 2 (n 2 − |k| 2 ) + 1 = 1 2 (n 2 − |k ′ | 2 ) + 1 = M(ℓ, n) and the proof is complete.
Now suppose r = s. Since k i , k ′ i ∈ {r, r − 1} for each i ∈ {1, 2, . . . , ℓ} and since ℓ i=1 k i = ℓ i=1 k ′ i ,
it is easily seen that k and k ′ are equal to within permutation of their coordinates, that is to say, there exists a permutation σ ∈ S ℓ such that k ′ i = k σ(i) for all i ∈ {1, 2, . . . , ℓ}. Clearly, in such a situation |k| = |k ′ | and 1 2 (n 2 − |k| 2 ) + 1 = 1 2 (n 2 − |k ′ | 2 ) + 1 = M(ℓ, n). (a) Suppose ℓ n. If k j = 0 for some j ∈ {1, 2, . . . , ℓ}, then k i ∈ {0, 1} for all i ∈ {1, 2, . . . , ℓ}, whence n = ℓ i=1 k i < ℓ n, a contradiction.
If ℓ = n, then clearly k i = 1 for all i ∈ {1, 2, . . . , ℓ}, so |k| 2 = n and M(ℓ, n) = M(n, n) = 1 2 (n 2 − n) + 1. (b) Suppose ℓ > n. Since n = ℓ i=1 k i , we must have k j = 0 for some j ∈ {1, 2, . . . , ℓ}. Thus k i ∈ {0, 1} for all i ∈ {1, 2, . . . , ℓ}, so |k| 2 = n and M(ℓ, n) = M(n, n).
Proposition 28. Let ℓ be an integer satisfying ℓ 2 and n 1 , n 2 , . . . , n k any sequence of positive integers. Then
M ℓ, k i=1 n i k i=1 M(ℓ, n i ).
Proof. We provide a proof in the case k = 2; the arguments used can be applied mutatis-mutandis to establish the inductive step in a proof by induction on k. Choose k = (k 1 , k 2 , . . . , k ℓ ) ∈ N ℓ 0 such that
ℓ i=1 k i = n 1(30)
and M(ℓ, n 1 ) = 1 2 n 2 1 − |k| 2 + 1,
and choose k = (k 1 ,k 2 , . . . ,k ℓ ) ∈ N ℓ 0 such that ℓ i=1k i = n 2(31)
and M(ℓ, n 2 ) = 1 2 n 2 2 − |k| 2 + 1.
If |supp k| = |supp k| = 1, then it follows from (30) that M(ℓ, n 1 ) = 1 2 (n 2 1 − n 2 1 ) + 1 = 1, and from (32) that M(ℓ, n 2 ) = 1 2 (n 2 2 − n 2 2 ) + 1 = 1. Since ℓ, n 1 +n 2 2, it is clear that we can choose k * = (k * 1 , k * 2 , . . . , k * ℓ ) ∈ N ℓ 0 such that |supp k * | 2 and ℓ i=1 k * i = n 1 + n 2 . Then M(ℓ, n 1 + n 2 ) 1 2 ((n 1 + n 2 ) 2 − |k * | 2 ) + 1
= 1 2 ℓ i=1 k * i 2 − ℓ i=1 (k * i ) 2 + 1 = ℓ i,j=1, i<j k * i k * j + 1 2 = M(ℓ, n 1 ) + M(ℓ, n 2 ),
as required. Now suppose |supp k| 2 or |supp k| 2. Put k = (k 1 ,k 2 , . . . ,k ℓ ) = k + k. By (30) and (32)
ℓ i=1k i = n 1 + n 2 .(34)
Then M(ℓ, n 1 + n 2 ) 1 2 (n 1 + n 2 ) 2 − |k| 2 + 1 [by (34) and the definition of M(ℓ, n 1 + n 2 )] (31) and (33) (30) Proof. That M(ℓ, n) is increasing in its second argument is an immediate consequence of Proposition 28.
= 1 2 n 2 1 + n 2 2 − |k| 2 − |k| 2 + n 1 n 2 − ℓ i=1 k iki + 1 = M(ℓ, n 1 ) + M(ℓ, n 2 ) + n 1 n 2 − ℓ i=1 k iki − 1 [by] = M(ℓ, n 1 ) + M(ℓ, n 2 ) + ℓ i=1 k i ℓ i=1k i − ℓ i=1 k iki − 1 [by
To show that M(ℓ, n) is increasing in its first argument, it suffices to show that M(ℓ, n) M(ℓ + 1, n). Choose k = (k 1 , k 2 , . . . , k ℓ ) ∈ N ℓ 0 such that ℓ i=1 k i = n and M(ℓ, n) = 1 2 (n 2 − |k| 2 ) + 1. Putting k ′ = (k 1 , k 2 , . . . , k ℓ , 0) ∈ N ℓ+1 0 , we see that M(ℓ, n) = 1 2 (n 2 − |k| 2 ) + 1 = 1 2 (n 2 − |k ′ | 2 ) + 1 M(ℓ + 1, n), as required.
We attempt now an explicit description of the function M(ℓ, n). This is achieved in Theorem 31. If ℓ and n are positive integers with ℓ n, then Corollary 27 exhibits the simple formula M(ℓ, n) = 1 2 (n 2 − n) + 1.
We shall therefore restrict our attention to the case ℓ n. For such integers ℓ and n we denote by n (mod ℓ) the nonnegative remainder on dividing n by ℓ, that is, the unique integer r < ℓ that satisfies n = n ℓ ℓ + r.
Let r = n (mod ℓ) and define d = (d 1 , d 2 , . . . , d ℓ ) ∈ N ℓ 0 by
d i def = n ℓ , for 1 i ℓ − r n ℓ + 1, for ℓ − r < i ℓ.(36)
We omit the proof of the following routine lemma.
Lemma 30. Let ℓ and n be positive integers with ℓ n and r = n (mod ℓ). If d is defined as in (36), then
|d| 2 = (ℓ − r) n ℓ 2 + r n ℓ + 1 2 = n 2 − r 2 ℓ + r.
Theorem 31. Let ℓ and n be positive integers with ℓ n and r = n (mod ℓ). If d is defined as in (36), then
M(ℓ, n) = 1 2 n 2 − |d| 2 + 1 = 1 2 n 2 − (ℓ − r) n ℓ 2 − r n ℓ + 1 2 + 1 = n 2 (ℓ − 1) 2ℓ + 1 2 r 2 ℓ − r + 1.
Proof. It is clear from the definition of d in (36) that ℓ i=1 d i = n and |d i − d j | 1 for all i, j ∈ {1, 2, . . . , ℓ}. Hence, by Proposition 26 ((b)⇒(a)),
M(ℓ, n) = 1 2 n 2 − |d| 2 + 1 = 1 2 n 2 − (ℓ − r) n ℓ 2 − r n ℓ + 1 2 + 1 [by Lemma 30] = 1 2 n 2 − n 2 − r 2 ℓ + r + 1 [by Lemma 30] = 1 2 n 2 − n 2 ℓ + r 2 ℓ − r + 1 = n 2 (ℓ − 1) 2ℓ + 1 2 r 2 ℓ − r + 1.
Suppose F is any field and R the algebra of n × n matrices over F of type d = ( (ℓ−r) times n ℓ , n ℓ , . . . , n ℓ , r times n ℓ + 1, n ℓ + 1, . . . , n ℓ + 1), with n ℓ 2. Figure 1 is a pictorial representation of the radical J of R. Inasmuch as R has the form R = F I n + J with J satisfying J ℓ = 0, it follows that R is Lie nilpotent of index ℓ − 1. (This assertion is explained in more detail in the discussion following the statement of the Conjecture (7).) Moreover,
dim F R = ℓ−1 j=1 d j n − j i=1 d i + 1 = 1 2 n 2 − |d| 2 + 1 = M(ℓ, n) [by Theorem 31].
Thus R is an F -subalgebra of M n (F ) whose dimension is maximal amongst F -subalgebras of M n (F ) with Lie nilpotence index ℓ − 1.
If 1 2 (n 2 − |k| 2 ) + 1 is interpreted as a real-valued function of real variables k = (k 1 , k 2 , . . . , k ℓ ) ∈ R ℓ , the methods of multivariable Calculus show that the function 1 2 (n 2 − |k| 2 ) + 1, subject to the constraint ℓ i=1 k i = n, attains a maximum of n 2 (ℓ−1) 2ℓ +1 at k = ( n ℓ , n ℓ , . . . , n ℓ ) ∈ R ℓ . Thus
· · · · · · · · · · · · · · · · · · · · · · · · · · J = d 1 = n ℓ d 2 = n ℓ d ℓ−r = n ℓ d ℓ−r+1 = n ℓ + 1 d ℓ−r+2 = n ℓ + 1 d ℓ−1 = n ℓ + 1 d ℓ = n ℓ + 1 n 2 (ℓ − 1) 2ℓ + 1 ≥ M(ℓ, n).(37)
We explore now instances in which (37) is an equation, a situation that arises precisely when D < 1, where
D def = n 2 (ℓ − 1) 2ℓ + 1 − M(ℓ, n).
It follows from Theorem 31 that
D = 1 2 r − r 2 ℓ (38)
where r = n (mod ℓ). Observe that D = D(r, ℓ) is a function only of r and ℓ. Figure 2 is a sketch of the level curve D(r, ℓ) = 1 in the rℓ-plane, interpreting r and ℓ as real-valued variables. A simple calculation shows that the curve has equation
ℓ = r 2 r − 2 .
Its essential features are obtained using elementary Calculus. Remark 33. The reader will observe with reference to Theorem 32(b)(i), that if, amongst others, 1 ℓ 7, we have the simplified formula M(ℓ, n) = n 2 (ℓ − 1) 2ℓ + 1.
In particular, if ℓ = 2, then M(2, n) = n 2 4 + 1,
which corresponds with the upper bound in Schur's classical result.
An illustrative example
The main body of theory developed in Section 5 is based on the triple of sequences {R k } k∈N , {J k } k∈N and {U k } k∈N defined in (14). In this section we show that the terms in these sequences are easily visualized in the case where R is the algebra of n × n matrices over field F of type (d 1 , d 2 , . . . , d ℓ ). Indeed, this special case provides the germ for our proof strategy.
Let F be any field and (d 1 , d 2 , . . . , d ℓ ) any sequence of positive integers satisfying ℓ i=1 d i = n with ℓ 2. Let R be the algebra of n × n matrices over F of type (d 1 , d 2 , . . . , d ℓ ). We saw in Section 1 that the radical J of R has pictorial representation · · · · · · · · · · · · · · ·
J 1 = J = d 1 d 2 d ℓ−1 d ℓ
Observe that dim F J 1 corresponds with the sum of the dimensions (to be visualized as areas) of each of the ℓ − 1 blocks that make up J 1 .
With this perspective we see that
dim F J 1 = 1st block d 1 (n − d 1 ) + 2nd block d 2 (n − d 1 − d 2 ) + · · ·+ (ℓ−1)th block d ℓ−1 (n − d 1 − · · · − d ℓ−1 ) .
Note also that J ℓ 1 = 0, from which we infer that R 1 is Lie nilpotent of index ℓ − 1. (This inference is explained in the discussion following the statement of the Conjecture (7).) Take V = F n , which in this context is to be visualized as a 1 × n block thus V = Given the above pictorial representations of V and J 1 , we see that
V J 1 = d 1 (zero entries) Choosing U 1 = d 1
we see that · · · · · · · · · · · · · · · J 2 = (0 :
R 1 U 1 ) = d 1 (zero rows) d 2 d ℓ−1 d ℓ
Here:
⊲ dim F J 2 = 2nd block d 2 (n − d 1 − d 2 ) + · · · + (ℓ−1)th block d ℓ−1 (n − d 1 − · · · − d ℓ−1 ); ⊲ J ℓ−1 2 = 0; ⊲ R 2 is Lie nilpotent of index ℓ − 2; ⊲ V J 1 J 2 = d 1 + d 2 ; ⊲ U 2 = d 1 d 2 .
Continuing in this manner, we arrive at a smallest F -subalgebra of R properly containing F I n , namely R ℓ−1 , and this has radical comprising a single block · · · · · · · · · J ℓ−1 = (14) is not unique. This has the consequence that the sequence {R k } k∈N of F -subalgebras of R is not uniquely determined by R. Are the R k unique to within isomorphism perhaps? Or failing this, are the dimensions (over F ) of the R k unique?
(2) Recall that if g is a Lie algebra, then the Derived Series {g [m] } m∈N for g is defined recursively as follows for all m ∈ N, from which it follows that every ring R that is Lie nilpotent of index m, is also Lie solvable of index m. This being so, it is natural to ask whether the main theorems of this paper remain valid if the condition 'Lie nilpotent of index m' is substituted with the weaker 'Lie solvable of index m'.
(3) Expressed in terms that make no explicit reference to the overlying matrix ring, a key result in this paper asserts that if R is an F -algebra with Lie nilpotence index m, and V is any faithful right R-module, then dim F R M(m + 1, dim F V ). (This is Theorem 24 with dim F V in place of n.) We ask whether the same inequality holds if the requirement that R be a finite dimensional F -algebra is weakened to R being merely a (two-sided) artinian ring. In such a situation, 'R-module length' takes the place of 'F -dimension' thus yielding the conjecture If R is a (two-sided) artinian ring with Lie nilpotence index m and V is any faithful right R-module with finite composition length, then length R R M(m + 1, length V R ).
In the case where m = 1, the above reduces to the question [7, Section 5, Open problem (a), p. 562] that is answered in [2] 2 .
Theorem 2 .
2Let F be any field, and m and n arbitrary positive integers. Then there exists an F -subalgebra R of M n (F ) with Lie nilpotence index m such that dim F R = M(m + 1, n).
( 0 :
0I X) def = {a ∈ I : Xa = 0} = I ∩ (0 : R X).
Proof. (a) is standard theory -see for example [1, Exercise 19.3, p. 231]. (b) The hypothesis entails R⊗ F K is a K-subalgebra of M n (F )⊗ F K. The result follows noting that M n (F ) ⊗ F K ∼ = M n (K) as K-algebras (see [12, Chapter 9, Exercise 10, p. 94]).
Theorem 5 .
5Let C be a nonempty class of fields and C the class of all subfields of fields in C. The following statements are equivalent: (a) The Conjecture holds in respect of all fields in C; (b) The Conjecture holds in respect of all fields in C.
[[ 1
1− e, (1 − e)a], e] = (1 − e)ae. Putting α = (1 − e)ae we see that αe = α and eα = 0 from which it follows that [α, e] = α. Iterating, we obtain [[α, e], e] = α, [[[α, e], e], e] = α, and in general [α, m e, e, . . . , e] * = α, for each m ∈ N. If R satisfies the Engel condition of index m, then we have α = [α, m e, e, . . . , e] * = 0, and so (1 − e)ae = 0.
Lemma 8 .
8Let F be a field and e an idempotent of M n (F ). If rank e = r, then eM n (F )e ∼ = M r (F ) as F -algebras.
Theorem 9 .
9The following statements are equivalent for a field F : (a) The Conjecture holds in respect of F ; (b) For all positive integers m and n, if R is any local F -subalgebra of M n (F ) with Lie nilpotence index m, then
Remark 11 .
11(a) The observation that the factor ring R/J(R) is commutative, is key in the proof of Proposition 10(b). We point out that this property is possessed by all Lie nilpotent rings. Indeed, [17, Proposition 3.1(3), p. 4790] asserts that if rad(R) denotes the prime radical of a Lie nilpotent ring R, then R/rad(R) is commutative. Since rad(R) ⊆ J(R), the commutativity of R/J(R) follows. (b) If F is an algebraically closed field of characteristic zero (the latter assumption is not made in Proposition 10) and R is any Lie nilpotent F -subalgebra of M n (F ), then R can be shown to be simultaneously upper triangularizable in M n (F ) as a consequence of Lie's Theorem which asserts that if g is a finite dimensional solvable Lie algebra with representation M n (F ), then g is simultaneously upper triangularizable in M n (F ). Lie's Theorem applies inasmuch as every Lie nilpotent ring is a nilpotent Lie algebra with respect to the commutator, and nilpotent Lie algebras are solvable. (This latter fact is explained in the second open question of Section 9.) Theorem 12. The following statements are equivalent for an algebraically closed field F : (a) The Conjecture holds in respect of F ; (b) For all positive integers m and n, if R is any local F -subalgebra of M n (F ) with Lie nilpotence index m, then dim F R M(m + 1, n); (c) For all positive integers m and n, if R is any F -subalgebra of U * n (F ) with Lie nilpotence index m, then dim F R M(m + 1, n).
∀L M, N + L = M ⇒ L = M. Lemma 13. If I is a nilpotent ideal of an arbitrary ring R and M is any right R-module, then MI is a superfluous submodule of M. Proof. Suppose MI + L = M with L M. Multiplying by I we obtain MI 2 + LI = MI, so MI 2 + LI + L = MI + L = M. Continuing in this way, we obtain MI k + L = M for all k ∈ N. Since I is nilpotent this yields, for k sufficiently large, the equation MI k + L = M · 0 + L = L = M. Important properties of the chain (19) are established in the next lemma. Lemma 14. Let the sequences {R k } k∈N , {J k } k∈N and {U k } k∈N be defined as in
This has the consequence thatℓ = min{k ∈ N : J 0 J 1 . . . J k = 0} = min {k ∈ N : J k = 0}.Proposition 16. Let the sequences {R k } k∈N , {J k } k∈N and {U k } k∈N be defined as in
from which (a) follows. (b) is an immediate consequence of (22) and (24). (c) If ℓ = 1, then J = J ℓ = 0, so dim F R = 1 = M(1, dim F V ) and there is nothing further to prove. Suppose ℓ 2.
Lemma 22 .
22Let R be an arbitrary ring and {r i : 1 i m} ⊆ R. Then [r 1 , r 2 , . . . , r m ] * = σ∈Sm c σ r σ(1) r σ(2) . . . r σ(m)
Corollary 27 .
27Let ℓ and n be positive integers. Then: (a) If ℓ n, then M(ℓ, n) = 1 2 (n 2 − |k| 2 )+1 for some k = (k 1 , k 2 , . . . , k ℓ ) ∈ N ℓ 0 with k i 1 for all i ∈ {1, 2, . . . , ℓ}. In particular, M(n, n) = 1 2 (n 2 − n) + 1. (b) If ℓ > n, then M(ℓ, n) = M(n, n) = 1 2 (n 2 − n) + 1. Proof. By Proposition 26, we can choose k = (k 1 , k 2 , . . . , k ℓ ) ∈ N ℓ 0 such that ℓ i=1 k i = n, M(ℓ, n) = 1 2 (n 2 − |k| 2 ) + 1 and |k i − k j | 1 for all i, j ∈ {1, 2, . . . , ℓ}.
by hypothesis, |supp k| 2 or |supp k| 2, we must have ℓ i,j=1, i =j k ikj 1, hence by (35), M(ℓ, n 1 +n 2 ) M(ℓ, n 1 )+M(ℓ, n 2 ), as required.Proposition 29. The function M(ℓ, n) is increasing in both its arguments.
Figure 1 .
1Pictorial representation of the radical J of R
Figure 2 .=
2The level curve D(r, ℓ) {(r, ℓ) ∈ R 2 : 0 r ℓ − 1 and D(r, ℓ) < 1}. The content of Theorem 32 below is easily gleaned fromFigure 2by assembling together points (r, ℓ) belonging to S that have integral coordinates.Theorem 32. Let ℓ and n be positive integers with ℓ n and r = n (mod ℓ). Then the following statements are equivalent:(a) M(ℓ, n) = n 2 (ℓ − 1) 2ℓ + 1;(b) (r, ℓ) belongs to one of the following (disjoint) sets:(i) {(r, ℓ) : 0 r ℓ − 1 and 1 ℓ 7};(ii) {(r, ℓ) : 0 r 2 and ℓ 8};
(
iii) {(r, r + 1) : r 7} ∪ {(r, r + 2) : r 7}; (iv) {(3, 8), (5, 8)}.
sequence {U k } k∈N of F -subspace complements defined in
8, Theorems 1 and 2, p. 434] extended Schur's result by showing that the upper bound holds for commutative subalgebras of M n (F ) for all fields F .
], we call a ring R Lie nilpotent [resp. Lie nilpotent of index m] if R, considered as a Lie algebra via the commutator, is nilpotent [resp. nilpotent of index m]. The reader will observe that the commutative rings are precisely the rings that are Lie nilpotent of index 1.A ring R is said to satisfy the Engel condition of index m if the identity[x, m times y, . . . , y] * = 0, holds in R. A ring is said to satisfy the Engel condition if it satisfies the Engel condition of index m for some m ∈ N. Clearly a ring that is Lie nilpotent of index m satisfies the Engel condition of index m. The following result of Riley and Wilson [14, p. 974] establishes a partial converse. Proposition 1. If F is any field and R an F -algebra that is generated by a finite number d of elements, and R satisfies the Engel condition of index m, then R is Lie nilpotent of index f (d, m) m, where the index f (d, m) depends only on d and m.
7, 2. Proof of Schur's Inequality, p. 558]. Theorem 17. Let the sequences {R k } k∈N , {J k } k∈N and {U k } k∈N be defined as in
g [1] def = g, and g [m] def = [g [m−1] , g [m−1] ], for m > 1. We say g is solvable if g [m] = 0 for some m ∈ N, m > 1, and more specifically, solvable of index m, if g [m+1] = 0. We call a ring R Lie solvable [resp. Lie solvable of index m] if R, considered as a Lie algebra via the commutator, is solvable [resp. solvable of index m]. If {g [m] } m∈N denotes the Lower Central Series for g, it is easily seen that g [m] ⊆ g [m]
Our Lie algebras are over a commutative ring that is not necessarily a field. No harm shall come of this more general interpretation since, in the few instances where results about standard Lie algebras are used, the underlying commutative ring is a field.
Although the ring R is assumed to be local in Gustafson's formulation, Cowsik's proof does not assume localness.
Rings and Categories of Modules. F W Anderson, K R Fuller, Graduate Texts in Mathematics, Series. 13Springer-Verlag2nd EditionF. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Series No. 13, 2nd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 1992.
A short note on the Schur-Jacobson Theorem. R C Cowsik, Proc. Amer. Math. Soc. 1182R. C. Cowsik, A short note on the Schur-Jacobson Theorem, Proc. Amer. Math. Soc. 118(2) (1993), 675-676.
Cayley-Hamilton theorem for 2 × 2 matrices over the Grassmann algebra. M Domokos, J. Pure Appl. Algebra. 133M. Domokos, Cayley-Hamilton theorem for 2 × 2 matrices over the Grassmann algebra, J. Pure Appl. Algebra 133 (1998), 69-81.
On the dimension of faithful modules over finite dimensional basic algebras. M Domokos, Linear Algebra Appl. 365M. Domokos, On the dimension of faithful modules over finite dimensional basic algebras, Linear Algebra Appl. 365 (2003), 155-157.
Free Algebras and PI-Algebras. V Drensky, Springer-VerlagV. Drensky, Free Algebras and PI-Algebras, Springer-Verlag, 2000.
. V Drensky, E Formanek, Polynomial Identity Rings. Birkhäuser-VerlagV. Drensky and E. Formanek, Polynomial Identity Rings, Birkhäuser-Verlag, 2004.
On Maximal Commutative Algebras of Linear Transformations. W H Gustafson, J. Algebra. 42W. H. Gustafson, On Maximal Commutative Algebras of Linear Transforma- tions, J. Algebra, 42 (1976), 557-563.
Schur's theorems on commutative matrices. N Jacobson, Bull. Amer. Math. Soc. 50N. Jacobson, Schur's theorems on commutative matrices, Bull. Amer. Math. Soc. 50 (1944), 431-436.
Schur's inequality for the dimension of commuting families of matrices. N S Karamzadeh, Math. Inequal. Appl. 133N. S. Karamzadeh, Schur's inequality for the dimension of commuting families of matrices, Math. Inequal. Appl. 13(3) (2010), 625-628.
Ideals of Identities of Associative Algebras. A R Kemer, Translations of Math. Monographs. 87AMSA. R. Kemer, Ideals of Identities of Associative Algebras, Translations of Math. Monographs, Vol. 87 (1991), AMS, Providence, Rhode Island.
A simple proof of a theorem of Schur. M Mirzakhani, Amer. Math. Monthly. 1053M. Mirzakhani, A simple proof of a theorem of Schur, Amer. Math. Monthly 105(3) (1998), 260-262.
A Course in Ring Theory. D S Passman, Wadsworth & Brooks/Cole Advanced Books & SoftwareD. S. Passman, A Course in Ring Theory, Wadsworth & Brooks/Cole Ad- vanced Books & Software, 1991.
H Radjavi, P Rosenthal, Simultaneous Triangularization. New York, IncH. Radjavi and P. Rosenthal, Simultaneous Triangularization, Springer-Verlag, New York, Inc., 2012.
Associative rings satisfying the Engel condition. D M Riley, M C Wilson, Proc. Amer. Math. Soc. 1274D. M. Riley and M. C. Wilson, Associative rings satisfying the Engel condition, Proc. Amer. Math. Soc. 127(4) (1999), 973-976.
Zur Theorie der vertauschbaren Matrizen. J Schur, J. Reine Angew. Math. 130J. Schur, Zur Theorie der vertauschbaren Matrizen, J. Reine Angew. Math. 130 (1905), 66-76.
New determinants and the Cayley-Hamilton theorem for matrices over Lie nilpotent rings. J Szigeti, Proc. Amer. Math. Soc. 1258J. Szigeti, New determinants and the Cayley-Hamilton theorem for matrices over Lie nilpotent rings, Proc. Amer. Math. Soc. 125(8) (1997), 2245-2254.
J Szigeti, L Van Wyk, On Lie nilpotent rings and Cohen's Theorem. 43J. Szigeti and L. van Wyk, On Lie nilpotent rings and Cohen's Theorem, Comm. Algebra 43(11) (2015), 4783-4796.
The two theorems of Schur on commutative matrices. Z Wan, G Li, Acta Math. Sinica. 14Chinese Math.Z. Wan and G. Li, The two theorems of Schur on commutative matrices, Acta Math. Sinica 14, 143-150 (Chinese); translated as Chinese Math. 5 (1964), 156-164.
| [] |
[
"ON ELLIPTIC SOLUTIONS OF THE ASSOCIATIVE YANG-BAXTER EQUATION",
"ON ELLIPTIC SOLUTIONS OF THE ASSOCIATIVE YANG-BAXTER EQUATION"
] | [
"Igor Burban ",
"Andrea Peruzzi "
] | [] | [] | We give a direct proof of the fact that elliptic solutions of the associative Yang-Baxter equation arise from an appropriate spherical order on an elliptic curve. | 10.1016/j.geomphys.2022.104499 | [
"https://arxiv.org/pdf/2110.02853v1.pdf"
] | 238,407,863 | 2110.02853 | 1f123a03fe59d4327407a685531a1477bcda914c |
ON ELLIPTIC SOLUTIONS OF THE ASSOCIATIVE YANG-BAXTER EQUATION
6 Oct 2021
Igor Burban
Andrea Peruzzi
ON ELLIPTIC SOLUTIONS OF THE ASSOCIATIVE YANG-BAXTER EQUATION
6 Oct 2021
We give a direct proof of the fact that elliptic solutions of the associative Yang-Baxter equation arise from an appropriate spherical order on an elliptic curve.
Introduction
Let A = Mat n (C) be the algebra of square matrices of size n ∈ N and C 3 , 0) r − A ⊗ A be the germ of a meromorphic function. The following version of the associative Yang-Baxter equation (AYBE) with spectral parameters was introduced by Polishchuk in [9]:
(1) r(u; x 1 , x 2 ) 12 r(u + v; x 2 , x 3 ) 23 = r(u + v; x 1 , x 3 ) 13 r(−v; x 1 , x 2 ) 12 + r(v; x 2 , x 3 ) 23 r(u; x 1 , x 3 ) 13 .
The upper indices in this equation indicate the corresponding embeddings of A ⊗ A into
A ⊗ A ⊗ A. For example, the germ r 13 is defined as
r 13 : C 3 r − A ⊗ A ı 13 − A ⊗ A ⊗ A,
where ı 13 (x ⊗ y) = x ⊗ 1 ⊗ y. Two other germs r 12 and r 23 are defined in a similar way. We are interested in those solutions of AYBE, which are non-degenerate, skew-symmetric (meaning that r(v; x 1 , x 2 ) = −r 21 (−v; x 2 , x 1 )) and which admit a Laurent expansion of the form (2) r(v; x 1 , x 2 ) = ½ ⊗ ½ v + r 0 (x 1 , x 2 ) + vr 1 (x 1 , x 2 ) + v 2 r 2 (x 1 , x 2 ) + . . .
All elliptic and trigonometric solutions of AYBE satisfying (2) were classified in [9,10].
Recall the description of elliptic solutions of AYBE. Let ε = exp 2πid n , where 0 < d < n is such that gcd(d, n) = 1. We put For any (k, l) ∈ I := 1, . . . , n × 1, . . . , n denote Z (k,l) = Y k X −l and Z ∨ (k,l) = 1 n X l Y −k . Then the following expression (4) r ((n,d),τ ) (v; x 1 , x 2 ) = (k,l)∈I exp 2πid n kx σ v + d n kτ + l , x Z ∨ (k,l) ⊗ Z (k,l)
is a solution of AYBE satisfying (2), where x = x 2 − x 1 and
(5) σ(a, z) = 2πi n∈Z exp(−2πinz) 1 − exp −2πi(a − 2πinτ )
is the Kronecker elliptic function [12] for τ ∈ C such that Im(τ ) > 0. See also [8, Section III] for a direct proof of this fact.
In his recent work [11] Polishchuk showed that non-degenerate skew-symmetric solutions of AYBE satisfying (2) can be obtained from appropriate triple Massey products in the perfect derived category of coherent sheaves Perf(E) on a non-commutative projective curve E = (E, A), where E is an irreducible projective curve over C of arithmetic genus one and A is a symmetric spherical order on E. A simplest example of such an order is given by A = End E (F), where F is a simple vector bundle on E. Let E = E τ := C/ 1, τ be the elliptic curve determined by τ ∈ C and F be a simple vector bundle of rank n and degree d on E. It follows from results of Atiyah [1] that such F exists and the sheaf of algebras A = A (n,d) := End E (F) does not depend on the choice of F. We show that the solution of AYBE arising from the non-commutative projective curve E τ , A (n,d) is given by the formula (4). In [9,Section 2], the corresponding computations were performed using the homological mirror symmetry and explicit formulae for triple Massey products in the Fukaya category of a torus. The expression for the resulted solution of AYBE (see [9, formula (2.3)]) was different from (4). Our computations are straightforward and based by techniques developed in the articles [6,5].
Acknowledgement. This work was supported by the DFG project Bu-1866/5-1 as well as by CRC/TRR 191 project "Symplectic Structures in Geometry, Algebra and Dynamics" of German Research Council (DFG). We are grateful to Raschid Abedin for discussions of results of this paper.
Symmetric spherical orders on curves of genus one and AYBE
In this section we make a brief review of Polishchuk's construction [11]. Let E be an irreducible projective curve over C of arithmetic genus one,Ȇ its smooth part, O its structure sheaf, K the sheaf of rational functions on E and Ω the sheaf of regular differential one-forms on E. There exists a regular differential one-form ω ∈ Γ(E, Ω) such that Γ(E, Ω) = Cω. Such ω also defines an isomorphism O ∼ = Ω. If E is singular then it is rational. In this case, let P 1 ν E be the normalization morphism and O = ν * O P 1 .
Let A be a sheaf of orders on E. By definition, A is a torsion free coherent sheaf of O-algebras on E such that A ⊗ O K ∼ = Mat n (K) for some n ∈ N. For any order A we have the canonical trace morphism A t O, which coincides with the restriction of the trace and Perf(E) be the corresponding perfect derived category. Recall that Ω E := Hom E (A, Ω) is a dualising bimodule of E. If A is symmetric then Ω E ∼ = A as A-bimodules and Perf(E) is a triangulated 1-Calabi-Yau category. The last assertion means that for any pair of objects G • , H • in Perf(E) there is an isomorphism of vector spaces (6) Hom
morphism A ֒ A ⊗ O K ∼ = Mat n (K) t K (if E is smooth then O = O).E G • , H • ∼ = Hom E H • , G • [1] *
which is functorial in both arguments. Let P = Pic 0 (E) be the Jacobian of E and L ∈ Pic(P × E) be a universal line bundle. For any v ∈ P , let L v := L {v}×E ∈ Pic 0 (E) and
A v := A ⊗ O L v ∈ Coh(E).
Lemma 2.1. The coherent sheaf A is semi-stable of slope zero. Moreover,
(7) Γ(E, A v ) = 0 = H 1 (E, A v )
for all but finitely many points v ∈ P .
Proof. Let B be the kernel of the trace morphism A t O. It follows from the long exact cohomology sequence of 0
B A t O 0 that H 0 (E, B) = 0 = H 1 (E, B).
Hence, B is a semi-stable coherent sheaf on E of slope zero and A ∼ = B ⊕ O. It follows that A is semi-stable, too. The latter fact also implies the vanishing Γ(E, A v ) = 0 = H 1 (E, A v ) for all but finitely many v ∈ P .
Corollary 2.2. There exists a proper closed subset D ⊂ P × P such that
(8) Hom E A v 1 , A v 2 = 0 = Ext 1 E A v 1 , A v 2 for all v 1 , v 2 ∈ (P × P ) \ D.
Proof. This statement follows from the isomorphisms
(9) Ext i E A v 1 , A v 2 ∼ = H i E, End E (A v 1 , A v 2 ) ∼ = H i E, A v 2 −v 1 , i = 0, 1 and the vanishing (7).
Recall that for any x, y ∈Ȇ we have the following standard short exact sequences
(10) 0 − Ω − Ω(x) res x − − C x − 0 and 0 − O(−y) − O ev y − − C y − 0,
where res x and ev y are the residue and evaluation morphisms, respectively. Using the isomorphism O ω Ω, we can rewrite the first short exact sequence as
(11) 0 − O − O(x) res ω x −− C x − 0. For any H ∈ Coh(E) denote H x := H ⊗ O C x ∈ Coh(E). Tensoring (11) by A v (where v ∈ P
is an arbitrary point), we get the following short exact sequence in Coh(E):
0 − A v − A v (x) − A v
x − 0. Next, for any (u, v) ∈ P × P we have the induced long exact sequence of vector spaces
0 − Hom E (A u , A v − Hom E (A u , A v (x) − Hom E (A u , A v x − Ext 1 E (A u , A v .
It follows from (7) that the linear map (12) Hom
E (A u , A v (x) res A (u,v;x) − −−−−−− Hom E (A v x , A v x is an isomorphism if (u, v) ∈ (P × P ) \ D.
Similarly, for any v ∈ P and x = y ∈Ȇ we have the following short exact sequence
0 − A v (x − y) − A v (x) − A v (x) y − 0 in Coh(E)
, which is obtained by tensoring the evaluation sequence (10) by A v (x). After applying to it the functor Hom E (A u , − ) and using a canonical isomorphism
A v y ∼ = A v (x) y ,
we obtain a linear map
(13) Hom E (A u , A v (x) ev A (u,v;x,y) −−−−−−− Hom E (A u y , A v y .
Let
Hom E A u x , A v x α(u,v;x,y) − −−−−− Hom E A u y ,
A v y be (the unique) linear map making the following diagram of vector spaces (14)
Hom E A u , A v (x) res A (u,v;x) x x q q q q q q q q q q q q q q ev A (u,v;x,y) & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ Hom E A u x , A v x α(u,v;x,y) / / Hom E A u y , A v y commutative. Let γ(u, v; x, y) ∈ Hom E A v x , A u x ⊗ Hom E A u y , A v y be the image of α(u, v; x, y)
under the composition of the following canonical isomorphisms of vector spaces:
Lin Hom E A u x , A v x , Hom E A u y , A v y ∼ = Hom E A u x , A v x * ⊗ Hom E A u y , A v y ∼ = Hom E A v x , A u x ⊗ Hom E A u y , A v y ,
where the last isomorphism is induced by the trace morphism t.
Let P × P η − P be the group operation on P and o ∈ P be the corresponding neutral element (i.e. O ∼ = L o ). Consider the canonical projections P ×P ×E
π i − P ×E, (x 1 , x 2 ; x) (x i , x) for i = 1, 2 and P × P × E π• − P × E, (x 1 , x 2 ; x) (x 1 , x 2 )
. Then there exists S ∈ Pic(P × P ) such that
(15) (η × ½) * L ∼ = π * 1 L ⊗ π * 2 L ⊗ π * • S. In particular, L v 1 ⊗ L v 2 ∼ = L v 1 +v 2 , where v 1 + v 2 = η(v 1 , v 2 ).
For any type of E (elliptic, nodal or cuspidal) there exists a complex analytic covering map (C, +) χ − (P, η), which is also a group homomorphism. In this way we get a local coordinate on P in a neighbourhood of o. Next, we put: L := (χ × ½) * L. Since any line bundle on C × C is trivial, we get from (15) an induced isomorphism
(16) (η × ½) * L ∼ =π * 1 L ⊗π * 2 L, whereη (respectively,π i ) is the composition of η (respectively, π i ) with χ × χ. It follows that we have isomorphisms (17) O E α − L 0×E and O C×C×E β − η * L ∨ ⊗π * 1 L ⊗π * 2 L.
Let U ⊂Ȇ be an open subset for which there exists an isomorphism of Γ(U, O E )-algebras
(18) Γ(U, A) ξ − A ⊗ C Γ(U, O E )
as well as a trivialization
(19) Γ(C × U, L)ζ − Γ(C × U, O C×E ),
which identify the sections α and β from (17) with the identity section. Since η is a complex analytic covering map, we get fromζ a local trivialization ζ of the universal family L. Then such trivializations ξ and ζ allow to identify γ(u, v; x, y) with a tensor ρ(u, v; x, y) ∈ A ⊗ A. Note that by the construction the tensor ρ(u, v; x, y) depends only the difference w := u − v ∈ P with respect to the group law on the Jacobian P .
Theorem 2.3 (Polishchuk [11]). The constructed tensor ̺(w; x, y) = ρ(u, v; x, y) is a non-degenerate skew-symmetric solution of the associatiove Yang-Baxter equation (1).
Recall the key steps of the proof of this result. For any x ∈Ȇ, let S x ∈ Coh(E) be a simple object of finite length supported at x (which is unique, up to an isomorphism). For any (u, v) ∈ (P × P ) \ D and (x, y) ∈ (Ȇ ×Ȇ), x = y consider the triple Massey product (6)), we get from m 3 (u, v; x, y) a linear map (20)
Hom E A u , S x ) ⊗ Ext 1 E (S x , A v ) ⊗ Hom E A v , S y ) m 3 (u,v;x,y) − −−−−−− Hom E A u , S y ) in the triangulated category Perf(E). Since Ext 1 E (S x , A v ) * ∼ = Hom E A v , S x ) (seeHom E A u , S x ) ⊗ Hom E A v , S y ) m u,v x,y − −− Hom E A v , S x ) ⊗ Hom E A u , S y ).
The constructed family of maps m u,v x,y satisfies the identity
(21) (m v 3 ,v 2 x 1 ,x 2 ) 12 (m v 1 ,v 3 x 1 ,x 3 ) 13 − (m v 1 ,v 3 x 2 ,x 3 ) 23 (m v 1 ,v 2 x 1 ,x 2 ) 12 + (m v 1 ,v 2 x 1 ,x 3 ) 13 (m v 2 ,v 3 y 2 ,yHom E (A v 1 , S x 1 ) ⊗ Hom E (A v 2 , S x 2 ) ⊗ Hom E (A v 3 , S x 3 ) − − Hom E (A v 2 , S x 1 ) ⊗ Hom E (A v 3 , S x 2 ) ⊗ Hom E (A v 1 , S x 3 ). Moreover, m u,v
x,y is non-degenerate and skew-symmetric:
(22) ι(m u,v x,y ) = −m v,u y,x , where ι is the isomorphism Hom E A u , S x ) ⊗ Hom E A v , S y ) − Hom E A v , S y ) ⊗ Hom E A u , S x )
given by ι(f ⊗ g) = g ⊗ f . Both identities (21) and (22) are consequences of existence of an A ∞ -structure on Perf(E) which is cyclic with respect to the Serre duality (6). Applying appropriate canonical isomorphisms, one can identify m u,v
x,y with the linear map α(u, v; x, y) from the commutative diagram (14). See also [6, Theorem 2.2.17] for a detailed exposition in a similar setting.
Solutions of AYBE as a section of a vector bundle
Following the work [6], we provide a global version of the commutative diagram (14). Let
B := P × P ×Ȇ ×Ȇ \ D ×Ȇ ×Ȇ ∪ P × P × Ξ ,
where D ⊂ P × P is the locus defined by (8) and Ξ ⊂Ȇ ×Ȇ is the diagonal. Let X := B × E. Then the canonical projection X π − B admits two canonical sections B
σ i − X given by σ i (v 1 , v 2 ; x 1 , x 2 ) := (v 1 , v 2 ; x 1 , x 2 ; x i ) for i = 1, 2. Let Σ i := σ i (B) ⊂ X be the corresponding Cartier divisor. Note that Σ 1 ∩ Σ 2 = ∅.
Similarly to (11), we have the following short exact sequence in the category Coh(X):
(23) 0 − O X − O X (Σ 1 ) res ω Σ 1 − −− O Σ 1 − 0.
Here, for a local section g
(v 1 , v 2 ; x 1 , x 2 ; x) = f (v 1 , v 2 ; x 1 , x 2 ; x) x − x 1 of the line bundle O X (Σ 1 ) we put: res ω Σ 1 (g) = res x=x 1 gω x ),
where ω x is the pull-back of ω under the canonical projection X π 5 − E. Consider the non-commutative scheme X = X, π * 5 (A) as well as coherent sheaves
A (i) := π * 5 (A) ⊗ π * i,5 (L) ∈ Coh(X), where X π i,5
− P × E is the canonical projection for i = 1, 2. Tensoring (23) by A (2) , we get a short exact sequence
(24) 0 − A (2) − A (2) (Σ 1 ) − A (2) Σ 1 − 0
in the category Coh(X). Since A (1) is a locally projective O X -module, applying the functor Hom X (A (1) , −) to (24), we get an induced short exact sequence
(25) 0 Hom X A (1) , A (2) − Hom X A (1) , A (2) (Σ 1 ) − Hom X A (1) , A (2) Σ 1 0
in the category Coh(X). Base-change isomorphism combined with the vanishing (8) imply that Rπ * Hom X A (1) , A (2) = 0, where Rπ * : D b Coh(X) − D b Coh(X) is the derived direct image functor. Applying the functor π * to the short exact sequence (25), we get the following isomorphism
π * Hom X A (1) , A (2) (Σ 1 ) ∼ = − π * Hom X A (1) , A (2) Σ 1 of coherent sheaves on B. Since Hom X A (1) , A (2) Σ 1 ∼ = Hom X A (1) Σ 1 , A (2) Σ 1
, we get an isomorphism res A Σ 1 of coherent sheaves on B (which are even locally free) given as the composition
π * Hom X A (1) , A (2) (Σ 1 ) ∼ = − π * Hom X A (1) , A (2) Σ 1 ∼ = − π * Hom X A (1) Σ 1 , A (2) Σ 1 .
Next, we have the following short exact sequence of coherent sheaves on X:
(26) 0 − O X (−Σ 2 ) − O X − O Σ 2 − 0. Since Σ 1 ∩Σ 2 = ∅, the canonical morphism O Σ 2 O(Σ 1 ) Σ 2 is an isomorphism. Tensoring (26) by A (2) (Σ 1 ), we get a short exact sequence (27) 0 − A (2) (Σ 1 − Σ 2 ) − A (2) (Σ 1 ) − A (2) Σ 2 − 0
in the category Coh(X). Applying to (27) the functor Hom X (A (1) , −), we get an induced short exact sequence 0
Hom X A (1) , A (2) (Σ 1 −Σ 2 ) − Hom X A (1) , A (2) (Σ 1 ) − Hom X A (1) , A (2) Σ 1 0
in the category Coh(X). Applying the functor π * , we get a morphism of locally free sheaves ev A Σ 2 on B given as the composition
π * Hom X A (1) , A (2) (Σ 1 ) − π * Hom X A (1) , A (2) Σ 2 ∼ = − π * Hom X A (1) Σ 2 , A (2) Σ 2 .
In other words, we get the following global version
π * Hom X A (1) , A (2) (Σ 1 ) res A Σ 1 v v ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ♠ ev A Σ 2 ( ( ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ π * Hom X A (1) Σ 1 , A (2) Σ 1 α A / / π * Hom X A (1) Σ 2 , A (2) Σ 2 of the the commutative diagram (14), where α A := ev A Σ 2 • res A Σ 1 −1 .
For any 1 ≤ i, j ≤ 2, consider the canonical projection
P × P ×Ȇ ×Ȇ κ ij − P × E, (v 1 , v 2 ; x 1 , x 2 ) (v j , x i ).
Then we have the following canonical isomorphism of coherent sheaves on B:
π * Hom X A (1) Σ i , A (2) Σ i ∼ = A i ⊗ Hom B κ * 1i (L), κ * 2i (L) ,
where A i is the pull-back of A on B via the projection morphism
P × P ×Ȇ ×Ȇ − E, (v 1 , v 2 ; x 1 , x 2 ) x i for i = 1, 2. The morphism of locally free O B -modules A 1 ⊗ Hom B κ * 11 (L), κ * 21 (L) α A − A 2 ⊗ Hom B κ * 12 (L), κ * 22 (L) determines a distinguished section (28) γ A ∈ Γ B, A 1 ⊗ A 2 ⊗ κ * 11 (L) ⊗ κ * 21 (L ∨ ) ⊗ κ * 22 (L) ⊗ κ * 12 (L ∨ ) .
For i = 1, 2 consider the canonical projections P × P × E
ψ i − P × E, (v 1 , v 2 ; x) (v i ; x)
as well as
P × P × E ψ − P × P, (v 1 , v 2 ; x) (v 1 , v 2 )
. Then there exists S ∈ Pic(P × P ) such that
ψ * 1 (L) ⊗ ψ * where P × P µ − P, (v 1 , v 2 ) v 1 − v 2 .
Finally, for i = 1, 2 consider the morphism
P × P ×Ȇ ×Ȇ µ i − P ×Ȇ, (v 1 , v 2 ; x 1 , x 2 ) (v 1 − v 2 ; x i ).
Then we have an isomorphism of locally free sheaves
κ * 11 (L) ⊗ κ * 21 (L ∨ ) ⊗ κ * 22 (L) ⊗ κ * 12 (L ∨ ) ∼ = µ * 1 (L) ⊗ µ * 2 (L ∨ )
. In these terms, we can regard γ A from (28) as a section
(29) γ A ∈ Γ B, A 1 ⊗ A 2 ⊗ µ * 1 (L) ⊗ µ * 2 (L ∨ ) .
Applying trivialisations ξ of A (see (18)) and ζ of L (see (19)), we obtain from γ A a tensor-valued function
ρ A ξ,ζ : V × V × U × U − A ⊗ A, which satisfies the translation property ρ A ξ,ζ (v 1 + u, v 2 + u; x 1 , x 2 ) = ̺ A ξ,ζ (v 1 , v 2 ; x 1 , x 2 ).
Recall that for all types of the genus one curve E (smooth, nodal or cuspidal) we have a group homomorphism (C, +) − (P, +), which is locally a biholomorphic map. After making these identifications, we get the germ of a meromorphic function
(30) (C 3 , 0) ̺ − A ⊗ A, where ̺(v 1 − v 2 ; x 1 , x 2 ) := ρ A ξ,ζ (v 1 , v 2 ; x 1 , x 2 )
. This function is a non-degenerate skew-symmetric solution of AYBE.
Summary. Let E = (E, A) be a non-commutative projective curve, where E is an irreducible projective curve of arithmetic genus one and A be a symmetric spherical order on E. Let P be the Jacobian of E and L be a universal family of degree zero line bundles on E. Then we have a distinguished section γ A ∈ Γ B, A 1 ⊗ A 2 ⊗ µ * 1 (L) ⊗ µ * 2 (L ∨ ) . Choosing trivializations ξ of A (see (18)) and ζ of L (see (19)), we get the germ of a meromorphic function (C 3 , 0) ̺ − A ⊗ A, which is a non-degenerate skew-symmetric solution of AYBE. A different trivializationξ of A leads to a gauge-equivalent solution (ϕ(x 1 ) ⊗ ϕ(x 1 ) ̺(v; x 1 , x 2 ), where (C, 0) ϕ − Aut C (A) is the germ ofξξ −1 . Analogously, another choice of a trivialization ζ leads to an equivalent solution exp v(β(
x 1 )− β(x 2 )) ̺(v; x 1 , x 2 ) for some holomorphic (C, 0) β − C.
Remark 3.1. The simplest example of a symmetric spherical order is A = End E (F), where F is a simple vector bundle on E of rank n and degree d. It follows from [1,4,3] that such F exists if and only if n and d are coprime and the sheaf of algebras A = A (n,d) = End E (F) does not depend on the choice of F. Moreover, according to [11,Proposition 1.8.1], any symmetric spherical order on an elliptic curve E is isomorphic to A (n,d) for some 0 < d < n mutually prime.
Remark 3.2. Let (u, v) ∈ P × P \ D and (x, y) ∈Ȇ ×Ȇ \ Ξ. Then we have canonical isomorphisms
Hom E A u x , A v x ∼ = H 0 (E, Hom E A u x , A v x ∼ = H 0 E, A v−u ([x]) .
Analogously, we have canonical isomorphisms
Hom E A u x , A v x ∼ = A v−u x and Hom E A u y , A v y ∼ = A v−u y
such that the following diagram (31)
Hom E A u x , A v x ∼ = Hom E A u , A v (x) ∼ = res A (u,v;x) o o ev A (u,v;x,y) / / Hom E A u y , A v y ∼ = A v−u x H 0 E, A v−u ([x]) res v−u x o o ev v−u y / / A v−
Elliptic solutions of AYBE
Let τ ∈ C be such that Im(τ ) > 0, C ⊃ Λ = 1, τ ∼ = Z 2 and E = E τ = C/Λ. Recall some standard techniques to deal with holomorphic vector bundles on complex tori. An automorphy factor is a pair (A, V ), where V is a finite dimensional vector space over C and A : Λ × C − GL(V ) is a holomorphic function such that A(λ + µ, z) = A(λ, z + µ)A(µ, z) for all λ, µ ∈ Λ and z ∈ C. Such (A, V ) defines the following holomorphic vector bundle on the torus E: Let Φ : C − GL n (C) be a holomorphic function such that Φ(z + 1) = Φ(z) for all z ∈ C. Then one can define the automorphy factor (A, C n ) in the following way.
E(A, V ) := C × V / ∼, where (z, v) ∼ z + λ, A(λ, z)v for all (λ, z, v) ∈ Λ × C × V.
− A(0, z) = I n (the identity n × n matrix).
− For any k ∈ N 0 we set: Proposition 4.1. Let 0 < d < n be coprime. Then the sheaf of orders A = A (n,d) has the following description:
A(kτ, z) = Φ z + (k − 1)τ . . . Φ(z) and A(−kτ, z) = A(kτ, z − kτ ) −1 .(32) A ∼ = C × A / ∼, where (z, Z) ∼ (z + 1, Ad X (Z)) ∼ (z + τ, Ad Y (Z)),
X and Y are matrices given by (3) and Ad T (Z) = T ZT −1 for T ∈ X, Y and Z ∈ A.
For any (k, l) ∈ I := 1, . . . , n × 1, . . . , n denote Z (k,l) = Y k X −l and Z ∨ (k,l) = 1 n X l Y −k . Note that the operators Ad X , Ad Y ∈ End C (A) commute. Moreover, Ad X (Z (k,l) ) = ε k Z (k,l) and Ad Y (Z (k,l) ) = ε l Z (k,l) for any (k, l) ∈ I. As a consequence Z (k,l) (k,l)∈I is a basis of A.
Let can : A ⊗ A − End C (A) be the canonical isomorphism sending a simple tensor Z ′ ⊗Z ′′ to the linear map Z tr(Z ′ · Z) · Z ′′ . Then we have:
(33) can(Z ∨ (k,l) ⊗ Z (k,l) )(Z (k ′ ,l ′ ) ) = Z (k,l) if (k ′ , l ′ ) = (k, l) 0 otherwise.
Recall the expressions for the first and third Jacobian theta-functions (see e.g. [7]):
(34) θ (z) = θ 1 (z|τ ) = 2q 1 4 ∞ n=0 (−1) n q n(n+1) sin (2n + 1)πz , θ(z) = θ 3 (z|τ ) = 1 + 2 ∞ n=1 q n 2 cos(2πnz),
where q = exp(πiτ ). They are related by the following identity: In these terms we also get a description of a universal family L of degree zero line bundles on E.
(35) θ z + 1 + τ 2 = i exp −πi z + τ 4 θ (z).
A proof of these statements can be for instance found in [7]
A x, y) be the solution of AYBE corresponding to the datum E τ , A (n,d) with respect to the trivializations ξ (respectively, ζ) of A (respectively, L) introduced above. Then it is given by the expression (4).
v x v x H 0 A v (x) res v x o o ev v y / / A v y v y A Sol (n,
Following [11], the order A is called symmetric spherical if the following conditions are fulfilled: • The image of the trace morphism t is O and the induced morphism of coherent sheaves A t ♯ − A ∨ := Hom E (A, O) is an isomorphism. • We have: Γ(E, A) ∼ = C. Consider the non-commutative projective curve E = (E, A). Let Coh(E) be the category of coherent sheaves on E (these are sheaves of A-modules which are coherent as O-modules)
3 ) 23 = 0, both sides of which are viewed as linear maps
Given two automorphy factors (A, V ) and (B, V ), the corresponding vector bundles E(A, V ) and E(B, V ) are isomorphic if and only if there exists a holomorphic function H : C GL(V ) such that B(λ, z) = H(z + λ)A(λ, z)H(z) −1 for all (λ, z) ∈ Λ × C.
a proof of the following result, see [5, Proposition 5.1].
Lemma 4. 2 .
2For any x ∈ C consider the function ϕ x (w) = − exp −2πi(w + τ − x) . Then the following results are true.• The vector space one-dimensional and generated by θ x (w) := θ w + 1+τ2 − x . • We have: E(ϕ x ) ∼ = O E [x] . • For a, b ∈ R let v = aτ + b ∈ C and [v] = υ(v) ∈ E.Then we have:(37) E(exp(−2πiv)) ∼ = O E [0] − [v] .
or [ 6 ,F
6Section 4.1]. Let U ⊂ C be a small open neighborhood of 0 and O = Γ(U, O C ) be the ring of holomorphic functions on U . Let z be a coordinate on U , C η − E be the canonical covering map,ω = dz ∈ H 0 (E, Ω) and Γ(U, A) ξ − A ⊗ C O bethe standard trivialization induced by the automorphy data (Ad X , Ad Y ). One can also define a trivialization ζ of the universal family L of degree zero line bundles on E compatible with the isomorphisms (37). Consider the following vector spaceSol (n, d), v, x = is holomorphic F (w + 1) = Ad X F (w) F (w + τ ) = ϕ x−v (w)Ad Y F (w)
Proposition 4. 3 .
3The following diagram
.
commutative, where for F ∈ Sol (n, d), v, x we have: The isomorphisms of vector spaces v x , v y and are induced by the trivializations ξ and ζ as well as the pull-back functor η * . Comment on the proof. Since an analogous result is proven in [6, Corollary 4.2.1], we omit details here. Now we are prepared to prove the main result of this work.
Theorem 4 . 4 .
44Let r ((n,d),τ ) (v;
(L ∨ ) ∼ = (µ × ½) * ⊗ ψ * (S),
Proof. We first compute an explicit basis of the vector space Sol (n, d), v, x . Let The condition F ∈ Sol (n, d), v, x yields the following constraints on the coefficients f (k,l) :It follows from Lemma 4.2 that the vector space of holomorphic solutions of the system (39) is one-dimensional and generated by the functionFrom Proposition 4.3 and formula (33) it follows that r ((n,d),τ ) (v; x, y) is given by the following expression:where z = y − x andRelation (35) implies thatMoreover, it follows from (35) that θ ′ 1 + τ 2 = i exp −πi τ 4 θ ′ (0). Hence, we get:Here we use the fact that the Kronecker elliptic function σ(u, z) defined by(5)Remark 4.5. Let r(u; x 1 , x 2 ) be a non-degenerate skew-symmetric solution of AYBE (1) satisfying(2). Let g = sl n (C) and A π − g, Z Z − 1 n tr(Z)I n . Then r(x 1 , x 2 ) = (π ⊗ π)(r 1 (x 1 , x 2 )) is a solution of the classical Yang-Baxter equation r 12 (x 1 , x 2 ),r 13 (x 1 , x 3 ) + r 13 (x 1 , x 3 ),r 23 (x 2 , x 3 ) + r 12 (x 1 , x 2 ),r 23 (x 2 , x 3 ) = 0 r 12 (x 1 , x 2 ) = −r 21 (x 2 , x 1 ), see[9,Lemma 1.2]. Under certain additional assumptions (which are fulfilled provided r(x 1 , x 2 ) is elliptic or trigonometric), the function R(x 1 , x 2 ) = r(u • ; x 1 , x 2 ) (where u = u • from the domain of definition of r is fixed) satisfies the quantum Yang-Baxter equationsee[10,Theorem 1.5]. In fact, the expression (4) is a well-known elliptic solution of Belavin of the quantum Yang-Baxter equation; see[2].
Vector bundles over an elliptic curve. M Atiyah, Proc. Lond. Math. Soc. 73M. Atiyah, Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. (3) 7 (1957) 414-452.
Dynamical symmetry of integrable quantum systems. A Belavin, Nuclear Phys. B. 1802A. Belavin, Dynamical symmetry of integrable quantum systems, Nuclear Phys. B 180 (1981), no. 2, FS 2, 189-200.
Stable vector bundles over cuspidal cubics, Cent. L Bodnarchuk, Yu Drozd, Eur. J. Math. 14L. Bodnarchuk, Yu. Drozd, Stable vector bundles over cuspidal cubics, Cent. Eur. J. Math. 1 (2003), no. 4, 650-660.
Stable vector bundles on a rational curve with one node. I Burban, Ukraïn. Mat. Zh. 557I. Burban, Stable vector bundles on a rational curve with one node, Ukraïn. Mat. Zh., 55 (2003), no. 7, 867-874.
Vector bundles on plane cubic curves and the classical Yang-Baxter equation. I Burban, T Henrich, Journal of the European Math. Society. 173I. Burban, T. Henrich, Vector bundles on plane cubic curves and the classical Yang-Baxter equation, Journal of the European Math. Society 17, no. 3, 591-644 (2015).
Vector bundles on degenerations of elliptic curves and Yang-Baxter equations. I Burban, B Kreußler, Memoirs of the AMS. 2201035131I. Burban, B. Kreußler, Vector bundles on degenerations of elliptic curves and Yang-Baxter equations, Memoirs of the AMS 220 (2012), no. 1035, vi+131 pp.
Tata lectures on theta I, Progress in Math. D Mumford, Birkhäuser28D. Mumford, Tata lectures on theta I, Progress in Math. 28, Birkhäuser, 1983.
Odd supersymmetrization of elliptic R-matrices. A Levin, M Olshanetsky, A Zotov, J. Phys. A. 5318ppA. Levin, M. Olshanetsky, A. Zotov, Odd supersymmetrization of elliptic R-matrices, J. Phys. A 53 (2020), no. 18, 185202, 16 pp.
Classical Yang-Baxter equation and the A∞-constraint. A Polishchuk, Adv. Math. 1681A. Polishchuk, Classical Yang-Baxter equation and the A∞-constraint, Adv. Math. 168 (2002), no. 1, 56-95.
Massey products on cycles of projective lines and trigonometric solutions of the Yang-Baxter equations, Algebra, arithmetic, and geometry: in honor of Yu. A Polishchuk, Progr. Math. IIBirkhäuserI. Manin.A. Polishchuk, Massey products on cycles of projective lines and trigonometric solutions of the Yang- Baxter equations, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 573-617, Progr. Math. 270, Birkhäuser, 2009.
A Polishchuk, arXiv:2006.06101Geometrization of trigonometric solutions of the associative and classical Yang-Baxter equations. A. Polishchuk, Geometrization of trigonometric solutions of the associative and classical Yang-Baxter equations, arXiv:2006.06101.
Elliptic functions according to Eisenstein and Kronecker. A Weil, Ergebnisse der Mathematik und ihrer Grenzgebiete. 88SpringerA. Weil, Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Mathematik und ihrer Grenzgebiete 88, Springer, 1976.
Periods of modular forms and Jacobi theta functions. D Zagier, Invent. Math. 1043D. Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math. 104 (1991), no. 3, 449-465.
. address: [email protected] Straße. 100Universität Paderborn, Institut für MathematikUniversität Paderborn, Institut für Mathematik, Warburger Straße 100, 33098 Pader- born, Germany Email address: [email protected]
. Warburger Straße. 100deUniversität Paderborn, Institut für MathematikGermany Email address: [email protected]ät Paderborn, Institut für Mathematik, Warburger Straße 100, 33098 Pader- born, Germany Email address: [email protected]
| [] |
[
"Small-Scale Challenges to the ΛCDM Paradigm",
"Small-Scale Challenges to the ΛCDM Paradigm"
] | [
"James S Bullock [email protected] \nDepartment of Physics and Astronomy\nUniversity of California\n92697IrvineCAUSA\n",
"Michael Boylan-Kolchin \nDepartment of Astronomy\nThe University of Texas at Austin\nStop C14002515, 78712Speedway, AustinTXUSA\n"
] | [
"Department of Physics and Astronomy\nUniversity of California\n92697IrvineCAUSA",
"Department of Astronomy\nThe University of Texas at Austin\nStop C14002515, 78712Speedway, AustinTXUSA"
] | [] | 6Bullock • Boylan-Kolchin | 10.1146/annurev-astro-091916-055313 | [
"https://arxiv.org/pdf/1707.04256v1.pdf"
] | 59,499,272 | 1707.04256 | e3db770c0d6bc603b4aac88f2f9e9b0cd84178a0 |
Small-Scale Challenges to the ΛCDM Paradigm
James S Bullock [email protected]
Department of Physics and Astronomy
University of California
92697IrvineCAUSA
Michael Boylan-Kolchin
Department of Astronomy
The University of Texas at Austin
Stop C14002515, 78712Speedway, AustinTXUSA
Small-Scale Challenges to the ΛCDM Paradigm
10.1146/annurev-astro-091916-055313This article's Draft version. Posted with permission from the Annual Review of Astronomy and Astrophysics, Volume 55 by Annual Reviews, http://www.annualreviews.orgcosmologydark matterdwarf galaxiesgalaxy formationLocal Group
6Bullock • Boylan-Kolchin
The dark energy plus cold dark matter (ΛCDM) cosmological model has been a demonstrably successful framework for predicting and explaining the large-scale structure of Universe and its evolution with time. Yet on length scales smaller than ∼ 1 Mpc and mass scales smaller than ∼ 10 11 M , the theory faces a number of challenges. For example, the observed cores of many dark-matter dominated galaxies are both less dense and less cuspy than naively predicted in ΛCDM. The number of small galaxies and dwarf satellites in the Local Group is also far below the predicted count of low-mass dark matter halos and subhalos within similar volumes. These issues underlie the most well-documented problems with ΛCDM: Cusp/Core, Missing Satellites, and Too-Big-to-Fail. The key question is whether a better understanding of baryon physics, dark matter physics, or both will be required to meet these challenges. Other anomalies, including the observed planar and orbital configurations of Local Group satellites and the tight baryonic/dark matter scaling relations obeyed by the galaxy population, have been less thoroughly explored in the context of ΛCDM theory. Future surveys to discover faint, distant dwarf galaxies and to precisely measure their masses and density structure hold promising avenues for testing possible solutions to the small-scale challenges going forward. Observational programs to constrain or discover and characterize the number of truly dark low-mass halos are among the most important, and achievable, goals in this field over then next decade. These efforts will either further verify the ΛCDM paradigm or demand a substantial revision in our understanding of the nature of dark matter.
INTRODUCTION
Astrophysical observations ranging from the scale of the horizon (∼ 15,000 Mpc) to the typical spacing between galaxies (∼ 1 Mpc) are all consistent with a Universe that was seeded by a nearly scale-invariant fluctuation spectrum and that is dominated today by dark energy (∼ 70%) and Cold Dark Matter (∼ 25%), with baryons contributing only ∼ 5% to the energy density (Planck Collaboration et al. 2016;Guo et al. 2016). This cosmological model has provided a compelling backbone to galaxy formation theory, a field that is becoming increasingly successful at reproducing the detailed properties of galaxies, including their counts, clustering, colors, morphologies, and evolution over time (Vogelsberger et al. 2014;Schaye et al. 2015). As described in this review, there are observations below the scale of ∼ 1 Mpc that have proven more problematic to understand in the ΛCDM framework. It is not yet clear whether the small-scale issues with ΛCDM will be accommodated by a better understanding of astrophysics or dark matter physics, or if they will require a radical revision of cosmology, but any correct description of our Universe must look very much like ΛCDM on large scales. It is with this in mind that we discuss the small-scale challenges to the current paradigm. For concreteness, we assume that the default ΛCDM cosmology has parameters h = H0/(100 km s −1 Mpc −1 ) = 0.6727, Ωm = 0.3156, ΩΛ = 0.6844, Ω b = 0.04927, σ8 = 0.831, and ns = 0.9645 (Planck Collaboration et al. 2016).
Given the scope of this review, we must sacrifice detailed discussions for a more broad, high-level approach. There are many recent reviews or overview papers that cover, in more depth, certain aspects of this review. These include Frenk & White (2012), Peebles (2012), and Primack (2012) on the historical context of ΛCDM and some of its basic predictions; Willman (2010) and McConnachie (2012) on searches for and observed properties of dwarf galaxies in the Local Group; Feng (2010), Porter, Johnson &Graham (2011), andStrigari (2013) on the nature of and searches for dark matter; Kuhlen, Vogelsberger & Angulo (2012) on numerical simulations of cosmological structure formation; and Brooks (2014), Weinberg et al. (2015) and Del Popolo & Le Delliou (2017) on small-scale issues in ΛCDM. Additionally, we will not discuss cosmic acceleration (the Λ in ΛCDM) here; that topic is reviewed in Weinberg et al. (2013). Finally, space does not allow us to address the possibility that the challenges facing ΛCDM on small scales reflects a deeper problem in our understanding of gravity. We point the reader to reviews by Milgrom (2002), McGaugh (2012), andMcGaugh (2015), which compare Modified Newtonian Dynamics (MOND) to ΛCDM and provide further references on this topic.
Preliminaries: how small is a small galaxy?
This is a review on small-scale challenges to the ΛCDM model. The past ∼ 12 years have seen transformative discoveries that have fundamentally altered our understanding of "small scales" -at least in terms of the low-luminosity limit of galaxy formation.
Prior to 2004, the smallest galaxy known was Draco, with a stellar mass of M 5×10 5 M . Today, we know of galaxies 1000 times less luminous. While essentially all Milky Way satellites discovered before 2004 were found via visual inspection of photographic plates (with the exceptions of the Carina and Sagittarius dwarf spheroidal galaxies), the advent of large-area digital sky surveys with deep exposures and accurate star-galaxy separation algorithms has revolutionized the search for and discovery of faint stellar systems in the Milky Way (see Willman 2010 for a review of the search for faint satellites). The Sloan Digital Sky Survey (SDSS) ushered in this revolution, doubling the number of known Milky Way satellites in the first five years of active searches. The PAndAS survey discovered a similar population of faint dwarfs around M31 (Richardson et al. 2011). More recently the DES survey has continued this trend (Koposov et al. 2015;Drlica-Wagner et al. 2015). All told, we know of ∼ 50 satellite galaxies of the Milky Way and ∼ 30 satellites of M31 today (McConnachie 2012, updated on-line catalog), most of which are fainter than any galaxy known at the turn of the century. They are also extremely dark-matter-dominated, with mass-to-light ratios within their stellar radii exceeding ∼ 1000 in some cases (Walker et al. 2009;Wolf et al. 2010).
Given this upheaval in our understanding of the faint galaxy frontier over the last decade or so, it is worth pausing to clarify some naming conventions. In what follows, the term "dwarf" will refer to galaxies with M 10 9 M . We will further subdivide dwarfs into three mass classes: Bright Dwarfs (M ≈ 10 7−9 M ), Classical Dwarfs (M ≈ 10 5−7 M ), and Ultra-faint Dwarfs (M ≈ 10 2−5 M ). Note that another common classification for dwarf galaxies is between dwarf spheroidals (dSphs) and dwarf drregulars (dIrrs). Dwarfs with gas and ongoing star formation are usually labeled dIrr. The term dSph is reserved for dwarfs that lack gas and have no ongoing star formation. Note that the vast majority of field dwarfs (meaning that they are not satellites) are dIrrs. Most dSph galaxies are satellites of larger systems. Figure 1 illustrates the morphological differences among galaxies that span these stellar mass ranges. From top to bottom we see three dwarfs each that roughly correspond to Bright, Classical, and Ultra-faint Dwarfs, respectively. Approaching the threshold of galaxy formation. Shown are images of dwarf galaxies spanning six orders of magnitude in stellar mass. In each panel, the dwarf's stellar mass is listed in the lower-left corner and a scale bar corresponding to 200 pc is shown in the lower-right corner. The LMC, WLM, and Pegasus are dwarf irregular (dIrr) galaxies that have gas and ongoing star formation. The remaining six galaxies shown are gas-free dwarf spheroidal (dSph) galaxies and are not currently forming stars. The faintest galaxies shown here are only detectable in limited volumes around the Milky Way; future surveys may reveal many more such galaxies at greater distances. Image credits: Eckhard Slawik (LMC); ESO/Digitized Sky Survey 2 (Fornax); Massey et al. (2007;WLM, Pegasus, Phoenix); ESO (Sculptor); Mischa Schirmer (Draco), Vasily Belokurov and Sergey Koposov (Eridanus II, Pictoris I).
LMC
ADOPTED DWARF GALAXY NAMING CONVENTION
Bright Dwarfs: M ≈ 10 7−9 M -the faint galaxy completeness limit for field galaxy surveys Classical Dwarfs: M ≈ 10 5−7 M -the faintest galaxies known prior to SDSS Ultra-faint Dwarfs: M ≈ 10 2−5 M -detected within limited volumes around M31 and the Milky Way With these definitions in hand, we move to the cosmological model within which we aim to explain the counts, stellar masses, and dark matter content of these dwarfs.
Overview of the ΛCDM model
The ΛCDM model of cosmology is the culmination of century of work on the physics of structure formation within the framework of general relativity. It also indicates the confluence of particle physics and astrophysics over the past four decades: the particle nature of dark matter directly determines essential properties of non-linear cosmological structure. While the ΛCDM model is phenomenological at present -the actual physics of dark matter and dark energy remain as major theoretical issues -it is highly successful at explaining the large-scale structure of the Universe and basic properties of galaxies that form within dark matter halos.
In the ΛCDM model, cosmic structure is seeded by primordial adiabatic fluctuations and grows by gravitational instability in an expanding background. The primordial power spectrum as a function of wavenumber k is nearly scale-invariant 1 , P (k) ∝ k n with n 1. Scales that re-enter the horizon when the Universe is radiation-dominated grow extremely slowly until the epoch of matter domination, leaving a scale-dependent suppression of the primordial power spectrum that goes as k −4 at large k. This suppression of power is encapsulated by the "transfer function" T (k), which is defined as the ratio of amplitude of a density perturbation in the post-recombination era to its primordial value as a function of perturbation wavenumber k. This processed power spectrum is the input for structure formation calculations; the dimensionless processed power spectrum, defined by
∆ 2 (k, a) = k 3 2π 2 P (k) T 2 (k) d 2 (a) ,(1)
therefore rises as k 4 for scales larger than the comoving horizon at matter-radiation equality (corresponding to k = 0.008 Mpc −1 ) and is approximately independent of k for scales that re-enter the horizon well before matter-radiation equality. Here, d(a) is the linear growth function, normalized to unity at a = 1. The processed z = 0 (a = 1) linear power spectrum for ΛCDM is shown by the solid line in Figure 2. The asymptotic shape behavior is most easily seen in the bottom panel, which spans the wave number range of cosmological interest. For a more complete discussion of primordial fluctuations and the processed power spectrum we recommend that readers consult Mo, van den Bosch & White (2010). It is useful to associate each wavenumber with a mass scale set by its characteristic length r l = λ/2 = π/k. In the early Universe, when δ 1, the total amount of matter contained within a sphere of comoving Lagrangian radius r l at z = 0 is
M l = 4 π 3 r 3 l ρm = Ωm H 2 0 2 G r 3 l (2) = 1.71 × 10 11 M Ωm 0.3 h 0.7 2 r l 1 Mpc 3 .
(
The mapping between wave number and mass scale is illustrated by the top and bottom axis in Figure 2. The processed linear power spectrum for ΛCDM shown in the bottom panel (solid line) spans the horizon scale to a typical mass cutoff scale for the most common cold dark matter candidate (∼ 10 −6 M ; see discussion in Section 1.6). A line at ∆ = 1 is plotted for reference, showing that fluctuations born on comoving length scales smaller than r l ≈ 10 h −1 Mpc ≈ 14 Mpc have gone non-linear today. The top panel is zoomed in on the small scales of relevance for this review (which we define more precisely below). Typical regions on these scales have collapsed into virialized objects today. These collapsed objects -dark matter halos -are the sites of galaxy formation.
1.3. Dark matter halos 1.3.1. Global properties. Soon after overdense regions of the Universe become non-linear, they stop expanding, turn around, and collapse, converting potential energy into kinetic energy in the process. The result is virialized dark matter halos with masses given by
Mvir = 4 π 3 R 3 vir ∆ ρm ,(4)
where ∆ ∼ 300 is the virial over-density parameter, defined here relative to the background matter density. As discussed below, the value of Mvir is ultimately a definition that requires some way of defining a halo's outer edge (Rvir). This is done via a choice for ∆. The numerical value for ∆ is often chosen to match the over-density one predicts for a virialized dark matter region that has undergone an idealized spherical collapse (Bryan & Norman 1998), and we will follow that convention here. Note that given a virial mass Mvir, the virial radius, Rvir, is uniquely defined by Equation 4. Similarly, the virial velocity
Vvir ≡ GMvir Rvir ,(5)
is also uniquely defined. The parameters Mvir, Rvir, and Vvir are equivalent mass labelsany one determines the other two, given a specified over-density parameter ∆. One nice implication of Equation 4 is that a present-day object with virial mass Mvir can be associated directly with a linear perturbation with mass M l . Equating the two gives
Rvir = 0.15 ∆ 300 −1/3 r l .(6)
We see that a collapsed halo of size Rvir is approximately 7 times smaller in physical dimension than the comoving linear scale associated with that mass today. The ΛCDM dimensionless power spectrum (solid lines, Equation 1) plotted as a function of linear wavenumber k (bottom axis) and corresponding linear mass M l (top axis). The bottom panel spans all physical scales relevant for standard CDM models, from the particle horizon to the free-streaming scale for dark matter composed of standard 100 GeV WIMPs on the far right. The top panel zooms in on the scales of interest for this review, marked with a rectangle in the bottom panel. Known dwarf galaxies are consistent with occupying a relatively narrow 2 decade range of this parameter space -10 9 − 10 11 M -even though dwarf galaxies span approximately 7 decades in stellar mass. The effect of WDM models on the power spectrum is illustrated by the dashed, dotted, and dash-dotted lines, which map to the (thermal) WDM particle masses listed. See Section 3.2.1 for a discussion of power suppression in WDM.
With this in mind, Equations 3-6 allow us to self-consistently define "small scales" for both the linear power spectrum and collapsed objects: M 10 11 M . As we will discuss, potential problems associated with galaxies inhabiting halos with Vvir 50 km s −1 may point to a power spectrum that is non-CDM-like at scales r l 1 Mpc.
WE DEFINE "SMALL SCALES" AS THOSE SMALLER THAN:
M ≈ 10 11 M ↔ k ≈ 3 Mpc −1 ↔ r l ≈ 1 Mpc ↔ Rvir ≈ 150 kpc ↔ Vvir ≈ 50 km s −1 .
As alluded to above, a common point of confusion is that the halo mass definition is subject to the assumed value of ∆, which can vary by a factor of ∼ 3 depending on the author. For the spherical collapse definition, ∆ 333 at z = 0 (for our fiducial cosmology) and asymptotes to ∆ = 178 at high redshift (Bryan & Norman 1998). Another common choice is a fixed ∆ = 200 at all z (often labeled M200m in the literature). Finally, some authors prefer to define the virial overdensity as 200 times the critical density, which, according to Equation 4 would mean ∆(z) = 200ρc(z)/ρm(z). Such a mass is commonly labeled "M200" in the literature. For most purposes (e.g., counting halos), the precise choice does not matter, as long as one is consistent with the definition of halo mass throughout an analysis: every halo has the same center, but its outer radius (and mass contained within that radius) shifts depending on the definition. In what follows, we use the spherical collapse definition (∆ = 333 at z = 0) and adhere to the convention of labeling that mass "Mvir".
Before moving on, we note that it is also possible (and perhaps even preferable) to give a halo a "mass" label that is directly tied to a physical feature associated with a collapsed dark matter object rather than simply adopting a ∆. More, have advocated the use of a "splash-back" radius , where the density profile shows a sharp break (this typically occurs at ∼ 2Rvir). Another common choice is to tag halos based not on a mass but on Vmax, which is the peak value of the circular velocity Vc(r) = GM (< r)/r as one steps out from the halo center. For any individual halo, the value of Vmax ( Vvir) is linked to the internal mass profile or density profile of the system, which is the subject of the next subsection. As discussed below, the ratio Vmax/Vvir increases as the halo mass decreases.
1.3.2. Abundance. In principle, the mapping between the initial spectrum of density fluctuations at z → ∞ and the mass spectrum of collapsed (virialized) dark matter halos at later times could be extremely complicated: as a given scale becomes non-linear, it could affect the collapse of nearby regions or larger scales. In practice, however, the mass spectrum of dark matter halos can be modeled remarkably well with a few simple assumptions. The first of these was taken by Press & Schechter (1974), who assumed that the mass spectrum of collapsed objects could be calculated by extrapolating the overdensity field using linear theory even into the highly non-linear regime and using a spherical collapse model (Gunn & Gott 1972). In the Press-Schechter model, the dark matter halo mass function -the abundance of dark matter halos per unit mass per unit volume at redshift z, often written as n(M, z) -depends only on the rms amplitude of the linear dark matter power spectrum, smoothed using a spherical tophat filter in real space and extrapolated to redshift z using linear theory. Subsequent work has put this formalism on more rigorous mathematical footing (Bond
ROBUST PREDICTIONS FROM CDM-ONLY SIMULATIONS
A defining characteristic of CDM-based hierarchical structure formation is that the smallest scales collapse first -a fact that arises directly from the shape of the power spectrum ( Figure 1) and that lies at the heart of many robust predictions for the counts and structure of dark matter halos today. As discussed below, baryonic processes can alter these predictions to various degrees, but pure dark matter simulations have provided a well-defined set of basic predictions used to benchmark the theory.
The dark matter profiles of individual halos are cuspy and dense [ Figure 3]
The density profiles of individual ΛCDM halos increase steadily towards small radii, with an overall normalization and detailed shape that reflects the halo's mass assembly. At fixed mass, early-forming halos tend to be denser than later-forming halos. As with the mass function, both the shape and normalization of dark matter halo density structure is predicted by ΛCDM, with a well-quantified prediction for the scatter in halo concentration at fixed mass.
There are many more small halos than large ones [ Figure 4] The comoving number density of dark matter halos rises steeply towards small masses, dn/dM ∝ M α with α −1.9. At large halo masses, counts fall off exponentially above the mass scale that is just going nonlinear today. Importantly, both the shape and normalization of the mass function is robustly predicted by the theory.
Substructure is abundant and almost self-similar [Figure 5]
Dark matter halos are filled with substructure, with a mass function that rises as dN/dm ∝ m αs with αs −1.8 down to the low-mass free-streaming scale (m 1M for canonical models). Substructure reflects the high-density cores of smaller merging halos that survive the hierarchical assembly process. Substructure counts are nearly self-similar with host mass, with the most massive subhalos seen at mmax ∼ 0.2M host . Cole 1991;Sheth, Mo & Tormen 2001), and this extended Press-Schechter (EPS) theory yields abundances of dark matter halos that are perhaps surprisingly accurate (see Zentner 2007 for a comprehensive review of EPS theory). This accuracy is tested through comparisons with large-scale numerical simulations.
Simulations and EPS theory both find a universal form for n(M, z): the comoving number density of dark matter halos is a power law with log slope of α −1.9 for M M * and is exponentially suppressed for M M * , where M * = M * (z) is the characteristic mass of fluctuations going non-linear at the redshift z of interest 2 . Importantly, given an initial power spectrum of density fluctuations, it is possible to make highly accurate predictions within ΛCDM for the abundance, clustering, and merger rates of dark matter halos at any cosmic epoch.
1.3.3. Internal structure. Dubinski & Carlberg (1991) were the first to use N -body simu-lations to show that the internal structure of a CDM dark matter halo does not follow a simple power-law, but rather bends from a steep outer profile to a mild inner cusp obeying ρ(r) ∼ 1/r at small radii. More than twenty years later, simulations have progressed to the point that we now have a fairly robust understanding of the structure of ΛCDM halos and the important factors that govern halo-to-halo variance (e.g., Navarro et al. 2010;Klypin et al. 2016), at least for dark-matter-only simulations.
To first approximation, dark matter halo profiles can be described by a nearly universal form over all masses, with a steep fall-off at large radii transitioning to mildly divergent cusp towards the center. A common way to characterize this is via the NFW functional form (Navarro, Frenk & White 1997), which provides a good (but not perfect) description dark matter profiles:
ρ(r) = 4ρ−2 (r/r−2)(1 + r/r−2) 2 .(7)
Here, r−2 is a characteristic radius where the log-slope of the density profile is −2, marking a transition point from the inner 1/r cusp to an outer 1/r 3 profile. The second parameter, ρ−2, sets the value of ρ(r) at r = r−2. In practice, dark matter halos are better described the three-parameter Einasto (1965) profile (Navarro et al. 2004;Gao et al. 2008). However, for the small halos of most concern for this review, NFW fits do almost as well as Einasto in describing the density profiles of halos in simulations (Dutton & Macciò 2014). Given that the NFW form is slightly simpler, we have opted to adopt this approximation for illustrative purposes in this review. As Equation 7 makes clear, two parameters (e.g., ρ−2 and r−2) are required to determine a halo's NFW density profile. For a fixed halo mass Mvir (which fixes Rvir), the second parameter is often expressed as the halo concentration: c = Rvir/r−2. Together, a Mvir − c combination completely specifies the profile. In the median, and over the mass and redshift regime of interest to this review, halo concentrations increase with decreasing mass and redshift: c ∝ M −a vir (1 + z) −1 , with a 0.1 (Bullock et al. 2001). Though halo concentration correlates with halo mass, there is significant scatter (∼ 0.1 dex) about the median at fixed Mvir (Jing 2000;Bullock et al. 2001). Some fraction of this scatter is driven by the variation in halo mass accretion history (Wechsler et al. 2002;Ludlow et al. 2016), with early-forming halos having higher concentrations at fixed final virial mass.
The dependence of halo profile on a mass-dependent concentration parameter and the correlation between formation time and concentration at fixed virial mass are caused by the hierarchical build-up of halos in ΛCDM: low-mass halos assemble earlier, when the mean density of the Universe is higher, and therefore have higher concentrations than high-mass halos (e.g., Navarro, Frenk & White 1997;Wechsler et al. 2002). At the very smallest masses, the concentration-mass relation likely flattens, reflecting the shape of the dimensionless power spectrum (see our Figure 1 and the discussion in Ludlow et al. 2016); at the highest masses and redshifts, characteristic of very rare peaks, the trend seems to reverse (a < 0; Klypin et al. 2016).
The right panel of Figure 3 summarizes the median NFW density profiles for z = 0 halos with masses that span those of large galaxy clusters (Mvir = 10 15 M ) to those of the smallest dwarf galaxies (Mvir = 10 8 M ). We assume the c−Mvir relation from Klypin et al. (2016). These profiles are plotted in physical units (unscaled to the virial radius) in order to emphasize that higher mass halos are denser at every radius than lower mass halos (at least in the median). However, at a fixed small fraction of the virial radius, small halos are slightly denser than larger ones. This is a result of the concentration-mass relation. Under Right: The density profiles of median NFW dark matter halos at z = 0 with masses that span galaxy clusters (M vir = 10 15 M , black) to the approximate HI cooling threshold that is expected to correspond to the smallest dwarf galaxies (M vir ≈ 10 8 M , yellow). The lines are color coded by halo virial mass according to the bar on the right and are separated in mass by 0.5 dex. We see that (in the median) massive halos are denser than low-mass halos at a fixed physical radius. However, at a fixed small fraction of the virial radius, smaller halos are typically slightly denser than larger halos, reflecting the concentration-mass relation. This is demonstrated by the dotted line which connects ρ(r) evaluated at r = R vir for halos over a range of masses. We have chosen = 0.015 because this value provides a good match to observed galaxy half-light radii over a wide range of galaxy luminosities under the assumption that galaxies occupy halos according to abundance matching (see Section 1.5 and Figure 6). Interestingly, the characteristic dark matter density at this 'galaxy radius' increases only by a factor of ∼ 6 over almost seven orders of magnitude in halo virial mass. Left: The equivalent circular velocity curves Vc(r) ≡ GM (< r)/r for the same halos plotted on the right. The dashed line connects the radius Rmax where the circular velocity is maximum (Vmax) for each rotation curve. The dotted line tracks the R vir -V vir relation. The ratio Rmax/R vir decreases towards smaller halos, reflecting the mass-concentration relation. The ratio Vmax/V vir also increases with decreasing concentration. the ansatz of abundance matching (Section 1.5, Figure 6), galaxy sizes (half-mass radii) track a fixed fraction of their host halo virial radius: r gal 0.015Rvir (Kravtsov 2013). This relation is plotted as a dotted line such that the dotted line intersects each solid line at that r = 0.015 Rvir, where Rvir is that particular halo's virial radius. We see that small halos are slightly denser at the typical radii of the galaxies they host than are larger halos. Interestingly, however, the density range is remarkably small, with a local density of dark matter increasing by only a factor of ∼ 6 over the full mass range of halos that are expected to host galaxies, from the smallest dwarfs to the largest cD galaxies in the universe.
On the left we show the same halos, now presented in terms of the implied circular velocity curves: Vc ≡ GM (< r)/r. The dotted line in left panel intersects Vvir at Rvir for each value of Mvir. The dashed line does the same for Vmax and its corresponding radius Rmax. Higher mass systems, with lower concentrations, typically have Vmax Vvir, but for smaller halos the ratio is noticeably different than one and can be as large as ∼ 1.5 for high-concentration outliers. Note also that the lowest mass halos have Rmax Rvir and thus it is the value of Vmax (rather than Vvir) that is more closely linked to the observable Steep mass functions. The black solid line shows the z = 0 dark matter halo mass function (M halo = M vir ) for the full population of halos in the universe as approximated by Sheth, Mo & Tormen (2001). For comparison, the magenta lines show the subhalo mass functions at z = 0 (defined as M halo = M sub = M peak , see text) at the same redshift for host halos at four characteristic masses (M vir = 10 12 , 10 13 , 10 14 , and 10 15 M ) with units given along the right-hand axis. Note that the subhalo mass functions are almost self-similar with host mass, roughly shifting to the right by 10× for every decade increase in host mass. The low-mass slope of subhalo mass function is similar than the field halo mass function. Both field and subhalo mass functions are expected to rise steadily to the cutoff scale of the power spectrum, which for fiducial CDM scenarios is 1M .
"flat" region of a galaxy rotation curve. For our "small-scale" mass of Mvir = 10 11 M , typically Vmax 1.2 Vvir 60 km s −1 .
Dark matter substructure
It was only just before the turn of the century that N -body simulations set within a cosmological CDM framework were able to robustly resolve the substructure within individual dark matter halos (Ghigna et al. 1998;Klypin et al. 1999a). It soon became clear that the dense centers of small halos are able to survive the hierarchical merging process: dark matter halos should be filled with substructure. Indeed, subhalo counts are nearly selfsimilar with host halo mass. This was seen as welcome news for cluster-mass halos, as the substructure could be easily identified with cluster galaxies. However, as we will discuss in the next section, the fact that Milky-Way-size halos are filled with substructure is less clearly consistent with what we see around the Galaxy. Quantifying subhalo counts, however, is not so straightforward. Counting by mass is tricky because the definition of "mass" for an extended distribution orbiting within a collapsed halo is even more fraught with subjective decisions than virial mass. When a small halo is accreted into a large one, mass is preferentially stripped from the outside. Typically, the standard virial overdensity "edge" is subsumed by the ambient host halo. One option is to compute the mass that is bound to the subhalo, but even these masses vary from halo finder to halo finder. The value of a subhalo's Vmax is better defined, and often serves as a good tag for quantifying halos.
Another option is to tag bound subhalos using the maximum virial mass that the halos had at the time they were first accreted 3 onto a host, M peak . This is a useful option because stars in a central galaxy belonging to a halo at accretion will be more tightly bound than the dark matter. The resultant satellite's stellar mass is most certainly more closely related to M peak than the bound dark matter mass that remains at z = 0. Moreover, the subsequent mass loss (and even Vmax evolution) could change depending on the baryonic content of the host because of tidal heating and other dynamical effects (D'Onghia et al. 2010). For these reasons, we adopt M sub = M peak for illustrative purposes here.
The magenta lines in Figure 4 show the median subhalo mass functions (M sub = M peak ) for four characteristic host halo masses (Mvir = 10 12−14 M ) according to the results of Rodríguez-Puebla et al. (2016). These lines are normalized to the right-hand vertical axis. Subhalos are counted only if they exist within the virial radius of the host, which means the counting volume increases as ∝ Mvir ∝ R 3 vir for these four lines. For comparison, the black line (normalized to the left vertical axis) shows the global halo mass function (as estimated via the fitting function from Sheth, Mo & Tormen 2001). The subhalo mass function rises with a similar (though slightly shallower) slope as the field halo mass function and is also roughly self-similar in host halo mass.
Linking dark matter halos to galaxies
How do we associate dark matter halos with galaxies? One simple approximation is to assume that each halo is allotted its cosmic share of baryons f b = Ω b /Ωm ≈ 0.15 and that those baryons are converted to stars with some constant efficiency : M = f b Mvir. Unfortunately, as shown in Figure 5, this simple approximation fails miserably. Galaxy stellar masses do not scale linearly with halo mass; the relationship is much more complicated. Indeed, the goal of forward modeling galaxy formation from known physics within the ΛCDM framework is an entire field of its own (galaxy evolution; Somerville & Davé 2015). Though galaxy formation theory has progressed significantly in the last several decades, many problems remain unsolved.
Other than forward modeling galaxy formation, there are two common approaches that give an independent assessment of how galaxies relate to dark matter halos. The first involves matching the observed volume density of galaxies of a given stellar mass (or other observable such as luminosity, velocity width, or baryon mass) to the predicted abundance of halos of a given virial mass. The second way is to measure the mass of the galaxy directly and to infer the dark matter halo properties based on this dynamical estimator.
1.5.1. Abundance matching. As illustrated in Figure 5, the predicted mass function of collapsed dark matter halos has a considerably different normalization and shape than the
d n/d log 10 M [Mpc −3 ] α = − 1 .6 2 α = − 1. 32 z = 0 M halo (Sheth et al. ) f b M halo M (GAMA 2017) M (Bernardi + 2013) Figure 5
The thick black line shows the global dark matter mass function. The dotted line is shifted to the left by the cosmic baryon fraction for each halo M vir → f b M vir . This is compared to the observed stellar mass function of galaxies from Bernardi et al. (2013, magenta stars) and Wright et al. (2017;cyan squares). The shaded bands demonstrate a range of faint-end slopes αg = −1.62 to −1.32. This range of power laws will produce dramatic differences at the scales of the classical Milky Way satellites (M 10 5−7 M ). Pushing large sky surveys down below 10 6 M in stellar mass, where the differences between the power law range shown would exceed a factor of ten, would provide a powerful constraint on our understanding of the low-mass behavior. Until then, this mass regime can only be explored with without large completeness corrections in vicinity of the Milky Way.
observed stellar mass function of galaxies. The difference grows dramatically at both large and small masses, with a maximum efficiency of 0.2 at the stellar mass scale of the Milky Way (M ≈ 10 10.75 M ). This basic mismatch in shape has been understood since the earliest galaxy formation models set within the dark matter paradigm (White & Rees 1978) and is generally recognized as one of the primary constraints on feedback-regulated galaxy formation (White & Frenk 1991;Benson et al. 2003;Somerville & Davé 2015).
At the small masses that most concern this review, dark matter halo counts follow dn/dM ∝ M α with a steep slope α dm −1.9 compared to the observed stellar mass function slope of αg = −1.47 (Baldry et al. 2012, which is consistent with the updated GAMA results shown in Figure 5). Current surveys that cover enough sky to provide a global field stellar mass function reach a completeness limit of M ≈ 10 7.5 M . At this mass, galaxy counts are more than two orders of magnitude below the naive baryonic mass function f b Mvir. The shaded band illustrates how the stellar mass function would extrapolate to the faint regime spanning a range of faint-end slopes α that are marginally consistent with observations at the completeness limit.
One clear implication of this comparison is that galaxy formation efficiency ( ) must vary in a non-linear way as a function of Mvir (at least if ΛCDM is the correct underlying model). Perhaps the cleanest way to illustrate this is adopt the simple assumption of Abundance Matching (AM): that galaxies and dark matter halos are related in a one-to- Abundance matching relation from Behroozi et al. (in preparation). Gray (magenta) shows a scatter of 0.2 (0.5) dex about the median relation. The dashed line is power-law extrapolation below the regime where large sky surveys are currently complete. The cyan band shows how the extrapolation would change as the faint-end slope of the galaxy stellar mass function (α) is varied over the same range illustrated by the shaded gray band in Figure 5. Note that the enumeration of M = 10 5 M galaxies could provide a strong discriminator on faint-end slope, as the ±0.15 range in α shown maps to an order of magnitude difference in the halo mass associated with this galaxy stellar mass and a corresponding factor of ∼ 10 shift in the galaxy/halo counts shown in Figure 4.
one way, with the most massive galaxies inhabiting the most massive dark matter halos (Frenk et al. 1988;Kravtsov et al. 2004;Conroy, Wechsler & Kravtsov 2006;Moster et al. 2010;Behroozi, Wechsler & Conroy 2013). The results of such an exercise are presented in Figure 6 (as derived by Behroozi et al., in preparation). The gray band shows the median M − Mvir relation with an assumed 0.2 dex of scatter in M at fixed Mvir. The magenta band expands the scatter to 0.5 dex . This relation is truncated near the completeness limit in Baldry et al. (2012). The central dashed line in Figure 6 shows the median relation that comes from extrapolating the Baldry et al. (2012) mass function with their best-fit αg = −1.47 down to the stellar mass regime of Local Group dwarfs. The cyan band brackets the range for the two other faint-end slopes shown in Figure 5: αg = −1.62 and −1.32. Figure 6 allows us to read off the virial mass expectations for galaxies of various sizes. We see that Bright Dwarfs at the limit of detection in large sky surveys (M ≈ 10 8 M ) are naively associated with Mvir ≈ 10 11 M halos. Galaxies with stellar masses similar to the Classical Dwarfs at M ≈ 10 6 M are associated with Mvir ≈ 10 10 M halos. As we will discuss in Section 3, galaxies at this scale with M /Mvir ≈ 10 −4 are at the critical scale where feedback from star formation may not be energetic enough to alter halo density profiles significantly. Finally, Ultra-faint Dwarfs with M ≈ 10 4 M , Mvir ≈ 10 9 M , and M /Mvir ≈ 10 −5 likely sit at the low-mass extreme of galaxy formation. Measures. An alternative way to connect to the dark matter halo hosting a galaxy is to determine the galaxy's dark matter mass kinematically. This, of course, can only be done within a central radius probed by the baryons. For the small galaxies of concern for this review, extended mass measurements via weak lensing or hot gas emission is infeasible. Instead, masses (or mass profiles) must be inferred within some inner radius, defined either by the stellar extent of the system for dSphs and/or the outer rotation curves for rotationally-supported gas disks.
Bright dwarfs, especially those in the field, often have gas disks with ordered kinematics. If the gas extends far enough out, rotation curves can be extracted that extend as far as the flat part of the galaxy rotation curve V flat . If care is taken to account for non-trivial velocity dispersions in the mass extraction (e.g., Kuzio de Naray, McGaugh & de Blok 2008), then we can associate V flat ≈ Vmax.
Owing to the difficulty in detecting them, the faintest galaxies known are all satellites of the Milky Way or M31 and are dSphs. These lack rotating gas components, so rotation curve measurements are impossible. Instead, dSphs are primarily stellar dispersion-supported systems, with masses that are best probed by velocity dispersion measurements obtained star-by-star for the closest dwarfs (e.g., Walker et al. 2009;Simon et al. 2011;Kirby et al. 2014). For systems of this kind, the mass can be measured accurately within the stellar half-light radius (Walker et al. 2009). The mass within the de-projected (3D) half-light radius (r 1/2 ) is relatively robust to uncertainties in the stellar velocity anisotropy and is given by M (< r 1/2 ) = 3 σ 2 r 1/2 /G, where σ is the measured, luminosity-weighted line-ofsight velocity dispersion (Wolf et al. 2010). This formula is equivalent to saying that the circular velocity at the half-light radius is V 1/2 = Vc(r 1/2 ) = √ 3 σ . The value of V 1/2 (≤ Vmax) provides a one-point measurement of the host halo's rotation curve at r = r 1/2 .
Connections to particle physics
Although the idea of "dark" matter had been around since at least Zwicky (1933), it was not until rotation curve measurements of galaxies in the 1970s revealed the need for significant amounts of non-luminous matter (Freeman 1970;Rubin, Thonnard & Ford 1978;Bosma 1978;Rubin, Ford & Thonnard 1980) that dark matter was taken seriously by the broader astronomical community (and shortly thereafter, it was recognized that dwarf galaxies might serve as sensitive probes of dark matter; Aaronson 1983;). Very quickly, particle physicists realized the potential implications for their discipline as well. Dark matter candidates were grouped into categories based on their effects on structure formation. "Hot" dark matter (HDM) particles remain relativistic until relatively late in the Universe's evolution and smooth out perturbations even on super-galactic scales; "warm" dark matter (WDM) particles have smaller initial velocities, become non-relativistic earlier, and suppress perturbations on galactic scales (and smaller); and CDM has negligible thermal velocity and does not suppress structure formation on any scale relevant for galaxy formation. Standard Model neutrinos were initially an attractive (hot) dark matter candidate; by the mid-1980s, however, this possibility had been excluded on the basis of general phase-space arguments (Tremaine & Gunn 1979), the large-scale distribution of galaxies (White, Frenk & Davis 1983), and properties of dwarf galaxies . The lack of a suitable Standard Model candidate for particle dark matter has led to significant work on particle physics extensions of the Standard Model. From a cosmology and galaxy formation perspective, the unknown particle nature of dark matter means that cosmologists must make assumptions about dark matter's origins and particle physics properties and then investigate the resulting cosmological implications.
Cold Dark Matter (CDM): m ∼ 100 GeV, v z=0 th ≈ 0 km s −1 Warm Dark Matter (WDM): m ∼ 1 keV, v z=0 th ∼ 0.03 km s −1 Hot Dark Matter (HDM): m ∼ 1 eV, v z=0 th ∼ 30 km s −1
A general class of models that are appealing in their simplicity is that of thermal relics. Production and destruction of dark matter particles are in equilibrium so long as the temperature of the Universe kT is larger than the mass of the dark matter particle mDMc 2 . At lower temperatures, the abundance is exponentially suppressed, as destruction (via annihilation) dominates over production. At some point, the interaction rate of dark matter particles drops below the Hubble rate, however, and the dark matter particles "freeze out" at a fixed number density (see, e.g., Kolb & Turner 1994; this is also known as chemical decoupling). Amazingly, if the annihilation cross section is typical of weak-scale physics, the resulting freeze-out density of thermal relics with m ∼ 100 GeV is approximately equal to the observed density of dark matter today (e.g., Jungman, Kamionkowski & Griest 1996). This subset of thermal relics is referred to as weakly-interacting massive particles (WIMPs). The observation that new physics at the weak scale naturally leads to the correct abundance of dark matter in the form of WIMPs is known as the "WIMP miracle" (Feng & Kumar 2008) and has been the basic framework for dark matter over the past 30 years.
WIMPs are not the only viable dark matter candidate, however, and it is important to note that the WIMP miracle could be a red herring. Axions, which are particles invoked to explain the strong CP problem of quantum chromodynamics (QCD), and right-handed neutrinos (often called sterile neutrinos), which are a minimal extension to the Standard Model of particle physics that can explain the observed baryon asymmetry and why neutrino masses are so small compared to other fermions, are two other hypothetical particles that may be dark matter (among a veritable zoo of additional possibilities; see Feng 2010 for a recent review). While WIMPs, axions, and sterile neutrinos are capable of producing the observed abundance of dark matter in the present-day Universe, they can have very different effects on the mass spectrum of cosmological perturbations.
While the cosmological perturbation spectrum is initially set by physics in the very early universe (inflation in the standard scenario), the microphysics of dark matter affects the evolution of those fluctuations at later times. In the standard WIMP paradigm, the lowmass end of the CDM hierarchy is set by first collisional damping (subsequent to chemical decoupling but prior to kinetic decoupling of the WIMPs), followed by free-streaming (e.g., Hofmann, Schwarz & Stöcker 2001;Bertschinger 2006). For typical 100 GeV WIMP candidates, these processes erase cosmological perturbations with M 10 −6 M (i.e., Earth mass; Green, Hofmann & Schwarz 2004). Free-streaming also sets the low-mass end of the mass spectrum in models where sterile neutrinos decouple from the plasma while relativistic. In this case, the free-streaming scale can be approximated by the (comoving) size of the horizon when the sterile neutrinos become non-relativistic. The comoving horizon size at z = 10 7 , corresponding to m ≈ 2.5 keV, is approximately 50 kpc, which is significantly smaller than the scale derived above for L * galaxies. keV-scale sterile neutrinos are therefore observationally-viable dark matter candidates (see Adhikari et al. 2016 for a recent, comprehensive review). QCD axions are typically ∼ µeV-scale particles but are produced out of thermal equilibrium (Kawasaki & Nakayama 2013). Their free-streaming scale is significantly smaller than that of a typical WIMP (see Section 3.2.1).
The previous paragraphs have focused on the effects of collisionless damping and freestreaming -direct consequences of the particle nature of dark matter -in the linear regime of structure formation. Dark matter microphysics can also affect the non-linear regime of structure formation. In particular, dark matter self-interactions -scattering between two dark matter particles -will affect the phase space distribution of dark matter. Within observational constraints, dark matter self-interactions could be relevant in the dense centers of dark matter halos. By transferring kinetic energy from high-velocity particles to lowvelocity particles, scattering transfers "heat" to the centers of dark matter halos, reducing their central densities and making their velocity distributions nearly isothermal. This would have a direct effect on galaxy formation, as galaxies form within the centers of dark matter halos and the motions of their stars and gas trace the central gravitational potential. These effects are discussed further in Section 3.2.2.
The particle nature of dark matter is therefore reflected in the cosmological perturbation spectrum, in the abundance of collapsed dark matter structures as a function of mass, and in the density and velocity distribution of dark matter in virialized dark matter halos.
OVERVIEW OF PROBLEMS
The CDM paradigm as summarized in the previous section emerged among other dark matter variants in the early 1980s (Peebles 1982;Blumenthal et al. 1984;Davis et al. 1985) with model parameters gradually settling to their current precise state (including Λ) in the wake of overwhelming evidence from large-scale galaxy clustering, supernovae measurements of cosmic acceleration, and cosmic microwave background studies, among other data. The 1990s saw the first N -body simulations to resolve the internal structure of CDM halos on small scales. Almost immediately researchers pinpointed the two most wellknown challenges to the theory: the cusp-core problem (Flores & Primack 1994;Moore 1994) and the missing satellites problem (Klypin et al. 1999b;Moore et al. 1999). This section discusses these two classic issues from a current perspective goes on to describe a third problem, too-big-to-fail (Boylan-Kolchin, Bullock & Kaplinghat 2011), which is in some sense is a confluence of the first two. Finally, we conclude this section with a more limited discussion of two other challenges faced by ΛCDM on small scales: the apparent planar distributions seen for Local Group satellites and the dynamical scaling relations seen in galaxy populations.
Missing Satellites
The highest-resolution cosmological simulations of MW-size halos in the ΛCDM paradigm have demonstrated that dark matter (DM) clumps exist at all resolved masses, with no break in the subhalo mass function down to the numerical convergence limit (e.g., Springel et al. 2008;Kuhlen, Madau & Silk 2009;Stadel et al. 2009;Garrison-Kimmel et al. 2014;Griffen et al. 2016). We expect thousands of subhalos with masses that are (in principle) large enough to have supported molecular cooling (M peak 10 7 M ). Meanwhile, only
THREE CHALLENGES TO BASIC ΛCDM PREDICTIONS
There are three classic problems associated with the small-scale predictions for dark matter in the ΛCDM framework. Other anomalies exist, including some that we discuss in this review, but these three are important because 1) they concern basic predictions about dark matter that are fundamental to the hierarchical nature of the theory; and 2) they have received significant attention in the literature.
Missing Satellites and Dwarfs [Figures 4-8]
The observed stellar mass functions of field galaxies and satellite galaxies in the Local Group is much flatter at low masses than predicted dark matter halo mass functions: dn/dM ∝ M αg with αg −1.5 (vs. α −1.9 for dark matter). The issue is most acute for Galactic satellites, where completeness issues are less of a concern. There are only ∼ 50 known galaxies with M > 300M within 300 kpc of the Milky Way compared to as many as ∼ 1000 dark subhalos (with M sub > 10 7 M ) that could conceivably host galaxies. One solution to this problem is to posit that galaxy formation becomes increasingly inefficient as the halo mass drops. The smallest dark matter halos have simply failed to form stars altogether.
Low-density Cores vs. High-density Cusps [ Figure 9]
The central regions of dark-matter dominated galaxies as inferred from rotation curves tend to be both less dense (in normalization) and less cuspy (in inferred density profile slope) than predicted for standard ΛCDM halos (such as those plotted in Figure 3). An important question is whether baryonic feedback alters the structure of dark matter halos.
Too-Big-to-Fail [ Figure 10]
The local universe contains too few galaxies with central densities indicative of Mvir 10 10 M halos. Halos of this mass are generally believed to be too massive to have failed to form stars, so the fact that they are missing is hard to understand. The stellar mass associated with this halo mass scale (M 10 6 M , Figure 6) may be too small for baryonic processes to alter their halo structure (see Figure 13).
∼ 50 satellite galaxies down to ∼ 300 M in stars are known to orbit within the virial radius of the Milky Way (Drlica-Wagner et al. 2015). Even though there is real hope that future surveys could bring the census of ultra-faint dwarf galaxies into the hundreds (Tollerud et al. 2008;Hargis, Willman & Peter 2014), it seems unlikely there are thousands of undiscovered dwarf galaxies to this limit within the virial volume of the Milky Way. The current situation is depicted in Figure 7, which shows the dark matter distribution around a Milky Way size galaxy as predicted by a ΛCDM simulation next to a map of the known galaxies of the Milky Way on the same scale.
Given the discussion of abundance matching in Section 1.5 and the associated Figure 6, it is reasonable to expect that dark matter halos become increasingly inefficient at making galaxies at low masses and at some point go completely dark. Physical mass scales of interest in this regard include the mass below which reionization UV feedback likely suppresses gas accretion Mvir ≈ 10 9 M (Vmax 30 km s −1 ; e.g., Efstathiou 1992; Bullock, Kravtsov & Weinberg 2000;Benson et al. 2002;Bovill & Ricotti 2009;Sawala et al. 2016) The Missing Satellites Problem: Predicted ΛCDM substructure (left) vs. known Milky Way satellites (right). The image on the left shows the ΛCDM dark matter distribution within a sphere of radius 250 kpc around the center of a Milky-Way size dark matter halo (simulation by V. Robles and T. Kelley in collaboration with the authors). The image on the right (by M. Pawlowski in collaboration with the authors) shows the current census of Milky Way satellite galaxies, with galaxies discovered since 2015 in red. The Galactic disk is represented by a circle of radius 15 kpc at the center and the outer sphere has a radius of 250 kpc. The 11 brightest (classical) Milky Way satellites are labeled by name. Sizes of the symbols are not to scale but are rather proportional to the log of each satellite galaxy's stellar mass. Currently, there are ∼ 50 satellite galaxies of the Milky Way compared to thousands of predicted subhalos with M peak 10 7 M . see, e.g., Rees & Ostriker 1977). According to Figure 6, these physical effects are likely to become dominant in the regime of ultra-faint galaxies M 10 5 M . The question then becomes: can we simply adopt the abundance-matching relation derived from field galaxies to "solve" the Missing Satellites Problem down to the scale of the classical MW satellites (i.e., Mvir 10 10 M ↔ M 10 6 M )? Figure 8 (modified from Garrison-Kimmel et al. 2017a) shows that the answer is likely "yes." Shown in magenta is the cumulative count of Milky Way satellite galaxies within 300 kpc of the Galaxy plotted down to the stellar mass completeness limit within that volume. The shaded band shows the 68% range predicted stellar mass functions from the dark-matter-only ELVIS simulations ) combined with the AM relation shown in Figure 6 with zero scatter. The agreement is not perfect, but there is no over-prediction. The dashed lines show how the predicted satellite stellar mass functions would change for different assumed (field galaxy) faint-end slopes in the calculating the AM relation. An important avenue going forward will be to push these comparisons down to the ultra-faint regime, where strong baryonic feedback effects are expected to begin shutting down galaxy formation altogether.
Cusp, Cores, and Excess Mass
As discussed in Section 1, ΛCDM simulations that include only dark matter predict that dark matter halos should have density profiles that rise steeply at small radius ρ(r) ∝ r −γ , with γ 0.8 − 1.4 over the radii of interest for small galaxies (Navarro et al. 2010). This is in contrast to many (though not all) low-mass dark-matter-dominated galaxies with wellmeasured rotation curves, which prefer fits with constant-density cores (γ ≈ 0 − 0.5; e.g., McGaugh, Rubin & de Blok 2001;Marchesini et al. 2002;Simon et al. 2005;de Blok et al. 2008;Kuzio de Naray, McGaugh & de Blok 2008). A related issue is that fiducial ΛCDM simulations predict more dark matter in the central regions of galaxies than is measured for the galaxies that they should host according to AM. This "central density problem" is an issue of normalization and exists independent of the precise slope of the central density profile (Alam, Bullock & Weinberg 2002;Oman et al. 2015). While these problems are in principle distinct issues, as the second refers to a tension in total cumulative mass and the first is an issue with the derivative, it is likely that they point to a common tension. Dark-matter-only ΛCDM halos are too dense and too cuspy in their centers compared to many observed galaxies. Figure 9 summarizes the basic problem. Shown as a dashed line is the typical circular velocity curve predicted for an NFW ΛCDM dark matter halo with Vmax ≈ 40km s −1 compared to the observed rotation curves for two galaxies with the same asymptotic velocity from Oh et al. (2015). The observed rotation curves rise much more slowly than the ΛCDM expectation, reflecting core densities that are lower and more core-like than the fiducial The Cusp-Core problem. The dashed line shows the naive ΛCDM expectation (NFW, from dark-matter-only simulations) for a typical rotation curve of a Vmax ≈ 40 km s −1 galaxy. This rotation curve rises quickly, reflecting a central density profile that rises as a cusp with ρ ∝ 1/r. The data points show the rotation curves of two example galaxies of this size from the LITTLE THINGS survey (Oh et al. 2015)), which are more slowly rising and better fit by a density profile with a constant density core (Burkert 1995, cyan line).
prediction.
Too-Big-To-Fail
As discussed above, a straightforward and natural solution to the missing satellites problem within ΛCDM is to assign the known Milky Way satellites to the largest dark matter subhalos (where largest is in terms of either present-day mass or peak mass) and attribute the lack of observed galaxies in in the remaining smaller subhalos to galaxy formation physics. As pointed out by Boylan-Kolchin, Bullock & Kaplinghat (2011) The Too-Big-to-Fail Problem. Left: Data points show the circular velocities of classical Milky Way satellite galaxies with M 10 5−7 M measured at their half-light radii r 1/2 . The magenta lines show the circular velocity curves of subhalos from one of the (dark matter only) Aquarius simulations. These are specifically the subhalos of a Milky Way-size host that have peak maximum circular velocities Vmax > 30 km s −1 at some point in their histories. Halos that are this massive are likely resistant to strong star formation suppression by reionization and thus naively too big to have failed to form stars (modified from Boylan-Kolchin, Bullock & Kaplinghat 2012). The existence of a large population of such satellites with greater central masses than any of the Milky Way's dwarf spheroidals is the original Too-Big-to-Fail problem. Right: The same problem -a mismatch between central masses of simulated dark matter systems and observed galaxiespersists for field dwarfs (magenta points), indicating it is not a satellite-specific process (modified from Papastergis & Ponomareva 2017). The field galaxies shown all have stellar masses in the range 5.75 ≤ log 10 (M /M ) ≤ 7.5. The gray curves are predictions for ΛCDM halos from the fully self-consistent hydrodynamic simulations of Fitts et al. (2016) that span the same stellar mass range in the simulations as the observed galaxies.
While there are subhalos with central masses comparable to the Milky Way satellites, these subhalos were never among the ∼ 10 most massive ( Figure 10). Why would galaxies fail to form in the most massive subhalos, yet form in dark matter satellites of lower mass? The most massive satellites should be "too big to fail" at forming galaxies if the lower-mass satellites are capable of doing so (thus the origin of the name of this problem). In short, while the number of massive subhalos in dark-matter-only simulations matches the number of classical dwarfs observed (see Figure 8), the central densities of these simulated dwarfs are higher than the central densities observed in the real galaxies (see Figure 10).
While too-big-to-fail was originally identified for satellites of the Milky Way, it was subsequently found to exist in Andromeda (Tollerud, Boylan-Kolchin & Bullock 2014) and field galaxies in the Local Group (those outside the virial radius of the Milky Way and M31; Kirby et al. 2014). Similar discrepancies were also pointed out for more isolated lowmass galaxies, first based on HI rotation curve data (Ferrero et al. 2012) and subsequently using velocity width measurements (Papastergis et al. 2015;Papastergis & Shankar 2016). This version of too-big-to-fail in the field is also manifested in the velocity function of Macciò et al. 2016 andBrooks et al. 2017 for arguments that no discrepancy exists). The generic observation in the low-redshift Universe, then, is that the inferred central masses of galaxies with 10 5 M /M 10 8 are ∼ 50% smaller than expected from dissipationless ΛCDM simulations.
The too-big-to-fail and core/cusp problems would be naturally connected if low-mass galaxies generically have dark matter cores, as this would reduce their central densities relative to CDM expectations 5 . However, the problems are, in principle, separate: one could imagine galaxies that have large constant-density cores yet still with too much central mass relative to CDM predictions (solving the core/cusp problem but not too-big-to-fail), or having cuspy profiles with overall lower density amplitudes than CDM (solving too-bigto-fail but not core/cusp). Kunkel & Demers (1976) and Lynden-Bell (1976) pointed out that satellite galaxies appeared to lie in a polar great circle around the Milky Way. Insofar as this cannot be explained in a theory of structure formation, this observation pre-dates all other small-scale interpreted as a "missing dwarfs" problem if one considers the discrepancy as one in numbers at fixed V halo . We believe, however, that the more more plausible interpretation is a discrepancy in V halo at fixed number density.
Satellite Planes
5 For a sense of the problem, the amount of mass that would need to be removed to alleviate the issue on classical dwarf scales is ∼ 10 7 M within ∼ 300 pc structure issues in the Local Group by approximately two decades. The anisotropic distribution of Galactic satellites received scant attention until a decade ago, when Kroupa, Theis & Boily (2005) argued that it proved that satellite galaxies cannot be related to dark matter substructures (and thereby constituted another crisis for CDM). Kroupa et al. examined classical, pre-SDSS dwarf galaxies in and around the Milky Way and found that the observed distribution was strongly non-spherical. From this analysis, based on the distribution of angles between the normal of the best-fitting plane of dwarfs and the position vector of each MW satellite in the Galacto-centric reference frame, Kroupa et al. argued that "the mismatch between the number and spatial distribution of MW dwarves compared to the theoretical distribution challenges the claim that the MW dwarves are cosmological sub-structures that ought to populate the MW halo."
This claim was quickly disputed by Zentner et al. (2005), who investigated the spatial distribution of dark matter subhalos in simulated CDM halos and determined that it was highly inconsistent with a spherical distribution. They found that the planar distribution of MW satellites was marginally consistent with being a random sample of the subhalo distributions in their simulations, and furthermore, the distribution of satellites they considered likely to be luminous (corresponding to the more massive subhalos) was even more consistent with observations. A similar result was obtained at roughly the same time by Kang et al. (2005). Slightly later, Metz, Kroupa & Jerjen (2007) argued that the distribution of MW satellite galaxies was inconsistent, at the 99.5% level, with isotropic or prolate substructure distributions (as might be expected in ΛCDM).
Related analysis of Milky Way satellite objects has further supported the idea that the configuration is highly unusual compared to ΛCDM subhalo distributions (Pawlowski, Pflamm-Altenburg & Kroupa 2012), with the 3D motions of satellites suggesting that there is a preferred orbital pole aligned perpendicular to the observed spatial plane (Pawlowski & Kroupa 2013). The left hand side of Figure 11 shows the current distribution of satellites (galaxies and star clusters) around the Milky Way looking edge-on at the planar configuration. Note that the disk of the Milky Way could, in principle, bias discoveries away from the MW disk axis, but it is not obvious that the orbital poles would be biased by this effect. Taken together, the orbital poles and spatial configuration of MW satellites is highly unusual for a randomly drawn sample of ΛCDM subhalos .
As shown in the right-hand panel of Figure 11, the M31 satellite galaxies also show evidence for having a disk-like configuration (Metz, Kroupa & Jerjen 2007). Following the discovery of new M31 satellites and the characterization of their velocities, Conn et al. (2013) and Ibata et al. (2013) presented evidence that 15 of 27 Andromeda dwarf galaxies indeed lie in a thin plane, and further, that the southern satellites are mostly approaching us with respect to M31, while the northern satellites are mostly receding (as coded by the direction of the red triangles in Figure 11). This suggests that the plane could be rotationally supported. Our view of this plane is essentially edge-on, meaning we have excellent knowledge of in-plane motions and essentially no knowledge of velocities perpendicular to the plane. Nevertheless, even a transient plane of this kind would be exceedingly rare for ΛCDM subhalos (e.g., Ahmed, Brooks & Christensen 2017).
Work in a similar vein has argued for the existence of planar structures in the Centaurus A group and for rotationally-supported systems of satellites in a statistical sample of galaxies from the SDSS (Ibata et al. 2015). Libeskind et al. (2015) have used ΛCDM simulations to suggest that some alignment of satellite systems in the local Universe may be naturally explained by the ambient shear field, though they cannot explain thin Creasey et al. (2017). For comparison, the gray band shows expectations from dark matter only ΛCDM simulations. There is much more scatter at fixed Vmax than predicted by the simulations. Note that the galaxies used in the RAR in left-hand panel have Vmax values that span the range shown on the right. The tightness of the acceleration relation is remarkable (consistent with zero scatter given observational error, red cross), especially given the variation in central densities seen on the right. planes this way. Importantly, Phillips et al. (2015) have re-analyzed the SDSS data and argued that it is not consistent with a ubiquitous co-rotating satellite population and rather more likely a statistical fluctuation. More data that enables a statistical sample of hosts down to fainter satellites will be needed to determine whether the configurations seen in the Local Group are common.
Regularity in the Face of Diversity
Among the more puzzling aspects of galaxy phenomenology in the context of ΛCDM are the tight scaling relations between dynamical properties and baryonic properties, even within systems that are dark matter dominated. One well-known example of this is the baryonic Tully-Fisher relation (McGaugh 2012), which shows a remarkably tight connection between the total baryonic mass of a galaxy (gas plus stars) and its circular velocity V flat ( Vmax):
M b ∝ V 4
flat . Understanding this correlation with ΛCDM models requires care for the lowmass galaxies of most concern in this review (Brook, Santos-Santos & Stinson 2016).
A generalization of the baryonic Tully-Fisher relation known as the radial acceleration relation (RAR) was recently introduced by McGaugh, Lelli & Schombert (2016). Plotted in left-hand Figure 12, the RAR shows a tight correlation between the radial acceleration traced by rotation curves (g obs = V 2 /r) and that predicted solely by the observed dis-tribution of baryons (g bar ) 6 . The upper right "high-acceleration" portion of the relation correspond to baryon-dominated regions of (mostly large) galaxies. Here the relation tracks the one-to-one line, as it must. However, rotation curve points begin to peel away from the line, towards an acceleration larger than what can be explained by the baryons alone below a characteristic acceleration of a0 10 −10 m s −2 . It is this additional acceleration that we attribute to dark matter. The outer parts of large galaxies contribute to this region, as do virtually all parts of small galaxies. It is surprising, however, that the dark matter contribution in the low-acceleration regime tracks the baryonic distribution so closely, particularly in light of the diversity in galaxy rotation curves seen among galaxies of at a fixed V flat , as we now discuss.
The right-hand panel of Figure 12 illustrates the diversity in rotation curve shapes seen from galaxy to galaxy. Shown is a slightly modified version of a figure introduced by Oman et al. (2015) and recreated by Creasey et al. (2017). Each data point corresponds to a single galaxy rotation curve. The horizontal axis shows the observed value of V flat (≈ Vmax) for each galaxy and the vertical axis plots the value of the circular velocity at 2 kpc from the galaxy center. Note that at fixed V flat , galaxies demonstrate a huge diversity in central densities. Remarkably, this diversity is apparently correlated with the baryonic content in such a way as to drive the tight relation seen on the left. The gray band in the right panel shows the expected relationship between Vmax and Vc(2kpc) for halos in ΛCDM darkmatter-only simulations. Clearly, the real galaxies demonstrate much more diversity than is naively predicted.
The real challenge, as we see it, is to understand how galaxies can have so much diversity in their rotation curve shapes compared to naive ΛCDM expectations while also yielding tight correlations with baryonic content. The fact that there is a tight correlation with baryonic mass and not stellar mass (which presumably correlates more closely with total feedback energy) makes the question all the more interesting.
SOLUTIONS
Solutions within ΛCDM
In this subsection, we explore some of the most popular and promising solutions to the problems discussed above. We take as our starting point the basic ΛCDM model plus reionization, i.e., we take it as a fundamental prediction of ΛCDM that the heating of the intergalactic medium to ∼ 10 4 K by cosmic reionization will suppress galaxy formation in halos with virial temperatures below ∼ 10 4 K (or equivalently, with Vvir 20 km s −1 ) at z 6.
3.1.1. Feedback-induced cores. Many of the most advanced hydrodynamic simulations today have shown that it is possible for baryonic feedback to erase the central cusps shown in the density profiles in Figure 3 and produce core-like density profiles as inferred from rotation curves such as those shown in Figure 9 (Mashchenko, Wadsley & Couchman 2008;Madau, Shen & Governato 2014;Oñorbe et al. 2015;Read, Agertz & Collins 2016). One key prediction is that the effect of core creation will vary with The impact of baryonic feedback on the inner profiles of dark matter halos. Plotted is the inner dark matter density slope α at r = 0.015R vir as a function of M /M vir for simulated galaxies at z = 0. Larger values of α ≈ 0 imply core profiles, while lower values of α 0.8 imply cusps. The shaded gray band shows the expected range of dark matter profile slopes for NFW profiles as derived from dark-matter-only simulations (including concentration scatter). The filled magenta stars and shaded purple band (to guide the eye) show the predicted inner density slopes from the NIHAO cosmological hydrodynamic simulations by Tollet et al. (2016). The cyan stars are a similar prediction from an entirely different suite of simulations from the FIRE-2 simulations (Fitts et al. 2016;Hopkins et al. 2017, Chan et al., in preparation). Note that at dark matter core formation peaks in efficiency at M /M vir ≈ 0.005, in the regime of the brightest dwarfs. Both simulations find that for M /M vir 10 −4 , the impact of baryonic feedback is negligible. This critical ratio below which core formation via stellar feedback is difficult corresponds to the regime of classical dwarfs and ultra-faint dwarfs.
the mass in stars formed Di Cintio et al. 2014). If galaxies form enough stars, there will be enough supernovae energy to redistribute dark matter and create significant cores. If too many baryons end up in stars, however, the excess central mass can compensate and drag dark matter back in. At the other extreme, if too few stars are formed, there will not be enough energy in supernovae to alter halo density structure and the resultant dark matter distribution will resemble dark-matter-only simulations. While the possible importance of supernova-driven blowouts for the central dark matter structure of dwarf galaxies was already appreciated by Navarro, Eke & Frenk (1996) and Gnedin & Zhao (2002), an important recent development is the understanding that even low-level star formation over an extended period can drive gravitational potential fluctuations that lead to dark matter core formation. This general behavior is illustrated in Figure 13, which shows the impact of baryonic Dark matter density profiles from full hydrodynamic FIRE-2 simulations (Fitts et al. 2016). Shown are three different galaxy halos, each at mass M vir ≈ 10 10 M . Solid lines show the hydro runs while the dashed show the same halos run with dark matter only. The hatched band at the left of each panel marks the region where numerical relaxation may artificially modify density profiles and the vertical dotted line shows the half-light radius of the galaxy formed. The stellar mass of the galaxy formed increases from left to right: M ≈ 5 × 10 5 , 4 × 10 6 , and 10 7 M , respectively. As M increases, so does the effect of feedback. The smallest galaxy has no effect on the density structure of its host halo.
feedback on the inner slopes of dark matter halos α measured at 1 − 2% of the halo virial radii. Core-like density profiles have α → 0. The magenta stars show results from the NIHAO hydrodynamic simulations as a function of M /Mvir, the ratio of stellar mass to the total halo mass (Tollet et al. 2016). The cyan stars show results from an entirely different set of simulations from the FIRE-2 collaboration (Wetzel et al. 2016;Fitts et al. 2016;Garrison-Kimmel et al. 2017b, Chan et al., in preparation). The shaded gray band shows the expected slopes for NFW halos with the predicted range of concentrations from dark-matter-only simulations. We see that both sets of simulations find core formation to be fairly efficient M /Mvir ≈ 0.005. This "peak core formation" ratio maps to M 10 8−9 M , corresponding to the brightest dwarfs. At ratios below M /Mvir ≈ 10 −4 , however, the impact of baryonic feedback is negligible. The ratio below which core formation is difficult corresponds to M ≈ 10 6 M -the mass-range of interest for the too-big-to-fail problem.
The effect of feedback on density profile shapes as a function of stellar mass is further illustrated in Figure 14. Here we show simulation results from Fitts et al. (2016) for three galaxies (from a cosmological sample of fourteen), all formed in halos with Mvir(z = 0) ≈ 10 10 M using the FIRE-2 galaxy formation prescriptions (Hopkins et al. 2014 and in preparation). The dark matter density profiles of the resultant hydrodynamical runs are shown as solid black lines in each panel, with stellar mass labeled, increasing from left to right. The dashed lines in each panel show dark-matter-only versions of the same halos. We see that only in the run that forms the most stars (M ≈ 10 7 M , M /Mvir ≈ 10 −3 ) does the feedback produce a large core. Being conservative, for systems with M /Mvir 10 −4 , feedback is likely to be ineffective in altering dark matter profiles significantly compared to dark-matter-only simulations.
SCALE WHERE FEEDBACK BECOMES INEFFECTIVE IN PRODUCING CORES
M /Mvir ≈ 10 −4 ↔ M ≈ 10 6 M ↔ Mvir ≈ 10 10 M It is important to note that while many independent groups are now obtaining similar results in cosmological simulations of dwarf galaxies Munshi et al. 2013;Madau, Shen & Governato 2014;Chan et al. 2015;Oñorbe et al. 2015;Tollet et al. 2016;Fitts et al. 2016) -indicating a threshold mass of M ∼ 10 6 M or Mvir ∼ 10 10 M -this is not an ab initio ΛCDM prediction, and it depends on various adopted parameters in galaxy formation modeling. For example, Sawala et al. (2016) do not obtain cores in their simulations of dwarf galaxies, yet they still produce systems that match observations well owing to a combination of feedback effects that lower central densities of satellites (thereby avoiding the too-big-to-fail problem). On the other hand, the very high resolution, non-cosmological simulations presented in Read, Agertz & Collins (2016) produce cores in galaxies of all stellar masses. We note that Read et al.'s galaxies have somewhat higher M at a given Mvir than the cosmological runs described cited above; this leads to additional feedback energy per unit dark matter mass, likely explaining the differences with cosmological simulations and pointing to a testable prediction for dwarf galaxies' M /Mvir.
3.1.2.
Resolving too-big-to-fail. The baryon-induced cores described in Section 3.1.1 have their origins in stellar feedback. The existence of such cores for galaxies above the critical mass scale of M ≈ 10 6 M would explain why ∼half of the classical Milky Way dwarfs -those above this mass -have low observed densities. However, about half of the MW's classical dwarfs have M < 10 6 M , so the scenario described in Section 3.1.1 does not explain these systems' low central masses. Several other mechanisms exist to reconcile ΛCDM with the internal structure of low-mass halos, however.
Interactions between satellites and the Milky Way -tidal stripping, disk shocking, and ram pressure stripping -all act as additional forms of feedback that can reduce the central masses of satellites. Many numerical simulations of galaxy formation point to the importance of such interactions (which are generally absent in dark-matter-only simulations 7 ), and these environmental influences are often invoked in explaining too-big-to-fail (e.g., Zolotov et al. 2012;Arraki et al. 2014;Brooks & Zolotov 2014;Brook & Di Cintio 2015;Wetzel et al. 2016;Tomozeiu, Mayer & Quinn 2016;Sawala et al. 2016;Dutton et al. 2016). In many of these papers, environmental effects are limited to 1-2 virial radii from the host galaxy. Several Local Group galaxies reside at greater distances. While only a handful of these systems have M < 10 6 M (most are M ∼ 10 7 M ), these galaxies provide an initial test of the importance of external feedback: if environmental factors are key in setting the central densities of low-mass systems, satellites should differ systematically from field galaxies. The results of Kirby et al. (2014) find no such difference; further progress will likely have to await the discovery of fainter systems an larger optical telescopes to provide spectroscopic samples for performing dynamical analyses. Other forms of feedback may persist to larger distances. For example, Benítez-Llambay et al. (2013) note that "cosmic web stripping" (ram pressure from large-scale filaments or pancakes) may be important in dwarf galaxy evolution.
None of these solutions would explain too-big-to-fail in isolated field dwarfs. However, a number of factors could influence the conversion between observed HI line widths and the underlying gravitational potential, complicating the interpretation of systematically low densities (for a discussion of some of these issues, see Papastergis & Ponomareva 2017). Some examples are: (1) gas may not have the radial extent necessary to reach the maximum of the dark matter halo rotation curve; (2) the contribution of non-rotational support (pressure from turbulent motions) may be non-negligible and not correctly handled; and (3) determinations of inclinations angles of galaxies may be systematically wrong. Macciò et al. (2016) find good agreement between their simulations and the observed abundance of field dwarf galaxies in large part because the gas distributions in the simulated dwarfs do not extend to the peak of the dark matter rotation curve (see also Kormendy & Freeman 2016 for a similar conclusion reached via different considerations). A more complete understanding of observational samples and very careful comparisons between observations and simulations are crucial for quantifying the magnitude of any discrepancies between observations and theory.
3.1.3. Explaining planes. Even prior to the Ibata et al. (2013) result on the potential rotationally-supported plane in M31, multiple groups continued to study the observed distribution of satellite galaxies, their orbits, and the consistency of these with ΛCDM. Libeskind et al. (2009) andLovell et al. (2011) argued that the MW satellite configuration and orbital distribution are consistent with predictions from ΛCDM simulations, while Metz, Kroupa & Jerjen (2009) and Pawlowski, Pflamm-Altenburg & Kroupa (2012) argued that evidence of a serious discrepancy had only become stronger. A major point of disagreement was whether or not filamentary accretion within ΛCDM is sufficient to explain satellite orbits. Given that SDSS only surveyed about 1/3 of the northern sky (centered on the North Galactic Pole, thereby focusing on the portion of the sky where the polar plane was claimed to lie), areal coverage was a serious concern when trying to understand the significance of the polar distribution of satellites. DES has mitigated this concern somewhat, but it is also surveying near the polar plane. Pawlowski (2016) has recently argued that incomplete sky coverage is not the driving factor in assessing phase-space alignments in the Milky Way; future surveys with coverage nearer the Galactic plane should definitively test this assertion.
Following Ibata et al. (2013), the question of whether the M31 configuration (right-hand side of Figure 11) is expected in ΛCDM also became a topic of substantial interest. The general consensus of work rooted in ΛCDM is that planes qualitatively similar (though not as thin) as the M31 plane are not particularly uncommon in ΛCDM simulations, but that these planes are not rotationally-supported structures (e.g., Bahl & Baumgardt 2014;Gillet et al. 2015;Buck, Dutton & Macciò 2016). Since we view the M31 plane almost perfectly edge-on, proper motions of dwarf galaxies in the plane would provide a clean test of its nature. Should this plane turn out to be rotationally supported, it would be extremely difficult to explain with our current understanding of the ΛCDM model. These proper motions may be possible with a combination of Hubble and James Webb Space Telescope data. Skillman et al. (2017) presented preliminary observations of three plane and three non-plane galaxies, finding no obvious differences between the two sets of galaxies. Future observations of this sort could help shed light on the M31 plane and its nature.
Explaining the radial acceleration relation. Almost immediately after McGaugh,
Lelli & Schombert (2016) published their RAR relation paper, Keller & Wadsley (2017) responded by demonstrating that a similar relation can be obtained using ΛCDM hydrodynamic simulations of disk galaxies. Importantly, however, the systems simulated did not include low-mass galaxies, which are dark-matter-dominated throughout. The smallest galaxies are the ones with low acceleration in their centers as well as in their outer parts, and they remain the most puzzling to explain (see Milgrom 2016 for a discussion related to this issue).
More recently, Navarro et al. (2016) have argued that ΛCDM can naturally produce an acceleration relation similar to that shown in Figure 12. A particularly compelling section of their argument follows directly from abundance-matching ( Figure 6): the most massive disk galaxies that exist are not expected to be in halos much larger than 5 × 10 12 M . This sets a maximum acceleration scale (∼ 10 −10 m s −2 ) above which any observed acceleration must track the baryonic acceleration. The implication is that any mass-discrepancy attributable to dark matter will only begin to appear at accelerations below this scale. Stated slightly differently, any successful model of galaxy formation set within a ΛCDM context must produce a relation that begins to peel above the one-to-one only below the characteristic scale observed.
It remains to be seen whether the absolute normalization and shape of the RAR in the low-acceleration regime can be reproduced in ΛCDM simulations that span the full range of galaxy types that are observed to obey the RAR. As stated previously, these same simulations must also simultaneously reproduce the observed diversity of galaxies at fixed Vmax that is seen in the data (e.g., as shown in the right-panel of Figure 12).
Solutions requiring modifications to ΛCDM
3.2.1. Modifying linear theory predictions. As discussed in Section 1.6, the dominant impact of dark matter particle nature on the linear theory power spectrum for CDM models is in the high-k cut-off (see labeled curves in Figure 2). This cut-off is set by the free-streaming or collisional damping scale associated with CDM and is of order 1 comoving pc (corresponding to perturbations of 10 −6 M ) for canonical WIMPs (Green, Hofmann & Schwarz 2004) or 0.001 comoving pc (corresponding to 10 −15 M ) for a m ≈ 10µeV QCD axion (Nambu & Sasaki 1990). In these models, the dark matter halo hierarchy should therefore extend 18 to 27 orders of magnitude below the mass scale of the Milky Way (10 12 M ; see Fig. 2).
A variety of dark matter models result in a truncation of linear perturbations at much larger masses, however. For example, WDM models have an effective free-streaming length λ fs that scales inversely with particle mass (Bode, Ostriker & Turok 2001;Viel et al. 2005); in the Planck (2016) cosmology, this relation is approximately
λ fs = 33 mWDM 1 keV −1.11 kpc(8)
and the corresponding free-streaming mass is
M fs = 4π 3 ρm λ fs 2 3 = 2 × 10 7 mWDM 1 keV −3.33 M .(9)
The effects of power spectrum truncation are not limited to the free-streaming scale, however: power is substantially suppressed for significantly larger scales (smaller wavenumbers k). A characterization of the scale at which power is significantly affected is given by the half-mode scale k hm = 2 π/λ hm , where the transfer function is reduced by 50% relative to CDM. The half-mode wavelength λ hm is approximately fourteen times larger than the free-streaming length (Schneider et al. 2012), meaning that structure below ∼ 5 × 10 10 M is significantly different from CDM in a 1 keV thermal dark matter model:
M hm = 5.5 × 10 10 mWDM 1 keV −3.33 M .(10)
Examples of power suppression for several thermal WDM models are shown by the dashed, dotted, and dash-dotted lines in Fig. 2.
The lack of small-scale power in models with warm (or hot) dark matter is a testable prediction. As the free-streaming length is increased and higher-mass dark matter substructure is erased, the expected number of dark matter satellites inside of a Milky Way-mass halo decreases. The observed number of dark-matter-dominated satellites sets a lower limit on the number of subhalos within the Milky Way, and therefore, a lower limit on the warm dark matter particle mass. Polisensky & Ricotti (2011) find this constraint is m > 2.3 keV (95% confidence) while Lovell et al. (2014) find m > 1.6 keV; these differences come from slightly different cosmologies, assumptions about the mass of the Milky Way's dark matter halo, and modeling of completeness limits for satellite detections.
It is important to note that particle mass and the free-streaming scale are not uniquely related: the free-streaming scale depends on the particle production mechanism and is set by the momentum distribution of the dark matter particles. For example, a resonantlyproduced sterile neutrino can have a much "cooler" momentum distribution than a particle of the same mass that is produced by a process in thermal equilibrium (Shi & Fuller 1999). Constraints therefore must be computed separately for each production mechanism (Merle & Schneider 2015;Venumadhav et al. 2016). As an example, the effects of Dodelson-Widrow (1994) sterile neutrinos, which are produced through non-resonant oscillations from active neutrinos, can be matched to effects of thermal relics via the following relation: m(νs) = 3.9 keV m thermal 1 keV
1.294
ΩDMh 2 0.1225 −1/3 (11) (Abazajian 2006;Bozek et al. 2016). The effects of power spectrum suppression are not limited to pure number counts of dark matter halos: since cosmological structure form hierarchically, the erasure of small perturbations affects the collapse of more massive objects. The primary result of this effect is to delay the assembly of halos of a given mass relative to the case of no power spectrum suppression. Since the central densities of dark matter halos reflect the density of the Universe at the time of their formation, models with reduced small-scale power also result in shallower central gravitational potentials at fixed total mass for halos within 2-3 dex of the free-streaming mass. This effect is highlighted in the lower-middle panel of Figure 15. It compares Vmax values for a CDM simulation and a WDM simulation that assumes a thermal-equivalent mass of 2 keV but is otherwise identical to the CDM simulation. Massive halos (Vmax 50 km s −1 ) have identical structure; at lower masses, WDM halos have systematically lower Vmax values than their CDM counterparts. This effect comes from a reduction of Vmax for a given halo in the WDM runs, not from there being fewer objects. The reduction in central density due to power spectrum suppression for halos near or just below the half-mode mass (but significantly more massive than the (Bozek et al. 2016). This effect can explain the bulk of the differences seen in the Vmax functions (bottom right panel). Note that SIDM does not reduce the abundance of substructure (unless the power spectrum is truncated) but it does naturally produce large constant-density cores in the dark matter distribution. WDM does not produce large constant-density cores at Milky Way-mass scales but does result in fewer subhalos near the free-streaming mass and reduces Vmax of a given subhalo (through reduced concentration) near the half-mode mass (M halo 10 10 M for the plotted 2 keV thermal equivalent model).
free-streaming mass) is how WDM can solve the too-big-to-fail problem (Anderhalden et al. 2013).
Modifying non-linear predictions.
The non-linear evolution of CDM is described by the Collisionless Boltzmann equation. Gravitational interactions are the only ones that are relevant for CDM particles, and these interactions operate in the mean field limit (that is, gravitational interactions between individual DM particles are negligible compared to interactions between a dark matter particle and the large-scale gravitational potential). The question of how strong the constraints are on non-gravitational interactions between individual dark matter particles is therefore crucial for evaluating non-CDM models. There has been long-standing interest in models that involve dark matter self-interactions (Carlson, Machacek & Hall 1992;Spergel & Steinhardt 2000). In its simplest form, self-interacting dark matter (SIDM, sometimes called collisional dark matter) is characterized by an energy-exchange interaction cross section σ. The mean free path λ of dark matter particles is then λ = (n σ) −1 , where n is the local number density of dark matter particles. Since the mass of the dark matter particle is not known, it is often useful to express the mean free path as (ρ σ/m) −1 and to quantify self-interactions in terms of the cross section per unit particle mass, σ/m. If λ(r)/r 1 at radius r from the center of a dark matter halo, many scattering events occur per local dynamical time and SIDM acts like a fluid, with conductive transport of heat. In the opposite regime, λ(r)/r 1, particles are unlikely to scatter over a local dynamical time and SIDM is effectively an optically thin (rarefied) gas, with elastic scattering between dark matter particles. Most work in recent years has been far from the fluid limit.
As originally envisioned by Spergel & Steinhardt (2000) in the context of solving the missing satellites and cusp/core problems, the mean free path for self-interactions is of order 1 kpc λ 1 Mpc at densities characteristic of the Milky Way's dark matter halo (0.4 Gev/cm 3 ; Read 2014), leading to self-interaction cross sections of 400 σ/m 0.4 cm 2 /g (800 σ/m 0.8 barn/GeV). This scale (∼barn/GeV) is enormous in particle physics terms -it is comparable to the cross-section for neutron-neutron scattering -yet it remains difficult to exclude observationally. It is important to emphasize that the dark matter particle self-interaction strength can, in principle, be completely decoupled from the dark matter's interaction strength with Standard Model particles and thus standard direct detection constraints offer no absolute model-independent limits on σ/m for the dark matter. Astrophysical constraints are therefore essential for understanding dark matter physics.
Though the SIDM cross section estimates put forth by Spergel & Steinhardt (2000) were based on analytic arguments, the interaction scale they proposed to alleviate the cusp/core problem does overlap (at the low end) with more modern results based on fully selfconsistent cosmological simulations. Several groups have now run cosmological simulations with dark matter self-interactions and have found that models with σ/m ≈ 0.5 − 10 cm 2 /g produce dark matter cores in dwarf galaxies with sizes ∼ 0.3 − 1.5 kpc and central densities 2 − 0.2 × 10 8 M kpc −3 = 7.4 − 0.74 GeV cm −3 that can alleviate the cusp/core and toobig-to-fail problems discussed above (e.g., Vogelsberger, Zavala & Loeb 2012;Peter et al. 2013;Fry et al. 2015;Elbert et al. 2015). SIDM does not, however, significantly alleviate the missing satellites problem, as the substructure counts in SIDM simulations are almost identical to those in CDM simulations ; see Figure 15).
One important constraint on possible SIDM models comes from galaxy clusters. The high central dark matter densities observed in clusters exclude SIDM models with σ/m 0.5 cm 2 /g, though SIDM with σ/m 0.1cm 2 /g may be preferred over CDM (e.g., Kaplinghat, Tulin & Yu 2016;Elbert et al. 2016). This means that in order for SIDM is to alleviate the small-scale problems that arise in standard CDM and also match constraints seen on the galaxy cluster scale, it needs to have a velocity-dependent cross section σ(v) that decreases as the rms speed of dark matter particles involved in the scattering rises from the scale of dwarfs (v ∼ 10 km s −1 ) to galaxy clusters (v ∼ 1000 km s −1 ). Velocity-dependent scattering cross sections are not uncommon among Standard Model particles. Figure 15 shows the results of three high-resolution cosmological simulations (performed by V. Robles, T. Kelley, and B. Bozek in collaboration with the authors) of the same Milky Way mass halo done with CDM, SIDM (σ/m = 1 cm 2 /g), and WDM (a 7 keV resonant model, with thermal-equivalent mass of 2 keV). The images show density maps spanning 600 kpc. It is clear that while WDM produces many fewer subhalos than CDM, the SIDM model yields a subhalo distribution that is very similar to CDM, with only slightly less substructure near the halo core, which itself is slightly lower density than the CDM case.
These visual impressions are quantified in the bottom three panels, which show the main halo density profiles (left) and the subhalo Vmax functions for all three simulations (right). The middle panel shows the relationship between the Vmax values of individual halos identified in both CDM and WDM simulations (Bozek et al. 2016). The left panel shows clearly that SIDM produces a large, constant-density core in the main halo, while the WDM profile is almost identical to the CDM case. However, for mass scales close to the half-mode suppression mass of the WDM model (M halo 10 10 M for this case), the density structure is affected significantly. This effect accounts for most of the difference seen in the right panel: WDM subhalos have Vmax values that are greatly reduced compared to their CDM counterparts, meaning there is a Vmax-dependent shift leftward at fixed number (i.e., subhalos at this mass scale are not being destroyed, which would result in a reduction in number at fixed Vmax).
Finally, we conclude by noting that it is possible to write down SIDM models that have both truncated power spectra and significant self-interactions. Such models produce results that are a hybrid between traditional WDM and SIDM with scale-invariant spectra (e.g. Vogelsberger et al. 2016). Specifically, it is possible to modify dark matter in such a way that it produces both fewer subhalos (owing to power spectra effects) and constant density cores (owing to particle self-interactions) and thus solve the substructure problem and core/cusp problem simultaneously without appealing to strong baryonic feedback.
Current Frontiers
Dwarf galaxy discovery space in the Local Group
The tremendous progress in identifying and characterizing faint stellar systems in the Local Group has led to a variety of new questions. For one, these discoveries have blurred what was previously a clear difference between dwarf galaxies and star clusters, leading to the question, "what is a galaxy?" (Willman & Strader 2012). DES has identified several new satellite galaxies, many of which appear to be clustered around the Large Magellanic Cloud (LMC; Drlica-Wagner et al. 2015). The putative association of these satellites with the LMC is intriguing (Jethwa, Erkal & Belokurov 2016;Sales et al. 2017), as the nearly self-similar nature of dark matter substructure implies that the LMC -which is likely to be hosted by a halo of M peak ∼ 10 11 M (Boylan-Kolchin et al. 2010) -could itself contain multiple dark matter satellites above the mass threshold required for galaxy formation. Satellites of the LMC and even fainter dwarfs will be attractive targets for ongoing and future observations to test basic predictions of ΛCDM (Wheeler et al. 2015).
The 800 pound gorilla in the dwarf discovery landscape is the Large Synoptic Survey Telescope (LSST). Currently under construction and set to begin operations in 2022, LSST has the potential to expand dwarf galaxy discovery space substantially: by the end of the survey, co-added LSST data will be sensitive to galaxies ten times more distant (at fixed luminosity) than SDSS, or equivalently, LSST will be able to detect galaxies that are one hundred times fainter than SDSS at the same distance. This means that LSST should be complete for galaxies with L 2 × 10 3 L within ∼ 1 Mpc of the Galaxy, dramatically increasing the census of very faint galaxies beyond ∼ 100 kpc from the Earth.
One of the unique features of LSST data sets will be the ability to explore the properties of low-mass, isolated dark matter halos (i.e., those that have not interacted with a more massive system such as the Milky Way), thereby separating out the effects of environment from internal feedback and dark matter physics. Given the predictions discussed in Sec. 3.1.1, any new discoveries with M 10 6 M at ∼ 1 Mpc from the Milky Way and M31 will be attractive targets for discriminating between baryonic feedback and dark matter physics. At this distance, spectrographs on 10m-class telescopes will not be sufficient to measure kinematics of resolved stars; planned 30m-class telescopes will be uniquely suited to this task.
In addition to hosting surviving satellites, galactic halos also act as a graveyard for satellite galaxies that have been disrupted through tidal interactions. These disrupted satellites can form long-lived tidal streams; more generally, the stars from these satellites are part of a galaxy's stellar halo (which may also encompass stars from globular clusters or other sources). Efforts are underway to disentangle disrupted satellites from other stars in the Milky Way halo via chemistry and kinematics (see Bland-Hawthorn & Gerhard 2016 for a recent review).
Dwarfs beyond the Local Group
An alternate avenue to probing deeper within the Local Group is to search for low-mass galaxies further away (but still in the very local Universe). The Dark Energy Camera (DECam) and Subaru/Hyper Suprime-Cam are being used by several groups to search for very faint companions in a variety of systems (from NGC 3109, itself a dwarf galaxy at ∼ 1.3 Mpc, to Centaurus A, a relatively massive elliptical galaxy at ≈ 3.8 Mpc (Sand et al. 2015b;Crnojević et al. 2016;Carlin et al. 2016). Searches for the gaseous components of galaxies that would otherwise be missed by surveys have also proven fruitful, with a number of individual discoveries (Giovanelli et al. 2013;Sand et al. 2015a;Tollerud et al. 2016).
Recently, the rediscovery of ultra-diffuse dwarf galaxies (Impey, Bothun & Malin 1988;Dalcanton et al. 1997;Koda et al. 2015;van Dokkum et al. 2015) has led to significant interest in these odd systems, which have sizes comparable to the Milky Way but luminosities comparable to bright dwarf galaxies. Ultra-diffuse dwarfs have been discovered predominantly in galaxy clusters, but if similar systems -perhaps with even lower luminosities -exist near the Local Group, they could have escaped detection. Understanding the formation and evolution of ultra-diffuse dwarfs, as well as their dark matter content and connection to the broader galaxy population, has the potential to alter our current understanding of faint stellar systems.
Searches for starless dwarfs
Very low mass dark matter halos must be starless, should they exist. Detecting starless halos would represent a strong confirmation of the ΛCDM model (and would place stringent constraints on the possible solutions to problems covered in this review); accordingly, astronomers and physicists are exploring a variety of possibilities for detecting such halos.
A promising technique for inferring the presence of the predicted population of lowmass, dark substructure within the Milky Way is through subhalos' effects on very cold low velocity dispersion stellar streams (Ibata et al. 2002;Carlberg 2009;Yoon, Johnston & Hogg 2011). Dark matter substructure passing through a stream will perturb the orbits of the stars, creating gaps and bunches in the stream. Although many physical phenomena may produce similar effects, and the very existence of gaps themselves remains a matter of debate, large samples of cold streams would likely provide the means to test the abundance of low-mass (Mvir ∼ 10 5−6 M ) substructure in the Milky Way. We note that the streams from disrupting satellite galaxies discussed above are not suitable for this technique, as they are produced with large enough stellar velocity dispersions that subhalos' effects will go undetectable. Blind surveys for HI gas provide yet another path to searching for starless (or extremely faint) substructure in the very nearby Universe. Some ultra-compact highvelocity clouds (UCHVCs) may be gas-bearing "mini-halos" that are devoid of stars (e.g., Blitz et al. 1999).
Most of the probes we have discussed so far rely on electromagnetic signatures of dark matter. Gravitational lensing is unique in that it is sensitive to mass alone, potentially providing a different window into low-mass dark matter halos. Vegetti et al. (2010Vegetti et al. ( , 2012 have detected two relatively low-mass dark matter subhalos within lensed galaxies using this technique. The galaxies are at cosmological distances, making it difficult to identify any stellar component associated with the subhalos; Vegetti et al. quote upper limits on the luminosities of detected subhalos of ∼ 5 × 10 6−7 L , comparable to classical dwarfs in the Local Group. The inferred dynamical masses are much higher, however: within 300 pc, Milky Way satellites all have M300 ≈ 10 7 M , while the detected subhalos have M300 ≈ (1 − 10) × 10 8 M . It remains to be seen whether this is related to the lens modeling or if the substructure in lensing galaxies is fundamentally different from that in the Local Group.
More recently, ALMA has emerged as a promising tool for detecting dark matter halo substructure via spatially-resolved spectroscopy of lensed galaxies. This technique was discussed in Hezaveh et al. (2013), and recently, a subhalo with a total mass of ∼ 10 9 M within ∼ 1 kpc was detected with ALMA (Hezaveh et al. 2016). At present, the detected substructure is significantly more massive than the hosts of dwarf galaxies in the Local Group: the velocity dispersion of the substructure is σDM ∼ 30 km s −1 as opposed to σ ≈ 5 − 10 km s −1 for Local Group dwarf satellites. This value of σDM is indicative of a galaxy similar to the Small Magellanic Cloud, which has M ∼ 5 × 10 8 M and Mvir ∼ (5 − 10) × 10 10 M . The discovery of additional lens systems, and the enhanced resolution and sensitivity of ALMA in its completed configuration, promise to reveal lower-mass substructure, perhaps down to scales similar to Local Group satellites but at cosmological distances and in very different host galaxies.
Indirect signatures of dark matter
If dark matter is indeed a standard WIMP, two dark matter particles can annihilate into Standard Model particles with electromagnetic signatures. This process is exceedingly rare, on average; as discussed in Section 1.6, the freeze-out of dark matter annihilations is what sets the relic density of dark matter in the WIMP paradigm. Nevertheless, the annihilation rate is proportional to the local value ρ 2 DM , meaning that the centers of dark matter halos are potential sites for annihilations. While the brightest source of such annihilations in the sky should be the Galactic Center, foregrounds make unambiguous detection of annihilating dark matter toward the Galaxy challenging. Dwarf spheroidal galaxies have somewhat lower predicted annihilation fluxes owing both to their greater distances and lower masses, but they have the significant advantage of being free of foreground contamination. The Fermi γ-ray telescope has surveyed MW dwarfs extensively, with no conclusive evidence for dark matter annihilation products. The upper limits on combined dwarf data from Fermi are already placing moderate tension on the most basic "WIMP miracle" predictions for the annihilation cross section for wimps with m 100 GeV (Ackermann et al. 2015). Searches for annihilation from starless dark matter subhalos within the Milky Way via the Fermi point source catalog have not yielded any detections to date (Calore et al. 2016).
On cosmic scales, dark matter annihilations may contribute to the extragalactic gammaray background (Zavala, Springel & Boylan-Kolchin 2010). The expected contributions of dark matter depend sensitively on the spectrum of dark matter halos and subhalos, as well as the relation between concentration and mass for very low mass systems. These relations can be estimated by a variety of methods (though generally not simulated directly, owing to the enormous range of scales that contribute), with uncertainties being grouped into a "boost factor" that describes unresolved annihilations.
If dark matter is a sterile neutrino rather than a WIMP-like particle, self-annihilation will not be seen. Sterile neutrinos decay radiatively to an active neutrino and a photon, however; for all of the relevant sterile neutrino parameter space, this decay is effectively at rest and a clean signature is therefore a spectral line at half the rest mass energy of the dark matter particle, Eγ = mDM/2. While there is no a priori expectation for the mass of the sterile neutrino, arguments from Section 3.2.1 point to Eγ 1 keV, so searches in the soft X-ray band are constraining. The most promising recent result in this field is the detection of a previously unknown X-ray line near 3.51 keV in the spectra of individual galaxy clusters, stacked galaxy clusters, and the halo of M31 (Bulbul et al. 2014;Boyarsky et al. 2014). X-ray observations and satellite counts in M31 rule out an oscillation (Dodelson & Widrow 1994) origin for this line if it indeed originates from sterile neutrino dark matter (Horiuchi et al. 2014), leaving heavy scalar decay and possibly resonant conversion as possible production mechanisms (Merle & Schneider 2015). A definitive test of the origin of the 3.5 keV line was expected from the Hitomi satellite, as it had the requisite energy resolution to see the thermal broadening of the line due to virial motions (i.e., the line width from a halo with mass Mvir should be ∼ Vvir/c). With Hitomi's untimely demise, tests of the line's origin may have to wait for Athena.
The high-redshift Universe
While studies of low-mass dark matter halos are most easily conducted in the very nearby Universe owing to the faintness of the galaxies they host, there are avenues at higher redshifts that may provide alternate windows in to the spectrum of density perturbations. One potentially powerful probe at z ∼ 2 − 6 is the Lyman-α forest of absorption lines produced by neutral hydrogen in the intergalactic medium between us and high-redshift quasars (see McQuinn 2016 for a recent review and further details). This hydrogen probes the density field in the quasi-linear regime (i.e., it is in perturbations that are just starting to collapse) and can constrain the dark matter power spectrum to wavenumbers as large as k ∼ 10 h Mpc −1 . Any model that reduces the power on this scale relative to ΛCDM expectations will predict different absorption patterns. In particular, WDM will suppress power on these scales. Viel et al. (2013) used Lyman-α flux power spectra from 25 quasar sightlines to constrain the mass of thermal relic WDM particle to m WDM,th > 3.3 keV at 95% confidence. This translates into a density perturbation spectrum that must be very close to ΛCDM down to M ∼ 10 8 M (Schneider et al. 2012) and would rule out the possibility that free-
The challenge of detecting "empty" dark matter halos
The detection of abundant, baryon-free, low-mass dark matter halos would be an unambiguous validation of the particle dark matter paradigm, would strongly constrain particle physics models, and would eliminate many of the dark matter candidates for the origin of the small-scale issues described in this review. Why is this such a challenging task?
The answer lies in the densities of low-mass dark matter halos compared to other astrophysical objects. From Equation (4), the average density within a halo's virial radius is 200 times the cosmic matter density. For the most abundant low-mass halos in standard WIMP models -those just above the free-streaming scale of ∼ 10 −6 M -the virial radius is approximately 0.1 pc. This is the equivalent of the mass of the Earth spread over a a distance that is significantly larger than the Solar System (the mean distance between Pluto and the Sun is ∼ 2 × 10 −4 pc). Even the lowest-mass, earliest-collapsing CDM structures are incredibly diffuse compared to typical astrophysical objects. Although there may be O(10 17 ) Earth-mass dark matter subhalos within the Milky Way's ≈ 300 kpc virial radius, detecting them is a daunting challenge.
streaming has direct relevance for the scales of classical dwarfs (and larger-mass systems). The potential complication with this interpretation is the relationship between density and temperature in the intergalactic medium, as pressure or thermal motions can mimic the effects of dark matter free-streaming.
Counts of galaxies in the high-redshift Universe also trace the spectrum of collapsed density perturbations at low masses, albeit in a non-trivial manner. The mere existence of galaxies at high redshift places an upper limit on the free-streaming length of dark matter (so long as all galaxies form within dark matter halos) in much the same way that the existence of substructure in the local Universe does (Schultz et al. 2014). Menci et al. (2016) have placed limits on the masses of thermal relic WDM particles of 2.4 keV (2.1 keV) at 68% (95%) confidence based on the detection of a single galaxy in the Hubble Frontier Fields at z ∼ 6 with absolute UV magnitude of MUV = −12.5 (Livermore, Finkelstein & Lotz 2017). While this stated constraint is very strong, and the technique is promising, correctly modeling faint, high-redshift galaxies -particularly lensed ones -at can be very challenging. Furthermore, the true redshift of the galaxy can only be localized to ∆z ∼ 1; the rapid evolution of the halo mass function at high redshift further complicates constraints. With the upcoming James Webb Space Telescope, the high-redshift frontier will be pushed fainter and to higher redshifts, raising the possibility of placing strong constraints on the free-streaming length of dark matter through structures in the early Universe.
Summary and Outlook
Small-scale structure sits at the nexus of astrophysics, particle physics, and cosmology. Within the standard ΛCDM model, most properties of small-scale structure can be modeled with high precision in the limit that baryonic physics is unimportant. And yet, the level of agreement between theory and observations remains remarkably hard to assess, in large part because of hard-to-model effects of baryonic physics on first-principles predictions. Given the stakes -absent direct detection of dark matter on Earth, indirect evidence from astrophysics provides the strongest clues to dark matter's nature -it is essential to take potential discrepancies seriously and to explore all avenues for their resolution.
We have discussed three main classes of problems in this review: (1) counts and (2) densities of low-mass objects, and (3) tight scaling relations between the dark and luminous components of galaxies. All of these issues may have their origin in baryonic physics, but they may also point to the need for a phenomenological theory that goes beyond ΛCDM. Understanding which of these two options is correct is pressing for both astrophysics and particle physics.
In our opinion, the search for abundant dark matter halos with inferred virial masses substantially lower than the expected threshold of galaxy formation (Mvir ∼ 10 8 M ) is the most urgent calling in this field today. The existence of these structures is an unambiguous prediction of all WIMP-based dark matter models (though it is not unique to WIMP models), and confirmation of the existence of dark matter halos with M ∼ 10 6 M or less would strongly constrain particle physics of dark matter and effectively rule out any role of dark matter free-streaming in galaxy formation. Here, too, accurate predictions for the number of expected dark subhalos will require an honest accounting of baryon physics -specifically the destructive effects of central galaxies themselves (e.g., Garrison-Kimmel et al. 2017b). Of nearly equal importance is characterizing the central dark matter density structure of very faint (M 10 6 M ) galaxies, as a prediction of many recent high-resolution cosmological simulations within the ΛCDM paradigm is that stellar feedback from galaxies below this threshold mass should not modify their host dark matter halos' cuspy density profile shape. The detection of ubiquitous cores in very low-mass galaxies therefore has the potential to falsify the ΛCDM paradigm.
While some of the tests of the paradigm are clear, their implementation is difficult. Dark matter substructure is extremely diffuse compared to baryonic matter, making its detection highly challenging. The smallest galaxies have very few stars to base accurate dynamical studies upon. Nevertheless, a variety of independent probes of the small-scale structure of dark matter are now feasible, and the LSST era will likely provide a watershed for our understanding of the nature of dark matter and the threshold of galaxy formation. It is not far-fetched to think that improved astrophysical data, theoretical understanding, and numerical simulations will provide a definitive test of ΛCDM within the next decade, even without the direct detection of particle dark matter on Earth.
DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.
Figure 1
1Figure 1
Figure 2
2Figure 2
Figure 3
3Figure 3
Figure 4
Figure 6
and the minimum mass for atomic cooling in the early Universe, Mvir ≈ 10 8 M (Vmax 15 km s
Figure 7
7Figure 7
Figure 8
"
Solving" the Missing Satellites Problem with abundance matching. The cumulative count of dwarf galaxies around the Milky Way (magenta) plotted down to completeness limits fromGarrison-Kimmel et al. (2017a). The gray shaded region shows the predicted stellar mass function from the dark-matter-only ELVIS simulations) combined with the fiducial AM relation shown inFigure 6, assuming zero scatter. If the faint end slope of the stellar mass function is shallower (dashed) or steeper (dotted), the predicted abundance of satellites with M > 10 4 M throughout the Milky Way's virial volume differs by a factor of 10. Local Group counts can therefore serve as strong constraints on galaxy formation models.
Figure 9
Figure 10
field galaxies 4(Zavala et al. 2009;Klypin et al. 2015;Trujillo-Gomez et al.
Planes of Satellites. Left: Edge-on view of the satellite distribution around the Milky Way (updated from Pawlowski, McGaugh & Jerjen 2015) with the satellite galaxies in yellow, young halo globular clusters and star clusters in blue, and all other newly-discovered objects (unconfirmed dwarf galaxies or star clusters) are shown as green triangles. The red lines in the center dictate the position and orientation of streams in the MW halo. The gray wedges span 24 degrees about the plane of the MW disk, where satellite discovery might be obscured by the Galaxy. Right: The satellite distribution around Andromeda (modified by M. Pawlowski from Ibata et al. 2013) where the red points are satellites belonging to the identified kinematic plane. Triangles pointing up are receding relative to M31. Triangles pointing down are approaching. et al. 2016, though see
Figure 12
Regularity vs. Diversity. Left: The radial acceleration relation from McGaugh, Lelli & Schombert (2016, slightly modified) showing the centripetal acceleration observed in rotation curves, g obs = V 2 /r, plotted versus the expected acceleration from observed baryons g bar for 2700 individual data points from 153 galaxy rotation curves. Large squares show the mean and the dashed line lines show the rms width. Right: Green points show the circular velocities of observed galaxies measured at 2 kpc as a function of Vmax from Oman et al. (2015) as re-created by
Figure 13
Figure 14
Figure 15
15Dark matter phenomenology in the halo of the Milky Way. The three images in the upper row show the same Milky-Way-size dark matter halo simulated with CDM, SIDM (σ/m = 1 cm 2 /g), and WDM (a Shi-Fuller resonant model with a thermal equivalent mass of 2 keV). The left panel in the bottom row shows the dark matter density profiles of the same three halos while the bottom-right panel shows the subhalo velocity functions for each as well. The middle panel on the bottom shows that while the host halos have virtually identical density structure in WDM and CDM, individual subhalos identified in both simulations smaller Vmax values in WDM
Galaxy Clusters: M vir ≈ 10 15 M V vir ≈ 1000 km s −1Milky Way:
M vir ≈ 10 12 M
V vir ≈ 100 km s −1
Smallest Dwarfs:
M vir ≈ 10 9 M
V vir ≈ 10 km s −1
www.annualreviews.org • Challenges to the ΛCDM Paradigm
Recent measurements find n = 0.968 ± 0.006 (Planck Collaboration et al. 2016), i.e., small but statistically different from true scale invariance. www.annualreviews.org • Challenges to the ΛCDM Paradigm
The black line inFigure 4illustrates the mass function of ΛCDM dark matter halos.www.annualreviews.org • Challenges to the ΛCDM Paradigm
This maximum mass is similar to the virial mass at the time of accretion, though infalling halos can begin losing mass prior to first crossing the virial radius.www.annualreviews.org • Challenges to the ΛCDM Paradigm
We note that the mismatch between the observed and predicted velocity function can also be www.annualreviews.org • Challenges to the ΛCDM Paradigm
Bullock • Boylan-Kolchin
This type of relation is what is generally expected in MOND, though the precise shape of the relation depends on the MOND interpolation function assumed (see McGaugh, Lelli & Schombert 2016 for a brief discussion).www.annualreviews.org • Challenges to the ΛCDM Paradigm
We note that capturing these effects is extremely demanding numerically, and it is not clear that any published cosmological hydrodynamical simulation of a Milky Way-size system can resolve the mass within 300 − 500 pc of satellite galaxies with the accuracy required to address this issue.
ACKNOWLEDGMENTSIt is a pleasure to thank our collaborators and colleagues for helpful discussions and for making important contributions to our perspectives on this topic. We specifically thank Peter Behroozi, Brandon Bozek, Peter Creasey, Sandy Faber, Alex Fitts, Shea
which is operated by AURA, Inc., under NASA contract NAS5-26555. JSB was supported by NSF grant AST-1518291 and by NASA through HST theory grants (programs AR-13921, AR-13888, and AR-14282) awarded by STScI. This work used computational resources granted by the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575 and ACI-1053575. -14282 from the Space Telescope Science Institute (STScI). HST-AR-13888, HST-AR-13896, and HST-ARResources supporting this work were also provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NASHST-AR-13888, HST-AR-13896, and HST-AR-14282 from the Space Telescope Science In- stitute (STScI), which is operated by AURA, Inc., under NASA contract NAS5-26555. JSB was supported by NSF grant AST-1518291 and by NASA through HST theory grants (programs AR-13921, AR-13888, and AR-14282) awarded by STScI. This work used compu- tational resources granted by the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575 and ACI-1053575. Resources supporting this work were also provided by the NASA High- End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS)
. M Aaronson, ApJ. 266Aaronson M. 1983. ApJ 266:L11-L15
. K Abazajian, Phys. Rev. D. 7363506Abazajian K. 2006. Phys. Rev. D 73:063506
. M Ackermann, A Albert, B Anderson, W B Atwood, L Baldini, Phys. Rev. Lett. 115231301Ackermann M, Albert A, Anderson B, Atwood WB, Baldini L, et al. 2015. Phys. Rev. Lett. 115:231301
. R Adhikari, M Agostini, N A Ky, T Araki, M Archidiacono, arXiv:1602.04816astro-phAdhikari R, Agostini M, Ky NA, Araki T, Archidiacono M, et al. 2016. arXiv:1602.04816 [astro-ph]
. S H Ahmed, A M Brooks, C R Christensen, MNRAS. 466Ahmed SH, Brooks AM, Christensen CR. 2017. MNRAS 466:3119-3132
. Smk Alam, J S Bullock, D H Weinberg, ApJ. 572Alam SMK, Bullock JS, Weinberg DH. 2002. ApJ 572:34-40
. D Anderhalden, A Schneider, A V Macciò, J Diemand, G Bertone, JCAP. 314Anderhalden D, Schneider A, Macciò AV, Diemand J, Bertone G. 2013. JCAP 3:014
. K S Arraki, A Klypin, S More, S Trujillo-Gomez, MNRAS. 438Arraki KS, Klypin A, More S, Trujillo-Gomez S. 2014. MNRAS 438:1466-1482
. H Bahl, H Baumgardt, MNRAS. 438Bahl H, Baumgardt H. 2014. MNRAS 438:2916-2923
. I K Baldry, S P Driver, J Loveday, E N Taylor, L S Kelvin, MNRAS. 421Baldry IK, Driver SP, Loveday J, Taylor EN, Kelvin LS, et al. 2012. MNRAS 421:621-634
. P S Behroozi, R H Wechsler, C Conroy, ApJ. 76231Behroozi PS, Wechsler RH, Conroy C. 2013. ApJ 762:L31
. A Benítez-Llambay, J F Navarro, M G Abadi, S Gottlöber, G Yepes, ApJ. 763ApJBenítez-Llambay A, Navarro JF, Abadi MG, Gottlöber S, Yepes G, et al. 2013. ApJ 763:L41 Benson AJ, Bower RG, Frenk CS, Lacey CG, Baugh CM, Cole S. 2003. ApJ 599:38-49
. A J Benson, C G Lacey, C M Baugh, S Cole, C S Frenk, MNRAS. 333Benson AJ, Lacey CG, Baugh CM, Cole S, Frenk CS. 2002. MNRAS 333:156-176
. M Bernardi, A Meert, R K Sheth, V Vikram, M Huertas-Company, MNRAS. 436Bernardi M, Meert A, Sheth RK, Vikram V, Huertas-Company M, et al. 2013. MNRAS 436:697-704
. E Bertschinger, Phys. Rev. D. 063509 Bland-Hawthorn J, Gerhard O74ARA&ABertschinger E. 2006. Phys. Rev. D 74:063509 Bland-Hawthorn J, Gerhard O. 2016. ARA&A 54:529-596
. L Blitz, D N Spergel, P J Teuben, D Hartmann, W B Burton, ApJ. 514Blitz L, Spergel DN, Teuben PJ, Hartmann D, Burton WB. 1999. ApJ 514:818-843
. G R Blumenthal, S M Faber, J R Primack, M J Rees, Nature. 311Blumenthal GR, Faber SM, Primack JR, Rees MJ. 1984. Nature 311:517-525
. P Bode, J P Ostriker, N Turok, ApJ. 556Bode P, Ostriker JP, Turok N. 2001. ApJ 556:93-107
. J R Bond, S Cole, G Efstathiou, N Kaiser, ApJ. 379Bond JR, Cole S, Efstathiou G, Kaiser N. 1991. ApJ 379:440-460
The distribution and kinematics of neutral hydrogen in spiral galaxies of various morphological types. A Bosma, Groningen Univ.PhD ThesisBosma A. 1978. The distribution and kinematics of neutral hydrogen in spiral galaxies of various morphological types. Ph.D. thesis, PhD Thesis, Groningen Univ., (1978)
. M S Bovill, M Ricotti, ApJ. 693Bovill MS, Ricotti M. 2009. ApJ 693:1859-1870
. A Boyarsky, O Ruchayskiy, D Iakubovskyi, J Franse, M Boylan-Kolchin, J S Bullock, M Kaplinghat, Phys. Rev. Lett. 113MNRASBoyarsky A, Ruchayskiy O, Iakubovskyi D, Franse J. 2014. Phys. Rev. Lett. 113:251301 Boylan-Kolchin M, Bullock JS, Kaplinghat M. 2011. MNRAS 415:L40-L44
. M Boylan-Kolchin, J S Bullock, M Kaplinghat, MNRAS. 422Boylan-Kolchin M, Bullock JS, Kaplinghat M. 2012. MNRAS 422:1203-1218
. M Boylan-Kolchin, V Springel, Sdm White, A Jenkins, MNRAS. 406Boylan-Kolchin M, Springel V, White SDM, Jenkins A. 2010. MNRAS 406:896-912
. B Bozek, M Boylan-Kolchin, S Horiuchi, S Garrison-Kimmel, K Abazajian, J S Bullock, 459MN-RASBozek B, Boylan-Kolchin M, Horiuchi S, Garrison-Kimmel S, Abazajian K, Bullock JS. 2016. MN- RAS 459:1489-1504
. C B Brook, Di Cintio, A , MNRAS. 450Brook CB, Di Cintio A. 2015. MNRAS 450:3920-3934
. C B Brook, I Santos-Santos, G Stinson, MNRAS. 459Brook CB, Santos-Santos I, Stinson G. 2016. MNRAS 459:638-645
. A Brooks, Annalen der Physik. 526Brooks A. 2014. Annalen der Physik 526:294-308
. A M Brooks, E Papastergis, C R Christensen, F Governato, A Stilp, arXiv:1701.07835astro-phBrooks AM, Papastergis E, Christensen CR, Governato F, Stilp A, et al. 2017. arXiv:1701.07835 [astro-ph]
. A M Brooks, A Zolotov, ApJ. 78687Brooks AM, Zolotov A. 2014. ApJ 786:87
. G L Bryan, M L Norman, ApJ. 495Bryan GL, Norman ML. 1998. ApJ 495:80-99
. T Buck, A A Dutton, A V Macciò, MNRAS. Buck T, Dutton AA, Macciò AV. 2016. MNRAS
. E Bulbul, M Markevitch, A Foster, R K Smith, M Loewenstein, S W Randall, ApJ. 78913Bulbul E, Markevitch M, Foster A, Smith RK, Loewenstein M, Randall SW. 2014. ApJ 789:13
. J S Bullock, T S Kolatt, Y Sigad, R S Somerville, A V Kravtsov, MNRAS. 321Bullock JS, Kolatt TS, Sigad Y, Somerville RS, Kravtsov AV, et al. 2001. MNRAS 321:559-575
. J S Bullock, A V Kravtsov, D H Weinberg, ApJ. 539Bullock JS, Kravtsov AV, Weinberg DH. 2000. ApJ 539:517-521
. A Burkert, ApJ. 447Burkert A. 1995. ApJ 447:L25-L28
. F Calore, V De Romeri, Di Mauro, M Donato, F Marinacci, F Carlberg, R G , arXiv:1611.03503ApJ. 705astro-phCalore F, De Romeri V, Di Mauro M, Donato F, Marinacci F. 2016. arXiv:1611.03503 [astro-ph] Carlberg RG. 2009. ApJ 705:L223-L226
. J L Carlin, D J Sand, P Price, B Willman, A Karunakaran, ApJ. 828ApJCarlin JL, Sand DJ, Price P, Willman B, Karunakaran A, et al. 2016. ApJ 828:L5 Carlson ED, Machacek ME, Hall LJ. 1992. ApJ 398:43-52
. T K Chan, D Kereš, J Oñorbe, P F Hopkins, A L Muratov, MNRAS. 454Chan TK, Kereš D, Oñorbe J, Hopkins PF, Muratov AL, et al. 2015. MNRAS 454:2981-3001
. S Cole, ApJ. 367Cole S. 1991. ApJ 367:45-53
. A R Conn, G F Lewis, R A Ibata, Q A Parker, D B Zucker, ApJ. 766120Conn AR, Lewis GF, Ibata RA, Parker QA, Zucker DB, et al. 2013. ApJ 766:120
. C Conroy, R H Wechsler, A V Kravtsov, ApJ. 647Conroy C, Wechsler RH, Kravtsov AV. 2006. ApJ 647:201-214
. P Creasey, O Sameie, L V Sales, H B Yu, M Vogelsberger, J Zavala, MNRAS. 468Creasey P, Sameie O, Sales LV, Yu HB, Vogelsberger M, Zavala J. 2017. MNRAS 468:2283-2295
. D Crnojević, D J Sand, D Zaritsky, K Spekkens, B Willman, Jr ; L14 Hargis, F Y Cyr-Racine, K Sigurdson, J Zavala, T Bringmann, M Vogelsberger, C Pfrommer, Phys. Rev. D. 824123527ApJCrnojević D, Sand DJ, Zaritsky D, Spekkens K, Willman B, Hargis JR. 2016. ApJ 824:L14 Cyr-Racine FY, Sigurdson K, Zavala J, Bringmann T, Vogelsberger M, Pfrommer C. 2016. Phys. Rev. D 93:123527
. J J Dalcanton, D N Spergel, J E Gunn, M Schmidt, D P Schneider, AJ. 114Dalcanton JJ, Spergel DN, Gunn JE, Schmidt M, Schneider DP. 1997. AJ 114:635-654
. M Davis, G Efstathiou, C S Frenk, Sdm White, ApJ. 292Davis M, Efstathiou G, Frenk CS, White SDM. 1985. ApJ 292:371-394
. Wjg De Blok, F Walter, E Brinks, C Trachternach, S H Oh, KennicuttJr, Rc, AJ. 136de Blok WJG, Walter F, Brinks E, Trachternach C, Oh SH, Kennicutt Jr. RC. 2008. AJ 136:2648- 2719
. Del Popolo, A , Le Delliou, M , Galaxies. 517Del Popolo A, Le Delliou M. 2017. Galaxies 5:17
. Di Cintio, A Brook, C B Macciò, A V Stinson, G S Knebe, A , MNRAS. 437Di Cintio A, Brook CB, Macciò AV, Stinson GS, Knebe A, et al. 2014. MNRAS 437:415-423
. J Diemand, M Kuhlen, P Madau, M Zemp, B Moore, Nature. 454Diemand J, Kuhlen M, Madau P, Zemp M, Moore B, et al. 2008. Nature 454:735-738
. B Diemer, A V Kravtsov, ApJ. 799108Diemer B, Kravtsov AV. 2015. ApJ 799:108
. S Dodelson, L M Widrow, Phys. Rev. Lett. 72Dodelson S, Widrow LM. 1994. Phys. Rev. Lett. 72:17-20
. E D'onghia, V Springel, L Hernquist, D Keres, ApJ. 709D'Onghia E, Springel V, Hernquist L, Keres D. 2010. ApJ 709:1138-1147
. A Drlica-Wagner, K Bechtol, E S Rykoff, E Luque, A Queiroz, ApJ. 813109Drlica-Wagner A, Bechtol K, Rykoff ES, Luque E, Queiroz A, et al. 2015. ApJ 813:109
. J Dubinski, R G Carlberg, ApJ. 378Dubinski J, Carlberg RG. 1991. ApJ 378:496-503
. A A Dutton, A V Macciò, MNRAS. 441Dutton AA, Macciò AV. 2014. MNRAS 441:3359-3374
. A A Dutton, A V Macciò, J Frings, L Wang, G S Stinson, MNRAS. L74-L78 Efstathiou G457MNRASDutton AA, Macciò AV, Frings J, Wang L, Stinson GS, et al. 2016. MNRAS 457:L74-L78 Efstathiou G. 1992. MNRAS 256:43P-47P
. J Einasto, Trudy Astrofizicheskogo Instituta Alma-Ata. 5Einasto J. 1965. Trudy Astrofizicheskogo Instituta Alma-Ata 5:87-100
. O D Elbert, J S Bullock, S Garrison-Kimmel, M Rocha, J Oñorbe, Ahg Peter, MNRAS. 453Elbert OD, Bullock JS, Garrison-Kimmel S, Rocha M, Oñorbe J, Peter AHG. 2015. MNRAS 453:29- 37
. O D Elbert, J S Bullock, M Kaplinghat, S Garrison-Kimmel, A S Graus, M Rocha, arXiv:1609.08626astro-phElbert OD, Bullock JS, Kaplinghat M, Garrison-Kimmel S, Graus AS, Rocha M. 2016. arXiv:1609.08626 [astro-ph]
. S M Faber, Dnc Lin, ApJ. 266Faber SM, Lin DNC. 1983. ApJ 266:L17-L20
. B Famaey, S S Mcgaugh, Living Reviews in Relativity. 1510Famaey B, McGaugh SS. 2012. Living Reviews in Relativity 15:10
. J L Feng, ARA&A. 48Feng JL. 2010. ARA&A 48:495-545
. J L Feng, J Kumar, Physical Review Letters. 101231301Feng JL, Kumar J. 2008. Physical Review Letters 101:231301
. I Ferrero, M G Abadi, J F Navarro, L V Sales, S Gurovich, MNRAS. 425Ferrero I, Abadi MG, Navarro JF, Sales LV, Gurovich S. 2012. MNRAS 425:2817-2823
. A Fitts, M Boylan-Kolchin, O D Elbert, J S Bullock, P F Hopkins, arXiv:1611.02281astro-phFitts A, Boylan-Kolchin M, Elbert OD, Bullock JS, Hopkins PF, et al. 2016. arXiv:1611.02281 [astro-ph]
. R A Flores, J R Primack, ApJ. 427Flores RA, Primack JR. 1994. ApJ 427:L1-L4
. K C Freeman, ApJ. 160811Freeman KC. 1970. ApJ 160:811
. C S Frenk, Sdm White, Annalen der Physik. 524Frenk CS, White SDM. 2012. Annalen der Physik 524:507-534
. C S Frenk, Sdm White, M Davis, G Efstathiou, ApJ. 327Frenk CS, White SDM, Davis M, Efstathiou G. 1988. ApJ 327:507-525
. A B Fry, F Governato, A Pontzen, T Quinn, M Tremmel, MNRAS. 452Fry AB, Governato F, Pontzen A, Quinn T, Tremmel M, et al. 2015. MNRAS 452:1468-1479
. L Gao, J F Navarro, S Cole, C S Frenk, Sdm White, MNRAS. 387Gao L, Navarro JF, Cole S, Frenk CS, White SDM, et al. 2008. MNRAS 387:536-544
. S Garrison-Kimmel, M Boylan-Kolchin, J S Bullock, K Lee, MNRAS. 438Garrison-Kimmel S, Boylan-Kolchin M, Bullock JS, Lee K. 2014. MNRAS 438:2578-2596
. S Garrison-Kimmel, J S Bullock, M Boylan-Kolchin, E Bardwell, MNRAS. 464Garrison-Kimmel S, Bullock JS, Boylan-Kolchin M, Bardwell E. 2017a. MNRAS 464:3108-3120
. S Garrison-Kimmel, A R Wetzel, J S Bullock, P F Hopkins, M Boylan-Kolchin, arXiv:1701.03792astro-phGarrison-Kimmel S, Wetzel AR, Bullock JS, Hopkins PF, Boylan-Kolchin M, et al. 2017b. arXiv:1701.03792 [astro-ph]
. S Ghigna, B Moore, F Governato, G Lake, T Quinn, J Stadel, MNRAS. 300Ghigna S, Moore B, Governato F, Lake G, Quinn T, Stadel J. 1998. MNRAS 300:146-162
. N Gillet, P Ocvirk, Aubert D Knebe, A Libeskind, N , ApJ. 80034Gillet N, Ocvirk P, Aubert D, Knebe A, Libeskind N, et al. 2015. ApJ 800:34
. R Giovanelli, M P Haynes, Eak Adams, J M Cannon, K L Rhode, AJ. 14615Giovanelli R, Haynes MP, Adams EAK, Cannon JM, Rhode KL, et al. 2013. AJ 146:15
. O Y Gnedin, H Zhao, MNRAS. 333Gnedin OY, Zhao H. 2002. MNRAS 333:299-306
. F Governato, A Zolotov, A Pontzen, C Christensen, S H Oh, MNRAS. 422Governato F, Zolotov A, Pontzen A, Christensen C, Oh SH, et al. 2012. MNRAS 422:1231-1240
. A M Green, S Hofmann, D J Schwarz, MNRAS. 353Green AM, Hofmann S, Schwarz DJ. 2004. MNRAS 353:L23-L27
. B F Griffen, A P Ji, G A Dooley, F A Gómez, M Vogelsberger, ApJ. 81810Griffen BF, Ji AP, Dooley GA, Gómez FA, Vogelsberger M, et al. 2016. ApJ 818:10
. J E Gunn, Jri Gott, ApJ. 1761Gunn JE, Gott JRI. 1972. ApJ 176:1
. H Guo, Z Zheng, P S Behroozi, I Zehavi, C H Chuang, MNRAS. 459Guo H, Zheng Z, Behroozi PS, Zehavi I, Chuang CH, et al. 2016. MNRAS 459:3040-3058
. J R Hargis, B Willman, Ahg Peter, ApJ. 79513Hargis JR, Willman B, Peter AHG. 2014. ApJ 795:L13
. Y Hezaveh, N Dalal, G Holder, M Kuhlen, D Marrone, ApJ. 7679Hezaveh Y, Dalal N, Holder G, Kuhlen M, Marrone D, et al. 2013. ApJ 767:9
. Y D Hezaveh, N Dalal, D P Marrone, Y Y Mao, W Morningstar, ApJ. 82337Hezaveh YD, Dalal N, Marrone DP, Mao YY, Morningstar W, et al. 2016. ApJ 823:37
. S Hofmann, D J Schwarz, H Stöcker, Phys. Rev. D. 6483507Hofmann S, Schwarz DJ, Stöcker H. 2001. Phys. Rev. D 64:083507
. P F Hopkins, D Kereš, J Oñorbe, C A Faucher-Giguère, E Quataert, MNRAS. 445Hopkins PF, Kereš D, Oñorbe J, Faucher-Giguère CA, Quataert E, et al. 2014. MNRAS 445:581-603
. P F Hopkins, A Wetzel, D Keres, C A Faucher-Giguere, E Quataert, arXiv:1702.06148astro-phHopkins PF, Wetzel A, Keres D, Faucher-Giguere CA, Quataert E, et al. 2017. arXiv:1702.06148 [astro-ph]
. S Horiuchi, P J Humphrey, J Oñorbe, K N Abazajian, M Kaplinghat, S Garrison-Kimmel, Phys. Rev. D. 8925017Horiuchi S, Humphrey PJ, Oñorbe J, Abazajian KN, Kaplinghat M, Garrison-Kimmel S. 2014. Phys. Rev. D 89:025017
. R A Ibata, B Famaey, G F Lewis, N G Ibata, N Martin, ApJ. 80567Ibata RA, Famaey B, Lewis GF, Ibata NG, Martin N. 2015. ApJ 805:67
. R A Ibata, G F Lewis, A R Conn, M J Irwin, A W Mcconnachie, Nature. 493Ibata RA, Lewis GF, Conn AR, Irwin MJ, McConnachie AW, et al. 2013. Nature 493:62-65
. R A Ibata, G F Lewis, M J Irwin, T Quinn, MNRAS. 332Ibata RA, Lewis GF, Irwin MJ, Quinn T. 2002. MNRAS 332:915-920
. C Impey, G Bothun, D Malin, ApJ. 330Impey C, Bothun G, Malin D. 1988. ApJ 330:634-660
. P Jethwa, D Erkal, V Belokurov, MNRAS. 461Jethwa P, Erkal D, Belokurov V. 2016. MNRAS 461:2212-2233
. Y P Jing, ApJ. 535Jing YP. 2000. ApJ 535:30-36
. G Jungman, M Kamionkowski, K Griest, Phys. Rep. 267Jungman G, Kamionkowski M, Griest K. 1996. Phys. Rep. 267:195-373
. X Kang, S Mao, L Gao, Y P Jing, A&A. 437Kang X, Mao S, Gao L, Jing YP. 2005. A&A 437:383-388
. M Kaplinghat, S Tulin, H B Yu, Physical Review Letters. 11641302Kaplinghat M, Tulin S, Yu HB. 2016. Physical Review Letters 116:041302
. M Kawasaki, K Nakayama, Annual Review of Nuclear and Particle Science. 63Kawasaki M, Nakayama K. 2013. Annual Review of Nuclear and Particle Science 63:69-95
. B W Keller, J W Wadsley, ApJ. 83517Keller BW, Wadsley JW. 2017. ApJ 835:L17
. E N Kirby, J S Bullock, M Boylan-Kolchin, M Kaplinghat, J G Cohen, MNRAS. 439Kirby EN, Bullock JS, Boylan-Kolchin M, Kaplinghat M, Cohen JG. 2014. MNRAS 439:1015-1027
. A Klypin, S Gottlöber, A V Kravtsov, A M Khokhlov, ApJ. 516Klypin A, Gottlöber S, Kravtsov AV, Khokhlov AM. 1999a. ApJ 516:530-551
. A Klypin, I Karachentsev, D Makarov, O Nasonova, MNRAS. 454Klypin A, Karachentsev I, Makarov D, Nasonova O. 2015. MNRAS 454:1798-1810
. A Klypin, A V Kravtsov, O Valenzuela, F Prada, ApJ. 522Klypin A, Kravtsov AV, Valenzuela O, Prada F. 1999b. ApJ 522:82-92
. A Klypin, G Yepes, S Gottlöber, F Prada, S Heß, MNRAS. 457Klypin A, Yepes G, Gottlöber S, Prada F, Heß S. 2016. MNRAS 457:4340-4359
. J Koda, M Yagi, H Yamanoi, Y Komiyama, ApJ. 8072Koda J, Yagi M, Yamanoi H, Komiyama Y. 2015. ApJ 807:L2
E W Kolb, M S Turner, The Early Universe. Boulder, CO; SE, Belokurov V, Torrealba G, Evans NWWestview Press Koposov69130ApJKolb EW, Turner MS. 1994. The Early Universe. vol. 69 of Frontiers in Physics. Boulder, CO: Westview Press Koposov SE, Belokurov V, Torrealba G, Evans NW. 2015. ApJ 805:130
. J Kormendy, K C Freeman, ApJ. 81784Kormendy J, Freeman KC. 2016. ApJ 817:84
. A V Kravtsov, ApJ. 76431Kravtsov AV. 2013. ApJ 764:L31
. A V Kravtsov, A A Berlind, R H Wechsler, A A Klypin, S Gottlöber, ApJ. 609Kravtsov AV, Berlind AA, Wechsler RH, Klypin AA, Gottlöber S, et al. 2004. ApJ 609:35-49
. P Kroupa, C Theis, C M Boily, A&A. 431Kroupa P, Theis C, Boily CM. 2005. A&A 431:517-521
. M Kuhlen, P Madau, J Silk, M ; -Kuhlen, M Vogelsberger, R Angulo, Physics of the Dark Universe. 325ScienceKuhlen M, Madau P, Silk J. 2009. Science 325:970- Kuhlen M, Vogelsberger M, Angulo R. 2012. Physics of the Dark Universe 1:50-93
The Magellanic Plane. W E Kunkel, S Demers, Dickens, Perry, Smith, King, The Galaxy and the Local Group. McGaugh SS, de Blok WJG182ApJKunkel WE, Demers S. 1976. The Magellanic Plane. In The Galaxy and the Local Group, eds. RJ Dickens, JE Perry, FG Smith, IR King, vol. 182 of Royal Greenwich Observatory Bulletins Kuzio de Naray R, McGaugh SS, de Blok WJG. 2008. ApJ 676:920-943
. N I Libeskind, C S Frenk, S Cole, A Jenkins, J C Helly, MNRAS. 399Libeskind NI, Frenk CS, Cole S, Jenkins A, Helly JC. 2009. MNRAS 399:550-558
. N I Libeskind, Y Hoffman, R B Tully, H M Courtois, D Pomarède, MNRAS. 452Libeskind NI, Hoffman Y, Tully RB, Courtois HM, Pomarède D, et al. 2015. MNRAS 452:1052-1059
. Dnc Lin, S M Faber, ApJ. 266Lin DNC, Faber SM. 1983. ApJ 266:L21-L25
. R C Livermore, S L Finkelstein, J M Lotz, ApJ. 835113Livermore RC, Finkelstein SL, Lotz JM. 2017. ApJ 835:113
. M R Lovell, V R Eke, C S Frenk, A Jenkins, MNRAS. 413Lovell MR, Eke VR, Frenk CS, Jenkins A. 2011. MNRAS 413:3013-3021
. M R Lovell, C S Frenk, V R Eke, A Jenkins, L Gao, T Theuns, MNRAS. 439Lovell MR, Frenk CS, Eke VR, Jenkins A, Gao L, Theuns T. 2014. MNRAS 439:300-317
. A D Ludlow, S Bose, R E Angulo, L Wang, W A Hellwing, MNRAS. 460Ludlow AD, Bose S, Angulo RE, Wang L, Hellwing WA, et al. 2016. MNRAS 460:1214-1232
. D Lynden-Bell, MNRAS. 174Lynden-Bell D. 1976. MNRAS 174:695-710
. A V Macciò, S M Udrescu, A A Dutton, A Obreja, L Wang, MNRAS. 463Macciò AV, Udrescu SM, Dutton AA, Obreja A, Wang L, et al. 2016. MNRAS 463:L69-L73
. P Madau, S Shen, F Governato, ApJ. 78917Madau P, Shen S, Governato F. 2014. ApJ 789:L17
. D Marchesini, D 'onghia, E Chincarini, G Firmani, C Conconi, P , ApJ. 575Marchesini D, D'Onghia E, Chincarini G, Firmani C, Conconi P, et al. 2002. ApJ 575:801-813
. S Mashchenko, J Wadsley, Hmp Couchman, Science. 319174Mashchenko S, Wadsley J, Couchman HMP. 2008. Science 319:174-
. P Massey, Kag Olsen, P W Hodge, G H Jacoby, R T Mcneill, AJ. 133Massey P, Olsen KAG, Hodge PW, Jacoby GH, McNeill RT, et al. 2007. AJ 133:2393-2417
. A W Mcconnachie, AJ. 1444McConnachie AW. 2012. AJ 144:4
. S S Mcgaugh, AJ. 14340McGaugh SS. 2012. AJ 143:40
. S S Mcgaugh, Canadian Journal of Physics. 93McGaugh SS. 2015. Canadian Journal of Physics 93:250-259
. S S Mcgaugh, F Lelli, J M Schombert, Physical Review Letters. 117McGaugh SS, Lelli F, Schombert JM. 2016. Physical Review Letters 117:201101
. S S Mcgaugh, V C Rubin, Wjg De Blok, AJ. 122McGaugh SS, Rubin VC, de Blok WJG. 2001. AJ 122:2381-2395
. M Mcquinn, ARA&A. 54McQuinn M. 2016. ARA&A 54:313-362
. N Menci, A Grazian, M Castellano, N G Sanchez, ApJ. 8251Menci N, Grazian A, Castellano M, Sanchez NG. 2016. ApJ 825:L1
. A Merle, A Schneider, Physics Letters B. 749Merle A, Schneider A. 2015. Physics Letters B 749:283-288
. M Metz, P Kroupa, H Jerjen, MNRAS. 374Metz M, Kroupa P, Jerjen H. 2007. MNRAS 374:1125-1145
. M Metz, P Kroupa, H Jerjen, MNRAS. 394Metz M, Kroupa P, Jerjen H. 2009. MNRAS 394:2223-2228
. M Milgrom, New Astron. Rev. 46Milgrom M. 2002. New Astron. Rev. 46:741-753
. M Milgrom, arXiv:1610.07538astro-phMilgrom M. 2016. arXiv:1610.07538 [astro-ph]
Galaxy Formation and Evolution Moore B. H Mo, Van Den Bosch, Fc, S White, Nature. 370Mo H, van den Bosch FC, White S. 2010. Galaxy Formation and Evolution Moore B. 1994. Nature 370:629-631
. B Moore, S Ghigna, F Governato, G Lake, T Quinn, ApJ. 52436ApJMoore B, Ghigna S, Governato F, Lake G, Quinn T, et al. 1999. ApJ 524:L19-L22 More S, Diemer B, Kravtsov AV. 2015. ApJ 810:36
. B P Moster, R S Somerville, C Maulbetsch, F C Van Den Bosch, A V Macciò, ApJ. 710Moster BP, Somerville RS, Maulbetsch C, van den Bosch FC, Macciò AV, et al. 2010. ApJ 710:903- 923
. F Munshi, F Governato, A M Brooks, C Christensen, S Shen, Phys. Rev. D. 766ApJMunshi F, Governato F, Brooks AM, Christensen C, Shen S, et al. 2013. ApJ 766:56 Nambu Y, Sasaki M. 1990. Phys. Rev. D 42:3918-3924
. J F Navarro, A Benítez-Llambay, A Fattahi, C S Frenk, A D Ludlow, arXiv:1612.06329astro-phNavarro JF, Benítez-Llambay A, Fattahi A, Frenk CS, Ludlow AD, et al. 2016. arXiv:1612.06329 [astro-ph]
. J F Navarro, V R Eke, C S Frenk, MNRAS. 283Navarro JF, Eke VR, Frenk CS. 1996. MNRAS 283:L72-L78
. J F Navarro, C S Frenk, Sdm White, ApJ. 490Navarro JF, Frenk CS, White SDM. 1997. ApJ 490:493-508
. J F Navarro, E Hayashi, C Power, A R Jenkins, C S Frenk, MNRAS. 349Navarro JF, Hayashi E, Power C, Jenkins AR, Frenk CS, et al. 2004. MNRAS 349:1039-1051
. J F Navarro, A Ludlow, V Springel, J Wang, M Vogelsberger, MNRAS. 402Navarro JF, Ludlow A, Springel V, Wang J, Vogelsberger M, et al. 2010. MNRAS 402:21-34
. J Oñorbe, M Boylan-Kolchin, J S Bullock, P F Hopkins, D Kereš, MNRAS. 454Oñorbe J, Boylan-Kolchin M, Bullock JS, Hopkins PF, Kereš D, et al. 2015. MNRAS 454:2092-2106
. S H Oh, D A Hunter, E Brinks, B G Elmegreen, A Schruba, AJ. 149180Oh SH, Hunter DA, Brinks E, Elmegreen BG, Schruba A, et al. 2015. AJ 149:180
. K A Oman, J F Navarro, A Fattahi, C S Frenk, T Sawala, MNRAS. 452Oman KA, Navarro JF, Fattahi A, Frenk CS, Sawala T, et al. 2015. MNRAS 452:3650-3665
E Papastergis, R Giovanelli, M P Haynes, F Shankar, E Papastergis, A A Ponomareva, A1 Papastergis E, Shankar F. 2016. A&A 591:A58 Pawlowski MS. 574A&APapastergis E, Giovanelli R, Haynes MP, Shankar F. 2015. A&A 574:A113 Papastergis E, Ponomareva AA. 2017. A&A 601:A1 Papastergis E, Shankar F. 2016. A&A 591:A58 Pawlowski MS. 2016. MNRAS 456:448-458
. M S Pawlowski, B Famaey, D Merritt, P Kroupa, ApJ. 81519Pawlowski MS, Famaey B, Merritt D, Kroupa P. 2015. ApJ 815:19
. M S Pawlowski, P Kroupa, MNRAS. 435Pawlowski MS, Kroupa P. 2013. MNRAS 435:2116-2131
. M S Pawlowski, S S Mcgaugh, H Jerjen, MNRAS. 453Pawlowski MS, McGaugh SS, Jerjen H. 2015. MNRAS 453:1047-1061
. M S Pawlowski, J Pflamm-Altenburg, P Kroupa, MNRAS. 423Pawlowski MS, Pflamm-Altenburg J, Kroupa P. 2012. MNRAS 423:1109-1126
. Pje Peebles, ApJ. 263Peebles PJE. 1982. ApJ 263:L1-L5
. Pje Peebles, ARA&A. 50Peebles PJE. 2012. ARA&A 50:1-28
. Ahg Peter, M Rocha, J S Bullock, M Kaplinghat, MNRAS. 430Peter AHG, Rocha M, Bullock JS, Kaplinghat M. 2013. MNRAS 430:105-120
. J I Phillips, M C Cooper, J S Bullock, M Boylan-Kolchin, MNRAS. 453Phillips JI, Cooper MC, Bullock JS, Boylan-Kolchin M. 2015. MNRAS 453:3839-3847
. Par Ade, Planck CollaborationN Aghanim, Planck CollaborationM Arnaud, Planck CollaborationM Ashdown, Planck CollaborationPhys. Rev. D. 59443506A&APlanck Collaboration, Ade PAR, Aghanim N, Arnaud M, Ashdown M, et al. 2016. A&A 594:A13 Polisensky E, Ricotti M. 2011. Phys. Rev. D 83:043506
. A Pontzen, F Governato, MNRAS. 421Pontzen A, Governato F. 2012. MNRAS 421:3464-3471
. T A Porter, R P Johnson, P W Graham, ARA&A. 49Porter TA, Johnson RP, Graham PW. 2011. ARA&A 49:155-194
. W H Press, P Schechter, ApJ. 187Press WH, Schechter P. 1974. ApJ 187:425-438
. J R Primack, Annalen der Physik. 524Primack JR. 2012. Annalen der Physik 524:535-544
. J I Read, Journal of Physics G Nuclear Physics. 4163101Read JI. 2014. Journal of Physics G Nuclear Physics 41:063101
. J I Read, O Agertz, Mlm Collins, MNRAS. 459Read JI, Agertz O, Collins MLM. 2016. MNRAS 459:2573-2590
. M J Rees, J P Ostriker, MNRAS. 179Rees MJ, Ostriker JP. 1977. MNRAS 179:541-559
. J C Richardson, M J Irwin, A W Mcconnachie, N F Martin, A L Dotter, ApJ. 73276Richardson JC, Irwin MJ, McConnachie AW, Martin NF, Dotter AL, et al. 2011. ApJ 732:76
. M Rocha, Ahg Peter, J S Bullock, M Kaplinghat, S Garrison-Kimmel, MNRAS. 430Rocha M, Peter AHG, Bullock JS, Kaplinghat M, Garrison-Kimmel S, et al. 2013. MNRAS 430:81- 104
. A Rodríguez-Puebla, P Behroozi, J Primack, A Klypin, C Lee, D Hellinger, MNRAS. 462Rodríguez-Puebla A, Behroozi P, Primack J, Klypin A, Lee C, Hellinger D. 2016. MNRAS 462:893- 916
. Rubin Vc, Ford WkJr, N Thonnard, ApJ. 238Rubin VC, Ford Jr. WK, Thonnard N. 1980. ApJ 238:471-487
. V C Rubin, N Thonnard, FordJr, Wk, ApJ. 225Rubin VC, Thonnard N, Ford Jr. WK. 1978. ApJ 225:L107-L111
. L V Sales, J F Navarro, N Kallivayalil, C S Frenk, MNRAS. 465Sales LV, Navarro JF, Kallivayalil N, Frenk CS. 2017. MNRAS 465:1879-1888
. D J Sand, D Crnojević, P Bennet, B Willman, J Hargis, ApJ. 80695Sand DJ, Crnojević D, Bennet P, Willman B, Hargis J, et al. 2015a. ApJ 806:95
. D J Sand, K Spekkens, D Crnojević, J R Hargis, B Willman, MNRAS. 812ApJSand DJ, Spekkens K, Crnojević D, Hargis JR, Willman B, et al. 2015b. ApJ 812:L13 Sawala T, Frenk CS, Fattahi A, Navarro JF, Bower RG, et al. 2016. MNRAS 457:1931-1943
. J Schaye, R A Crain, R G Bower, M Furlong, M Schaller, MNRAS. 446Schaye J, Crain RA, Bower RG, Furlong M, Schaller M, et al. 2015. MNRAS 446:521-554
. A Schneider, R E Smith, A V Macciò, B Moore, MNRAS. 424Schneider A, Smith RE, Macciò AV, Moore B. 2012. MNRAS 424:684-698
. A Schneider, S Trujillo-Gomez, E Papastergis, D S Reed, G Lake, arXiv:1611.09362astro-phSchneider A, Trujillo-Gomez S, Papastergis E, Reed DS, Lake G. 2016. arXiv:1611.09362 [astro-ph]
. C Schultz, J Oñorbe, K N Abazajian, J S Bullock, MNRAS. 442Schultz C, Oñorbe J, Abazajian KN, Bullock JS. 2014. MNRAS 442:1597-1609
. R K Sheth, H J Mo, G Tormen, MNRAS. 323Sheth RK, Mo HJ, Tormen G. 2001. MNRAS 323:1-12
. X Shi, G M Fuller, Phys. Rev. Lett. 82Shi X, Fuller GM. 1999. Phys. Rev. Lett. 82:2832-2835
. J D Simon, A D Bolatto, A Leroy, L Blitz, E L Gates, ApJ. 621Simon JD, Bolatto AD, Leroy A, Blitz L, Gates EL. 2005. ApJ 621:757-776
. J D Simon, M Geha, Q E Minor, G D Martinez, E N Kirby, ApJ. 73346Simon JD, Geha M, Minor QE, Martinez GD, Kirby EN, et al. 2011. ApJ 733:46
. E D Skillman, M Monelli, D R Weisz, S L Hidalgo, A Aparicio, ApJ. 837102Skillman ED, Monelli M, Weisz DR, Hidalgo SL, Aparicio A, et al. 2017. ApJ 837:102
. R S Somerville, R Davé, ARA&A. 53Somerville RS, Davé R. 2015. ARA&A 53:51-113
. D N Spergel, P J Steinhardt, Physical Review Letters. 84Spergel DN, Steinhardt PJ. 2000. Physical Review Letters 84:3760-3763
. V Springel, J Wang, M Vogelsberger, A Ludlow, A Jenkins, MNRAS. 391Springel V, Wang J, Vogelsberger M, Ludlow A, Jenkins A, et al. 2008. MNRAS 391:1685-1711
. J Stadel, D Potter, B Moore, J Diemand, P Madau, Phys. Rep. 398MNRASStadel J, Potter D, Moore B, Diemand J, Madau P, et al. 2009. MNRAS 398:L21-L25 Strigari LE. 2013. Phys. Rep. 531:1-88
. L E Strigari, J S Bullock, M Kaplinghat, J D Simon, M Geha, Nature. 454Strigari LE, Bullock JS, Kaplinghat M, Simon JD, Geha M, et al. 2008. Nature 454:1096-1097
. E J Tollerud, M Boylan-Kolchin, J S Bullock, MNRAS. 440Tollerud EJ, Boylan-Kolchin M, Bullock JS. 2014. MNRAS 440:3511-3519
. E J Tollerud, J S Bullock, L E Strigari, B Willman, ApJ. 688Tollerud EJ, Bullock JS, Strigari LE, Willman B. 2008. ApJ 688:277-289
. E J Tollerud, M C Geha, J Grcevich, M E Putman, D R Weisz, A E Dolphin, ApJ. 82789Tollerud EJ, Geha MC, Grcevich J, Putman ME, Weisz DR, Dolphin AE. 2016. ApJ 827:89
. E Tollet, A V Macciò, A A Dutton, G S Stinson, L Wang, MNRAS. 456Tollet E, Macciò AV, Dutton AA, Stinson GS, Wang L, et al. 2016. MNRAS 456:3542-3552
. M Tomozeiu, L Mayer, T Quinn, ApJ. 82715Tomozeiu M, Mayer L, Quinn T. 2016. ApJ 827:L15
. S Tremaine, J E Gunn, Phys. Rev. Lett. 42Tremaine S, Gunn JE. 1979. Phys. Rev. Lett. 42:407-410
. S Trujillo-Gomez, A Schneider, E Papastergis, D S Reed, G Lake, arXiv:1610.09335astro-phTrujillo-Gomez S, Schneider A, Papastergis E, Reed DS, Lake G. 2016. arXiv:1610.09335 [astro-ph]
. R B Tully, N I Libeskind, I D Karachentsev, V E Karachentseva, L Rizzi, Ej ; L25 Shaya, P G Van Dokkum, R Abraham, A Merritt, J Zhang, M Geha, C ; L45 Conroy, S Vegetti, Lve Koopmans, A Bolton, T Treu, R Gavazzi, MNRAS. 802ApJTully RB, Libeskind NI, Karachentsev ID, Karachentseva VE, Rizzi L, Shaya EJ. 2015. ApJ 802:L25 van Dokkum PG, Abraham R, Merritt A, Zhang J, Geha M, Conroy C. 2015. ApJ 798:L45 Vegetti S, Koopmans LVE, Bolton A, Treu T, Gavazzi R. 2010. MNRAS 408:1969-1981
. S Vegetti, D J Lagattuta, J P Mckean, M W Auger, C D Fassnacht, Lve Koopmans, Nature. 481Vegetti S, Lagattuta DJ, McKean JP, Auger MW, Fassnacht CD, Koopmans LVE. 2012. Nature 481:341-343
. T Venumadhav, F Y Cyr-Racine, K N Abazajian, C M Hirata, Phys. Rev. D. 9443515Venumadhav T, Cyr-Racine FY, Abazajian KN, Hirata CM. 2016. Phys. Rev. D 94:043515
. M Viel, G D Becker, J S Bolton, M G Haehnelt, Phys. Rev. D. 8843502Viel M, Becker GD, Bolton JS, Haehnelt MG. 2013. Phys. Rev. D 88:043502
. M Viel, J Lesgourgues, M G Haehnelt, S Matarrese, A Riotto, Phys. Rev. D. 7163534Viel M, Lesgourgues J, Haehnelt MG, Matarrese S, Riotto A. 2005. Phys. Rev. D 71:063534
. M Vogelsberger, S Genel, V Springel, P Torrey, D Sijacki, MNRAS. 444Vogelsberger M, Genel S, Springel V, Torrey P, Sijacki D, et al. 2014. MNRAS 444:1518-1547
. M Vogelsberger, J Zavala, F Y Cyr-Racine, C Pfrommer, T Bringmann, K Sigurdson, MNRAS. 460Vogelsberger M, Zavala J, Cyr-Racine FY, Pfrommer C, Bringmann T, Sigurdson K. 2016. MNRAS 460:1399-1416
. M Vogelsberger, J Zavala, A Loeb, MNRAS. 423Vogelsberger M, Zavala J, Loeb A. 2012. MNRAS 423:3740-3752
. M G Walker, M Mateo, E W Olszewski, J Peñarrubia, Wyn Evans, N Gilmore, G , ApJ. 704Walker MG, Mateo M, Olszewski EW, Peñarrubia J, Wyn Evans N, Gilmore G. 2009. ApJ 704:1274- 1287
. R H Wechsler, J S Bullock, J R Primack, A V Kravtsov, A Dekel, ApJ. 568Wechsler RH, Bullock JS, Primack JR, Kravtsov AV, Dekel A. 2002. ApJ 568:52-70
. D H Weinberg, J S Bullock, F Governato, Kuzio De Naray, R Peter, Ahg , Proceedings of the National Academy of Science. 112Weinberg DH, Bullock JS, Governato F, Kuzio de Naray R, Peter AHG. 2015. Proceedings of the National Academy of Science 112:12249-12255
. D H Weinberg, M J Mortonson, D J Eisenstein, C Hirata, A G Riess, E Rozo, Phys. Rep. 530Weinberg DH, Mortonson MJ, Eisenstein DJ, Hirata C, Riess AG, Rozo E. 2013. Phys. Rep. 530:87- 255
. A R Wetzel, P F Hopkins, Kim Jh, C A Faucher-Giguère, D Kereš, E Quataert, ApJ. 82723Wetzel AR, Hopkins PF, Kim Jh, Faucher-Giguère CA, Kereš D, Quataert E. 2016. ApJ 827:L23
. C Wheeler, J Oñorbe, J S Bullock, M Boylan-Kolchin, O D Elbert, MNRAS. 453Wheeler C, Oñorbe J, Bullock JS, Boylan-Kolchin M, Elbert OD, et al. 2015. MNRAS 453:1305- 1316
. Sdm White, C S Frenk, ApJ. 379White SDM, Frenk CS. 1991. ApJ 379:52-79
. Sdm White, C S Frenk, M Davis, ApJ. 274White SDM, Frenk CS, Davis M. 1983. ApJ 274:L1-L5
. Sdm White, M J Rees, MNRAS. 183White SDM, Rees MJ. 1978. MNRAS 183:341-358
. B Willman, Advances in Astronomy. 2010285454Willman B. 2010. Advances in Astronomy 2010:285454
. B Willman, J Strader, AJ. 14476Willman B, Strader J. 2012. AJ 144:76
. J Wolf, G D Martinez, J S Bullock, M Kaplinghat, M Geha, MNRAS. 406Wolf J, Martinez GD, Bullock JS, Kaplinghat M, Geha M, et al. 2010. MNRAS 406:1220-1237
. A H Wright, Asg Robotham, S P Driver, M Alpaslan, S K Andrews, MNRAS. 470Wright AH, Robotham ASG, Driver SP, Alpaslan M, Andrews SK, et al. 2017. MNRAS 470:283-302
. J H Yoon, K V Johnston, D W Hogg, ApJ. 73158Yoon JH, Johnston KV, Hogg DW. 2011. ApJ 731:58
. J Zavala, Y P Jing, A Faltenbacher, G Yepes, Y Hoffman, ApJ. 700Zavala J, Jing YP, Faltenbacher A, Yepes G, Hoffman Y, et al. 2009. ApJ 700:1779-1793
. J Zavala, V Springel, M Boylan-Kolchin, MNRAS. 405Zavala J, Springel V, Boylan-Kolchin M. 2010. MNRAS 405:593-612
. A R Zentner, International Journal of Modern Physics D. 16Zentner AR. 2007. International Journal of Modern Physics D 16:763-815
. A R Zentner, A V Kravtsov, O Y Gnedin, A A Klypin, ApJ. 629Zentner AR, Kravtsov AV, Gnedin OY, Klypin AA. 2005. ApJ 629:219-232
. A Zolotov, A M Brooks, B Willman, F Governato, A Pontzen, ApJ. 76171Zolotov A, Brooks AM, Willman B, Governato F, Pontzen A, et al. 2012. ApJ 761:71
. F Zwicky, Helvetica Physica Acta. 6Zwicky F. 1933. Helvetica Physica Acta 6:110-127
| [] |
[
"Scanning magnetoresistance microscopy of atom chips",
"Scanning magnetoresistance microscopy of atom chips"
] | [
"M Volk \nARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy\nSwinburne University of Technology\n3122HawthornVictoriaAustralia\n",
"S Whitlock \nARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy\nSwinburne University of Technology\n3122HawthornVictoriaAustralia\n",
"B V Hall \nARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy\nSwinburne University of Technology\n3122HawthornVictoriaAustralia\n",
"A I Sidorov \nARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy\nSwinburne University of Technology\n3122HawthornVictoriaAustralia\n"
] | [
"ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy\nSwinburne University of Technology\n3122HawthornVictoriaAustralia",
"ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy\nSwinburne University of Technology\n3122HawthornVictoriaAustralia",
"ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy\nSwinburne University of Technology\n3122HawthornVictoriaAustralia",
"ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy\nSwinburne University of Technology\n3122HawthornVictoriaAustralia"
] | [] | Surface based geometries of microfabricated wires or patterned magnetic films can be used to magnetically trap and manipulate ultracold neutral atoms or Bose-Einstein condensates. We investigate the magnetic properties of such atom chips using a scanning magnetoresistive (MR) microscope with high spatial resolution and high field sensitivity. We show that MR sensors are ideally suited to observe small variations of the magnetic field caused by imperfections in the wires or magnetic materials which ultimately lead to fragmentation of ultracold atom clouds. Measurements are also provided for the magnetic field produced by a thin current-carrying wire with small geometric modulations along the edge. Comparisons of our measurements with a full numeric calculation of the current flow in the wire and the subsequent magnetic field show excellent agreement. Our results highlight the use of scanning MR microscopy as a convenient and powerful technique for precisely characterizing the magnetic fields produced near the surface of atom chips. | 10.1063/1.2839015 | [
"https://arxiv.org/pdf/0704.3137v1.pdf"
] | 308,550 | 0704.3137 | 7938464efa6453489bc14cf296a9e8ac32caf8db |
Scanning magnetoresistance microscopy of atom chips
24 Apr 2007 (Dated: February 1, 2008)
M Volk
ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy
Swinburne University of Technology
3122HawthornVictoriaAustralia
S Whitlock
ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy
Swinburne University of Technology
3122HawthornVictoriaAustralia
B V Hall
ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy
Swinburne University of Technology
3122HawthornVictoriaAustralia
A I Sidorov
ARC Centre of Excellence for Quantum-Atom Optics and Centre for Atom Optics and Ultrafast Spectroscopy
Swinburne University of Technology
3122HawthornVictoriaAustralia
Scanning magnetoresistance microscopy of atom chips
24 Apr 2007 (Dated: February 1, 2008)arXiv:0704.3137v1 [physics.atom-ph]PACS numbers: 3925+k,0755Ge
Surface based geometries of microfabricated wires or patterned magnetic films can be used to magnetically trap and manipulate ultracold neutral atoms or Bose-Einstein condensates. We investigate the magnetic properties of such atom chips using a scanning magnetoresistive (MR) microscope with high spatial resolution and high field sensitivity. We show that MR sensors are ideally suited to observe small variations of the magnetic field caused by imperfections in the wires or magnetic materials which ultimately lead to fragmentation of ultracold atom clouds. Measurements are also provided for the magnetic field produced by a thin current-carrying wire with small geometric modulations along the edge. Comparisons of our measurements with a full numeric calculation of the current flow in the wire and the subsequent magnetic field show excellent agreement. Our results highlight the use of scanning MR microscopy as a convenient and powerful technique for precisely characterizing the magnetic fields produced near the surface of atom chips.
I. INTRODUCTION
Surface based potentials for manipulating neutral atoms on a micron scale have attracted widespread interest in recent years. Atom chips [1,2] consisting of planar geometries of microfabricated wires or patterned magnetic materials provide intricate magnetic potentials and have become a practical and robust tool for producing, trapping and manipulating Bose-Einstein condensates. Atoms chips have recently been used to precisely position Bose-Einstein condensates [3], realize trapped atom interferometers [4,5] and have provided new and sensitive techniques for detecting tiny forces on a small spatial scale [6]. The fabricated wires or magnetic materials used for atom chips have been the topic of several recent studies, finding that their quality must be exceptionally high since even the smallest imperfections, for example roughness of the wire edge, can lead to uncontrolled magnetic field variations. These variations subsequently corrugate the bottom of the trapping potential [7,8]. Recently, fragmentation of ultracold atoms has also been observed in close proximity to magnetic materials [9,10,11] and has been traced to long range spatial variations in the film magnetization [10]. As the energy scales associated with ultracold atoms and Bose-Einstein condensates are in the nanokelvin regime, even the smallest magnetic field variations of only a few nanotesla can dramatically alter their properties [12].
Until now, characterizing the smoothness of the potentials produced by atom chips has relied on the atom clouds themselves, through either the equilibrium atomic The sample is placed on a computer controlled x-y translation stage. The magnetoresistive probe is connected to a preamplifier and the signal is filtered and digitized by a lock-in amplifier. A CMOS camera is used to determine the distance between the sensor tip and the sample.
density distribution [8] or radio frequency spectroscopy of trapped atom clouds [10]. With the increasing complexity of atom chips, however, it is necessary to obtain fast and reliable methods of characterizing the magnetic potentials prior to installing the atom chips in ultrahigh vacuum and trapping ultracold atoms. In this paper we describe the application of a micron sized magnetoresistance (MR) sensor to accurately profile the magnetic fields generated by magnetic film and current-carrying wire atom chips. Our home-built magnetoresistance microscope (Fig. 1) is used to measure small magnetic field variations above a permanent magnetic film atom chip which causes fragmentation of ultracold atom clouds. The MR measurements support independent measurements performed using trapped ultracold atoms as the magnetic field probe [10]. The study indicates the variations occur predominately near the edge of the film and are associated with heating of the film during vacuum bake-out. In addition, we have fabricated a new current-carrying wire atom chip using femtosecond laser ablation of a gold film [13]. A wire is sculpted with a periodically modulated edge to produce a complex magnetic potential for ultracold atoms. Two dimensional images of the field produced by the wire are obtained and are in excellent agreement with numeric calculations of the expected field strength. The measurements show that it is possible to fabricate and characterize a linear array of magnetic potentials produced by modifying the edge of a straight current carrying wire.
II. APPARATUS
Analysis of the atom chips is performed using an ultrasensitive low-field magnetoresistive sensor based on magnetic tunneling junction technology [14,15]. A magnetic tunneling junction sensor consists of two ferromagnetic layers separated by an ultra-thin insulating interlayer. One magnetic layer has fixed 'pinned' magnetization while the other responds to the local magnetic field. The interlayer resistance depends on the relative magnetization of the neighboring magnetic layers. These devices provide an absolute measure of the magnetic field with high sensitivity and high spatial resolution. They provide a linear response over a large field range (typically about 0.5 mT) and are ideal for studying the magnetic fields produced by microfabricated current-carrying wires or patterned magnetic materials on atom chips. Here the sensor is incorporated into a home-built scanning magnetic field microscope, schematically depicted in Figure 1, and used to study the corrugated field produced by the atom chip.
The microscope ( Fig. 1) consists of the MR sensor probe, the preamplification electronics, a lock-in amplifier, a motorized x-y translation stage and a computer interfaced via LabView to both stage and lock-in amplifier. The probe tip is manually positioned above the sample using a micrometer stage and a CMOS camera for height calibration. This setup allows us to acquire one-dimensional scans as well as two-dimensional maps of the z-component, i.e. the out-of-plane component, of the magnetic field at variable heights above the sample surface.
Our scanning magnetoresistance microscope incorporates a commercially available magnetic tunnel junction probe (MicroMagnetics STJ-020), polished to allow very close approaches to the surface (∼ 10 µm). The active area of the sensor is approximately 5 × 5 µm 2 and it detects the magnetic field oriented along the sensor tip (z direction). The sensor is interfaced using an Anderson loop [16] to convert small changes in the sensor resistance to a signal voltage. The output is then amplified using a signal amplification board (MicroMagnetics AL-05) with a gain of 2500 and a bandwidth of 1 MHz. The sensor and preamplifier are calibrated to give an output of 20 V/mT. Due to its small size the sensor exhibits significant 1/f noise which can be overcome by reducing the bandwidth of the output signal.
To increase the signal-to-noise ratio we use an AC modulation technique. In the case of current-carrying wires this is simply done by modulating the wire current at kHz frequencies and detecting the signal with a lock-in amplifier (Stanford Research Systems SR830). When studying permanent magnetic films we use mechanical modulation of the probe. The tip of the probe is oscillated along the scanning direction at its mechanical resonance frequency (18 kHz) using a piezo actuator. At this frequency the noise level of the sensor is reduced to less than 15% compared to DC; however the output of the lock-in amplifier is now proportional to the first derivative of the magnetic field. This output is calibrated against a known magnetic field gradient by first measuring the field in DC mode 200 µm above the edge of the film. The field is large enough to provide good signal-to-noise and features a large gradient of 1 Tm −1 . We then compare the numerical derivative of this measurement to the data obtained while oscillating the tip. This allows us to determine the oscillation amplitude of the probe and hence to reconstruct the magnetic field up to a constant offset by numerical integration of the data. The oscillation amplitude and subsequently the spatial resolution of this measurement is approximately 50 µm. The AC modulation technique reduces the noise levels to about 0.1 µT, equivalent to that obtained using ultracold atoms as a probe [10] and a factor of 5 lower than what is obtained for an equivalent measurement time using just low-pass filtering.
III. PERMANENT MAGNET ATOM CHIP
As a first application of the magnetic field microscope we investigated the random variations in the magnetic potential created near the surface of a magnetic film atom chip used in previous experiments to trap ultracold atoms and Bose-Einstein condensates and is described in detail elsewhere [17]. It uses a multilayer Tb 6 Gd 10 Fe 80 Co 4 film which exhibits strong perpendicular anisotropy. The film is deposited on a 300 µm thick glass substrate where one edge is polished to optical quality prior to film deposition. At this edge the magnetic film produces a field that is analogous to that of a thin current-carrying wire aligned with the edge (I eff = 0.2 A). A magnetic microtrap is formed by the field from the film, a uniform magnetic bias field, and two current-carrying end-wires. To account for the need of a reflecting surface for the mirror magnetooptical trap the chip is completed by a second glass slide and both sides are coated with gold.
Due to their narrow energy distribution, ultracold atoms are very sensitive to small fluctuations of the magnetic trapping potential. In a recent paper [10] we used radio frequency (rf) spectroscopy of trapped atoms to measure the absolute magnetic field strength above the We also developed a model describing the spatial decay of random magnetic fields from the surface due to inhomogeneity in the film magnetization. After removing the atom chip from the vacuum chamber we used the magnetoresistance microscope to further characterize the film properties. Our first measurement consists of a series of scans of the magnetic field parallel to the film edge over a region of 3.5 mm at various heights ranging from 500 µm down to 60 µm, the minimum distance limited by the adjacent protruding gold coated glass slide. Four of these profiles are depicted in Figure 2. Due to the large field gradient produced at the film edge it was necessary to carefully align the measurement direction and subtract a third order polynomial from the data. Also plotted in the same figure are the corresponding profiles previously measured by rf spectroscopy of ultracold atoms. The results from the two different methods are in remarkable agreement. It should be noted however that a quantitative comparison is difficult as the two methods are sensitive to different components of the corrugated magnetic field: the magnetoresistive sensing direction is perpendicular to the surface while the trap bottom probed by the rf spectroscopy is defined by the in-plane component of the magnetic field.
The results of the complete series of magnetoresistance scans as well as the rf spectroscopy measurements are summarized in Figure 3 where the root mean square (rms) noise is plotted as a function of distance to the surface. For random white noise fluctuations of the film The solid line is a power-law fit to the magnetoresistance microscope data. The inset shows the dependence of the field roughness on the transverse distance from the film edge for a fixed height of z0 = 60 µm above the film surface. magnetization our model described in [10] predicts a z −2 decay of the field roughness. A power law fit to the data obtained by the MR scans gives ∆B rms ∝ z −1.9±0.2 in excellent agreement with this prediction.
We have also performed a series of scans at constant height (z 0 = 60 µm) above the film surface but variable transverse distance to the film edge. The rms noise levels of these scans are depicted in the inset of Figure 3 (circles) together with the prediction of the random magnetization model (lines). While the model describes the results adequately above the non-magnetic half plane of the atom chip the measured inhomogeneity decreases away from the edge above the magnetic film side, whereas in the case of homogeneous magnetization fluctuations ∆B rms is expected to stay constant (dotted line in Fig. 3).
IV. TAILORED MAGNETIC MICROTRAPS
Section III of this paper focused on MR studies of the corrugated potential produced by a partially inhomogeneous magnetic film atom chip. In this section, we describe the analysis of a current-carrying wire atom chip fabricated using micron-scale femtosecond laser ablation of a thin metal film. We have produced a tailored magnetic potential by sculpting the shape of a wire to create a linear array of magnetic traps for cold atoms. Two-dimensional magnetoresistance microscopy provides an image of the perpendicular magnetic field component produced by the wire at a fixed distance to the surface. A solution to the magnetostatic inverse problem is then applied to obtain the remaining two field components, allowing a complete reconstruction of the magnetic trapping potential. Of particular interest is the field component parallel to the wire, which defines the bottom of the trapping potential. A comparison of the measured and reconstructed field components with full numeric calculations of the field produced by the sculptured wire shows excellent agreement.
A. Sculptured wire atom chip
Femtosecond laser ablation can be used to pattern micron and submicron scale structures on a wide variety of materials [18] and can be used to produce atom chips [13]. In this work we use the technique to directly fabricate complex wire patterns in an evaporatively deposited gold film to form a current-carrying wire atom chip [2]. The chip consists of a glass slide substrate with a 25 nm thick Cr bonding layer and a 150 nm thick Au layer. The wire structure is patterned by cutting three 3 µm wide insulating channels into the Au film.
We have patterned two parallel Au wires with widths of 20 and 30 µm and lengths of 10 mm which can be used to create a magnetic potential for trapping Bose-Einstein condensates (Fig. 4). Each wire has been sculptured with one periodically modulated boundary with a period of 200 µm. Deliberately modulating the wire boundary slightly modifies the current path and produces a small field component oriented parallel to the wire, which modulates the corresponding longitudinal magnetic potential experienced by the trapped atoms [19]. This is used to realize a linear array of asymmetric double wells which are separated by potential barriers with small amplitudes which can be precisely controlled by varying the wire current or the distance of the trap to the wire surface [4,6,20].
The 30 µm wire is chosen for the magnetoresistance measurements. We use the reference source of the lockin amplifier to drive a small AC current of 37 mA rms through the wire at a frequency of 1 kHz. The output of the lock-in amplifier is recorded by a computer. Two computer controlled translations stages are used to position the wire sample with respect to the MR probe. The probe is calibrated against the expected field produced by the wire calculated using Biot-Savart's law, neglecting the effect of the small modulations. We record an image of the perpendicular magnetic field component produced by the wire over a 2×1.5 mm 2 spatial region at a distance of z = 30 µm above the wire. The spatial resolution is 10 µm which corresponds to 150 × 200 data points. The lock-in integration time is set to 300 ms and each line of the image is scanned twice and averaged, which results in a measurement time of approximately 5 hours for the whole two-dimensional magnetic field image. Figure 5a shows the result of this measurement (only the central part of the full image is shown). The field amplitude produced across the wire at this height is ±100 µT. Directly above the wire the perpendicular field is nearly zero apart from a small modulated field component with amplitude of about ±2.5 µT. The noise level for this measurement determined from a region about 0.7 mm away from the wire was as low as 50 nT.
B. Reconstruction of the in-plane field components
With a two-dimensional image of the out-of-plane field component at a given height it is possible to convert to a uniquely defined in-plane current distribution [21] and subsequently back to any other field component. Given that the height of the wire is small compared to the measurement distance above the surface, the current density can be considered as a two-dimensional distribution. The Fourier transforms of the magnetic field components b x and b y are then simply related to b z :
b x (k x , k y ) = i k x k b z (k x , k y ) b y (k x , k y ) = i k y k b z (k x , k y )(1)
where k = k 2 x + k 2 y . Shown in Figure 5 (b) and (c) are the reconstructed inplane field components B x and B y derived from the measured B z component. The B y field image clearly shows the modulated component along the length of the wire which defines the bottom of the trapping potential.
C. Numeric calculations of magnetic fields
We have also performed detailed numeric calculations of the field produced by the sculptured wire to compare them with our measurements. The current density distribution of the wire is computed from the solution of Laplace's equation ∇ · (σ∇V ) = 0 satisfied by the electrostatic potential V . We assume that the conductivity σ is uniform throughout the wire and, since we are interested in the field at distances much larger than the wire thickness, we assume that V depends only on x and y. Exact analytical solutions for this problem can be obtained for particular geometries; however in general one has to rely on numerical methods.
Here, solutions of Laplace's equation were computed using the finite element method which provides an approximate solution of partial differential equations with defined boundary conditions. For this problem we have used the Matlab Partial Differential Equation (PDE) toolbox. The boundary conditions are specified such that the normal component of the current density on the wire edge is zero (Neumann conditions). The wire geometry is then decomposed into a set of triangular elements which define a mesh of nodes, for each of which the electrostatic potential is solved. From this it is straight forward to compute the current distribution and the associated magnetic field. The results of these calculations are depicted in Figure 5 (d) to (f) next to the corresponding measurements.
The y component of the magnetic field produced by the atom chip is of particular interest because it determines the potential minimum experienced by the trapped atoms. Figure 6 compares field profiles of B y along the wire extracted from the measurement and the simulation. We note that the two profiles differ by about 10% in amplitude and attribute this to a systematic error in the calibration of the sensor which was done assuming that the measured B z profile was produced by an infinitely thin wire. In addition to that, the measured field amplitude decreases slightly over the 2 mm scan region which is most likely due to a tilt between the sample surface and the measurement plane on the order of 2 mrad.
V. CONCLUSION
We have demonstrated a scanning magnetic microscopy technique for characterizing atom chips. The microscope is based on a commercially available magnetoresistive probe. It has been used to scan the corrugation of the magnetic field produced by a permanent magnet atom chip as well as to investigate the field produced by a sculptured current-carrying wire.
The spatial resolution of the device is in principle limited by the size of the active area of the probe, i.e., about 5 µm for the sensor used in this work; however submicron resolution has been demonstrated in similar applications [22]. For our demonstration the smallest measurable feature sizes were determined by the minimum distance to the surface (≥ 10 µm) and the scaling laws for magnetic fields. The scan range is limited only by the computer controlled translation stages which can easily be extended to several centimeters.
By simple low pass filtering of the output signal and averaging we were able to achieve a sensitivity of 0.5 µT when measuring a permanent magnetic film. Using AC modulation techniques we could reduce this down to 0.1 µT in case of stationary magnetic fields and even 50 nT for current carrying wires.
In conclusion, the high field sensitivity, large scan range, ease of use and low cost makes the magnetoresistance microscope the quintessential tool for ex-situ characterization of cold atom magnetic microtraps.
FIG. 1 :
1(color online) Schematic of the scanning magnetoresistance microscope.
FIG. 2 :
2Magnetic field profiles at various distances above the magnetic film edge of a permanent magnetic atom chip measured with the magnetoresistance microscope sensitive to the Bz field component (solid lines). The dotted lines correspond to measurements of the magnetic field at approximately the same distance using an ultracold atom cloud sensitive to the By field component. The profiles have been offset for clarity. The relative longitudinal offset between the two measurements is initially unknown and is adjusted for optimum agreement.edge of the film. This provided an accurate measurement of the corrugation of the longitudinal component of the magnetic field produced by the permanent magnetic atom chip, i.e. the component parallel to the film edge.
FIG. 3 :
3Behavior of the magnetic field roughness ∆Brms above the film edge measured using the magnetoresistance microscope (filled circles) and rf spectroscopy of ultracold atoms (open circles), as a function of distance from the film surface.
FIG. 4 :
4Optical microscope image of the current-carrying wire atom chip. The two sculptured wires are formed by cutting three 3 µm wide insulating channels, visible as black lines, into a 150 nm thick Au layer using fs laser ablation.
online) a-c: measured out-of-plane component Bz and reconstructed in-plane components Bx, By of the magnetic field above the current-carrying wire atom chip. df: corresponding results of the numerical simulation of the current distribution and the associated magnetic field, based on the geometric dimensions of the wire structure.
FIG. 6 :
6Line profile of the magnetic field component parallel to the wire (By) at x = 0, i.e. directly above the wire. The solid line represents the field data reconstructed from the MR measurement while the dotted line shows the simulated values.
AcknowledgmentsThe authors would like to thank J. Wang for the deposition of the films. This project is supported by the ARC Centre of Excellence for Quantum-Atom Optics.
. R Folman, P Krüger, J Schmiedmayer, J Denschlag, C Henkel, Adv. At. Mol. Opt. Phys. 48263R. Folman, P. Krüger, J. Schmiedmayer, J. Denschlag, and C. Henkel, Adv. At. Mol. Opt. Phys. 48, 263 (2002).
. J Fortágh, C Zimmermann, Rev. Mod. Phys. 79235J. Fortágh and C. Zimmermann, Rev. Mod. Phys. 79, 235 (2007).
. A Günther, M Kemmler, S Kraft, C J Vale, C Zimmermann, J Fortágh, Phys. Rev. A. 7163619A. Günther, M. Kemmler, S. Kraft, C. J. Vale, C. Zim- mermann, and J. Fortágh, Phys. Rev. A 71, 063619 (2005).
. T Schumm, S Hofferberth, L M Andersson, S Wildermuth, S Groth, I Bar-Joseph, J Schmiedmayer, P Krüger, Nature. 157T. Schumm, S. Hofferberth, L. M. Andersson, S. Wil- dermuth, S. Groth, I. Bar-Joseph, J. Schmiedmayer, and P. Krüger, Nature 1, 57 (2005).
. G.-B Jo, Y Shin, S Will, T A Pasquini, M Saba, W Ketterle, D E Pritchard, Phys. Rev. Lett. 9830407G.-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba, W. Ketterle, and D. E. Pritchard, Phys. Rev. Lett. 98, 030407 (2007).
. B V Hall, S Whitlock, R Anderson, P Hannaford, A Sidorov, Phys. Rev. Lett. 9830402B. V. Hall, S. Whitlock, R. Anderson, P. Hannaford, and A. Sidorov, Phys. Rev. Lett. 98, 030402 (2007).
. D W Wang, M D Lukin, E Demler, Phys. Rev. Lett. 9276802D. W. Wang, M. D. Lukin, and E. Demler, Phys. Rev. Lett. 92, 076802 (2004).
. J Estève, C Aussibal, T Schumm, C Figl, D Mailly, I Bouchoule, C I Westbrook, A Aspect, Phys. Rev. A. 7043629J. Estève, C. Aussibal, T. Schumm, C. Figl, D. Mailly, I. Bouchoule, C. I. Westbrook, and A. Aspect, Phys. Rev. A 70, 043629 (2004).
. C D J Sinclair, J A Retter, E A Curtis, B V Hall, I Llorente-Garcia, S Eriksson, B E Sauer, E A Hinds, Eur. Phys. J. D. 35105C. D. J. Sinclair, J. A. Retter, E. A. Curtis, B. V. Hall, I. Llorente-Garcia, S. Eriksson, B. E. Sauer, and E. A. Hinds, Eur. Phys. J. D 35, 105 (2005).
. S Whitlock, B V Hall, T Roach, R Anderson, M Volk, P Hannaford, A I Sidorov, Phys. Rev. A. 7543602S. Whitlock, B. V. Hall, T. Roach, R. Anderson, M. Volk, P. Hannaford, and A. I. Sidorov, Phys. Rev. A 75, 043602 (2007).
. M Boyd, E W Streed, P Medley, G K Campbell, J Mun, W Ketterle, D E Pritchard, cond- mat/0608370M. Boyd, E. W. Streed, P. Medley, G. K. Camp- bell, J. Mun, W. Ketterle, and D. E. Pritchard, cond- mat/0608370 (2006).
. S Wildermuth, S Hofferberth, I Lesanovsky, E Haller, L M Andersson, S Groth, I Bar-Joseph, P Krüger, J Schmiedmayer, Nature. 435440S. Wildermuth, S. Hofferberth, I. Lesanovsky, E. Haller, L. M. Andersson, S. Groth, I. Bar-Joseph, P. Krüger, and J. Schmiedmayer, Nature (London) 435, 440 (2005).
. S Whitlock, unpublishedS. Whitlock et al., unpublished (2007).
. J S Moodera, L R Kinder, T M Wong, R Meservey, Phys. Rev. Lett. 743273J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meser- vey, Phys. Rev. Lett. 74, 3273 (1995).
. D Lacoura, H Jaffre's, F N V Dau, F Petroff, A Vaure's, J Humbert, J. Appl. Phys. 914655D. Lacoura, H. Jaffre's, F. N. V. Dau, F. Petroff, A. Vaure's, and J. Humbert, J. Appl. Phys. 91, 4655 (2002).
K F Anderson, NASA TM-104260. K. F. Anderson, NASA TM-104260 (1992).
. B V Hall, S Whitlock, F Scharnberg, P Hannaford, A Sidorov, J. Phys. B: At. Mol. Opt. Phys. 3927B. V. Hall, S. Whitlock, F. Scharnberg, P. Hannaford, and A. Sidorov, J. Phys. B: At. Mol. Opt. Phys. 39, 27 (2006).
. S Nolte, C Momma, H Jacobs, A Tünnermann, B N Chichkov, B Wellegehausen, H Welling, J. Opt. Soc. Am. B. 142716S. Nolte, C. Momma, H. Jacobs, A. Tünnermann, B. N. Chichkov, B. Wellegehausen, and H. Welling, J. Opt. Soc. Am. B 14, 2716 (1997).
. L D Pietra, S Aigner, C V Hagen, H J Lezec, J Schmiedmayer, J. Phys.: Conf. Ser. 1930L. D. Pietra, S. Aigner, C. v. Hagen, H. J. Lezec, and J. Schmiedmayer, J. Phys.: Conf. Ser. 19, 30 (2005).
. A I Sidorov, B J Dalton, S M Whitlock, F Scharnberg, Phys. Rev. A. 7423612A. I. Sidorov, B. J. Dalton, S. M. Whitlock, and F. Scharnberg, Phys. Rev. A 74, 023612 (2006).
. B J Roth, N G Sepulveda, J P Wilkswo, J. Appl. Phys. 65361B. J. Roth, N. G. Sepulveda, and J. P. Wilkswo, J. Appl. Phys. 65, 361 (1989).
. B D Schrag, G Xiao, Appl. Phys. Lett. 823272B. D. Schrag and G. Xiao, Appl. Phys. Lett. 82, 3272 (2003).
| [] |
[
"XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE Towards the next generation of exergames: Flexible and personalised assessment-based identification of tennis swings",
"XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE Towards the next generation of exergames: Flexible and personalised assessment-based identification of tennis swings"
] | [
"Boris Bačić [email protected] \nSchool of Engineering, Computing & Mathematical Sciences\nAuckland University of Technology Auckland\nNew Zealand\n"
] | [
"School of Engineering, Computing & Mathematical Sciences\nAuckland University of Technology Auckland\nNew Zealand"
] | [] | Current exergaming sensors and inertial systems attached to sports equipment or the human body can provide quantitative information about the movement or impact e.g. with the ball. However, the scope of these technologies is not to qualitatively assess sports technique at a personalised level, similar to a coach during training or replay analysis. The aim of this paper is to demonstrate a novel approach to automate identification of tennis swings executed with erroneous technique without recorded ball impact. The presented spatiotemporal transformations relying on motion gradient vector flow and polynomial regression with RBF classifier, can identify previously unseen erroneous swings (84.5-94.6%). The presented solution is able to learn from a small dataset and capture two subjective swing-technique assessment criteria from a coach. Personalised and flexible assessment criteria required for players of diverse skill levels and various coaching scenarios were demonstrated by assigning different labelling criteria for identifying similar spatiotemporal patterns of tennis swings.Keywords-Feature extraction technique (FET), motion gradient vector flow, Radial Basis Function (RBF), human motion modelling and analysis (HMMA), computational sport science, augmented coaching systems and technology (ACST). | 10.1109/ijcnn.2018.8489602 | [
"https://arxiv.org/pdf/1804.06948v2.pdf"
] | 4,945,981 | 1804.06948 | e351318cf4a15d13357f7323e7694aa6f49bba87 |
XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE Towards the next generation of exergames: Flexible and personalised assessment-based identification of tennis swings
Boris Bačić [email protected]
School of Engineering, Computing & Mathematical Sciences
Auckland University of Technology Auckland
New Zealand
XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE Towards the next generation of exergames: Flexible and personalised assessment-based identification of tennis swings
Current exergaming sensors and inertial systems attached to sports equipment or the human body can provide quantitative information about the movement or impact e.g. with the ball. However, the scope of these technologies is not to qualitatively assess sports technique at a personalised level, similar to a coach during training or replay analysis. The aim of this paper is to demonstrate a novel approach to automate identification of tennis swings executed with erroneous technique without recorded ball impact. The presented spatiotemporal transformations relying on motion gradient vector flow and polynomial regression with RBF classifier, can identify previously unseen erroneous swings (84.5-94.6%). The presented solution is able to learn from a small dataset and capture two subjective swing-technique assessment criteria from a coach. Personalised and flexible assessment criteria required for players of diverse skill levels and various coaching scenarios were demonstrated by assigning different labelling criteria for identifying similar spatiotemporal patterns of tennis swings.Keywords-Feature extraction technique (FET), motion gradient vector flow, Radial Basis Function (RBF), human motion modelling and analysis (HMMA), computational sport science, augmented coaching systems and technology (ACST).
I. INTRODUCTION
Current wearables and sports equipment with inertial sensors can detect specific motion patterns and provide a range of quantitative analyses, but such technology cannot teach endusers how to improve punch, kick or (golf) swing technique [1,2]. For an exergame to emulate a broadcasted TV sport event experience, it would be a desired feature for the participant(s) to see the replay of a good or bad movement with running commentary similar to a broadcaster's expert panel providing subjective opinions, and strategic and coaching advice. For augmented coaching and rehabilitation monitoring system design, it would be a desired feature to capture personalised assessments emulating a physiotherapist assisting and monitoring an athlete's progress with injury/sport-specific exercises before returning to the sport. The systems and technology that could quantify qualitative assessment of human movement would be applicable to several fields such as exergames, technology-mediated coaching practice and team selection, as well as activity, health and rehabilitation monitoring technologies.
To design the next generation of exergames and augmented coaching systems and technology (ACST), a common obstacle is to find a solution for how to distinguish between good or bad movement patterns based on qualitative and subjective criteria as opposed to quantitative criteria relying on measured results of the movement. Inspired to overcome this obstacle, this study provides an investigation for the related questions: (1) Can a machine detect good and bad tennis swing technique? (2) Is it possible for a machine to capture a subjective expert's swing technique assessment from replay using a small training dataset? and (3) Can anonymised replay such as a 3D animated stick figure be used for expert assessment of tennis swings?
Early investigation of the swing plane concept in golf [2] demonstrates that it is possible to quantify common-sense descriptive rules that guide coaching feedback and provide validity using a data-driven AI approach for such rules. Given that there is little work available on the use of AI and specifically computational intelligence for human motion modelling and analysis (HMMA), this multi-disciplinary work aims to contribute to computational sport science and advancements of the next generation of exergames and ACST.
II. BACKGROUND
A. Exergames, Augmented Coaching Systems and eSports
Since the inception of exergaming consoles with Nintendo Wii and later with Microsoft Kinect, it seems that there have been no substantial advancements in this particular genre when compared to the rising popularity of video games included in international eSports tournaments (e.g. www.espn.com/esports/ and http://dailyesports.tv/). While both versions of Kinect sensors provide unobtrusive marker-less depth data acquisition streamed at 30 fps and are considered by the scientific community as an open-source hardware with Windows-based proprietary SDK and open-source software libraries, exergames are not considered as part of the eSports community, which according to a CNN projection will grow in revenue to $1 billion by 2019 [3]. In spite of the health benefits of physical movement, there is no obvious inclusion of exergaming in eSports. The possible reasons for the exclusion of exergaming in eSports are: privacy preservation concerns and a lack of competitive, strategic team play (including sense of belonging and level of emotions), challenges associated with player's movement goals (including reaction times and proprioceptive feel) and perceived 'fairness' of movement assessment.
In the physical world, the ball flight as an outcome of a swing, represents quantitative knowledge of results (KR), which in exergaming is typically not recorded. While the advancements in commercial golf immersive reality applications may be the best candidate for inclusion in eSports, there is still an open question of the 'fairness' of movement assessment for approach shot and judging the putting trajectory for each green. Knowing that in gymnastics, ice skating, and other stylistic-execution sports there is a panel of judges assigned to evaluate performance it seems obvious that qualitative evaluation based on hard-to-define assessment criteria is a significant challenge for AI. One of the first scientific investigations on the effects of augmented coaching feedback on elements of performance was reported in 1976 [4]. From Hatze's research in biomechanics [4], it is possible to generalise that if training for a given motor learning task is based only on KR feedback, the participant's performance will plateau due to his/her natural adaptation. However, it is also possible to further improve the subject's performance by providing qualitative feedback that is based on knowledge of performance (KP) i.e. knowledge about the elements of performance associated with the goal(s) of the movement.
B. Motion Acquisition Technology, Privacy and Data
Ownership Modern inertial systems attached to sports equipment and the human body use proprietary algorithms to report quantitative data. Unfortunately, raising legal concerns [5,6], such systems do not provide end-users with option to own the recorded raw data on expressed motion patterns, which are typically processed on third party's cloud and are not shared with the broader scientific community. In tennis, for example various sensors attached to the racquet's handle (http://en.babolatplay.com, www.smarttennissensor.sony.net, and www.zepp.com), can provide training statistics including the number of particular shots hit, the estimated ball rotation, speed at point of impact and even whether the player has missed the optimal impact zone of the racquet's string bed. Although such information can be used for swing-quality assessment or to produce other similar assessments based on quantitative criteria, mishitting the ball is still possible regardless of a good or bad swing technique and no available systems so far are able to index swings based on swing technique that would be based on qualitative personalised and flexible assessment criteria.
C. Tennis, Coaching, and Sports Analysis: Neural Signal
Processing Perspective Tennis is considered an open-skill sport where opponents influence each other's choices. Stylistic execution of tennis swings is subject to skill-level, game situation (e.g. defending or attacking), and other personal idiosyncrasies. Unlike coaching at an elite-level, when coaching beginners, a coach is expected to recognise, prioritise, produce feedback and recommend intervention for common errors that beginners typically show during their play. Furthermore, in tennis, it is considered as general knowledge that: (1) racquet and string technology have evolved since the times of wooden racquets and (2) advancedlevel players are sensitive about the subtle differences between seemingly identical racquets and about when to swap the racquets during the match.
At an advanced and professional competitive level, tennis players often describe their state of mind as being in the zone and racquet interaction with the ball as feel. The area on the string bed surface that has the best feel is known as the sweet spot and is typically included in a racquet's specification.
"A coach can teach many things, but they cannot teach feel. That is something you must master on your own."
[Nick Bolletieri, tennis coach]
How the sweet spot influences racquet choice and the racquet's feel and what happens in the brain of an experienced player when swinging the racquet through the air outside of the tennis court may be interesting questions for neuroscience, but what is pertinent to computer science is the question of whether we can model and emulate this feel.
III. METHODOLOGY
The 3D motion dataset utilised in this study was recorded in a biomechanics laboratory using eMotion (BTS) SMART-e 900, a nine-camera optoelectronic motion system. The capture volume where the tennis swings were recorded was approximately 3x2x2 m. The utilised minimalistic retroreflective marker set to produce a 3D stick figure ( Fig. 1) is similar to Kinect sensor, but with additional markers attached to a tennis racquet. Compared to MS Kinect™ sensors capturing depth video at 30 Hz, the captured dataset was recorded at 50 Hz with high (sub-millimetre) 3D resolution and was also able to produce additional information about the racquet movement and forearm internal and external rotation as shown in Fig. 1. The tennis swing experimental dataset contains common errors typical for novice to intermediate-level tennis players. The set of forehands were executed as a mix of fast and slow swings from diverse stances (Fig. 2). Due to the variety of forehands, the selected action zones' temporal region of interest (ROI) are not all of the same duration, but last between 7 and 13 frames.
A. Design Decisions, Insights and Rationale of the Study
As a design decision, only captured motion data was used in processing without any synthetic data derived from the experimental dataset. For an expert to consistently select a swing's action zone ROI that is 0.14 -0.26 seconds in duration, a stand-alone 3D stick-figure player [7] was used. The 3D stickfigure player allows pixel-accurate interactive replays using virtual camera 360° view with panning, zooming and other features such as selected A-B sequence replay and variable slow-motion. Accurate and anonymised 3D replay capabilities are considered one of the key tools for visual analysis and human motion modelling and analysis.
The utilised dataset size was considered sufficient for the intended purposes of: (i) feature extraction algorithm development; (ii) model design; (iii) supervised machine learning experiments relying on flexible assessment criteria based on observed common errors and their similarity-based grouping; and (iv) small expert-based training data requirement.
To quantify the internalised phenomenon of 'feel' through the impact and action zone, feature extraction technique (FET) is represented in this work as a single gradient vector flow of the racquet's sweet spot motion. The FET approach is intended to capture in data the previous state of the racquet and the change in direction and velocity.
In the area of machine learning and neural information processing, the use of a gradient function is typically associated with the well-known gradient descent algorithm. Applied to this study, HMMA and computational sport science in general, gradient function allows: (i) visualisation of two-and threedimensional curvatures (as contours and vector fields) pointing out the direction of highest changes in space as the gradient vector field where the vector magnitude is associated with the steepness of the slope at a particular point; (ii) computing displacement of a point or a position of a virtual marker in 3D plane; (iii) representing the directional derivative of a function in the direction of computed unit vector(s); (iv) kinematic motion data processing; (v) linear approximation of a function value at a given point; and (vi) mathematical transformations of temporal and spatial region of interest into spatial feature (or spatial pattern) space that can be further transformed and processed by a machine.
For a function F of three variables (x,y,z) in 3D Cartesian orthogonal space with Euclidean metrics, gradient F (1) is denoted as:
, , = ∇ , , = , ,(1)
Similarly, in biomechanics and computational sport science, it is common to use the notations (2) for kinematic data processing and analysis:
.
(2)
The gradient derivative of a function ÑF (3) in the direction of , , unit vectors relative to x,y,z coordinate is computed as:
∇ = + +(3)
Gradient vector (4) can also be used for changing relative position of a virtual marker on the given plane -as a linear function approximation at the point x 0 :
≈ ( 5 ) + ∇ 7 8 ⋅ ( − 5 )(4)
Where the gradient vector at the specific 3D point p(x,y,z) (5) is noted as:
∇ ;(7,<,=)(5)
Considering the applications of the gradient function for motion data processing, the idea of producing a representation of a tennis swing or other sport-specific movement patterns is based on combining the state or position of the initial point x 0 and its subsequent states (5) with the resulting gradient motion vector field (3).
B. Data Pre-Processing
For modelling and analytical purposes using external software, the captured 3D motion dataset ℝ ? was exported as an ASCII file containing right-handed XYZ marker locations in columns and the samples organised as rows. For visualisation, modelling and analysis in Mathwork's Matlab™, the righthanded XYZ motion dataset was converted into Matlab's default left-handed XZY internal data format, noted as :
ℝ ? → ℝ ? , where B , B , B = B , B , − B .
The stick figure (Fig. 1) is considered as a set M of interconnected markers (n=22). Each marker m i ∈ ℝ ? is comprised of the three (x i ,y i ,z i ) time series (6) sampled at regular time intervals.
Intended ball impact is a subset of swing action zone ROI.
Visualisation and expert labelling ( Fig. 3) show examples of good swings and common errors using overlaid stick figure frames. The racquet path and the estimated sweet spot trajectories are shown as swing volume through action zonethe temporal region of interest. Subjective decision boundaries are depicted with diverse expert assessment decisions that were associated with flexible skill-level criteria of the same swing (Fig. 3b). Furthermore (as in Fig. 3), some bad swings contain multiple issues that a coach would need to prioritise in his/her feedback.
C. Data Transformation and Feature Extraction
Body and racquet movement are represented by 22 markers resulting in 66 time-series data. Problem space and dimensionality reduction was partially guided by an empirical approach and expert insight. Given the large number of forehand variations, including 'good' and 'bad' technique and problem space dimensionality, the reduction of redundant data was based on expert insight. The insight and rationale here is that swing technique is linked to cognitive activity associated with the racquet's feel, which is also linked to proprioception of balance, movement fluidity and timely swing execution as a response to the opponent's activity. The full set of 22 markers of a swing S j were used for expert visual assessment or swing labelling using an animated 3D stick figure. Selecting a subset of markers on empirical basis is considered as feature reduction and it is also aligned with sports technology, where an inertial sensor is attached to sports equipment. The chosen representation of a sensor would represent the racquet's sweet spot movement. While attaching a marker to the racquet's sweet spot (or other sports equipment in general e.g. golf club) would be obtrusive and impractical, the workaround was to compute a virtual marker (Table I -Algorithm 1). To compute the racquet's 'feel' at impact and through the action zone and produce related discriminative feature set for machine learning purpose, the swing motion data are represented as single motion gradient flow of the racquet's sweet spot. To produce a sweet spot virtual marker's data, a minimum of three markers were needed to compute the racquet plane which must be aligned with the racquet's string bed. The locations of the virtual sweet-spot's marker are to be combined with the spatiotemporal pattern of a computed motion gradient vector flow comprising displacement and changes in marker velocity. Information about location, direction and change in displacement magnitude at regular time intervals is visualised as a vector flow, which was transformed and expressed as a spatial pattern at later processing stage. (side and top views). Spatial patterns converted from curvature shapes are provided as input to the Radial Basis Function (RBF) connectionist system for classification purposes.
D. Visualisation of Intermediate Results
The following figures show evidence of computational steps involved in the feature extraction algorithm (Table I - The curved shapes of the sweet spot trajectory through the action zone were transformed using polynomial interpolation (7), where the polynomial parameters become variables or features.
Visual inspection of one of the non-linear marker trajectories of the racquet (Fig. 5) and common knowledge of the racquets' mass (typically over 300 g) suggests that using a second-degree polynomial is the best option for this curve fitting model. The shape of the curve (Fig. 5) shows how the racquet's string bed is producing a forward motion (direction x: left to right) and top-spin rotation (direction z(x): low to high) throughout the impact zone. Top-spin vs. ball speed depends on a player's court position but also on their personal style of play.
The benefits of the employed polynomial curve fitting approach include:
• Temporal to spatial pattern transformation
• Data compression
• High frequency noise filtering (data smoothing), which may be useful for subsequent analysis (e.g. acceleration and up-sampling).
The produced output vectors Y (Table I -Algorithm 1) were linearly normalised between (-0.8 … 0.8) for RBF classifier input. Such single-marker computation is considered relatively fast, computationally inexpensive, simple and sufficient for feature extraction representing the racquet motion through the intended ball impact and swing action zone.
The chosen RBF classifier model is intended for fast classification operation requiring low computational resources. Along with other traditional ANN models (e.g. SVM and MLP), RBF is typically used for benchmarking purposes. Unlike MLP and more recent deep learning ANNs (also requiring larger training datasets), RBF processing architecture that has only one hidden layer. For future advancements, the traditional RBF is still considered a good candidate for modifications that would allow adaptive and evolving operation for incremental learning such as [8,9]. Future advancements are likely to investigate the underlying KNN responsible for multivariate Gaussian parameter settings from data that could be modified with the evolving clustering function (ECF) or other adaptive and evolving classification model alternatives [10].
IV. RESULTS
Capturing tacit expert assessment via supervised learning into a computer model has practical applications for the next generation of inertial sensors, wearables and optoelectronic systems. With a small training dataset, such systems could quantify the number of errors during the recorded training session for many sports disciplines beyond tennis. The key evidence of feature extraction concepts was provided as intermediate results while the performance of the produced RBF model provided the insight into flexible assessment criteria for the motion dataset.
The developed feature extraction technique (Table I -Algorithm 1), for spatiotemporal data transformation into spatial patterns is robust to the varying durations of (visually) selected tennis swings' action zone, and for each swing it produces a vector consisting of 12 variables. Where, for example, the selected swing's ROI duration is: 10 samples x 66-time series = 660.
Algorithm 1 (Table I) Regarding expert labelling and decision boundaries related to whether the observed swing was good or bad for the skill level of the player, the small dataset also reduced the time required to manually produce multiple assessments for novice and advanced players. Given the challenge posed by the small-sized dataset and random initialisation of RBF model, the LOO crossvalidation was repeated 12 times and the mean classification Accuracy (8) was reported in Table III. Given the small dataset, the results (Table III) include sub-optimal RBF solutions to indicate possible overfitting and model performance with s suboptimal number of hidden-layer's processing units.
The classification accuracy for each LOO cross-validation was calculated as:
= 1 − ε o BpI ⋅ 100%(8)
Where: N … the number of input vectors, and ε … is 1 for a misclassified input vector, or 0 otherwise. Table III shows for optimal and sub-optimal RBF modelling solutions. All processing units (artificial neurons) of RBF models used in experiments have a Gaussian activation function, and model training was based on KNN clustering. The results (Table III) show the difference (approx. 10%) in classification accuracy for diverse skill assessment training data on the same motion dataset. Better classification accuracy for the intermediate skill level than for novices, suggests that more follow-up research is needed to investigate potential RBF model 'awareness' on single and compound technique errors found in 'bad' swings compared to more consistent 'good' swings.
V. DISCUSSION
The human body and racquet was modelled as a set of interconnected rigid segments that should be sufficient for visual analysis of human motion without the potential for human bias (e.g. potentially caused by liking/disliking a player, prior knowledge or recent observation of a player, clothing, height, body shape, or gender). As part of our motion perception, the majority of people are able to sense whether a movement is 'natural' or not, demonstrated, for example, by the ability to distinguish between animations created by animation artists only and those created by using motion capture. For such reasons, the experimental design did not include synthetic data or attempts to reconstruct incomplete marker trajectories needed to compute a virtual sweet spot from the captured motion dataset. Further swing reduction from the captured motion data was limited to forehands only. Visual examinations of the captured forehand swings have shown better coverage and variations than the backhand swings (e.g. stance coverage, swing width, top-spin variations and swing durations). In addition, for novice players, forehand is typically easier to learn than the backhand (whose learning may progress at different pace than forehand). The high classification accuracy produced by using a relatively small dataset for modelling purposes provides an advantage for practical applications, such as where a coach would like to automate the tagging of erroneous or good shots for analytical replay purposes or for the next generation of exergames. Furthermore, it would be beneficial to initially use small datasets and later to employ adaptive and incremental learning capabilities that would still rely on occasional human expert labelling.
One of the obstacles for research rigour was stick figure 3D replay for expert visual assessment, consistent selection of action zone ROI for swings of varying durations, and consistent data labelling based on qualitative analysis of human movement and coaching practise. As some swings were harder to assess than others, a standalone 3D player was developed with a proprietary graphics library that provided smooth and pixelaccurate interactive virtual camera movement during the replay [7]. For the research community using Matlab, Octave or similar, the 3D stick figure player code can be implemented using plot() or plot3() functions with fewer lines of code than if implemented in C++, Java or Object Pascal (Delphi or Lazarus). For computational sport science, HMMA and expert labelling video and 3D replay tools are considered essential, since feature space and internal workings of an ANN are typically not comprehensive for human learning or understanding. Extended functionality for augmented coaching using video and 3D stick figure replay was reported in [11].
Stick figure replay and silhouette filtering [12] may be used for coaching, on-line coaching and also encourage participation in on-line exergaming by facilitating privacy preservation, such as that needed for healthcare/elderly-care monitoring systems.
The presented novel approach and ideas were driven to support model design that can operate on initially small to large datasets and for spatiotemporal motion patterns for which there are no statistical ground truth available. Using relatively small training data from the coach, the aim was to solve the 'curse of dimensionality' associated with kinematic 3D data analysis of diverse forehand swings [13]. More and less rigid criteria for swing technique assessments for novices and intermediate skill levels reflect subjective and qualitative nature of coaching practise, where feedback is focused on performance elements rather than on the knowledge of the outcome, that is ball flight.
Single virtual marker computation concepts are transferrable for practical use with for example, smart watch inertial sensors or an inertial sensor attached to the racquet or other sports equipment. Using computer vision (video or depth video), it would be possible to combine or fuse data from diverse sources to enable advancements of HMMA with a high sampling frequency around ROI and video replay for the next generation of ACST and exergames.
VI. CONCLUSION
The next generation of exergames, and augmented coaching/rehabilitation systems and technology are expected to provide analytical capabilities that will help an end user to improve his/her sport-specific technique or (re)gain motor skills. This multidisciplinary paper presents concepts and solutions that viably automate the classification of good or bad movement patterns based on qualitative, flexible and subjective criteria similar to a coach.
The presented artificial neural network-based solution can, in part, mimic a coach, who can immediately tell if the observed swing (or other movement pattern) does not 'feel right' before providing subsequent analysis with descriptive/qualitative feedback to improve element(s) of performance. Relying on 3D stick figure replay of the recorded dataset, it was possible to capture two different subjective assessment criteria. The use of two different assessment criteria reflects expected swing techniques for novices and more advanced skill-level players. The achieved classification of tennis swings (with accuracy of: 84.5% for novices and 94.6% for intermediate-level players) demonstrates a flexible and personalised machine learning solution for designing exergaming and augmented coaching systems and technology.
The presented mathematical transformation concepts involved in the presented feature extraction technique, insights and neural data processing techniques have practical aspects that could be transferred to a number of sport disciplines or rehabilitation scenarios. Potential examples include: (i) keeping track of person-specific sub-standard movements at the end of a race, game or training session -a system could quantify a number of motion patterns executed with poor technique using small initial system-training data from a coach who is familiar with the player's idiosyncrasies; (ii) autonomously tagging irregular spatiotemporal patterns for replay purposes; (iii) recording intellectual property into a machine that could capture an expert's tacit knowledge by combining swing replays (or other sport-specific motion patterns) with output labelling; (iv) healthcare monitoring, to automate finding of irregular signal patterns from logging devices (e.g. a holter cardiac monitoring system and other auscultation expert systems); and (v) an augmented coaching system to supervise, in a controlled environment, a patient's activities by monitoring for erroneous movement patterns that could adversely affect rehabilitation time.
The concepts and technique presented in this paper utilised a motion dataset that was captured at 50 Hz without ball impact information. Considering Microsoft Kinect™ sensor's streaming capabilities (streaming at 30 Hz), recent mobile and sport camera video capabilities (mono and stereo vision at 120 and 240 Hz) and the ability of some inertial sensors to capture motion data close to 1000 Hz, it is likely that this work will be compatible with further motion capture technology advancements. The concepts of the racquet's feel and sweet spot expressed as single virtual marker data processing techniques are generally applicable to neural information processing and utilisation of data fusion from diverse sources (e.g. sport equipment-attached, smartwatch or other wearable sensors with computer vision).
Future work will be focused on advancements of machine learning approaches for technique assessment for augmented coaching systems, wearables, and rehabilitation devices. Another broader avenue to pursue is computational sport science that will also include the implementation of evolving and adaptive systems, deep learning and the third generation of ANN to advance human motion modelling and analysis applicable to human motor learning, skill and technique (re) acquisition, and knowledge-discovery from diverse disciplines datasets.
Fig. 1 .
1Stick figure model produced from a minimalistic retro-reflective marker set attached to the tennis racquet and human body.
Fig. 2 .
2Action zone durations in experimental dataset.
The spatiotemporal patterns are computed from the previous state and movement vector flow of the virtual sweet spot. Temporal patterns of motion trajectories and vector flow within the action zone of a swing were projected in Sagittal and Transverse
Fig. 3 .
3Stick figure overlay for the three examples of forehand swing and qualitative nature of possible coach's feedback. Racquet movement through the action zone can be seen as swing volume covered by the string bed. The sweet spot virtual marker trajectory (in orange) is close to the middle of the racquet's string bed. The swing can be classified as good a) or bad c), or both b) depending on the player' skill level.
Fig. 4 .
4Algorithm 1). Fig. 4 shows a swing with the gradient vector flow originating from the computed virtual sweet spot of the racquet. Gradient vector flow of the racquet's sweet spot with (a) and without (b) player's stick figure visualisation.
Fig. 5 .
5Example of curve fitting model for the racquet's marker trajectory within swing's action zone on the sagittal plane (side view).
output Y is: 3 (parameters/curve fitting) x (sagittal plane + transverse plane) x (sweet spot virtual marker trajectory + top of gradient vectors' curve) = 12.Resulting in: [1-(12/660)] x 100% = 98.2% data reduction.
TABLE I .
IPSEUDO CODE FOR FEATURE EXTRACTION TECHNIQUE REPRESENTING A SINGLE 3D MARKER MOTION PATTERN OF A TENNIS SWINGAlgorithm 1 Motion Gradient Vector Flow of the Projected Racquet's Sweet SpotRequire: Swing's ROI S j : Note: a. For simplicity, the virtual sweet-spot marker was calculated as equidistant from the racquet's markers F HI ,∩ F HI ,
JJJJJJJJ⃗ HL ,
JJJJJJJJ⃗ M
JJJJJJ⃗ N = ∅
Ensure: : Q →
1:
{* Compute racquet's 3D sweet spot marker *} a
SS
JJJJJJJ⃗ ←
( HI ,
JJJJJJJJ⃗ HL ,
JJJJJJJJ⃗ M
JJJJJJ⃗ )
2:
{* Compute vector array of the sweet spot's movements *} b
_ aa ,
JJJJJJJJJJ⃗ aa
JJJJJJJJ⃗ ,
aa
JJJJJJJJJJ⃗ c ←
( SS
JJJJJJJ⃗ )
3:
{* Compute vector array tips as virtual marker *}
_
aa ,
JJJJJJJJJJJJJ⃗
aa
JJJJJJJJJJJJ⃗ ,
aa
JJJJJJJJJJJJJ⃗ c ←
( aa ,
JJJJJJJJJJ⃗ aa
JJJJJJJJ⃗ ,
aa
JJJJJJJJJJ⃗ , SS
JJJJJJJ⃗ )
4:
{* Convert spatiotemporal vectors flow into temporal patterns *} c
Sagittal_Plane ← _
g
aa ,
JJJJJJJJJJJJJ⃗
aa
JJJJJJJJJJJJJ⃗ h,
g aa
JJJJJJJJJ⃗ , aa
JJJJJJJJJ⃗ hc
5: Transverse_Plane ← _
g
,
JJJJJJJJJJJJ⃗ JJJJJJJJJJJJ⃗ h,
gJJJJJJJJ⃗, JJJJJJJJ⃗ hc
6: Y¬ [Sagittal_Plane, Transverse_Plane]
7: return Y
JJJJJJJJ⃗ HL ,
JJJJJJJJ⃗ M
JJJJJJ⃗ N.
b. Gradient vector at the specific 3D point: (∇ ) j kk =
( aa ).
c. Second-degree polynomial curve fitting: [p2, p1, p0] = polyFit().
Table II shows
IIthe achieved efficiency of the problem space dimensionality reduction by Motion Gradient Vector Flow (Table I -Algorithm 1).
TABLE II .
IISUMMARY OF ALGORITHM 1 INPUT SPACE DIMENSIONALITY
REDUCTION
Swing's ROI
duration
Input
Dimensionality
Output
Dimensionality
Space reduction
13
858
12
98.6%
10
660
12
98.2%
7
462
12
97.4%
TABLE III .
IIIFLEXIBLE SKILL-LEVEL ASSESSMENT CRITERIA AND CLASSIFICATION RESULTS USING RBF CLASSIFIER MODEL Reported errors -where the RBF model training may not converge towards the intended solution.Repeated Leave-one-out Cross-validations for Two Assessment
Criteria (Novice and Intermediate) using RBF Classifier Model
Number of RBF training epochs: 100
Number of input vectors: 14
Number of repeated LOO cross-validations: 12
Observation
Number of RBF
Hidden -Layer
Processing Units
Novice Skill
Assessment
Bad swings
portion: 28.6 %
Intermediate
Skill Assessment
Bad swings
portion: 71.4 %
Sub-optimal
solution
2
71.4%
85.7%
3
83.3%
91.7%
Optimal
4
84.5%
94.6% a
Potential
overfitting
5
85.1%
93.4%
6+
N/A b
N/A b
a. Intended RBF model would converge with minimal epochs (5-12) out of 100 training epochs limit.
b.
ACKNOWLEDGEMENTSThe author wishes to thank Prof. Ian Nabney and Dr. Christopher Bishop for sharing and updating their source code for Netlab and RBF for the past two decades. This multidisciplinary work was also supported by New Zealand tennis coaches Shelley Bryce (née Stephens) and Kevin Woolcott who took their time and efforts by participating in 3D replay assessment of the tennis dataset utilised in this study.The dataset has been recorded in Peharec Polyclinic (Pula, Croatia) with help of MSc. Petar Bačić, who provided valuable multidisciplinary assistance with motion capture, biomechanics support and tennis coaching critiques during data capture. Free laboratory access and resource-sharing was granted by Dr. Stanislav Peharec.
Silicon gets sporty. K Lightman, IEEE Spectr. 55K. Lightman, "Silicon gets sporty," IEEE Spectr., vol. 55, pp. 48-53, 2016.
Predicting golf ball trajectories from swing plane: An artificial neural networks approach. B Bačić, Expert. Syst. Appl. 65B. Bačić, "Predicting golf ball trajectories from swing plane: An artificial neural networks approach," Expert. Syst. Appl., vol. 65, pp. 423-438, 2016.
ESports: Global revenue expected to smash $1 billion by. D Riddell, 27D. Riddell. (2016, 27 Jan.). ESports: Global revenue expected to smash $1 billion by 2019. Available: https://edition.cnn.com/2016/05/29/sport/esports-revolution-revenue- audience-growth/index.html
Biomechanical aspects of a successful motion optimisation. H Hatze, Biomechanics V-B, BaltimoreH. Hatze, "Biomechanical aspects of a successful motion optimisation " in Biomechanics V-B, Baltimore, 1976.
Game-changing wearable devices that collect athlete data raise data ownership issues. B Socolow, World Sports Advocate. 15B. Socolow, "Game-changing wearable devices that collect athlete data raise data ownership issues," World Sports Advocate, vol. 15, 2017.
Wearables technology data use in professional sports. B Socolow, World Sports Law Report. 14B. Socolow, "Wearables technology data use in professional sports," World Sports Law Report, vol. 14, pp. 12-15, 2016.
Prototyping and user interface design for augmented coaching systems with MATLAB and Delphi: Implementation of personal tennis coaching system. B Bačić, Auckland, New ZealandB. Bačić, "Prototyping and user interface design for augmented coaching systems with MATLAB and Delphi: Implementation of personal tennis coaching system," presented at the MATLAB Conference 2015, Auckland, New Zealand, 2015.
An evolving radial basis neural network with adaptive learning of its parameters and architecture. Y V Bodyanskiy, A K Tyshchenko, A A Deineko, Aut. Control Comp. Sci. 49Y. V. Bodyanskiy, A. K. Tyshchenko, and A. A. Deineko, "An evolving radial basis neural network with adaptive learning of its parameters and architecture," Aut. Control Comp. Sci., vol. 49, pp. 255-260, 4 Nov. 2015.
Adaptive training of radial basis function networks based on cooperative evolution and evolutionary programming. E P Topchy, O A Lebedko, V V Miagkikh, N K Kasabov, Progress in Connectionist-Based Information Systems. SpringerE. P. Topchy, O. A. Lebedko, V. V. Miagkikh, and N. K. Kasabov, "Adaptive training of radial basis function networks based on cooperative evolution and evolutionary programming," in Progress in Connectionist- Based Information Systems, Springer, 1998, pp. 253-258.
Evolving connectionist systems: Methods and applications in bioinformatics, brain study and intelligent machines. N Kasabov, Springer VerlagLondonN. Kasabov, Evolving connectionist systems: Methods and applications in bioinformatics, brain study and intelligent machines. London: Springer Verlag, 2002.
Open-source video players for coaches and sport scientists. B Bačić, XXXIII International Symposium on Biomechanics in Sports. Poitiers, FranceB. Bačić, "Open-source video players for coaches and sport scientists," in XXXIII International Symposium on Biomechanics in Sports, Poitiers, France, 2015, pp. 515-518.
Privacy preservation for eSports: A case study towards augmented video golf coaching system. B Bačić, Q Meng, K Y Chan, Tenth International Conference on Developments in e-Systems Engineering -DeSE. Paris, FranceB. Bačić, Q. Meng, and K. Y. Chan, "Privacy preservation for eSports: A case study towards augmented video golf coaching system," in Tenth International Conference on Developments in e-Systems Engineering - DeSE 2017, Paris, France, 2017, pp. 169-174.
Forehands. M Crespo, J Higueras, World-Class Tennis Technique, P. Roetert and J. GroppelHuman KineticsChampaign, IL, USM. Crespo and J. Higueras, "Forehands," in World-Class Tennis Technique, P. Roetert and J. Groppel, Eds., Champaign, IL, US: Human Kinetics, 2001, pp. 147-171.
| [] |
[
"Observation of quantum spin Hall states in Ta",
"Observation of quantum spin Hall states in Ta",
"Observation of quantum spin Hall states in Ta",
"Observation of quantum spin Hall states in Ta"
] | [
"Pd ",
"Te \nHiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan\n\nDepartment of Materials Science and Engineering\nStanford University\n94305StanfordCaliforniaUSA\n\nCenter of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Xuguang Wang \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Daiyu Geng \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Dayu Yan \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Wenqi Hu \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Hexu Zhang \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Shaosheng Yue \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Zhenyu Sun \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Shiv Kumar \nHiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan\n",
"Kenya Shimada \nHiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan\n",
"Peng Cheng \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Lan Chen \nSongshan Lake Materials Laboratory\n523808DongguanGuangdongChina\n",
"Simin Nie \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nDepartment of Materials Science and Engineering\nStanford University\n94305StanfordCaliforniaUSA\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Zhijun Wang \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Youguo Shi \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nCenter of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nSongshan Lake Materials Laboratory\n523808DongguanGuangdongChina\n",
"Yi-Qi Zhang \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Kehui Wu \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nSongshan Lake Materials Laboratory\n523808DongguanGuangdongChina\n",
"Baojie Feng \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Pd ",
"Te \nHiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan\n\nDepartment of Materials Science and Engineering\nStanford University\n94305StanfordCaliforniaUSA\n\nCenter of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Xuguang Wang \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Daiyu Geng \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Dayu Yan \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Wenqi Hu \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Hexu Zhang \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Shaosheng Yue \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Zhenyu Sun \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Shiv Kumar \nHiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan\n",
"Kenya Shimada \nHiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan\n",
"Peng Cheng \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Lan Chen \nSongshan Lake Materials Laboratory\n523808DongguanGuangdongChina\n",
"Simin Nie \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nDepartment of Materials Science and Engineering\nStanford University\n94305StanfordCaliforniaUSA\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Zhijun Wang \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Youguo Shi \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nCenter of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nSongshan Lake Materials Laboratory\n523808DongguanGuangdongChina\n",
"Yi-Qi Zhang \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Kehui Wu \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n\nSongshan Lake Materials Laboratory\n523808DongguanGuangdongChina\n",
"Baojie Feng \nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nSchool of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n"
] | [
"Hiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan",
"Department of Materials Science and Engineering\nStanford University\n94305StanfordCaliforniaUSA",
"Center of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Hiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan",
"Hiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Songshan Lake Materials Laboratory\n523808DongguanGuangdongChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"Department of Materials Science and Engineering\nStanford University\n94305StanfordCaliforniaUSA",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Center of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Songshan Lake Materials Laboratory\n523808DongguanGuangdongChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Songshan Lake Materials Laboratory\n523808DongguanGuangdongChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Hiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan",
"Department of Materials Science and Engineering\nStanford University\n94305StanfordCaliforniaUSA",
"Center of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Hiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan",
"Hiroshima Synchrotron Radiation Center\nHiroshima University\nHigashi-Hiroshima2-313, 739-0046KagamiyamaJapan",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Songshan Lake Materials Laboratory\n523808DongguanGuangdongChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"Department of Materials Science and Engineering\nStanford University\n94305StanfordCaliforniaUSA",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Center of Materials Science and Optoelectronics Engineering\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Songshan Lake Materials Laboratory\n523808DongguanGuangdongChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Songshan Lake Materials Laboratory\n523808DongguanGuangdongChina",
"Institute of Physics\nChinese Academy of Sciences\n100190BeijingChina",
"School of Physical Sciences\nUniversity of Chinese Academy of Sciences\n100049BeijingChina"
] | [] | Two-dimensional topological insulators (2DTIs), which host the quantum spin Hall (QSH) effect, are one of the key materials in next-generation spintronic devices. To date, experimental evidence of the QSH effect has only been observed in a few materials, and thus, the search for new 2DTIs is at the forefront of physical and materials science. Here, we report experimental evidence of a 2DTI in the van der Waals material Ta2Pd3Te5. First-principles calculations show that each monolayer of Ta2Pd3Te5 is a 2DTI with weak interlayer interactions. Combined transport, angleresolved photoemission spectroscopy, and scanning tunneling microscopy measurements confirm the existence of a band gap at the Fermi level and topological edge states inside the gap. These results demonstrate that Ta2Pd3Te5 is a promising material for fabricating spintronic devices based on the QSH effect.Two-dimensional topological insulators (2DTIs), also known as quantum spin Hall (QSH) insulators, feature a bulk band gap and helical in-gap states at the material boundaries [1-3]. The edge states of a 2DTI can serve as one-dimensional conducting channels in which backscattering is forbidden by time-reversal symmetry. Therefore, 2DTIs provide an ideal platform to fabricate low-dissipation spintronic devices. To date, 2DTIs have only been realized in two types of materials. The first type is quantum well systems, including HgTe/CdHgTe[4]and InAs/GaSb[5][6][7].However, the material synthesis of these quantum wells is extremely challenging, and experimental signatures of the QSH effect have only been observed by a few research groups. The second type of 2DTI has been reported in several two-dimensional (2D) materials, such as bilayer Bi[8,9], monolayer 1T ′ WTe 2 [10-12], and graphene-like honeycomb lattices[13][14][15][16][17]. However, despite the large number of 2D materials as candidate 2DTIs, transport evidence of the QSH effect has only been reported in monolayer 1T ′ WTe 2 [18-20].Realizing QSH states in van der Waals materials, such as WTe 2 , offers great opportunities to fabricate quantum transport devices, as monolayer or few-layer materials for realizing the QSH effect are very easy to obtain. However, as a prototypical van der Waals material, 1T ′ WTe 2 was confirmed to be a 2DTI only in the monolayer limit[19]. In bulk form, WTe 2 becomes a metal with zero energy gap[21,22]. As a result, QSH states disappear in multilayer WTe 2 because of the hybridization of the bulk and edge states. Therefore, QSH devices based on WTe 2 suffer from easy degradation of monolayer samples under ambient conditions. Since multilayer materials are typically more inert and have higher tunability via the thickness or twist angles, realizing QSH states in multilayer materials is highly desirable. This requires a semiconducting van der Waals material that hosts a similar inverted gap as the monolayer.In this work, we report the observation of QSH states in the van der Waals material Ta 2 Pd 3 Te 5 [23], which hosts a band gap in both the bulk and monolayer forms.We synthesize Ta 2 Pd 3 Te 5 single crystals and investigate their electronic structures by combined first-principles calculations, transport, angle-resolved photoemission spectroscopy (ARPES), and scanning tunneling microscopy/spectroscopy (STM/STS) measurements. We prove that Ta 2 Pd 3 Te 5 hosts a band gap at the Fermi level. Because of the weak interlayer coupling, the topmost layer of Ta 2 Pd 3 Te 5 can be viewed as a monolayer material placed on a singlecrystal substrate. As expected, we directly observe topological edge states using STS. These results provide strong evidence for QSH states in Ta 2 Pd 3 Te 5 . The discovery of QSH states in van der Waals materials with a significant band gap could pave the way to realizing practical QSH devices.Ta 2 Pd 3 Te 5 single crystals were synthesized by the selfflux method. The starting materials of Ta (99.999%), Pd (99.9999%), and Te (99.9999%) were mixed in an Arfilled glove box at a molar ratio of Ta:Pd:Te=2:4.5:7.5. The mixture was placed in an alumina crucible and | 10.1103/physrevb.104.l241408 | [
"https://arxiv.org/pdf/2012.07293v1.pdf"
] | 229,156,054 | 2012.07293 | 19749e86f0d93cd20d2680f9cbf5fb1bfb76ef37 |
Observation of quantum spin Hall states in Ta
14 Dec 2020
Pd
Te
Hiroshima Synchrotron Radiation Center
Hiroshima University
Higashi-Hiroshima2-313, 739-0046KagamiyamaJapan
Department of Materials Science and Engineering
Stanford University
94305StanfordCaliforniaUSA
Center of Materials Science and Optoelectronics Engineering
University of Chinese Academy of Sciences
100049BeijingChina
Xuguang Wang
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Daiyu Geng
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Dayu Yan
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Wenqi Hu
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Hexu Zhang
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Shaosheng Yue
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Zhenyu Sun
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Shiv Kumar
Hiroshima Synchrotron Radiation Center
Hiroshima University
Higashi-Hiroshima2-313, 739-0046KagamiyamaJapan
Kenya Shimada
Hiroshima Synchrotron Radiation Center
Hiroshima University
Higashi-Hiroshima2-313, 739-0046KagamiyamaJapan
Peng Cheng
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Lan Chen
Songshan Lake Materials Laboratory
523808DongguanGuangdongChina
Simin Nie
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
Department of Materials Science and Engineering
Stanford University
94305StanfordCaliforniaUSA
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Zhijun Wang
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Youguo Shi
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Center of Materials Science and Optoelectronics Engineering
University of Chinese Academy of Sciences
100049BeijingChina
Songshan Lake Materials Laboratory
523808DongguanGuangdongChina
Yi-Qi Zhang
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Kehui Wu
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Songshan Lake Materials Laboratory
523808DongguanGuangdongChina
Baojie Feng
Institute of Physics
Chinese Academy of Sciences
100190BeijingChina
School of Physical Sciences
University of Chinese Academy of Sciences
100049BeijingChina
Observation of quantum spin Hall states in Ta
14 Dec 2020(Dated: December 15, 2020)7 These authors contributed equally to this work.
Two-dimensional topological insulators (2DTIs), which host the quantum spin Hall (QSH) effect, are one of the key materials in next-generation spintronic devices. To date, experimental evidence of the QSH effect has only been observed in a few materials, and thus, the search for new 2DTIs is at the forefront of physical and materials science. Here, we report experimental evidence of a 2DTI in the van der Waals material Ta2Pd3Te5. First-principles calculations show that each monolayer of Ta2Pd3Te5 is a 2DTI with weak interlayer interactions. Combined transport, angleresolved photoemission spectroscopy, and scanning tunneling microscopy measurements confirm the existence of a band gap at the Fermi level and topological edge states inside the gap. These results demonstrate that Ta2Pd3Te5 is a promising material for fabricating spintronic devices based on the QSH effect.Two-dimensional topological insulators (2DTIs), also known as quantum spin Hall (QSH) insulators, feature a bulk band gap and helical in-gap states at the material boundaries [1-3]. The edge states of a 2DTI can serve as one-dimensional conducting channels in which backscattering is forbidden by time-reversal symmetry. Therefore, 2DTIs provide an ideal platform to fabricate low-dissipation spintronic devices. To date, 2DTIs have only been realized in two types of materials. The first type is quantum well systems, including HgTe/CdHgTe[4]and InAs/GaSb[5][6][7].However, the material synthesis of these quantum wells is extremely challenging, and experimental signatures of the QSH effect have only been observed by a few research groups. The second type of 2DTI has been reported in several two-dimensional (2D) materials, such as bilayer Bi[8,9], monolayer 1T ′ WTe 2 [10-12], and graphene-like honeycomb lattices[13][14][15][16][17]. However, despite the large number of 2D materials as candidate 2DTIs, transport evidence of the QSH effect has only been reported in monolayer 1T ′ WTe 2 [18-20].Realizing QSH states in van der Waals materials, such as WTe 2 , offers great opportunities to fabricate quantum transport devices, as monolayer or few-layer materials for realizing the QSH effect are very easy to obtain. However, as a prototypical van der Waals material, 1T ′ WTe 2 was confirmed to be a 2DTI only in the monolayer limit[19]. In bulk form, WTe 2 becomes a metal with zero energy gap[21,22]. As a result, QSH states disappear in multilayer WTe 2 because of the hybridization of the bulk and edge states. Therefore, QSH devices based on WTe 2 suffer from easy degradation of monolayer samples under ambient conditions. Since multilayer materials are typically more inert and have higher tunability via the thickness or twist angles, realizing QSH states in multilayer materials is highly desirable. This requires a semiconducting van der Waals material that hosts a similar inverted gap as the monolayer.In this work, we report the observation of QSH states in the van der Waals material Ta 2 Pd 3 Te 5 [23], which hosts a band gap in both the bulk and monolayer forms.We synthesize Ta 2 Pd 3 Te 5 single crystals and investigate their electronic structures by combined first-principles calculations, transport, angle-resolved photoemission spectroscopy (ARPES), and scanning tunneling microscopy/spectroscopy (STM/STS) measurements. We prove that Ta 2 Pd 3 Te 5 hosts a band gap at the Fermi level. Because of the weak interlayer coupling, the topmost layer of Ta 2 Pd 3 Te 5 can be viewed as a monolayer material placed on a singlecrystal substrate. As expected, we directly observe topological edge states using STS. These results provide strong evidence for QSH states in Ta 2 Pd 3 Te 5 . The discovery of QSH states in van der Waals materials with a significant band gap could pave the way to realizing practical QSH devices.Ta 2 Pd 3 Te 5 single crystals were synthesized by the selfflux method. The starting materials of Ta (99.999%), Pd (99.9999%), and Te (99.9999%) were mixed in an Arfilled glove box at a molar ratio of Ta:Pd:Te=2:4.5:7.5. The mixture was placed in an alumina crucible and
Two-dimensional topological insulators (2DTIs), which host the quantum spin Hall (QSH) effect, are one of the key materials in next-generation spintronic devices. To date, experimental evidence of the QSH effect has only been observed in a few materials, and thus, the search for new 2DTIs is at the forefront of physical and materials science. Here, we report experimental evidence of a 2DTI in the van der Waals material Ta2Pd3Te5. First-principles calculations show that each monolayer of Ta2Pd3Te5 is a 2DTI with weak interlayer interactions. Combined transport, angleresolved photoemission spectroscopy, and scanning tunneling microscopy measurements confirm the existence of a band gap at the Fermi level and topological edge states inside the gap. These results demonstrate that Ta2Pd3Te5 is a promising material for fabricating spintronic devices based on the QSH effect.
Two-dimensional topological insulators (2DTIs), also known as quantum spin Hall (QSH) insulators, feature a bulk band gap and helical in-gap states at the material boundaries [1][2][3]. The edge states of a 2DTI can serve as one-dimensional conducting channels in which backscattering is forbidden by time-reversal symmetry. Therefore, 2DTIs provide an ideal platform to fabricate low-dissipation spintronic devices. To date, 2DTIs have only been realized in two types of materials. The first type is quantum well systems, including HgTe/CdHgTe [4] and InAs/GaSb [5][6][7].
However, the material synthesis of these quantum wells is extremely challenging, and experimental signatures of the QSH effect have only been observed by a few research groups. The second type of 2DTI has been reported in several two-dimensional (2D) materials, such as bilayer Bi [8,9], monolayer 1T ′ WTe 2 [10][11][12], and graphene-like honeycomb lattices [13][14][15][16][17]. However, despite the large number of 2D materials as candidate 2DTIs, transport evidence of the QSH effect has only been reported in monolayer 1T ′ WTe 2 [18][19][20].
Realizing QSH states in van der Waals materials, such as WTe 2 , offers great opportunities to fabricate quantum transport devices, as monolayer or few-layer materials for realizing the QSH effect are very easy to obtain. However, as a prototypical van der Waals material, 1T ′ WTe 2 was confirmed to be a 2DTI only in the monolayer limit [19]. In bulk form, WTe 2 becomes a metal with zero energy gap [21,22]. As a result, QSH states disappear in multilayer WTe 2 because of the hybridization of the bulk and edge states. Therefore, QSH devices based on WTe 2 suffer from easy degradation of monolayer samples under ambient conditions. Since multilayer materials are typically more inert and have higher tunability via the thickness or twist angles, realizing QSH states in multilayer materials is highly desirable. This requires a semiconducting van der Waals material that hosts a similar inverted gap as the monolayer.
In this work, we report the observation of QSH states in the van der Waals material Ta 2 Pd 3 Te 5 [23], which hosts a band gap in both the bulk and monolayer forms.
We synthesize Ta 2 Pd 3 Te 5 single crystals and investigate their electronic structures by combined first-principles calculations, transport, angle-resolved photoemission spectroscopy (ARPES), and scanning tunneling microscopy/spectroscopy (STM/STS) measurements. We prove that Ta 2 Pd 3 Te 5 hosts a band gap at the Fermi level. Because of the weak interlayer coupling, the topmost layer of Ta 2 Pd 3 Te 5 can be viewed as a monolayer material placed on a singlecrystal substrate. As expected, we directly observe topological edge states using STS. These results provide strong evidence for QSH states in Ta 2 Pd 3 Te 5 . The discovery of QSH states in van der Waals materials with a significant band gap could pave the way to realizing practical QSH devices. Ta 2 Pd 3 Te 5 single crystals were synthesized by the selfflux method. The starting materials of Ta (99.999%), Pd (99.9999%), and Te (99.9999%) were mixed in an Arfilled glove box at a molar ratio of Ta:Pd:Te=2:4.5:7.5. The mixture was placed in an alumina crucible and sealed in an evacuated quartz tube. The tube was heated to 950 • C over 10 h and maintained at this temperature for 2 days. Then, the tube was slowly cooled to 800 • C at a rate of 0.5 • C/h. Finally, the extra flux was removed by centrifugation at 800 • C. After centrifugation, single crystals of Ta 2 Pd 3 Te 5 could be selected from the remnants in the crucible. To investigate the crystalline structure, single-crystal X-ray diffraction (XRD) was carried out at 273 K using Mo Kα radiation (λ = 0.71073Å). The crystalline structure was refined by the full-matrix least-squares method on F 2 by using the SHELXL-2018/3 program. Electrical resistivity (ρ) measurements were carried out on a physical property measurement system (PPMS, Quantum Design Inc.) using a standard dc four-probe technique.
ARPES experiments were performed at beamline BL-1 of the Hiroshima synchrotron radiation center [26]. The clean surfaces required for the ARPES measurements were obtained by cleaving the samples in an ultrahigh vacuum chamber with a base pressure of 1.0×10 −9 Pa. Both the cleavage process and ARPES measurements were performed at 30 K. The energy resolution of the ARPES measurements was approximately 15 meV. STM/STS experiments were carried out in a homebuilt low-temperature (∼5 K) STM system with a base pressure of 2×10 −8 Pa. The clean surfaces for STM/STS measurements were also obtained by cleaving the samples in situ at low temperature.
First-principles calculations were performed within the framework of the projector augmented wave (PAW) method [27,28] implemented in the Vienna ab initio simulation (VASP) package [29,30]. The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) exchange-correlation functional [31] was implemented in the calculations. The cutoff energy for plane wave expansion was 500 eV. Spinorbit coupling was self-consistently taken into account within the second variational method. A 4-unit-cell slab structure (with 20Å vacuum) was built to simulate the surface spectrum. Ta 2 Pd 3 Te 5 crystallizes in an orthorhombic structure with the space group Pnma (No. 62). Schematic drawings of the atomic structure and Brillouin zones (BZs) of Ta 2 Pd 3 Te 5 are shown in Figs. 1(a) and 1(b), respectively. Each unit cell contains two Ta 2 Pd 3 Te 5 monolayers, which are stacked along the a direction via weak van der Waals interactions. Each monolayer contains a Ta-Pd mixed layer sandwiched between two Te layers. Figure 1(c) shows the XRD spectrum on a flat surface of Ta 2 Pd 3 Te 5 , whereby only (h00) peaks are observed. A photograph of a typical Ta 2 Pd 3 Te 5 crystal is displayed in the inset of Fig. 1(c). The picture shows that the crystal is as large as 1 mm and has shiny surfaces, indicating the high crystallinity of our samples. The lattice parameters of Ta 2 Pd 3 Te 5 determined from the XRD data are a = 13.9531(6) A, b = 3.7038(2)Å, and c = 18.5991(8)Å. Figure 1(d) shows a large-scale STM image of Ta 2 Pd 3 Te 5 (the bc plane). The surface is slightly corrugated, forming periodic stripes along the b direction. A zoomed-in STM image with atomic resolution is displayed in Fig. 1(e). Each bright protrusion corresponds to a Te atom, which well matches the structure model of Ta 2 Pd 3 Te 5 . From the STM image, the rectangular structure of the bc plane of Ta 2 Pd 3 Te 5 can also be identified in the fast Fourier transformed image in Fig. 1(e). The lattice constants determined from our STM results are ∼3.6 A and ∼18.7Å, respectively, which agree well with the lattice constants along the b and c directions.
The temperature dependence of the electrical resistivity of Ta 2 Pd 3 Te 5 is displayed in Fig. 1(g). When the temperature is decreased from 300 K to 2 K, the resistivity increases monotonically, indicating semiconductor behavior. The temperature-dependent resistivity can be fitted with the Arrhenius model ρ ∼ exp(ǫ act /k B T ), where k B and ǫ act are the Boltzmann constant and thermal activation energy, respectively. The fitting results are shown by the red line in Fig. 1(g). The fitted ǫ act is approximately 14 meV. Therefore, bulk Ta 2 Pd 3 Te 5 is a narrow-gap semiconductor with a global band gap of ∼14 meV.
Before showing further experimental results of Ta 2 Pd 3 Te 5 , we briefly discuss the topological properties based on our first-principles calculation results. For bulk Ta 2 Pd 3 Te 5 , the symmetry indicators (Z 2 × Z 2 × Z 2 × Z 4 ) are 0. However, it has a nontrivial mirror Chern number in the k y = 0 plane due to the band inversion at the Γ point [23], which indicates the topological nature of bulk Ta 2 Pd 3 Te 5 . In the monolayer limit, Ta 2 Pd 3 Te 5 becomes a 2DTI [23] with a similar band inversion. Its nontrivial topology has also been confirmed by the one-dimensional Wilson loop method. Figure 2(a) shows the band structure of monolayer Ta 2 Pd 3 Te 5 , where an inverted band gap near the Fermi level can be identified. The calculated gap along the Γ-Z direction is approximately 5 meV. Notably, the gap fitted based on our transport measurements (∼14 meV) on bulk samples is larger than the calculated value. We will later show that our STS measurements also indicate a significantly larger gap compared to the calculation results. This inconsistency probably originates from the fact that density functional theory (DFT) calculations may underestimate the band gap of materials.
To study the electronic structure of Ta 2 Pd 3 Te 5 , we performed ARPES measurements on a freshly cleaved surface. An ARPES intensity map of the Fermi surface is displayed in Fig. 2(c), which shows a weak spectral weight along theΓ-Ȳ direction. Because of the large lattice constant along the c direction, we observed four BZs alongΓ-Z. With increasing binding energy, an oval- like pocket appears at theΓ point, as shown in Fig. 2(d). The band structure along theΓ-Ȳ direction is shown in Fig. 2(e), which agrees well with our slab calculation results (see Fig. 2(f)). These ARPES results, combined with the DFT calculations, provide strong evidence of the topological band structure of Ta 2 Pd 3 Te 5 . Now that we have shown the existence of a topological band structure and a band gap in Ta 2 Pd 3 Te 5 , we proceed to studying the topological edge states, which are a key signature of the QSH state in monolayer Ta 2 Pd 3 Te 5 . Because of the weak interlayer coupling, topological edge states are expected to exist at the periphery of the topmost layers. An ideal technique to study the edge states is STS because the tunneling conductance is proportional to the local density of states (LDOS). Figure 3(a) shows an STM image that contains a step edge. From the line profile in Fig. 3(b), the step height is 1.4 nm, which corresponds to the lattice constant along the a direction. Figure 3(c) shows the dI/dV curves taken on the flat terrace and near the step edge. On the flat terrace, we observe a band gap at the Fermi level, in agreement with the semiconductor behavior of Ta 2 Pd 3 Te 5 . The estimated gap size is approximately 43 meV, with the valence band top and conduction band bottom at -33 meV and 10 meV, respectively. The band gap shows negligible variation across the flat terrace, despite the height variation of the Te chains, as shown in Supplementary Fig. S1. This indicates the high spatial homogeneity of the surface, which provides further evidence for the global nature of the band gap. Notably, the gap estimated from our STS data is larger than that fitted based on the transport measurements. This may originate from the surface sensitivity of the STS technique, which indicates a larger band gap in monolayer Ta 2 Pd 3 Te 5 than in the bulk material. Near the step edge, however, the LDOS is dramatically enhanced, featuring a V-shape in the energy range between -40 and 40 meV. This indicates the existence of edge states inside the band gap. To better visualize the evolution of the edge states, we present a series of dI/dV curves across the step edge in Fig. 3(d). When the tip approaches the step edge, the tunneling conductance inside the gap gradually increases, as indicated by the red dashed line in Fig. 3(d).
To confirm the spatial distribution of the edge states, we performed real space dI/dV mapping, as shown in Fig. 3(e). When the bias voltage is set to be within the band gap (e.g., 0 and -20 mV), the tunneling conductance is dramatically enhanced near the step edge. The enhancement of the tunneling conductance vanishes at bias voltages outside the band gap, resulting in a uniform LDOS over the entire surface (e.g., at ±80 mV). Figure 3(f) shows an averaged dI/dV map at several bias voltages that span the band gap, where the enhancement of the LDOS near the step edge can be clearly seen. The fact that these edge states are located inside the gap agrees well with their topological nature. Notably, the step edge is not straight and contains several different terminations. Nevertheless, edge states always exist, despite the slight variation in the details of the STS spectra. This provides strong evidence for the robustness of the edge states, which is also consistent with their topological nature [23]. Similar topological edge states have been reported in several other topological materials, such as ZrTe 5 [24] and TaIrTe 4 [25].
In summary, our results support the existence of significant inverted gap and topological edge states in the van der Waals material Ta 2 Pd 3 Te 5 , thus providing strong evidence for QSH states in Ta 2 Pd 3 Te 5 . In stark contrast to WTe 2 , multilayer and even bulk Ta 2 Pd 3 Te 5 hosts a similar inverted gap at the Fermi level. This is beneficial for device applications because multilayer samples are typically more inert and have higher tunability. Therefore, we expect that Ta 2 Pd 3 Te 5 will become a promising material for fabricating QSH devices.
FIG. 1 :
1Structure and characterization of Ta2Pd3Te5 single crystals. (a) Schematic drawing of the atomic structure of Ta2Pd3Te5. (b) Schematic drawing of the Brillouin zones and high-symmetry points of monolayer Ta2Pd3Te5. (c) X-ray diffraction spectrum on a flat surface of Ta2Pd3Te5. The inset shows a photograph of typical Ta2Pd3Te5 single crystals. (d) Large-scale STM topographic image of the Ta2Pd3Te5 surface (VB=1 V; I=0.05 nA). (e) Zoomed-in STM image showing atomic resolution (VB=50 mV; I=0.05 nA). The atomic structure of the surface Te atoms (red balls) is superimposed on the left part of the image. (f) Fast Fourier transformed STM image, which shows a rectangular reciprocal lattice. Red arrows indicate the reciprocal lattice vectors. (g) In-plane resistivity of Ta2Pd3Te5 single crystals as a function of temperature. The red line is the fitting result obtained using the Arrhenius formula.
FIG. 2 :
2Topological band structure of Ta2Pd3Te5. (a) Calculated band structure of monolayer Ta2Pd3Te5. (b) Magnified view of the blue-shaded area in (a), showing the existence of a band gap. The calculated gap is approximately 5 meV. (c) and (d) ARPES intensity plots at the Fermi level and EB=0.35 eV, respectively. The blue lines indicate the surface BZs of Ta2Pd3Te5. (e) ARPES intensity plot of the band structure along theΓ-Ȳ direction. (f) Slab calculation results of the band structure along theΓ-Ȳ direction.
FIG. 3 :
3STM characterization of the topological edge states in Ta2Pd3Te5. (a) STM topographic image containing a step edge. Dashed black lines indicate the position of the step edge. (b) Line profile along the solid black line in (a). (c) dI/dV curves taken on the flat terrace (blue) and near the step edge (black). (d) Series of dI/dV spectra taken along the red arrow in (a). The red dashed line indicates the emergence of in-gap states near the step edge. (e) dI/dV maps taken in the same area as (a). The bias voltage is indicated on the left side of each map. Pronounced edge states appear when the bias voltage is in the range of -40 to 20 meV, as indicated by the black dashed lines. (f) Averaged dI/dV map at four different bias voltages: -40, -20, 0, and 20 mV.
Colloquium: topological insulators. M Z Hasan, C L Kane, Rev. Mod. Phys. 82M. Z. Hasan and C. L. Kane, Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045-3067 (2010).
Topological insulators and superconductors. X.-L Qi, S.-C Zhang, Rev. Mod. Phys. 83X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057-1110 (2011).
Topological insulator materials. Y Ando, J. Phys. Soc. Jpn. 82102001Y. Ando, Topological insulator materials. J. Phys. Soc. Jpn. 82, 102001 (2013).
Quantum spin Hall insulator state in HgTe quantum wells. M König, S Wiedmann, C Brüne, A Roth, H Buhmann, L W Molenkamp, X L Qi, S C Zhang, Science. 318M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X. L. Qi, and S. C. Zhang, Quantum spin Hall insulator state in HgTe quantum wells. Science 318, 766-770 (2007).
. C Liu, T L Hughes, X.-L Qi, K Wang, S.-C , C. Liu, T. L. Hughes, X.-L. Qi, K. Wang, and S.-C.
Quantum spin Hall effect in inverted type-II semiconductors. Zhang, Phys. Rev. Lett. 100236601Zhang, Quantum spin Hall effect in inverted type-II semiconductors. Phys. Rev. Lett. 100, 236601 (2008).
Evidence for helical edge modes in inverted InAs/GaSb quantum wells. I Knez, R.-R Du, G Sullivan, Phys. Rev. Lett. 107136603I. Knez, R.-R. Du, and G. Sullivan, Evidence for helical edge modes in inverted InAs/GaSb quantum wells. Phys. Rev. Lett. 107, 136603 (2008).
Robust helical edge transport in gated InAs/GaSb bilayers. L Du, I Knez, G Sullivan, R.-R Du, Phys. Rev. Lett. 11496802L. Du, I. Knez, G. Sullivan, and R.-R. Du, Robust helical edge transport in gated InAs/GaSb bilayers. Phys. Rev. Lett. 114, 096802 (2015).
. F Yang, L Miao, Z F Wang, M.-Y Yao, F Zhu, Y R Song, M.-X Wang, J.-P Xu, A V Fedorov, Z Sun, G B Zhang, C Liu, F Liu, D Qian, C L Gao, J.-F , F. Yang, L. Miao, Z. F. Wang, M.-Y. Yao, F. Zhu, Y. R. Song, M.-X. Wang, J.-P. Xu, A. V. Fedorov, Z. Sun, G. B. Zhang, C. Liu, F. Liu, D. Qian, C. L. Gao, and J.-F.
Spatial and energy distribution of topological edge states in single Bi(111) bilayer. Jia, Phys. Rev. Lett. 10916801Jia, Spatial and energy distribution of topological edge states in single Bi(111) bilayer. Phys. Rev. Lett. 109, 016801 (2012).
One-dimensional topological edge states of bismuth bilayers. I K Drozdov, A Alexandradinata, S Jeon, S Nadj-Perge, H Ji, R J Cava, B Andrei Bernevig, A Yazdani, Nat. Phys. 10I. K. Drozdov, A. Alexandradinata, S. Jeon, S. Nadj- Perge, H. Ji, R. J. Cava, B. Andrei Bernevig, and A. Yazdani, One-dimensional topological edge states of bismuth bilayers. Nat. Phys. 10, 664-669 (2014).
Quantum spin Hall state in monolayer 1T ′ -WTe2. S Tang, C Zhang, D Wong, Nat. Phys. 13S. Tang, C. Zhang, D. Wong, et al. Quantum spin Hall state in monolayer 1T ′ -WTe2. Nat. Phys. 13, 683-687 (2017).
. Z Jia, Y.-H Song, X.-B Li, K Ran, P Lu, H.-J Zheng, X.-Y Zhu, Z.-Q Shi, J Sun, J Wen, D Xing, S.-C , Z. Jia, Y.-H. Song, X.-B. Li, K. Ran, P. Lu, H.-J. Zheng, X.-Y. Zhu, Z.-Q. Shi, J. Sun, J. Wen, D. Xing, and S.-C.
Direct visualization of a two-dimensional topological insulator in the single-layer 1T ′ -WTe2. Li, Phys. Rev. B. 96R41108Li, Direct visualization of a two-dimensional topological insulator in the single-layer 1T ′ -WTe2. Phys. Rev. B 96, 041108(R) (2017).
Observation of topological states residing at step edges of WTe2. L Peng, Y Yuan, G Li, X Yang, J.-J Xian, C.-J Yi, Y.-G Shi, Y.-S Fu, Nat. Commun. 865L. Peng, Y. Yuan, G. Li, X. Yang, J.-J. Xian, C.-J. Yi, Y.- G. Shi, and Y.-S. Fu, Observation of topological states residing at step edges of WTe2. Nat. Commun. 8, 65 (2017).
. F Zhu, W Chen, Y Xu, C Gao, D Guan, C , F.-f. Zhu, W.-j. Chen, Y. Xu, C.-l. Gao, D.-d. Guan, C.-h.
Epitaxial growth of two-dimensional stanene. D Liu, S.-C Qian, J.-F Zhang, Jia, Nat. Mater. 14Liu, D. Qian, S.-C. Zhang, and J.-f. Jia, Epitaxial growth of two-dimensional stanene. Nat. Mater. 14, 1020-1025 (2015).
Bismuthene on a SiC substrate: A candidate for a high-temperature quantum spin Hall material. F Reis, G Li, L Dudy, M Bauernfeind, S Glass, W Hanke, R Thomale, J Schäfer, R Claessen, Science. 357F. Reis, G. Li, L. Dudy, M. Bauernfeind, S. Glass, W. Hanke, R. Thomale, J. Schäfer, R. Claessen, Bismuthene on a SiC substrate: A candidate for a high-temperature quantum spin Hall material. Science 357, 287-290 (2017).
Evidence of topological edge states in buckled antimonene monolayers. S Y Zhu, Y Shao, E Wang, L Cao, X.-Y Li, Z.-L Liu, C Liu, L.-W Liu, J.-O Wang, K Ibrahim, J.-T Sun, Y.-L Wang, S Du, H.-J Gao, Nano Lett. 19S. Y. Zhu, Y. Shao, E. Wang, L. Cao, X.-Y. Li, Z.-L. Liu, C. Liu, L.-W. Liu, J.-O. Wang, K. Ibrahim, J.-T. Sun, Y.- L. Wang, S. Du, and H.-J. Gao, Evidence of topological edge states in buckled antimonene monolayers. Nano Lett. 19 6323-6329 (2019).
Epitaxial growth of ultraflat stanene with topological band inversion. J Deng, B Xia, X Ma, H Chen, H Shan, X Zhai, B Li, A Zhao, Y Xu, W Duan, S.-C Zhang, B Wang, J G Hou, Nat. Mater. 17J. Deng, B. Xia, X. Ma, H. Chen, H. Shan, X. Zhai, B. Li, A. Zhao, Y. Xu, W. Duan, S.-C. Zhang, B. Wang, and J. G. Hou, Epitaxial growth of ultraflat stanene with topological band inversion. Nat. Mater. 17, 1081-1086 (2018).
. C.-Z Xu, Y.-H Chan, P Chen, X Wang, D Flötotto, J A Hlevyack, G Bian, S.-K Mo, M.-Y Chou, T.-C , C.-Z. Xu, Y.-H. Chan, P. Chen, X. Wang, D. Flötotto, J. A. Hlevyack, G. Bian, S.-K. Mo, M.-Y. Chou, and T.-C.
Gapped electronic structure of epitaxial stanene on InSb(111). Chiang, Phys. Rev. B. 9735122Chiang, Gapped electronic structure of epitaxial stanene on InSb(111). Phys. Rev. B 97, 035122 (2018).
Edge conduction in monolayer WTe2. Z Fei, T Palomaki, S Wu, W Zhao, X Cai, B Sun, P Nguyen, J Finney, X Xu, D H Cobden, Nat. Phys. 13Z. Fei, T. Palomaki, S. Wu, W. Zhao, X. Cai, B. Sun, P. Nguyen, J. Finney, X. Xu, and D. H. Cobden, Edge conduction in monolayer WTe2. Nat. Phys. 13, 677-682 (2017).
Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal. S Wu, V Fatemi, Q D Gibson, K Watanabe, T Taniguchi, R J Cava, P Jarillo-Herrero, Science. 359S. Wu, V. Fatemi, Q. D. Gibson, K. Watanabe, T. Taniguchi, R. J. Cava, and P. Jarillo-Herrero, Observation of the quantum spin Hall effect up to 100 kelvin in a monolayer crystal. Science 359, 76-79 (2018).
Imaging quantum spin Hall edges in monolayer WTe2. Y Shi, J Kahn, B Niu, Z Fei, B Sun, X Cai, B A Francisco, D Wu, Z.-X Shen, X Xu, D H Cobden, Y.-T Cui, Sci. Adv. 58799Y. Shi, J. Kahn, B. Niu, Z. Fei, B. Sun, X. Cai, B. A. Francisco, D. Wu, Z.-X. Shen, X. Xu, D. H. Cobden, and Y.-T. Cui, Imaging quantum spin Hall edges in monolayer WTe2. Sci. Adv. 5, eaat8799 (2019).
. M N Ali, J Xiong, S Flynn, J Tao, Q D Gibson, L M Schoop, T Liang, N Haldolaarachchige, M , M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gibson, L. M. Schoop, T. Liang, N. Haldolaarachchige, M.
Large, nonsaturating magnetoresistance in WTe2. N P Hirschberger, R J Ong, Cava, Nature. 514Hirschberger, N. P. Ong, and R. J. Cava, Large, non- saturating magnetoresistance in WTe2. Nature 514, 205- 208 (2014).
Type-II Weyl semimetals. A A Soluyanov, D Gresch, Z Wang, Q Wu, M Troyer, X Dai, B A Bernevig, Nature. 527A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, Type-II Weyl semimetals. Nature 527, 495-498 (2015).
Z Guo, D Yan, H Sheng, S Nie, Y Shi, Z Wang, arXiv:2012.05917Quantum spin Hall effect in monolayer Ta2M3Te5 (M=Pd, Ni). Z. Guo, D. Yan, H. Sheng, S. Nie, Y. Shi, and Z. Wang, Quantum spin Hall effect in monolayer Ta2M3Te5 (M=Pd, Ni). arXiv: 2012.05917 (2020).
Evidence for topological edge states in a large energy gap near the step edges on the surface of ZrTe5. R Wu, J.-Z Ma, S.-M Nie, L.-X Zhao, X Huang, J.-X Yin, B.-B Fu, P Richard, G.-F Chen, Z Fang, X Dai, H.-M Weng, T Qian, H Ding, S H Pan, Phys. Rev. X. 621017R. Wu, J.-Z. Ma, S.-M. Nie, L.-X. Zhao, X. Huang, J.-X. Yin, B.-B. Fu, P. Richard, G.-F. Chen, Z. Fang, X. Dai, H.-M. Weng, T. Qian, H. Ding, and S. H. Pan, Evidence for topological edge states in a large energy gap near the step edges on the surface of ZrTe5. Phys. Rev. X 6, 021017 (2016).
Observation of topological edge states at the step edges on the surface of Type-II Weyl semimetal TaIrTe4. X Dong, M Wang, D Yan, X Peng, J Li, W Xiao, Q Wang, J Han, J Ma, Y Shi, Y Yao, ACS Nano. 13X. Dong, M. Wang, D. Yan, X. Peng, J. Li, W. Xiao, Q. Wang, J. Han, J. Ma, Y. Shi, and Y. Yao, Observation of topological edge states at the step edges on the surface of Type-II Weyl semimetal TaIrTe4. ACS Nano 13, 9571- 9577 (2018).
Rotatable highresolution ARPES system for tunable linear-polarization geometry. H Iwasawa, K Shimada, E F Schwier, M Zheng, Y Kojima, H Hayashi, J Jiang, M Higashiguchi, Y Aiura, H Namatame, M Taniguchi, J. Synchrotron Radiat. 24H. Iwasawa, K. Shimada, E. F. Schwier, M. Zheng, Y. Kojima, H. Hayashi, J. Jiang, M. Higashiguchi, Y. Aiura, H. Namatame, and M. Taniguchi, Rotatable high- resolution ARPES system for tunable linear-polarization geometry. J. Synchrotron Radiat. 24, 836-841 (2017).
Projector augmented-wave method. P E Blöchl, Phys. Rev. B. 50P. E. Blöchl, Projector augmented-wave method. Phys. Rev. B 50, 17953-17979 (1994).
From ultrasoft pseudopotentials to the projector augmented-wave method. G Kresse, D Joubert, Phys. Rev. B. 59G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758-1775 (1999).
Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. G Kresse, J Furthmüller, Comput. Mater. Sci. 6G. Kresse, and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15- 50 (1996).
Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. G Kresse, J Furthmüller, Phys. Rev. B. 54G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169-11186 (1996).
. J P Perdew, K Burke, M Ernzerhof, Phys. Rev. Lett. 77J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865-3868 (1996).
| [] |
[
"Decomposition of the static potential in the Maximal Abelian gauge",
"Decomposition of the static potential in the Maximal Abelian gauge"
] | [
"Vitaly Bornyakov \nInstitute for High Energy Physics NRC Kurchatov Institute\n142281ProtvinoRussia\n\nPacific Quantum Center\nFar Eastern Federal University\n690950VladivostokRussia\n\nInstitute of Theoretical and Experimental Physics\nNRC Kurchatov Institute\n117218MoscowRussia\n",
"Vladimir Goy \nPacific Quantum Center\nFar Eastern Federal University\n690950VladivostokRussia\n\nInstitut Denis Poisson CNRS/UMR 7013\nUniversité de Tours\n37200France\n",
"Ilya Kudrov \nInstitute of Theoretical and Experimental Physics\nNRC Kurchatov Institute\n117218MoscowRussia\n",
"Roman Rogalyov \nInstitute for High Energy Physics NRC Kurchatov Institute\n142281ProtvinoRussia\n"
] | [
"Institute for High Energy Physics NRC Kurchatov Institute\n142281ProtvinoRussia",
"Pacific Quantum Center\nFar Eastern Federal University\n690950VladivostokRussia",
"Institute of Theoretical and Experimental Physics\nNRC Kurchatov Institute\n117218MoscowRussia",
"Pacific Quantum Center\nFar Eastern Federal University\n690950VladivostokRussia",
"Institut Denis Poisson CNRS/UMR 7013\nUniversité de Tours\n37200France",
"Institute of Theoretical and Experimental Physics\nNRC Kurchatov Institute\n117218MoscowRussia",
"Institute for High Energy Physics NRC Kurchatov Institute\n142281ProtvinoRussia"
] | [] | Decomposition of SU(2) gauge field into the monopole and monopoleless components is studied in the Maximal Abelian gauge using Monte-Carlo simulations in lattice SU(2) gluodynamics as well as in two-color QCD with both zero and nonzero quark chemical potential. The interaction potential between static charges is calculated for each component and their sum is compared with the non-Abelian static potential. A good agreement is found in the confinement phase. Implications of this result are discussed. * | 10.1051/epjconf/202225802009 | [
"https://arxiv.org/pdf/2201.04035v1.pdf"
] | 245,853,952 | 2201.04035 | 9f251dd232f62fe7c3e2e99d9e00def78c159f01 |
Decomposition of the static potential in the Maximal Abelian gauge
Vitaly Bornyakov
Institute for High Energy Physics NRC Kurchatov Institute
142281ProtvinoRussia
Pacific Quantum Center
Far Eastern Federal University
690950VladivostokRussia
Institute of Theoretical and Experimental Physics
NRC Kurchatov Institute
117218MoscowRussia
Vladimir Goy
Pacific Quantum Center
Far Eastern Federal University
690950VladivostokRussia
Institut Denis Poisson CNRS/UMR 7013
Université de Tours
37200France
Ilya Kudrov
Institute of Theoretical and Experimental Physics
NRC Kurchatov Institute
117218MoscowRussia
Roman Rogalyov
Institute for High Energy Physics NRC Kurchatov Institute
142281ProtvinoRussia
Decomposition of the static potential in the Maximal Abelian gauge
Decomposition of SU(2) gauge field into the monopole and monopoleless components is studied in the Maximal Abelian gauge using Monte-Carlo simulations in lattice SU(2) gluodynamics as well as in two-color QCD with both zero and nonzero quark chemical potential. The interaction potential between static charges is calculated for each component and their sum is compared with the non-Abelian static potential. A good agreement is found in the confinement phase. Implications of this result are discussed. *
Introduction
We study the decomposition of the non-Abelian gauge field in the Maximal Abelian gauge (MAG) [1,2] into the sum of the monopole component and the monopoleless component. For this purpose we employ the SU(2) lattice gauge theory.
In terms of vector potentials A µ (x), the decomposition has the form
A µ (x) = A mod µ (x) + A mon µ (x)(1)
where A mon µ (x) is the monopole component and A mod µ (x) is the monopoleless component defined below and referred to as the modified gauge field.
In the MAG, the Abelian dominance for the string tension has long been known [3][4][5][6][7] (for a review see e.g. [8,9]). Moreover, it was found [5,10,11] that the properly determined monopole component of the gauge field produces the string tension close to its exact value in agreement with conjecture that the monopole degrees of freedom are responsible for confinement [12,13].
In Refs. [14,15] it was shown that the topological charge, chiral condensate and effects of chiral symmetry breaking in quenched light hadron spectrum disappear after removal of the monopole contribution from the relevant operators. Similar computations were made within the scope of the Z 2 projection studies [16]. In particular it was shown that after removal of Pvortices the confinement property disappears. We perform a similar removal of monopoles. We consider the following types of the static potential: V mod (r) obtained from the Wilson loops of the modified gauge field U mod µ (x), V mon (r) obtained from the Wilson loops of the monopole gauge field u mon µ (x) and the sum of these two static potentials.
For one value of lattice spacing it was shown [17] that V mod (r) can be well approximated by purely Coulomb fit function and the sum V mod (r) + V mon (r) provides a good approximation to the original non-Abelian static potential V(r) at all distances.
Here we study this phenomenon at three lattice spacings using the Wilson action and thus we can draw conclusions about the continuum limit. We also present the results for one lattice spacing obtained with the improved lattice field action thus checking the universality. Furthermore, we present results for the SU(2) theory with dynamical quarks, i.e. for QC 2 D. These results were partially presented in [18].
Details of simulations
We study the SU(2) lattice gauge theory. Vector potentials of the gauge field can be defined in terms of the link variables by the formula
A µ (x) = 1 2iag U xµ − U † xµ ,(2)
where a is the lattice spacing. Up to terms of the order O(a 2 ), the decomposition (1) can be rearranged to the form
U µ (x) = U mod µ (x)u mon µ (x)(3)
which furnishes the subject of our research.
To fix the MAG we use the simulated annealing algorithm [5] with one gauge copy per configuration. Usually after fixing MAG the following decomposition of the non-Abelian lattice gauge field U µ (x) is made
U µ (x) = C µ (x)u µ (x) ,(4)
where u µ (x) is the Abelian field and C µ (x) is the non-Abelian coset field. The Abelian gauge field u µ (x) is further decomposed [19] into the monopole (singular) part u mon µ (x) and the photon (regular) part u ph µ (x):
u µ (x) = u mon µ (x)u ph µ (x) .(5)
Then it follows from eq. (1) that
U mod µ (x) = C µ (x)u ph µ (x).(6)
Note that u ph (x) is the Abelian projection of U mod µ (x) and involves no monopoles. We need to compute the usual Wilson loops
W(C) = 1 2 TrW(C) , W(C) = l∈C U(l) ,(7)
the monopole Wilson loops
W mon (C) = 1 2 Tr l∈C u mon (l) ,(8)
and the non-Abelian Wilson loops with removed monopole contribution
W mod (C) = 1 2 TrW mod (C) , W mod (C) = l∈C U mod (l) ,(9)
It is known that MAG fixing leaves U(1) gauge symmetry unbroken. The monopole Wilson loop W mon (C) is invariant under respective residual gauge transformations. This is not true for W mod (C) [17]. Thus we need to fix the Landau U(1) gauge by finding the maximum of the gauge-fixing functional,
max ω x,µ Re(ω(x)u µ (x)ω † (x + aμ)) ,(10)
where ω ∈ U(1) is the gauge transformation. To imrove the noise-to-signal ratio for the static potential we use the APE smearing [20] in computations of the Wilson loops. We generate 100 statistically independent gauge field configurations with the Wilson lattice action at β = 2.4, 2.5 on the 24 4 lattice and at β = 2.6 on the 32 4 lattice. The respective values of lattice spacing are a = 0.118, 0.085 and 0.062 fm, which are determined from a fit to the lattice data [21] on the string tension, whose "experimental" value is set to √ σ = 440 MeV. As a check of universality the computations were done with the tadpole improved action at β = 3.4 on 24 4 lattices. Additionally we present our results [22] obtained in QC 2 D on 32 4 lattice at zero and nonzero quark chemical potential µ q .
It is worth to note that another decomposition, namely eq. (4) was investigated in Ref. [7] in the case of SU(3) gluodynamics. Good agreement between the static potenial V(r) and the sum V abel (r) + V o f f (r) was found. We believe that this decomposition also deserves further study.
Results
In Fig. 1 we show the usual non-Abelian static potential V(r) denoted as 'full' and compare it with the sum V sum (r) = V mon (r) + V mod (r). One can see that approximate equality is satisfied for all three lattice spacings and the approximation improves toward the continuum limit. To give further support to this statement we plot in Fig. 2 the relative deviation ∆(r) defined as
V(r) ≈ V sum (r)(11)∆(r) = V(r) − V sum (r) V(r) .(12)
It is clear that ∆(r) decreases with decreasing lattice spacing. We also studied the universality of the decomposition of the static potential eq. (11). The simulations were made with the tadpole improved action at β = 3.4 with lattice spacing approximately equal to that of the Wilson action at β = 2.5. The results of our computations are presented in Fig. 3. It is seen that the agreement between V(r) and V sum (r) is nearly as good as in Fig. 1 for β = 2.5.
Furthermore, we completed the same computations in QC 2 D on 32 4 lattices with small lattice spacing at zero and nonzero quark chemical potential µ q (details of simulations in QC 2 D can be found, e.g., in [22]). The results of these computations are presented in Fig. 4 for µ q = 0 and in Fig. 5 for aµ q = 0.19. It can be seen clearly that approximate decomposition (11) is fulfilled with rather high both at zero and nonzero µ q .
Conclusions
We have studied the decomposition of the static potential into the linear term produced by the monopole (Abelian) gauge field U mon (x) and the Coulomb term produced by the monopoleless non-Abelian gauge field U mod (x). In the case of Wilson action we have presented the results (Figs. 1 and 2) for three values of lattice spacing to demonstrate that the agreement becomes better with a decrease of lattice spacing. Thus our results imply that the relation (1) becomes exact in the continuum limit. Further work is needed to provide more evidence for this conclusion. Next, we have demonstrated that the decomposition (1) holds true also in the case of improved lattice action (see Fig. 3). Furthermore, we have found that it works also in QC 2 D for both zero and nonzero quark chemical potential. It should be noticed that in Ref. [18] we also presented results for the static potential in the adjoint representation and found that the decomposition (1) works quite well in this case although not so well as in the case of the fundamental representation. It is of course interesting to verify how decomposition (1) works for other observables, e.g. action density and energy density of the flux tube. Also the decomposition should be checked in the case of SU(3) gauge group.
One can draw the following conclusions from the decomposition (1). The monopole part of the gauge field U mon (x) is responsible for the classical part of the energy of the hadronic string, whereas the monopoleless part U mod (x) is associated with the fluctuating part of its energy, i.e. U mod (x) reproduces perturbative results at small distances and contributes to the nonperturbative physics at large distances.
Figure 1 .
1Non-Abelian static potential V(r) (filled symbols) is compared with the sum V mon (r) + V mod (r) (empty symbols) for Wilson action at β = 2.4 (squares), β = 2.5 (circles), β = 2.6 (triangles).
Figure 2 .Figure 3 .
23The relative deviation ∆(r) vs. distance r for three values of the lattice spacing. Non-Abelian static potential V(r) (filled symbols) is compared with the sum of the monopole and modified potentials V sum (r) (empty symbols) for the improved action at β = 3.4.
Figure 4 .Figure 5 .
45The non-Abelian static potential V(r) (filled squares) is compared with the sum of the monopole and modified potentials V sum (r) (filled circles) for QC 2 D at zero chemical potential. The potentials V mon (r) (empty circles) and V mod (r) (empty squares) are also shown. The non-Abelian static potential V(r) (filled squares) is compared with the sum of the monopole and modified potentials V sum (r) (filled circles) in QC 2 D at aµ q = 0.19. The potentials V mon (r) (empty circles) and V mod (r) (empty squares) are also shown.
. A S Kronfeld, M L Laursen, G Schierholz, U J Wiese, Phys. Lett. B. 198516A. S. Kronfeld, M. L. Laursen, G. Schierholz and U. J. Wiese, Phys. Lett. B 198, 516 (1987).
. G Hooft, 10.1016/0550-3213(81)90442-9Nucl. Phys. B. 190G. 't Hooft, Nucl. Phys. B 190 (1981), 455-478 doi:10.1016/0550-3213(81)90442-9
. T Suzuki, I Yotsuyanagi, Phys. Rev. D. 424257T. Suzuki and I. Yotsuyanagi, Phys. Rev. D 42 (1990) 4257.
. S Hioki, S Kitahara, S Kiura, Y Matsubara, O Miyamura, S Ohno, T Suzuki, Phys. Lett. B. 272326Erratum-ibid. B 281, 416 (1992)S. Hioki, S. Kitahara, S. Kiura, Y. Matsubara, O. Miyamura, S. Ohno and T. Suzuki, Phys. Lett. B 272, 326 (1991) [Erratum-ibid. B 281, 416 (1992)].
. G S Bali, V Bornyakov, M Muller-Preussker, K Schilling, Phys. Rev. D. 542863G. S. Bali, V. Bornyakov, M. Muller-Preussker and K. Schilling, Phys. Rev. D 54 (1996) 2863.
. V Bornyakov, M Muller-Preussker, Nucl. Phys. Proc. Suppl. 106646V. Bornyakov and M. Muller-Preussker, Nucl. Phys. Proc. Suppl. 106, 646 (2002).
. N Sakumichi, H Suganuma, 10.1103/PhysRevD.90.111501arXiv:1406.2215Phys. Rev. D. 9011hep-latN. Sakumichi and H. Suganuma, Phys. Rev. D 90 (2014) no.11, 111501 doi:10.1103/PhysRevD.90.111501 [arXiv:1406.2215 [hep-lat]].
. M N Chernodub, M I Polikarpov, arXiv:hep-th/9710205hep-thM. N. Chernodub and M. I. Polikarpov, [arXiv:hep-th/9710205 [hep-th]].
. R W Haymaker, arXiv:hep-lat/9809094Phys. Rept. 315hep-latR. W. Haymaker, Phys. Rept. 315 (1999), 153-173 [arXiv:hep-lat/9809094 [hep-lat]].
. H Shiba, T Suzuki, Phys. Lett. B. 333461H. Shiba and T. Suzuki, Phys. Lett. B 333, 461 (1994).
. J D Stack, S D Neiman, R J Wensley, Phys. Rev. D. 503399J. D. Stack, S. D. Neiman and R. J. Wensley, Phys. Rev. D 50, 3399 (1994).
G Hooft, Proceedings of the EPS International. A. Zichichithe EPS International1225G. 't Hooft, in Proceedings of the EPS International, edited by A. Zichichi, p. 1225, 1976.
. S Mandelstam, Phys. Rept. 23S. Mandelstam, Phys. Rept. 23 (1976), 245-249
. O Miyamura, Phys. Lett. B. 35391O. Miyamura, Phys. Lett. B 353, 91 (1995).
. S Kitahara, O Miyamura, T Okude, F Shoji, T Suzuki, Nucl. Phys. B. 533576S. Kitahara, O. Miyamura, T. Okude, F. Shoji and T. Suzuki, Nucl. Phys. B 533, 576 (1998)
. P De Forcrand, M D'elia, arXiv:hep-lat/9901020Phys. Rev. Lett. 824582P. de Forcrand and M. D'Elia, Phys. Rev. Lett. 82, 4582 (1999) [arXiv:hep-lat/9901020].
. V G Bornyakov, M I Polikarpov, G Schierholz, T Suzuki, S N Syritsyn, arXiv:hep-lat/0512003Nucl. Phys. B Proc. Suppl. 153hep-latV. G. Bornyakov, M. I. Polikarpov, G. Schierholz, T. Suzuki and S. N. Syritsyn, Nucl. Phys. B Proc. Suppl. 153, 25-32 (2006) [arXiv:hep-lat/0512003 [hep-lat]].
. V G Bornyakov, I Kudrov, R N Rogalyov, arXiv:2101.04196hep-latV. G. Bornyakov, I. Kudrov and R. N. Rogalyov, [arXiv:2101.04196 [hep-lat]].
. J Smit, A Van Der Sijs, Nucl. Phys. B. 355603J. Smit and A. van der Sijs, Nucl. Phys. B 355, 603 (1991).
. M , APE CollaborationPhys. Lett. B. 192163M. Albanese et al. [APE Collaboration], Phys. Lett. B 192, 163 (1987).
. J Fingberg, Urs M Heller, F Karsch, Nucl. Phys., B. 392J. Fingberg, Urs M. Heller, F. Karsch, Nucl. Phys., B 392, 493-517 (1993).
. V G Bornyakov, V V Braguta, E M Ilgenfritz, A Y Kotov, A V Molochkov, A A Nikolaev, 10.1007/JHEP03arXiv:1711.01869JHEP. 03161hep-latV. G. Bornyakov, V. V. Braguta, E. M. Ilgenfritz, A. Y. Kotov, A. V. Molochkov and A. A. Nikolaev, JHEP 03 (2018), 161 doi:10.1007/JHEP03(2018)161 [arXiv:1711.01869 [hep-lat]].
| [] |
[
"Theory of metallic double perovskites with spin orbit coupling and strong correlations; application to ferrimagnetic Ba 2 FeReO 6",
"Theory of metallic double perovskites with spin orbit coupling and strong correlations; application to ferrimagnetic Ba 2 FeReO 6"
] | [
"Ashley Cook \nDepartment of Physics\nUniversity of Toronto\nM5S 1A7TorontoOntarioCanada\n",
"Arun Paramekanti \nDepartment of Physics\nUniversity of Toronto\nM5S 1A7TorontoOntarioCanada\n\nCanadian Institute for Advanced Research\nM5G 1Z8TorontoOntarioCanada\n"
] | [
"Department of Physics\nUniversity of Toronto\nM5S 1A7TorontoOntarioCanada",
"Department of Physics\nUniversity of Toronto\nM5S 1A7TorontoOntarioCanada",
"Canadian Institute for Advanced Research\nM5G 1Z8TorontoOntarioCanada"
] | [] | We consider a model of the double perovskite Ba2FeReO6, a room temperature ferrimagnet with correlated and spin-orbit coupled Re t2g electrons moving in the background of Fe moments stabilized by Hund's coupling. We show that for such 3d/5d double perovskites, strong correlations on the 5d-element (Re) are essential in driving a half-metallic ground state. Incorporating both strong spin-orbit coupling and the Hubbard repulsion on Re leads to a band structure consistent with ab initio calculations. Using our model, we find a large spin polarization at the Fermi level, and obtain a semi-quantitative understanding of the saturation magnetization of Ba2FeReO6, as well as X-ray magnetic circular dichroism data indicating a significant orbital magnetization. Based on the orbital populations obtained in our theory, we predict a specific doping dependence to the tetragonal distortion accompanying ferrimagnetic order. Finally, the combination of a net magnetization and spin-orbit interactions is shown to induce Weyl nodes in the band structure, and we predict a significant intrinsic anomalous Hall effect in hole-doped Ba2FeReO6. The uncovered interplay of strong correlations and spin-orbit coupling lends partial support to our previous work, which used a local moment description to capture the spin wave dispersion found in neutron scattering measurements. Our work is of interest in the broader context of understanding metallic double perovskites which are of fundamental importance and of possible relevance to spintronic applications. | 10.1103/physrevb.88.235102 | [
"https://arxiv.org/pdf/1308.3701v2.pdf"
] | 119,255,866 | 1308.3701 | 626f7fc8931ceedf6a03523485ba1467e883fa81 |
Theory of metallic double perovskites with spin orbit coupling and strong correlations; application to ferrimagnetic Ba 2 FeReO 6
Ashley Cook
Department of Physics
University of Toronto
M5S 1A7TorontoOntarioCanada
Arun Paramekanti
Department of Physics
University of Toronto
M5S 1A7TorontoOntarioCanada
Canadian Institute for Advanced Research
M5G 1Z8TorontoOntarioCanada
Theory of metallic double perovskites with spin orbit coupling and strong correlations; application to ferrimagnetic Ba 2 FeReO 6
We consider a model of the double perovskite Ba2FeReO6, a room temperature ferrimagnet with correlated and spin-orbit coupled Re t2g electrons moving in the background of Fe moments stabilized by Hund's coupling. We show that for such 3d/5d double perovskites, strong correlations on the 5d-element (Re) are essential in driving a half-metallic ground state. Incorporating both strong spin-orbit coupling and the Hubbard repulsion on Re leads to a band structure consistent with ab initio calculations. Using our model, we find a large spin polarization at the Fermi level, and obtain a semi-quantitative understanding of the saturation magnetization of Ba2FeReO6, as well as X-ray magnetic circular dichroism data indicating a significant orbital magnetization. Based on the orbital populations obtained in our theory, we predict a specific doping dependence to the tetragonal distortion accompanying ferrimagnetic order. Finally, the combination of a net magnetization and spin-orbit interactions is shown to induce Weyl nodes in the band structure, and we predict a significant intrinsic anomalous Hall effect in hole-doped Ba2FeReO6. The uncovered interplay of strong correlations and spin-orbit coupling lends partial support to our previous work, which used a local moment description to capture the spin wave dispersion found in neutron scattering measurements. Our work is of interest in the broader context of understanding metallic double perovskites which are of fundamental importance and of possible relevance to spintronic applications.
We consider a model of the double perovskite Ba2FeReO6, a room temperature ferrimagnet with correlated and spin-orbit coupled Re t2g electrons moving in the background of Fe moments stabilized by Hund's coupling. We show that for such 3d/5d double perovskites, strong correlations on the 5d-element (Re) are essential in driving a half-metallic ground state. Incorporating both strong spin-orbit coupling and the Hubbard repulsion on Re leads to a band structure consistent with ab initio calculations. Using our model, we find a large spin polarization at the Fermi level, and obtain a semi-quantitative understanding of the saturation magnetization of Ba2FeReO6, as well as X-ray magnetic circular dichroism data indicating a significant orbital magnetization. Based on the orbital populations obtained in our theory, we predict a specific doping dependence to the tetragonal distortion accompanying ferrimagnetic order. Finally, the combination of a net magnetization and spin-orbit interactions is shown to induce Weyl nodes in the band structure, and we predict a significant intrinsic anomalous Hall effect in hole-doped Ba2FeReO6. The uncovered interplay of strong correlations and spin-orbit coupling lends partial support to our previous work, which used a local moment description to capture the spin wave dispersion found in neutron scattering measurements. Our work is of interest in the broader context of understanding metallic double perovskites which are of fundamental importance and of possible relevance to spintronic applications.
I. INTRODUCTION
Double perovskite (DP) materials A 2 BB'O 6 , where the transition metal ions B and B' reside on the two sublattices of a cubic lattice, can realize many complex phases. 1 Metallic variants, such as Sr 2 FeMoO 6 , provide us with the simplest multi-orbital examples of ferrimagnetic order 2 kinetically stabilized by the Pauli exclusion principle. [3][4][5][6][7][8][9][10][11] Insulating variants where only the B'-site ion is magnetic, such as Ba 2 YMoO 6 and La 2 LiMoO 6 , provide material examples of quantum mechanical moments living on the geometrically frustrated face-centered cubic lattice. [12][13][14][15][16] Metallic DPs, such as Sr 2 FeMoO 6 , are also of significant technological importance, being room temperature ferrimagnets with half-metallic band structures and a large spin polarization which is useful for spintronic applications. 17,18 Metallic 3d/5d DPs are of particular interest in this regard since they appear to have strongly reduced B/B' site mixing; samples of Ba 2 FeReO 6 studied in previous work 19 have low < 1% anti-site disorder. Such anti-site disorder, which is common in other DPs and which is detrimental to spintronic applications, appears to be alleviated in 3d/5d DPs by the B/B' ionic size mismatch suggesting that they might be better suited for applications. However, such 3d/5d DPs require us to confront the twin aspects of strong correlations and strong spin-orbit coupling, topics at the forefront of fundamental research 20 motivated by the possibility of stabilizing states such as fractionalized topological insulators (TIs), [21][22][23][24] or Weyl semimetals. [25][26][27][28] In this paper, we focus on metallic ordered DPs with mixed 3d/5d transition metal ions on the B/B' sites, specifically the Ba 2 FeReO 6 material, 29,30 with the structure as shown in Fig.1. we obtain the following main results. (i) We consider a model of the ordered dou- ble perovskite Ba 2 FeReO 6 (see Fig.1 electron itinerancy (∼ 0.3eV). In addition, we include weaker terms such as inter-orbital mixing and second neighbor hopping which are required to reproduce the band degeneracies at high symmetry points in the Brillouin zone found in earlier ab initio studies. (ii) Our theory accounts semi-quantitatively for the measured saturation magnetization 32 , as well as X-ray magnetic circular dichroism (XMCD) experiments which find a significant orbital contribution to the Re magnetization in the ordered state. 33,34 (iii) Based on the orbital occupations in the magnetically ordered state, we predict a tetragonal distortion, with c-axis compression accompanying magnetic order, in agreement with experimental data. 33,34 We also predict a specific doping dependence to this orbital order and distortion which could be tested in future experiments. (iv) The strong correlations on Re, inferred from our study, lends partial support to earlier work which showed that a local moment description of the ferrimagnetic state provides a reasonably good description of the magnetic dynamic structure factor obtained using inelastic neutron scattering experiments. 19 This importance of strong correlation effects and local moment physics on the 5d element is in agreement with previous ab initio studies 11 that discussed the emergence of local moments of closely related Cr-based 3d/5d DPs Sr 2 CrB'O 6 upon progressing through the series with B'=W,Re,Os. (v) From our computed band dispersion, we show the appearance of Weyl nodes in such metallic ferrimagnetic DPs. This is in line with the general understanding that in the presence of spin-orbit coupling, such Weyl nodes are expected to be induced by breaking of time-reversal symmetry or inversion symmetry. 35,36 (vi) Using the Kubo formula for the spin-orbit coupled bands, we find that Ba 2 FeReO 6 itself appears to have only a small intrinsic anomalous Hall effect (AHE) in the ordered ferrimagnetic state at low temperature, but the AHE is significant in hole doped systems, and we speculate that it might also be significant at intermediate temperatures below the ferrimagnetic T c in Ba 2 FeReO 6 . Taking a broader viewpoint, Re-based layered quasitwo-dimensional oxides or heterostructures may be more strongly correlated than the three-dimensional DPs, and may lead to interesting Mott physics 14,16 beyond the iridates due to the local competition between interactions and spin-orbit coupling due to the d 2 configuration of Re 5+ . Furthermore, one can carry out detailed inelastic neutron scattering studies in Re-based oxides, thus allowing for the possibility to explore the magnetism in more detail than in the iridates. This may prove to be useful in future studies of exotic variants of Re-based oxides.
II. MODEL
The simple charge counting for Ba 2 FeReO 6 suggests Re 5+ and Fe 3+ valence states on the transition metal ions. In this state, the five 3d-electrons on Fe are expected to be locked into a spin-5/2 moment due to strong Hund's coupling in the half-filled d-shell. Here, we will treat this magnetic moment as a classical vector. The two 5d-electrons in the Re t 2g orbital are mobile, able to hop on and off the Fe sites subject to a charge transfer energy ∆ = E Fe − E Re > 0, and Pauli exclusion which constrains electrons arriving on Fe to be antiparallel to the direction of the local Fe moment. For a general direction of the Fe moment, F = (sin θ cos φ, sin θ sin φ, cos θ) at a given site, we must project the added electrons onto the allowed direction to satisfy Pauli exclusion, locally setting f ↑ = sin θ 2 e −iφ/2 f and f ↓ = − cos θ 2 e iφ/2 f , effectively "stripping" the electron of its spin degree of freedom. Such models have been proposed for other DP materials, [3][4][5][6][7][8]10,11 and shown to capture the phenomenology of Sr 2 FeMoO 6 including thermal phase transitions and disorder effects. [37][38][39] However, most of these previous studies, with the notable exception of Ref. 11 have ignored spin-orbit coupling effects, which are expected to be extremely important for 5d transition metal oxides.
Our model does not explicitly account for additional superexchange interactions between the Fe local moment and the emerging local moments on the Re sites which is explicitly taken into account as a separate term in some previous studies (for example, Ref. 11); however, we think such terms should emerge more naturally from an effective tight-binding model when strong correlations are incorporated, as might be relevant to Mott insulating oxides like Sr 2 CrOsO 6 . Fe-Fe superexchange terms which we omit, since they are not necessary to drive the ferrimagnetic state observed in Ba2FeReO6, may prove to be important in understanding the complete magnetic phase diagram as a function of doping which is not addressed in this paper. However, they are likely to be small given the Fe-Fe separation in the DP structure. Further differences between the results of Ref. 11 and our work stem from the fact that their model is for d 3 configuration of Cr, as opposed to our d 5 state on Fe; while both spin components of the itinerant electrons are permitted on Cr (since the e g orbital is available), only one spin projection is allowed for itinerant electrons on Fe due to the Pauli exclusion.
A. Non-interacting tight binding model
The model describing Re electrons moving in the presence of Fe moments then takes the form H 0 = H hop + H so + H ct . Here, the Hamiltonian H hop describes intraorbital hopping of electrons on the lattice, from Re to Fe (nearest-neighbor) and from Re to Re (next-neighbor), as well as inter-orbital hopping of electrons between nextneighbor Re sites; H so is the atomic spin-orbit coupling on Re, projected to the t 2g manifold, of strength λ; finally, H ct describes the charge transfer energy offset ∆ between Re and Fe sites. For simplicity, we only focus on the case of a uniform magnetization on the Fe site, assuming (θ, φ) which describe the Fe moment to be site-independent; it is straightforward to generalize our work to a nonuniform spatially varying magnetization. We use the simple triclinic unit cell, with one Re and one Fe atom, as shown in Fig.1 to study the model Hamiltonian; however in order to facilitate a comparison with published ab initio electronic structure calculations, we will later assume a body-centered tetragonal unit cell containing two Re and two Fe atoms, with lattice con- Fig. 1, and use orthorhombic notation to plot the band dispersion of the eighteen bands in the Brillouin zone.
stants d a = d b = d c / √ 2 as shown in
We label the electrons on the Fe and Re sites by f and d σ respectively, with =(1 ≡ yz, 2 ≡ xz, 3 ≡ xy) denoting the orbital, and σ =↑, ↓ being the spin. The Hamiltonian takes the following form in momentum space, where we assume implicit summation over repeated spin and orbital indices,
H hop = k (η (k)g σ (θ, φ)d † σ (k)f (k) + h.c.) + k (k)(d † σ (k)d σ (k) + α f f † (k)f (k)) + k( = ) γ (k)(d † σ (k)d σ (k)+α f f † (k)f (k)) (1) H so =i λ 2 k ε mn τ n σσ d † σ (k)d mσ (k) (2) H ct = ∆ k f † (k)f (k) (3)
Here, in light of our previous discussion, we have only retained a single spin projection on the Fe site, with g ↑ (θ, φ) = sin θ 2 e −iφ/2 and g ↓ (θ, φ) = − cos θ 2 e iφ/2 . The various hopping processes are schematically illustrated in Fig. 2. The first term in H hop describes nearest-neighbor intra-orbital hopping from Re to Fe, parameterized by t π , t δ . The next two terms in H hop characterize nextneighbor hopping processes, with the ratio of Fe-Fe hoppings to Re-Re hoppings being α f ; we will fix α f = 0.5. While the second term captures intra-orbital hopping between closest pairs of Re atoms or Fe atoms (parameterized by t , t ), the third term captures inter-orbital hopping between closest pairs of Re atoms or Fe atoms (parameterized by t m ). Many of these hopping processes (t δ , t m , t ) have a small energy scale; however they are important to reproduce the band degeneracies found in ab initio calculations at high symmetry points in the Brillouin zone. The explicit momentum dependence of the dispersion coefficients appearing in H hop is given in Appendix A.
B. Interaction effects
Electron-electron interactions are partially accounted for by H 0 in the previous section -in part, by the charge transfer gap ∆, and, in part, by the implicit Hund's coupling which locks the Fe electrons into a high-spin state. However, electronic interactions on Re have been omitted in H 0 . We next include these local Hubbard interactions on Re. The interaction Hamiltonian in the t 2g orbitals of Re takes the form 40
H int =U i α n i ↑ n i ↓ + (U −5 J H 2 ) < n i n i −2J H < S i · S i + J H = d † i ↑ d † i ↓ d i ↓ d i ↑ (4)
where i labels the Re sites, and S i = 1 2 d † i α σ αβ d i β is the spin at site i in orbital . We wish to then study the full Hamiltonian H = H 0 + H int . For simplicity, we only retain only the dominant intra-orbital Coulomb repulsion, treating it at mean field (Hartree) level, as
H int ≈U i ρ 2 (n i ↑ +n i ↓ )−2 m · S i − ρ 2 4 + m · m (5)
where ρ = n i ↑ + n i ↓ , m = S i , and we set m = −m (sin θ cos φ, sin θ sin φ, cos θ), with m > 0, so that m is anti-parallel to the Fe moment F . For simplicity, we only focus on the case θ = φ = 0, so the Fe sites can only accommodate itinerant spin-↓ electrons. We then numerically determine m and ρ in a self-consistent fashion, using the non-interacting ground state as the starting point for the iterative solution, while ensuring that the choice of the chemical potential lead to a total of two electrons per unit cell (i.e., per Re atom). Such a mean field treatment of electron-electron interactions does not capture all aspects of the strong correlation physics, e.g. bandwidth renormalization and mass enhancement. Nevertheless, recognizing this caveat, we use the self-consistent solution of the mean field equations to study the effects of interactions and spin-orbit coupling on the reorganization of the nine electronic bands, compare the physical properties with experimental results, and make qualitative predictions for future experiments.
III. PHYSICAL PROPERTIES
We begin by discussing the effect of electronic correlations in the DPs in the absence of spin-orbit coupling. We show that such correlation effects appear to be crucial to stabilize a half-metallic state with complete polarization in the 5d perovskites, due to the large second-neighbor Re-Re hopping which otherwise prevents a half-metallic state. We then turn to the effect of spin-orbit coupling, and show that it reorganizes the band structure, yielding results which are in reasonable agreement with previous ab initio electronic structure studies. 31 (As pointed out earlier, the band dispersions discussed below are plotted using the orthorhombic notation with an enlarged unit cell containing two Fe and two Re atoms, leading to eighteen electronic bands instead of nine.) Finally, we compare the mean field result for the saturation magnetization with experiments, and the spin and orbital magnetization on the Re site with previous XMCD data, and discuss other physical properties such as tetragonal lattice distortion and predictions for the AHE. Throughout this discussion, we will assume a ferromagnetic order of the Fe moments -a more complete study of the magnetic phase diagram as a function of doping and temperature will be the subject of future numerical investigations.
A. Correlations stabilize a half-metal
If we ignore Re correlations entirely, setting U = 0, and also ignore spin-orbit coupling by setting λ = 0, the band structure shown in Fig.3(a) has decoupled spin-↑ and spin-↓ bands. The twelve spin-↓ bands corresponding to electrons which can delocalize on Re and Fe. By contrast, the six spin-↑ bands corresponds to purely Re states. Working in units where t π = 1, we find that to make a reasonable comparison with the ab initio calculations, we have to choose a significant t = 0.3 (Re-Re hopping), but all other hoppings can be assumed to be small; for simplicity, we fix t δ = t = t m = 0.1. Finally, we have to assume a moderate charge transfer energy ∆ = 3 which splits the spin-↓ states into two groups: 6 lower energy Re-Fe hybridized spin-↓ states (dominant Re character) which form a broad band, and 6 higher energy dominantly Re-Fe hybridized spin-↓ states (dom- inant Fe character) which form a narrow band. Finally, the remaining 6 Re-↑ states form a narrow dispersing band, crossing the chemical potential and overlapping in energy with the broad spin-↓ band. For U = 0, the system thus contains both spin states at the Fermi level. When we incorporate a Hubbard repulsion U = 8t π at mean field level, we see from Fig. 3(b) that its main effect is to self-consistently shift the spin-↑ bands higher in energy, leaving only spin-↓ states at the Fermi level. The resulting band dispersion is in reasonably good agreement with LDA+U calculations. Although we have not attempted a detailed quantitative fitting to the LDA+U band structure, the features noted below are robust. (i) A rough comparison with the overall bandwidth in the ab initio calculations without spin-orbit coupling 31 suggests that t π ≈ 330meV. This is somewhat larger than estimates for Sr 2 FeMoO 6 in the literature 3,4,10 (∼270 meV).
Z Γ X S Y Γ Z Γ X S Y Γ Z Γ X S Y Γ Z Γ (a) (b) (c) X (d)
(ii) We estimate the interaction energy scale on Re to be U ≈ 2.5eV, smaller by a factor of two compared with typical values for 3d transition metals. (iii) There is a significant Re-Re hopping, t /t π ∼ 0.3, we need to include in order to be able to capture the bandwidths of the spin-↑ and spin-↓ bands. All these observations are reasonable given the more extended nature of Re orbitals when compared with 3d or 4d transition metal ions. The presence of appreciable Re-Re hoppings has been pointed out in previous work, 6,41 although they did not take correlation effects on Re into account. More recent work has also arrived at similar conclusions regarding significant Re-Re hoppings. 11 To summarize, we have obtained a tight-binding description including interactions of DPs with spin-orbit coupling.
In contrast to 3d/4d DP materials like Sr 2 FeMoO 6 , we find that 3d/5d DPs have a significant second neighbor hopping; strong correlations on the 5d element (Re) therefore play a crucial role in stabilizing a half-metallic ground state in the 3d/5d DPs.
B. Spin-orbit coupling: Band reconstruction, spin/orbital magnetization, and comparison with magnetization and XMCD experiments
We next turn to the effect of incorporating both spinorbit coupling and Hubbard interactions on Re, solving the mean field equations in case of a nonzero U . From Fig.3(c) and (d), where we have set λ = 2t π (∼660meV for our estimated t π ), we see that spin-orbit coupling clearly eliminates the degeneracies occurring at the Γpoint for λ = 0. It also significantly reconstructs the dispersion of the eighteen bands, leading to reasonably good agreement with published ab initio calculations which include spin-orbit coupling. 31 In the next section, we will discuss the resulting appearance of Weyl nodes in the band dispersion and the intrinsic anomalous Hall effect in the ordered state. Here, we will use the mean field solution to estimate the average Fe valence, the Fe ordered moment, and the spin and orbital contributions to the Re moment. In the ground state with correlations, we find that the average valence of Fe shifts from the naive charge counting value Fe 3+ to Fe 2.6+ , and the Fe moment is lowered to an effective value F z ≈ 2.3 (corresponding to 4.6µ B ). Quantum spin fluctuations beyond the mean field result might further slightly suppress this value. On Re, we find an ordered spin moment S z ≈ 0.78 and an orbital moment L z ≈ 0.48; taking the g-factor into account, and undoing the sign change of the orbital angular momentum which appears upon projection to the t 2g Hamiltonian, this implies a ratio of magnetic moments µ orb Re /µ spin Re ≈ −0.31, remarkably close to the experimentally measured XMCD result ≈ −0.29. We find that the actual value of the spin magnetic moment, µ spin Re ≈ 1.56µ B , is larger than the experimentally reported XMCD value ≈ 1.08µ B . This discrepancy might be partly due to the fact that (i) the experimental results are on powder samples, and hence might appear to be smaller simply due to averaging over grain orientations, and (ii) the method to extract the individual spin or orbital magnetic moments relies on additional assump-tions, while the ratio is apparently more reliable. 33 We must contrast these results with the case where we ignore Re correlations entirely; in that case, the Fe moment is not much affected, F U =0 z ≈ 2.4, but the Re moments are strongly suppressed, yielding S U =0 z ≈ 0.15 and an orbital moment L U =0 z ≈ 0.09 which would lead to a much smaller µ spin Re (U = 0) ≈ 0.3µ B than is experimentally estimated, as well as a much larger saturation magnetization, 4.6µ B , than the measured value 30,32 which is ≈ 3.2-3.3µ B . Our estimates in the presence of correlations, by contrast, yield m sat ≈ 3.5µ B , in much better agreement with the data. Finally, we use our solution to estimate the polarization, defined as the degree of magnetization for states near the Fermi level. We find that while the correlated half-metal state in the absence of spin-orbit coupling exhibits (obviously) 100% polarization, using λ = 2t π reduces the polarization to ∼ 90%. However, if we only take spin-orbit coupling into account and ignore strong correlations, the states near the Fermi level are nearly unpolarized.
In 3d/5d DP materials, spin orbit coupling and strong correlations are both crucial to obtain the experimentally observed spin and orbital magnetization and their locking, and to explain the experimentally observed saturation magnetization and XMCD signal. Spin-orbit coupling leads to a slight decrease of the correlation-induced spin polarization at the Fermi level.
C. Orbital order, tetragonal distortion in ferrimagnetic state, and doping dependence
In the converged mean field state, with the magnetization along the z-axis, we find that the density on Re in the three orbitals are different, with ρ xy ≈ 0.60 and ρ xz = ρ yz ≈ 0.53. This orbital imbalance is induced in the z-ferrimagnetic state due the spin-orbit coupling. The larger extent of the xy-orbital in the xy-plane, compared with its smaller extent along the z-direction, implies that this orbital charge imbalance would lead to a tetragonal distortion of the lattice, to occur coincident with ferrimagnetic ordering and with a shrinking of the c-axis, as has indeed been observed to occur experimentally. The precise extent of this distortion, which is observed 33 to be ∼ 0.1%, depends on details such as the lattice stiffness, and is beyond the scope of our calculation.
When we solve the self-consistent equations at various dopings δ (excess electrons per Re) assuming persistent ferrimagnetic order, the extent of this orbital imbalance, characterized by a tetragonal order parameter η tet = 1 ρ (ρ xy − ρ xz /2 − ρ yz /2), changes systematically as shown in Fig. 4(a). Light electron doping leads to a slightly larger orbital population imbalance and should enhance the c-axis compression, while a larger electron doping leads to a gradual decrease of η tet . Hole doping beyond 0.25 holes/Re leads to η tet < 0, which should cause elongation along the c-axis. The spin contribution to the magnetization on Re, arising from the different orbitals, also shows a similar doping trend as seen from Fig. 4(b), while the orbital contribution to the magnetization on Re has the largest magnitude at zero doping. These results could be possibly be explored experimentally by partially substituting Ba by trivalent La (electron doping), or by Cs or other monovalent ions (hole doping).
Thus, in 3d/5d DP materials, spin orbit coupling and the ferrimagnetic order of itinerant electrons leads to orbital ordering. This, in turn, should lead to a compression along the c-axis, consistent with the experimentally observed tetragonal distortion, and we predict a specific doping dependence to this structural distortion.
D. Doping-dependent anomalous Hall effect
We next turn to the intrinsic AHE in the ferrimagnetic state of such 3d/5d DPs. As pointed out in recent work, for pyrochlore iridates with all-in-all-out order under uniaxial pressure, 27 as well the ferromagnetic infinitelayer ruthenate SrRuO 3 , 42 this intrinsic AHE contains two contributions: (i) a surface contribution arising from Fermi arc states 25 associated with Weyl nodes in the dispersion, and (ii) a bulk contribution from carriers near the Fermi surface. A pair of such Weyl nodes for Ba 2 FeReO 6 is shown in Fig. 5 obtained from the interacting band dispersion. 46 Both contributions to the intrinsic AHE are captured by the momentum-dependent Berry curvature 43,44 of the spin-orbit coupled bands, which is, in turn, obtained from the Kubo formula
σ xy = e 2 d 3 k (2π) 3 m =n f (ε km ) − f (ε kn ) (ε km − ε kn ) 2 Im(v x mn v y nm ).(6)
Here, ε km is the single-particle energy at momentum k and band m, v α = 1 (∂H mf /∂k α ) are components of the velocity operator with H mf being the self-consistently determined mean-field Hamiltonian matrix, and f (.) is the Fermi function. 47 From the mean field solution corresponding to two electrons per Re, as appropriate for Ba 2 FeReO 6 , we find that σ xy at zero temperature is small, σ xy ∼ 10 −3 e 2 dc where d c ≈ 8Å is the lattice constant in Fig. 1. This translates into σ xy ∼ 5Ω −1 cm −1 .
In order to explore σ xy over a larger space of parameters, we consider its variation with doping. Rather than simply shifting the chemical potential, we solve the Hartree mean field equations over a range of electron densities, and then compute σ xy in the resulting selfconsistent band structure. We find that electron doping does not significantly enhance the AHE, but a hole doping of about 0.5-0.8 holes/Re leads to a larger AHE σ xy ∼ −100Ω −1 cm −1 to −250Ω −1 cm −1 . Even this significant AHE is small in natural units (∼ 0.1 e 2 dc ) at T = 0, which we attribute to the large spin polarization in the completely ordered ferromagnet. It is possible that the AHE is a non-monotonic function of temperature, peaked at some intermediate temperature below the magnetic T c even in the undoped compound.
Thus, in 3d/5d DP materials, spin orbit coupling and the ferrimagnetic order is expected to lead to an intrinsic AHE. The AHE appears likely to be larger for hole doped systems compared to an expected small value for Ba 2 FeReO 6 and is likely, in Ba 2 FeReO 6 , to be peaked at intermediate temperatures below T c .
IV. CONCLUSION
We have obtained a tight-binding description of the metallic DPs, including spin-orbit coupling and strong correlation effects. Although we have here only applied it to Ba 2 FeReO 6 , finding good agreement with a broad variety of experiments and with electronic structure calculations, our work should be broadly applicable to other 3d/4d and 3d/5d DP materials as well. Our finding that strong correlation effects are needed to explain many of the experimental observations also lends partial justification to our previous theoretical work which modelled the measured spin wave spectrum using a local moment model. Further theoretical work is needed to study the thermal fluctuation effects of the Fe moments, clarify what factors control the doping dependence of σ xy , and to separate the bulk and surface contributions to the AHE. Furthermore, it would be useful to investigate if ferrimagnetic order in fact survives over a wide range of doping using an unbiased numerical approach. In future experiments, it would be useful to test our predictions for the doping dependence of the structural distortion and the AHE. Given that most DP materials are in the form of powder samples, measuring the AHE and separating the intrinsic contribution from extrinsic contributions would be experimentally challenging; nevertheless systematic doping studies of the various properties of such 3d/5d DPs would be valuable. Finally, it appears extremely important to find ways to synthesize bulk single crystals or high quality thin films of such DP materials which would greatly open up the exploration of their physical properties and applications.
FIG. 1 :
1Crystal structure of Ba2FeReO6 showing the choice of axes, the unit vectors for the elementary triclinic unit cell (green arrows), and magnetic moments in the ferrimagnetic ground state. Also shown is the enlarged body-centered tetragonal unit cell with lattice parameters da and dc = da √ 2.
FIG. 2 :
2Symmetry-allowed hopping matrix elements for double perovskites A2BB'O6 (e.g., Ba2FeReO6), indicated for a few orbitals. tπ, t δ are B-B' (Fe-Re) intraorbital hoppings, t , t are B'-B' (Re-Re) intraorbital hoppings, and tm denotes the interorbital B'-B' (Re-Re) hopping. All processes related to these by cubic symmetry are allowed. The Fe-Fe hoppings are identical to Re-Re hoppings, but scaled by a factor α f = 0.5. Also shown are the rotated axes (compass) for the tetragonal unit cell of Ba2FeReO6, with x -y (dashed lines) being the original cubic axes for defining the orbitals.
FIG. 3 :
3Band dispersion in the orthorhombic notation for the Re and Fe electronic states for different choices of Hubbard interaction U and spin-orbit coupling λ, with energy on the yaxis in units of tπ. The solid black line indicates the chemical potential. For no spin-orbit coupling, (a) U = 0 and λ = 0 and (b) U = 8tπ and λ = 0, we find decoupled spin-↓ (red, solid) and spin-↑ (blue, dashed) states. Comparing (a) and (b), we see that correlations on Re push the spin-↑ states to higher energy, leading to the stabilization of a half-metal ground state. A nonzero spin-orbit coupling, (c) U = 0 and λ = 2tπ, and (d) U = 8tπ and λ = 2tπ, leads to mixed-spin states and splits degeneracies, but for a physically reasonable value U = 8tπ preserves significant spin polarization ∼ 90% for states at the Fermi level.
orbital occupancy in the ferrimagnetic state of Ba2FeReO6 incorporating mean-field interactions and strong spin-orbit coupling at a doping of δ excess electrons per Re. The parameters used are the same as those forFig. 3(b), namely U = 8tπ and λ = 2tπ. This orbital order implies a tetragonal distortion of the lattice, with c-axis compression for δ −0.25, and c-axis elongation for δ −0.25. (b) Doping dependence of orbital magnetization on Re, and the various orbital components of the spin magnetization on Re.
FIG. 5 :
5(a) Band dispersion of Ba2FeReO6 in the presence of interactions and spin-orbit coupling plotted along kz at fixed (kx = 0.0393, ky = 1.9242), showing the nine bands and a pair of Weyl nodes (with energy on y-axis in units of tπ). The parameters used are the same as those forFig. 3(b), namely U = 8tπ and λ = 2tπ. (b) Doping dependence of the intrinsic anomalous Hall conductivity σxy.
D. Serrate, J. M. D. Teresa, and M. R. Ibarra, Journal of Physics: Condensed Matter 19, 023201 (2007).
AcknowledgmentsThis research was supported by NSERC of Canada. We acknowledge useful discussions with Anton Burkov, Patrick Clancy, Young-June Kim, Priya Mahadevan, Kemp Plumb, Mohit Randeria, Nandini Trivedi, and Roser Valenti. AP acknowledges the support and hospitality of the Max-Planck-Institut für Physik komplexer Systeme (Dresden), and discussions with the participants of the SPORE13 workshop, where part of this work was completed.Appendix A: Tight binding coefficientsWhen we work with the triclinic unit cell, there is one Fe atom and one Re atom in each unit cell. Going to mo-mentum space, the coefficients of the tight-binding hopping Hamiltonian H hop in Eq. 1 have intra-orbital terms given byandThe intra-orbital terms take the form γ xz,yz = −2t m (cos k x a − cos k y a) (A10) γ xy,yz = −4t m sin( k x a + k y a 2 ) sin k z c 2 (A11) γ xy,xz = 4t m sin( k x a − k y a 2 ) sin k z c 2 .(A12)
. K I Kobayashi, T Kimura, H Sawada, K Terakura, Y Tokura, Nature. 395677K. I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and Y. Tokura, Nature 395, 677 (1998).
. D D Sarma, P Mahadevan, T Saha-Dasgupta, S Ray, A Kumar, Phys. Rev. Lett. 852549D. D. Sarma, P. Mahadevan, T. Saha-Dasgupta, S. Ray, and A. Kumar, Phys. Rev. Lett. 85, 2549 (2000).
. T Saha-Dasgupta, D D Sarma, Phys. Rev. B. 6464408T. Saha-Dasgupta and D. D. Sarma, Phys. Rev. B 64, 064408 (2001).
. G Jackeli, Phys. Rev. B. 6892401G. Jackeli, Phys. Rev. B 68, 092401 (2003).
. A Chattopadhyay, A J Millis, Phys. Rev. B. 6424424A. Chattopadhyay and A. J. Millis, Phys. Rev. B 64, 024424 (2001).
. K Phillips, A Chattopadhyay, A J Millis, Phys. Rev. B. 67125119K. Phillips, A. Chattopadhyay, and A. J. Millis, Phys. Rev. B 67, 125119 (2003).
. L Brey, M J Calderón, S Das Sarma, F Guinea, Phys. Rev. B. 7494429L. Brey, M. J. Calderón, S. Das Sarma, and F. Guinea, Phys. Rev. B 74, 094429 (2006).
. P Sanyal, P Majumdar, Phys. Rev. B. 8054411P. Sanyal and P. Majumdar, Phys. Rev. B 80, 054411 (2009);
. V N Singh, P Majumdar, Europhys. Lett. 9447004V. N. Singh and P. Majumdar, Europhys. Lett. 94, 47004 (2011).
. O Erten, O N Meetei, A Mukherjee, M Randeria, N Trivedi, P Woodward, Phys. Rev. Lett. 107257201O. Erten, O. N. Meetei, A. Mukherjee, M. Randeria, N. Trivedi, and P. Woodward, Phys. Rev. Lett. 107, 257201 (2011).
. H Das, P Sanyal, T Saha-Dasgupta, D D Sarma, Phys. Rev. B. 83104418H. Das, P. Sanyal, T. Saha-Dasgupta, and D. D. Sarma, Phys. Rev. B 83, 104418 (2011).
. T Aharen, J E Greedan, C A Bridges, A A Aczel, J Rodriguez, G Macdougall, G M Luke, T Imai, V K Michaelis, S Kroeker, Phys. Rev. B. 81224409T. Aharen, J. E. Greedan, C. A. Bridges, A. A. Aczel, J. Rodriguez, G. MacDougall, G. M. Luke, T. Imai, V. K. Michaelis, S. Kroeker, et al., Phys. Rev. B 81, 224409 (2010).
. J P Carlo, J P Clancy, T Aharen, Z Yamani, J P C Ruff, J J Wagman, G J Van Gastel, H M L Noad, G E Granroth, J E Greedan, Phys. Rev. B. 84100404J. P. Carlo, J. P. Clancy, T. Aharen, Z. Yamani, J. P. C. Ruff, J. J. Wagman, G. J. Van Gastel, H. M. L. Noad, G. E. Granroth, J. E. Greedan, et al., Phys. Rev. B 84, 100404 (2011).
. G Chen, R Pereira, L Balents, Phys. Rev. B. 82174440G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82, 174440 (2010).
. T Dodds, T.-P Choy, Y B Kim, Phys. Rev. B. 84104439T. Dodds, T.-P. Choy, and Y. B. Kim, Phys. Rev. B 84, 104439 (2011).
. G Chen, L Balents, Phys. Rev. B. 8494420G. Chen and L. Balents, Phys. Rev. B 84, 094420 (2011).
. I Žutić, J Fabian, S. Das Sarma, Rev. Mod. Phys. 76323I.Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
L Alff, Electron Correlation in New Materials and Nanosystems. K. Scharnberg and S. KruchininNetherlandsSpringerL. Alff, in Electron Correlation in New Materials and Nanosystems, edited by K. Scharnberg and S. Kruchinin (Springer Netherlands, 2007), vol. 241 of NATO Science Series, pp. 393-400.
. K W Plumb, A M Cook, J P Clancy, A I Kolesnikov, B C Jeon, T W Noh, A Paramekanti, Y.-J Kim, Phys. Rev. B. 87184412K. W. Plumb, A. M. Cook, J. P. Clancy, A. I. Kolesnikov, B. C. Jeon, T. W. Noh, A. Paramekanti, and Y.-J. Kim, Phys. Rev. B 87, 184412 (2013).
. W Witczak-Krempa, G Chen, Y B Kim, L Balents, 1305.2193ArXiv e-printsW. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, ArXiv e-prints (2013), 1305.2193.
. M Levin, A Stern, Phys. Rev. Lett. 103196803M. Levin and A. Stern, Phys. Rev. Lett. 103, 196803 (2009).
. D Pesin, L Balents, Nature Physics. 6376D. Pesin and L. Balents, Nature Physics 6, 376 (2010).
. J Maciejko, X.-L Qi, A Karch, S.-C Zhang, Phys. Rev. Lett. 105246809J. Maciejko, X.-L. Qi, A. Karch, and S.-C. Zhang, Phys. Rev. Lett. 105, 246809 (2010).
. B Swingle, M Barkeshli, J Mcgreevy, T Senthil, Phys. Rev. B. 83195139B. Swingle, M. Barkeshli, J. McGreevy, and T. Senthil, Phys. Rev. B 83, 195139 (2011).
. X Wan, A M Turner, A Vishwanath, S Y Savrasov, Phys. Rev. B. 83205101X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011).
. A A Burkov, L Balents, Phys. Rev. Lett. 107127205A. A. Burkov and L. Balents, Phys. Rev. Lett. 107, 127205 (2011).
. K.-Y Yang, Y.-M Lu, Y Ran, Phys. Rev. B. 8475129K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B 84, 075129 (2011).
. W Witczak-Krempa, Y B Kim, Phys. Rev. B. 8545124W. Witczak-Krempa and Y. B. Kim, Phys. Rev. B 85, 045124 (2012).
. A W Sleight, J F Weiher, J. Phys. Chem. Solids. 33679A. W. Sleight and J. F. Weiher, J. Phys. Chem. Solids 33, 679 (1972).
. W Prellier, V Smolyaninova, A Biswas, C Galley, R L Greene, K Ramesha, J Gopalakrishnan, J. Phys.: Condens. Matter. 12965W. Prellier, V. Smolyaninova, A. Biswas, C. Galley, R. L. Greene, K. Ramesha, and J. Gopalakrishnan, J. Phys.: Condens. Matter 12, 965 (2000).
. B C Jeon, C H Kim, S J Moon, W S Choi, H Jeong, Y S Lee, J Yu, C J Won, J H Jung, N Hur, J. Phys.: Condens. Matter. 22345602B. C. Jeon, C. H. Kim, S. J. Moon, W. S. Choi, H. Jeong, Y. S. Lee, J. Yu, C. J. Won, J. H. Jung, N. Hur, et al., J. Phys.: Condens. Matter 22, 345602 (2010).
. J M D Teresa, J M Michalik, J Blasco, P A Algarabel, M R Ibarra, C Kapusta, U Zeitler, Applied Physics Letters. 90252514J. M. D. Teresa, J. M. Michalik, J. Blasco, P. A. Algarabel, M. R. Ibarra, C. Kapusta, and U. Zeitler, Applied Physics Letters 90, 252514 (2007).
. C Azimonte, J C Cezar, E Granado, Q Huang, J W Lynn, J C P Campoy, J Gopalakrishnan, K Ramesha, Phys. Rev. Lett. 9817204C. Azimonte, J. C. Cezar, E. Granado, Q. Huang, J. W. Lynn, J. C. P. Campoy, J. Gopalakrishnan, and K. Rame- sha, Phys. Rev. Lett. 98, 017204 (2007).
. A Winkler, N Narayanan, D Mikhailova, K G Bramnik, H Ehrenberg, H Fuess, G Vaitheeswaran, V Kanchana, A. Winkler, N. Narayanan, D. Mikhailova, K. G. Bramnik, H. Ehrenberg, H. Fuess, G. Vaitheeswaran, V. Kanchana,
. F Wilhelm, A Rogalev, New Journal of Physics. 1173047F. Wilhelm, A. Rogalev, et al., New Journal of Physics 11, 073047 (2009).
. A A Burkov, M D Hook, L Balents, Phys. Rev. B. 84235126A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev. B 84, 235126 (2011).
. G B Halász, L Balents, Phys. Rev. B. 8535103G. B. Halász and L. Balents, Phys. Rev. B 85, 035103 (2012).
. O Erten, O N Meetei, A Mukherjee, M Randeria, N Trivedi, P Woodward, Phys. Rev. Lett. 107257201O. Erten, O. N. Meetei, A. Mukherjee, M. Randeria, N. Trivedi, and P. Woodward, Phys. Rev. Lett. 107, 257201 (2011).
. O Erten, O N Meetei, A Mukherjee, M Randeria, N Trivedi, P Woodward, Phys. Rev. B. 87165105O. Erten, O. N. Meetei, A. Mukherjee, M. Randeria, N. Trivedi, and P. Woodward, Phys. Rev. B 87, 165105 (2013).
. O N Meetei, O Erten, A Mukherjee, M Randeria, N Trivedi, P Woodward, Phys. Rev. B. 87165104O. N. Meetei, O. Erten, A. Mukherjee, M. Randeria, N. Trivedi, and P. Woodward, Phys. Rev. B 87, 165104 (2013).
P Fazekas, Lectures Notes on Electron Correlation and Magnetism. World ScientificP. Fazekas, Lectures Notes on Electron Correlation and Magnetism (World Scientific, 1999).
. J Gopalakrishnan, A Chattopadhyay, S B Ogale, T Venkatesan, R L Greene, A J Millis, K Ramesha, B Hannoyer, G Marest, Phys. Rev. B. 629538J. Gopalakrishnan, A. Chattopadhyay, S. B. Ogale, T. Venkatesan, R. L. Greene, A. J. Millis, K. Ramesha, B. Hannoyer, and G. Marest, Phys. Rev. B 62, 9538 (2000).
. Y Chen, D L Bergman, A A Burkov, 1305.0183ArXiv e-printsY. Chen, D. L. Bergman, and A. A. Burkov, ArXiv e-prints (2013), 1305.0183.
. G Sundaram, Q Niu, Phys. Rev. B. 5914915G. Sundaram and Q. Niu, Phys. Rev. B 59, 14915 (1999).
. N Nagaosa, J Sinova, S Onoda, A H Macdonald, N P Ong, Rev. Mod. Phys. 821539N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010).
. T Fukui, Y Hatsugai, H Suzuki, Journal of the Physical Society of Japan. 741674T. Fukui, Y. Hatsugai, and H. Suzuki, Journal of the Phys- ical Society of Japan 74, 1674 (2005).
The full set of Weyl nodes -their location, charges, and dependence on the direction of the magnetization vectorwill be discussed elsewhere. A. M. Cook, A. A. Burkov, and A. Paramekantiwork in progressThe full set of Weyl nodes -their location, charges, and dependence on the direction of the magnetization vector - will be discussed elsewhere. (A. M. Cook, A. A. Burkov, and A. Paramekanti, work in progress)
Each such 2D band can have a momentum dependent Berry curvature and a nonzero Chern number, thus yielding a quantum Hall insulator for a filled band. Weyl nodes, which act as 'monopoles' in momentum space -sources or sinks of integer quanta of Berry flux 25-27 -correspond to quantum Hall transitions in momentum space where the Chern number jumps as a function of k3. Integrating the Berry curvature, obtained by using gauge invariant plaquette products of wavefunction overlaps. 42An equivalent route to computing σxy is to view the 3D band dispersion ε k as a sequence of 2D band structures parameterized by the momentum k3 along one direction, 26,42 writing it as ε [k 3 ] (k1, k2). 45 over all the k3 slices yields the total σxyAn equivalent route to computing σxy is to view the 3D band dispersion ε k as a sequence of 2D band structures pa- rameterized by the momentum k3 along one direction, 26,42 writing it as ε [k 3 ] (k1, k2). Each such 2D band can have a momentum dependent Berry curvature and a nonzero Chern number, thus yielding a quantum Hall insulator for a filled band. Weyl nodes, which act as 'monopoles' in mo- mentum space -sources or sinks of integer quanta of Berry flux 25-27 -correspond to quantum Hall transitions in mo- mentum space where the Chern number jumps as a func- tion of k3. Integrating the Berry curvature, obtained by using gauge invariant plaquette products of wavefunction overlaps, 42,45 over all the k3 slices yields the total σxy.
| [] |
[
"Spatial analysis and prediction of COVID-19 spread in South Africa after lockdown",
"Spatial analysis and prediction of COVID-19 spread in South Africa after lockdown"
] | [
"Mohammad Arashi ",
"Andriette Bekker [email protected] \nDepartment of Statistics\nFaculty of Natural and Agricultural Science\nUniversity of Pretoria\nPretoriaSouth Africa\n",
"Mahdi Salehi [email protected] \nDepartment of Mathematics and Statistics\nFaculty of Basic Sciences\nUniversity of Neyshabur\nIran\n",
"Sollie Millard [email protected] \nDepartment of Statistics\nFaculty of Natural and Agricultural Science\nUniversity of Pretoria\nPretoriaSouth Africa\n",
"Barend Erasmus [email protected] \nOffice of the Dean\nFaculty of Natural and Agricultural Science\nUniversity of Pretoria\nPretoriaSouth\n",
"Tanita Cronje [email protected] \nDepartment of Statistics\nFaculty of Natural and Agricultural Science\nUniversity of Pretoria\nPretoriaSouth Africa\n",
"Mohammad Golpaygani \nDepartment of Statistics\nFaculty of Sirjan School of Medical Sciences\nSirjanIran\n",
"\nDepartment of Statistics\nFaculty of Mathematical Science\nShahrood University of Technology\nShahroodIran\n"
] | [
"Department of Statistics\nFaculty of Natural and Agricultural Science\nUniversity of Pretoria\nPretoriaSouth Africa",
"Department of Mathematics and Statistics\nFaculty of Basic Sciences\nUniversity of Neyshabur\nIran",
"Department of Statistics\nFaculty of Natural and Agricultural Science\nUniversity of Pretoria\nPretoriaSouth Africa",
"Office of the Dean\nFaculty of Natural and Agricultural Science\nUniversity of Pretoria\nPretoriaSouth",
"Department of Statistics\nFaculty of Natural and Agricultural Science\nUniversity of Pretoria\nPretoriaSouth Africa",
"Department of Statistics\nFaculty of Sirjan School of Medical Sciences\nSirjanIran",
"Department of Statistics\nFaculty of Mathematical Science\nShahrood University of Technology\nShahroodIran"
] | [] | What is the impact of COVID-19 on South Africa? This paper envisages assisting researchers and decision-makers in battling the COVID-19 pandemic focusing on South Africa. This paper focuses on the spread of the disease by applying heatmap retrieval of hotspot areas and spatial analysis is carried out using the Moran index. For capturing spatial autocorrelation between the provinces of South Africa, the adjacent, as well as the geographical distance measures, are used as a weight matrix for both absolute and relative counts. Furthermore, generalized logistic growth curve modeling is used for the prediction of the COVID-19 spread. We expect this data-driven modeling to provide some insights into hotspot identification and timeous action controlling the spread of the virus.Figure 9. Observed cumulative confirmed cases (the black points), the fitted logistic growth model (the red line) given by(2)and the corresponding 95% confidence interval for EC province. | null | null | 218,684,818 | 2005.09596 | 21134e38ac3734206970297aff3f9ab6a6290e83 |
Spatial analysis and prediction of COVID-19 spread in South Africa after lockdown
Mohammad Arashi
Andriette Bekker [email protected]
Department of Statistics
Faculty of Natural and Agricultural Science
University of Pretoria
PretoriaSouth Africa
Mahdi Salehi [email protected]
Department of Mathematics and Statistics
Faculty of Basic Sciences
University of Neyshabur
Iran
Sollie Millard [email protected]
Department of Statistics
Faculty of Natural and Agricultural Science
University of Pretoria
PretoriaSouth Africa
Barend Erasmus [email protected]
Office of the Dean
Faculty of Natural and Agricultural Science
University of Pretoria
PretoriaSouth
Tanita Cronje [email protected]
Department of Statistics
Faculty of Natural and Agricultural Science
University of Pretoria
PretoriaSouth Africa
Mohammad Golpaygani
Department of Statistics
Faculty of Sirjan School of Medical Sciences
SirjanIran
Department of Statistics
Faculty of Mathematical Science
Shahrood University of Technology
ShahroodIran
Spatial analysis and prediction of COVID-19 spread in South Africa after lockdown
Adjacent distanceCOVID-19Geographical distanceLogistic modellingMoran's ISouth Africa
What is the impact of COVID-19 on South Africa? This paper envisages assisting researchers and decision-makers in battling the COVID-19 pandemic focusing on South Africa. This paper focuses on the spread of the disease by applying heatmap retrieval of hotspot areas and spatial analysis is carried out using the Moran index. For capturing spatial autocorrelation between the provinces of South Africa, the adjacent, as well as the geographical distance measures, are used as a weight matrix for both absolute and relative counts. Furthermore, generalized logistic growth curve modeling is used for the prediction of the COVID-19 spread. We expect this data-driven modeling to provide some insights into hotspot identification and timeous action controlling the spread of the virus.Figure 9. Observed cumulative confirmed cases (the black points), the fitted logistic growth model (the red line) given by(2)and the corresponding 95% confidence interval for EC province.
Introduction
During December 2019, several cases of pneumonia of an unknown aetiology were reported in Wuhan, a city within the Hubei province of China ( [1]). Within a week investigators found that the initial cases where all associated with a seafood market where live poultry and wild animals were being sold ( [2]). Since then the disease has been registered, and become known, as the coronavirus or COVID-19 which is caused by the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV2). This disease has shown that in early stages of infection, symptoms of severe acute respiratory infection can occur. These may include a cough, fever and shortness of breath ( [3]). Some patients may then develop acute respiratory distress syndrome (ARDS) and other serious complications which may potentially lead to multiple organ failure ( [1]). Since mid-December, COVID-19 has spread to all seven continents, increasing its prevalence throughout the entire world, and was declared a pandemic, by the World Health Organization (WHO) on the 11th of March ( [4]). This rapid spread has been fuelled by the fact that the majority of infected people do not experience severe symptoms, which makes it more likely to remain mobile and hence infect others ( [5]). The transmission primarily occurs through contact from person to person, coughing or sneezing and touching of contaminated surfaces ( [6]).
On the 14th of February the first case of COVID-19 was reported in Africa, in the City of Cairo, by the Egyptian Ministry of Health and Population. The individual, who travelled between China and Cairo on a business trip was identified through contact screening ( [7]). The first South African case was confirmed by The National Institute for Communicable Diseases (NICD) on the morning of the 5th of March 2020. The patient, a 38 year old male, was part of a group of 10 people, including his wife, who arrived back in South Africa on the 1st of March from Europe. Since then the number of infections and deaths have risen drastically. President Cyril Rhamaphosa was praised by the director-general of the WHO, Dr Tedros Adganom Ghemreyesus, for his leadership and approach to protecting South Africans during these trying times ( [8]). The British Broadcasting Corporation (BBC) also commended President Cyril Ramaphosa for his leadership and for South Africa's "ruthlessly efficient" response to the coronavirus ( [9]). On the 15th of March President Cyril Ramaphosa declared a national state of disaster, the terms of the Disaster Management Act which enable the focus to be put on preventing and reducing the risk of the virus spreading ( [10]), and only a few days later on the 23rd of March the President declared a national lockdown commencing on midnight of the 26th of March.
In South Africa, these extreme measures are absolutely necessary, as the country contains a high risk population combined with low-income country characteristics. The main concerns, which are thought to escalate the spread of the coronavirus, are the large and densely populated areas and townships, including a high level of poverty and movement within these areas. Combined with existing epidemics such as the human immunodeficiency virus (HIV), tuberculosis (TB) and malaria, this might lead to an increase in morbidity and mortality. Since the wide spread of non-communicable diseases, such as chronic obstructive pulmonary disease (COPD), heart disease, hypertension and diabetes, in Africa are known risk factors for severe cases of COVID-19 these may also increase the death rate in these lowerincome countries ( [11]). As winter is approaching, overcrowded houses and the large immunocompromised population, will contribute to the increase in the number of COVID-19 cases rising ( [12]).
To date, as reported by the Coronavirus disease 2019 [13], South Africa has the largest number of COVID-19 cases in Africa. Although it has been shown that South Africans are generally complying during the lockdown, by investigating insights from vehicle-tracking data, is was shown that vehicle activity dropped by 20% even before the lockdown and reduced by 75% after the lockdown was implemented ( [14]). This decline in movement directly indicates the effect on the economy, with the closure of businesses like manufacturing, retail and restaurants, to name only a few. With numerous businesses no longer operating, many South Africans are no longer receiving an income. The Human Science Research Council also released a note on the mental health of South Africans, stating that the stage is already set for major mental health implications, and noting that that failure to put measures into place to mitigate the psychological impacts of quarantine, is likely to lead to an ineffective and slow economic recovery ( [15]).
In this study, we make use of the Moran index to spatially identify the spread in South Africa with respect to the provinces for the COVID-19 infections. Finding these hotspots will provide insights in identifying, and assist in tracking, the COVID-19 spread. With this information South Africa will be better able to predict local outbreaks and develop public health policies to better manage and update medical procedures currently set in place. Since the strict lockdown (level 5) is phased out from the 1st of May (moved to level 4 lockdown), the location of these hotspots could assist in guiding the riskadjusted strategy and the economic activity plan, set out by the South African government. It would thus be important to know where these hotspots are and if they are statistically significant. Further, a generalized logistic model of the growth trend will be employed to show the difference between the hotspot areas and the areas outside of it. With the continuing growth and development of COVID-19 in South Africa, this analysis might be helpful to guide political leaders and health authorities to manage the allocation of resources and prepare for future virus control.
The effect of COVID-19 is still in early stages in South Africa but different tendencies have already been observed when compared to the US and other European countries ( [13]). Understanding these tendencies will be very important in guiding the fight against COVID-19 in South African as well as the rest of Africa. This is an initial study from which many other interesting studies will follow and it will be very important to continue with analyses as more cases are reported and more data becomes available. South Africa, with the most confirmed COVID-19 cases, will need to be the leader in guiding the fight against COVID-19 in Africa.
Study Area and Materials
South Africa, formally known as the Republic of South Africa, is situated at the southernmost tip of Africa and covers a surface area of 1 219 602 km. With a coastline stretching more than 3000km from the desert border of Namibia touching the Atlantic Ocean, around the tip of Africa to the northern bordered of Mozambique on the Indian Ocean side. South Africa shares common boundaries with Namibia, Botswana, Zimbabwe, Swaziland, with the Mountain Kingdom of Lesotho landlocked by SA. The Prince Edward and Marion islands lie some 1 920km south-east of Cape Town ( [16]).
With a population of more than 59 million, South Africa is the world's 25'th most populated nation consisting of nine different provinces. South Africa has three designated capital cities; executive Pretoria, judicial Bloemfontein and legislative Cape Town. The largest city and main economic hub being Johannesburg, which is also the main entry point for visitors from other countries via OR Thambo International Airport ( [16]).
The following timeline of the major interventions in South Africa for the COVID-19 outbreak and the statistics are shown in Figures 1 and 2 respectively.
March 2020
April 2020 May 2020 Distances between the provinces are determined using the main city from each of the provinces. The main city is the city most likely to be the highest risk for COVID-19 infection. In most cases, the main city is also the capital city of the province. The main cities are indicated in Table 1.
Methods
The spatial correlation between the 9 provinces, Northern Cape, Eastern Cape, Free State, Western Cape, Limpopo, North West, KwaZulu-Natal, Mpumalanga, and Gauteng in South Africa, we use the Moran's autocorrelation coefficient, also known as the Moran index (denoted by I) in geographic health science. Furthermore, we make use of generalized logistic function (GLF) for identifying an appropriate growth curve of COVID-19. Hence, this section is devoted to the definition of Moran index (Moran's I) as well as the GLF.
Spatial correlation coefficient
The Moran index, originally defined by [17], is a measure of spatial association or spatial autocorrelation which can be used to find spatial hotspots or clusters and is available in many software applications. This index has been defined as the measure of choice for scientists, specifically in environmental sciences, ecology and public health ( [18]). Some other indices include the Getis' G index, Geary's C, local Ii and Gi, spatial scan statistics and Tango's C index ( [19] and [18]). The Moran index has both a local and global representation. The global Moran's I is a global measure for spatial autocorrelation while the local Moran's I index examines the individual locations, enabling hotspots to be identified based on comparisons to the neighbouring locations ( [19]). This local Moran's I has been successfully applied to hotspot identification for infection clusters such as those investigated by [20], who researched the bovine tuberculosis breakdowns (bTB )in Northern Irish cattle herds in order to access the spatial association in the number and prevalence of chronic bTB across Northern Ireland. Other areas where this index has been successfully applied and commonly used are diseases, mortality rates, environmental planning and environmental sciences. It's important to note that the result can be affected by the definition of the weight function, data transformation and existence of outliers ( [19]).
Until now, not many COVID-19 related research has made use of the Moran Index and no research was found for South African specific cases. Some studies that include the use of this index are: 1) [21] explored the spatial epidemic dynamic of COVID-19 in mainland China in order to determine whether a spatial association of the COVID-19 infection existed; 2) [22] applied the Moran index to a spatial panel which showed that COVID-19 infection is spatially dependent and mainly spread from Hubei Province in Central Chine to neighbouring areas; 3) [23] used a global dataset of COVID-19 cases as well as a global climate database and investigated how climate parameters could contribute to the growth rate of COVID-19 cases while simultaneously controlling for potential confounding effects using spatial analysis; 4) [24] used data on all mobile phone users to examine the impact of the Coronavirus outbreak under the Swedish mild recommendations and restrictions regime on individual mobility and if the changes in geographical mobility vary over different socio-economic strata and 5) [25] investigated the influence of spatial proximities and travel patterns from Italy on the further spread of the SARS-CoV-2 around the globe.
This index is an extension of the Pearson's product-moment correlation coefficient for spatial pattern recognition. Observations in close proximity are more likely to be similar than those far apart ( [26][27]). In order to formulate the Moran index for our purpose, assume we have provinces and the pair ( , ) is for the attribute (variable) in provinces , = 1, ⋯ , , respectively. Then, the spatial weight quantifies the level of closeness between and and the Moran index is defined by
ℐ = × ∑ ∑ ( − ̅ )( − ̅ ) =1 =1 ∑ ( − ̅ ) 2 =1 ,(1)
where ̅ = −1 ∑ =1 and = ∑ ∑ =1
=1
; ≠ .
The Moran's ℐ takes value on [−1,1] and ℐ = 0 shows no spatial correlation between the provinces for the underlying attribute. According to [28], there are two ways to identify the weights. In our context, we identify the ( , )-th element of the weight matrix , from taxonomic level classification viewpoint, as = { 1 if the provinces i and j are connected 0 otherwise
Using the phylogenetic tree classification (geographical distance), we assign the weights following
= { 1 ≤ 0 >
where is the distance between the province centre and province centre , is a distance threshold, and is a power level parameter. See [29] for more detail and comparison between different weights.
Modeling population growth
In this section, we predict some attributes via logistic growth curve modeling. The logistic function/curve is commonly used for dynamic modeling in many branches of science including chemistry, physics, material science, forestry, disease progression, sociology, etc. For our purpose and generality, we follow the Richards' differential equation (RDE) due to [30] given by
( ) = 1 (1 − ( ( ) ) ) ( )
with initial condition ( ) = , is the carrying capacity, the maximum capacity or total population here, , > 0 to obtain the generalized logistic curve (GLC)
( ) = (1 + − ( − ) ) 1 (2) with = ( ) − 1.
The typical logistic curve which is widely used in modeling, is the special of the GLC for = 1. Further, the Gompertz curve can be obtained for the limiting case → 0 + . See [31] for more details and applications of the GLC.
While only a few studies applied the logistic growth models to COVID-19 specific research questions, only one combined the model with the use of the Moran index to show that the infection is spatially dependent ( [22]), with no studies for South African data. Some of these studies, which applied only the logistic growth model include; 1) [32] who uses the logistic growth equation to describe the process on a macroscopic level and 2) [33] who reviews the epidemic virus growth and decline curves in China using the phenomenological logistic growth model.
Results and Discussions
In this section, we start off with a general inspection on the provincial distribution of the total confirmed and death cases given in Figures 3 and 4, respectively. From these figures it is observed that the heatmaps of confirmed and death cases agree, and therefore more confirmed cases are followed by more deaths. Furthermore, it is observed that the hotspots are Western Cape and Gauteng, with the former the highest risk of infection. In order to test the spatial autocorrelation of COVID-19 in South Africa, the interaction between provinces is estimated using Moran's I from March 21, 2020 to April 25, 2020 based on absolute counts by using Eq. (1) and the results are reported in Table 2.
Based on the adjacent 0-1 weight matrix, not all Moran coefficients are significant at the significance level of 5% (see the fifth column in Table 2), and the values are around the interval of [0, 1], since there is a positive correlation among the confirmed cases according to the geographical structure. Comparatively, no significant spatial correlation is tested out based on spatial geographic distance (the last column of Table 2), which indicates the spreading direction in South Africa is mainly based on adjacent areas to neighbors, and doesn't matter how far the distance to the infectious center. So main cities adjacent to are at higher risk. To extend the analysis, we calculated the corresponding p-values of Moran's test given by Table 2 over the time, shown in Figure 5. Comparing the corresponding p-values of Moran's test, some deviation exist in the statistical timeliness in main cities of provinces from March 21, 2020 to April 25 so it is inevitable that maybe bias occurred in our results. From April 20 to 25, the spatial autocorrelation is significantly different in terms of adjacency to main cities. This means that the prevalence of COVID-19 varies in main cities. Additional analysis has been taken into account to validate the results based on the Moran index characteristics. The observed Moran index along with its expected value and standard error are tabulated in Table 3. The expected value of Moran index is -1/(N-1)=-1/8=-0.125 in our case. Using the adjacency weights, we obtain the same result as discussed based on Table 2. However, since the null hypothesis that there is no spatial autocorrelation between the provinces is not rejected for the geographical distance, we can argue there is no evidence of negative auto-correlation here, as with random data you would expect it to be a negative value more often than positive.
The impact of President Cyril Ramaphosa's decision in containing the outbreak by strict lockdown regulations is supported by the p-values in Figure 6 (see dates 20 April and onwards).
In addition, the spatial autocorrelation of COVID-19 in South Africa based on the relative counts, has been estimated using I from March 21, 2020 to April 25, 2020, and the results are reported in Table 3.
In this table, for the adjacent 0-1 weights, more coefficients are significant at the level of 5% (see the fifth column in Table 4), with values in the interval [0, 1], since there is no positive correlation among the confirmed cases according to the geographical structure. Comparatively, very few significant spatial correlations are measured based on the spatial geographic distance (the last column of Table 3), which indicates that the spread direction in South Africa is mainly based on adjacent areas to neighbours, and less so on the distance to the infection centre. In conclusion, as it is identified in heatmaps Figures 3 and 4 the two main cities are at higher risk. We also calculated the corresponding p-values of Moran's test given by Table 3 over the time, shown in Figure 7.
Comparing the corresponding p-values of Moran's test, some deviation exist in the statistical timeliness in main cities of provinces from March 21, 2020 to April 25 so it is inevitable that bias occurs in the results. April 20 to 25 is not significantly different in terms of adjacency to main cities. This means that the prevalence of COVID-19 based on relative counts, is the similar in the main cities. Table 4 over the time, according to the relative counts (absolute counts divided by 1M residents). Table 2 over the time, according to the relative counts. Figure 9 displays observed cumulative confirmed cases, the fitted logistic growth model given by Eq.
(2) with ν=1 and the corresponding 95% confidence interval for each province. However, the model in Eq. (2) was not fitted on the data associated with the WC province. As it is seen from Figure 9, the cumulative number of confirmed cases in all provinces is described very well by a logistic growth model. The high values of the R-Squared, shown at the bottom-right of each figure, also confirms the goodness of fit of all 8 models, with FS having the lowest R-Square (0.939). Hence, we can rely on predictions given as red lines until June 5 proportionally to the magnitude of the given R-Squared for each province.
Conclusions
Despite the inaccuracies associated with medical predictions, identifying hot spots and logistic modelling is still invaluable for better understanding of the spread in South Africa. The results of the Moran index showed the impact of President Cyril Ramaphosa's decision in containing the outbreak by strict lockdown regulations and the inter provisional travelling prohibition has a positive role in tapering the counts. The results indicated that the spreading direction in South Africa is mainly based on adjacent areas to neighbours, and doesn't matter how far the distance is to the infectious centre.
The logistic growth models show a good fit to the provincial data, with R-Square values above 0.9. Visually however it is clear that for certain provinces a different modelling strategy could yield even better results ( [36]). These provinces are GP, FS and NC and likely also WC.
With South Africa phasing out the lockdown as of the beginning of May, implementing the riskadjusted strategy and economic activity plan, South African will be seeing workers returning to their workplace and COVID-19 cases are expected to increase. This initial study highlights the importance of continued analysis and showcases the valuable input that can be obtained from these analyses results.
Figure 1 .
1Timeline of major interventions.
Figure 2 .
2Covid-19 epidemic in South Africa. The Data Science for Social Impact research (DSFSI) group at the University of Pretoria captures the COVID-19 data, number of cases, on national and provincial level. Missing values and anomalies in the provincial data are adjusted or imputed using data from the University of Cape Town COVID-19 dashboard. The demographic data used are the provincial population numbers published by. Number of recoveries and deaths used are those published by the NICD.
Figure 3 .
3The heat map of the provincial distribution of total confirmed cases of COVID-19 up to April 25 in South Arica.
Figure 4 .
4The heat map of the provincial distribution of total deaths of COVID-19 up to 25 April in South Arica.
Figure 5 .
5The corresponding p-values of Moran's test given byTable 2over the time, on the basis of the absolute counts.
Figure 6 .
6The difference between Moran's observed and expected values given byTable 2over the time, on the basis of the absolute counts.
Figure 7 .
7The corresponding p-values of Moran's test given by
Figure 8 .
8The difference between Moran's observed and expected values given by
Table 1 .
1Main cities in each of the provinces of South AfricaProvince
Main City
Eastern Cape (EC)
Port Elizabeth
Free State (FS)
Bloemfontein
Gauteng (GP)
Johannesburg
Kwazulu Natal (KZ)
Durban
Limpopo (LP)
Polokwane
Mpumalanga (MP)
Mbombela
North Cape (NC)
Kimberley
North West (NW)
Klerksdorp
Western Cape (WC)
Cape Town
Table 2 .
2Moran's ℐ (observed), its expected value and the corresponding test for different days based on absolute
counts.
Measurement by adjacency
Measurement by geographical distance
Date
Obs. Exp. sd p-value
Obs. Exp. sd p-value
Table 3 .
3Moran's ℐ (observed), its expected value and the corresponding test for different days based on relative counts.Measurement by adjacency
Measurement by geographical distance
Date
Obs. Exp. sd p-value
Obs. Exp. sd p-value
Mathematical description of biological growth (i.e. population models) is very important in many research disciplines. Amongst these population models, especially noteworthy are the clear analytical solutions of the generalized logistic functions ([34]).
Conflicts of Interest:The authors declare no conflict of interest.
Diagnostic utlity of clinical laboratory data daterminations for patients with the severe COVID-19. Y Gao, T Li, M Han, X Li, D Wu, J Med Virol. Gao, Y.; Li, T.; Han, M.; Li, X.; Wu, D. Diagnostic utlity of clinical laboratory data daterminations for patients with the severe COVID-19. J Med Virol 2020.
COVID-19: a fast evolving pandemic. J Whitworth, Trans R Soc Trop Med Hyg. Whitworth, J. COVID-19: a fast evolving pandemic. Trans R Soc Trop Med Hyg 2020, pp. 227-228.
COVID-19 Outbreak on Malawi Perspective. G C Munthali, W Xuelian, Electron J Gen Med. 17Munthali, G.C.; Xuelian, W. COVID-19 Outbreak on Malawi Perspective. Electron J Gen Med 2020, pp. 17.
WHO Director-General's opening remarks at the media briefing on COVID-19-11. T A Ghebreyesus, World Heal Organ. 2020Ghebreyesus, T.A. WHO Director-General's opening remarks at the media briefing on COVID-19- 11 March 2020. World Heal Organ Speeches 2020, Available: https://www.who.int/dg/speeches/detail/who-director-general-s-opening-remarks-at-the-media- briefing-on-covid-19---11-march-2020. Accessed 28 April 2020.
Covert coronavirus infections could be seeding new outbreaks. J Qiu, Nature. 2020Qiu, J. Covert coronavirus infections could be seeding new outbreaks. Nature 2020, Available: https://www.nature.com/articles/d41586-020-00822-x. Accessed 27 April 2020.
Three Emerging Coronaviruses in Two Decades: The Story of SARS, MERS, and Now COVID-19. J Guarner, Am J Clin Pathol. 153Guarner, J. Three Emerging Coronaviruses in Two Decades: The Story of SARS, MERS, and Now COVID-19. Am J Clin Pathol 2020 Volume 153, pp. 420-421.
Imaging clinic operations in the times of COVID-19: Strategies, Precautions and Experiences. J Czernin, S Fanti, P T Meyer, M Allen-Auerbach, M Hacker, J Nucl Med. Czernin, J.; Fanti, S.; Meyer, P.T.; Allen-Auerbach, M.; Hacker, M. et al. Imaging clinic operations in the times of COVID-19: Strategies, Precautions and Experiences. J Nucl Med 2020, pp. 1-5.
OIL 2020. OIL 2020, Available:https://www.iol.co.za/news/politics/coronavirus-in-sa-who-boss-praises- south-africas response-to-covid-19-pandemic-45923836. Accessed 27 April 2020.
Ruthlessly Efficient' Fight Against COVID-19. R Pitjeng, Bbc Commends, Sa's ', 2020Pitjeng, R. BBC Commends SA'S 'Ruthlessly Efficient' Fight Against COVID-19. EWN 2020, Available: https://ewn.co.za/2020/04/03/coronavirus-sa-acted-faster-more-efficiently-than-other-countries- bbc. Accessed 27 April 2020.
SANews President Ramaphosa Announces A Nationwide Lockdown 2020. SANews President Ramaphosa Announces A Nationwide Lockdown 2020, Available:https://www.sanews.gov.za/south-africa/president-ramaphosa-announces-nationwide- lockdown. Accessed 27 April 2020.
. F Thienemann, F Pinto, D E Grobbee, M Boehm, N Bazargani, World Heart. 2020Thienemann, F.; Pinto, F.; Grobbee, D.E.; Boehm, M.; Bazargani, N. World Heart Federation 2020.
World Health Oranization Update on COVID-19 in the Eastern Mediterranean Region. Media Cent. 2020 Available. World Health Oranization Update on COVID-19 in the Eastern Mediterranean Region. Media Cent. 2020 Available:http://www.emro.who.int/media/news/update-on-covid-19-in-the-eastern- mediterranean-region.html. Accessed 27 April 2020.
World Health Organization (2020) Coronavirus disease 2019 (COVID-19) Situation Report. 97World Health Organization (2020) Coronavirus disease 2019 (COVID-19) Situation Report, pp. 97.
DispatchLive Tracker Reveals Whether Motorists Have Kept To Lockdown Rules. DispatchLive 2020. DispatchLive Tracker Reveals Whether Motorists Have Kept To Lockdown Rules. DispatchLive 2020, Available: https://www.dispatchlive.co.za/news/2020-04-03-tracker-reveals-whether- motorists-have-kept-to-lockdown-rules/. Accessed 28 April 2020.
Living through global trauma: Mental-health implications of COVID-19 from a developing country perspective. A Teagle, Humarn Sci Res. 2020Teagle, A. Living through global trauma: Mental-health implications of COVID-19 from a developing country perspective. Humarn Sci Res Counc 2020, Available:http://www.hsrc.ac.za/en/news/general/mental-health-covid-19. Accessed 28 April 2020.
Government Communications 2020, United Nations: Department of Economic and Social Affairs. E Tibane, South Africa Yearbook, World26th Editi. (GCIS) GC, editor (GCIS. Population DevisionTibane, E. South Africa Yearbook 2018/19, 26th Editi. (GCIS) GC, editor (GCIS), Government Communications 2020, United Nations: Department of Economic and Social Affairs (Population Devision) (2019) World.
. P A Moran, P. Notes on Continuous Stochastic Phenomena. Biometrika. 37Moran, P.A.P. Notes on Continuous Stochastic Phenomena. Biometrika 2018 Volume 37, pp. 17- 23.
Comparing implementations of global and local indicators of spatial association. R S Bivand, D W Wong, Test. 27Bivand, R.S.; Wong, D.W.S. Comparing implementations of global and local indicators of spatial association. Test 2018 Volume 27, pp. 716-748.
Use of local Moran's I and GIS to identify pollution hotspots of Pb in urban soils of Galway. C Zhang, L Luo, W Xu, V Ledwith, Ireland. Sci Total Environ. 398Zhang, C.; Luo, L.; Xu, W.; Ledwith, V. Use of local Moran's I and GIS to identify pollution hotspots of Pb in urban soils of Galway, Ireland. Sci Total Environ 2008, Volume 398, pp. 212-221.
Spatiotemporal analysis of prolonged and recurrent bovine tuberculosis breakdowns in Northern Irish cattle herds reveals a new infection hotspot. G M Milne, J Graham, A Allen, A Lahuerta-Marin, C Mccormick, Spat Spatiotemporal Epidemiol. 28Milne, G.M.; Graham, J.; Allen, A.; Lahuerta-Marin, A.; McCormick, C. Spatiotemporal analysis of prolonged and recurrent bovine tuberculosis breakdowns in Northern Irish cattle herds reveals a new infection hotspot. Spat Spatiotemporal Epidemiol 2019 Volume 28, pp. 33-42.
Spatial epidemic dynamics of the COVID-19 outbreak in China. D Kang, H Choi, J H Kim, J Choi, Int J Infect Dis. 94Kang, D.; Choi, H.; Kim, J.H.; Choi, J. Spatial epidemic dynamics of the COVID-19 outbreak in China. Int J Infect Dis 2020 Volume 94, pp. 96-102.
Spatial-temporal distribution of COVID-19 in China and its prediction: A data-driven modeling analysis. R Huang, M Liu, Y Ding, J Infect Dev Ctries. 143Huang, R.; Liu, M.; Ding, Y. Spatial-temporal distribution of COVID-19 in China and its prediction: A data-driven modeling analysis, J Infect Dev Ctries, 2020 Volume 14(3), pp. 246-253.
Global COVID-19 transmission rate is influenced by precipitation seasonality and the speed of climate temperature warming 2020. K Chiyomaru, K Takemoto, medRxiv: 2020.04.10Chiyomaru, K.; Takemoto, K. Global COVID-19 transmission rate is influenced by precipitation seasonality and the speed of climate temperature warming 2020, medRxiv: 2020.04.10.20060459.
Effects of the COVID-19 Pandemic on Population Mobility under Mild Policies: Causal Evidence from Sweden. M Dahlberg, P.-A Edin, E Grönqvist, J Lyhagen, J Östh, 2004.09087arXiv PreprDahlberg, M.; Edin, P.-A.; Grönqvist, E.; Lyhagen, J.; Östh, J. Effects of the COVID-19 Pandemic on Population Mobility under Mild Policies: Causal Evidence from Sweden 2020. arXiv Prepr 2004.09087.
Change in outbreak epicenter and its impact on the importation rikss of COVID-19 progression: a modelling study 2020. O Adegboye, A Adekunle, A Pak, E Gayawan, D Leung, MedRxiv: 2020.03.17Adegboye, O.; Adekunle, A.; Pak, A.; Gayawan, E.; Leung, D. et al. Change in outbreak epicenter and its impact on the importation rikss of COVID-19 progression: a modelling study 2020, MedRxiv: 2020.03.17.20036681.
Spatial and temporal analysis: autocorrelationin space and time. A D Cliff, J K Ord, Quant Geography: A British View. Routledge & Kegan PaulLondonCliff, A.D.; Ord, J.K. Spatial and temporal analysis: autocorrelationin space and time. In E. N. Wrigley and R. J. Bennett, editors, Quant Geography: A British View, 1981, pp. 104-110. Routledge & Kegan Paul, London.
Moran's autocorrelation coefficient in comparative methods, Package ape, cran.rproject.org. E Paradis, Paradis, E. Moran's autocorrelation coefficient in comparative methods, Package ape, cran.r- project.org, 2019.
Adaptation: statistics and a null model for estimating phylogenetic effects. J L Gittleman, M Kot, Syst Zoology. 39Gittleman, J.L.; Kot, M. Adaptation: statistics and a null model for estimating phylogenetic effects, Syst Zoology 1990 Volume 39, pp. 227-241.
On the four types of weight functions for spatial contiguity matrix. Y Chen, Lett Spatial Res Sci. 5Chen, Y. On the four types of weight functions for spatial contiguity matrix, Lett Spatial Res Sci 2012 Volume 5, pp. 65-72.
A flexible growth function for empirical use. F J Richards, J. Experimental Botany. 102Richards, F.J. A flexible growth function for empirical use, J. Experimental Botany 1959 Volume 10(2), pp. 290-300.
Features and partial derivatives of Bertalanffy-Richards growth model in forestry. Y C Lei, S Y Zhang, Nonlinear Anal. Model. Cont. 91Lei, Y.C.; Zhang, S.Y. Features and partial derivatives of Bertalanffy-Richards growth model in forestry, Nonlinear Anal. Model. Cont. 2004 Volume 9(1), pp. 65-73.
Predicting the ultimate outcome of the COVID-19 outbreak in Italy. G Vattay, 07912arXiv PreprVattay, G. Predicting the ultimate outcome of the COVID-19 outbreak in Italy 2020, arXiv Prepr 2003.07912: 1-4.
Outbreak analysis with a logistic growth model shows 2020 COVID-19 suppression dynamics in China. Y Zou, S Pan, P Zhao, L Han, X Wang, medRxiv: 2020.03.25Zou. Y.; Pan, S.; Zhao, P.; Han, L.; Wang, X. Outbreak analysis with a logistic growth model shows 2020 COVID-19 suppression dynamics in China. medRxiv: 2020.03.25.20043539.
Generalized logistic functions in modelling emergence of Brassica napus L. A Szparaga, S Kocira, PLoS One. 13Szparaga, A.; Kocira, S. Generalized logistic functions in modelling emergence of Brassica napus L. PLoS One 2018, pp. 13.
Coronavirus In SA: WHO Boss Praises South Africa's Response To. J Maromo, Maromo, J. Coronavirus In SA: WHO Boss Praises South Africa's Response To Covid-19 Pandemic 2020.
Modelling the South African 14-day COVID-19 infection rate. To be submitted. S Millard, F Kanfer, Millard, S.; Kanfer F. Modelling the South African 14-day COVID-19 infection rate. To be submitted. 2020.
| [] |
[
"I n i t i a l D r a f t Noised Consistency Training for Text Summarization",
"I n i t i a l D r a f t Noised Consistency Training for Text Summarization"
] | [
"Junnan Liu \nBeihang University\n\n",
"Qianren Mao \nBeihang University\n\n",
"Bang Liu [email protected] \nUniversity of Montreal\n\n",
"Hao Peng \nBeihang University\n\n",
"Hongdong Zhu \nBeihang University\n\n",
"Jianxin Li \nBeihang University\n\n"
] | [
"Beihang University\n",
"Beihang University\n",
"University of Montreal\n",
"Beihang University\n",
"Beihang University\n",
"Beihang University\n"
] | [] | Neural abstractive summarization methods often require large quantities of labeled training data. However, labeling large amounts of summarization data is often prohibitive due to time, financial, and expertise constraints, which has limited the usefulness of summarization systems to practical applications. In this paper, we argue that this limitation can be overcome by a semi-supervised approach: consistency training which is to leverage large amounts of unlabeled data to improve the performance of supervised learning over a small corpus. The consistency regularization semi-supervised learning can regularize model predictions to be invariant to small noise applied to input articles. By adding noised unlabeled corpus to help regularize consistency training, this framework obtains comparative performance without using the full dataset. In particular, we have verified that leveraging large amounts of unlabeled data decently improves the performance of supervised learning over an insufficient labeled dataset. | null | [
"https://arxiv.org/pdf/2105.13635v1.pdf"
] | 235,248,420 | 2105.13635 | 7a605700341f3148affa7f2503bfe90e9c7c3189 |
I n i t i a l D r a f t Noised Consistency Training for Text Summarization
Junnan Liu
Beihang University
Qianren Mao
Beihang University
Bang Liu [email protected]
University of Montreal
Hao Peng
Beihang University
Hongdong Zhu
Beihang University
Jianxin Li
Beihang University
I n i t i a l D r a f t Noised Consistency Training for Text Summarization
10.1145/nnnnnnn.nnnnnnnText SummarizationSemi-supervised LearningConsistency Train- ingLanguage ModelData-augmentation
Neural abstractive summarization methods often require large quantities of labeled training data. However, labeling large amounts of summarization data is often prohibitive due to time, financial, and expertise constraints, which has limited the usefulness of summarization systems to practical applications. In this paper, we argue that this limitation can be overcome by a semi-supervised approach: consistency training which is to leverage large amounts of unlabeled data to improve the performance of supervised learning over a small corpus. The consistency regularization semi-supervised learning can regularize model predictions to be invariant to small noise applied to input articles. By adding noised unlabeled corpus to help regularize consistency training, this framework obtains comparative performance without using the full dataset. In particular, we have verified that leveraging large amounts of unlabeled data decently improves the performance of supervised learning over an insufficient labeled dataset.
INTRODUCTION
Automatic text summarization [15] is a challenging task which generates a condensed version of an input text that captures the original's core meaning. A fundamental requirement of automatic text summarization is that it typically requires a lot of labeled data to work well. However, acquiring handcrafted labels is a costly process, which motivates research on methods that can effectively utilize abundant unlabeled data to improve text summarization performance. We aim to provide an empirical answer to the following research question: what is the efficient way to leverage a large amount of unlabeled data to address the weakness of insufficient annotation for text summarization tasks?
Towards this goal, we first revisit an effective semi-supervised method, consistency training [16], to address the above issues. The consistency training leverages voluminous unlabeled data and employs data augmentation methods to generate diverse and realistic noisy source text, forcing the model to be consistent with these noises. The consistency regularization of semi-supervised learning has been extensively studied on the classification problems, such as text classification [5], image recognition [17], automatic speech recognition [9]. However, it is still unclear how semi-supervised learning works on text summarization tasks.
We believe that a good summarization model should be robust to any articles' small changes. No prior research has ever tested this or a similar idea, so we try to perform consistency training in text summarization tasks. We first transform unlabeled articles into original samples and noisy samples to augment the supervised learning for text summarization with a sizeable unlabeled corpus. Then, consistency learning is used to ensure similar semantics of articles can be mapped to the same or similar output in prediction distributions. Thus, consistent regularization of semi-supervised learning can take effect on supervised learning performance over insufficient labeled data.
We evaluate text summarization on two full labeled datasets: CNN/DailyMail [4], and BBC XSum [8]. We split the full dataset into unlabeled and labeled data, e.g., a small-scale labeled dataset and a large-scale unlabeled dataset. We use two data augmentation methods: back-translation and word replacement with TF-IDF, to transform unlabeled data into noise-injected unlabeled data. The semi-supervised learning framework's joint optimization includes two parts: supervised learning using labeled data, unsupervised consistency learning using unlabeled data, and noise-injected unlabeled data. We show that consistency-regularization semi-supervised learning could lead to a competitive result on a dataset with partial labeled and partial unlabeled data, compared with the performance on a full labeled dataset. In particular, we empirically observed performance gains for consistency training framework, compared with supervised baselines only using the labeled corpus. In summary, the main contributions of our method 1 are:
• Discussing the feasibility of consistency-regularization semisupervised learning in automatic text summarization tasks. • Extensive evaluations demonstrate that consistency training with unsupervised corpus could greatly improve the performance of the text summarization model on a limited dataset.
Related Work
Abstractive text summarization is typically based on sequence-tosequence (seq2seq) neural networks. The emergence of pre-training models for seq2seq learning [12] has extensively promoted the development of sequence generation tasks. Rothe et al. [13] develops a Transformer based seq2seq model that is compatible with publicly available pre-trained BERT [2], GPT-2 [11], and RoBERTa [7] checkpoints. These models result in new salient performances on single document text summarization. In recent work, consistency regularization methods for semisupervised learning [1] have been shown to work well on many classification tasks [16]. The consistency training methods regularize model predictions to be invariant to noise applied to unlabeled examples. Tarvainen and Valpola [14] prove that a model trained with noisy labeled data learns to give consistent predictions around labeled data points. Additionally, high-quality data augmentation methods [16] can replace traditional noise injection methods to improve consistency learning performance. Their work can match and even outperform purely supervised learning that uses affluent labeled data.
Though pre-trained seq2seq learning and consistency training have recently achieved impressive gains in specific tasks respectively, it remains unclear to what extent combining pre-trained language models with consistency training can be beneficial to text summarization tasks.
FRAMEWORK
The consistent regularization of semi-supervised learning leverages unlabeled data and employs data augmentation methods to inject noisy data, and then enforces the summarization model to regularize semi-supervised learning by encouraging consistent predictions. We provide more details on our framework in what follows.
Data Augmentation.
We first transform unlabeled articles into original samples and noisy samples to improve the supervised learning for text summarization with a large unlabeled corpus. We use two advanced augmentation methods: back-translation and word replacement with TF-IDF.
Back-translation. Back-translation [3] refers to translating an existing document from its original language A to another language B, and then translating it back to A to obtain an augmented docu-ment^. As stated by Yu et al. [18], back-translation can produce different expressions while retaining the original sentence's semantics, thereby achieving significant performance improvements in question answering tasks. In our work, we use back-translation to rewrite documents without changing the original intention 2 .
Word replacement with TF-IDF. This method tends to replace words with low TF-IDF scores while keeping high TF-IDF values [16]. Specifically, it would like to preserve keywords whose TF-IDF values are usually higher and replace non-informative words with other non-informative words.
Backbone Model: BERT2BERT
We focus on leveraging BERT [2] for summarization because BERT is often used as a benchmark model. We use BERT2BERT implemented by Rothe et al. [13] as the backbone model of the summarization framework. The BERT2BERT uses the BERT checkpoint to initialize the encoder for better input understanding and mapping the inputs to a context. Additionally, it uses the BERT checkpoint as the decoder to get better text generation from this context. We tokenize text using WordPiece 3 to match the BERT pre-trained vocabulary.
Consistency Learning
A key aspect of consistent training to work well is to add noise to the input data. One assumption as to why noise is beneficial is that it enforces local smoothness for this task. This assumption supports our proposal for the following framework to improve the performance of text summarization models. In the text summarization task, the model generates a summary based on the overall understanding of a document so that a good model should also be invariant to documents with similar intentions. Hence, consistency learning can be used to ensure similar semantics of documents to be mapped to the same or similar output in prediction distributions. Figure 1 gives an overview of our consistency training framework for text summarization. The inputs of the framework are labeled texts , unlabeled texts ′ , and noise injected unlabeled texts^. We use * to denote the gold summaries of labeled texts. Then we use to represent the distributions of data generated by the model, where refers to the model's parameters. Our framework can be summarized as follows:
Firstly, we feed the labeled text into the model to get the distribution ( | ) and calculate the supervised loss:
1 ∑︁ =1 ( * , ( | ))(1)
We then add noises to the texts to perform the consistency training in text summarization task. We generate a noised version^of unlabeled texts ′ using data augmentation methods. Both unlabeled texts and augmented unlabeled texts are fed to the summarization model, and then we get the output distribution of original unlabeled data˜( ′ | ′ ) and an additional noised version of augmented unlabeled data (^|^).
We then calculate the semi-supervised loss between unlabeled texts and augmented unlabeled texts.
1 ∑︁ =1 (˜( ′ | ′ ), (^|^)),(2)
where˜is just a copy of the current parameters indicating that the back-propagation of the gradient is truncated. We use KL divergence loss to perform consistency training; Finally, we combine supervised loss and semi-supervised loss, and train the model * by minimizing the combined loss:
1 ∑︁ =1 ( * , ( | )) + 1 ∑︁ =1 (˜( ′ | ′ ), (^|^)).(3)
EXPERIMENTAL SETUP 3.1 Datasets
We conducted experiments on two datasets: CNN/DailyMail [4], and BBC XSum [8]. CNN/DailyMail includes news articles and corresponding extractive highlights. We used the standard splits [4] for training, validation, and testing, which contains training samples with 287,227 pairs, 13,368 pairs for the development, and 11,490 pairs for the test. BBC XSum's summaries provide a high level of abstraction. The model requires document-level inference and paraphrasing to generate them. It includes 226,711 news articles and corresponding one-sentence summaries. We followed the splits [8] for training, validation, and testing with 20,404, 11,332, 11,334 articlesummary pairs, respectively. We obtained both labeled and unlabeled data from the full dataset. Specifically, we divided parts of the original dataset into labeled data. For the rest data, we deleted the labels and treated them as unlabeled data. For a concise comparison, we set the proportion of labeled data to three versions: 30% labeled data + 70% unlabeled data, 50% labeled data + 50% unlabeled data, and 70% labeled data + 30% unlabeled data.
Training details
During training on CNN/DailyMail dataset, the documents were truncated to 512 tokens, and the length of the summaries was limited to 128 tokens. For the BBC XSum dataset, the documents and summaries were truncated to 512 tokens and 64 tokens, respectively. We used a batch size of 16 for labeled data by default. Generally, semi-supervised learning performs a larger batch size on unlabeled data than labeled data to make full use of large quantities of unlabeled texts, refer to [16]. We implemented different batch sizes for unlabeled data and found that using a batch size of 32 leads to better performance.
All models were trained for 50,000 steps on 3 Tesla V100 GPUs. The learning rate started at 2e-5 and decayed every 1000 steps. We also performed a linear warmup method to increase the learning rate smoothly from 0 to 2e-5 during 2000 steps at the beginning of training. In the procedure of decoding, we used beam search (size 4) and set the 2.0 for the length penalty. The length of predicted summaries was limited to 162 for CNN/DailyMail and 62 for XSum. We also used trigram blocking to reduce repetition [10].
RESULTS ANALYSIS
Our experiments address the following research questions.
• RQ1: How does our proposed consistency training framework on single document text summarization perform? • RQ2: Which of the two data aggregation methods has a greater impact on the overall performance?
Main results of summarization (RQ1)
On CNN/DailyMail dataset. We evaluated the summarization predictions by ROUGE [6] in this paper. Table 1 shows the results on the CNN/DailyMail dataset. The first block of Table 1 contains the baseline for different proportions of labeled data. The second block includes the consistency training models' results. As shown, consistency training models obtain improvements in ROUGE-1 points from 37.85 to 39.06, ROUGE-2 points from 15.58 to 17.16, and ROUGE-L points from 25.07 to 26.51 compared with the baseline on 30% labeled data. In the case where 50% of the data is labeled data, the model using consistency training achieves +0.64 point improvement in ROUGE-1, +0.67 point improvement in ROUGE-2, and +0.60 point improvement in ROUGE-L compared with baseline trained on 50% labeled data. We conclude that:
• The consistent training model trained on the dataset with partially labeled data (30%) and partial unlabeled data (70%) could achieve competitive results, compared to the model trained on the full dataset (100%). • The model trained on the dataset with insufficient labeled data (30% ) and large augmented unlabeled data (70%) performs performance gains with a large margin (+1.58 on R2, for instance) compared to the model only trained using 30% labeled data. The results indicate that consistent training with large unlabeled data could improve the supervised model's performance trained on a small labeled corpus.
On BBC XSum dataset. We also conducted experiments to verify if consistent training would be equally effective on the abstractive BBC XSum dataset. It has one-sentence summaries and is more abstractive than the CNN/DailyMail dataset. As shown in Table 2, the first block shows the baseline of the different proportioned labeled data on the XSum dataset. The results on the XSum dataset using consistency training are summarized in the second block. Unexpectedly, the model with consistency training trained by 70% labeled data and 30% unlabeled data achieves similar results over the model trained on the full dataset. The results on XSum represent that our consistency-training method is also effective in generating extreme abstractive summaries.
Data augmentation methods study (RQ2)
We performed experiments to discuss the impact of different data augmentation methods on the performance of consistency training. As shown in Table 3, the models using back-translation for data augmentation are superior to models using TF-IDF replacement. The former models outperform the latter ones by about +0.7 ROUGE-L points gains, +0.75 ROUGE-2 points gains, and +0.61 ROUGE-1 points gains on average. These results show that back-translation leads to a better performance than TF-IDF replacement. We guess that back-translation could add more diversity to the text. On the other hand, backtranslation can maintain the global semantics of sentences so as to maintain the input distribution.
Unlabeled data batch size study (RQ3)
Generally, the batch size of unlabeled data in semi-supervised learning should be larger than the batch size of labeled data in order to make full use of unlabeled data. We performed experiments to investigate whether a larger batch size leads to better performance in text summarization tasks. Figure ?? shows Rouge-L results for different batch sizes of unlabeled data. We find that setting the batch size to 32 is better than setting it to 16, but setting it to 64 is slightly worse than setting it to 32. Sub-figure (b) shows the trend of loss curves under different batch sizes when validation. The batch size of 32 obtains optimal convergence speed.
CONCLUSIONS
Our work demonstrates that it is practical to use unlabeled data to improve the abstractive summarization model's performance. We present a new perspective on effectively using consistency training to improve supervised text summarization over an insufficiently labeled dataset. By substituting simple noise injection operations with advanced data augmentation methods such as back-translation, our method brings substantial improvements across datasets with partial labeled and partial unlabeled data under the same consistency training framework. Our method obtains comparative performance without using the full dataset. Future works include migrating our consistency training framework to other natural language generation tasks, such as Q&A and dialogue generation.
Figure 1 :
1Illustration of noised Consistency Training for Text Summarization. In the figure, refers to the parameters of the model, and˜means it is just a copy of and the gradient will not propagate through it.
Figure 2 :
2The training loss curves under different batch size during validation.
Table 1 :
1ROUGE F1 results on the CNN/DailyMail dataset.Labeled
ROUGE-1 ROUGE-2 ROUGE-L
30%
37.85
15.58
25.07
50%
39.14
17.22
26.62
70%
39.81
17.95
27.32
100%
40.03
18.27
27.61
Consistency training
Labeled + Unlabeled ROUGE-1 ROUGE-2 ROUGE-L
30% + 70%
39.06
17.16
26.51
50% + 50%
39.78
17.89
27.22
70% + 30%
39.95
18.13
27.52
• RQ3: How does the main hyperparameter of batch size affect the
model's performance?
Table 2 :
2ROUGE F1 results on the XSum dataset.Labeled
ROUGE-1 ROUGE-2 ROUGE-L
30%
33.32
12.08
26.29
50%
35.31
13.85
28.20
70%
36.17
14.55
29.04
100%
36.90
15.28
29.71
Consistency training
Labeled + Unlabeled ROUGE-1 ROUGE-2 ROUGE-L
30% + 70%
35.06
13.79
28.15
50% + 50%
36.09
14.41
28.93
70% + 30%
36.92
15.17
29.70
Table 3 :
3ROUGE F1 results on the CNN/DailyMail using backtranslation or TF-IDF replacement on our consistency training framework. 83(↓ 0.12%) 18.05(↓ 0.08%) 27.54(↓ 0.02%) Only using TF-IDF replacement 30% + 70% 37.72(↓ 1.34%) 15.64(↓ 1.52%) 25.07(↓ 1.44%) 50% + 50% 39.19(↓ 0.59%) 17.25(↓ 0.64%) 26.68(↓ 0.54%) 70% + 30% 39.67(↓ 0.28%) 17.89(↓ 0.24%) 27.33(↓ 0.19%)Labeled + Unlabeled
ROUGE-1
ROUGE-2
ROUGE-L
Only using Back-translation
30% + 70%
38.83(↓ 0.23%) 17.11(↓ 0.23%) 26.28(↓ 0.23%)
50% + 50%
39.74(↓ 0.04%) 17.87(↓ 0.02%) 27.13(↓ 0.09%)
70% + 30%
39.
ͷͲͲͲ ͳͲͲͲͲ ͳͷͲͲͲ ʹͲͲͲͲ ʹͷͲͲͲ ͵ͲͲͲͲ ͵ͷͲͲͲ ͶͲͲͲͲ ͶͷͲͲͲͳǤͷ
ʹǤͲ
ʹǤͷ
͵ǤͲ
͵Ǥͷ
ͶǤͲ
ͶǤͷ
ͷǤͲ
ͷǤͷ
ǤͲ
αͳ
α͵ʹ
αͶ
Code and data available at: [url redacted for blind review]. arXiv:2105.13635v1 [cs.CL] 28 May 2021
We use WMT'19 English-German translation models (in both directions) to perform back-translation on each article provided by Facebook-FAIR ? ]. This model is the state-of-the-art language translation model.
https://github.com/google-research/bert/blob/master/tokenization.py
Learning with Pseudo-Ensembles. Philip Bachman, Ouais Alsharif, Doina Precup, NIPS. Philip Bachman, Ouais Alsharif, and Doina Precup. 2014. Learning with Pseudo- Ensembles. In NIPS. 3365-3373.
BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. Jacob Devlin, Ming-Wei Chang, Kenton Lee, Kristina Toutanova, NAACL-HLT, Jill Burstein, Christy Doran, and Thamar SolorioAssociation for Computational LinguisticsJacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019. BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. In NAACL-HLT, Jill Burstein, Christy Doran, and Thamar Solorio (Eds.). Association for Computational Linguistics, 4171-4186.
Understanding Back-Translation at Scale. Sergey Edunov, Myle Ott, Michael Auli, David Grangier, EMNLP. Sergey Edunov, Myle Ott, Michael Auli, and David Grangier. 2018. Understanding Back-Translation at Scale. In EMNLP. 489-500.
Teaching Machines to Read and Comprehend. Karl Moritz Hermann, Tomás Kociský, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Suleyman, Phil Blunsom, NIPS. Karl Moritz Hermann, Tomás Kociský, Edward Grefenstette, Lasse Espeholt, Will Kay, Mustafa Suleyman, and Phil Blunsom. 2015. Teaching Machines to Read and Comprehend. In NIPS. 1693-1701.
Learning a Deep Hybrid Model for Semi-Supervised Text Classification. Alexander G Ororbia, I I , C Lee Giles, David Reitter, Alexander G. Ororbia II, C. Lee Giles, and David Reitter. 2015. Learning a Deep Hybrid Model for Semi-Supervised Text Classification. In EMNLP. 471-481.
Automatic evaluation of summaries. Lin Chin Yew, Workshop on Text Summarization Branches Out at ACL. Chin Yew Lin. 2010. Automatic evaluation of summaries. In Workshop on Text Summarization Branches Out at ACL, 2004.
RoBERTa: A Robustly Optimized BERT Pretraining Approach. Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, Veselin Stoyanov, CoRR abs/1907.11692Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019. RoBERTa: A Robustly Optimized BERT Pretraining Approach. CoRR abs/1907.11692 (2019).
Don't Give Me the Details, Just the Summary! Topic-Aware Convolutional Neural Networks for Extreme Summarization. Shashi Narayan, Shay B Cohen, Mirella Lapata, Shashi Narayan, Shay B. Cohen, and Mirella Lapata. 2018. Don't Give Me the Details, Just the Summary! Topic-Aware Convolutional Neural Networks for Extreme Summarization. In ACL. 1797-1807.
Improved Noisy Student Training for Automatic Speech Recognition. Daniel S Park, Yu Zhang, Ye Jia, Wei Han, Chung-Cheng Chiu, Bo Li, Yonghui Wu, V Quoc, Le, INTERSPEECH. ISCA. Daniel S. Park, Yu Zhang, Ye Jia, Wei Han, Chung-Cheng Chiu, Bo Li, Yonghui Wu, and Quoc V. Le. 2020. Improved Noisy Student Training for Automatic Speech Recognition. In INTERSPEECH. ISCA, 2817-2821.
A Deep Reinforced Model for Abstractive Summarization. Romain Paulus, Caiming Xiong, Richard Socher, ICLR. OpenReview.net. Romain Paulus, Caiming Xiong, and Richard Socher. 2018. A Deep Reinforced Model for Abstractive Summarization. In ICLR. OpenReview.net.
Language Models are Unsupervised Multitask Learners. Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, Ilya Sutskever, Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019. Language Models are Unsupervised Multitask Learners. (2019).
Unsupervised Pretraining for Sequence to Sequence Learning. Prajit Ramachandran, Peter J Liu, V Quoc, Le, EMNLP. Prajit Ramachandran, Peter J. Liu, and Quoc V. Le. 2017. Unsupervised Pretraining for Sequence to Sequence Learning. In EMNLP. 383-391.
Leveraging Pretrained Checkpoints for Sequence Generation Tasks. Sascha Rothe, Shashi Narayan, Aliaksei Severyn, Trans. Assoc. Comput. Linguistics. 8Sascha Rothe, Shashi Narayan, and Aliaksei Severyn. 2020. Leveraging Pre- trained Checkpoints for Sequence Generation Tasks. Trans. Assoc. Comput. Linguistics 8 (2020), 264-280.
Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. Antti Tarvainen, Harri Valpola, NIPS. Antti Tarvainen and Harri Valpola. 2017. Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. In NIPS. 1195-1204.
Automatic Text Summarization. Juan-Manuel Torres-Moreno, WileyJuan-Manuel Torres-Moreno. 2014. Automatic Text Summarization. Wiley.
Unsupervised Data Augmentation for Consistency Training. Qizhe Xie, Zihang Dai, Eduard Hovy, Minh Thang Luong, Quoc V Le, Qizhe Xie, Zihang Dai, Eduard Hovy, Minh Thang Luong, and Quoc V Le. 2019. Unsupervised Data Augmentation for Consistency Training. (2019).
Self-Training With Noisy Student Improves ImageNet Classification. Qizhe Xie, Minh-Thang Luong, Eduard H Hovy, V Quoc, Le, CVPR. Qizhe Xie, Minh-Thang Luong, Eduard H. Hovy, and Quoc V. Le. 2020. Self- Training With Noisy Student Improves ImageNet Classification. In CVPR.
QANet: Combining Local Convolution with Global Self-Attention for Reading Comprehension. Adams Wei Yu, David Dohan, Minh-Thang Luong, Rui Zhao, Kai Chen, Mohammad Norouzi, Quoc V Le, ICLR. Adams Wei Yu, David Dohan, Minh-Thang Luong, Rui Zhao, Kai Chen, Moham- mad Norouzi, and Quoc V. Le. 2018. QANet: Combining Local Convolution with Global Self-Attention for Reading Comprehension. In ICLR.
| [
"https://github.com/google-research/bert/blob/master/tokenization.py"
] |
[
"A Database of Dorsal Hand Vein Images",
"A Database of Dorsal Hand Vein Images"
] | [
"Senior Member, IEEEFelipe Wilches-Bernal ",
"Bernardo Núñez-Álvarez ",
"Senior Member, IEEEPedro Vizcaya "
] | [] | [] | The dorsal hand vein has been demonstrated as a useful biometric for identity verification. This work details the procedure taken to collect two databases of dorsal hand veins in a biometric recognition project. The purpose of this work is to serve as a reference for the databases that are being shared with the public. | null | [
"https://arxiv.org/pdf/2012.05383v1.pdf"
] | 228,083,394 | 2012.05383 | 26205c2634158faac753edb6f354278ddf866cbf |
A Database of Dorsal Hand Vein Images
Senior Member, IEEEFelipe Wilches-Bernal
Bernardo Núñez-Álvarez
Senior Member, IEEEPedro Vizcaya
A Database of Dorsal Hand Vein Images
1Index Terms-Databaseveinsbiometricsdorsal hand veinsimagesimage processing
The dorsal hand vein has been demonstrated as a useful biometric for identity verification. This work details the procedure taken to collect two databases of dorsal hand veins in a biometric recognition project. The purpose of this work is to serve as a reference for the databases that are being shared with the public.
I. INTRODUCTION
Biometrics are physical or behavioral characteristics that be used for identity verification [1]. Hence, these characteristics are expected to be unique for an individual. Commonly used physical biometrics are fingerprints and the iris but other exits such as those relate to the vascular network of the person. Behavioral biometrics include the signature or the gait of an individual. Combined biometrics, that have both physical and behavioral components, such as the voice also exist [2].
The vascular network form due to the fundamental processes of vasculogenesis and angiogenesis and the terminations of these networks are determined to be unique among individuals [3]. The vascular network can then be used as a biometric. In practice the vein configuration of: fingers [4], [5], hand palms [6], [7], hand dorsum [8]- [12], wrist [13]- [15], retina in the eye [16], [17] and sclera in the eye [18], have all been used as biometrics.
The dorsal hand vein have been used as a biometric for over two decades [8]. Several studies have been conducted in different countries and even patents have been granted in this regard [19]. This work documents the effort to collect two databases of dorsal hand vein images. The databases were collected as part of a larger biometric project performed in 2007 and 2008 at Pontificia Universidad Javeriana in Bogota, Colombia [9]. The paper describes the image acquisition system used for the image capture and the final databases obtained. The authors of the original biometric work decided to make the databases publicly available in order to foster research in the subject. Novel techniques such as those based on machine learning or artificial intelligence could potentially be used to the shared datasets. The databases can also be used for educational purposes.
The remaining of this paper is organized as follows. Section II presents the image acquisition system. Section III F. Wilches-Bernal is an electrical engineer and researcher ([email protected]). B. Núñez-Álvarez is is an electrical engineer and tech sales consultant ([email protected]). P. Vizcaya is with Pontificia Universidad Javeriana, Bogota, Colombia ([email protected]). At the moment of the development of this work, all authors were with Pontificia Universidad Javeriana.
describes the databases and how they can be found online. Finally, Section IV presents the conclusions and future work.
II. IMAGE ACQUISITION SYSTEM
This section presents the physical structure developed to capture images with increased visibility of the dorsal veins of the hand.
The box-like structure in Fig. 1 was built for the purpose of taking dorsal hand vein images. This structure has dimensions of 32 × 29 × 35 cm (L × W × H). This structure is built out of wood and its internal walls, except those at the top and bottom, are coated with extruded polystyrene foam (styrofoam) as it can be observed in Fig. 2. The bottom of the structure was covered with black foam and the top of the structure had a custom illumination mechanism and a hole to include the camera, both of which will be explained below. The structure also had a rod-line piece that was meant to be grasped by the person whose images were being captured. This rod-like piece had a small stick (or rod) in the middle that was intended to be placed between the middle and ring fingers. Both the rod-like piece and the rod were selected so that the images taken where in a consistent position for different individuals. Note also that these components induce the individual to clench their fist which is a position that tends to increase the visibility of the dorsal hand veins. The box-like structure had different slots where the rod-like structure could be placed. This was done because the distance of the hand to the camera was a parameter of study of the overall project.
This study is outside of the scope of this document and all the images collected in the database were at the same distance (of approximately 20 cm).
A. Illumination
The top face of the box-like structure was removable. The inner wall of the top face was equipped with four custommade LED-lamps such as the one shown in Fig. 3. Each one of the four lamps was composed of 25 LEDs for 100 LEDs in total. The LEDs of each lamp were connected in series and supplied with a constant current source. The current sources were designed and controller to provide the same current. Because the light emitted by the LED is dependent in the amount of current, the setup of the project is intended to guarantee that each LED emits roughly the same amount of light.
The LED lamps were furnished with the QED 223 [20] LED device. The QED 223 is an infrared light emitting diode that was selected for the reasons that follow:
• The medium wide emission angle has a value of 30 • which tend to be a high value for LEDs and is suitable for its purpose of being an illumination source • The wavelength (λ) of the light emitted has a peak at 880 nm. The wavelength range is between 840 and 950 nm [20]. At these wavelengths the absorption coefficient of the blood in the veins which carry deoxygenated hemoglobin is much higher than that of surrounding tissue. • The price of this component was low which made it suitable for a self-funded graduation project. On top of the LED lamps a layer of parchment paper was added in order to scatter the light produced by the lamps.
B. Camera
The ISG LW-1.3-S-1394 is the camera used in this project [21]. This device can be connected to a computer using the IEEE-1394 interface. This device was selected because it was already available at the laboratory of the Electronics Engineering Department of Pontificia Universidad Javeriana. Also, this device has an excellent response in the infrared frequency range. In addition this device could be easily configured by software from the computer is connected to. This project used the Image Acquisition Toolbox from MATLAB to configure the camera. The parameters that could be configured were: integration time, frame-rate, gain, brightness, and exposure time. An important feature was that these parameters could be adjusted in real-time from software. Image capture was also easily available from software.
Because we wanted to capture images only in the infrared spectrum an infrared transmitting filter was added to the top of the lens of the camera. This filter was made from a black processed photographic film (also called a negative).
C. Constrast-enhancing Control System
The project implemented a simple control system to enhance the contrast of the images captured by the camera. The intent of the controller was to increase the contrast of the dorsal hand veins in the captured image as well. The control was also useful to homogenize the color distribution of the image taken for all the people. That is, if the parameters of the acquisition system are kept constant for all people the inherent differences in people's skin colors and their veins would produce images with extremely different contrasts. Fig. 4 the schematic of the control system implemented to enhance the contrast of the captured image. The variable to control was the camera integration time. Controlling this parameter had a similar effect to controlling the amount of illumination in the hand. The output of the camera is a frame noted I(x, y, t k ) (i.e. a time-varying image or twodimensional signal). The images have a defined size of n r rows and n c columns. The i th row and the j th column of the image are noted, respectively, as r i (x, t k ) and c j (y, t k ) and are one-dimensional signals. In order to determine if the frame (image) was suitable for the project, a function to determine the contrast of it was implemented. This function transformed the frame I(x, y, t k ) to a contrast value s(t k ). The function is described by
s(t k ) = f (I(x, y, t k )) = 1 N val i∈S max(r i (x, t k )) (1)
where S is the set of rows with a maximum above zero 1 and N val is the number of elements in S. As seen in Fig. 4 the a PI controller was implemented for the task of contrast enhancement. The integral part was selected so the error in steadystate was zero. The reference value s ref was experimentally determined from a set of images that had reasonable contrasts.
PI Controller
Camera Σ
D. Image Capture -Averaging
The output of the image acquisition system was an image of dorsal hand vein of the individual who was holding the rod. Once the control system in II-C have settled a burst of ten consecutive images (frames) were captured. The final captured image corresponded to the average of those ten frames. An image of variances was also computed and used as a marker to determine whether the captured image was acceptable, this procedure however was not automated. It is important to note that because the frame rate of the camera was set to 15 fps, capturing the ten frames only took two thirds of a second (i.e. 0.6667 seconds). This time was empirically verified as a suitable time for people to hold still the rod.
III. DORSAL HAND VEIN IMAGE DATABASE
This section presents general information on the two dorsal hand vein image databases generated as part of a biometric project. The project intended to demonstrate that dorsal hand vein images could be used as a reliable biometric for identity verification. This section also presents links to where these images can be retrieved online.
A. Image Collection Procedure
This project asked people to voluntary participate in data collection sessions.The participants were students and personnel of Pontificia Universidad Javeriana in the year 2007. The ages of the participants vary from 18 to 29 years old. An image collection session was intended to capture several images of the right and left dorsal hand veins. In a session, the participant was asked to grab the rod as shown in Fig. 2. At that point an image was captured as explained in Section II-D. Because more than one image per hand was captured per session, the participant was asked to remove the hand from the data acquisition structure, wait for at least 45 seconds, and grab again the rod. This procedure was performed for both the left and right-hands.
The image resolution of the images collected is 752 × 560 with 16-bit quantization. The format of the images is .tif. It 1 in practice this meant that the maximum value of the respective column was above a user defined threshold that close to zero (black color).
is important to note that in the year 2020 the images taken in 2007 were parsed and fully anonymized in a Ptyhon algorithm before being rewritten in the same .tif format. These latter images are those shared in the Github repositories.
As explained in Section II the rod to be grabbed by the person had a rod intended to standardize the position of the image. However rotations around the position of the rod were still possible. Fig. 5 shows the position of the pixel location of the rod in an image without a hand. This location is roughly at row 363 and column 412. An example of the images available in the first database is presented in Fig. 6. Note the images where cropped from its original format to fit better the two-column format of this document.
C. Dorsal Hand Veins Image Database 2
The second database comprises 113 people, and has 3 images per person per hand for a total of 678 images. All the people in the 113 people in the second database are included in the third database. Due to practical reasons not all the people that participated in the first session of data collection were available for the second session. Each data collection session spanned several days and the the time difference between the data collected in the session for Database 1 and the session for Database 2 is two months. [y] is either L or R indicating respectively the left or right hand, and [z] is a number from 1 to 3.
An example of the images available in Database 2 is presented in Fig. 7. Note the images where cropped from its original format to fit better the two-column format of this document.
IV. CONCLUSIONS AND FUTURE WORK
This work presents two databases of dorsal hand veins that can be used for human recognition. This work details the steps taken to generate the databases. Even though the databases were collected over a decade ago, they are being made publicly available in order to foster research and discussion in the area.
Fig. 1 :
1Box-like structure for dorsal hand vein image acquisition.
Fig. 2 :
2Data acquisition structure: view from above.
Fig. 3 :
3A LED-lamp designed to provide the inner illumination to the data acquisition structure.
Fig. 4 :
4Schematic of the control enhancing system.
Fig. 5 :
5Image captured of the rod. B. Dorsal Hand Veins Image Database 1
Dorsal left hand vein image of person 18. (b) Dorsal left hand vein image of person 54.
Fig. 6 :
6Image examples of Database 1. Database 2 can be retrieved at: https://github.com/ wilchesf/dorsalhandveins. The naming of each image, in Database 2, is as follows: • person_[xxx]_db2_[yz].tif where [xxx] is a three digit number from 001 to 113 and indicates the number of the person for which the image correspond.
Dorsal left hand vein image of person 18. (b) Dorsal left hand vein image of person 54.
Fig. 7 :
7Image examples of Database 2.
The first database, namedDatabase 1, comprises 138 people, and has 4 images per person per hand for a total of 1,104 images. Database 1 can be retrieved at: https://github.com/ wilchesf/dorsalhandveins. The naming of each image, in Database 1, is as follows: • person_[xxx]_db1_[yz].tif where [xxx] is a three digit number from 001 to 138 and indicates the number of the person for which the image correspond. [y] is either L or R indicating respectively the left or right hand, and [z] is a number from 1 to 4.
Introduction to biometrics. A K Jain, A A Ross, K Nandakumar, Springer Science & Business MediaA. K. Jain, A. A. Ross, and K. Nandakumar, Introduction to biometrics. Springer Science & Business Media, 2011.
An introduction to biometric recognition. A K Jain, A Ross, S Prabhakar, IEEE Transactions on circuits and systems for video technology. 14A. K. Jain, A. Ross, and S. Prabhakar, "An introduction to biometric recognition," IEEE Transactions on circuits and systems for video technology, vol. 14, no. 1, pp. 4-20, 2004.
Handbook of vascular biometrics. A Uhl, C Busch, S Marcel, R Veldhuis, Springer NatureA. Uhl, C. Busch, S. Marcel, and R. Veldhuis, Handbook of vascular biometrics. Springer Nature, 2020.
A systematic review of finger vein recognition techniques. K Shaheed, H Liu, G Yang, I Qureshi, J Gou, Y Yin, Information. 99213K. Shaheed, H. Liu, G. Yang, I. Qureshi, J. Gou, and Y. Yin, "A systematic review of finger vein recognition techniques," Information, vol. 9, no. 9, p. 213, 2018.
Finger vein recognition based on a personalized best bit map. G Yang, X Xi, Y Yin, Sensors. 122G. Yang, X. Xi, and Y. Yin, "Finger vein recognition based on a personalized best bit map," Sensors, vol. 12, no. 2, pp. 1738-1757, 2012.
Literature survey on contactless palm vein recognition. R C Rahul, M Cherian, M Mohan, Int J Comput Sci Trends Technol (IJCST). 35R. C. Rahul, M. Cherian, and M. Mohan, "Literature survey on contact- less palm vein recognition," Int J Comput Sci Trends Technol (IJCST), vol. 3, no. 5, 2015.
Palm vein recognition with local binary patterns and local derivative patterns. L Mirmohamadsadeghi, A Drygajlo, 2011 International Joint Conference on Biometrics (IJCB). IEEEL. Mirmohamadsadeghi and A. Drygajlo, "Palm vein recognition with local binary patterns and local derivative patterns," in 2011 International Joint Conference on Biometrics (IJCB). IEEE, 2011, pp. 1-6.
Biometric authentication by hand vein patterns. T Tanaka, N Kubo, SICE 2004 Annual Conference. IEEE1T. Tanaka and N. Kubo, "Biometric authentication by hand vein pat- terns," in SICE 2004 Annual Conference, vol. 1. IEEE, 2004, pp. 249-253.
Verificación de personal por el patrón de las venas del dorso de la mano. B Nuñez-Alvarez, F Wilches-Bernal, Pontificia Universidad JaverianaB. Nuñez-Alvarez and F. Wilches-Bernal, "Verificación de personal por el patrón de las venas del dorso de la mano," 2008, Pontificia Universidad Javeriana.
Hand dorsal vein recognition: Sensor, algorithms and evaluation. R Raghavendra, J Surbiryala, C Busch, 2015 IEEE international conference on imaging systems and techniques (IST). IEEER. Raghavendra, J. Surbiryala, and C. Busch, "Hand dorsal vein recog- nition: Sensor, algorithms and evaluation," in 2015 IEEE international conference on imaging systems and techniques (IST). IEEE, 2015, pp. 1-6.
Hand-dorsa vein recognition by matching local features of multisource keypoints. D Huang, Y Tang, Y Wang, L Chen, Y Wang, IEEE transactions on cybernetics. 459D. Huang, Y. Tang, Y. Wang, L. Chen, and Y. Wang, "Hand-dorsa vein recognition by matching local features of multisource keypoints," IEEE transactions on cybernetics, vol. 45, no. 9, pp. 1823-1837, 2014.
A recognition system for partially occluded dorsal hand vein using improved biometric graph matching. F Liu, S Jiang, B Kang, T Hou, IEEE Access. 8F. Liu, S. Jiang, B. Kang, and T. Hou, "A recognition system for partially occluded dorsal hand vein using improved biometric graph matching," IEEE Access, vol. 8, pp. 74 525-74 534, 2020.
Combining left and right wrist vein images for personal verification. C Mohamed, Z Akhtar, B N Eddine, T H Falk, 2017 Seventh International Conference on Image Processing Theory, Tools and Applications (IPTA). IEEEC. Mohamed, Z. Akhtar, B. N. Eddine, and T. H. Falk, "Combining left and right wrist vein images for personal verification," in 2017 Seventh International Conference on Image Processing Theory, Tools and Applications (IPTA). IEEE, 2017, pp. 1-6.
A new wrist vein biometric system. A Das, U Pal, M A F Ballester, M Blumenstein, 2014 IEEE Symposium on Computational Intelligence in Biometrics and Identity Management (CIBIM). IEEEA. Das, U. Pal, M. A. F. Ballester, and M. Blumenstein, "A new wrist vein biometric system," in 2014 IEEE Symposium on Computational Intelligence in Biometrics and Identity Management (CIBIM). IEEE, 2014, pp. 68-75.
Vein biometric recognition on a smartphone. R Garcia-Martin, R Sanchez-Reillo, IEEE Access. R. Garcia-Martin and R. Sanchez-Reillo, "Vein biometric recognition on a smartphone," IEEE Access, 2020.
Retina Recognition: Secure Biometric Authentication System An Approach to Implement the Eye Recognition System using Artificial Neural Networks. R Islam, M Khan, LAP Lambert Academic PublishingR. Islam and M. Khan, Retina Recognition: Secure Biometric Authen- tication System An Approach to Implement the Eye Recognition System using Artificial Neural Networks. LAP Lambert Academic Publishing, 2012.
Retina based biometric authentication system: a review. J B Mazumdar, S Nirmala, International Journal of Advanced Research in Computer Science. 91J. B. Mazumdar and S. Nirmala, "Retina based biometric authentication system: a review," International Journal of Advanced Research in Computer Science, vol. 9, no. 1, 2018.
Sclera recognition-a survey. A Das, U Pal, M Blumenstein, M A F Ballester, 2013 2nd IAPR Asian Conference on Pattern Recognition. IEEEA. Das, U. Pal, M. Blumenstein, and M. A. F. Ballester, "Sclera recognition-a survey," in 2013 2nd IAPR Asian Conference on Pattern Recognition. IEEE, 2013, pp. 917-921.
Portable biometric identification device using a dorsal hand vein pattern. Y Yabov, D Broydo, U.S. Patent. 9Y. yabov and D. Broydo, "Portable biometric identification device using a dorsal hand vein pattern," U.S. Patent 9,289,160, Aug. 4, 2015.
QED223 plastic infrared light emitting diode. On SemiconductorOn Semiconductor, "QED223 plastic infrared light emitting diode," https://www.onsemi.com/pub/Collateral/QED223-D.PDF, online: ac- cessed July 10 2020.
| [] |
[
"**************************************** BANACH CENTER PUBLICATIONS, VOLUME ** INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 201* THE GELFAND-NAIMARK-SEGAL CONSTRUCTION FOR UNITARY REPRESENTATIONS OF Z n 2 -GRADED LIE SUPERGROUPS",
"**************************************** BANACH CENTER PUBLICATIONS, VOLUME ** INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 201* THE GELFAND-NAIMARK-SEGAL CONSTRUCTION FOR UNITARY REPRESENTATIONS OF Z n 2 -GRADED LIE SUPERGROUPS"
] | [
"Mohammad Mohammadi [email protected] \nDepartment of Mathematics\nDepartment of Mathematics and Statistics\nInstitute for Advanced Studies in Basic Sciences (IASBS)\nNo. 444, Prof. Yousef Sobouti BlvdP. O. Box45195-1159ZanjanIran\n",
"Hadi Salmasian [email protected] \nUniversity of Ottawa\n585 King Edward AveK1N 6N5OttawaONCanada\n"
] | [
"Department of Mathematics\nDepartment of Mathematics and Statistics\nInstitute for Advanced Studies in Basic Sciences (IASBS)\nNo. 444, Prof. Yousef Sobouti BlvdP. O. Box45195-1159ZanjanIran",
"University of Ottawa\n585 King Edward AveK1N 6N5OttawaONCanada"
] | [] | We establish a Gelfand-Naimark-Segal construction which yields a correspondence between cyclic unitary representations and positive definite superfunctions of a general class of Z n 2 -graded Lie supergroups.2010 Mathematics Subject Classification: 17B75; 22E45. | 10.4064/bc113-0-14 | [
"https://arxiv.org/pdf/1709.06546v1.pdf"
] | 119,157,196 | 1709.06546 | b0fdfcc2c57d74584fe6c6c82f4f0ad3811a3582 |
**************************************** BANACH CENTER PUBLICATIONS, VOLUME ** INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 201* THE GELFAND-NAIMARK-SEGAL CONSTRUCTION FOR UNITARY REPRESENTATIONS OF Z n 2 -GRADED LIE SUPERGROUPS
19 Sep 2017
Mohammad Mohammadi [email protected]
Department of Mathematics
Department of Mathematics and Statistics
Institute for Advanced Studies in Basic Sciences (IASBS)
No. 444, Prof. Yousef Sobouti BlvdP. O. Box45195-1159ZanjanIran
Hadi Salmasian [email protected]
University of Ottawa
585 King Edward AveK1N 6N5OttawaONCanada
**************************************** BANACH CENTER PUBLICATIONS, VOLUME ** INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 201* THE GELFAND-NAIMARK-SEGAL CONSTRUCTION FOR UNITARY REPRESENTATIONS OF Z n 2 -GRADED LIE SUPERGROUPS
19 Sep 2017arXiv:1709.06546v1 [math-ph]
We establish a Gelfand-Naimark-Segal construction which yields a correspondence between cyclic unitary representations and positive definite superfunctions of a general class of Z n 2 -graded Lie supergroups.2010 Mathematics Subject Classification: 17B75; 22E45.
1. Introduction. It is by now well known that unitary representations of Lie supergroups appear in various areas of mathematical physics related to supersymmetry (see [8], [10], [9], [17] as some important examples among numerous references). A mathematical approach to analysis on unitary representations of Lie supergroups was pioneered in [3], where unitary representations are defined based on the notion of the Harish-Chandra pair associated to a Lie supergroup.
More recently, there is growing interest in studying generalized supergeometry, that is, geometry of graded manifolds where the grading group is not Z 2 , but Z n 2 := Z 2 × · · ·× Z 2 . The foundational aspects of the theory of Z n 2 -supermanifolds were recently established in the works of Covolo-Grabowski-Poncin (see [5], [6]) and Covolo-Kwok-Poncin (see [7]).
In this short note, we make a first attempt to extend the theory of unitary representations to the Z n 2 -graded setting. To this end, we use the concept of a Z n 2 -graded Harish-Chandra pair, which consists of a pair (G 0 , g) where G 0 is a Lie group and g is a Z n 2 -graded generalization of a Lie superalgebra, sometimes known as a Lie color algebra. Extending one of the main results of [15], in Theorem 5.4 we prove that under a "perfectness" condition on g, there exists a Gelfand-Naimark-Segal (GNS) construction which yields a correspondence between positive definite smooth super-functions and cyclic unitary representations of (G 0 , g). Theorem 5.4 is applicable to interesting examples, such as Z n 2 -Lie supergroups of classical type, e.g., the Harish-Chandra pair corresponding to gl(V ) defined in Example 2.5, where V is a Z n 2 -graded vector space. The key technical tool in the proof of Theorem 5.4 is a Stability Theorem (see Theorem 4.3) which guarantees the existence of a unique unitary representation associated to a weaker structure, called a pre-representation. Such a Stability Theorem holds unconditionally when n = 1. But for n > 1, it is not true in general. We are able to retrieve a variation of the Stability Theorem under the aforementioned extra condition on g. However, this still leaves the question of a general GNS construction open for further investigation. We defer the latter question, presentation of explicit examples, as well as some proof details, to a future work.
Acknowledgements. We thank Professor Karl-Hermann Neeb for illuminating conversations related to Theorem 4.3 and Remark 2.2, and for many useful comments on a preliminary draft of this article, which improved our presentation substantially. This work was completed while the first author was visiting the University of Ottawa using a grant from the Iranian Ministry of Science, Research, and Technology. During this project, the second author was supported by an NSERC Discovery Grant.
2. Z n 2 -supergeometry. We begin by reviewing the basic concepts of Z n 2 -graded supergeometry, in the sense of [6]. Let Γ := Z n 2 := Z 2 × · · · × Z 2 where Z 2 := {0, 1}, and let b : Γ × Γ → Z 2 be a non-degenerate symmetric Z 2 -bilinear map. By a result of Albert (see [1,Thm 6] or [11,), if b(·, ·) is of alternate type (i.e., b(a, a) = 0 for every a ∈ Γ), then b(·, ·) is equivalent to the standard "symplectic" form b − (a, b) := n j=1 a 2j−1 b 2j + a 2j b 2j−1 for every a = (a 1 , . . . , a n ) and b = (b 1 , . . . , b n ) ∈ Γ, whereas if b(a, a) = 0 for some a ∈ Γ, then b(·, ·) is equivalent to the standard symmetric form b + (a, b) := m j=1 a j b j for every a = (a 1 , . . . , a n ) and b = (b 1 , . . . , b n ) ∈ Γ.
Henceforth we assume that b(·, ·) is not of alternate type. Since an equivalence of b(·, ·) and b + (·, ·) is indeed an automorphism of the finite abelian group Γ, without loss of generality from now on we can assume that b = b + . In particular, from now on we represent an element a ∈ Γ by a := n j=1 a j e j , where {e j } n j=1 is an orthonormal basis of THE GNS CONSTRUCTION FOR Z n 2 -GRADED LIE SUPERGROUPS 3 Γ with respect to b(·, ·). We now define β : Γ × Γ → {±1} by β(a, b) := (−1) b(a,b) for a, b ∈ Γ. We equip the category Vec Γ of Γ-graded complex vector spaces with the symmetry operator
S V,W : V ⊗ W → W ⊗ V , S V,W (v ⊗ w) := β(|v|, |w|)w ⊗ v,(1)
where |v| ∈ Γ denotes the degree of a homogeneous vector v ∈ V .
Remark 2.1. As it is customary in supergeometry, equality (1) should be construed as a relation for homogeneous vectors that is subsequently extended by linearity to nonhomogeneous vectors. In the rest of the manuscript we will stick to this convention.
Equipped with S and the usual Γ-graded tensor product of vector spaces, Vec Γ is a symmetric monoidal category. This fact was also observed in [2, Prop. 1.5].
Remark 2.2. Note that β : Γ × Γ → {±1} is a 2-cocycle. In fact it represents the obstruction to the lifting of the map Γ → {±1}, (a 1 , . . . , a n ) → (−1) n i=1 ai , with respect to the exact sequence
1 / / {±1} / / {±1, ±i} t →t 2 / / {±1} / / 1 , where i := √ −1.
More precisely, the lifting obstruction cocycle is naturally represented by δ : Γ × Γ → {±1} , (a 1 , . . . , a n ) → (−1) 1≤i,j≤n aibj , but β − δ = dη where η(a 1 , . . . , a n ) := (−1) 1≤i =j≤n aiaj . We thank Professor Karl-Hermann Neeb for letting us know about this property of β.
Let ≺ denote the lexicographic order on elements of Γ. That is, for a := n j=1 a j e j and b := n j=1 b j e j , we set a ≺ b if and only if there exists some 1 ≤ j ≤ n such that a j = 0, b j = 1, and a k = b k for all k < j, . Thus we can express Γ as Γ = {γ 0 ≺ γ 1 ≺ · · · ≺ γ 2 n −1 }, where γ 0 = 0.
Let p ∈ N ∪ {0} and let q := (q 1 , . . . , q 2 n −1 ) be a (2 n − 1)-tuple such that q j ∈ N ∪ {0} for all j. We set |q| := 2 n −1 j=1 q j . By a Γ-superdomain of dimension p|q we mean a locally ringed space (U, O U ) such that U ⊂ R p is an open set, and the structure sheaf O U is given by
O U (U ′ ) := C ∞ (U ′ ; C)[[ξ 1 , · · · , ξ |q| ]] for every open set U ′ ⊆ U,
where the right hand side denotes the Γ-graded algebra of formal power series in variables ξ 1 , . . . , ξ |q| with coefficients in C ∞ (U ′ ; C), such that |ξ r | = γ k for k−1 j=1 q j < r ≤ k j=1 q j , subject to relations ξ r ξ s = β(|ξ r |, |ξ s |)ξ s ξ r for every 1 ≤ r, s ≤ |q|.
By a smooth Γ-supermanifold M of dimension p|q we mean a locally ringed space (M, O M ) that is locally isomorphic to a p|q-dimensional Γ-superdomain. From this viewpoint, a Γ-Lie supergroup is a group object in the category of smooth Γ-supermanifolds.
As one expects, to a Γ-Lie supergroup one can canonically associate a Γ-Lie superalgebra, which is an object of Vec Γ of the form g = a∈Γ g a , equipped with a Γ-superbracket [·, ·] : g × g → g that satisfies the following properties:
(i) [·, ·] is bilinear, and [g a , g b ] ⊂ g ab for a, b ∈ Γ. (ii) [x, y] = −β(a, b)[y, x] for x ∈ g a , y ∈ g b , where a, b ∈ Γ. (iii) [x, [y, z]] = [[x, y], z]+ β(a, b)β(a, c)[y, [z, x]] for x ∈ g a , y ∈ g b , z ∈ g c , where a, b, c ∈ Γ.
Remark 2.3. We remark that a Γ-Lie superalgebra is more commonly known as a Lie color algebra. Nevertheless, in order to keep our nomenclature compatible with [6], we use the term Γ-Lie superalgebra instead.
From classical supergeometry (that is, when n = 1) one knows that the category of Lie supergroups can be replaced by another category with a more concrete structure, known as the category of Harish-Chandra pairs (see [12], [13]). A similar statement holds in the case of Γ-Lie supergroups.
Definition 2.4. A Γ-Harish-Chandra pair is a pair (G 0 , g)
where G 0 is a Lie group, and g = a∈Γ g a is a Γ-Lie superalgebra equipped with an action Ad : G 0 × g → g of G 0 by linear operators that preserve the Γ-grading, which extends the adjoint action of G 0 on g 0 ∼ = Lie(G 0 ).
Example 2.5. Let V := a∈Γ V a be a Γ-graded vector space. The Γ-Lie superalgebra gl(V ) is the vector space of linear transformations on V , with superbracket [S, T ] := ST − β(|S|, |T |)T S. One can also consider the Γ-Harish-Chandra pair (G 0 , g), where G 0 ∼ = a∈Γ GL(V a ).
Indeed the Γ-Harish-Chandra pairs form a category in a natural way. The morphisms of this category are pairs of maps (φ, ϕ) : (G, g) → (H, h), where φ : G → H is a Lie group homomorphism and ϕ : g → h is a morphism in the category of Γ-Lie superalgebras such that ϕ g0 = dφ. The following statement plays a key role in the study of Γ-Lie supergroups.
Proposition 2.6. The category of Γ-Lie supergroups is isomorphic to the category of Γ-Harish-Chandra pairs.
Proof. The proof is a straightforward but lengthy modification of the argument for the analogous result in the case of ordinary supergeometry (see [12], [13], or [4]). Therefore we only sketch an outline of the proof. The functor from Γ-Lie supergroups to Γ-Harish-Chandra pairs is easy to describe: a Γ-Lie supergroup (G 0 , O G0 ) is associated to the Harish-Chandra pair (G 0 , g), where g is the Γ-Lie superalgebra of (G 0 , O G0 ). As in the classical super case, the functor associates a homomorphism of Γ-Lie su-
pergroups (G 0 , O G0 ) → (H 0 , O H0 )
to the pair of underlying maps G 0 → H 0 and the tangent map at identity g → h. Conversely, from a Γ-Harish-Chandra pair (G 0 , g) we construct a Lie supergroup (G 0 , O G0 ) as follows. For every open set U ⊆ G 0 , we set O G0 (U ) := Hom g0 U(g C ), C ∞ (U ; C) , where U(g C ) denotes the universal enveloping algebra of g C := g ⊗ R C. The Γ-superalgebra structure on O G0 (U ) is defined using the algebra structure of C ∞ (U ; C) and the Γ-coalgebra structure of U(g C ), exactly as in the classical super case. Using the PBW Theorem for Γ-Lie superalgebras (see [18]) one can see that (G 0 , O G0 ) is indeed a Γ-supermanifold (see [4,Prop. 7.4.9]). The definition of the Γ-Lie supergroup structure of (G 0 , O G0 ) is similar to the classical case as well (see [4,
O G0 (U ) → O G0 (U ) , s → D → β(|D|, |s|) L D s for D ∈ U(g C ),
where |D|, |s| ∈ Γ are the naturally defined degrees, L (2)
Definition 3.1. Let H ∈ Obj(Vec Γ ).
We call H a Γ-inner product space if and only if it is equipped with a non-degenerate sesquilinear form ·, · that satisfies the following properties:
i) H a , H b = 0, for a, b ∈ Γ such that a = b. ii) w, v = β(a, a) v, w , for v, w ∈ H a where a ∈ Γ. iii) α(a) v, v ≥ 0, for every v ∈ H a where a ∈ Γ.
Remark 3.2. The choice of α in (2) is made as follows. The most natural property that one expects from Hilbert spaces is that the tensor product of (pre-)Hilbert spaces is a (pre-)Hilbert space. Thus, we are seeking α : Γ → C × such that the tensor product of two Γ-inner product spaces is also a Γ-inner product space. Clearly after scaling the values of α by positive real numbers we can assume that α(Γ) ⊆ {±1, ±i}. For two Γ-inner product spaces H and K , one has
(H ⊗ K ) a := bc=a H b ⊗ H c , equipped with the canonically induced sesquilinear form v ⊗ w, v ′ ⊗ w ′ H ⊗K := β(|w|, |v ′ |) v, v ′ H w, w ′ K .
Closedness under tensor product implies that
α(bc) = β(b, c)α(b)α(c), for all b, c ∈ Γ.(3)
In the language of group cohomology, this means that the 2-cocycle β satisfies β = dα.
Consequently, up to twisting by a group homomorphism Γ → {±1}, there exists a unique α which satisfies the latter relation. The vector space H , equipped with (·, ·), is indeed a pre-Hilbert space in the usual sense. Thus we can consider the completion of H with respect to the norm v := (v, v) 1 2 . By reversing the process of obtaining (·, ·) from ·, · , we obtain a Γ-inner product on the completion of H .
Definition 3.4. Let H be a Γ-inner product space, and let T ∈ End C (H ). We define the adjoint T † of T as follows. If T ∈ End C (H ) a for some a ∈ Γ, we define
T † by v, T w = β(a, b) T † v, w ,
where v ∈ H b . We then extend the assignment T → T † to a conjugate-linear map on End C (H ). Now let H be a Γ-inner product space, and let (·, ·) be the ordinary inner product associated to H as in Remark 3.3. Then for a linear map T : H → H we can define an adjoint with respect to (·, ·), by the relation
(T v, w) = (v, T * w)
for every v, w ∈ H .
A straightforward calculation yields
T * = α(|T |)T † . It follows that T † † = T . Furthermore, (ST ) † = β(|S|, |T |)T † S † .
We are now ready to define unitary representations of Γ-Harish-Chandra pairs. The definition of a unitary representation of a Γ-Harish-Chandra pair is a natural extension of the one for the Z 2 -graded case. First recall that for a unitary representation (π, H ) of a Lie group G on a Hilbert space H , we denote the space of C ∞ vectors by H ∞ . Thus the vector space H ∞ consists of all vectors v ∈ H for which the map G → H , g → π(g)v is smooth. Furthermore, given x ∈ Lie(G) and v ∈ H , we set dπ(x) := lim
t→0 1 t (π(exp(tx))v − v) ,
whenever the limit exists. We denote the domain of the unbounded operator dπ(x) by D(dπ(x)). The unbounded operator −idπ(x) is the self-adjoint generator of the oneparameter unitary representation t → π(exp(tx)). For a comprehensive exposition of the theory of unitary representations and relevant facts from the theory of unbounded operators, see [19].
Definition 3.5. A smooth unitary representation of a Γ-Harish-Chandra pair (G 0 , g) is a triple (π, ρ π , H ) that satisfies the following properties:
(R1) (π, H ) is a smooth unitary representation of the Lie group G 0 on the Γ-graded Hilbert space H by operators π(g), g ∈ G 0 , which preserve the Γ-grading. (R2) ρ π : g → End C (H ∞ ) is a representation of the Γ-Lie superalgebra g.
(R3) ρ π (x) = dπ(x) H ∞ for x ∈ g 0 . (R4) ρ π (x) † = −ρ π (x) for x ∈ g.
(R5) π(g)ρ π (x)π(g) −1 = ρ π (Ad(g)x) for g ∈ G 0 and x ∈ g.
Unitary representations of a Γ-Harish-Chandra pair form a category Rep = Rep(G 0 , g). A morphism in this category from (π, ρ π , H ) to (σ, ρ σ , K ) is a bounded linear map THE GNS CONSTRUCTION FOR Z n 2 -GRADED LIE SUPERGROUPS 7 T : H → K which respects the Γ-grading and satisfies T π(g) = σ(g)T for g ∈ G 0 (from which it follows that T H ∞ ⊆ K ∞ ) and T ρ π (x) = ρ σ (x)T for x ∈ g.
Remark 3.6. At first glance, it seems that the definition of a unitary representation of a Γ-Harish-Chandra pair depends on the choice of α. Nevertheless, it is not difficult to verify that for two coboundaries α, α ′ which satisfy (3), the corresponding categories Rep α and Rep α ′ are isomorphic. Indeed if χ : Γ → {±1} is a group homomorphism such that α ′ = χα, then we can define a functor F :
Rep α → Rep α ′ which maps (π, ρ π , H ) to (π ′ , ρ π ′ , H ′ ) where H := H ′ (but the Γ-inner product of H ′ is defined by v, v H ′ = χ(a) v, v H for v ∈ H a
where a ∈ Γ), π ′ := π, and ρ π ′ (x) := χ(a)ρ π (x) for x ∈ g a and a ∈ Γ. The functor F is defined to be identity on morphisms. That is, for a morphism T : (π, ρ π , H ) → (σ, ρ σ , K ) we set F (T ) := T . It is straightforward to see that with (π ′ , ρ π ′ , H ′ ) and (σ ′ , ρ σ ′ , K ′ ) defined as above, the map T :
(π ′ , ρ π ′ , H ′ ) → (σ ′ , ρ σ ′ , K ′ )
is still an intertwining map. The main point is that ρ π ′ and ρ σ ′ are obtained from ρ π and ρ σ via scaling by the same scalar. The inverse of F is defined similarly.
4. The stability theorem. We now proceed towards the statment and proof of the Stability Theorem. In what follows, we will need the following technical definition (see [14]).
Definition 4.1. Let (G 0 , g) be a Γ-Harish-Chandra pair. By a pre-representation of (G 0 , g), we mean a 4-tuple (π, H , B, ρ B ) that satisfies the following conditions:
(PR1) (π, H ) is a smooth unitary representation of the Lie group G 0 on the Γ-graded Hilbert space H by operators π(g), g ∈ G 0 , which preserve the Γ-grading. (PR2) B is a dense, G 0 -invariant, and Γ-graded subspace of H that is contained in (π, H , B, ρ B ) be a pre-representation of a Γ-Harish-Chandra pair (G 0 , g). Assume that for every a ∈ Γ 0 \{0} we have
x∈g0 D(dπ(x)). (PR3) ρ B : g → End C (B) is a representation of the Γ-Lie superalgebra g. (PR4) ρ B (x) = dπ(x) B and ρ B (x) is essentially skew adjoint for x ∈ g 0 . (PR5) ρ B (x) † = −ρ B (x) for x ∈ g. (PR6) π(g)ρ B (x)π(g) −1 = ρ B (Ad(g)x) for g ∈ G 0 and x ∈ g.g a = b,c∈Γ 1 ,bc=a [g b , g c ].
Then there exists a unique extension of ρ B to a linear map ρ π : g → End C (H ∞ ) such that (π, ρ π , H ) is a smooth unitary representation of (G 0 , g).
Proof. For every a ∈ Γ 1 the direct sum g 0 ⊕ g a is a Lie superalgebra (in the Z 2 -graded sense). We define a Z 2 -grading of H by
H 0 := b(a,b)=0 H b and H 1 := b(a,b)=1 H b ,
and then we use the Stability Theorem in the Z 2 -graded case (see [14,Thm 6.14]) to obtain that there exists a unique Γ-Lie superalgebra homomorphism ρ π,a : g 0 ⊕ g a → End C (H ∞ ), such that (π, ρ π,a , H ) is a unitary representation of the (Z 2 -graded) Harish-Chandra pair (G 0 , g 0 ⊕ g a ). Since the maps ρ π,a agree on g 0 , they give rise to a linear map
g 0 ⊕ ( a∈Γ 1 g a ) → End C (H ∞ ).
It remains to obtain a suitable extension of the latter map to g a for every a ∈ Γ 0 \{0}. Fix x ∈ g a , a ∈ Γ 0 \{0}. Then we can write x = j [y j , z j ] such that y j ∈ g bj , z j ∈ g cj , where b j , c j ∈ Γ 1 , and b j c j = a. For every v ∈ H ∞ , we define
ρ π (x)v := j ρ π,bj (y j )ρ π,cj (z j )v − β(b j , c j )ρ π,cj (z j )ρ π,bj (y j )v .(4)
Let us first verify that ρ π (x)v is well-defined. To this end, it suffices to show that if j [y j , z j ] = 0, then the right hand side of (4) vanishes. To verify the latter assertion, observe that for every w ∈ B s , s ∈ Γ, we have
w, ρ π (x)v = j β(b j , s)β(c j , sb j ) ρ π,cj (z j ) † ρ π,bj (y j ) † w, v − j β(c j , s)β(b j , sc j )β(b j , c j ) ρ π,bj (y j ) † ρ π,cj (z j ) † w, v = β(a, s) j β(b j , c j )ρ π,cj (z j ) † ρ π,bj (y j ) † w − j ρ π,bj (y j ) † ρ π,cj (b j ) † w , v = −β(a, s) j ρ B ([y j , z j ])w, v = 0.
The assertion now follows from density of B in H .
It remains to show that given any x a ∈ g a and y b ∈ g b , where a, b ∈ Γ, we have
ρ π ([x a , y b ])v = ρ π (x a )ρ π (y b ) − β(a, b)ρ π (y b )ρ π (x a ) v for v ∈ H ∞ .(5)
To verify the last equality note that for every w ∈ B s where s ∈ Γ,
w, ρ π (x a )ρ π (y b )v − β(a, b)ρ π (y b )ρ π (x a )v = β(a, s)β(as, b) ρ π (y b ) † ρ π (x a ) † w, v − β(b, s)β(s, a) ρ π (x a ) † ρ π (y b ) † w, v = −β(a, s)β(b, s) ρ B (x a )ρ B (y b )w − β(a, b)ρ B (y b )ρ B (x a )w, v = β(a, s)β(b, s) ρ B [x a , y b ] † w, v = w, ρ π [x a , y b ] v .
Again density of B in H implies that both sides of (5) are equal. Finally, uniqueness of ρ π (x) can be proved by the same technique. Example 4.4. For n > 1, one cannot expect the Stability Theorem to hold without any condition on g. For example, let Γ = Z 2 × Z 2 , let G 0 be the trivial group, and let g = g 00 ⊕ g 01 ⊕ g 10 ⊕ g 11 where g 11 := R and g 00 := g 01 := g 10 := {0}.
Then a pre-representation of (G 0 , g) is the same (up to scaling) as a symmetric operator defined on a dense subspace of a Hilbert space, whereas a unitary representation of (G 0 , g) is the same (up to scaling) as a bounded self-adjoint operator. Therefore a prerepresentation does not necessarily extend to a unitary representation.
The GNS representation.
Our goal in this section is to extend the GNS construction of [15] to the setting of Γ-Harish-Chandra pairs. We begin by outlining some generalities about this construction. Let (G 0 , g) be a Γ-Harish-Chandra pair, and let G := (G 0 , O G0 ) denote the Γ-Lie supergroup corresponding to (G 0 , g). Our main goal, to be established in Theorem 5.4, is to construct a correspondence between unitary representations of (G 0 , g) with a cyclic vector, and smooth positive definite functions on G.
The suitable definition of a unitary representation with a cyclic vector is given in Definition 5.3. The main subtlety is to define positive definite functions on G. To this end, we use the method developed in [15] for the classical super case, which we describe below.
Recall that the Γ-superalgebra C ∞ (G) of smooth functions on G is isomorphic to
Hom g0 U(g C ), C ∞ (G 0 ; C) ,
where g 0 acts on C ∞ (G 0 ; C) by left invariant differential operators. We realize elements of the latter algebra as functions on a semigroup S which is equipped with an involution. Then we use the abstract definition of a positive definite function on a semigroup with an involution (see Definition 5.1). Given a unitary representation of (G 0 , g) with a cyclic vector, the matrix coefficient of that cyclic vector will be a positive definite element of C ∞ (G). Conversely, from a positive definite f ∈ C ∞ (G), we construct a unitary representation of (G 0 , g) by considering the reproducing kernel Hilbert space associated to f , now realized as a function on S. The Lie group G 0 has a canonical action on the reproducing kernel Hilbert space. However, the Γ-Lie superalgebra g acts on a dense subspace of the latter Hilbert space, which is in general strictly smaller than the space of smooth vectors for the G 0 -action. The main part of the proof of Theorem 5.4 is to show that the action of g is well defined on the entire space of smooth vectors. For this we need Theorem 4.3, which is the reason for presence of the condition (8) in Theorem 5.4. Set g C := g ⊗ R C. Let U(g C ) be the universal enveloping algebra of g C , that is, the quotient T (g C )/I, where T (g C ) denotes the tensor algebra of g C in the category Vec Γ , and I denotes the two-sided ideal of T (g C ) generated by elements of the form
x ⊗ y − β(|x|, |y|)y ⊗ x − [x, y],
for homogeneous x, y ∈ g C .
See [16,Sec. 2.1] for further details. Let x → x * be the (unique) conjugate-linear map on g C that is defined by the relation
x * := −α(a)x,
for every x ∈ g a , a ∈ Γ.
We extend the map x → x * to a conjugate-linear anti-automorphism of the algebra U(g C ).
Thus (D 1 D 2 ) * = D * 2 D * 1 for every D 1 , D 2 ∈ U(g C ). Such an extension is possible because of * -invariance of I. We now define a monoid
S := G 0 ⋉ U(g C ),
with a multiplication given by (g 1 , D 1 )(g 2 , D 2 ) := g 1 g 2 , (Ad(g −1 2 )(D 1 ))D 2 . The neutral element of S is 1 S := (1 G0 , 1 U(g C ) ). The map
S → S , (g, D) → (g, D) * := g −1 , Ad(g)(D * )
is an involution of S.
The proof of the latter assertion is similar to the one in the Z 2 -graded case (see [13], [12], [15,Thm 5.5.2]).
Next, for every f ∈ C ∞ (G), we define a map
f : S → C , (g, D) → f (D)(g).
Also, for any a ∈ Γ we set S a := {(g, D) ∈ S : |D| = a}.
Definition 5.
1. An f ∈ C ∞ (G) is called positive definite if it satisfies the following two conditions:
(i)f (g, D) = 0 unless (g, D) ∈ S 0 . (ii) 1≤i,j≤n c i c jf (s * i s j ) ≥ 0,
for all n ≥ 1, c 1 , · · · , c n ∈ C, s 1 , · · · , s n ∈ S.
Given a unitary representation (π, ρ π , H ) of G, for any two vectors v, w ∈ H we define the matrix coefficient ϕ v,w to be the map ϕ v,w : S → C , ϕ v,w (g, D) := (π(g)ρ π (D)v, w).
Proposition 5.2. Let (π, ρ π , H ) be a smooth unitary representation of (G 0 , g), and let v, w ∈ H ∞ be homogeneous vectors such that |v| = |w|. Then there exists an f ∈ C ∞ (G) such thatf = ϕ v,w . Furthermore,f (s) = 0 unless s ∈ S 0 . If v = w thenf is positive definite.
Proof. Similar to [15,Prop. 6.5.2].
We are now ready to describe the GNS construction for unitary representations of Γ-Harish-Chandra pairs. It is an extension of the one given in [15,Sec. 6] in the Z 2 -graded case, and therefore we will skip the proof details.
Let (π, ρ π , H ) be a smooth unitary representation of (G 0 , g). One can construct a * -representation ρ π of the monoid S by setting ρ π : S → End C (H ∞ ) , (g, D) → π(g)ρ π (D).
Being a * -representation means that ρ π (s * ) = ρ π (s) * for every s ∈ S, and in particular ( ρ π (s)v, w) = (v, ρ π (s * )w).
By Proposition 5.2, for every matrix coefficient ϕ v,v , where v ∈ H 0 , there exists a positive definite f ∈ C ∞ (G) such that ϕ v,v =f . Conversely, given a positive definite function f ∈ C ∞ (G), one can associate a *representation of S to f as follows. Set ψ :=f , and for every s ∈ S let ψ s : S → C be the map given by ψ s (t) := ψ(ts). We also set D ψ := Span C {ψ s : s ∈ S} .
Note that D ψ is a Γ-graded vector space of complex-valued functions on S, where the homogeneous parts of the Γ-grading are defined by D ψ,a := {h ∈ D ψ : h(s) = 0 unless s ∈ S a }.
The space D ψ can be equipped with a sesquilinear form that is uniquely defined by the relation (ψ t , ψ s ) := ψ(s * t). The completion H ψ of the resulting pre-Hilbert space is the reproducing kernel Hilbert space that corresponds to the kernel K : S × S → C , (t, s) → ψ(ts * ).
In other words, with respect to the inner product (·, ·) on H ψ , we have h(s) = (h, K s ) for h ∈ H ψ , s ∈ S.
There is a natural * -representation ρ ψ : S → End C (D ψ ) by right translation, given by ( ρ ψ (s)h)(t) := h(ts) for s, t ∈ S, h ∈ D ψ .
If s ∈ S satisfies ss * = s * s = 1 S , then ρ ψ (s) : D ψ → D ψ is an isometry and extends uniquely to a unitary operator on H ψ . Using the latter fact for elements of the form (g, 1 U(g C ) ) where g ∈ G 0 , one obtains a unitary representation of G 0 on H ψ (in fact the vectors K s , s ∈ S, have smooth G 0 -orbits). Setting B := D ψ , H := H ψ , ρ B (x) := ρ ψ (1 G0 , x) for x ∈ g, and π(g) := ρ ψ (g, 0), we obtain a pre-representation of (G 0 , g). Theorem 4.3 implies that this pre-representation corresponds to a unique unitary representation of (G 0 , g). Definition 5.3. By a cyclic vector in a unitary representation (π, ρ π , H ) we mean a vector v ∈ H such that ρ π (S)v is dense in H .
The above construction results in Theorem 5.4 below.
Theorem 5.4. Let (G 0 , g) be a Γ-Harish-Chandra pair such that
g a = b,c∈Γ 1 ,bc=a [g b , g c ].(8)
Also, let f ∈ C ∞ (G) be positive definite.
(i) There exists a unitary representation (π, ρ π , H ) of (G 0 , g) with a cyclic vector v 0 ∈ H 0 such thatf = ϕ v0,v0 . (ii) Let (σ, ρ σ , K ) be another unitary representation of (G 0 , g) with a cyclic vector w 0 ∈ K 0 such thatf = ϕ w0,w0 . Then (π, ρ π , H ) and (σ, ρ σ , K ) are unitarily equivalent via an intertwining operator that maps v 0 to w 0 .
Proof. The proof is an extension of the argument of [15, Thm 6.7.5].
D denotes the left invariant differential operator on (G 0 , O G0 ) corresponding to D, and L D s means evaluation of the section at points of U . The proof of bijective correspondence of morphisms in the two categoies is similar to [4, Prop. 7.4.12]. 3. Γ-Hilbert superspaces and unitary representations. In order to define a unitary representation of a Γ-Harish-Chandra pair, one needs to obtain a well-behaved definition of Hilbert spaces in the category Vec Γ . This is our first goal in this section. For any a = n j=1 a j e j ∈ Γ, set u(a) := |{1 ≤ j ≤ n : a j = 1}| (for example, u(e 1 + e 3 + e 4 ) = 3) and define α(a) := e πi 2 u(a) .
Remark 3. 3 .
3Associated to any Γ-inner product space H , there is an inner product (in the ordinary sense) defined by (v, w) := 0 if |v| = |w|, α(a) v, w if |v| = |w| = a where a ∈ Γ.
Remark 4. 2 .
2It is shown in [14, Rem. 6.5] that (PR2) and (PR3) imply that B ⊆ H ∞ . Set Γ 0 ; = {a ∈ Γ : b(a, a) = 0} and Γ 1 := {a ∈ Γ : b(a, a) = 1} Theorem 4.3. (Stability Theorem) Let
THE GNS CONSTRUCTION FOR Z n 2 -GRADED LIE SUPERGROUPS Prop. 7.4.10]). Finally, to show that the two functors are inverse to each other, we need a sheaf isomorphism O G0 ∼ = O G0 . This sheaf isomorphism is given by the maps5
Symmetric and alternate matrices in an arbitrary field. A , Adrian Albert, I. Trans. Amer. Math. Soc. 433A. Adrian Albert, Symmetric and alternate matrices in an arbitrary field. I. Trans. Amer. Math. Soc. 43 (1938), no. 3, 386-436.
Tensor product representations for orthosymplectic Lie superalgebras. Georgia Benkart, Chanyoung Lee Shader, Arun Ram, J. Pure Appl. Algebra. 1301Georgia Benkart and Chanyoung Lee Shader and Arun Ram, Tensor product represen- tations for orthosymplectic Lie superalgebras. J. Pure Appl. Algebra 130 (1998), no. 1, 1-48.
Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles. Claudio Carmeli, Gianni Cassinelli, Alessandro Toigo, Veeravalli S Varadarajan, Comm. Math. Phys. 2631Claudio Carmeli and Gianni Cassinelli and Alessandro Toigo and Veeravalli S. Varadara- jan, Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles. Comm. Math. Phys. 263 (2006), no. 1, 217-258.
Mathematical foundations of supersymmetry. Claudio Carmeli, Lauren Caston, Rita Fioresi, EMS Series of Lectures in Mathematics. European Mathematical SocietyClaudio Carmeli, Lauren Caston, Rita Fioresi, Mathematical foundations of supersym- metry. EMS Series of Lectures in Mathematics. European Mathematical Society (EMS), Zrich, 2011. xiv+287 pp.
Splitting theorem for Z n 2 -supermanifolds. Tiffany Covolo, Janusz Grabowski, Norbert Poncin, J. Geom. Phys. 110Tiffany Covolo and Janusz Grabowski and Norbert Poncin, Splitting theorem for Z n 2 - supermanifolds. J. Geom. Phys. 110 (2016), 393-401.
The category of Z n 2 -supermanifolds. Tiffany Covolo, Janusz Grabowski, Norbert Poncin, J. Math. Phys. 577ppTiffany Covolo and Janusz Grabowski and Norbert Poncin, The category of Z n 2 - supermanifolds. J. Math. Phys. 57 (2016), no. 7, 073503, 16 pp.
Tiffany Covolo, Stephen Kwok, Norbert Poncin, arXiv:1608.00961Differential calculus on Z n 2 -supermanifolds. Tiffany Covolo and Stephen Kwok, and Norbert Poncin, Differential calculus on Z n 2 - supermanifolds. arXiv:1608.00961.
Superconformal invariance in two dimensions and the tricritical Ising model. Daniel Friedan, Zongan Qiu, Stephen Shenker, Phys. Lett. B. 1511Daniel Friedan and Zongan Qiu and and Stephen Shenker, Superconformal invariance in two dimensions and the tricritical Ising model. Phys. Lett. B 151 (1985), no. 1, 37-43.
Savoy and Bruno Zumino General massive multiplets in extended supersymmetry. Sergio Ferrara, Carlos A , Phys. Lett. B. 10081085MRSergio Ferrara and Carlos A. Savoy and Bruno Zumino General massive multiplets in extended supersymmetry. Phys. Lett. B 100:5 (1981), 393-398. MR 82f:81085
Unitary representations of the Virasoro and super-Virasoro algebras. Peter Goddard, Adrian Kent, David I Olive, Phys. 1031Peter Goddard and Adrian Kent and David I. Olive, Unitary representations of the Virasoro and super-Virasoro algebras. Phys. 103 (1986), no. 1, 105-119.
Linear algebra and geometry. A second course. Irving Kaplansky, Allyn and Bacon, IncBoston, MassIrving Kaplansky, Linear algebra and geometry. A second course. Allyn and Bacon, Inc., Boston, Mass. 1969 xii+139 pp.
Bertram Kostant, Graded manifolds, graded Lie theory, and prequantization. Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn. Bonn; BerlinSpringer570Bertram Kostant, Graded manifolds, graded Lie theory, and prequantization. Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), pp. 177-306. Lecture Notes in Math., Vol. 570, Springer, Berlin, 1977.
Graded manifolds and graded Lie algebras. Jean-Louis Koszul, Proceedings of the international meeting on geometry and physics. the international meeting on geometry and physicsBolognaFlorenceJean-Louis Koszul, Graded manifolds and graded Lie algebras. Proceedings of the inter- national meeting on geometry and physics (Florence, 1982), 71-84, Pitagora, Bologna, 1983.
Differentiable vectors and unitary representations of Fréchet-Lie supergroups. Karl-Hermann Neeb, Hadi Salmasian, Math. Z. 2751-2Karl-Hermann Neeb and Hadi Salmasian, Differentiable vectors and unitary represen- tations of Fréchet-Lie supergroups. Math. Z. 275 (2013), no. 1-2, 419-451.
Positive definite superfunctions and unitary representations of Lie supergroups. Karl-Hermann Neeb, Hadi Salmasian, Transform. Groups. 183Karl-Hermann Neeb and Hadi Salmasian, Positive definite superfunctions and unitary representations of Lie supergroups. Transform. Groups 18 (2013), no. 3, 803-844.
Oscillator representations for orthosymplectic algebras. Kyo Nishiyama, J. Algebra. 1291Kyo Nishiyama, Oscillator representations for orthosymplectic algebras. J. Algebra 129 (1990), no. 1, 231-262.
Unitary representations of super-gauge symmetries. Abdus Salam, J A Strathdee, Nuclear Phys. B. 80Abdus Salam and J. A. Strathdee, Unitary representations of super-gauge symmetries. Nuclear Phys. B 80:3 (1974), 499-505.
Generalized Lie algebras. Manfred Scheunert, J. Math. Phys. 204Manfred Scheunert, Generalized Lie algebras. J. Math. Phys. 20 (1979), no. 4, 712-720.
Harmonic analysis on semi-simple Lie groups. I. Die Grundlehren der mathematischen Wissenschaften. Garth Warner, Springer-Verlag188New York-Heidelbergxvi+529 ppGarth Warner, Harmonic analysis on semi-simple Lie groups. I. Die Grundlehren der mathematischen Wissenschaften, Band 188. Springer-Verlag, New York-Heidelberg, 1972. xvi+529 pp.
| [] |
[
"Collective states of the odd-mass nuclei within the framework of the Interacting Vector Boson Model The odd-mass nuclei in IVBM 2",
"Collective states of the odd-mass nuclei within the framework of the Interacting Vector Boson Model The odd-mass nuclei in IVBM 2"
] | [
"H G Ganev [email protected] \nInstitute for Nuclear Research and Nuclear Energy\nBulgarian Academy of Sciences\n1784SofiaBulgaria\n"
] | [
"Institute for Nuclear Research and Nuclear Energy\nBulgarian Academy of Sciences\n1784SofiaBulgaria"
] | [] | A supersymmetric extension of the dynamical symmetry group Sp B (12, R) of the Interacting Vector Boson Model (IVBM), to the orthosymplectic group OSp(2Ω/12, R) is developed in order to incorporate fermion degrees of freedom into the nuclear dynamics and to encompass the treatment of odd mass nuclei. The bosonic sector of the supergroup is used to describe the complex collective spectra of the neighboring even-even nuclei and is considered as a core structure of the odd nucleus. The fermionic sector is represented by the fermion spin group SO F (2Ω) ⊃ SU F (2).The so obtained, new exactly solvable limiting case is applied for the description of the nuclear collective spectra of odd mass nuclei. The theoretical predictions for different collective bands in three odd mass nuclei, namely 157 Gd, 173 Y b and 163 Dy from rare earth region are compared with the experiment. The B(E2) transition probabilities for the 157 Gd and 163 Dy between the states of the ground band are also studied. The important role of the symplectic structure of the model for the proper reproduction of the B(E2) behavior is revealed. The obtained results reveal the applicability of the models extension. | 10.1088/0954-3899/35/12/125101 | [
"https://arxiv.org/pdf/0705.2169v2.pdf"
] | 119,243,664 | 0705.2169 | 432de41c871e60d7642f54aef8d976d91584ec34 |
Collective states of the odd-mass nuclei within the framework of the Interacting Vector Boson Model The odd-mass nuclei in IVBM 2
11 Apr 2008
H G Ganev [email protected]
Institute for Nuclear Research and Nuclear Energy
Bulgarian Academy of Sciences
1784SofiaBulgaria
Collective states of the odd-mass nuclei within the framework of the Interacting Vector Boson Model The odd-mass nuclei in IVBM 2
11 Apr 2008arXiv:0705.2169v2 [nucl-th]
A supersymmetric extension of the dynamical symmetry group Sp B (12, R) of the Interacting Vector Boson Model (IVBM), to the orthosymplectic group OSp(2Ω/12, R) is developed in order to incorporate fermion degrees of freedom into the nuclear dynamics and to encompass the treatment of odd mass nuclei. The bosonic sector of the supergroup is used to describe the complex collective spectra of the neighboring even-even nuclei and is considered as a core structure of the odd nucleus. The fermionic sector is represented by the fermion spin group SO F (2Ω) ⊃ SU F (2).The so obtained, new exactly solvable limiting case is applied for the description of the nuclear collective spectra of odd mass nuclei. The theoretical predictions for different collective bands in three odd mass nuclei, namely 157 Gd, 173 Y b and 163 Dy from rare earth region are compared with the experiment. The B(E2) transition probabilities for the 157 Gd and 163 Dy between the states of the ground band are also studied. The important role of the symplectic structure of the model for the proper reproduction of the B(E2) behavior is revealed. The obtained results reveal the applicability of the models extension.
Introduction
Symmetry is an important concept in nuclear physics. In finite many-body systems of this type, it appears as time reversal, parity, and rotational invariance, but also in the form of dynamical symmetries.
Many collective properties of the nuclei have been investigated using models based on dynamical groups. One of the most popular and widely used models of this type are the Interacting Boson Model (IBM) [1] and its extensions [2], [3] as well as the symplectic model [4] based on the group Sp(6, R). In these, algebraic models the bands of collective states are classified by the irreducible representations (irreps) of the corresponding chains of groups and their corresponding properties, such as energy levels and electromagnetic transition strengths, are determined by algebraic methods.
It is well known that nucleons have intrinsic spin and that there are strong spinorbit interactions. Moreover, the experiment reveals, that the presence of spin does not prevent the appearance of rotational bands. It also establishes similarity in the rotational character of the different collective bands for neighboring even-even and oddeven nuclei far from closed shells. For the description of the nuclear spectra of such even-even nuclei the above mentioned variety of boson models is used. This is possible because in the even-even nuclei the pairs of nucleons are usually considered as coupled to integer angular momentum. However, this is not the case for odd mass nuclei. Thus, the following question naturally arises: how to incorporate fermion degrees of freedom into the nuclear dynamics in a way that the rotational character of the collective bands is preserved.
In general, it is believed that the collective states of odd nuclei can be described by using particle-core coupled-type models. The natural extension of IBM, the Interacting Boson-Fermion Model (IBFM) [5], which includes single-particle (fermion) degrees of freedom in addition to the collective (boson) ones, have provided in the last decays a unified framework for the description of even-even and odd-even nuclei distant from closed shell configurations, at least in the low-angular momentum domain.
For the description of odd−A nuclei, a fermion needs to be coupled to the N boson system. This can be done by a semimicroscopical approach which relies on seniority in the nuclear shell model [2]. As an alternative to this, in the IBFM approach, Hamiltonians exhibiting dynamical Bose-Fermi symmetries, that are analytically solvable [5] are constructed. Thus, the extension of the IBM for the case of odd mass nuclei leads to the group structure U B (6) ⊗ U F (m) (IBFM-1) or U B π (6) ⊗ U B ν (6) ⊗ U F (m) (IBFM-2), where m = j (2j + 1) is the dimension of the single-particle space. Obviously, in the general case for arbitrary m− values, analytical expressions for the nuclear levels would be too cumbersome and will contain too many parameters. Moreover, orbitals higher in energy than those of the valence shell might play a role and have to be considered (for example, in the Sp(6, R) model), thus breaking the symmetric scheme. Therefore, numerical calculations have to be performed with schematic Hamiltonians. These deficiencies, motivate the development of the new extension of the IVBM, which will be based on the success of the boson description of the even-even nuclei, but will include the fermion degrees of freedom in a simple and straightforward way, that still leads to exact analytic solutions.
In the early 1980s, a boson-number-preserving version of the phenomenological algebraic Interacting Vector Boson Model (IVBM) [6] was introduced and applied successfully [7] to a description of the low-lying collective rotational spectra of the eveneven medium and heavy mass nuclei. With the aim of extending these applications to incorporate new experimental data on states with higher spins and to incorporate new excited bands, we explored the symplectic extension of the IVBM [8], for which the dynamical symmetry group is Sp(12, R). This extension is realized from, and has its physical interpretation over basis states of its maximal compact subgroup U (6) ⊂ Sp(12, R), and resulted in the description of various excited bands of both positive and negative parity of complex systems exhibiting rotation-vibrational spectra [9]. With the present work we extend the earlier applications of IVBM for the description of the ground and first excited positive and/or negative bands of odd mass nuclei. In order to do this we propose a new dynamical symmetry which is applied to real odd nuclear systems.
Thus, it is the purpose of this paper to bring spin explicitly into the symplectic IBVM. We approach the problem by considering the simplest physical picture in which a particle (or quasiparticle) with intrinsic spin taking a single j−value j is coupled to an even-even nucleus whose states belong to an Sp(12, R) irrep. Nevertheless, the results for the energy spectra and the intraband transitions between the states of the ground state band obtained in this simplified version of the model agree rather well with the experimental data.
The even-even core
The IVBM is based on the introduction of two kinds of vector bosons (called p and n bosons), that "built up" the collective excitations in the nuclear system. The creation operators of these bosons are assumed to be SO(3) vectors and they transform according to two independent fundamental representations (1,0) of the group SU(3) . These bosons form a "pseudospin" doublet of the U(2) group and differ in their "pseudospin" projection α = ± 1 2 . We want to point out that these vector bosons should be considered as "building blocks" generating appropriate algebraic structures rather than real correlated fermion pairs coupled to angular momentum l = 1.
The algebraic structure of the IVBM is realized in terms of creation and annihilation operators u + m (α), u m (α) (m = 0, ±1). The later are related to the cyclic coordinates x ±1 (α) = ∓ 1 √ 2 (x 1 (α) ± ix 2 (α)), x 0 (α) = x 3 (α), and their associated momenta q m (α) = −i∂/∂x m (α) in the standard way
u + m (α) = 1 √ 2 (x m (α) − iq m (α)),(1)u m (α) = (u + m (α)) † ),
where x i (α) i = 1, 2, 3 are Cartesian coordinates of a quasi-particle vectors with an additional index -the projection of the "pseudo-spin" α = ± 1 2 . The bilinear products of the creation and annihilation operators of the two vector bosons (1) generate the boson representations of the non-compact symplectic group Sp(12, R) [6]:
F L M (α, β) = k,m C LM 1k1m u + k (α)u + m (β), G L M (α, β) = k,m C LM 1k1m u k (α)u m (β),(2)A L M (α, β) = k,m C LM 1k1m u + k (α)u m (β),(3)
where C LM 1k1m , which are the usual Clebsch-Gordon coefficients for L = 0, 1, 2 and M = −L, −L + 1, ...L, define the transformation properties of (2) and (3) under rotations. The commutation relations between the pair creation and annihilation operators (2) and the number preserving operators (3) are given in [6].
Being a noncompact group, the representations of Sp(12, R) are of infinite dimension, which makes it impossible to diagonalize the most general Hamiltonian. When restricted to the group U B (6), each irrep of the group Sp B (12, R) decomposes into irreps of the subgroup characterized by the partitions [8], [10]:
[N, 0 5 ] 6 ≡ [N] 6 ,
where N = 0, 2, 4, . . . (even irrep) or N = 1, 3, 5, . . . (odd irrep). The subspaces [N] 6 are finite dimensional, which simplifies the problem of diagonalization. Therefore the complete spectrum of the system can be calculated through the diagonalization of the Hamiltonian in the subspaces of all the unitary irreducible representations (UIR) of U(6), belonging to a given UIR of Sp(12, R), which further clarifies its role of a group of dynamical symmetry. Since N is the number of collective excitations (phonons) rather than real nucleon pairs, in the present paper we consider only the even irrep of Sp(12, R).
The most general one and two-body Hamiltonian can be expressed in terms of symplectic generators. In general, such rather general Hamiltonian has to be diagonalized numerically to obtain the energy eigenvalues and wave functions. There exist, however, special situations in which the eigenvalues can be obtained in closed, analytical form. These special solutions provide a framework in which energy spectra and other nuclear properties can be interpreted in a qualitative way. These situations correspond to dynamical symmetries of the Hamiltonian.
The Hamiltonian, corresponding to the unitary limit of IVBM [8] Sp
(12, R) ⊃ U(6) ⊃ U(3) ⊗ U(2) ⊃ O(3) ⊗ (U(1) ⊗ U(1)),(4)
expressed in terms of the first and second order invariant operators of the different subgroups in the chain (4) is [8]:
H = aN + bN 2 + α 3 T 2 + β 3 L 2 + α 1 T 2 0 .(5)
H (5) is obviously diagonal in the basis
| [N] 6 ; (λ, µ); KLM; T 0 ≡ | (N, T ); KLM; T 0 ,(6)
labelled by the quantum numbers of the subgroups of the chain (4). Its eigenvalues are the energies of the basis states of the boson representations of Sp(12, R):
E((N, T ), L, T 0 ) = aN + bN 2 + α 3 T (T + 1) + β 3 L(L + 1) + α 1 T 2 0 .(7)
The non-compact group Sp(12, R) has a Jordan (three grading) decomposition with respect to its maximal compact subgroup U(6). Its Lie algebra g can be decomposed as a vector space direct sum:
g = g − ⊕ g 0 ⊕ g + .
Every unitary lowest weight representation of Sp(12, R) can be constructed by acting consequently on the boson lowest weight state (LWS) | Ω B , transforming in a definite U(6) representation, with the raising generators F L M (α, β) which belong to the g + space. This action generates an infinite set of states (6) that form the basis of a unitary lowest weight representation of Sp(12, R). If the LWS | Ω B transforms irreducibly under U(6), then the corresponding unitary representation of Sp(12, R) is also irreducible. The unitary lowest weight irreducible representation of Sp(12, R) can therefore be uniquely labeled by the U(6) labels of their lowest weight states. In the boson space there are only two nonequivalent irreducible lowest weight states, namely, the (boson) vacuum
| Ω B =| 0 B(8)
and the "one-particle" state
| Ω B = u † k (α) | 0 B .(9)
The construction of the symplectic basis for the even IR of Sp(12, R), which can be obtained by action of the fully symmetric coupled powers of raising operators F L M (α, β) on the on vacuum state (8), is given in details in [8]. The Sp(12, R) classification scheme for the SU(3) boson representations for even value of the number of bosons N is shown on Table I in Ref. [8] (see also Table 1).
The most important application of the U B (6) ⊂ Sp B (12, R) limit of the theory is the possibility it affords for describing both even and odd parity bands up to very high angular momentum [8]. In order to do this we first have to identify the experimentally observed bands with the sequences of basis states of the even Sp(12, R) irrep ( Table 1). As we deal with the symplectic extension of the boson representations of the number preserving U B (6) symmetry we are able to consider all even eigenvalues of the number of vector bosons N with the corresponding set of pseudospins T , which uniquely define the SU B (3) irreps (λ, µ). The multiplicity index K appearing in the final reduction to the SO(3) is related to the projection of L on the body fixed frame and is used with the parity (π) to label the different bands (K π ) in the energy spectra of the nuclei. For the even-even nuclei we have defined the parity of the states as π core = (−1) T [8]. This allowed us to describe both positive and negative bands.
Further, we use the algebraic concept of "yrast" states, introduced in [8]. According to this concept we consider as yrast states the states with given L, which minimize the The presented mapping of the experimental states onto the SU(3) basis states, using the algebraic notion of yrast states, is a particular case of the so called "stretched" states [11]. The latter are defined as the states with
(λ 0 + 2k, µ 0 ) or (λ 0 , µ 0 + k), where N i = λ 0 + 2µ 0 and k = 0, 1, 2, 3, . . ..
It was established [12] that the correct placement of the bands in the spectrum strongly depends on their bandheads configuration, and in particular, on the minimal or initial number of bosons, N = N i , from which they are built. The latter determines the starting position of each excited band.
Thus, for the description of the different excited bands, we first determine the N i of the band head structure and develop the corresponding excited band over the stretched SU(3) multiplets. This corresponds to the sequence of basis states with N = N i , N i +4, N i +8, . . . (∆N = 4). The values of T for the first type of stretched states (λ−changed) are changed by step ∆T = 2, whereas for the second type (µ−changed) −T is fixed so that in both cases the parity is preserved even or odd, respectively. For all presented even-even nuclei, the states of the corresponding β− and γ− bands are associated with the stretched states of the first type (λ− changed).
The odd-A nuclei 157 Gd, 173 Y b and 163 Dy, to which we apply our model, can be considered as a particle coupled to the even-even cores 156 Gd, 172 Y b and 162 Dy, respectively. We determine the values of the five phenomenological model parameters a, b, α 3 , β 3 , α 1 by fitting the energies of the ground and few excited bands (γ− and/or β− bands) of the even-even nuclei to the experimental data [13], using a χ 2 procedure. The theoretical predictions for the even core nuclei are presented in the Figures 1−3. For comparison, the predictions of IBM (with 4 adjustable parameters) are also shown. The IBM results for 156 Gd and 162 Dy, 172 Y b are extracted from Refs. [14] and [15], respectively. From the figures one can see that the calculated energy levels agree rather well up to very high angular momenta with the observed data. One can see also that for high spins (L ≥ 10 − 14), where the deviations of the IBM predictions become more significant, the structure of the energy levels of the GSB (β− and γ−bands) is reproduced rather well.
The inclusion of spin
Underlying the conventional nuclear shell model is the idea that the low-lying states of nuclei can be restricted to a valence space of states obtained by putting nucleons into a finite set of single-particle states indexed i = 1, . . . , Ω; i. e. the M valenceparticle Hilbert space is the anti-symmetrized (exterior) product of M copies of an Ω−dimensional single-nucleon Hilbert space. This space carries a sum of two irreducible representations of the fermion pair algebra SO(2Ω) [16]. The set of all even fermion states span an irreducible representation of the SO(2Ω) algebra and the set of all states of odd fermion number span another irreducible representation.
Thus, in order to incorporate the intrinsic spin degrees of freedom into the symplectic IVBM, we extend the dynamical algebra of Sp(12, R) to the orthosymplectic algebra of OSp(2Ω/12, R). For this purpose we introduce a particle (quasiparticle) with spin j and consider a simple core plus particle picture. Thus, in addition to the boson collective degrees of freedom (described by dynamical symmetry group Sp(12, R)) we introduce creation and annihilation operators a † m and a m (m = −j, . . . , j), which satisfy the anticommutation relations
{a † m , a † m ′ } = {a m , a m ′ } = 0, {a m , a † m ′ } = δ mm ′ .(10)
All bilinear combinations of a + m and a m ′ , namely
f mm ′ = a † m a † m ′ , m = m ′ g mm ′ = a m a m ′ , m = m ′ ;(11)C mm ′ = (a † m a m ′ − a m ′ a † m )/2(12)
generate the (Lie) fermion pair algebra of SO F (2Ω). Their commutation relations are:
[g mn , C m ′ n ′ ] = δ nm ′ g mn ′ − δ mm ′ g nn ′ , [C mn , f n ′ n ′ ] = δ nm ′ g mn ′ − δ nm ′ g mn ′ , [g mn , f m ′ n ′ ] = − δ mm ′ C n ′ n − δ nn ′ C n ′ m + δ n ′ m C n ′ n + δ m ′ n C n ′ m ,
The number preserving operators (12) generate maximal compact subalgebra of SO F (2Ω), i.e. U F (Ω). The upper (lower) script B or F denotes the boson or fermion degrees of freedom, respectively. Making use of the embedding SU F (2) ⊂ SO F (2Ω), we make orthosymplectic (supersymmetric) extension of the IVBM which is defined through the chain:
OSp(2Ω/12, R) ⊃ SO F (2Ω) ⊗ Sp B (12, R) ⇓ ⇓ ⊗ U B (6) N ⇓ SU F (2) ⊗ SU B (3) ⊗ U B T (2) j (λ, µ) ⇐⇒ (N, T ) ց ⇓ ⊗ SO B (3) ⊗ U(1) L T 0 ⇓ Spin BF (3) ⊃ Spin BF (2), J J 0(13)
where bellow the different subgroups the quantum numbers characterizing their irreducible representations are given. Here with Spin BF (n) (n = 2, 3) is denoted the universal covering group of the SO(n). From (13) it can be seen that the coupling of the boson and fermion degrees of freedom is done on the level of the angular momenta. We want to stress, however, that although the formal "coupling" is done at the "final" stage, the present situation is not identical to that of IBFM. In the latter the eveneven core, to which an odd unpaired nucleon is coupled to, is considered as "inert". In the present approach since the (ortho)symplectic structures are taken into account (allowing for the change of number of phonon excitations N), the core is not anymore inert. Physically, this does not correspond to the weak coupling limit (as should be if N was fixed) between the core and particle as it is in the case of IBFM (on this level of coupling).
Application of the new dynamical symmetry
The energy spectrum
In this paper we expand the earlier application of the IVBM [8], developed for the description of the collective bands of even-even nuclei, in order to include in our considerations the case of odd mass nuclei. We can label the basis states according to the chain (13) as:
where [N] 6 − is the U(6) labeling quantum number, (λ, µ)− are the SU(3) quantum numbers characterizing the core excitations, K is the multiplicity index in the reduction SU(3) ⊂ SO(3), L is the core angular momentum, j−the spin of the odd particle, J, J 0 are the total (coupled) angular momentum and its third projection, and T ,T 0 are the pseudospin and its third projection, respectively. Since the SO(2Ω) label is irrelevant for our application, we drop it in the states (14).
The Hamiltonian can be written as linear combination of the Casimir operators of the different subgroups in (13):
H = aN + bN 2 + α 3 T 2 + β ′ 3 L 2 + α 1 T 2 0 + ηj 2 + γ ′ J 2 + ζJ 2 0(15)
and it is obviously diagonal in the basis (14) labeled by the quantum numbers of their representations. Then the eigenvalues of the Hamiltonian (15), that yield the spectrum of the odd mass system are:
E(N; T, T 0 ; L, j; J, J 0 ) = aN + bN 2 + α 3 T (T + 1) + β ′ 3 L(L + 1) + α 1 T 2 0 + ηj(j + 1) + γ ′ J(J + 1) + ζJ 2 0 .(16)
We note that only the last three terms of (15) come from the orthosymplectic extension. But since only one fermion (M = 1) is considered (and j is fixed), the j−term in (16) is just additive constant and can be dropped. (The presence of the latter should only rescale the values of the rest model parameters.) Thus, for the description of the excitation spectra of odd-mass nuclei only two new parameters are involved in the fitting procedure. We choose parameters β ′ 3 = 1 2 β 3 and γ ′ = 1 2 γ instead of β 3 and γ in order to obtain the Hamiltonian form of ref. [8] (setting β 3 = γ), when for the case j = 0 (hence J = L) we recover the symplectic structure of the IVBM.
The infinite set of basis states classified according to the reduction chain (13) are schematically shown in Table 1. The fourth and fifth columns show the SO B (3) content of the SU B (3) group, given by the standard Elliott's reduction rules [17], while in the next column are given the possible values of the common angular momentum J, obtained by coupling of the orbital momentum L with the spin j. The latter is vector coupling and hence all possible values of the total angular momentum J should be considered. For simplicity, only the maximally aligned (J = L + j) and maximally antialigned (J = L − j) states are illustrated in Table 1.
The basis states (14) can be considered as a result of the coupling of the orbital | (N, T ); KLM; T 0 (6) and spin φ jm wave functions. Then, if the parity of the single particle is π sp , the parity of the collective states of the odd−A nuclei will be π = π core π sp . Thus, the description of the positive and/or negative parity bands requires only the proper choice of the core band heads, on which the corresponding single particle is coupled to, generating in this way the different odd−A collective bands. Our choice is based on the fact, which has been always understood in nuclear physics, that well defined rotational bands can exist only when they are adiabatic relative to other degrees of freedom. In this way (in adiabatic approximation) the single particle is dragged around in the core field (which corresponds to the "strong" coupling limit as is in our case) and the combined system is essentially a new rotor with slightly different bulk properties, such as moment of inertia, etc.
Further in the present considerations, the yrast conditions yield relations between the number of bosons N and the coupled angular momentum J that characterizes each Table 1. Classification scheme of basis states (14) according the decompositions given by the chain (13).
N T (λ, µ) K L J = L ± j 0 0 (0, 0) 0 0 j 2 1 (2, 0) 0 0, 2 j; 2 ± j 0 (0, 1) 0 1 1 ± j 2 (4, 0) 0 0, 2, 4 j; 2 ± j; 4 ± j 4 1 (2, 1) 1 1, 2, 3 1 ± j; 2 ± j; 3 ± j 0 (0, 2) 0 0, 2 j; 2 ± j
3 (6, 0) 0 0, 2, 4, 6 j; 2 ± j; 4 ± j; 6 ± j 2 (4, 1) 1 1, 2, 3, 4, 5 1 ± j; 2 ± j; 3 ± j;
4 ± j; 5 ± j 6 1 (2, 2) 2 2, 3, 4 2 ± j; 3 ± j; 4 ± j 0 0, 2 j; 2 ± j 0 (0, 3) 0 1, 3 1 ± j; 3 ± j 4 (8, 0) 0 0, 2, 4, 6, 8 j; 2 ± j; 4 ± j;
6 ± j; 8 ± j 3 (6, 1) 1 1, 2, 3, 4, 5, 6, 7 1 ± j; 2 ± j; 3 ± j; 4 ± j; 5 ± j; 6 ± j; 7 ± j; 8 ± j 2 (4, 2) 2 2, 3, 4, 5, 6 2 ± j; 3 ± j; 4 ± j;
5 ± j; 6 ± j 8 0 0, 2, 4 j; 2 ± j; 4 ± j 1 (2, 3) 2 2, 3, 4, 5 2 ± j; 3 ± j; 4 ± j; 5 ± j Here it is assumed that the single particle has j = 3/2 and parity π sp = (−), so that the common parity π is also negative.
For the description of the different excited bands, we first determine the N i of the band head structure and then we map the states of the corresponding band onto the sequence of basis states with N = N i , N i +2, N i +4, . . . (∆N = 2) and T = even = f ixed or T = odd = f ixed, respectively. This choice corresponds to the stretched states of the second type (µ−changed).
We will point out that the (ortho)symplectic structure of the model space gives us rather rich possibilities to map experimentally observed states onto the basis states.
Thus, another possibility of developing the sequence of band's states is to take again N = N i , N i + 4, N i + 8, . . . (∆N = 4) but to change T = T i , T i + 2, T i + 4, ... (∆T = 2) in such a way, that the parity is preserved even or odd, respectively. Such correspondence takes place for the first type of the stretched states (λ−changed). In the present application, all the collective bands under consideration are associated with the stretched states of second type (µ−changed).
The number of adjustable parameters needed for the complete description of the collective spectra of the odd-A nuclei is two, namely γ and ζ. They are evaluated by a fit to the experimental data [13] of the GSB of the corresponding odd-A nucleus. The comparison between the experimental spectra for the GSB and first few excited bands and our calculations using the values of the model parameters given in Table 2 for the nuclei 157 Gd, 173 Y b and 163 Dy is illustrated in Figures 4−6. The last single particle, which for all of these rare earth nuclei is a neutron, occupies the major shell N = 82 − 126, where the relevant single particle levels are 2f7 having odd parity (π sp = −) (excluding the intruder from the upper shell with opposite parity). In our considerations we take into account only the first available single particle orbit j (generating the group SO(2Ω) with Ω = (2j + 1)), which for the first nucleus implies j = 3 2 , while for the other two − j = 5 2 . The Nilsson asymptotic quantum numbers Ω[Nn 3 Λ] are written bellow each band. One can see from the figures that the calculated energy levels agree rather well in general with the experimental data up to very high angular momenta. For comparison, in the Figures 4−6 the IBFM results (obtained by total 7 adjustable parameters) are also shown. They are extracted from Refs. [14] and [15], respectively. Note that all calculated levels, for the bands considered, are in correct order in contrast to IBFM results (for 157 Gd). Another difference between the IVBM and IBFM predictions is that in the former the correct placement of all the band heads is reproduced quite well.
In the Table 2, the values of N i , T , T 0 , J, J 0 and χ 2 for each band under consideration are also given.
Electromagnetic transition probabilities
A successful nuclear model must yield a good description not only of the energy spectrum of the nucleus but also of its electromagnetic properties. Calculation of the latter is a good test of the nuclear model functions. The most important electromagnetic features are the E2 transitions. In this subsection we discuss the calculation of the E2 transition strengths and compare the results with the available experimental data.
As was mentioned, in the symplectic extension of the IVBM the complete spectrum of the system is obtained in all the even subspaces with fixed N-even of the UIR [N] 6 of U(6), belonging to a given even UIR of Sp(12, R). The classification scheme of the SU(3) boson representations for even values of the number of bosons N was presented in Table 1.
In this paper we give as an example the evaluation of the E2 transition probabilities Using the tensorial properties of the Sp(12, R) generators with respect to (4) it is easy to define the E2 transition operator [18] between the states of the considered band as: The tensor product As was mentioned, the basis states (14) can be considered as a result of the coupling of the orbital | (N, T ); KLM; T 0 (6) and spin φ jm wave functions. Since the spin j is simply vector coupled to the orbital momentum L, the action of the transition operator T E2 concerns only the orbital part of the basis functions (14). In Ref. [18] it is shown that the two main types of B(E2) behavior -the enhancement or the reduction of the B(E2) values within the GSB K π = 0 + , can be reproduced simply by the change of the sign of θ. The strongly enhanced values which are an indication for increased collectivity in the high angular momentum domain are easily obtained for positive values of the parameter θ. For negative values of the parameter θ we obtain behavior similar to that of the standard SU(3) one and it can be used to reproduce the well known cutoff effect. Such saturation effect is also characteristic feature of the IBM based calculations in its SU(3) limit. It is shown also that although the coefficient in front of symplectic term is some orders of magnitude smaller than the SU(3) contribution to the transition operator its role in reproducing the correct behavior (with or without cutoff) of the transition probabilities between the states of the GSB band is very important. For more details concerning discussed behavior of the B(E2) values see [18].
T E2 = e A [1−1] 6 20 [210] 3 [0] 2 00 + θ([F × F ][F × F ] [4] 6 20 (0,2)[0] 2 00 = C [2] 6 [2] 6 [4] 6 (2,0)[2] 2 (2,0)[2] 2 (0,2)[0] 2 C (2,0) (2,0) (0,2) (2) 3 (2) 3 (2) 3(18)B(E2; J i → J f ) = 1 2J i + 1 | f T E2 i | 2 .(19)
In order to prove the correct predictions following from our theoretical results we apply the theory to real nuclei for which there is available experimental data for the transition probabilities [20] between the states of the ground bands up to very high angular momenta. The application actually consists of fitting the two parameters of the transition operator T E2 (17) to the experiment for each of the considered bands. The B(E2) strengths between the negative parity states of the GSB, as were attributed to the SU(3) symmetry-adapted basis states of the model, are calculated. For this SU(3) multiplets, the procedure for their calculations actually coincides with that given in [18]. The theoretical predictions for the nuclei 157 Gd and 163 Dy are compared with the experimental data in Figures 7 and 8. From the figures one can see that the experimental values are reproduced quite well for the both typical examples − with enhanced B(E2) values ( 157 Gd) and with cutoff ( 163 Dy).
Conclusions
In this work we extended the dynamical symmetry group Sp(12, R) of the IVBM to the orthosymplectic one OSp(2Ω/12, R). We introduced the fermion degrees of freedom by means of including a particle (quasiparticle) with spin j and exploiting the corresponding reduction SO F (2Ω) ⊃ SU F (2).
Further, the basis states of the odd systems are classified by the new dynamical symmetry (13) and the model Hamiltonian is written in terms of the first and second order invariants of the groups from the corresponding reduction chain. Hence the problem is exactly solvable within the framework of the IVBM which, in turn, yields a simple and straightforward application to real nuclear systems.
We present results that were obtained through a phenomenological fit of the models' predictions for the spectra of collective states to the experimental data for odd−A nuclei and their even-even neighbors, used as a core for the formers. The good agreement between the theoretical and the experimental band structures confirms the applicability of the newly proposed dynamical symmetry of the IVBM. The success is based on the (ortho)symplectic structures of the model which allow the mixing of the basic collective modes −rotational and vibrational ones arising from the yrast conditions. This allows for the proper reproduction of the high spin states of the collective bands and the correct placement of the different band heads.
For two of the three isotopes considered, the B(E2) transition probabilities are calculated and compared with the experimental data. The important role of the symplectic extension of the model for the correct reproduction of the B(E2) behavior, observed at high angular momenta, is revealed.
The supersymmetry group OSp(2Ω/12, R) which is natural generalization of the dynamical symmetry group Sp(12, R) of the IVBM could be further used to examine the correlations between the spectroscopic properties of the neighboring eveneven, odd-even and odd-odd spectra of the neighboring nuclei and the underlying supersymmetry which might be considered in nuclear physics as proved experimentally [21]. These investigations are the subject of the forthcoming paper, but our preliminary results obtained in this work already suggest the typical signatures of the nuclear supersymmetry.
Figure 1 .
1Comparison of the theoretical and experimental energies for the ground and first excited bands of 156 Gd.
Figure 2 .
2The same asFig. 1, but for 172Yb.
energy(7) with respect to the number of vector bosons N that build them. Thus the states of the ground state band (GSB) were identified with the SU(3) multiplets (0, µ)[8]. In terms of (N, T ) this choice corresponds to (N = 2µ, T = 0) and the sequence of states with different numbers of bosons N = 0, 4, 8, . . . and pseudospin T = 0 (and also T 0 = 0). Hence the minimum values of the energies (7) are obtained at N = 2L.
Figure 3 .
3The same asFig. 1, but for 162 Dy.
|
[N] 6 ; (λ, µ); KL; j; JJ 0 ; T 0 ≡ | [N] 6 ; (N, T ); KL; j; JJ 0 ; T 0 ,
2 ± j; 4 ± j . . . . . . . . . . . . . . . . . .collective state. For example, the collective states of the GSB K π J = 3 2 − are identified with the SU(3) multiplets (0, µ) which yield the sequence N = 2(J − j) = 0, 2, 4, . . . for the corresponding values J = 3 2 , 5 2 , 7 2 , .... The pseudospin for the SU(3) multiplets (0, µ) is T = 0 and hence π core = (−1) T = (+).
−
. For both cases, the states of the GSB are identified with the SU(3) multiplets (0, µ) and µ = L. This yields the sequence N = 2(J −j) = 0, 2, 4, . . . for the corresponding values J = 3 2 , 5 2 , 7 2 , .... In terms of (N, T ) this corresponds to (N = 2µ, T = 0).
part of (17) is a SU(3) generator and actually changes only the angular momentum with ∆L = 2.
Figure 4 .
4Comparison of the theoretical and experimental energies for the ground and first excited negative parity bands of 157 Gd.
Figure 5 .
5The same asFig. 4, but for 173 Yb.of bosons by ∆N = 4 and ∆L = 2. It is obvious that this term in T E2 (17) comes from the symplectic extension of the model. In(17) e is the effective boson charge.The transition probabilities are by definition SO(3) reduced matrix elements of transition operators T E2 (17) between the |i −initial and |f −final collective states(14)
Figure 6 .
6Comparison of the theoretical and experimental energies for the ground and first excited positive and negative parity bands of 163 Dy.
Figure 7 .
7(Color online) Comparison of the theoretical and experimental values for the B(E2) transition probabilities for the 157 Gd.
Figure 8 .
8(Color online) The same asFig. 7, but for 163 Dy.
Table 2 .
2Values of the model parameters.
AcknowledgmentsThe author is grateful to Dr. A. I. Georgieva and Dr. V. P. Garistov for the helpful discussions. This work was supported by the Bulgarian National Foundation for scientific research under Grant Number Φ − 1501.
F Iachello, A Arima, The Interacting Boson Model. CambridgeCambridge University PressF. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, Cambridge, 1987).
. F Iachello, O Scholten, Phys. Rev. Lett. 43679F. Iachello and O. Scholten, Phys. Rev. Lett. 43, 679 (1979).
. F Iachello, Phys. Rev. Lett. 44772F. Iachello, Phys. Rev. Lett. 44, 772 (1980).
. G Rosensteel, D J Rowe, Phys. Rev. Lett. 3810G. Rosensteel and D.J. Rowe, Phys. Rev. Lett. 38, 10 (1977);
. Ann. Phys. 126343Ann. Phys. 126, 343 (1980);
. D J Rowe, Rep. Prog. Phys. 481419D.J. Rowe, Rep. Prog. Phys. 48, 1419 (1985).
F Iachello, P Van Isacker, The Interacting Boson Fermion Model. CambridgeCambridge University PressF. Iachello and P. Van Isacker, The Interacting Boson Fermion Model (Cambridge University Press, Cambridge, 1991).
. A Georgieva, P Raychev, R Roussev, J. Phys. G: Nucl. Phys. 81377A. Georgieva, P. Raychev, and R. Roussev, J. Phys. G: Nucl. Phys., 8, 1377 (1982).
. A Georgieva, P Raychev, R Roussev, J. Phys. G: Nucl. Phys. 9521A. Georgieva, P. Raychev, and R. Roussev, J. Phys. G: Nucl. Phys., 9, 521 (1983).
. H Ganev, V P Garistov, A I Georgieva, Phys. Rev. C. 6914305H. Ganev, V. P. Garistov, and A. I. Georgieva, Phys. Rev. C 69, 014305 (2004).
A I Georgieva, H Ganev, J P Draayer, Proceedings of the 5th International Symposium on Quantum Theory and Symmetries. the 5th International Symposium on Quantum Theory and SymmetriesValladolid, SpainA. I. Georgieva, H. Ganev, and J. P. Draayer, Proceedings of the 5th International Symposium on Quantum Theory and Symmetries, 22-28 July, Valladolid, Spain (2007).
. C Quesne, J. Math. Phys. 14366C. Quesne, J. Math. Phys. 14, 366 (1973).
. D J Rowe, Rep. Prog. Phys. 481419D. J. Rowe, Rep. Prog. Phys. 48, 1419 (1985).
. H G Ganev, A I Georgieva, J P Draayer, Phys. Rev. C. 7214314H. G. Ganev, A. I. Georgieva, and J. P. Draayer, Phys. Rev. C 72, 014314 (2005).
. Level Retrieval Parameters. Level Retrieval Parameters, http://iaeand.iaea.or.at/nudat/levform.html.
. D S Chuu, S T Hsieh, Nucl. Phys. 49645D. S. Chuu and S. T. Hsieh, Nucl. Phys. A496, 45 (1989).
. N Yoshida, H Sagawa, T Otsuka, A Arima, Nucl. Phys. 50390N. Yoshida, H. Sagawa, T. Otsuka and A. Arima, Nucl. Phys. A503, 90 (1989).
B G Wybourne, Classical Groups for Physicists. New YorkWileyB. G. Wybourne, Classical Groups for Physicists (Wiley, New York, 1974).
. J P Elliott, Proc. R. Soc. A245. 128J. P. Elliott, Proc. R. Soc. A245, 128, 562 (1958).
. H G Ganev, A I Georgieva, Phys. Rev. C. 7654322H. G. Ganev and A. I. Georgieva, Phys. Rev. C 76, 054322 (2007).
. D J Rowe, R Le Blanc, J Repka, J. Phys. A: Math. Gen. 22309D. J. Rowe, R. Le Blanc, and J. Repka, J. Phys. A: Math. Gen. 22, L309 (1989).
. H Kusakari, Phys. Rev. C. 461257H. Kusakari et. al., Phys. Rev. C 46, 1257 (1992);
. M Oshima, Phys. Rev. C. 39645M. Oshima et. al., Phys. Rev. C 39, 645 (1989).
. F Iachello, Phys. Rev Lett. 44772F. Iachello, Phys. Rev Lett. 44, 772 (1980);
. A B Balantekin, I Bars, R Bijker, F Iachello, Phys. Rev. C. 271761A. B. Balantekin, I. Bars, R. Bijker, and F. Iachello, Phys. Rev. C 27, 1761 (1983);
. P Van Isacker, J Jolie, K Heyde, A Frank, Phys. Rev Lett. 54653P. Van Isacker, J. Jolie, K. Heyde, and A. Frank, Phys. Rev Lett. 54, 653 (1985);
. A Metz, Phys. Rev Lett. 831542A. Metz et. al., Phys. Rev Lett. 83, 1542 (1999);
. J Barea, R Bijker, A Frank, G Loyola, Phys. Rev. C. 6464313J. Barea, R. Bijker, A. Frank, and G. Loyola, Phys. Rev. C 64, 064313 (2001);
. J Barea, R Bijker, A Frank, J. Phys. A: Math. Gen. 3710251J. Barea, R. Bijker, and A. Frank, J. Phys. A: Math. Gen. 37, 10251 (2004);
. J Barea, R Bijker, A Frank, Phys. Rev Lett. 94152501J. Barea, R. Bijker, and A. Frank, Phys. Rev Lett. 94, 152501 (2005).
| [] |
[
"Theoretical prediction of perfect spin filtering at interfaces between close-packed surfaces of Ni or Co and graphite or graphene",
"Theoretical prediction of perfect spin filtering at interfaces between close-packed surfaces of Ni or Co and graphite or graphene"
] | [
"V M Karpan \nFaculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands\n",
"P A Khomyakov \nFaculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands\n",
"A A Starikov \nFaculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands\n",
"G Giovannetti \nFaculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands\n\nInstituut-Lorentz for Theoretical Physics\nUniversiteit Leiden\nP. O. Box 95062300 RALeidenThe Netherlands\n",
"M Zwierzycki \nInstitute of Molecular Physics\nSmoluchowskiego 17P.A.N60-179PoznańPoland\n",
"M Talanana \nFaculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands\n",
"G Brocks \nFaculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands\n",
"J Van Den Brink \nInstituut-Lorentz for Theoretical Physics\nUniversiteit Leiden\nP. O. Box 95062300 RALeidenThe Netherlands\n\nInstitute for Molecules and Materials\nRadboud Universiteit Nijmegen\nP. O. Box 90106500 GLNijmegenThe Netherlands\n",
"P J Kelly \nFaculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands\n"
] | [
"Faculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands",
"Faculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands",
"Faculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands",
"Faculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands",
"Instituut-Lorentz for Theoretical Physics\nUniversiteit Leiden\nP. O. Box 95062300 RALeidenThe Netherlands",
"Institute of Molecular Physics\nSmoluchowskiego 17P.A.N60-179PoznańPoland",
"Faculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands",
"Faculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands",
"Instituut-Lorentz for Theoretical Physics\nUniversiteit Leiden\nP. O. Box 95062300 RALeidenThe Netherlands",
"Institute for Molecules and Materials\nRadboud Universiteit Nijmegen\nP. O. Box 90106500 GLNijmegenThe Netherlands",
"Faculty of Science and Technology\nMESA + Institute for Nanotechnology\nUniversity of Twente\nP.O. Box 2177500 AEEnschedeThe Netherlands"
] | [] | The in-plane lattice constants of close-packed planes of fcc and hcp Ni and Co match that of graphite almost perfectly so that they share a common two dimensional reciprocal space. Their electronic structures are such that they overlap in this reciprocal space for one spin direction only allowing us to predict perfect spin filtering for interfaces between graphite and (111) fcc or (0001) hcp Ni or Co. First-principles calculations of the scattering matrix show that the spin filtering is quite insensitive to amounts of interface roughness and disorder which drastically influence the spinfiltering properties of conventional magnetic tunnel junctions or interfaces between transition metals and semiconductors. When a single graphene sheet is adsorbed on these open d-shell transition metal surfaces, its characteristic electronic structure, with topological singularities at the K points in the two dimensional Brillouin zone, is destroyed by the chemical bonding. Because graphene bonds only weakly to Cu which has no states at the Fermi energy at the K point for either spin, the electronic structure of graphene can be restored by dusting Ni or Co with one or a few monolayers of Cu while still preserving the ideal spin injection property. | 10.1103/physrevb.78.195419 | [
"https://arxiv.org/pdf/0809.5168v2.pdf"
] | 119,251,197 | 0809.5168 | a7be925235c258680f490b066f8ac0a56b39ab32 |
Theoretical prediction of perfect spin filtering at interfaces between close-packed surfaces of Ni or Co and graphite or graphene
8 Dec 2008
V M Karpan
Faculty of Science and Technology
MESA + Institute for Nanotechnology
University of Twente
P.O. Box 2177500 AEEnschedeThe Netherlands
P A Khomyakov
Faculty of Science and Technology
MESA + Institute for Nanotechnology
University of Twente
P.O. Box 2177500 AEEnschedeThe Netherlands
A A Starikov
Faculty of Science and Technology
MESA + Institute for Nanotechnology
University of Twente
P.O. Box 2177500 AEEnschedeThe Netherlands
G Giovannetti
Faculty of Science and Technology
MESA + Institute for Nanotechnology
University of Twente
P.O. Box 2177500 AEEnschedeThe Netherlands
Instituut-Lorentz for Theoretical Physics
Universiteit Leiden
P. O. Box 95062300 RALeidenThe Netherlands
M Zwierzycki
Institute of Molecular Physics
Smoluchowskiego 17P.A.N60-179PoznańPoland
M Talanana
Faculty of Science and Technology
MESA + Institute for Nanotechnology
University of Twente
P.O. Box 2177500 AEEnschedeThe Netherlands
G Brocks
Faculty of Science and Technology
MESA + Institute for Nanotechnology
University of Twente
P.O. Box 2177500 AEEnschedeThe Netherlands
J Van Den Brink
Instituut-Lorentz for Theoretical Physics
Universiteit Leiden
P. O. Box 95062300 RALeidenThe Netherlands
Institute for Molecules and Materials
Radboud Universiteit Nijmegen
P. O. Box 90106500 GLNijmegenThe Netherlands
P J Kelly
Faculty of Science and Technology
MESA + Institute for Nanotechnology
University of Twente
P.O. Box 2177500 AEEnschedeThe Netherlands
Theoretical prediction of perfect spin filtering at interfaces between close-packed surfaces of Ni or Co and graphite or graphene
8 Dec 2008numbers: 7225Pi7215Gd7550Rr
The in-plane lattice constants of close-packed planes of fcc and hcp Ni and Co match that of graphite almost perfectly so that they share a common two dimensional reciprocal space. Their electronic structures are such that they overlap in this reciprocal space for one spin direction only allowing us to predict perfect spin filtering for interfaces between graphite and (111) fcc or (0001) hcp Ni or Co. First-principles calculations of the scattering matrix show that the spin filtering is quite insensitive to amounts of interface roughness and disorder which drastically influence the spinfiltering properties of conventional magnetic tunnel junctions or interfaces between transition metals and semiconductors. When a single graphene sheet is adsorbed on these open d-shell transition metal surfaces, its characteristic electronic structure, with topological singularities at the K points in the two dimensional Brillouin zone, is destroyed by the chemical bonding. Because graphene bonds only weakly to Cu which has no states at the Fermi energy at the K point for either spin, the electronic structure of graphene can be restored by dusting Ni or Co with one or a few monolayers of Cu while still preserving the ideal spin injection property.
I. INTRODUCTION
We recently predicted a perfect spin filtering effect for ultra-thin films of graphite sandwiched between two ferromagnetic leads. 1 This prediction emerged from two rapidly developing branches of condensed matter physics: magnetoelectronics 2 and graphene electronics. 3 Magneto-electronics exploits the additional degree of freedom presented by the intrinsic spin and associated magnetic moment of electrons while graphene electronics is based upon the unique electronic properties of twodimensional graphene sheets. Based on the giant magnetoresistance effect discovered twenty years ago, 4,5 magnetoelectronics was rapidly applied to making improved read head sensors for hard disk recording and is a promising technology for a new type of magnetic storage device, a magnetic random access memory. The giant magnetoresistance (GMR) effect is based on the spin dependence of the transmission through interfaces between normal and ferromagnetic metals (FM). The effect is largest when the current passes through each interface in a socalled current-perpendicular-to-the-plane (CPP) measuring configuration but the absolute resistance of metallic junctions is too small for practical applications and the current-in-plane (CIP) configuration with a much smaller MR is what is used in practice. Replacing the nonmagnetic metal spacer with a semiconductor 6 or insulator (I), such as Al 2 O 3 7,8 results in spin-dependent tunneling and much larger resistances are obtained with FM|I|FM magnetic tunnel junctions (MTJs). Substantial progress has been made in increasing the tunneling MR effect by replacing the amorphous Al 2 O 3 insulator with crystalline MgO. 9,10 Though there is a relatively large lattice mismatch of 3.8% between Fe and MgO, the tunneling magnetoresistance (TMR) in Fe|MgO|Fe junctions has been reported to reach values as high as 180% at room temperature. 11 Low temperature values as high as 1010% have been reported for FeCoB|MgO|FeCoB MTJs. 12,13 The sensitivity of TMR (and spin injection) to details of interface structure 14,15 makes it difficult to close the quantitative gap between theory and experiment so it is important for our understanding of TMR to be able to prepare interfaces where disorder does not dominate the spin filtering properties. This remains a challenge due to the high reactivity of the open-shell transition metal (TM) ferromagnets Fe, Co, and Ni with typical semiconductors and insulators.
With this in mind, we wish to draw attention to a quite different material system in which a thin graphite film is sandwiched between two ferromagnetic leads. Graphite is the ground state of carbon and as one of the most important elemental materials, its electronic structure has been studied in considerable detail. It consists of weakly interacting sheets of carbon atoms strongly bonded in a very characteristic honeycomb structure. Because of the weak interaction between these "graphene" or "monolayer graphite" sheets, the electronic structure of graphite is usually discussed in two steps: first, in terms of the electronic structure of a single sp 2 -bonded sheet, followed by consideration of the interaction be- tween sheets. 16,17,18 From these early, and many subsequent studies, it is known that graphene is a "zero-gap semiconductor" or a semimetal in which the Fermi surface is a point at the "K" point in the two-dimensional reciprocal space. The physical properties associated with this peculiar electronic structure have been studied theoretically in considerable detail, in particular in the context of carbon nanotubes which can be considered as rolled-up graphene sheets. 19 With the very recent discovery and development of an exceptionally simple procedure for preparing single and multiple graphene sheets, micromechanical cleavage, 20 it has became possible to probe these predictions experimentally. Single sheets of graphene turn out to have a very high mobility 21 that manifests itself in a variety of spectacular transport phenomena such as a minimum conductivity, anomalous quantum Hall effect (QHE), 22,23 bipolar supercurrent 24 and room-temperature QHE. 25 Spin injection into graphene using ferromagnetic electrodes has already been realized. 26,27 The weak spin-orbit interaction implied by the low atomic number of carbon should translate into very long intrinsic spin-flip scattering lengths, a very desirable property in the field of spin electronics or "spintronics", which aims to combine traditional semiconductor-based electronics with control over spin degrees of freedom. However, the room temperature two-terminal MR effect of ∼ 10% observed in lateral, current-in-plane (CIP) graphene-based devices with soft permalloy leads is still rather small. 26 Instead of a CIP geometry, we consider a CPP TM|Gr|TM (111) junction, where TM is a close-packed surface of fcc or hcp Ni or Co and Gr is graphite (or n sheets of graphene, Gr n ). We argue that such a junction should work as a perfect spin filter. The essence of the argument is given by Table I and Fig. 1. According to Table I, the surface lattice constants of (111) Ni, Co and Cu match the in-plane lattice constants of graphene and graphite almost perfectly. The lattice mismatch of 1.3% at the Ni(111)|Gr interface is, in fact, one of the smallest for the magnetic junctions that have been studied so far. This small lattice mismatch suggests that epitaxial TM|Gr|TM junctions might be realized experimentally, for example using chemical vapor deposition. 29,30,31 Assuming perfect lattice matching at the TM|Gr interface, it is possible to directly compare the Fermi surface projection of graphite with the projections of the Fermi surfaces (FS) of fcc Cu and of fcc and hcp Ni and Co onto close-packed planes, see Fig. 1.
The Fermi surface of graphene is a point at the highsymmetry K point in reciprocal space. The Fermi surfaces of graphite and of doped graphene are centred on this point and close to it. Figure 1 shows that there are no majority spin states for Ni and Co close to the K point whereas minority spin states exist (almost) everywhere in the surface BZ. Only the minority spin channel should then contribute to transmission from a close-packed TM surface into graphite. In a TM|Gr|TM junction, electrons in other regions of reciprocal space on the left electrode would have to tunnel through graphite to reach the right electrode. If the graphite film is taken thick enough to suppress tunneling, majority spin conductance will be quenched and only minority spin conductance through the graphite will survive i.e. perfect spin filtering will occur when the magnetizations are aligned in parallel (P). For antiparallel (AP) alignment, the conductance will vanish.
In this paper, we wish to study the effectiveness of this spin filtering quantitatively: how it depends on the thickness of the graphite film, the geometry of the clean metal-graphite interface, interface roughness and disorder, and lattice mismatch. While we will be mainly concerned with the CPP geometry, we will also comment on the applicability of some of our conclusions to the CIP geometry. The paper is organized as follows. In Sec. II we give a brief description of the computational method and outline the most important technical details of the calculations. The transport formalism we use is based upon a very efficient minimal basis of tightbinding muffin tin orbitals (TB-MTO) in combination with the atomic spheres approximation (ASA). 32 While the ASA works well for close-packed structures, some care is needed in using it for very open structures like that of graphite. In Sec. III we therefore benchmark the electronic structures calculated using the TB-MTO-ASA method with those obtained from plane-wave pseudopotential calculations. Section IV contains the results of spin-dependent electron transport calculations for specular interfaces (ideal junction) as well as for junctions with interface roughness and alloy disorder. A summary is given and some conclusions drawn in Sec. V.
II. COMPUTATIONAL METHOD
The starting point for our study is an atomic structure calculated by minimizing the total energy within the local spin density approximation (LSDA) of density functional theory (DFT). This was done using a planewave pseudopotential (PWP) method based upon projector augmented wave (PAW) pseudopotentials 33 as implemented in the VASP program. 34,35,36 The interaction between graphite and the TM surface is modelled using a repeated slab geometry of six metal layers with a graphene sheet on top and a vacuum thickness of ∼ 12 A. To avoid interactions between periodic images of the slab, a dipole correction is applied. 37 The surface Brillouin zone (SBZ) was sampled with a 36×36 k-point grid and the SBZ integrals carried out with the tetrahedron integration scheme. 38 A plane wave kinetic energy cutoff of 400 eV was used. The plane-wave pseudopotential calculations yield energy band structures, charge transfers, binding energies and work functions for single TM|Gr interfaces. 1,39 The equilibrium distances d 0 between the graphene sheet and the TM surfaces are summarized in Table I.
The equilibrium geometries are used as input for selfconsistent TB linearized MTO (TB-LMTO) 32 calculations for the TM|Gr n |TM junction. The resulting Kohn-Sham potentials are used to calculate spin-dependent transmission probabilities through the TM|Gr n |TM junction using a TB-MTO wave-function matching 40,41 scheme. 42,43,44 To do this, the junction is divided into three parts consisting of a scattering region sandwiched between semi-infinite left and right leads, all of which are divided into layers that are periodic in the lateral direction. The leads are assumed to be ideal periodic crystals in which the electron states (modes) are wave functions with Bloch translational symmetry. By making use of its Bloch symmetry, a semi-infinite lead can be represented as an energy-dependent non-Hermitian potential on the boundary of the scattering region so that the infinite system is made finite. According to the Landauer-Büttiker formalism of transport, the conductance can be calculated by summing up all the probabilities for transmitting an electron from the electron modes in the left lead through the junction into electron modes in the right leads. 43,45,46 The effect of various types of disorder on the transmission can be studied using the same formalism and computer codes by modelling the disorder within large lateral supercells 42,43 and averaging over many configurations of disorder generated by choosing positions of impurity atoms or imperfections randomly. We study three types of disorder: interface roughness, interface alloying and lattice mismatch. In the first two cases, averaging is performed over a minimum of ten configurations of disorder. To model interface roughness, some surface atoms are removed (replaced by "empty spheres" with nuclear charges that are zero in the ASA) and the ASA potentials are calculated self-consistently using a layer version 47 of the coherent potential approximation (CPA). 48 The effect of interface alloying which might occur if deposition of a thin layer of Cu on Ni or Co ("dusting") leads to intermixing is modelled in a similar fashion. Thirdly, the small lattice mismatch between graphite and TM is modelled by "cutting and pasting" AS potentials from selfconsistent calculations for TM|Gr n |TM junctions with two different in-plane lattice constants. The two systems are then combined using a supercell whose size is determined by the lattice mismatch. For self-consistent TB-LMTO-ASA calculations, the BZ of lateral supercells is sampled with a density roughly corresponding to a 24×24 k-point grid for a 1×1 interface unit cell. To converge the conductance, denser grids containing 800 × 800, 20 × 20 and 8 × 8 k-points are used for 1 × 1 (ideal junction), 5 × 5 and 20 × 20 lateral supercells, respectively.
III. GEOMETRY AND ELECTRONIC STRUCTURE OF TM|Grn|TM
In this section we describe in more detail how the electronic structure of TM|Gr n |TM junctions for TM=Cu, Ni or Co is calculated. These close-packed metals can be grown with ABC stacking in the (111) direction (fcc), or with AB stacking in the (0001) direction (hcp). We neglect the small lattice mismatch of 1.3%, 1.9% and 3.9% for the Ni|Gr, Co|Gr, and Cu|Gr interfaces, respectively, and assume the junction in-plane lattice constant to be equal to that of graphite, a Gr = 2.46Å. In the atomic spheres approximation, the atomic sphere radii of Ni, Co and Cu are then r TM = 2.574 a.u. The ASA works well for transition metals like Co, Ni or Cu which have close-packed structures. For materials like graphite which has a very open structure with an in-plane lattice constant a Gr = 2.46Å, and an out-of-plane lattice constant c Gr = 6.7Å, the unmodified ASA is not sufficient. Fortunately, a reasonable description of the crystal potential can be obtained by packing the interstitial space with empty spheres. 49 This procedure should satisfy the following criteria: (i) the total volume of all atomic spheres has to be equal to the volume of the entire system (space filling), and the (ii) overlap between the atomic spheres should be as small as possible.
A. Graphite and graphene
To see how this procedure works in practice, we benchmark the TB-MTO-ASA band structure of graphite against the "exact" band structure calculated with the PWP method. To preserve the graphite D 4 6h (P6 3 /mmc) space group symmetry, 50 the positions of the atomic spheres are chosen at Wyckoff positions. There are twelve different Wyckoff positions consistent with D 4 6h symmetry and the best choice of empty spheres is not immediately obvious. We construct two models that describe the band structure close to the Fermi energy well compared to the PWP results; this is what is most relevant for studying transport in the linear response regime. Model I with 32 empty spheres per unit cell and model II with only 4 empty spheres per unit cell both preserve the symmetry of graphite within the ASA. The crystal structures of graphite packed with empty spheres according to these two models is shown schematically in Fig. 2. Note that not all the empty spheres in a unit cell are shown in the figure. The Wyckoff labels, atomic sphere coordinates and radii are given in Table II. Figure 3 shows the band structure of graphite obtained with the TB-MTO-ASA for models I and II compared to the "exact" PWP band structure. Both models are seen to describe the graphite π bands around the Fermi energy very well. Model I provides a very good description of the bands within ±2 eV of the Fermi energy, while the smaller basis model II is quite good within ±1 eV. At the cost of including many more empty spheres, model I provides a better description of the crystal potential between the graphene planes than model II. For this reason we use model I to study the transport properties of ideal junctions, junctions with interface roughness and alloy disorder. To be able to handle the large 20 × 20 lateral supercells needed to model a lattice mismatch of 5% at the TM|Gr interface, we use model II.
B. Graphene on Ni(111) substrate
The next step is to put a monolayer of graphite (graphene) on top of the Ni(111) substrate at a distance d 0 from the metal surface. From our studies of the energetics of graphene on TM(111), we found 1,39 that the lowest energy configuration (with 3m symmetry) for TM=Ni or Co corresponds to an "AC" configuration in which one The other carbon atom, c2, is above a third layer TM atom on a "C" site. An equivalent c2c2 configuration in which the c2 atoms are on top of "A" site TM atoms can be realized by rotating the top and bottom electrodes by 180 • about a vertical axis through the second layer "B" sites; this effectively interchanges c1 and c2. Two other equivalent configurations c1c2 and c2c1 can be realized in an analogous fashion by rotating either the top or the bottom electrode through 180 • . For two sheets of graphene stacked as in graphite, a c2c2 configuration is sketched in (b). Interlayer distance is indicated as d0 and c/2 is the distance separating two neighbouring graphene sheets. carbon atom is positioned on top of a surface TM atom (an "A" site) while the second carbon atom is situated above a third layer TM atom (a "C" site), where A and C refer to the ABC stacking of fcc close-packed planes, see Fig. 4. This is in agreement with another recent firstprinciples calculations 51 as well as with experiments 30,31 for graphene on the Ni(111) surface. The electronic structure of a single graphene sheet will depend on d 0 and the details of such graphene-metallic substrate contacts can be expected to play an important role in current-in-plane (CIP) devices. 26,27 For the less strongly bound BC configuration of Gr on Ni, the equilibrium separation is rather large, d 0 ∼ 3.3Å and the characteristic band structure of an isolated graphene sheet is clearly recognizable; see Fig. 5. For the lowest energy AC configuration, the interaction between the graphene sheet and Ni surface is much stronger, a gap is opened in the graphene derived p z bands and at the Fermi energy there are no graphene states at the K-point in reciprocal space for the minority spin channel. This may complicate efficient spin injection into graphene in lateral, CIP devices. 26 The band structure calculated with the TB-MTO-AS approximation for the AC configuration is shown in the bottom panel of Fig. 5 and is seen to describe graphene on Ni(111) qualitatively quite well. However, the splitting of the graphene bands, which arises because the two carbons atoms are no longer equivalent when one is above a top layer A site Ni atom and the other is above a third layer C site Ni atom, is somewhat larger than that resulting from the PWP calculation.
C. Ni|Grn|Ni(111) junction
The transmission of electrons through a TM|Gr n |TM junction will obviously depend on the geometry of the metal-graphite contacts. Rather than carrying out a total energy minimization explicitly for every different value of n, we assume that the weak interaction between graphene sheets will not influence the stronger TM|Gr interaction and construct the junction using the "AC" configuration and the equilibrium separation d 0 = 2.03 A for each interface, as shown in Fig. 4. The interstitial space at the TM|Gr interfaces is filled with empty spheres using a procedure analogous to that described for bulk graphite. 52 Because the two carbon atoms c 1 and c 2 in the graphene unit cell are equivalent, either of them can be positioned above a surface Ni atom on an A site with the other on the C site in an "AC" configuration, without changing the total energy. Since this can be done for each TM|Gr interface separately, four different configurations of the TM|Gr|TM junction can be constructed by rotating one or both electrodes through 180 • about a vertical axis through the second layer B sites which interchanges electrode A and C sites in Fig. 4. We label these four different configurations c 1 c 1 , c 1 c 2 , c 2 c 1 and c 2 c 2 in terms of the carbon atoms which are bonded to A site TM atoms. For more than one graphene sheet, the second sheet breaks the symmetry between the c 1 and c 2 atoms. While we have not checked this explicitly, we expect the corresponding energy difference to be small and neglect it.
In Figure 5 we saw that the graphene π states interacted strongly with the nickel surface in the minimum energy "AC" configuration. The interaction with the metal substrate made the c 1 and c 2 carbon atoms inequivalent and led to the opening of an energy gap in the graphene π bands. Having constructed an interface geometry, we study the band structure of the Ni|graphene|Ni (111) junction as a function of k , the two dimensional Bloch vector, modelling it as a Ni 3 |graphene|Ni 3 junction repeated periodically in the (111) direction (which is equivalent to a Ni 6 |Gr 1 multilayer). The bands in the top panels of Fig. 6 were calculated using the benchmark planewave pseudopotential (PWP) method, those in the bottom panels with the TB-LMTO-ASA. We see that the Ni-related bands are described well by the TB-LMTO-ASA -as might be expected since the ASA is known to work well for close-packed solids. The second thing we see is that there is no gap in the graphene π bands. This is because the c 1 c 2 configuration used in the calculation has inversion symmetry and the equivalence between the two carbon atoms is restored; see Fig. 4(a). The third point to be made is that the charge transfer from graphene (work function: 4.5 eV) to Ni (work function: 5.5 eV) and strong chemisorption leads to the formation of a potential step at the interface and a significant shift of the graphene π bands with respect to the Fermi level 39 which is pinned at that of bulk Ni. We find similar results for Co|Gr|Co(111) and Co|Gr|Co(0001) junctions. There is a difference between the position of the graphene π-derived bands, most noticeably at the K point, in the PWP and TB-LMTO-ASA band structures shown in Fig. 6 for both spin channels. It appears that the interface dipole is not accurately described by the ASA. From the point of view of describing transmission of electrons through this junction, the electronic band structure is the most important measure of the quality of our basis, description of the potential, etc., so this discrepancy will most certainly have quantitative implications. Fortunately, our most important conclusions will be qualitative and will not depend on this aspect of the electronic structure.
IV. ELECTRON TRANSPORT THROUGH A FM|Grn|FM JUNCTION
Using the geometries and potentials described above, we proceed to study the spin-dependent transmission through ideal Ni|Gr|Ni junctions in the CPP geometry as a function of the thickness of the graphite spacer layer. We then discuss how interface roughness, alloy disorder and the lattice mismatch between graphite and the substrate affect the spin-filtering properties of the junctions using large lateral supercells to model the various types of disorder. Because very similar results are found for all the TMs shown in Fig. 1, we focus on fcc Ni as a substrate because it has the smallest lattice mismatch with graphite and graphene has been successfully grown on Ni using chemical vapour deposition. 29,30,31 A. Specular interface
The spin-dependent transmission through Ni|Gr n |Ni (111) junctions is shown in Fig. 7 for parallel (P) and antiparallel (AP) orientations of the magnetization in the nickel leads, in the form of the conductances G σ P and G σ AP with σ = min, maj. All the conductance values are averaged over the four interface configurations of the Ni|Gr n |Ni junction which are consistent with AC configurations of the Ni|Gr (111) interface. G maj P and G σ AP are strongly attenuated, while G min P saturates to an nindependent value. The magnetoresistance (MR) defined as (1) rapidly approaches its maximum possible value of 100%, as shown in the right inset in Fig. 7. This pessimistic definition of MR is more convenient here because G AP vanishes for large n. It is usually the optimistic version, that approaches 10 12 % in our calculations but does not saturate, that is quoted. 9,10,53,54 The left inset in Fig. 7 shows how the conductance depends on the particular configuration of the junction. The minority spin conductance in the parallel configuration, which dominates the magnetoresistance behaviour, is highest for the c 1 c 1 configuration with an asymptotic value of G min P ∼ 10 −2 G 0 . This is approximately an order of magnitude larger than G min P for the c 2 c 1 , c 1 c 2 and c 2 c 2 configurations. The c 1 c 2 and c 2 c 1 configurations are equivalent so the corresponding values of G min P should be identical. The small differences between these two configurations which can be seen in the figure are an indication of the overall accuracy of the numerical calculation. The points which are circled and connected with a dashed line are the oscillating values which were shown in Ref. 1.
MR = R AP − R P R AP × 100% ≡ G P − G AP G P × 100%,
To demonstrate that spin-filtering occurs due to high transmission of minority spin electrons around the K point, we plot the majority-and minority-spin transmission for the P configuration as a function of k for two graphite films of different thickness in Figure 8. A single sheet of graphene (a monolayer of graphite) is essentially transparent with a conductance of order G 0 in both spin channels. In the minority spin channel, the transmission is very low or vanishes close toΓ and close to K along the high symmetry Γ-K line, in spite of there being one or more sheets of Fermi surface in these regions of reciprocal space. This is a clear indication of the importance of matrix element effects: selection rules resulting from the incompatibility of wave functions on either side of the interface. 43 The majority transmission must be zero aroundΓ and around the K point because there are no states there in the Ni leads. For thicker graphite, the only contribution to the majority-spin conductance comes from tunneling through graphite in regions of the 2D-BZ where there are Ni states and the gap between graphite bonding and antibonding π states is small. This occurs close to the M point; 55 see Fig. 3. Because the gap decreases going from M to K, the transmission increases in this direction. At the edge of the Fermi surface projection, the velocity of the Bloch electrons in the leads is zero so that the maximum transmission occurs just on the M side of these edges.
The total minority transmission consist of two contributions. On the one hand there is a tunneling contribution from throughout the 2D-BZ which, depending on the particular k point, is determined by the gap in graphite as well as by the compatibility of the symmetries, at that point, of the wave functions in Ni and in graphite. On the other hand there is a large transmission from the neighbourhood of the K point coming from the Bloch states there in graphite. Once these have coupled to available states in Ni, this contribution does not change much as more layers of graphite are added. Perfect spin-filtering (100% magnetoresistance) occurs when the tunneling contributions are essentially quenched compared to the minority spin K point contribution. For four MLs of graphite the polarization is within a percent of 100% and for five MLs it is for all intents and purposes complete. The only discernible transmission in Figs. 8(ce) is found close to the K point. Magnification of this region in Fig. 8(e) shows a certain amount of structure in the transmission. This can be explained in terms of the multiple sheets of Ni minority spin Fermi surface in the vicinity of K (Fig. 1) and the small but finite dispersion of the graphite bands perpendicular to the basal plane. 55 The transmission is seen to have the threefold symmetry of the junction.
The spin-filtering does not depend on details of how graphite is bonded to the ferromagnetic leads as long as the translational symmetry parallel to the metal-graphite interfaces is preserved. We have verified this by performing explicit calculations (results not shown here) for junctions in the "AB" and "BC" configurations with different metal-graphite separations d.
B.
Ni|Cum|Grn|Cum|Ni (111) In Section III, we saw that the electronic structure of a sheet of graphene depends strongly on its separation from the underlying TM substrate. For Co and Ni, equilibrium separations of the order of 2.0Å were calculated for the lowest energy AC configuration (see Table I), the interaction was strong and the characteristic linear dispersion of the graphene electronic structure was destroyed, Fig. 5. For a separation of 3.3Å, the small residual interaction does not destroy the linear dispersion. Unlike Co and Ni, Cu interacts only weakly with graphene, there is only a small energy difference between the "asymmetric" AC configuration with d 0 = 3.3Å and the slightly more weakly bound "symmetric" BC configuration with d 0 = 3.4Å, and bonding to Cu preserves the characteristic graphene electronic structure, opening up only a very small gap of about 10 meV at the Dirac point. 56 Should it be desirable to avoid forming a strong bond between graphite and the TM electrode, then it should be a simple matter of depositing one or a few layers of Cu on e.g. Ni. Such a thin layer of Cu will adopt the in-plane lattice constant of Ni and graphite will bind to it weakly so that the electronic structure of the first layer of graphite will be only weakly perturbed. Because Cu oxidizes less readily than Ni or Co, it may be used as a protective layer. Cu has no states at or around the K point for either spin channel (Fig. 1) so it will simply attenuate the conductance of the minority spin channel at the K point. This is demonstrated in Fig 9 where the magnetoresistance of a Ni|Cu m |Gr n |Cu m |Ni junction is shown as a function of the number m of layers of Cu when there are 5 MLs and 7 MLs of graphite. As the thickness of Cu is increased reducing the transmission of the minority-spin K point channel, the MR decreases. The reduction of the MR can be compensated by increasing the thickness of graphite. These conclusions are consistent with the qualitative conclusions drawn above in connection with Fig 1(a).
Although the linear dispersion of the graphene bands is essentially unchanged by adsorption on Cu, application of an in-plane bias will destroy the translational symmetry parallel to the interface upon which our considerations have been based. The finite lateral size of a Ni|Cu electrode will also break the translational symmetry in a CIP measuring configuration and edge effects may destroy the spin-injection properties. So far, we have assumed TM and graphite lattices which are commensurate in-plane. In practice there is a lattice mismatch with graphite of 1.3% for Ni, 1.9% for Co and 3.9% for Cu which immediately poses the question of how this will affect the perfect spin-filtering. While lattice mismatch between lattices with lattice constants a 1 and a 2 can in principle be treated by using n 1 units of lattice 1 and n 2 units of lattice 2 with n 1 a 1 = n 2 a 2 , in practice we cannot perform calculations for systems with n much larger than 20 which limits us to treating a large lattice mismatch of 5%. To put an upper limit on the effect of a 1.3 − 1.9% lattice mismatch, we performed calculations for a Ni|Gr 5 |Ni junction matching 19 × 19 unit cells of Ni in-plane to 20 × 20 unit cells of graphite. The effect of this 5% lattice mismatch was to reduce the (pessimistic) magnetoresistance from 100% to 90% (or ∼ 900% in the optimistic definition). We conclude that the actual Ni|Gr mismatch of 1.3% should not be a serious limiting factor in practice.
Interface Roughness
Incommensurability is not the only factor that might reduce the magnetoresistance. Preparing atomically perfect interfaces is not possible and raises the question of how sensitive the perfect spin-filtering will be to interface roughness or disorder. Our studies of spin injection in Ref. 15 and TMR in Ref. 14 suggest they may be very important and can even dominate the spin transport properties.
The simplest way to prepare a CPP Ni|Gr|Ni junction would presumably be to begin with a (111) oriented Ni or Co crystal characterized on an atomic scale by STM or AFM, grow the required number of layers of graphene by e.g. chemical vapour deposition 29,30,31 and after characterization of the graphene layers to then deposit the second Ni electrode. To prepare a CIP junction, we envisage a procedure in which thin graphite layers are prepared by micromechanical cleavage of bulk graphite onto a SiO 2 covered Si wafer 57 into which TM (Ni or Co) electrodes have been embedded. We assume that the (111) electrodes can be prepared in ultrahigh vacuum and characterized on an atomic scale and that the surfaces are flat and defect free. Layers of graphene are peeled away until the desired value of n is reached.
Assuming it will be possible to realize one essentially perfect interface, we have studied the effect of roughness at the second interface, assuming it is prepared by evaporation or some similar method. The graphite is assumed to be atomically perfect and all of the roughness occurs in the metal interface layer. We model this roughness as in Ref. 14 by removing a certain percentage of the top layer atoms. The atomic sphere potentials are cal- culated using the layer version 47 of the coherent potential approximation (CPA). 48 The CPA AS potentials are then distributed at random with the appropriate concentration in 5 × 5 lateral supercells and the transmission is calculated in a CPP geometry for a number of such randomly generated configurations. The effect on the magnetoresistance of removing half a monolayer of Ni is shown in Fig. 10 as a function of the number of graphite layers. 50% roughness at one interface is seen to reduce the 100% magnetoresistance to about 70% (∼ 230% optimistic).
Interface Disorder
The last type of disorder we consider is a layer of interface alloy. We imagine that depositing a layer of Cu on Ni to prevent graphite bonding to the Ni has led to a layer of Ni and Cu mixing. In a worst case scenario, we assume all of the disorder is in the surface layer and assume this to be a Ni 50 Cu 50 random alloy. The potentials are once again calculated self-consistently using the layer CPA and the transmission calculated as for roughness. The effect on a monolayer of CuNi alloy is to reduce the MR to 90% (900% in the optimistic definition) for a thick graphite film, as shown in Fig. 10. These results indicate that the momentum transfer induced by the scattering due to imperfections is insufficient to bridge the large gap about the K point in the majority spin FS projections.
Ideally, we should avoid interface roughness and disorder altogether. Since metal surfaces can be prepared with very little disorder, what is required is to be able to perform micromechanical cleavage on a metal surface rather than on SiO 2 . If this were possible, two essentially perfect TM|Gr interfaces could perhaps be joined using a method analogous to vacuum bonding. 58 Alternatively, since graphite has a large c-axis resistivity 59 it may only be necessary to prepare one near-perfect Ni|graphite interface. If the graphite layer is sufficiently thick, then it should be possible to achieve 100% spin accumulation in a high resistivity material making it suitable for injecting spins into semiconductors. 60 Because carbon is so light, spin-flip scattering arising from spin-orbit interaction should be negligible.
V. DISCUSSION AND CONCLUSIONS
Motivated by the recent progress in preparing and manipulating discrete, essentially atomically perfect graphene layers, we have used parameter-free, materialsspecific electronic structure calculations to explore the spin transport properties of a novel TM|graphite system. Perfect spin-filtering is predicted for ideal TM|Gr n |TM junctions with TM = Co or Ni in both fcc and hcp crystal structures. The spin filtering stems from a combination of almost perfect matching of Gr and TM lattice constants and unique features of their electronic band structures. Graphite films have occupied states at the Fermi level only around the K-point in the first (interface) BZ. Close-packed fcc and hcp Ni and Co have only minority spin states in the vicinity of the same K point, at the Fermi energy. For a modest number of layers of graphite, transport from one TM electrode to the other can only occur via the graphite states close to the K point and perfect spin filtering occurs if the in-plane translational symmetry is preserved. For majority spins, the graphite film acts as a tunnel barrier while it is conducting for minority spin electrons, albeit with a small conductance.
Compared to a conventional magnetic tunnel junction, a TM|Gr n |TM CPP junction has several important advantages. Firstly, the lateral lattice mismatch is three times smaller than the 3.8% found for the now very popular Fe|MgO|Fe(001) MTJs. 13 This will reduce the number of defects caused by strain that otherwise limits the thickness of the tunnel barrier and degrades the efficiency of spin injection. Secondly, the spin polarization approaches 100% for an ideal junction with n > 3 graphene layers, and is only reduced to 70-90% for junctions with large interface roughness or disorder. Thirdly, the spin-filtering effect should not be very sensitive to temperature. From Fig. 8(a) and the corresponding figures for other thicknesses of graphite, we see that the largest contribution to the majority spin conduction comes from tunneling at the M point where bulk Co and Ni have propagating states at the Fermi level and the distance in energy to states in graphite with the same k vector is a minimum. From Fig. 3 we see that the energy gap is almost 1 eV between the Fermi level and the closest graphite band at this point. To bridge the horizontal gap between states close to the K-point in graphite and the closest states in Co or Ni requires an in-plane momentum transfer of order ∆k ∼ π/a. The corresponding energy would be (comparable to) that of an optical phonon which is large because of the stiffness of a graphene sheet.
To achieve perfect spin-injection into a single sheet of graphene is more troublesome. 26,27 The electronic structure calculations presented here show that the carbon π orbitals hybridize strongly with Ni (and also Co) surfaces leading to the destruction of graphene's characteristic electronic structure. We have already suggested that dusting Ni (or Co) with Cu will lead to near-complete restoration of the graphene electronic structure because of the weak interaction between graphene and Cu. Moreover Cu might also prevent rapid oxidation of the Ni(Co) (111) surfaces, which could be important for making practical devices. However, application of a bias would lead to a breaking of the translational symmetry responsible for the perfect spin filtering. The finite size of electrodes might also present a problem in practice especially if the potential drop occurs at the edges. The problem can be simply solved by forcing the electric field to be perpendicular to the TM|graphite interface as sketched in Fig. 11 where the right electrode could equally well be placed on top of the graphite.
In conclusion, we propose a new class of latticematched junctions, TM|Gr n |TM, that exhibit exceptionally high magnetoresistance effect which is robust with respect to interface disorder, roughness, and finite temperatures making them highly attractive for possible applications in spintronic devices.
FIG. 1 :
1Fermi surface projections onto close-packed planes for: (a) fcc Cu; (c) majority-and (d) minority-spin fcc Ni (111); (e) majority-and (f) minority-spin fcc Co (111); (g) majority-and (h) minority-spin hcp Ni (0001); (i) majority-and (j) minority-spin hcp Co (0001). For graphene and graphite, surfaces of constant energy are centred on the K point of the two dimensional interface Brillouin zone (b). The number of Fermi surface sheets is given by the colour bar.
FIG. 2 :
2Top and side perspective views (top and bottom panels) of graphite where the potential is represented in the atomic spheres approximation using additional, empty atomic spheres. Model I (left) contains 32 empty spheres in a unit cell containing 4 carbon atoms (red spheres). Model II (right), contains just 4 empty spheres. For model I, gray, green, blue and yellow spheres display the positions of the empty spheres E1, E2, E3 and E4, respectively. For model II, there is just one type of empty sphere (green).
FIG
. 3: (Color online) Band structure of graphite for model I (on the left), and model II (on the right). Gray (green) dots and black lines correspond to band structures calculated using the PWP and TB-MTO-ASA methods, respectively.
FIG. 4 :
4(Color online) "AC" model of TM|Grn|TM structure for (a) odd and (b) even numbers of graphene sheets. Carbon atoms are represented by small dark (red) spheres, TM atoms by larger gray spheres. The configuration shown in (a) is a c1c1 configuration with the carbon atom labelled c1 above an "A" site surface layer TM atom of the top and the bottom electrodes.
FIG. 5 :
5(Color online) The results of PWP (top and middle rows) and TB-MTO-ASA (bottom row) calculations of majority (left panels) and minority (right panels) spin band structures (green) of single graphene layers absorbed on both sides of a 13 layer (111) Ni slab for a BC configuration with d0 = 3.3Å, (top) and an AC configuration with d0 = 2.0Å (middle and bottom). The bands are replotted and superimposed in black using the carbon pz character as a weighting factor. The Fermi energy is indicated by the horizontal dashed line.
FIG. 6 :
6Energy band structures of an ideal Ni6|Gr (111) multilayer with 6 layers of fcc Ni sandwiching a single graphene sheet in a c1c2 configuration, for majority (left panels) and minority (right panels) spin channels. Plane wave pseudopotential calculations are shown on top (dotted lines), the TB-LMTO-ASA results on the bottom (solid lines).
), and G σ AP (×) averaged over the four configurations c1c1, c1c2, c2c1 and c2c2 of a Ni|Grn|Ni junction as a function of the number of graphene monolayers n for ideal junctions. Right inset: magnetoresistance MR as a function of n. Left inset: minority parallel conductance G min P (▽) given for four different configurations. The points which are circled and connected with a dashed line are the values which were shown in Ref. 1.
FIG. 8 :
8Transmission as a function of the transverse crystal momentum k in the two dimensional interface BZ for a c1c2 configuration of an ideal Ni|Grn|Ni (111) junction in a parallel state. (a) and (b) are for a single graphene sheet, n = 1; (c) and (d) are for n = 5; (e) shows the minority spin transmission in a small circle of radius r = 0.057 (2π/aGr) around the K point for 5 ML of graphite on an enlarged scale.
FIG. 9 :
9Magnetoresistance as a function of the number of Cu monolayers on both left and right Ni leads in case of 5 ML (dashed line) and 7 ML (solid line) of graphene.
FIG. 10 :
10(Color online) Magnetoresistance as a function of n for: ideal junctions (circles); Ni|Grn|Cu50Ni50|Ni junctions where the surface layer is a disordered alloy (diamonds); Ni|Grn|Ni junctions where the top layer of one of the electrodes is rough with only half of the top layer sites occupied (squares). For the rough surface layer, the error bars indicate the spread of MR obtained for different configurations. Inset: schematic representation of Ni|Grn|Ni junction with alloy disorder (roughness) at the right Ni|Gr interface. Ni atoms are given by large gray spheres while Cu (missing) atoms in the case of alloy disorder (roughness) are given by large dark (blue) spheres. Positions of carbon atoms are represented by small dark (red) spheres.
FIG. 11 :
11Schematic figure of a TM|Gr|TM CIP junction in which the electric field is forced to be essentially perpendicular to the TM|graphene interface. The dashed shaded box indicates an alternative configuration with the right-hand electrode on top of the graphene.
TABLE I :
ILattice constants of Co, Ni, Cu, and graphite, a hex ≡ a fcc / √ 2. Equilibrium separation d0 for a single graphene sheet on top of the graphite (0001) and Co, Ni or Cu fcc (111) surfaces as calculated within the framework of the DFT-LDA using the in-plane lattice constant a hex = 2.46 A.Graphite
Co
Ni
Cu
a expt
fcc (Å)
3.544 a
3.524 a
3.615 a
a expt
hex (Å)
2.46
2.506
2.492
2.556
a LDA
hex (Å)
2.45
2.42
2.42
2.49
d0 (Å)
3.32
2.04
2.03
3.18
a Ref.28
TABLE II :
IIWyckoff symbols, standardized position parameters and atomic sphere radii for carbon atoms, C, and empty spheres, E (with nuclear charge Z=0), for two structural models of graphite with space group D 4 6h (P63/mmc) Ref.50. Model I contains four different types of empty sphere: E1, E2, E3, E4; model II only one, E.Model
Atom
Wyckoff
position
radius
position
parameters
(a.u.)
I
C1
2b
1.56
C2
2c
1.56
E1
2a
1.4
E2
2d
1.6
E3
4f
z=0.5
1.4
E4
24l
x=1/3, y=0, z=0.38
0.9
II
C1
2b
1.56
C2
2c
1.56
E
4f
z=0.4
2.18
AcknowledgmentsThis work is supported by "NanoNed", a nanotechnology programme of the Dutch Ministry of Economic Affairs. It is part of the research programs of "Chemische Wetenschappen" (CW) and "Stichting voor Fundamenteel Onderzoek der Materie" (FOM) and the use of supercomputer facilities was sponsored by the "Stichting Nationale Computer Faciliteiten" (NCF), all financially supported by the "Nederlandse Organisatie voor Wetenschappelijk Onderzoek" (NWO). MZ wishes to acknowledge support from EU grant CARDEQ under contract IST-021285-2.
. V M Karpan, G Giovannetti, P A Khomyakov, M Talanana, A A Starikov, M Zwierzycki, J Van Den, G Brink, P J Brocks, Kelly, Phys. Rev. Lett. 99176602V. M. Karpan, G. Giovannetti, P. A. Khomyakov, M. Ta- lanana, A. A. Starikov, M. Zwierzycki, J. van den Brink, G. Brocks, and P. J. Kelly, Phys. Rev. Lett. 99, 176602 (2007).
. I Žutić, J Fabian, S D Sarma, Rev. Mod. Phys. 76323I.Žutić, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004).
. A H C Neto, F Guinea, N M R Peres, K S Novoselov, A K Geim, cond- mat/0709.1163Rev. Mod. Phys. A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. -, (2008), cond- mat/0709.1163.
. M N Baibich, J M Broto, A Fert, F Nguyen Van Dau, F Petroff, P Etienne, G Creuzet, A Friederich, J Chazelas, Phys. Rev. Lett. 612472M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).
. G Binasch, P Grünberg, F Saurenbach, W Zinn, Phys. Rev. B. 394828G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989).
. M Julliere, Phys. Lett. A. 54225M. Julliere, Phys. Lett. A 54, 225 (1975).
. J S Moodera, L R Kinder, T M Wong, R Meservey, Phys. Rev. Lett. 743273J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meser- vey, Phys. Rev. Lett. 74, 3273 (1995).
. T Miyazaki, N Tezuka, J. Magn. & Magn. Mater. 139231T. Miyazaki and N. Tezuka, J. Magn. & Magn. Mater. 139, L231 (1995).
. S Yuasa, T Nagahama, A Fukushima, Y Suzuki, K Ando, Nature Materials. 3868S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando, Nature Materials 3, 868 (2004).
. S S P Parkin, C Kaiser, A Panchula, P M Rice, B Hughes, M Samant, S H Yang, Nature Materials. 3862S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, and S. H. Yang, Nature Materi- als 3, 862 (2004).
. S Yuasa, T Katayama, T Nagahama, A Fukushima, H Kubota, Y Suzuki, K Ando, Appl. Phys. Lett. 87222508S. Yuasa, T. Katayama, T. Nagahama, A. Fukushima, H. Kubota, Y. Suzuki, and K. Ando, Appl. Phys. Lett. 87, 222508 (2005).
. Y M Lee, J Hayakawa, S Ikeda, F Matsukura, H Ohno, Appl. Phys. Lett. 90212507Y. M. Lee, J. Hayakawa, S. Ikeda, F. Matsukura, and H. Ohno, Appl. Phys. Lett. 90, 212507 (2007).
. S Yuasa, D D Djayaprawira, J. Phys. D: Appl. Phys. 40337S. Yuasa and D. D. Djayaprawira, J. Phys. D: Appl. Phys. 40, R337 (2007).
. P X Xu, V M Karpan, K Xia, M Zwierzycki, I Marushchenko, P J Kelly, Phys. Rev. B. 73180402P. X. Xu, V. M. Karpan, K. Xia, M. Zwierzycki, I. Marushchenko, and P. J. Kelly, Phys. Rev. B 73, 180402(R) (2006).
. M Zwierzycki, K Xia, P J Kelly, G E W Bauer, I Turek, Phys. Rev. B. 6792401M. Zwierzycki, K. Xia, P. J. Kelly, G. E. W. Bauer, and I. Turek, Phys. Rev. B 67, 092401 (2003).
. P R Wallace, Phys. Rev. 71622P. R. Wallace, Phys. Rev. 71, 622 (1947).
. J C Slonczewski, P R Weiss, Phys. Rev. 109272J. C. Slonczewski and P. R. Weiss, Phys. Rev. 109, 272 (1958).
. W M Lomer, Proc. R. Soc. London, Ser. A. 227330W. M. Lomer, Proc. R. Soc. London, Ser. A 227, 330 (1955).
. T Ando, J. Phys. Soc. Jpn. 74777T. Ando, J. Phys. Soc. Jpn. 74, 777 (2005).
. K S Novoselov, D Jiang, F Schedin, T J Booth, V V Khotkevich, S V Morozov, A K Geim, Proc. Natl. Acad. Sci. U.S.A. 10210451K. S. Novoselov, D. Jiang, F. Schedin, T. J. Booth, V. V. Khotkevich, S. V. Morozov, and A. K. Geim, Proc. Natl. Acad. Sci. U.S.A. 102, 10451 (2005).
. K S Novoselov, A K Geim, S V Morozov, D Jiang, Y Zhang, S V Dubonos, I V Grigorieva, A A Firsov, Science. 306666K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
. K S Novoselov, A K Geim, S V Morozov, D Jiang, M I Katsnelson, I V Grigorieva, S V Dubonos, A A Firsov, Nature. 438197K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005).
. Y B Zhang, Y W Tan, H L Stormer, P Kim, Nature. 438201Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature 438, 201 (2005).
. H B Heersche, P Jarillo-Herrero, J B Oostinga, L M K Vandersypen, A F Morpurgo, Nature. 44656H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K. Vandersypen, and A. F. Morpurgo, Nature 446, 56 (2007).
. K S Novoselov, Z Jiang, Y Zhang, S V Morozov, H L Stormer, U Zeitler, J C Maan, G S Boebinger, P Kim, A K Geim, Science. 3151379K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, Science 315, 1379 (2007).
. E W Hill, A K Geim, K Novoselov, F Schedin, P Blake, IEEE Trans. Mag. 422694E. W. Hill, A. K. Geim, K. Novoselov, F. Schedin, and P. Blake, IEEE Trans. Mag. 42, 2694 (2006).
. N Tombros, C Jozsa, M Popinciuc, H T Jonkman, B J Van Wees, Nature. 448571N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Nature 448, 571 (2007).
H Ibach, H Lüth, Solid-State Physics. Berlin, HeidelbergSpringer-Verlag2nd edH. Ibach and H. Lüth, Solid-State Physics (Springer- Verlag, Berlin, Heidelberg, 1995), 2nd ed.
. Y S Dedkov, M Fonin, C Laubschat, Appl. Phys. Lett. 9252506Y. S. Dedkov, M. Fonin, and C. Laubschat, Appl. Phys. Lett. 92, 052506 (2008).
. C Oshima, A Nagashima, J. Phys.: Condens. Matter. 91C. Oshima and A. Nagashima, J. Phys.: Condens. Matter. 9, 1 (1997).
. Y Gamo, A Nagashima, M Wakabayashi, M Terai, C Oshima, Surface Science. 37461Y. Gamo, A. Nagashima, M. Wakabayashi, M. Terai, and C. Oshima, Surface Science 374, 61 (1997).
O K Andersen, O Jepsen, D Glötzel, International School of Physics 'Enrico Fermi. F. Bassani, F. Fumi, and M. P. TosiAmsterdam; Varenna, ItalyNorth-HollandHighlights of Condensed Matter TheoryO. K. Andersen, O. Jepsen, and D. Glötzel, in Highlights of Condensed Matter Theory, edited by F. Bassani, F. Fumi, and M. P. Tosi (North-Holland, Amsterdam, 1985), Inter- national School of Physics 'Enrico Fermi', Varenna, Italy,, pp. 59-176.
. P E Blöchl, Phys. Rev. B. 5017953P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
. G Kresse, D Joubert, Phys. Rev. B. 591758G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
. G Kresse, J Hafner, Phys. Rev. B. 47558G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).
. G Kresse, J Furthmuller, Phys. Rev. B. 5411169G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996).
. J Neugebauer, M Scheffler, Phys. Rev. B. 4616067J. Neugebauer and M. Scheffler, Phys. Rev. B 46, 16067 (1992).
. P E Blöchl, O Jepsen, O K Andersen, Phys. Rev. B. 4916223P. E. Blöchl, O. Jepsen, and O. K. Andersen, Phys. Rev. B 49, 16223 (1994).
. G Giovannetti, P A Khomyakov, G Brocks, V M Karpan, J Van Den, P J Brink, Kelly, Phys. Rev. Lett. 10126803G. Giovannetti, P. A. Khomyakov, G. Brocks, V. M. Karpan, J. van den Brink, and P. J. Kelly, Phys. Rev. Lett. 101, 026803 (2008).
. T Ando, Phys. Rev. B. 448017T. Ando, Phys. Rev. B 44, 8017 (1991).
. P A Khomyakov, G Brocks, V Karpan, M Zwierzycki, P J Kelly, Phys. Rev. B. 7235450P. A. Khomyakov, G. Brocks, V. Karpan, M. Zwierzycki, and P. J. Kelly, Phys. Rev. B 72, 035450 (2005).
. K Xia, P J Kelly, G E W Bauer, I Turek, J Kudrnovský, V Drchal, Phys. Rev. B. 6364407K. Xia, P. J. Kelly, G. E. W. Bauer, I. Turek, J. Ku- drnovský, and V. Drchal, Phys. Rev. B 63, 064407 (2001).
. K Xia, M Zwierzycki, M Talanana, P J Kelly, G E W Bauer, Phys. Rev. B. 7364420K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and G. E. W. Bauer, Phys. Rev. B 73, 064420 (2006).
. M Zwierzycki, P A Khomyakov, A A Starikov, K Xia, M Talanana, P X Xu, V M Karpan, I Marushchenko, I Turek, G E W Bauer, phys. stat. sol. B. 245623M. Zwierzycki, P. A. Khomyakov, A. A. Starikov, K. Xia, M. Talanana, P. X. Xu, V. M. Karpan, I. Marushchenko, I. Turek, G. E. W. Bauer, et al., phys. stat. sol. B 245, 623 (2008).
. M Büttiker, Y Imry, R Landauer, S Pinhas, Phys. Rev. B. 316207M. Büttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B 31, 6207 (1985).
S Datta, Electronic Transport in Mesoscopic Systems. CambridgeCambridge University PressS. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, 1995).
I Turek, V Drchal, J Kudrnovský, M Šob, P Weinberger, Electronic Structure of Disordered Alloys, Surfaces and Interfaces. Boston-London-DordrechtKluwerI. Turek, V. Drchal, J. Kudrnovský, M.Šob, and P. Wein- berger, Electronic Structure of Disordered Alloys, Surfaces and Interfaces (Kluwer, Boston-London-Dordrecht, 1997).
. P Soven, Phys. Rev. 156809P. Soven, Phys. Rev. 156, 809 (1967).
. D Glötzel, B Segall, O K Andersen, Sol. State Comm. 36403D. Glötzel, B. Segall, and O. K. Andersen, Sol. State Comm. 36, 403 (1980).
T Hahn, International Tables for Crystallography. SpringerT. Hahn, ed., International Tables for Crystallography, vol. A (Springer, 2005).
. G Bertoni, L Calmels, A Altibelli, V Serin, Phys. Rev. B. 7175402G. Bertoni, L. Calmels, A. Altibelli, and V. Serin, Phys. Rev. B 71, 075402 (2005).
Ni within model I we use two ESs with a WSR r = 1.12 a.u. at B and C sites, positioned 1.57 a.u. above the metal surface. For model II we use one ES with a WSR r = 1.7 a.u. at a C site. at 2.42 a.u. above the surface. If the distance between graphite and the metal surface is larger, which is the case for graphite on Cu, we use three ESs with WSR r = 1.6 a.u., at A, B, and C sites, respectively, 3.33 a.u. above the surfaceTo describe the adsorption of graphite on Co and Ni within model I we use two ESs with a WSR r = 1.12 a.u. at B and C sites, positioned 1.57 a.u. above the metal surface. For model II we use one ES with a WSR r = 1.7 a.u. at a C site, at 2.42 a.u. above the surface. If the distance between graphite and the metal surface is larger, which is the case for graphite on Cu, we use three ESs with WSR r = 1.6 a.u., at A, B, and C sites, respectively, 3.33 a.u. above the surface.
. W H Butler, X.-G Zhang, T C Schulthess, J M Maclaren, Phys. Rev. B. 6354416W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M. MacLaren, Phys. Rev. B 63, 054416 (2001).
. J Mathon, A Umerski, Phys. Rev. B. 63220403J. Mathon and A. Umerski, Phys. Rev. B 63, 220403(R) (2001).
. J.-C Charlier, X Gonze, J.-P Michenaud, Phys. Rev. B. 434579J.-C. Charlier, X. Gonze, and J.-P. Michenaud, Phys. Rev. B 43, 4579 (1991).
. G Giovannetti, P A Khomyakov, G Brocks, P J Kelly, J Van Den, Brink, Phys. Rev. B. 7673103G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly, and J. van den Brink, Phys. Rev. B 76, 073103 (2007).
. A K Geim, K S Novoselov, Nature Materials. 6183A. K. Geim and K. S. Novoselov, Nature Materials 6, 183 (2007).
. D J Monsma, R Vlutters, J C Lodder, Science. 281407D. J. Monsma, R. Vlutters, and J. C. Lodder, Science 281, 407 (1998).
. K Matsubara, K Sugihara, T Tsuzuku, Phys. Rev. B. 41969K. Matsubara, K. Sugihara, and T. Tsuzuku, Phys. Rev. B 41, 969 (1990).
. G Schmidt, D Ferrand, L W Molenkamp, A T Filip, B J Van Wees, Phys. Rev. B. 624790G. Schmidt, D. Ferrand, L. W. Molenkamp, A. T. Filip, and B. J. van Wees, Phys. Rev. B 62, R4790 (2000).
| [] |
[
"IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOLUME/NUMBER/DATE 1 A Scalable Real-Time Architecture for Neural Oscillation Detection and Phase-Specific Stimulation",
"IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOLUME/NUMBER/DATE 1 A Scalable Real-Time Architecture for Neural Oscillation Detection and Phase-Specific Stimulation"
] | [
"Member, IEEEChristopher Thomas ",
"Thilo Womelsdorf "
] | [] | [] | Oscillations in the local field potential (LFP) of the brain are key signatures of neural information processing. Perturbing these oscillations at specific phases in order to alter neural information processing is an area of active research. Existing systems for phase-specific brain stimulation typically either do not offer real-time timing guarantees (desktop computer based systems) or require extensive programming of vendorspecific equipment. This work presents a real-time detection system architecture that is platform-agnostic and that scales to thousands of recording channels, validated using a proof-ofconcept microcontroller-based implementation. | null | [
"https://arxiv.org/pdf/2009.07264v1.pdf"
] | 221,703,240 | 2009.07264 | af89dcb83f8e78da1eff7509d8788fa3707a84f1 |
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOLUME/NUMBER/DATE 1 A Scalable Real-Time Architecture for Neural Oscillation Detection and Phase-Specific Stimulation
Member, IEEEChristopher Thomas
Thilo Womelsdorf
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOLUME/NUMBER/DATE 1 A Scalable Real-Time Architecture for Neural Oscillation Detection and Phase-Specific Stimulation
Index Terms-filteringlocal field potential (LFP)neuro- sciencetime-frequency analysis
Oscillations in the local field potential (LFP) of the brain are key signatures of neural information processing. Perturbing these oscillations at specific phases in order to alter neural information processing is an area of active research. Existing systems for phase-specific brain stimulation typically either do not offer real-time timing guarantees (desktop computer based systems) or require extensive programming of vendorspecific equipment. This work presents a real-time detection system architecture that is platform-agnostic and that scales to thousands of recording channels, validated using a proof-ofconcept microcontroller-based implementation.
I. INTRODUCTION
Recording of electrical signals from neurons in human and animal brains is a well-established field [1]. Processing these signals reveals two related components: "spikes", representing the firing of individual neurons near the pickup electrodes, and the "local field potential" (LFP), representing the aggregate activity of the larger population of neurons surrounding the electrode site [2]. Both of these signal components carry information: spikes via firing rate and timing [3] [4] [5] [6], and the LFP via the presence or absence of transient oscillations representing coherent activity of a large group of neurons [7] [8] [9] [10] [11]. The relative timing of spikes with respect to LFP oscillation phase has also been shown to encode information [12] [13] [6] [14] [15].
Artificial stimulation of human and animal brains (via electrical, optical, or other means) is also a field of active study [16] [17] [18] [19]. It has recently been shown that if LFP oscillations are present near a stimulation site, the timing of stimulation with respect to the LFP phase is important [20] [21] [22] [23]. In order to study this, it is necessary to perform "on-line" detection of transient LFP oscillations and to extract phase in real-time.
Existing experiments studying phase-specific stimulation can be divided into those that use a desktop computer to perform their signal processing [20] [24] [25] [21] and those which perform some or all of their signal processing on dedicated hardware [26] [27]. Both types of system have signal processing latency that must be compensated for (typically 20-100 ms) [26] [21] but desktop computer based systems typically have substantial random variation (jitter) in processing and communications latency (typically 5-10 ms) [21], which is avoided in systems that keep the stimulation trigger processing entirely in dedicated hardware.
Low-latency signal processing systems running on dedicated hardware may be implemented in software running on dedicated digital signal processing (DSP) platforms [26] [ 28] or implemented using a field-programmable gate array (FPGA) tightly coupled to the recording system [27] [29]. Signal processing on dedicated hardware is widely used for processing of neural signals but is typically implemented adhoc.
The goal of this work is to present an an open architecture for "on-line" LFP oscillation detection and for phase-aligned stimulation that is suitable for instantiation on conventional FPGA-based electrophysiology equipment and that is scalable to thousands of recording channels. The purpose of this architecture is to make experiment-specific FPGA-based closedloop stimulation systems easier and faster to implement, as most of the implementation and debugging will already have been done.
II. BACKGROUND
A diagram of a typical electrophysiology recording and stimulation setup is shown in Figure 1. One or more probes, typically containing multiple electrical contacts per probe, are inserted into the brain. A "headstage" and a recording controller amplify and digitize the analog signals and forward them to a host computer. Electrical stimulation is performed using either a dedicated controller and probes or auxiliary functions of the controller, headstage, and probes used for recording. Recording and stimulation are typically performed while the subject performs some consistently-structured activity.
A typical single-channel recording waveform is shown in Figure 2 [30]. Noteworthy features are spikes (sub-millisecond duration) [31] [32], local field potential oscillations (typically 4-50 Hz and lasting for a small number of cycles [33]), and background noise (typically 1 f 2 power-law noise at LFP frequencies [34] [35]). Spiking and LFP oscillation patterns vary widely depending on the region of the brain being measured [36] [37], and LFP oscillation duration (absolute and number of cycles) also depends strongly on the oscillation arXiv:2009.07264v1 [eess.SP] 15 Sep 2020 Typical wide-band-signal recorded from a primate brain using tungsten wire probes [30]. Noteworthy features in this signal are sub-millisecond spikes and 20-25 Hz oscillations. frequency [38]. At high frequencies (50-200 Hz), oscillations occur with durations of many cycles that are are modulated by co-occurring low-frequency oscillations [10]. A typical closed-loop phase-aligned stimulation setup based on a desktop computer is shown in Figure 3. Signals are acquired using the recording controller and processed using "on-line" algorithms that are intended to function in real-time. When an oscillation occurs and stimulation is commanded during the experiment, the desktop computer waits until the appropriate oscillation phase before commanding the stimulation controller to activate.
Typical "on-line" oscillation detection and characterization algorithms are variants of a widely-used "offline" algorithm. In this offline algorithm, the LFP frequency band of interest is isolated and the analytic signal is computed, with the imaginary component provided by the Hilbert transform of the band-pass-filtered signal. The analytic signal encodes the magnitude and phase of the original narrow-band signal [39] [25]. Oscillation events are identified by looking for magnitude excursions, with 2σ or 3σ from baseline magnitude being typical [20] [7] [26] [5]. Oscillation phase at any given instant is taken to be the analytic signal phase at the time of interest. For ease of reference, this will be referred to as the "offline Hilbert algorithm".
For "on-line" implementation, band-pass filtering is typically performed using a finite impulse response filter (FIR) [26]. Magnitude and phase may be extracted via templatefitting [20] or interpolation between peaks, troughs, and zerocrossings in the narrow-band signal [39] [40]. Oscillation period may be estimated from the peaks, troughs, and zerocrossings or by using a filter bank with densely-spaced center frequencies and looking for the filter with the strongest response [26].
"Offline" algorithms for oscillation detection and parameterization are more varied [41], as they do not need to meet time constraints and they can consider both the past and future signal around a point of interest. Typical approaches that do not use the Hilbert transform involve decomposing the signal using either a fixed dictionary such as Gabor wavelets [42] or an optimized dictionary via sparse coding approaches [43] [44].
Hardware-based signal processing of electrophysiology signals typically involves electrophysiology controllers that expose digital signal processors (DSPs) or field-programmable gate arrays (FPGAs) to the user. These are programmable (DSP) or configurable (FPGA) hardware devices capable of running specialized computing operations much faster than general-purpose microprocessors. A typical electrophysiology controller that exposes DSP features to the user is the Tucker-Davis RZ2 BioAmp Processor [28] (based on the SHARC series of DSP processors). Typical electrophysiology controllers that expose FPGA features to the user are the Open Ephys acquisition board [45] [46] and the related Intan RHD recording controller [47] (both based on the Spartan 6 LX45 FPGA), and the NeuraLynx Hardware Processing Platform expansion board [29] (based on the Zynq 7045 SoC which integrates a Kintex 7 FPGA).
The performance metric that determines filtering and signal processing capability is the number of multiply-accumulate operations (MACs) that a given platform can perform per second. For DSP-based systems, the number of MACs is usually equivalent to the number of floating point operations per second (FLOPS). The SHARC DSP processors used by Tucker-Davis can perform 2.4 GFLOPS per core (at 400 MHz), for an aggregate maximum processing power of about 77 GMAC/sec (8 quad-core boards). For Xilinxfamily FPGA-based systems, the number of MACs per second is determined by the number of "digital signal processing slices" and the rate at which these slices may be clocked. The XC6LX45 chip used in the Open Ephys and Intan controllers provides 5.8 GMAC/sec (58 units clocked at 100 MHz), some of which is used for the controller's existing filtering operations. These controllers can acquire data from up to 1024 recording channels. The XC7Z045 chip used in the NeuraLynx processing board provides 900 GMAC/sec (900 units clocked at 1 GHz), all of which is available for signal processing. The controller in which this board is installed can acquire data from up to 512 recording channels. While these examples are not exhaustive, it is reasonable to assume a processing budget of at least 2 MMAC/sec per channel, with up to 2 GMAC/sec per channel available in systems with more hardware resources available. LFP signal processing is typically performed at 1 ksps, with signals acquired at 25 ksps-40 ksps [26].
The desired goal is to detect local field potential oscillations while they are still happening (within 1-2 oscillation periods), and to accurately determine the oscillation phase so that phasespecific stimulation may be performed. The accuracy needed can be inferred from the number of phase bins used for spike-phase coding analyses; 4-10 phase bins are typical, with diminishing returns past 6 bins [5] [3]. This indicates that the full-width half-maximum of the phase error distribution should be 60 degrees or less.
III. IMPLEMENTATION
A block diagram of the oscillation detection architecture is shown in Figure 4. Full-rate data is passed through an antialiasing filter and downsampled. Downsampled data is passed through a filter bank that performs band-pass filtering, and an approximation of its instantaneous magnitude and phase is extracted. Event detection in each band is performed by magnitude thresholding, with inter-band logic building event triggers. A time shift is applied to correct the instantaneous phase estimate for filter delay, and trigger signals are generated using a target phase or a target time since zero-crossing. The output of the signal processing block is a set of narrow-band signal waveforms, estimates of instantaneous magnitude and phase associated with these waveforms, and event trigger logic signals suitable for driving neural stimulation equipment. Any of these outputs may be exposed to the host system.
The individual signal processing blocks in this architecture were implemented as modules, with the intention being that application-specific signal processing architectures would be built by assembling modules with a minimum of new code needed. Each of the modules was implemented in C++ and in Matlab, with FPGA-based implementations in development. The intention is to allow rapid prototyping via Matlab, embedded software implementations via C++, and full-scale hardware implementations via hardware description languages, with confidence that all three types of implementation would produce comparable output if given the same input. As FPGA implementation is the end-goal, the modules are written to operate in a pipelined manner on a sample-by-sample basis (to facilitate translation to hardware).
Three closed-loop systems were assembled as reference implementations: One Matlab-based, one C++-based running on a desktop workstation, and one embedded C++ implementation running on a proof-of-concept "Burst Box" prototype. The Matlab implementation was used to verify that the architecture is conceptually sound; it was not otherwise resource-constrained (no memory or processing time limits, double-precision arithmetic). The workstation-based C++ implementation was used to verify that the architecture's integerarithmetic implementation produced output acceptably close to that of the Matlab implementation. While the workstationbased implementation was not explicitly memory-constrained, care was taken to keep internal structure sizes small enough to be instantiated on FPGAs. The embedded "Burst Box" prototype was used to verify that the architecture was capable of performing closed-loop stimulation in real-time with limited memory and a limited amount of processing power available.
Module library code and the reference implementations were made freely available under an open-source license [48].
A. Embedded Microcontroller Implementation
A block diagram of the embedded microcontroller-based implementation of the oscillation detector architecture is shown in Figure 5. Processing is restricted to one channel and one frequency band. Anti-aliasing and band-pass filters are implemented as infinite impulse response filters (IIR filters), to minimize processing load. Input signals may optionally be sampled at the downsampled rate directly without software anti-aliasing, reducing processing load but increasing signal noise due to aliasing artifacts. Approximate instantaneous magnitude and phase were extracted using peak, trough, and zero-crossing detection. Calculations were performed using 32-bit integer arithmetic with a signal range of 14 bits (to ensure sufficient head-room during multiply-accumulate operations). IIR filters were implemented as cascaded biquads with Direct Form I implementation. The a 0 biquad denominator coefficient was required to be a power of two, so that the 1 a0 operation could be performed as a bit-shift.
A block diagram of the firmware for the embedded implementation of the oscillation detector architecture is shown in Figure 6. Three concurrent execution threads are running: an interrupt service thread, which handles events that must occur with every real-time clock tick and complete within that timeslice; a high-priority polling thread, which is woken up by the real-time clock (preempting low-priority polling) but which may take multiple timeslices to complete; and a low-priority polling thread, which handles operations which do not require timing guarantees.
The physical implementation of the microcontroller-based prototype is shown in Figure 7 (the "Burst Box"). The microcontroller used is an Atmel ATmega2560 (8-bit, running at 16 MHz, with 8 kiB of SRAM); DSP performance was b + b z + b z a + a z + a z 0 benchmarked at approximately 100 kMAC/sec with 32-bit operands. "Full-rate" signals are sampled at 2.5 ksps using an external analog to digital converter and an analog anti-aliasing filter, and then downsampled to 500 sps internally (the "DSP rate") after the application of a digital anti-aliasing filter. At 200 MAC/sample, this represents a worst-case lower bound to the processing budget available in real implementations.
A diagram of the hardware implementation of the "Burst Box" is shown in Figure 8. There is hardware support for up to 4 input channels, which constrains the processing budget further but allows testing of coincidence detection. The analog anti-aliasing filter in this prototype was implemented as an RC ladder filter for simplicity and to avoid any possibility of resonance from inductive components, with the tradeoff of having poor roll-off compared to a Butterworth implementation. For debugging purposes, the system can be configured to bypass the external analog-to-digital converter and use the microcontroller's internal analog-to-digital converter at 500 sps without anti-aliasing.
IV. VALIDATION
A. Datasets
Two datasets were used for testing and validation of oscillation detector implementations. The first (the "synthetic" dataset) consisted of 5 minutes of 1 f 2 noise ("red noise") with tones overlaid. Tones had weak frequency chirping and amplitude ramping (less than 5% and 10% respectively), with cosine roll-off (Tukey window roll-off), and durations of 3-5 periods between midpoints of the roll-off flanks. Tones had a signal-to-noise ratio of 20 dB with respect to in-band noise; frequency bands used for noise calculations are shown in Table I. The "red noise" spectrum spanned from 2-200 Hz, with power concentrated at lower frequencies, so per-band adjustment of tone amplitude was necessary in order to have consistent signal-to-noise ratios.
The second dataset ("biological" dataset) consisted of a concatenated selection of recordings from a primate dataset [49]. The raw dataset consisted of "epochs" that were typically less than 10 seconds long, taken during individual task trials within one extended recording session. Signals from individual epochs were trimmed to time periods within the task that showed consistent activity with few electrical artifacts. Signals were evaluated on an epoch-by-epoch basis to reject records that contained artifacts within the trimming interval (typically large step transients caused by physical contact with equipment or 60 Hz tones coupled from nearby equipment). Remaining "clean" epochs were normalized to have consistent average power and were concatenated with an overlap of 0.5 s with linear interpolation between signals within the overlap interval. The intention was to produce an artifact-free signal of several minutes' duration with biologically valid noise and oscillation features.
B. Test Procedure
Testing of the Matlab-based and workstation-based oscillation detectors was straightforward; both provide time series waveforms for all desired signals in their processing pipelines, with a common time reference between all signals. The challenge was to extract comparable information from the embedded microcontroller-based implementation during realtime tests.
The physical setup for real-time testing is shown in Figure 9. Signal waveforms were converted to sound files and played back to the "Burst Box" prototype via computer audio output. Volume settings for playback were adjusted until the output amplitude was approximately 3 V peak-to-peak, as measured using an oscilloscope. The "Burst Box" is capable of providing monitoring streams of two signals (typically one band-pass filtered waveform and one other signal derived from it). Tests with a given input waveform were run repeatedly, capturing different output waveform pairs, and these output waveform pares were time-aligned using the band-pass filtered waveform as a reference (which should remain consistent between successive trials).
Signals streamed from the "Burst Box" could be read via two methods: parallel output via a logic header (8 bits per sample, precise timing and no dropped samples), and diagnostic output via the USB serial command interface (16 bits per sample, some dropped samples). Both capture methods were used. Unless otherwise indicated, the logic header output was used to generate plots.
Functionality exists for inspecting and modifying the internal state of the "Burst Box" using the serial command interface for single-stepped testing. While this would provide all of the desired signals with high fidelity, it was not practical to use for full-duration test signals, due to being far slower than real-time testing.
C. Filtering
The purpose of filter validation is to confirm that the integer math C++ implementations of the oscillation detector's filters match the behavior of the Matlab implementation of the same filters. This tested by plotting the inferred filter transfer functions measured during functionality tests against the ideal transfer functions.
Filter gain, phase shift, phase delay, and group delay were characterized by taking the Fourier transform of the timealigned input and output waveforms for each filter under test. Dividing spectrum elements gives the frequency-domain transfer function directly, per Equation 1. This is smoothed, to reduce artifacts due to noise, and the phase is unwrapped. The phase delay and group delay are then computed per Equations 2 and 3, respectively. The derivative of φ(ω) is approximated by taking the first difference and performing additional smoothing.
H(ω) = F{y(t)} F{x(t)} G(ω) = ||H(ω)|| φ(ω) = arg(H(ω)) (1) τ φ (ω) = − φ(ω) ω (2) τ g (ω) = − dφ(ω) dω(3)
The filter configurations used by the "Burst Box" prototype are shown in Table II. These were Butterworth infinite impulse response filters implemented as cascaded biquad stages. A representative plot of the designed and measured transfer functions for the "beta band" filter is shown in Figure 10, using the "synthetic" dataset as the input signal.
Within the regions of interest (blue in the single-filter plots, dark in the multi-filter plots), the designed and measured transfer functions are virtually identical. As a result, the filter implementation can be considered sound, and the Matlab models of the filters may be used as proxies for the real filter implementations without significant discrepancies expected.
All causal filters introduce delay into the filtered signal. For FIR filters, this delay is constant, and for IIR filters, different frequency components are delayed by different amounts. To 11. Group delay calibration for the IIR beta band filter. The real group delay (red curve) is approximated by a lookup table of calibration delays (step-wise curve). The resulting delay error after calibration is shown in the blue curve. allow later processing stages to compensate for this, a calibration table of delay vs frequency is built. Figure 11 shows an example of the calibrated delay (step-wise curve), actual delay (red curve), and delay error after calibration (blue curve) for the beta-band infinite impulse response filter shown in Figure 10.
D. Feature Extraction
Feature extraction was performed by looking for zerocrossings in the band-pass-filtered waveform, inferring period and phase from those zero crossings, and taking the maximum or minimum value of the waveform between successive zerocrossings as the magnitude of the signal. Feature extraction accuracy was characterized by comparing the oscillation detector's estimates of instantaneous magnitude, phase, and frequency to the instantaneous magnitude, phase, and frequency computed from the band-pass filtered signal by using Hilbert transform to derive the imaginary component of the analytic signal. Fig. 12. Reconstructed vs analytic magnitude, phase, frequency, and waveform. Reconstruction was performed using peak, trough, and zero-crossing analysis of a short test waveform. Figure 12 shows a representative reconstruction of magnitude, phase, frequency, and waveform using the peak-trough-ZC feature extractor (blue) and using the analytic signal (orange). Reconstruction was performed in regions where the magnitude was above-threshold, where threshold was set to twice the average magnitude. The analytic signal features are shown in yellow outside of these regions. This figure shows the analysis performed for a short test waveform, for illustration purposes.
"Delayed" and "zero-shift" versions of the band-pass filtered signal are considered. The "delayed" signal is the version received from the filter bank: frequencies are delayed by a fixed amount for FIR filters and a frequency-dependent amount for IIR filters (the "group delay" from Section IV-C). A "zeroshift" signal is computed by using the gain component G(ω) of the filter's transfer function (from Equation 1) as a non-causal filter to transform the wideband signal into a "zero-shift" bandpass signal (Equation 4). Time shift from the hardware antialiasing filter, software anti-aliasing filter, and software bandpass filters can be compensated in this manner.
Y 0 (ω) = X(ω) · G(ω) y 0 (t) = F −1 {G(ω) · F {x(t)}}(4)
Comparison using the Hilbert transform of the original shifted signal y(t) shows whether the oscillation detector's approximation of the Hilbert transform is accurate. Comparison using the Hilbert transform of the "zero-shift" signal y 0 (t) shows whether the oscillation detector's internal calibration of filter delay is accurate. Accurate estimation of instantaneous phase with respect to the wideband input signal is vital for phase-aligned neural stimulation. Figure 13 shows histograms of magnitude error normalized to the analytic magnitude (relative error), and polar histograms of phase error with respect to analytic phase. The error distributions of the uncalibrated parameters with respect to the "delayed" waveform are shown in the top row and the error distributions of the calibrated parameters with respect to Fig. 13. Normalized reconstructed magnitude error (left) and absolute reconstructed phase error (right) with respect to analytic signal magnitude and phase for "delayed" (top) and "zero-shift" (bottom) band-pass filtered signals. Beta band IIR filter, "synthetic" dataset.
the "zero-shift" waveform are shown in the bottom row. This analysis was performed using the "synthetic" dataset.
Magnitude error distributions are broad in all cases. This is because the envelopes of event tones change on a timescale that is not substantially longer than the analysis timescale (one half-period of the event tone). As the magnitude estimate is out of date by half a period, there may be a considerable difference between the estimated and actual magnitudes. This can be seen in the bottom strip in Figure 12; the estimated envelope is time-shifted relative to the actual envelope.
Uncalibrated phase error with respect to the "delayed" wave is tightly clustered (35 • FWHM, +10 • offset). This represents the uncertainty in the phase estimate, caused by frequency shifts during the event, noise perturbing the detected locations of zero-crossings, and quantization of the detected half-period into an integer number of samples.
E. Phase-Aligned and Delay-Aligned Triggering
Trigger alignment was characterized by specifying a desired delay in milliseconds from the rising or falling zero-crossing, or a desired phase angle, and measuring the distribution of delays and phase angles at which stimulation trigger signals were actually generated. Histograms of the delay and phase error were generated with respect to the "delayed" band-pass signal without delay calibration and with respect to the "zeroshift" band-pass signal with delay calibration. Figure 14 shows representative plots of trigger delay (left) and of delay error (right) for triggers scheduled with respect to the rising zero-crossing (top) or falling zero-crossing (bottom) of the input signal, with respect to the "delayed" signal, without calibration. These measurements were taken using the beta band IIR filter and the "synthetic" dataset. Under these test conditions, delay is tightly clustered (4 ms FWHM), but the error distribution has a broad base which grows to dominate the distribution for small delay values (less than 10 ms). for triggers scheduled with respect to the rising zero-crossing (top) or falling zero-crossing (bottom) of the input signal. "Synthetic" dataset, beta band IIR filter, no calibration, "delayed" signal. Fig. 15. Representative plots of trigger delay error without calibration (left) and with calibration (right) for triggers scheduled with respect to the rising zero-crossing (top) or falling zero-crossing (bottom) of the input signal. "Synthetic" dataset, beta band IIR filter, "zero-shift" signal. Figure 15 shows representative plots of trigger delay error without calibration (left) and with calibration (right) for triggers scheduled with respect to the rising zero-crossing (top) or falling zero-crossing (bottom) of the input signal, with respect to the "zero-shift" signal. These measurements were taken using the beta band IIR filter and the "synthetic" dataset. Under these test conditions, calibration narrows the peak of the error distribution (to approx. 10 ms FWHM from approx. 20-30 ms FWHM), but a systematic delay of approx. 20 ms is applied, and a large secondary lobe at -20 ms results in mistimed triggers for a large fraction of cases. Figure 16 shows representative plots of trigger phase (top) and trigger phase error statistics (bottom) for triggers scheduled with respect to specific phases of the input signal. Plots on the left show phase measured with respect to the peak and trough detector's phase estimate, and plots on the right show phase measured with respect to the "delayed" signal, without calibration. These measurements were taken using the beta Fig. 16. Representative plots of trigger phase (top) and trigger phase error statistics (bottom) with respect to the peak and trough detector's phase estimate (left) and the phase of the analytic signal phase (right). "Synthetic" dataset, beta band IIR filter, "delayed" signal, no calibration. band IIR filter and the "synthetic" dataset. Under these test conditions, phase with respect to the peak and trough phase estimate is clustered within the desired 60 • FWHM for most target angles but shows scatter near 90 • and 270 • (phase targets near the zero-crossings). Phase with respect to the "delayed" band-pass filtered signal shows additional scatter, marginally meeting the 60 • FWHM target.
V. CONCLUSION
A modular, scalable signal processing framework has been presented that is capable of detecting and characterizing oscillations on the local field potential of neural signals, and of generating trigger signals to allow phase-aligned and delayaligned stimulation to be performed. As a case study, this framework was used to prototype a microcontroller-based oscillation detector that is capable of processing one signal channel and of responding to oscillation events within 3 4 period of onset. The prototype's real-time estimate of signal phase has an error distribution FWHM of 35 • with respect to the analytic signal. Triggers scheduled using a delay since a rising or falling zero-crossing have an error distribution FWHM of 4 ms, and triggers scheduled for a specific phase have an error distribution FWHM of approximately 60 • with respect to the band-pass filtered signal, which meets the design requirements for a phase-aligned neural stimulation system. The framework was designed to be readily adapted to FPGAbased implementation for rapid development of closed-loop stimulation experiments using FPGA-based electrophysiology controllers.
Fig. 1 .
1Typical electrophysiology recording and stimulation setup using wire probes or linear silicon probes.
Fig. 2 .
2Fig. 2. Typical wide-band-signal recorded from a primate brain using tungsten wire probes [30]. Noteworthy features in this signal are sub-millisecond spikes and 20-25 Hz oscillations.
Fig. 3 .
3Data processing flow within a typical electrophysiology recording and stimulation setup using a desktop computer for phase-aligned stimulation.
Fig. 4 .
4Top-level oscillation detector architecture.
Fig. 6 .
6Embedded oscillation detector firmware architecture.
Fig. 7 .
7Physical implementation of the microcontroller-based oscillation detector ("Burst Box").
ADC
Fig. 8 .
8Block diagram of the "Burst Box" hardware implementation.
Fig. 9 .
9Physical setup for validation tests.
Fig. 10 .
10Designed and measured transfer functions for the beta band filter.
Fig.
Fig. 11. Group delay calibration for the IIR beta band filter. The real group delay (red curve) is approximated by a lookup table of calibration delays (step-wise curve). The resulting delay error after calibration is shown in the blue curve.
Fig. 14 .
14Representative plots of trigger delay (left) and delay error (right)
Fig. 5. Microcontroller-based implementation of the oscillation detector architecture. − Read and queue new ADC data if present. High−Priority Poll Thread: − Queue USB serial reports. − Emit TTL and parallel outputs. − Process new ADC data if queued.− Check USB serial for commands. − Emit queued reports to USB serial.1
2
0
1
2
−1
−2
−1
−2
H(z) =
up to 4 stages
IIR biquads
b + b z + b z
a + a z + a z
0
1
2
0
1
2
−1
−2
−1
−2
H(z) =
up to 4 stages
IIR biquads
f del
Filter Delay
Correction
Desired Delay
Desired Phase
Cool−Down Time
Pulse Duration
Want Trigger?
Wideband
Downsampled
Downsampling
Low−Pass Filter
Anti−Aliasing
Full−Rate
Wideband
Wideband
Downsampled
Band−Pass Filter Bank
Narrow−Band
Magnitude
Threshold
Factor
1st order exp LPF
Average Mag.
Threshold
Event Detected
up to 4 bands
Peak/Trough/ZC
Magnitude
Period
Time Since ZC
Configuration
Stim. Trigger
Trig. Logic
Period
Time Since ZC
Interrupt Thread:
Low−Priority Poll Thread:
USB Serial I/O
Spin Loop
Real−Time Clock Interrupt
Hardware
Firmware
Real−Time Clock
TTL out
Analyzer
Logic
Host
Computer
ADC in
Stim Trigger
Analog Out
Ephys
Ephys
Parallel Out
TABLE I SYNTHETIC
IDATASET NOISE BANDSFreq (Hz)
4-7
7-12
12-21
21-36
36-63
63-108
Band
theta alpha
beta
gamma gamma
gamma
TABLE II INFINITE
IIIMPULSE RESPONSE FILTER CONFIGURATIONSLabel
Type
Biquad Stages
Corners
anti-alias
low-pass
2
100 Hz
theta
band-pass
2
3 Hz-8 Hz
alpha
band-pass
2
6 Hz-16 Hz
beta
band-pass
2
12 Hz-32 Hz
low gamma
band-pass
2
24 Hz-64 Hz
Scaling brain size, keeping timing: Evolutionary preservation of brain rhythms. G Buzsaki, N Logothetis, W Singer, Neuron. 803G. Buzsaki, N. Logothetis, and W. Singer, "Scaling brain size, keeping timing: Evolutionary preservation of brain rhythms," Neuron, vol. 80, no. 3, pp. 751-764, 2013.
Large-scale recording of neuronal ensembles. G Buzsaki, Nature Neuroscience. 7G. Buzsaki, "Large-scale recording of neuronal ensembles," Nature Neuroscience, vol. 7, pp. 446-451, 2004.
Orientation selectivity and noise correlation in awake monkey area v1 are modulated by the gamma cycle. T Womelsdorf, B Lima, M Vinck, R Oostenveld, W Singer, S Neuenschwander, P Fries, Proceedings of the National Academy of Sciences. 10911T. Womelsdorf, B. Lima, M. Vinck, R. Oostenveld, W. Singer, S. Neuen- schwander, and P. Fries, "Orientation selectivity and noise correlation in awake monkey area v1 are modulated by the gamma cycle," Proceedings of the National Academy of Sciences, vol. 109, no. 11, pp. 4302-4307, 2012.
Spikephase coding boosts and stabilized information carried by spatial and temporal spike patterns. C Kayser, M A Montemurro, N K Logothetis, S Panzeri, Neuron. 614C. Kayser, M. A. Montemurro, N. K. Logothetis, and S. Panzeri, "Spike- phase coding boosts and stabilized information carried by spatial and temporal spike patterns," Neuron, vol. 61, no. 4, pp. 597-608, 2009.
Phase of firing coding of learning variables across prefrontal cortex, anterior cingulate cortex and striatum during feature learning. B Voloh, M Oemisch, T Womelsdorf, Nature Communications. in pressB. Voloh, M. Oemisch, and T. Womelsdorf, "Phase of firing coding of learning variables across prefrontal cortex, anterior cingulate cortex and striatum during feature learning," Nature Communications, in press.
Categoryselective phase coding in the superior temporal sulcus. H K Turesson, N K Logothetis, K L Hoffman, Proceedings of the National Academy of Sciences. 10947H. K. Turesson, N. K. Logothetis, and K. L. Hoffman, "Category- selective phase coding in the superior temporal sulcus," Proceedings of the National Academy of Sciences, vol. 109, no. 47, pp. 19 438-19 443, 2012.
Gamma and beta bursts during working memory readout suggest roles in its volitional control. M Lundqvist, P Herman, M R Warden, S L Brincat, E K Miller, Nature Communications. 9394M. Lundqvist, P. Herman, M. R. Warden, S. L. Brincat, and E. K. Miller, "Gamma and beta bursts during working memory readout suggest roles in its volitional control," Nature Communications, vol. 9, no. 394, 2018.
Neural oscillations: Sustained rhythms or transient burst-events?. F Van Ede, A J Quinn, M W Wollrich, A C Nobre, Trends in Neurosciences. 417F. van Ede, A. J. Quinn, M. W. Wollrich, and A. C. Nobre, "Neural oscillations: Sustained rhythms or transient burst-events?" Trends in Neurosciences, vol. 41, no. 7, pp. 415-417, 2018.
Neural mechanisms of transient neocortical beta rhythms: Converging evidence from humans, computational modeling, monkeys, and mice. M A Sherman, S Lee, R Shaw, S Haegens, C A Thorn, M S Hamalainen, C I Moore, S R Jones, Proceedings of the National Academy of Sciences. the National Academy of Sciences113M. A. Sherman, S. Lee, R. Shaw, S. Haegens, C. A. Thorn, M. S. Hamalainen, C. I. Moore, and S. R. Jones, "Neural mechanisms of transient neocortical beta rhythms: Converging evidence from humans, computational modeling, monkeys, and mice," Proceedings of the Na- tional Academy of Sciences, vol. 113, no. 33, pp. E4885-E4894, 2016.
Thetagamma coordination between anterior cingulate and prefrontal cortex indexes correct attention shifts. B Voloh, T A Valiante, S Everling, T Womelsdorf, Proceedings of the National Academy of Sciences. 11227B. Voloh, T. A. Valiante, S. Everling, and T. Womelsdorf, "Theta- gamma coordination between anterior cingulate and prefrontal cortex indexes correct attention shifts," Proceedings of the National Academy of Sciences, vol. 112, no. 27, pp. 8457-8462, 2015.
Bursts of beta oscillation differentiate postperformance activity in the striatum and motor cortex of monkeys performing movement tasks. J Feingold, D J Gibson, B Depasquale, A M Graybiel, Proceedings of the National Academy of Sciences. 11244J. Feingold, D. J. Gibson, B. DePasquale, and A. M. Graybiel, "Bursts of beta oscillation differentiate postperformance activity in the striatum and motor cortex of monkeys performing movement tasks," Proceedings of the National Academy of Sciences, vol. 112, no. 44, pp. 12 687-13 692, 2015.
Theta phasespecific codes for two-dimensional position, trajectory and heading in the hippocampus. J R Huxter, T J Senior, K Allen, J Csicsvari, Nature Neuroscience. 115J. R. Huxter, T. J. Senior, K. Allen, and J. Csicsvari, "Theta phase- specific codes for two-dimensional position, trajectory and heading in the hippocampus," Nature Neuroscience, vol. 11, no. 5, pp. 587-594, May 2008.
Phase-dependent neuronal coding of objects in short-term memory. M Siegel, M R Warden, E K Miller, Proceedings of the National Academy of Sciences. 10650M. Siegel, M. R. Warden, and E. K. Miller, "Phase-dependent neuronal coding of objects in short-term memory," Proceedings of the National Academy of Sciences, vol. 106, no. 50, pp. 21 341-21 346, 2009.
Gamma-band synchronization and information transmission. M Vinck, T Womelsdorf, P Fries, Principles of Neural Coding. R. Q. Quiroga and S. PanzeriCRC PressM. Vinck, T. Womelsdorf, and P. Fries, "Gamma-band synchronization and information transmission," in Principles of Neural Coding, R. Q. Quiroga and S. Panzeri, Eds. CRC Press, 2013, ch. 23, pp. 449-469.
Gamma-phase shifting in awake monkey visual cortex. M Vinck, B Lima, T Womelsdorf, R Oostenveld, W Singer, S Neuenschwander, P Fries, Journal of Neuroscience. 304M. Vinck, B. Lima, T. Womelsdorf, R. Oostenveld, W. Singer, S. Neuen- schwander, and P. Fries, "Gamma-phase shifting in awake monkey visual cortex," Journal of Neuroscience, vol. 30, no. 4, pp. 1250-1257, 2010.
Studying and modifying brain function with non-invasive brain stimulation. R Polania, M A Nitsche, C C Ruff, Nature Neuroscience. 21R. Polania, M. A. Nitsche, and C. C. Ruff, "Studying and modifying brain function with non-invasive brain stimulation," Nature Neuro- science, vol. 21, pp. 174-187, 2018.
Brain-machine interfaces: From basic science to neuroprostheses and neurorehabilitation. M A Lebedev, M A L Nicolelis, Physiological Reviews. 97M. A. Lebedev and M. A. L. Nicolelis, "Brain-machine interfaces: From basic science to neuroprostheses and neurorehabilitation," Physiological Reviews, vol. 97, pp. 767-837, 2017.
A causal network analysis of neuromodulation in the mood processing network. S Qiao, J I Sedillo, K A Brown, B Ferrentino, B Pesaran, Neuron. 107S. Qiao, J. I. Sedillo, K. A. Brown, B. Ferrentino, and B. Pesaran, "A causal network analysis of neuromodulation in the mood processing network," Neuron, vol. 107, pp. 1-14, 2020.
Closed-loop and activity-guided optogenetic control. L Grosenick, J H Marshel, K Deisseroth, Neuron. 861L. Grosenick, J. H. Marshel, and K. Deisseroth, "Closed-loop and activity-guided optogenetic control," Neuron, vol. 86, no. 1, pp. 106- 139, 2015.
Phase-locked stimulation during cortical beta oscillations produces bidirectional synaptic plasticity in awake monkeys. S Zanos, I Rembado, D Chen, E E Fetz, Current Biology. 28S. Zanos, I. Rembado, D. Chen, and E. E. Fetz, "Phase-locked stim- ulation during cortical beta oscillations produces bidirectional synaptic plasticity in awake monkeys," Current Biology, vol. 28, pp. 1-12, 2018.
Enhancement of encoding and retrieval functions through theta phase-specific manipulation of hippocampus. J H Siegle, M A Wilson, 3061J. H. Siegle and M. A. Wilson, "Enhancement of encoding and retrieval functions through theta phase-specific manipulation of hippocampus," p. e03061, 2014.
Stimulating at the right time: phase-specific deep brain stimulation. H Cagnan, D Pedrosa, S Little, A Pogosyan, B Cheeran, T Aziz, A Green, J Fitzgerald, T Foltynie, P Limousin, L Zrinzo, M Hariz, K J Friston, T Denison, P Brown, Brain. 1401H. Cagnan, D. Pedrosa, S. Little, A. Pogosyan, B. Cheeran, T. Aziz, A. Green, J. Fitzgerald, T. Foltynie, P. Limousin, L. Zrinzo, M. Hariz, K. J. Friston, T. Denison, and P. Brown, "Stimulating at the right time: phase-specific deep brain stimulation," Brain, vol. 140, no. 1, pp. 132- 145, 2017.
Predicting the effects of deep brain stimulation using a reduced coupled oscillator model. G Weerasinghe, B Duchet, H Cagnan, P Brown, C Bick, R Bogacz, PLOS Computational Biology. 1581006575G. Weerasinghe, B. Duchet, H. Cagnan, P. Brown, C. Bick, and R. Bo- gacz, "Predicting the effects of deep brain stimulation using a reduced coupled oscillator model," PLOS Computational Biology, vol. 15, no. 8, p. e1006575, 2019.
Continuous phase estimation for phase-locked neural stimulation using an autoregressive model for signal prediction. E Blackwood, M Lo, A S Widge, Proc. 40th Conf. 40th ConfE. Blackwood, M. chen Lo, and A. S. Widge, "Continuous phase estimation for phase-locked neural stimulation using an autoregressive model for signal prediction," in Proc. 40th Conf. IEEE Engineering in Medicine and Biology Society, 2018, pp. 4736-4739.
Realtime brain oscillation detection and phase-locked stimulation using autoregressive spectral estimation and time-series forward prediction. L L Chen, R Madhaan, B I Rapoport, W S Anderson, IEEE Transactions on Biomedical Engineering. 603L. L. Chen, R. Madhaan, B. I. Rapoport, and W. S. Anderson, "Real- time brain oscillation detection and phase-locked stimulation using autoregressive spectral estimation and time-series forward prediction," IEEE Transactions on Biomedical Engineering, vol. 60, no. 3, pp. 753- 762, 2013.
Real-time detection of neural oscillation bursts allows behaviorally relevant neurofeedback. G Karvat, A Schneider, M Alyahyay, F Steenbergen, M Tangermann, I Diester, Communications Biology. 372G. Karvat, A. Schneider, M. Alyahyay, F. Steenbergen, M. Tangermann, and I. Diester, "Real-time detection of neural oscillation bursts allows behaviorally relevant neurofeedback," Communications Biology, vol. 3, no. 72, pp. 1-10, 2020.
Closed-loop interruption of hippocampal ripples through fornix stimulation in the non-human primate. O Talakoub, A G P Schjetnan, T A Valiante, M R Popovic, K L Hoffman, Brain Stimulation. 9O. Talakoub, A. G. P. Schjetnan, T. A. Valiante, M. R. Popovic, and K. L. Hoffman, "Closed-loop interruption of hippocampal ripples through fornix stimulation in the non-human primate," Brain Stimulation, vol. 9, pp. 911-918, 2016.
RZ2 BioAmp Processor. "RZ2 BioAmp Processor," https://www.tdt.com/files/manuals/ Sys3Manual/RZ2.pdf, accessed: 2020-08-26.
Hardware Processing Platform (HPP): Getting Started Guide. "Hardware Processing Platform (HPP): Getting Started Guide," https: //neuralynx.com/documents/HPP Getting Started Guide.pdf, accessed: 2020-08-26.
Feature-specific prediction errors and surprise across macaque fronto-striatal circuits. M Oemisch, S Westendorff, M Azimi, S A Hassani, S Ardid, P Tiesinga, T Womelsdorf, Nature Communications. 10176M. Oemisch, S. Westendorff, M. Azimi, S. A. Hassani, S. Ardid, P. Tiesinga, and T. Womelsdorf, "Feature-specific prediction errors and surprise across macaque fronto-striatal circuits," Nature Communica- tions, vol. 10, no. 176, 2019.
The action potential in mammalian central neurons. B P Bean, Nature Reviews Neuroscience. 8B. P. Bean, "The action potential in mammalian central neurons," Nature Reviews Neuroscience, vol. 8, pp. 451-465, 2007.
Adaptive spike-artifact removal from local field potentials uncovers prominent beta and gamma band neuronal synchronization. K B Boroujeni, P Tiesinga, T Womelsdorf, Journal of Neuroscience Methods. 108485K. B. Boroujeni, P. Tiesinga, and T. Womelsdorf, "Adaptive spike-artifact removal from local field potentials uncovers prominent beta and gamma band neuronal synchronization," Journal of Neuroscience Methods, p. 330:108485, 2020.
The origin of extracellular fields and currents -eeg, ecog, lfp and spikes. G Buzsaki, C A Anastassiou, C Koch, Nature Reviews Neuroscience. 13G. Buzsaki, C. A. Anastassiou, and C. Koch, "The origin of extracel- lular fields and currents -eeg, ecog, lfp and spikes," Nature Reviews Neuroscience, vol. 13, pp. 407-420, May 2012.
Powerlaw scaling in the brain surface electric potential. K J Miller, L B Sorensen, J G Ojemann, M Den Nijs, PLOS Computational Biology. 5121000609K. J. Miller, L. B. Sorensen, J. G. Ojemann, and M. den Nijs, "Power- law scaling in the brain surface electric potential," PLOS Computational Biology, vol. 5, no. 12, p. e1000609, 2009.
Neuronal shot noise and brownian 1/f 2 behavior in the local field potential. J Milstein, F Mormann, I Fried, C Koch, PLOS One. 424338J. Milstein, F. Mormann, I. Fried, and C. Koch, "Neuronal shot noise and brownian 1/f 2 behavior in the local field potential," PLOS One, vol. 4, no. 2, p. e4338, 2009.
Corticothalamic resonance, states of vigilance and mentation. M Steriade, Neuroscience. 1012M. Steriade, "Corticothalamic resonance, states of vigilance and men- tation," Neuroscience, vol. 101, no. 2, pp. 243-276, 2000.
Dynamic circuit motifs underlying rhythmic gain control, gating, and integration. T Womelsdorf, T Valiante, N T Sahin, K J Miller, P Tiesinga, Nature Neuroscience. 17T. Womelsdorf, T. Valiante, N. T. Sahin, K. J. Miller, and P. Tiesinga, "Dynamic circuit motifs underlying rhythmic gain control, gating, and integration," Nature Neuroscience, vol. 17, pp. 1031-1039, 2014.
Rhythms of the Brain. G Buzsaki, Oxford University PressG. Buzsaki, Rhythms of the Brain. Oxford University Press, 2006.
Instantaneous envelope and phase extraction from real signals: Theory, implementation, and an application to eeg analysis. P Y Ktonas, N Papp, Signal Processing. 24P. Y. Ktonas and N. Papp, "Instantaneous envelope and phase extraction from real signals: Theory, implementation, and an application to eeg analysis," Signal Processing, vol. 2, no. 4, pp. 373-385, 1980.
Cycle-by-cycle analysis of neural oscillations. S Cole, B Voytek, Journal of Neurophysiology. 1222S. Cole and B. Voytek, "Cycle-by-cycle analysis of neural oscillations," Journal of Neurophysiology, vol. 122, no. 2, pp. 849-861, 2019.
Representing and decomposing neural potential signals. J C Principe, A J Brockmeier, Current Opinion in Neurobiology. 31J. C. Principe and A. J. Brockmeier, "Representing and decomposing neural potential signals," Current Opinion in Neurobiology, vol. 31, pp. 13-17, 2015.
Multiscale adaptive gabor expansion (mage): Improved detection of transient oscillatory burst amplitude and phase. R T Canolty, T Womelsdorf, https:/www.biorxiv.org/content/10.1101/369116v4.abstractR. T. Canolty and T. Womelsdorf, "Multiscale adaptive gabor expansion (mage): Improved detection of transient oscillatory burst amplitude and phase," 2019, https://www.biorxiv.org/content/10.1101/369116v4. abstract.
Learning recurrent waveforms within eegs. A J Brockmeier, J C Príncipe, IEEE TRansactions on Biomedical Engineering. 631A. J. Brockmeier and J. C. Príncipe, "Learning recurrent waveforms within eegs," IEEE TRansactions on Biomedical Engineering, vol. 63, no. 1, pp. 43-54, 2016.
Adaptive waveform learning: A framework for modeling variability in neurophysiological signals. S Hitziger, M Clerc, S Saillet, C Bénar, T Papadopoulo, IEEE Transactions on Signal Processing. 6516S. Hitziger, M. Clerc, S. Saillet, C. Bénar, and T. Papadopoulo, "Adaptive waveform learning: A framework for modeling variability in neurophys- iological signals," IEEE Transactions on Signal Processing, vol. 65, no. 16, pp. 4324-4338, 2017.
Open ephys: an open-source plugin-based platform for multichannel electrophysiology. J H Siegle, A C Lopez, Y A Patel, K Abramov, S Ohayon, J Voigts, Journal of Neural Engineering. 144J. H. Siegle, A. C. Lopez, Y. A. Patel, K. Abramov, S. Ohayon, and J. Voigts, "Open ephys: an open-source plugin-based platform for multichannel electrophysiology," Journal of Neural Engineering, vol. 14, no. 4, pp. 1-14, 2017.
Acquisition board -Open Ephys. "Acquisition board -Open Ephys," https://open-ephys.org/acq-board, accessed: 2020-08-26.
" Rhd Usb/Fpga Interface, Rhythm USB3. "RHD USB/FPGA Interface: Rhythm USB3," http://intantech.com/ products RHD2000 Rhythm USB3.html, accessed: 2020-08-26.
Attention Circuits Control Laboratory -NeuroLoop. "Attention Circuits Control Laboratory -NeuroLoop," https://github. com/att-circ-contrl/NeuroLoop, accessed: 2020-09-14.
Fast spiking interneuron activity in primate striatum tracks learning of attention cues. K B Boroujeni, M Oemisch, S Hassani, T Womelsdorf, Proceedings of the National Academy of Sciences. 11730K. B. Boroujeni, M. Oemisch, S. alireza Hassani, and T. Womelsdorf, "Fast spiking interneuron activity in primate striatum tracks learning of attention cues," Proceedings of the National Academy of Sciences, vol. 117, no. 30, pp. 18 049-18 058, 2020.
| [] |
[] | [] | [] | [] | This paper presents a hybrid approach to spatial indexing of two dimensional (2D) data. It sheds new light on the age old problem by thinking of the traditional algorithms as working with images. Inspiration is drawn from an analogous situation that is found in machine and human vision. Image processing techniques are used to assist in the spatial indexing of the data. A fixed grid approach is used and bins with too many records are sub-divided hierarchically. Search queries are pre-computed for bins that do not contain any data records. This has the effect of dividing the search space up into non rectangular regions which are based on the spatial properties of the data. The bucketing quad tree can be considered as an image with a resolution of 2x2 for each layer. The results show that this method performs better than the quad tree if there are more divisions per layer. This confirms our suspicions that the algorithm works better if it gets to "look" at the data with higher resolution images. An elegant class structure is developed where the implementation of concrete spatial indexes for a particular data type merely relies on rendering the data onto an image. | 10.1109/icsmc.2006.385020 | [
"https://arxiv.org/pdf/0705.0204v1.pdf"
] | 783 | 0705.0204 | b318b417887218aa2a662eb9757cb43fb9240885 |
This paper presents a hybrid approach to spatial indexing of two dimensional (2D) data. It sheds new light on the age old problem by thinking of the traditional algorithms as working with images. Inspiration is drawn from an analogous situation that is found in machine and human vision. Image processing techniques are used to assist in the spatial indexing of the data. A fixed grid approach is used and bins with too many records are sub-divided hierarchically. Search queries are pre-computed for bins that do not contain any data records. This has the effect of dividing the search space up into non rectangular regions which are based on the spatial properties of the data. The bucketing quad tree can be considered as an image with a resolution of 2x2 for each layer. The results show that this method performs better than the quad tree if there are more divisions per layer. This confirms our suspicions that the algorithm works better if it gets to "look" at the data with higher resolution images. An elegant class structure is developed where the implementation of concrete spatial indexes for a particular data type merely relies on rendering the data onto an image.
I. INTRODUCTION
HIS paper sheds new light on the way in which spatial indexing is considered and performed. It begins by giving a brief overview of two common spatial indexing techniques. It then gives evidence to suggest that an analogous situation exists in the human vision system and therefore opens the door to using image processing techniques to deal with the spatial indexing problem. The design and implementation of this spatial index is discussed and the results are presented and analyzed. A brief description of the further work explains where this research is headed.
II. LITERATURE REVIEW
A. Spatial Indexing
With the advent of high performance computing, there are an increasing amount of datasets that contain a significant spatial component to them. Being able to perform spatial queries on this data is only feasible if there is a way to manage the large quantities of data. Database indexing is a technique used to speed up searches for data by creating a searchable catalogue of the data based on a unique key. Spatial indexing Manuscript received March 1, 2006. This work was supported in part by Storm Logistics, South Africa (www.profilerxp.com). L. A. Machowski is with Storm Logistics and currently doing his PhD at the School of Electrical and Information Engineering, University of the Witwatersrand, Republic of South Africa (e-mail: [email protected]; [email protected]).
T. Marwala, is with the School of Electrical and Information Engineering, University of the Witwatersrand, Republic of South Africa (e-mail: [email protected]).
uses the spatial coordinates of the data to create the searchable catalogue. This has the effect of prioritizing searches through the data, based on the spatial extent from the query point. Since one of the most recent developments in database technology is the addition of spatial data types, the spatial indexing methods that databases rely on are becoming increasingly more important [1].
The most typical spatial indexing scheme makes use of a divide-and-conquer approach where the original domain space is broken down into several regions. These regions are in turn divided up as necessary to form a hierarchical tree that can be traversed when searching for records. One of the most common examples of this spatial indexing technique is the quad tree (and region quad tree) for two dimensions and the oct tree for three dimensions [1]- [3] which create a recursive decomposition of space [2]. An alternative to the hierarchical data structure is the fixed grid or cell method, which is popular amongst cartographers. The advantage of this method is that it is easy to do lookups and adding or deleting records from the data structure is simple. The disadvantage of this method is that it is only suited to uniformly distributed data which is not typically the case when dealing with geographic information such as road network data for an entire country. The difficulty with hierarchical spatial indexes is in partitioning and grouping the records [1]. This can be done either as a batch process once all the records are available, or it can be done as the records are being added. Removing records from certain regions may get complicated and this limits the usefulness of the structure to dynamic information. There is also an overhead in memory and access time which must be taken into account when using hierarchical spatial indexes. An excellent review of the various spatial indexing techniques as well an extensive description and analysis are found in [2].
B. Spatial Indexing Methods
In order to simplify the problem domain for this paper, we only consider the spatial indexing of points in two-dimensional space. This delimitation is acceptable since most other geometric primitives can be expressed as points and typical real world datasets are generated from geospatial data which is approximately 2D. We also only deal with the updating of the spatial indexes as a batch process in order to illustrate the key concepts presented in this paper.
1) Fixed Grid Method
The fixed grid method is one of the simplest spatial indexing methods to implement. It simply divides up the extents of the data into regular sized grid bins. Each bin maintains a list of all the data records that fall within the bin.
Using Images to create a Hierarchical Grid Spatial Index
Lukasz A. Machowski, and Tshilidzi Marwala, Member, IEEE T Searching for the nearest record involves finding the bin at the query location and iterating through each record in the list to find the nearest match. Resolving the bin indices is an O(1) operation because all we have to do is divide the offset of the query point from the corner of the spatial index extents, by the width of each bin.
For a uniformly distributed set of points, we expect the average search cost to be:
( ) y x n c × = (1)
Where n is the total number of records, x is the number of divisions in the X dimension and y is the number of divisions in the Y dimension. Unfortunately, this equation is only valid near the centers of the bins because of the effect that occurs at the edges as illustrated in Fig 1. For points that are near the edge of a bin, one must consider the points that are in the adjacent bins in order to ensure that the search result is indeed the closest record to the query point. The negative effect of this is compounded once we attempt to index data that is highly non-uniform. If the data has regions where there are gaps, there is a good chance that several of the bins will be empty. Querying for the nearest record at these locations will incorrectly return no results. It is therefore necessary to apply some heuristic to search through adjacent bins that are near the query point. This paper presents a neat solution to this problem and is discussed later.
2) Quad Tree As discussed in [2], there are a number of different spatial indexing methods that can be classified as quad trees. The overall idea is that the region of interest is divided up into quadrants which are further sub-divided recursively until a set number of records are present in each quadrant. This is a hierarchical technique that can be thought of as processing at multiple resolutions, because the spatial extents of the quadrants are continually decreasing as the search depth in the tree increases. This paper mainly deals bucketing methods where the data records are added to buckets (or bins) that are defined by the extents of the quad tree quadrants [1]. An example quad tree is shown in Fig 2. The problem with the quad tree is that one needs to travel several layers deep before one reaches a high enough resolution that is suitable for indexing through large amounts of data. Also, it becomes slightly more complicated to perform range searches because of the tree structure that needs to be taken into account when looking at adjacent bins.
III. BACKGROUND
The inspiration for this spatial indexing technique comes from two sources. The first is that humans are very adept at visually searching through large sets of data while being able to filter out irrelevant details. The second is that most of the existing spatial indexing techniques can astonishingly be thought of as using images to perform the spatial searches visually. The techniques might not all use regular grid-like images, but their effect can definitely be considered as an image processing operation. A little more clarification is in order.
A. Humans as excellent spatial indexers
The most widely accepted theory of spatial vision is that of the multichannel model [4] developed by Enroth-Cugell and Robson [5] and Campbell [6]. This theory proposes that the visual system processes the retinal image simultaneously at several different spatial scales [4]. This is consistent with the type of data that needs to be processed in the real world, which is often made up of different levels of detail. The physiological evidence for this ability is in the size of a neuron's receptive field in each stage of early vision [4]. The neurons that process the raw signals from the photoreceptor cells have varying receptive field sizes and therefore we are able to detect a wide range of detail. It is believed that these various-scale outputs along with information from other channels are combined by the higher vision processes into our interpretation of a scene.
The impulse response of individual neurons falls-off as the signal moves further away from the centre of the neurons receptive field. This characteristic is highly desirable and essential for performing spatial queries on sets of data.
B. Traditional Techniques as Image Operations
One of the most common 2D spatial indexing techniques is the Quad-Tree as discussed earlier. If one considers that the quad tree is a pyramid of images, then one realizes that each layer is only represented by a 2x2 image. The same concept can be applied to other common spatial indexing techniques and one soon realizes that decisions on what search path to follow are based on a very limited view of the data. Imagine if all that we could see was 4 pixels at a time! In this paper, we propose that one considers the problem of spatial indexing as an image processing operation which gets performed on multi-resolution representations of the underlying data. The process of inserting data into the spatial index is thought of as rendering the data onto an image, which is the regular grid at each hierarchical level. This means that we can use the wealth of knowledge and the abundance of algorithms that are available for drawing geometric primitives in order to insert the data into the spatial index. One can also use the concept of alpha-blending as a way of creating a histogram of regions that contain a large number of records. If we render the data with an additive drawing mode, then regions with a bright (high) color are known to contain large amounts of data records.
Considering the spatial index as a set of images also helps us when performing range and nearest point queries. This is because we are able to use the vast amount of morphological operators and other techniques that are available for image processing [7]- [10]. One of the fundamental concepts of morphological operators is the idea of neighborhoods and connectivity. Given a 5x5 image shown in Fig. 3, we can describe the 1-neighbourhood of a pixel as the set of pixels that are touching the centre pixel. In our definition, we consider the diagonal pixels as being connected as well. The n-neighborhood is therefore, the set of pixels that are n pixels away from the centre pixel. The morphological operators make use of the neighboring pixels to decide on the value for the centre pixel. In fact, many 2D image filters are defined as kernels which are convolved with an image to perform complex operations such as blurring, sharpening, opening and closing [7][8].
Anderson and McCartney have shown that using images (or diagrams) can be very effective for performing several complex spatial database queries [10]. They use logical operators on 2D diagrams to perform the search queries. This paper extends their idea by using a hierarchical set of images to perform the spatial indexing of the data.
IV. METHOD
The spatial index described in this paper is designed as a set of Object-Oriented classes in C# for Microsoft .NET V2. The design makes use of inheritance, polymorphism and interfaces to achieve an elegant and extensible solution to the problem. The use of generics is not necessary to implement the spatial index.
The spatial index is implemented as a class hierarchy as shown in Fig. 4.
The AbstractGridSpatialIndex is the base class for all grid spatial indexes. This class contains all the common functionality for indexing data in a grid. It also performs the bulk of the indexing and spatial queries. This design allows us to have a solid and consistent implementation for the grid spatial index while allowing a variety of sub classes to implement different indexing behaviors. The class is declared as abstract so that sub classes are forced to implement the abstract methods. In order for the spatial index to be useful for indexing many types of data, it is necessary to make the abstract class accept a very general data type. For this reason, an interface is used instead of a concrete class and it is described in the next section.
A. IGridSpatiaIndexable
This is the interface that needs to be implemented by data collections in order for them to be indexed in a grid spatial index. The reason for using an interface is so that the spatial index does not limit the class structure that may be indexed. It allows arbitrary class hierarchies to exist for the data collections as long as the class implements the methods required for performing spatial indexing. The interface declaration also defines the minimal functionality required for indexing data in a grid spatial index. This interface is shown in Fig. 5.
Note that the interface includes properties. This is a feature of the .NET framework and it allows interfaces to declare field-like elements that are implemented with getter and setter methods. If your language does not support this feature then the interface would merely have the corresponding getter and setter methods to replace the properties.
The spatial index has been designed to index only integer values. This scheme provides a good trade-off between Fig. 5. IGridSpatialIndexable is the minimal interface required for spatial indexing in a grid. ISpatialQueriable2D is the minimal interface for data that can be indexed. generality for multiple applications and it also allows complex data access schemes to be spatially indexed. It is therefore the role of the IGridSpatialIndexable object to supply the mapping between an index and the actual record to be indexed. Concrete spatial index classes must implement the GetRecordToIndex() method of the interface. This method gets passed the index of the record to process and a temporary object of the type being indexed. The method must get the information for that specific record and return it to the spatial index. The temporary object that is passed to the method allows one to perform arbitrary calculations on the data (such as coordinate transformations) without having to create hundreds of transient objects for this process. This object reuse improves performance considerably, especially for cases where the record itself has to be converted into a form that can be indexed. The record to be indexed also has to implement the ISpatialQueriable2D interface, which defines the methods shown in Fig. 5.
It is necessary for the spatial index to get the extents of the data being indexed (GetSpatialIndexExtents()). This is so that the initial grid spatial index can be generated. A method that returns the total number of records (GetRecordCount()) is required for the spatial index to know how many records to index. The interface also has a Boolean property (Changed) which flags whether the data has changed. The spatial index uses this flag to rebuild itself whenever a spatial query is about to be run. With the IGridSpatialIndexable interface, we are able to represent a collection of data that can be indexed hierarchically or in only one layer of a grid index.
B. AbstractGridSpatialIndex
This class takes the IGridSpatialIndexable collection and the number of divisions for the initial spatial grid as parameters to its constructor. It contains two grids of integer lists. The grid is implemented as a 2D array of integer lists. The first grid holds in each bin, the record indices that fall inside that bin. This grid is the result of rendering all the data to an image and saving which records were rendered to the pixels. Any bins that do not have data are set to null. Every integer list that is unique for this grid is maintained in a dictionary where the integer list is the key and the corresponding grid coordinate is the value. This allows an efficient lookup of the grid coordinates for a particular integer list. The second grid holds a duplicate of the rendered list but all the null bins are set to point to the integer list that contains the nearest record to the centre of the bin. This is a method of pre-computing approximate results to the problem discussed above for empty bins in the fixed grid spatial index. Rebuilding the spatial index is done when the data is flagged as being changed and may be described as the following high level process:
1. Get the spatial data-extents. 2. Get the number of records to index. 3. Clear the unique-list dictionary. 4. Create the bins for the lists.
5. Calculate the bin sizes. 6. Allow descendant classes to perform extra processing before the index is rebuilt. 7. Render the records into the grid (nulls where there is no data). 8. Create a shallow copy of the rendered lists. 9. Fill in the gaps by finding the integer lists with the nearest record to the centre of the bin. 10. Allow descendant classes to perform extra processing after the index is rebuilt. 11. Flag that the data has been processed and watch for further changes. The class also has several protected helper methods to assist descendant classes to render their data correctly to the bins. These are in the form of efficient point, line and area rendering methods that add the indices of the records into the integer lists in the grid. This design means that concrete descendant classes only need to implement two methods for the spatial index to work, namely RenderRecordsToLists() and CreateRecordInstance() (which makes the temporary record described previously). It is evident from these two methods that we have successfully managed to abstract out all the spatial indexing functionality from the data rendering functionality. This means that the developer of concrete sub classes only has to program how to render the data to a grid (which is essentially the same as rendering the data to an image or the screen).
C. Searching
The efficiency of a spatial index lies in its role as a pruning device for searching that is done [2]. In order to solve the empty-bin problem discussed earlier, we propose a solution that pre-computes the bin with the nearest record to the centre of each of the empty bins (step 9 above). This has the effect of creating non-rectangular regions that all point to the same integer list. This is a very desirable effect because the grid is partitioned into arbitrary regions that depend entirely on the data. Most other algorithms partition the search space into strict rectangular segments which are not well suited to real-world data. Fig. 6 shows an example of this partitioning.
With these pre-computed bins, performing a search for the nearest record simply involves querying the bin at the query point and all the bins in the 1-neighbourhood. This guarantees that distant and adjacent records are searched and it solves the problem encountered with the fixed grid method. It is important to note that this is only valid if we are querying inside the extents of the spatial index. If this is not the case, then we have to query the entire edge (all the bins along the side) of the spatial index for the nearest record.
The naïve approach to the described search method will search every bin in the neighborhood or every bin along the edge of the data extents. This, however, is not always necessary. If the distance of the nearest point in a bin to the query point is shorter than the distance to any of the bin edges, then we have found the nearest record and we do not have to search additional bins. This allows the algorithm to short circuit after searching through the first bin.
D. Fixed Grid Spatial Index
By implementing the two abstract methods of the AbstractGridSpatialIndex to render the data, we would have a complete implementation for a fixed grid spatial index with no hierarchical sub-divisions. This is suitable for hand-tuned datasets or when a lightweight spatial index is required.
E. Hierarchical Grid Spatial Index
The hierarchical grid spatial index is implemented by introducing a proxy collection (SubGridDataCollection) that implements IGridSpatialIndexable, and by overriding steps 6 and 10 of the abstract class' RebuildIndex() method. An internal list of all the sub grids is maintained and another grid stores the indices of these sub lists for each bin. When step 6 (OnBeforeIndexRebuilt()) is called, it merely recreates the sub grid lists.
Step 10 (OnAfterIndexRebuilt()) does all of the actual work by going through all the unique integer lists and checking if their count exceeds MaxBinRecords. If this is the case then a clone of the current spatial index is made and it is passed a proxy to the integer list as its data source. This means that all the sub spatial indexes deal with a proxy to the original data source. This makes the implementation more efficient than making sub copies of the original data.
The hierarchical grid spatial index also overrides the GetNearestRecord() method in order to first check whether a sub grid needs to be queried. If this is the case then the query is passed down to the sub grid, otherwise the default implementation is used from the abstract base class.
A threshold parameter (MaxBinRecords) is used to decide when to sub divide a bin further with another Hierarchical Grid Spatial Index. Several schemes exist where the sub grids contain the same or varying amounts of sub divisions.
It is necessary to introduce a SmallestBinDimension parameter for this spatial index. This is because we need to limit the depth to which the spatial index will partition the search space. This is particularly important for the case when there are more than MaxBinRecords located at the exact same position. No matter how many times we sub divide the search space, we will never manage to partition the records any further. It is therefore important to have this threshold so that if either of the X or Y dimensions of the bins are smaller, then the partitioning stops.
V. RESULTS AND ANALYSIS
In order to evaluate the search cost for the spatial index, an extent twice the size of the data extent is evaluated at a regular interval. The performance of the spatial index is evaluated for a varying number of grid divisions. This demonstrates what effect the regular grid has on spatial indexing (remember that this spatial index can be thought of as a bucket quad tree when the divisions are set to 2x2). The search cost in terms of number of records is evaluated at each point and the results are shown as an image. Two types of coloring schemes are used to look at the results. A relative color range is normalized to the minimum and maximum search costs for the image. This highlights areas of interest in the performance of the spatial index. An absolute range for the color is used so that the performance at different grid divisions can be compared. Examples of uniform and Gaussian data points are given in Fig. 7 and they show the corresponding search costs. Fig. 8 shows the results for varying grid divisions for the grid spatial index. It is worthwhile looking at these results even though there is no hierarchical aspect to the algorithm, since the results can be thought of as a type of an impulse response for a particular layer.
When performing a nearest-point-search on a uniformly distributed set of data inside the data extents, our fixed grid spatial index should never exceed the maximum search cost given by (2):
( ) y x n c × × = 9 max (2)
where n is the total number of records, x is the number of X divisions and y is the number of Y divisions. The reasoning behind this equation is that we need to search through the current bin plus 8 of its neighbors. For a uniform distribution, the average search cost is described by (1). The empirical results obtained so far (as seen in Fig. 8) show that this relationship is true. Equation (2) puts an upper bound on the search cost for the non-hierarchical grid spatial index. It also predicts that the maximum search cost decreases exponentially as the number of grid divisions increases. This clearly explains the decreasing trend in the graph of Fig. 8. Fig. 9 shows the results for the hierarchical grid spatial index. The absolute value is scaled so that maximum color value corresponds to 1% of the total number of records. The MaxBinRecords threshold is set to 1 so that the grid is always sub-divided. We see that the performance of this spatial index is well under 1% so it is clear that it is effective at performing the spatial indexing tasks. The graph in Fig. 9 reaches the lower limit because of the SmallestBinDimension parameter.
In both hierarchical and non-hierarchical cases, the performance of spatial indexes with more than 2 divisions per dimension, the maximum search cost is always lower. This validates our previous expectation that better performance can be achieved by "looking" at the data with higher resolution images. There is an interesting memory trade off because having more division's means that the hierarchical tree will not be as deep as when there are only a few divisions per layer.
VI. FURTHER WORK
This paper has only analyzed the performance of this spatial indexing technique based on the number of records searched. Further work needs to be done to analyze the memory and time performance of the algorithm at varying grid divisions.
Since this spatial indexing method has roots in image processing, the algorithm is to be moved over to a hardware implementation where the indexing of the data is rendered by a hardware accelerated graphics card. This makes use of the card as a General Purpose Graphics Processing Unit (GPGPU) [11] [12]. The algorithm will make use of the GPU to render the data to images that represent the grid in the spatial index described in this paper. The recent advances in vertex and pixel shaders on the GPU will make it feasible to implement a part of this spatial index on the hardware [13] [14].
VII. CONCLUSIONS
This paper describes the design and implementation of a hierarchical grid spatial index. It shows that treating the spatial indexing as an image processing operation makes for an elegant solution to the spatial indexing problem. The design of the classes has lead to an attractive solution where implementers of specific spatial indexes merely need to render the data onto a grid. The search costs for various grid divisions were analyzed and the results show that using more than 2 divisions per dimension (more than a quad tree) provides better search performance.
Fig. 1 .
1Search required near the edge of a bin. Some bins are empty.Fig. 2. An example of a bucketing quad tree for a small set of points. The maximum number of records in a bucket is 1.
Fig. 4 .Fig. 3 .
43Class The 1-(light) and the 2-(dark) neighborhood of a pixel.
Fig. 6 .
6The partitioning of reused bins after gaps in the grid are filled.
Fig. 7 .
7Data points (top) and Search Costs (bottom). Left: 5000 Uniformly distributed points. Right: 5000 Gaussian points.
Fig. 8 .
8Uniform Data Points (5000). No Hierarchical Divisions.
Fig. 9 .
9Uniform Data Points (5000). Hierarchical Division, MaxBinRecords = 1
Decoupling Partitioning and Grouping: Overcoming Shortcomings of Spatial Indexing with Bucketing. H Samet, ACM Transactions on Database Systems. 294H. Samet, "Decoupling Partitioning and Grouping: Overcoming Shortcomings of Spatial Indexing with Bucketing," in ACM Transactions on Database Systems, Vol. 29, No. 4, Dec. 2004.
The Design and Analysis of Spatial Data Structures. H Samet, Addison-WesleyMassachusettsH. Samet, "The Design and Analysis of Spatial Data Structures," Addison-Wesley, Massachusetts, 1990.
A Multiresolution Representation for Massive Meshes. E Shafter, M Garland, IEEE Transactions on Visualization and Computer Graphics. 11E. Shafter, and M. Garland, "A Multiresolution Representation for Massive Meshes," in IEEE Transactions on Visualization and Computer Graphics, Vol. 11, No. 2, pp. 139-148; Mar./Apr. 2005.
Level of Detail for 3D Graphics. D Luebke, M Reddy, J Cohen, A Varshney, B Watson, R Huebner, Elsevier ScienceUSAD. Luebke, M. Reddy, J. Cohen, A. Varshney, B. Watson, and R. Huebner, "Level of Detail for 3D Graphics," Morgan Kaufmann Publishers, Elsevier Science, pp. 247-248, USA, 2003.
The Contrast Sensitivity of Retinal Ganglion Cells of the Cat. C Enroth-Cugell, J Robson, Journal of Physiology. 187C. Enroth-Cugell, and J. Robson, "The Contrast Sensitivity of Retinal Ganglion Cells of the Cat," in Journal of Physiology, vol. 187, pp. 517-552, 1966.
Application of Fourier Analysis to the Visibility of Gratings. F Campbell, J Robson, Journal of Physiology. 197F. Campbell, and J. Robson, "Application of Fourier Analysis to the Visibility of Gratings," in Journal of Physiology, vol. 197, pp. 551-556, 1968.
Machine Vision: Theory, Algorithms, Practicalities. E Davies, Academic Pressnd ed.E. Davies, "Machine Vision: Theory, Algorithms, Practicalities," 2 nd ed., Academic Press, 1997.
B Jähne, Digital Image Processing: Concepts, Algorithms and Scientific Applications. BerlinSpringer4th ed.B. Jähne, "Digital Image Processing: Concepts, Algorithms and Scientific Applications," 4th ed., Springer, Berlin, 1997.
Representing and Classifying 2D Shapes of Real-World Objects using Neural Networks. L Machowski, T Marwala, Proceedings of the IEEE Conference on Systems, Man and Cybernetics. the IEEE Conference on Systems, Man and CyberneticsThe Hague, NetherlandsL. Machowski, and T. Marwala, "Representing and Classifying 2D Shapes of Real-World Objects using Neural Networks," in Proceedings of the IEEE Conference on Systems, Man and Cybernetics, The Hague, Netherlands, pp. 6366-6372, 2004.
Diagram processing: Computing with diagrams. M Anderson, R Mccartney, Artificial Intelligence. 145ElsevierM. Anderson, and R. McCartney, "Diagram processing: Computing with diagrams," in Artificial Intelligence, No. 145, pp 181-226, Elsevier, 2003.
General-Purpose Computation Using Graphics Hardware. 25/10/2005GPGPUGPGPU, "General-Purpose Computation Using Graphics Hardware," 2005. Website Last Accessed: 25/10/2005; http://www.gpgpu.org/
GPU Gems 2: Programming Techniques for High-Performance Graphics and General-Purpose Computation. M Pharr, Addison-Wesley, NVIDIA CorporationM. Pharr, "GPU Gems 2: Programming Techniques for High-Performance Graphics and General-Purpose Computation," Addison-Wesley, NVIDIA Corporation, 2005.
Hardware Acceleration in Commercial Databases: A Case Study of Spatial Operations. N Bandiy, C Sunz, D Agrawaly, A El Abbadiy, Santa BarbaraComputer Science Department, University of CaliforniaTechnical ReportN. Bandiy, C. Sunz, D. Agrawaly, and A. El Abbadiy, "Hardware Acceleration in Commercial Databases: A Case Study of Spatial Operations," Technical Report, Computer Science Department, University of California, Santa Barbara, 2004.
Fast Computation of Database Operations using Graphics Processors. N Govindaraju, B Lloyd, W Wang, M Lin, D Manocha, Proceedings of the 2004 ACM SIGMOD international conference on Management of data. the 2004 ACM SIGMOD international conference on Management of dataN. Govindaraju, B. Lloyd, W. Wang, M. Lin, and D. Manocha, "Fast Computation of Database Operations using Graphics Processors," in Proceedings of the 2004 ACM SIGMOD international conference on Management of data, pp215-226, Jun. 2004.
| [] |
[
"Detailed comparative study and a mechanistic model of resuspension of spherical particles from rough and smooth surfaces",
"Detailed comparative study and a mechanistic model of resuspension of spherical particles from rough and smooth surfaces"
] | [
"Ron Shnapp \nSchool of Mechanical Engineering\nTurbulence Structure Laboratory\nTel Aviv University\n69978Tel AvivIsrael\n",
"Alex Liberzon \nSchool of Mechanical Engineering\nTurbulence Structure Laboratory\nTel Aviv University\n69978Tel AvivIsrael\n"
] | [
"School of Mechanical Engineering\nTurbulence Structure Laboratory\nTel Aviv University\n69978Tel AvivIsrael",
"School of Mechanical Engineering\nTurbulence Structure Laboratory\nTel Aviv University\n69978Tel AvivIsrael"
] | [] | Resuspension of solid particles by a tornado-like vortex from surfaces of different roughness is studied using a three-dimensional particle tracking velocimetry (3D-PTV) method. By utilizing the three-dimensional information on particle positions, velocities and accelerations before, during and after the resuspension (lift-off) event, we demonstrate that the resuspension efficiency is significantly higher from the rough surface, and propose a mechanistic model of this peculiar effect. The results indicate that for all Reynolds numbers tested, the resuspension rate, as well as particle velocities and accelerations, are higher over the rough surface, as compared to the smooth counterpart. The results and the model can help to improve modeling and analysis of resuspension rates in engineering and environmental applications. * * Electronic address: [email protected]; Corresponding author 1 arXiv:1411.5152v1 [physics.flu-dyn] 19 Nov 2014 study of resuspension demands a break-up of the mechanism as a whole into separate stages, so in the future, the broad picture may be better understood. In this work we want to focus on the stage of the freely moving particle lift-off from smooth or rough surfaces.(2014)show that the complexity of the resuspension phenomena is caused by two inherent features: particle interaction with the surface to which it is attached, and particle interaction with the fluid to which it is exposed. It is also evident that these two features of the system are affected by each other, as the surface roughness affects the flow regime, and the fluid properties affect the bonds that the particles have with the surface (cohesive and adhesive forces). Furthermore, studies have shown different (often opposite)Recent reviews of particle resuspension from surfaces by Ziskind (2006); Henry & Minier | 10.1016/j.ces.2015.03.048 | [
"https://arxiv.org/pdf/1411.5152v1.pdf"
] | 119,266,742 | 1411.5152 | ba37a19972ad36baeff32b5b6724c77ba5b0b20c |
Detailed comparative study and a mechanistic model of resuspension of spherical particles from rough and smooth surfaces
Ron Shnapp
School of Mechanical Engineering
Turbulence Structure Laboratory
Tel Aviv University
69978Tel AvivIsrael
Alex Liberzon
School of Mechanical Engineering
Turbulence Structure Laboratory
Tel Aviv University
69978Tel AvivIsrael
Detailed comparative study and a mechanistic model of resuspension of spherical particles from rough and smooth surfaces
Dated: November 20, 2014
Resuspension of solid particles by a tornado-like vortex from surfaces of different roughness is studied using a three-dimensional particle tracking velocimetry (3D-PTV) method. By utilizing the three-dimensional information on particle positions, velocities and accelerations before, during and after the resuspension (lift-off) event, we demonstrate that the resuspension efficiency is significantly higher from the rough surface, and propose a mechanistic model of this peculiar effect. The results indicate that for all Reynolds numbers tested, the resuspension rate, as well as particle velocities and accelerations, are higher over the rough surface, as compared to the smooth counterpart. The results and the model can help to improve modeling and analysis of resuspension rates in engineering and environmental applications. * * Electronic address: [email protected]; Corresponding author 1 arXiv:1411.5152v1 [physics.flu-dyn] 19 Nov 2014 study of resuspension demands a break-up of the mechanism as a whole into separate stages, so in the future, the broad picture may be better understood. In this work we want to focus on the stage of the freely moving particle lift-off from smooth or rough surfaces.(2014)show that the complexity of the resuspension phenomena is caused by two inherent features: particle interaction with the surface to which it is attached, and particle interaction with the fluid to which it is exposed. It is also evident that these two features of the system are affected by each other, as the surface roughness affects the flow regime, and the fluid properties affect the bonds that the particles have with the surface (cohesive and adhesive forces). Furthermore, studies have shown different (often opposite)Recent reviews of particle resuspension from surfaces by Ziskind (2006); Henry & Minier
Introduction
Particle resuspension is the process in which a submerged particle is being detached from a surface to the fluid medium above, after the break-up of the particle-surface bond. Resuspension is an ubiquitous process in many engineering and environmental applications, for instance in sediment transport (Wu & Chou, 2013), powder handling processes (Grzybowski & Gradon, 2007), and studies of Martian dust devils (Greeley et al., 2004). Previous studies have shown that resuspension is inherently related to the flow regimes near the wall, and in many cases may be related to turbulent flow velocity fluctuations or flow structures, such as sweeps and ejections (Henry & Minier, 2014). The dependency of the resuspension phenomena on diverse flow regimes makes it hard to study in a general fashion. Therefore, the detailed relations between the wall surface roughness, particle diameter and the efficiency of the resuspension process (e.g. Nino et al., 2003;Yanbin et al., 2008;Lee & Balachandar, 2012;Barth et al., 2014, among others).
Surface roughness effect on the resuspension rate can be modeled in many different ways and by different mechanisms. For instance, Henry et al. (2012) in their model of re-entrainment coupled surface roughness with the effect of particle-surface adhesion. Lee & Balachandar (2012) calculated a critical shear stress criterion for the initiation of particle movement, and determined that surface roughness may affect particle movement through the level of relative particle protrusion, or through a moment balance of hydrodynamical and resistive forces against an asperity. Using a channel airflow experiment, Yanbin et al. (2008) demonstrated that the effect surface roughness has on resuspension varies for particles of different sizes and for different scales of surface roughness. Hall (1988) measured and derived an expression for the lift force acting on a particle on smooth and rough surfaces, and found that the force can change by several orders of magnitude with surface roughness, and with the position of a particle relative to the roughness elements. In our study, we choose to focus on a single aspect of this diverse phenomena, namely on the way by which particle mobility (rolling or sliding motion along the surface) over the smooth and rough surfaces affects resuspension of relatively large spherical particles.
Many experimental methods have been introduced so far for measuring resuspension rates. The majority of studies were conducted through a wind tunnel or a duct flow with particles spread over the channel bed. The initial load of particles is measured and particles that leave the observation volume are counted, providing the fraction remaining (Ibrahim & Dunn, 2003;Nino et al., 2003;Yanbin et al., 2008, among others). Other experimental methods intended to study resuspension under specific flows, such as the air flow generated by the foot during walking, or through porous medium, have also been introduced -examples can be found in a recent review by Henry & Minier (2014). These methods allow to quantify the resuspension rates and test models of the resuspension problem at large. For our purpose of studying the basics of the resuspension mechanism, and focusing on a single major difference between the smooth and rough surfaces, an experiment with a confined flow and particle motion, along with the detailed three-dimensional measurements, is required.
In order to achieve a quasi-static state, a steady vortex flow type was chosen, mimicking several industrial and environmental applications such as mixers, bio-reactors, tornadoes or dust devils. The low pressure, found at the center of a vortex-like swirling flow, creates a "suction" effect that generates high lift forces over submerged bodies. As a result, vortex flows present higher resuspension rates at a low level of energy input to the system, as compared to the unidirectional boundary layer type of flows. Moreover, the low pressure zone at the vortex core keeps the initial group of particles within a observation volume, thus allowing high fidelity measurements and significant statistics based on long and detailed observations to be collected for relatively small groups of test particles. Using a three-dimensional particle tracking velocimetry (3D-PTV) system, the particles' locations and velocities can be measured in time, and thus different aspects of their instantaneous and statistical behavior can be put under examination (e.g. Traugott et al., 2011).
Materials and methods
Experimental methods
The experimental set up is shown in figure 1. A 300 × 300 × 400mm 3 glass tank filled with filter water at room temperature (density ρ = 1000 kg m −3 and kinematic viscosity ν = 10 −6 m 2 s −1 ) up to 230 mm height (1). A Four blade rotor (2) rotates on a shaft of a stirrer (3) equipped with an angular velocity control (RD-03, MRC Inc.). At the tank bottom wall, the different roughness surfaces (4) were attached. Four high speed digital CMOS cameras (5) were placed around the tank, along with two LED lights (6). The digital video data was recorded to the RAM of a video recording unit and processed later using an open source particle tracking velocimetry software, OpenPTV (www.openptv.net). We tested four different angular velocities of 70, 100, 130 and 160 rpm and the bulk Reynolds number of this vortex type of flow is defined using the stirrer radius R, and the motor angular velocity ω, Re = ωR 2 /ν.
The corresponding Reynolds numbers tested are in the range of 13, 000 ÷ 30, 000. Each experiment was repeated at the steady state conditions, after the motor was running for at least five minutes with the particles in the tank, so the flow was allowed to reach a stable steady state vortex flow. After establishing the steady flow and resuspension conditions, a digital video sequence from different view angles was taken at a rate of 500 frames per second with the cameras focused at the bottom surface under the vortex core.
We used soda lime glass spherical particles d p = 800 µm in diameter and with the density of 2.5 g cm −3 . The "smooth" surface was made of a thin sheet of PVC with a layer of PVDF coating. This configuration yields a negligible surface roughness of Ra = 0.65 µm. The "rough" surface was made out of a sheet of 240 grit aluminium oxide sand paper, which yields a surface roughness of Ra = 14.24 µm.
Thus, relative to the particle diameter, the roughness Ra/d p is 0.81 × 10 −3 and 18.0 × 10 −3 , respectively.
Using 3D-PTV, we measured v ≈ 0.5 m s −1 at y = 400 µm, a height of one particle radius above the floor, from which the friction velocity u τ = 0.042 m s −1 , and the thickness of the viscous sub-layer of y = 120 µm can be estimated. Although this is a very rough estimate, it is clear that for both surface Prior to the dynamic flow driven experiments, the effect of surface roughness on the incipient motion of the particles through rolling/sliding was tested using an inclined bed test. After a number of particles were positioned on the surface, an angle of inclination was gradually increased and the angle corresponding to the first motion was measured. The test results indicate that over the rough surface, the force required for the incipient motion (in this experiment we could not distinguish between a rolling or sliding motion) is about 3 times larger than for the smooth surface.
Data reduction
The captured videos were analyzed using OpenPTV, an open source particle tracking velocimetry software (www.openptv.net) that through a process of calibration and image processing, provides the 3D locations of the particle in each video frame (Dracos, 1996;Traugott et al., 2011). An example of the video frame, seen from one of the four cameras at different stages of a particle resuspension, is shown in the top panel of figure 2. The obtained dataset provides us with the information of particle position, velocity and acceleration in time as Lagrangian trajectories, shown for instance in the bottom panel of figure 2. There is a single particle trajectory during resuspension, shown in a three dimensional view, along with the quantitative information regarding the elevation above the wall y, the velocity magnitude
V = v 2
x + v 2 y + v 2 z , and acceleration magnitude a, at different time instances. A sharp increase of the vertical velocity of the particle, which is still located on the wall (y ≈ d p /2), is identified as a resuspension event, if at the following time interval the particle is lifted from the wall. Applying this condition, we can identify the particles freely moving along the bottom wall (either rolling or sliding) without vertical (against gravity direction) motion, and distinguish those from the particles that were resuspended. An evolution of the particle motion in the given flow, shown in figure 3 is the following. As the motor is started and the vortex starts to form, particles start moving on the bottom wall and gather at a location beneath the vortex core. After a certain time, when the vortex reaches a steady state, the particles behave in a repetitive cyclic manner of resuspension in a helical trajectory that is followed by a fall back to the surface (deposition). A more detailed observation reveals that the resuspension occurs mostly when the particle is close to the vortex core, while as the particles are swept up, their distance from the vortex core increases, their velocity decreases below their own settling velocity and they fall back to the bottom wall. This behavior of leaving the center of the vortex is due to the large centrifugal forces acting on the particles as they are being swept by the vortex in helical trajectories. This observation is evident in figures 2 and 3. Integral quantities that characterize overall resuspension efficiency, such as the potential and kinetic energy of the particles (denoted E p and E k , respectively) were calculated and presented here as normalized quantities per unit mass (in order to remove the effect of a slightly different number of particles at different runs). The total energy was estimated accordingly, combining the potential and kinetic energy of the particle i at position y i and velocity V i :
E T = E P + E K = g y i + V 2 i 2(1)
For the sake of resuspension efficiency comparison, the dimensionless resuspension rate n * was defined using a ratio of the number of resuspended particles n r , to the total number of the particles in the frame, n T :
n * = 1 − n r n T(2)
In the next section, probability density functions of the resuspension rate, as well as the energy components will be presented. Moreover, we will show how the results help us to understand the differences between the resuspension rates from smooth and rough surfaces.
Results
Energy distributions
As a first measure for the resuspension efficiency of the system, the total energy (E T ) of the particles, calculated using Eq. 1, is considered. The probability density functions (obtained by the least-square fit to the measured histogram) are presented in figure 4(a), for the 100 rpm case, as an example. It can be seen that over the smooth surface, the distribution is denser and with a slightly lower energy level of the mode compared to the rough surface energy distribution. Moreover, although they seem to behave in a similar manner, the particles above the rough surface exhibit a wider distribution of the total energy with a longer tail and with stronger probability of higher energy. This behavior was seen for all cases studied here, and it is in agreement with the results of Soltani & Ahmadi (1995) that showed a lower threshold of critical shear velocity for the resuspension of particles on a rough surface, with surface roughness much smaller that the particle diameter, as the case studied here.
The mean values of particle total energy distributions are plotted for the increasing vortex angular velocity in figure 4(b). From the graph, a monotonic increase in total particle energy is visible in the range of 70 to 130 rpm or Reynolds of 13,000 to 30,000, and slightly reduced values for the highest velocity tested of Re ≈ 30, 000 (or 160 rpm). Although this decrease is slightly counter-intuitive (as one would expect to see a rise in particle energy for increasing values of the input energy), an explanation is due to the type of the flow and the description of particle motion given above: as the motor velocity is increased, the rise in angular velocity of the particles is related to stronger centrifugal forces acting on the particles, driving them farther away from the vortex core where the gravity forces are dominant and faster deposition. Thus, it can be derived that the vortex flow structure can not efficiently sustain particle suspension for all cases. This effect is evident in our results where the mean total particle energy reduced, while motor velocity in increased. Figure 4: (a) PDF for the particle kinetic and potential energy at 100 RPM motor velocities over the smooth and the rough surfaces. (b) mean particle energy values over the two surfaces with the changes in motor velocity With regard to the particle total energy, it is instructive to visualize the distributions of the components, namely the kinetic and potential energy of the particles. The mean energy components are presented in figure 5 with respect to the previously defined Reynolds number. From these graphs, we understand that the change in total energy with the motor velocity is solely due to changes in particle kinetic energy, while the average particle potential energy remains independent of the angular velocity of the vortex, yet higher for the rough surfaces (suggesting higher resuspension efficiency). Moreover, the values of particle energy are higher above the rough surface for all cases. In addition, as the Reynolds number grows, the kinetic energy level rises with higher rate for the rough surfaces, meaning that more energy is transferred to the resuspended particles. The decline of growth for the highest tested Reynolds number is also seen in this result. The effect is again due to higher centrifugal forces as the motor speed has increased.
A more detailed view is shown in the following figure 6, in which we demonstrate the horizontal and vertical (against gravity) particle kinetic energy components. Figure 7(b) presents the mean acceleration values for the increasing Reynolds number over the two surfaces in the horizontal and the vertical directions. From these two graphs, it is obvious that in both directions and for all Reynolds numbers tested, the particles over the rough surface experience significantly higher accelerations, and thus much stronger forces. This is true for both vertical and horizontal components, but with values much higher in the horizontal (parallel to the wall) direction.
The peculiar result is that there is almost no change in the accelerations for the particles over the smooth surface. It is also seen that although the horizontal accelerations (and centrifugal forces) increase with the increasing vortex speed, the vertical (up-lifting) forces are almost constant. This is consistent with the previous results of the vertical kinetic energy, and supports the assertion that the lift forces in this flow configuration are supported mostly by the pressure differences which are insensitive to the vortex angular velocity. Nevertheless, the important result for the purpose of this work is that both components of the acceleration are higher for the rough surface as compared to the smooth surface. This means that larger drag and lift forces are acting on the particles above the rough surface.
Entrainment rate
The most typical measure of a system resuspension efficiency is the fraction of particles that are entrained, or its complementary part, the fraction of particles that remain on the floor. This integral measure allows to quantify relative tendency for a given type of particles to become resuspended under different flow conditions. For each run of the experiment, the number of resuspended particles (n * ), and the total number of particles (n T ), were counted, and using the Eq. 2 the fraction of the particles that remain on the smooth or rough surface was obtained. Figure 8 shows the fraction remaining for each experimental run as a function of the increasing Reynolds number of the flow. It is seen from the graph that the particles above the smooth surface are more likely to remain on the tank bottom wall for all the vortex velocities tested. It is also evident that with the increasing vortex angular speed, a slightly larger portion remains on the bed. This is explained by much higher centrifugal forces that remove particles from the vortex core, and thus decrease their chances to experience vertical lift forces which are strongest at the vortex core. The trends also show some resemblance to the mean potential energy curves plotted in figure 5a. In examining the differences between figures 8 and 5a, we infer that while the potential energy levels remain roughly the same above the two surfaces, it is clear that the number of resuspended particles is higher above the rough surface. This means that above the rough surface more particles are resuspended as compared to the smooth surface case, but to slightly lower heights. It is again linked to the higher horizontal accelerations that the particles experience above the rough surfaces. The following discussion section is devoted to a simplified mechanistic model that explains the differences between the two types of surfaces.
Summary and discussion
In this work, the 3D-PTV was used to study the effect of surface roughness on the different modes of resuspension. The study focused on the detailed motion analysis of single solid particles from smooth and rough surfaces. In a quasi static vortex flow, the energy levels, the velocities, and the amount of resuspended particles were measured. The results were obtained through a 3D-PTV system in a simple vortex flow that provided us with the detailed information regarding the positions, velocities, and accelerations of particles to be collected for a long period of time. The results presented in this work show unequivocally that in the case of spherical particles with a diameter larger than the viscous sublayer, roughness improves the resuspension rate. It is important to note that this result is new both at the level of the detailed information about trajectories of the freely moving particles as they are suspended or deposited, and at the direct comparative study of the smooth versus rough surfaces under the same flow conditions. This experiment allows us to single out the mechanism by which the resuspension rate increases for the mobile beds over the rough surfaces.
We would like to propose a simplified conceptual mechanistic model that explains the differences of the spherical particles resuspension from smooth and rough surfaces. The model is relevant only for the particles large enough, such that the adhesion to the smooth surface is negligible, as compared to hydrodynamic forces. In this case, the spherical particle on a smooth surface can roll or slide, whereas on a rough surface this ability is impaired. A conceptual cartoon is shown in figure 9. The central concept is the ability of the particles to roll/slide on the surface. The major difference and the increased resuspension rate over the rough surface is therefore due to the strongly increased relative velocity between the moving particle and the flow, i.e. W = U − V p ( figure 9 top panel). Since the major hydrodynamic forces of lift and drag are both proportional to some power of the relative velocity L, D ∝ W α , which, depending on the particle Reynolds number, can be in the range of α = 1 ÷ 2, the increase of the forces can be up to the W 2 .
As one can see from our results of resuspension efficiency (figure 8), velocity and acceleration of the particles that are lifted up to the flow (figure 5), these resemble the effects predicted by this mechanistic model. Moreover, the model also predicts that at some energy level, the roughness cannot sustain the energy transferred to the particle by the flow, and thus the effect of increasing suspension rate is less prominent. In our case, this happened at the highest rotation rate shown here of 160 rpm, or Re ∼ 30000.
The higher rotation rates have shown the same trend -the velocities were too high and as the particle mobility along the wall was recovered, the resuspension rate decreased towards the smooth wall case. Also similar results concerning particle mobility over the surface were observed by Ibrahim & Dunn (2003) where it was shown that particles with a less spherical shape were more easily resuspended than their perfectly spherical counterparts (with higher degree of mobility).
V p U U L L D D Figure 9: Cartoon of the large spherical particle resuspension from smooth (left panel) versus rough (right panel) surfaces: (a) spherical particle under the flow from the left can roll or slide at the velocity comparable to the flow velocity, (b) same particle on the rough surface cannot roll or slide due to the roughness, (c) the very small relative velocity of the particle decreases significantly the drag and lift forces responsible for the resuspension; (d) lift and drag forces are enhanced due to a very high relative velocity which in this model is equal to the flow velocity.
In further support of our mechanistic model of strongly different forces acting on the particles, where mobility along the wall is hindered on the rough surfaces, we provide the direct measurements of the horizontal accelerations of the conditionally sampled subset of particles where their centers are within one radius from the wall. The results of the horizontal accelerations, i.e. forces, are shown in the following figure 10. Obviously, the distributions of the forces experienced by the particles on the rough surfaces are several times higher than those on the smooth surface, despite the exact same flow and other conditions of the experiments. We see this result as a direct support of the proposed model. It is also clear that the effect saturates at high velocities, as predicted by the model. We also would like to discuss the reasons for differences found in the results of several studies concerning the effect of surface roughness. Nino et al. (2003) and Lee & Balachandar (2012) have concluded that surface roughness increases the threshold shear velocity for particle resuspension (and reducing resuspension efficiency). It is important to note that in those studies the surface roughness size was taken to be at the order of the size of the particle diameter, such that the surface elements are hiding the particles from the flow. Another study by Soltani & Ahmadi (1995) showed lower resuspension threshold for particles over a surface with roughness much smaller than the particle diameter. These different effects can be seen in the results of Jiang et al. (2008), concluding that surface roughness affects particles of different relative (to the surface roughness and to the viscous sub-layer thickness) diameters in different ways.
The results of this study add to the existing knowledge regarding the resuspension mechanisms of freely moving particles over smooth and rough surfaces. The new results, based on the detailed measurements of the resuspended particles, allow for a better modeling and analysis of suspensions in industrial applications of fluidized beds, bio-reactors, mixers of solids, and environmental problems of dust devils or tornadoes.
Beyond the flow types that resemble the vortex model used in this study, we can understand better the observations of strong, stochastic suspension events in the turbulent boundary layers. In addition to the commonly accepted role of sweeps and ejections, we present a simple mechanistic model based on the inhomogeneous roughness of the surface that can provide an alternative and simpler explanation of the
Figure 1 :
1Schematic view of the experimental set up: (1) water tank (2) four-blade rotor, (3) overhead stirrer, (4) smooth or rough bed with the particles, (5) four high speed digital cameras and (6) two LED line light sources types, the surface roughness is significantly smaller than the viscous sub-layer thickness for the tested velocities.
Figure 2 :
2(top) Single particle at different stages of the resuspension event in a vortex flow; (bottom left) -isometric view of a single particle trajectory experiencing the resuspension event, (bottom right) -the height above the surface (top), the velocity (center) and acceleration (bottom) magnitude of the particle in time.For the statistics, a larger dataset of the separate particles at different time instances, disregarding their Lagrangian trajectories, is used. An example of the graph demonstrating the gross picture of the particles at all times in two experimental cases is presented infigure 3. The two orthogonal views on the particle motion under a tornado-like vortex of the same angular velocity are shown. These views allow quantification of the probability density functions of particle horizontal (top panel) or height (bottom panel) positions, and the respective distributions of velocities or accelerations.
Figure 3 :
3Particle position with 160 RPM angular velocity over (a) smooth surface side view and (b) top view (c) rough surface side view (d) top view
Figure 5 :
5Mean values of (a) potential and (b) kinetic energies for the different runs of the experiment. Symbols represent the measured values and dashed lines are trend lines added for clarity.
Figure 6 (
6a) shows the distribution of velocities magnitude in the vertical and the horizontal directions for the 100 rpm motor velocity run over the smooth and the rough surfaces, while figure 6(b) shows the values of the mean particle kinetic energy in the vertical and the horizontal directions for increasing Reynolds numbers. Naturally, due to the type of the vortex flow, the particle kinetic energy is an order of magnitude larger in the horizontal relative to the vertical directions. It is also evident that over the rough surface the particles exhibit higher energy levels, with the trend similar for both vertical and horizontal components of energy. Moreover, the mean values of the velocity rise with the angular velocity of the vortex, again only up to 130 rpm. Although this behavior exists in both directions it is much more prominent in the horizontal direction.
Figure 6 :
6(a) kinetic energy distributions in the horizontal and the vertical directions for the 100 RPM motor velocity run over the smooth and the rough surfaces (b) mean kinetic energy in the vertical and horizontal directions over the smooth and the rough surfaces for the different values of motor velocity3.2 Particle accelerations3D-PTV allows us to analyze the particle material acceleration along trajectories in space and time, providing insight regarding the forces acting on the particles. Since the fluid velocity around the particles is unknown, it is impossible to decompose our results to the hydrodynamic forces of lift and drag.Nevertheless, it is useful to examine the particles accelerations and decompose it to the horizontal and the vertical components, since in the case of resuspension, the vertical component directly relates to the lift force, as long as the particle moves on the surface.Figure 7(a) shows the acceleration probability distribution for the 100 RPM run over the smooth and the rough surfaces in the vertical and the horizontal components.
Figure 7 :
7Acceleration PDF in the horizontal and vertical directions for the smooth and the rough surfaces for the 100 RPM experiment; (b) mean particle accelerations in the horizontal and the vertical directions over the two different surfaces for the changing values of motor velocity
Figure 8 :
8Fraction of particles remaining on the bottom wall for the different values of motor velocity over the smooth and the rough surfaces.
Figure 10 :
10Horizontal acceleration component of the particles on smooth and rough surfaces.
Single particle resuspension experiments in turbulent channel flows. T Barth, J Preuss, X Muller, X Hampel, Journal of Aerosol Science. 71Barth, T., Preuss, J., Muller, X., & Hampel, X. (2014). Single particle resuspension experiments in turbulent channel flows. Journal of Aerosol Science, 71, 4-51.
Three-Dimensional Velocity and Vorticity Measuring and Image Analysis Technique: Lecture Notes from the short course. T H Dracos, Kluwer Academic PublisherZurich, SwitzerlandDracos, T. H. (1996). Three-Dimensional Velocity and Vorticity Measuring and Image Analysis Technique: Lecture Notes from the short course held in Zurich, Switzerland. Kluwer Academic Publisher.
Martian dust devils: Directions of movement inferred from their tracks. R Greeley, P L Whelley, L D V Neakrase, Geophysical Research Letters. 31Greeley, R., Whelley, P. L., & Neakrase, L. D. V. (2004). Martian dust devils: Directions of movement inferred from their tracks. Geophysical Research Letters, 31.
Re-entrainment of particles from powder structure: experimental investigations. K Grzybowski, L Gradon, Advanced Powder Technology. 18Grzybowski, K. & Gradon, L. (2007). Re-entrainment of particles from powder structure: experimental investigations. Advanced Powder Technology, 18, 427-439.
Measurements of the mean force on a particle near a boundary in turbulent flow. D Hall, Journal of Fluid Mechanics. 187Hall, D. (1988). Measurements of the mean force on a particle near a boundary in turbulent flow. Journal of Fluid Mechanics, 187, 451-466.
Progress in particle resuspension from rough surfaces by turbulent flows. C Henry, J.-P Minier, Progress in Energy and Combustion Science. 45Henry, C. & Minier, J.-P. (2014). Progress in particle resuspension from rough surfaces by turbulent flows. Progress in Energy and Combustion Science, 45, 1-53.
Numerical study on the adhesion and reentrainment of nondeformable particles on surfaces: The role of surface roughness and electrostatic forces. C Henry, J.-P Minier, G Lefevre, Langmuir. 28Henry, C., Minier, J.-P., & Lefevre, G. (2012). Numerical study on the adhesion and reentrainment of nondeformable particles on surfaces: The role of surface roughness and electrostatic forces. Langmuir, 28, 438-452.
Microparticle detachment from surfaces exposed to turbulent air flow: controlled experiments and modeling. A Ibrahim, P Dunn, Aerosol Science. 34Ibrahim, A. & Dunn, P. (2003). Microparticle detachment from surfaces exposed to turbulent air flow: controlled experiments and modeling. Aerosol Science, 34, 765-782.
Characterizing the effect of substrate surface roughness on particlewall interaction with the airflow method. Y Jiang, S Matsusaka, H Masuda, Y Qian, Powder Technology. 1863Jiang, Y., Matsusaka, S., Masuda, H., & Qian, Y. (2008). Characterizing the effect of substrate surface roughness on particlewall interaction with the airflow method. Powder Technology, 186(3), 199-205.
Critical shear stress for incipient motion of a particle on a rough bed. H Lee, S Balachandar, Journal of Geophysical Research -Earth Surface. 117Lee, H. & Balachandar, S. (2012). Critical shear stress for incipient motion of a particle on a rough bed. Journal of Geophysical Research -Earth Surface, 117.
Threshold for particle entrainment into suspension. Y Nino, F Lopez, M Garcia, Sedimentology. 502Nino, Y., Lopez, F., & Garcia, M. (2003). Threshold for particle entrainment into suspension. Sedimen- tology, 50(2), 247-263.
Particle detachment from rough surfaces in turbulent flows. M Soltani, G Ahmadi, Adhesion. 51Soltani, M. & Ahmadi, G. (1995). Particle detachment from rough surfaces in turbulent flows. Adhesion, 51, 105-123.
Resuspension of particles in an oscillating grid turbulent flow using piv and 3d-ptv. H Traugott, T Hayse, A Liberzon, Journal of Physics: Conference Series. 318Traugott, H., Hayse, T., & Liberzon, A. (2011). Resuspension of particles in an oscillating grid turbulent flow using piv and 3d-ptv. Journal of Physics: Conference Series, 318.
Rolling and lifting probabilities for sediment entrainment. F Wu, Y Chou, Journal of Hydraulic Engineering. 129Wu, F. & Chou, Y. (2013). Rolling and lifting probabilities for sediment entrainment. Journal of Hydraulic Engineering, 129, 110-119.
Characterizing the effect of substrate surface roughness on particle-wall interaction with the airflow method. J Yanbin, S Matsusaka, H Masuda, Y Qian, Powder Technology. 186Yanbin, J., Matsusaka, S., Masuda, H., & Qian, Y. (2008). Characterizing the effect of substrate surface roughness on particle-wall interaction with the airflow method. Powder Technology, 186, 199-205.
Particle resuspension from surfaces: Revisited and re-evaluated. G Ziskind, Reviews in Chemical Engineering. 221-2Ziskind, G. (2006). Particle resuspension from surfaces: Revisited and re-evaluated. Reviews in Chemical Engineering, 22(1-2), 1-123.
| [] |
[
"arXiv:0711.4477v1 [quant-ph] Three-tangle for mixtures of generalized GHZ and generalized W states",
"arXiv:0711.4477v1 [quant-ph] Three-tangle for mixtures of generalized GHZ and generalized W states"
] | [
"C Eltschka \nInstitut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n",
"A Osterloh \nInstitut für Theoretische Physik\nLeibniz Universität Hannover\nD-30167HannoverGermany\n",
"J Siewert \nInstitut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany\n",
"A Uhlmann \nInstitut für Theoretische Physik\nUniversität Leipzig\nD-04109LeipzigGermany\n"
] | [
"Institut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany",
"Institut für Theoretische Physik\nLeibniz Universität Hannover\nD-30167HannoverGermany",
"Institut für Theoretische Physik\nUniversität Regensburg\nD-93040RegensburgGermany",
"Institut für Theoretische Physik\nUniversität Leipzig\nD-04109LeipzigGermany"
] | [] | We give a complete solution for the three-tangle of mixed three-qubit states composed of a generalized GHZ state, a |000 + b |111 , and a generalized W state, c |001 + d |010 + f |100 . Using the methods introduced by Lohmayer et al. we provide explicit expressions for the mixed-state three-tangle and the corresponding optimal decompositions for this more general case. Moreover, as a special case we obtain a general solution for a family of states consisting of a generalized GHZ state and an orthogonal product state. | 10.1088/1367-2630/10/4/043014 | [
"https://arxiv.org/pdf/0711.4477v1.pdf"
] | 67,769,361 | 0711.4477 | c8c6473078eb3753f419a0ebefff3f65127a90c7 |
arXiv:0711.4477v1 [quant-ph] Three-tangle for mixtures of generalized GHZ and generalized W states
28 Nov 2007
C Eltschka
Institut für Theoretische Physik
Universität Regensburg
D-93040RegensburgGermany
A Osterloh
Institut für Theoretische Physik
Leibniz Universität Hannover
D-30167HannoverGermany
J Siewert
Institut für Theoretische Physik
Universität Regensburg
D-93040RegensburgGermany
A Uhlmann
Institut für Theoretische Physik
Universität Leipzig
D-04109LeipzigGermany
arXiv:0711.4477v1 [quant-ph] Three-tangle for mixtures of generalized GHZ and generalized W states
28 Nov 2007numbers: 0367-a0367Mn0365Ud
We give a complete solution for the three-tangle of mixed three-qubit states composed of a generalized GHZ state, a |000 + b |111 , and a generalized W state, c |001 + d |010 + f |100 . Using the methods introduced by Lohmayer et al. we provide explicit expressions for the mixed-state three-tangle and the corresponding optimal decompositions for this more general case. Moreover, as a special case we obtain a general solution for a family of states consisting of a generalized GHZ state and an orthogonal product state.
Introduction
The occurrence of entanglement in multipartite systems is one of the most important and distinctive features in quantum theory [1,2]. With the ever-increasing number of applications of entanglement, its quantification has become one of the foremost topics in contemporary quantum information research.
While entanglement of pure and mixed states of two qubits is already well understood [3][4][5][6][7], to date there is no generally accepted theory for classification and quantification of entanglement in multipartite qubit systems. For three-qubit systems, numerous interesting results have been found [8][9][10][11][12][13][14][15][16][17][18][19]. A complete SLOCC characterization of three-qubit entanglement has been achieved only for pure states [9,10]. It leads to a schematic characterization for mixed states [14]. A crucial concept for this is the so-called three-tangle, a polynomial invariant for three-qubit states that quantifies the three-partite entanglement contained in a pure three-qubit state (the three-tangle is equal to the modulus of the hyperdeterminant [20,21]). However, even for the simplest case of rank-2 mixed states, no general expression is known for its three-tangle.
Recently, Lohmayer et al. [18] have provided an analytic quantification of the threetangle for a representative family of rank-2 three-qubit states, namely for mixtures of a symmetric GHZ state and an orthogonal symmetric W state. In this article we show that by applying the methods of [18,22] these results can be extended to rank-2 mixtures of a generalized GHZ state and an orthogonal generalized W state. This article is organized as follows. In Section 2, we introduce some basic terminology and give a precise formulation of the problem whose general solution we outline in Section 3. In Section 4 we discuss special cases of this solution, in particular we find the three-tangle for rank-2 mixtures of generalized GHZ states and certain orthogonal product states.
Notations and formulation of the problem
Consider the state |ψ in a three-qubit Hilbert space |ψ ∈ H A ⊗H B ⊗H C . Its coefficients with respect to a basis of product states (the 'computational basis') are ψ jkl = jkl|ψ , j, k, l ∈ {0, 1}. An important measure for the entanglement in pure three-qubit states is the three-tangle (or residual tangle) introduced in [9]. The three-tangle of |ψ is a so-called polynomial invariant [23,24] and can be written in terms of the coefficients ψ ijk as
τ 3 (ψ) = 4|d 1 − 2d 2 + 4d 3 |(1)
d 1 = ψ 2 000 ψ 2 111 + ψ 2 001 ψ 2 110 + ψ 2 010 ψ 2 101 + ψ 2 100 ψ 2 011 d 2 = ψ 000 ψ 111 ψ 011 ψ 100 + ψ 000 ψ 111 ψ 101 ψ 010 + ψ 000 ψ 111 ψ 110 ψ 001 + ψ 011 ψ 100 ψ 101 ψ 010 + ψ 011 ψ 100 ψ 110 ψ 001 + ψ 101 ψ 010 ψ 110 ψ 001 d 3 = ψ 000 ψ 110 ψ 101 ψ 011 + ψ 111 ψ 001 ψ 010 ψ 100 .
The three-tangle of a mixed state
ρ = j p j π j , π j = |φ j φ j | φ j |φ j(2)
can be defined as convex-roof extension [25] of the pure state three-tangle,
τ 3 (ρ) = min decompositions j p j τ 3 (π j ).(3)
A given decomposition {q k , π k : ρ = k q k π k } with τ 3 (ρ) = k q k τ 3 (π k ) is called optimal. We note that τ 3 (ρ) is a convex function on the convex (and compact) set Ω of density matrices ρ.
In this paper, we determine three-tangle and optimal decompositions for the family of mixed three-qubit states
ρ(p) = p |gGHZ a,b gGHZ a,b | + (1 − p) |gW c,d,f gW c,d,f |(4)
composed of a generalized GHZ state
|gGHZ a,b = a |000 + b |111 , |a| 2 + |b| 2 = 1(5)
and a generalized W state
|gW c,d,f = c |001 + d |010 + f |100 , |c| 2 + |d| 2 + |f | 2 = 1 .(6)
We note that τ 3 (gW c,d,f ) = 0 and τ gGHZ 3 := τ 3 (gGHZ a,b ) = 4|a 2 b 2 |. For the symmetric GHZ and W state (a = b = 1/ √ 2 and c = d = f = 1/ √ 3) the problem and results of [18] are recovered.
The generic case
In this section it is assumed that none of the coefficients is zero, i.e. a, b, c, d, f = 0. The opposite case corresponds to either a rank-2 mixture of a generalized GHZ and a biseparable state, or to a mixture of a generalized W and a completely factorized state and will be studied in the next section.
In the following, we will apply the methods developed in [18,22]. There it was shown that in order to find the convex roof of an entanglement measure for rank-2 mixed states it is useful to study the pure states that are superpositions of the eigenstates of ρ
|p, ϕ = √ p |gGHZ a,b − 1 − p e iϕ |gW c,d,f .(7)
The three-tangle of these states is
τ 3 (p, ϕ) = 4 p 2 a 2 b 2 − 4 p(1 − p) 3 e 3iϕ bcdf .(8)
The phases of the coefficients in |gGHZ a,b and |gW c,d,f merely produce different offsets for the relative phase ϕ in the expression for the three-tangle, Eq. (8). Therefore it suffices to consider the case where all coefficients are positive real numbers. In the following, it will be beneficial to introduce the definition
s = 4cdf a 2 b > 0 .(9)
If we factor out the three-tangle τ gGHZ 3 of the generalized GHZ state, the three-tangle of the superposition (7) can be written as
τ 3 (p, ϕ) = τ gGHZ 3 p 2 − p(1 − p) 3 e 3iϕ s .(10)
Since τ gGHZ 3 is just a constant factor, the behaviour of this function of p and ϕ is completely determined by the value of the parameter s.
As a first step, we identify the zero-simplex containing all mixed states ρ(p) with τ 3 (ρ(p)) = 0. Its corner states are obtained as the zeros of Eq. (10). One obvious solution is p = 0, which corresponds to a pure generalized W state. Therefore, in the calculation of the other solutions we can assume p > 0 and the zeros are determined by
p 3 = (1 − p) 3 e 3iϕ s .(11)
Since p and s are real and positive, this implies ‡ ϕ = n 2π 3 , n ∈ N .
(12) ‡ Note that the 2π/3-periodicity is due to the fact that this relative phase is induced by the local transformation diag{exp(i2π/3), 1} on each qubit.
For p, we then get the solution
p 0 = s 2/3 1 + s 2/3 = 3 16c 2 d 2 f 2 3 √ a 4 b 2 + 3 16c 2 d 2 f 2 .(13)
This means that in addition to the state |gW c,d,f the three-tangle vanishes for |p 0 , n · 2π/3 , n = 0, 1, 2. All mixed states whose density matrices are convex combinations of those four states have zero three-tangle. On the Bloch sphere with gGHZ and gW at its poles, this corresponds to a simplex with those four states at the corners. All ρ(p) with p < p 0 are inside this set, and therefore τ 3 (ρ(p)) = 0 for 0 ≤ p ≤ p 0 . In order to determine the mixed three-tangle of ρ(p) for p > p 0 , we note that for any fixed p, τ 3 (p, ϕ) takes a minimum at ϕ 0 = 0 which due to the symmetry of τ 3 is repeated at ϕ 1 = 2π/3 and ϕ 2 = 4π/3. Consequently, for any value of p the state ρ(p) can be decomposed into the three states |p, ϕ i , i = 0, 1, 2. Therefore the characteristic curve τ 3 (p, 0) is an upper bound to τ 3 (ρ(p)). Moreover is it known to give the correct values for the three-tangle at p = p 0 (at the top face of the zero simplex) and p = 1 (ρ(1) = |gGHZ a,b gGHZ a,b |). However, if there is a range of values where τ 3 (p, 0) is a concave function, there are decompositions for ρ(p) with a lower average threetangle [18]. Therefore it is important to examine where the function τ 3 (p, 0) is concave for p ≥ p 0 .
For ϕ = 0 and p ≥ p 0 , the term inside the absolute value bars in (10) is real and positive, and the characteristic curve τ 3 (p, 0) is equal to
t(p) = τ gGHZ 3 · (p 2 − p(1 − p) 3 s) .(14)
Concavity of t(p) is indicated by a negative sign of its second derivative
t ′′ (p) = τ gGHZ 3 2 − 8p 2 − 4p − 1 4p p(1 − p) s .(15)
The limit p → 1 (p = 1 − ε) in (15) gives
t ′′ (1 − ε) = − 3τ gGHZ 3 s 4 √ ε + 2τ gGHZ 3 + O(ε 1/2 ) ,(16)
that is, t(p) is concave close to p = 1. On the other hand, for small p
t ′′ (p) = τ gGHZ 3 s 4p 3/2 + O(p −1/2 ) .(17)
That is, close to p = 0 we find that t(p) is convex (note that due to the absolute value, τ 3 (p, 0) is actually concave close to p = 0). Due to continuity, there must be at least one zero of t ′′ (p) in between. Moreover we note that the third derivative
t ′′′ (p) = −3τ gGHZ 3 s 8p 2 p(1 − p) 3 ≤ 0(18)
is negative for all values of p. Thus t ′′ (p) is strictly monotonous and has precisely one zero, implying that t(p) is convex before and concave after that point. As the mixed state three-tangle is convex, the characteristic curve needs to be convexified where it is concave in the interval [p 0 , 1]. Since the concavity extends up to p = 1, corresponding to the state |gGHZ a,b , that state has to be part of the optimal decomposition [22] in this interval. The symmetry and the results in [18] suggest that a good ansatz for the optimal decomposition is
ρ(p) = α |gGHZ a,b gGHZ a,b | + 1 − α 3 2 k=0 p 1 , k · 2π 3 p 1 , k · 2π 3(19)
where p 1 is chosen such that the mixed-state three-tangle becomes minimal. The value of α is fixed by p and p 1 :
α = p − p 1 1 − p 1 .(20)
The average three-tangle for this decomposition is (p > p 0 )
τ conv 3 (p, p 1 ) = p − p 1 1 − p 1 · τ gGHZ 3 + 1 − p 1 − p 1 · t(p 1 ) .(21)
This describes a linear interpolation between τ 3 (p 1 , 0) and τ gGHZ 3 . Note that for p < p 1 , (19) ceases to be a valid decomposition because α becomes negative.
To find the minimum in p 1 for given p, we look for the zeros of the derivative ∂τ conv 3 /∂p 1 . The resulting equation has the solution p noabs
1 = 1 2 + 1 2 √ 1 + s 2 .(22)
Note that for s > 2 √ 2 we get p noabs 1 < p 0 . In that case the minimum is reached at the border p 1 = p 0 of the considered interval [p 0 , 1], and therefore
p 1 = max{ p 0 , 1 2 + 1 2 √ 1 + s 2 } .(23)
Putting it all together, we present the central result of this article
τ 3 (ρ(p)) = 0 for 0 ≤ p ≤ p 0 τ 3 (p, 0) for p 0 ≤ p ≤ p 1 τ conv 3 (p, p 1 ) for p 1 ≤ p ≤ 1(24)
where p 0 is given by (13), p 1 by (23), τ 3 (p, 0) by (8) and τ conv 3 (p, p 1 ) by (21). The corresponding optimal decompositions are
ρ(p) = p p 0 ρ ∆ (p 0 ) + p 0 − p p 0 π gW for 0 ≤ p ≤ p 0 ρ ∆ (p) for p 0 ≤ p ≤ p 1 1 − p 1 − p 1 ρ ∆ (p 1 ) + p − p 1 1 − p 1 π gGHZ for p 1 ≤ p ≤ 1 (25) where ρ ∆ (p) = 1 3 2 k=0 p, k · 2π 3 p, k · 2π 3(26)
and π j as defined in (2) The curve (24) is convex, and for all p and ϕ: τ 3 (ρ(p)) ≤ τ 3 (p, ϕ). Therefore it is a lower bound to the three-tangle of ρ(p). On the other hand, for each p we have given an explicit decomposition realizing this lower bound. Thus it represents also an upper bound and hence coincides with the three-tangle of ρ(p).
Special cases
In this section we will discuss various special cases of our general solution (24).
First, we briefly demonstrate that the results for the symmetric GHZ state and the symmetric W state in [18] are reproduced. Indeed, the general behaviour described in Section 3 (that is, analytic properties of the three-tangle, optimal decompositions) matches the one found in [18], so we only have to check the values of p 0 and p 1 . In the symmetric case we have
a = b = 1/ √ 2 and c = d = f = 1/ √ 3, resulting in s = 2 7/2 3 3/2 .(27)
Inserting this in (13) and (23) leads to
p 0 = 2 7/3 /3 1 + 2 7/3 /3 = 4 3 √ 2 3 + 4 3 √ 2 (28) p 1 = 1 2 + 1 2 1 + 2 7 /3 3 = 1 2 + 3 2 3 155
(29) as found in [18]. Next, we consider the limiting cases where at least one of the coefficients is 0. Those require extra care as the calculations above have been done under the assumption of non-vanishing coefficients. However, since we are dealing with continuous functions, one should expect that the results still apply, although possibly in a degenerate form.
The first case we consider is when the generalized GHZ state degenerates into a pure three-party product state. This corresponds to the limit s → ∞. However note that at the same time τ gGHZ 3 → 0 such that (10) remains regular. This can be seen by looking at the explicit form (8). It is clear that in this case τ 3 (ρ(p)) = 0 for all p.
There are two non-equivalent ways to achieve this. One possibility is b = 0 which reduces the generalized GHZ state to |000 . In this case, the three-tangle (8) vanishes for all superpositions (7), and therefore also all mixed states anywhere inside the Bloch sphere have vanishing three-tangle.
The other way to get s → ∞ is a = 0 where the generalized GHZ state is reduced to |111 . While ρ(p) as a mixture of product and gW state again has no three-tangle, unlike in the case b = 0 the three-tangle does not vanish everywhere on the Bloch sphere. Equation (8) reduces to
τ 3 (p, ϕ) = 16 p(1 − p) 3 cdf,(30)
which is independent of ϕ and concave for all p ∈ [0, 1]. Thus the zero simplex degenerates into a zero axis. As long as cdf > 0, outside of this axis the three-tangle never vanishes. If both a = 0 and cdf = 0, the three-tangle is zero everywhere inside the Bloch sphere.
The opposite limiting case is s = 0, that is, when at least one of the coefficients c, d, f vanishes. Note that for the three-tangle it does not matter whether only one of them vanishes, resulting in a product of a single qubit state with a generalized Bell state, or two of them, resulting in a product of three single-qubit states: In all cases (10) reduces to
τ 3 (p, ϕ) = τ gGHZ 3 · p 2 ,(31)
which is convex for all p ∈ [0, 1]; indeed, (13) and (23) yield p 0 = 0 and p 1 = 1 at s = 0. Consequently,
τ 3 (ρ(p)) = τ gGHZ 3 · p 2(32)
for all p. Even more, τ 3 (ρ) = τ gGHZ 3 p 2 for any mixed state ρ inside the Bloch sphere with gGHZ a,b | ρ |gGHZ a,b = p. We would like to point out that this result reminds of the situation both for two-qubit superpositions [26] and for two-qubit mixtures of an arbitrary entangled state and an orthogonal product state.
Conclusion
In this paper, we have given explicit expressions for the three-tangle of mixtures ρ(p) according to (4) of arbitrary generalized GHZ and orthogonal generalized W states, including the limiting cases where those states are reduced to product states. We have found that the qualitative pattern described in [18] for mixtures of symmetric GHZ and W states holds also more generally. Up to a certain value p 0 given by (13), the mixed three-tangle vanishes. The optimal decomposition for those states consists of the pure states (7) for which the three-tangle is zero. One is always the generalized W state at the bottom of the Bloch sphere. The other three form an equilateral horizontal triangle at the height of p 0 . Note that those states do not depend on p as long as p ≤ p 0 .
For p > p 0 , there may follow a region up to some value p 1 given by Eq. (23), where the mixed state three-tangle follows the minimal pure state three-tangle (10) with the same value for p (which for positive real coefficients is achieved at ϕ = 0). In this region, the optimal decomposition consists of the three states with this property, which form a horizontal eqilateral triangle with corners on the Bloch sphere and ρ(p) in the center. If s ≥ 2 √ 2, p 1 and p 0 coincide and this region with "leaves" of constant three-tangle in the convex roof (cf. [18], Figure 2) is absent. This can be viewed as contraction of this middle region into one point.
For p > p 1 , the three-tangle grows linearly up to its maximum value at p = 1. The optimal decomposition in this case consists of the three pure superposition states for p = p 1 with minimal three-tangle and the generalized GHZ state. That is, the convex roof in the Bloch sphere is affine for an entire simplex whose corners are given by the four pure states that form the optimal decomposition. Moreover, we have demonstrated how the results of this work connect to the findings for the special case of mixtures of a symmetric GHZ and a symmetric W state [18].
In principle, the scheme of three regions for p values as outlined above holds also in the limiting cases when some of the coefficients in the states vanish, except that in this situation the "outer regions" may shrink away. A common feature of these limits is a ϕ-independent characteristic curve. If the generalized GHZ state degenerates into a product state, τ 3 (ρ(p)) = 0 for all p. On the other hand, for s = 0 (i.e., at least one of the coefficients in the generalized W states vanishes), both "outer" affine regions disappear and the whole range of p is covered by the "middle region" with a strictly convex characteristic curve. This case corresponds to a mixture of a generalized GHZ state and an orthogonal product state and the exact convex roof of the three-tangle is obtained everywhere inside the Bloch sphere.
Figure 1 .
1Three0396. The solid line is the minimal pure state tangle, τ 3 (p, 0) (10). The short-dashed line is t(p) (14). The dotted vertical lines show the positions of p 0 (13), p noabs 1 (22) and p 1 (23), and the thick dashed line gived the resulting mixed threetangle τ 3 (ρ(p)) (24). In addition, the first figure shows as dotted line the curve which would result from using p noabs 1 instead of p 1 in (21).
AcknowledgmentsWe acknowledge interesting discussion with Géza Tóth. This work was supported by the Sonderforschungsbereich 631 of the German Research Foundation. JS. receives support from the Heisenberg Programme of the German Research Foundation.
. M Plenio, S Virmani, Quant. Inf. Comp. 71Plenio M and Virmani S 2007 Quant. Inf. Comp. 7 1
. R Horodecki, P Horodecki, M Horodecki, K Horodecki, ArXiv:quant-ph/0702225Horodecki R, Horodecki P, Horodecki M and Horodecki K 2007 ArXiv:quant-ph/0702225
. R Werner, Phys. Rev. A. 404277Werner R 1989 Phys. Rev. A 40 4277
. C H Bennett, H Bernstein, S Popescu, B Schumacher, Phys. Rev. A. 532046Bennett C H, Bernstein H, Popescu S and Schumacher B 1996 Phys. Rev. A 53 2046
. C H Bennett, D Divincenzo, J Smolin, W K Wootters, Phys. Rev. A. 543824Bennett C H, DiVincenzo D, Smolin J A and Wootters W K 1996 Phys. Rev. A 54 3824
. W Wootters, Phys. Rev. Lett. 80Wootters W 1998 Phys. Rev. Lett. 80 2245-2248
. W Wootters, Quant. Inf. Comp. 1Wootters W 2001 Quant. Inf. Comp. 1 27-44
. W Dür, J Cirac, R Tarrach, Phys. Rev. Lett. 833562Dür W, Cirac J and Tarrach R 1999 Phys. Rev. Lett. 83 3562
. V Coffman, J Kundu, W K Wootters, Phys. Rev. A. 6152306Coffman V, Kundu J and Wootters W K 2000 Phys. Rev. A 61 052306
. W Dür, G Vidal, J I Cirac, Phys. Rev. A. 6262314Dür W, Vidal G and Cirac J I 2000 Phys. Rev. A 62 062314
. A Acín, A A L Costa, Jane E Latorre, J Tarrach, R , Phys. Rev. Lett. 851560Acín A, Costa A A L, Jane E, Latorre J and Tarrach R 2000 Phys. Rev. Lett. 85 1560
. H Carteret, A Sudbery, J. Phys. A: Math. Gen. 334981Carteret H and Sudbery A 2000 J. Phys. A: Math. Gen. 33 4981
. A Acín, A Andrianov, Jane E Tarrach, R , J. Phys. A. 346725Acín A, Andrianov A, Jane E and Tarrach R 2001 J. Phys. A 34 6725
. A Acín, D Bruss, M Lewenstein, A Sanpera, Phys. Rev. Lett. 8740401Acín A, Bruss D, Lewenstein M and Sanpera A 2001 Phys. Rev. Lett. 87 040401
. T C Wei, P M Goldbart, Phys. Rev. A. 6842307Wei T C and Goldbart P M 2003 Phys. Rev. A 68 042307
. P Levay, Phys. Rev. A. 7112334Levay P 2005 Phys. Rev. A 71 012334
. C S Yu, H S Song, Phys. Rev. A. 7332322Yu C S and Song H S 2006 Phys. Rev. A , 73 032322
. R Lohmayer, A Osterloh, J Siewert, A Uhlmann, Phys. Rev. Lett. 97260502Lohmayer R, Osterloh A, Siewert J and Uhlmann A 2006 Phys. Rev. Lett. 97 260502
. B Röthlisberger, J Lehmann, D Saraga, P Traber, D Loss, ArXiv:0705.1710Röthlisberger B, Lehmann J, Saraga D, Traber P and Loss D 2007 ArXiv:0705.1710
. Caley A 1845, Cambridge Math. J. 4193Caley A 1845 Cambridge Math. J. 4 193
. A Miyake, Phys. Rev. A. 6712108Miyake A 2003 Phys. Rev. A 67 012108
. A Osterloh, J Siewert, A Uhlmann, ArXiv:0710.5909Osterloh A, Siewert J and Uhlmann A 2007 ArXiv:0710.5909
. F Verstraete, Dehaene J Moor, B D , Phys. Rev. A. 6812103Verstraete F, Dehaene J and Moor B D 2003 Phys. Rev. A 68 012103
. M S Leifer, Linden N Winter, A , Phys Rev A. 6952304Leifer M S, Linden N and Winter A 2004 Phys Rev A 69 052304
. A Uhlmann, Phys. Rev. A. 6232307Uhlmann A 2000 Phys. Rev. A 62 032307
. A F Abouraddy, B E A Saleh, A Sergienko, M C Teich, Phys. Rev. A. 6450101RAbouraddy A F, Saleh B E A, Sergienko A V and Teich M C 2001 Phys. Rev. A 64 050101(R)
| [] |
[
"Semi-fragile Tamper Detection and Recovery based on Region Categorization and Two-Sided Circular Block Dependency",
"Semi-fragile Tamper Detection and Recovery based on Region Categorization and Two-Sided Circular Block Dependency"
] | [
"Seyyed Hossein Soleymani \nDepartment of Computer Engineering\nFaculty of engineering\nFerdowsi University of Mashhad\nMashhadIran\n",
"Amir Hossein Taherinia \nDepartment of Computer Engineering\nFaculty of engineering\nFerdowsi University of Mashhad\nMashhadIran\n"
] | [
"Department of Computer Engineering\nFaculty of engineering\nFerdowsi University of Mashhad\nMashhadIran",
"Department of Computer Engineering\nFaculty of engineering\nFerdowsi University of Mashhad\nMashhadIran"
] | [] | This paper presents a new semi-fragile algorithm for image tamper detection and recovery, which is based on region attention and two-sided circular block dependency. This method categorizes the image blocks into three categories according to their texture. In this method, less information is extracted from areas with the smooth texture, and more information is extracted from areas with the rough texture. Also, the extracted information for each type of blocks is embedded in another block with the same type. So, changes in the smooth areas are invisible to Human Visual System. To increase the localization power a two-sided circular block dependency is proposed, which is able to distinguish partially destroyed blocks. Pairwise block dependency and circular block dependency, which are common methods in the block-based tamper detection, are not able to distinguish the partially destroyed blocks. Cubic interpolation is used in order to decrease the blocking effects in the recovery phase. The results of the proposed method for regions with different texture show that the proposed method is superior to non-region-attention based methods. | null | [
"https://arxiv.org/pdf/1804.02680v1.pdf"
] | 4,711,064 | 1804.02680 | 332d36af29ad2a5af2010212c80e917a8533b5ec |
Semi-fragile Tamper Detection and Recovery based on Region Categorization and Two-Sided Circular Block Dependency
Seyyed Hossein Soleymani
Department of Computer Engineering
Faculty of engineering
Ferdowsi University of Mashhad
MashhadIran
Amir Hossein Taherinia
Department of Computer Engineering
Faculty of engineering
Ferdowsi University of Mashhad
MashhadIran
Semi-fragile Tamper Detection and Recovery based on Region Categorization and Two-Sided Circular Block Dependency
WatermarkingSemi-fragileTamper detectionTamper recoveryRegion based
This paper presents a new semi-fragile algorithm for image tamper detection and recovery, which is based on region attention and two-sided circular block dependency. This method categorizes the image blocks into three categories according to their texture. In this method, less information is extracted from areas with the smooth texture, and more information is extracted from areas with the rough texture. Also, the extracted information for each type of blocks is embedded in another block with the same type. So, changes in the smooth areas are invisible to Human Visual System. To increase the localization power a two-sided circular block dependency is proposed, which is able to distinguish partially destroyed blocks. Pairwise block dependency and circular block dependency, which are common methods in the block-based tamper detection, are not able to distinguish the partially destroyed blocks. Cubic interpolation is used in order to decrease the blocking effects in the recovery phase. The results of the proposed method for regions with different texture show that the proposed method is superior to non-region-attention based methods.
Introduction
Watermarking can be categorized into fragile, semifragile and robust watermarking [1]. In fragile watermarking, the embedded watermark will be destroyed after both of intentional attacks (such as image cropping, image copymove forgery, and other image tampering operations) and unintentional attacks (such as image compression and image enhancement operations) [2,3,4,5,6]. So, this type of watermarking is suitable for authentication of image. Robust watermarking is robust against both of intentional and unintentional attacks [7,8,9]. So, this type of watermarking is suitable for copyright protection. Finally, semi-fragile watermarking is robust against unintentional attacks and also is fragile against intentional attacks and it reveals the tampered locations [10,11,12]. Given that on the Internet, operations such as image compression, image quality enhancement, communication noise and image tampering are common, so it is need to semi-fragile watermarking for authentication and recovery of the tampered image. For recovery of the tampered image, an image digest must be created and embedded as a watermark in the original image. The watermark embedding and tamper localizaion must be imperceptible and precise, respectively, and finally, the tampered regions must be recovered with high quality.
So far, many fragile watermarking methods have been proposed for tamper detection and recovery but the number of semi-fragile watermarking methods are not so much, because there are some constraints in semi-fragile watermarking such as the limitation of robust locations for embedding. The structure of this article is as follows. The related works and the proposed method are described in sections 2 and 3, respectively. Also, the experimental results and the conclusions are described in sections 4 and 5, respectively.
Related Works
In this section, state-of-the-art methods are reviewed that focus on the semi-fragile image watermarking for tamper detection and recovery.
In [13,14], two semi-fragile image watermarking methods for tamper detection and recovery are proposed, which the main embedding algorithm of them are similar. In [13], two watermarks are created for tamper detection and recovery, separately. But in [14], just one watermark is created for both purposes. Therefore, the watermarked image quality has improved because the size of watermark is smaller and the amount of change in the original image is less. In [13], the authentication watermark is created using a key randomly and the recovery watermark is created using some low frequency coefficients of DCT transform of the original image. These two created watermarks are compressed using Haffman codding and then to increase the robustness the BCH error correction coding is applied on the compressed watemarks. In both of [13] and [14] metods, the watermark information is embedded in detail sub-bands of Integer Wavelet Transform (IWT) of the original image. The group quantization of coefficients is used as the embedding method. The robustness and quality of the recovered image are not high in these two methods.
In [15,16], two methods are proposed that the foundation of them are similar. In other words, method [16] is an enhancement on the quality of method [15]. In [15], six bits are created randomly as the authentication watermark for each 8 × 8 block. Each 8 × 8 block is divided into four 4 × 4 sub-blocks, and the average values of gray levels of sub-blocks are used as recovery watermark. The authentication watermark is embedded into the low-frequency coefficients of DCT transform of a paired block, which the paired block is in a different place. The four average values of recovery watermark are embedded in the middle frequency coefficients of DCT transform of a paired block. Embedding method for authentication watermark is quantization and for recovery watermark is substitution in the coefficients. Novelty of method [15] is in recovery phase. The four average values are extracted and modified using linear regression in order to make the DC0 and three low-frequency coefficients of an 8 × 8 zero blocks. After that, the inverse DCT is applied on the created 8 × 8 block.
The robustness and quality of method [15] is high but it has the blocking effects. The mothod [16] has tried to solve the blocking effects using a linear optimization mechanism and the estimation of the lost coefficients in a DCT transform. The results of mothod [16] are much better than the results of mothod [15].
In [17], a block-based method is proposed, which the image is divided into 16 × 16 non-overlapping blocks and the average value of each block is calculated. Five most significant bits (5-MSB) of each average value are used as the recovery watermark for each block. In this method onesided circular block dependency is used in the embedding and tamper localization phases. Look back to the previous block in the circular dependency is used for increasing the localization power. The detail sub-bands coefficients of the second level of IWT transform are used as the place of embedding. The quantization of maximum value in a group of coefficients is used as the embedding method. The quality of recovered regions for the tampered image is about 20 dB based on the PSNR measure. The localization power is reported near to zero based on the false rejection (FR) and false acceptance (FA) measures. This method has a high robustness against to unintentional attacks.
In [18], a method is proposed that is based on compressive sensing. The compressive sensing is used for estimation of the missed coefficents. In this method, low frequency coefficients of 4 × 4 blocks are embedded into LH1 and HL1 sub-band of DWT transform using the substitution embedding method. Robustness and the quality of the recovered image are acceptable.
In [19], a method is proposed that is based on random sampling and image inpainting. In this method, some lines of pixels are selected in random directions and then the DCT transform of each line is calculated. Some lowfrequency coefficients are used as recovery watermark and are embedded in the middle-frequency coefficients of blocking DCT using the quantization based embedding method. In this method, inpainting is used for each pixel or region that there is no information about it. Robustness of this method is high but the quality of the recovered image is not high.
In [20], a method is proposed in which the halftone image of two-level IWT is used as recovery watermark. The authentication watermark is created using a key randomly. The recovery watermark is embedded in LH1 and HL1 sub-bands and the authentication watermark is embedded in LH2 and HL2 sub-bands, respectively using dither-QIM embedding method. Quality of recovered image is not high because the halftone image is created using two-level IWT and a Gaussian kernel convolution is used for calculation of inverse halftoning.
In [21], the average value of each block is calculated and then it is normalized using some calculations to be suitable for substitution embedding method in detail sub-bands of DWT transform. The novelty of this method is embedding the decimal values using the proposed calculations.
In [22,23,23], three methods are proposed for the protection of a special region of an image. In all these papers the image is categorized to the region of important (ROI) and the region of background (ROB). In paper [22], the ROI and ROB are determined manually. But, the papers [23,23] are used for protection of face or faces in the image and the detection of faces are automatically using face detection algorithms. Recovery watermark is calculated from the ROB region and it is embedded into the ROI region. Quality and robustness of these methods are high because the protection is just done on the ROI region.
As seen in the related works of this section, the amount of extracted bits for all regions of the image is equal. Although, some papers such as [22,23,24] have paid attention to the amount of extracted bits for the special ROI regions. Aside from the amount of extracted bits for different regions, the embedding regions is another important problem that is not taken into consideration by the state-of-the-art methods. So, in this paper, the original image is categorized into three regions automatically based on the texture of the image. The amount of extracted bits for each block in different regions is various and the extracted bits for each block in a region is embedded into a different block in its region. Also, a two-sided circular block dependency is proposed in front of the one-sided circular block dependency in [17], which the two-sided circular block dependency has some advantages over the one side circular block dependency.
Proposed Method
The proposed algorithm is made up of four subsections. Information extraction and embedding algorithms are described in subsection 3.1. Then, the tamper detection and tamper recovery algorithms are described in subsections 3.2 and 3.2, respectively. Diagram of the information extraction and the embedding phase of the proposed method are shown in Fig. 1, which each part of it will be explained in detail in the sub-sections.
Information Extraction and Embedding
In this method, the original image is divided into 16×16 non-overlapping blocks and the standard deviation of each block is calculated. Then, all the standard deviations are normalized and they are categorized into three types using two experimental thresholds T h 1 and T h 2 . For normalization of the standard deviation values the Eq.1 is used, which X, X min and X max are the current standard deviation, minimum value and maximum value between all of the standard deviation values, respectively. In this situation, all blocks of the image are categorized into one of the smooth, normal or rough types. Categorization result of blocks according to the texture is shown in Fig.2.
N orm(X) = X − X min X max − X min .(1)
In this method, each smooth, normal and rough block is divided into 1, 4 and 9 sub-blocks, respectively. Then, the average value of gray level for each sub-block is calculated. Also, regardless of the type of blocks, the average value of gray level is calculated for all 16 × 16 blocks and five the most significant bits (5-MSB) of the average values are calculated for use in embedding phase. The block dividing operation into its sub-bands is shown in Fig.3.
Pairwise block dependency is used for the blocks whose type is normal or rough. It is important that the information of a normal block must be embedded into another normal block and the information of a rough block must be embedded into another rough block. As seen in Fig.3, each normal and rough block must maintain 20 bits and 45 bits, respectively, for its pair dependent block. The pair dependent blocks must be far from each other. For this purpose, a random block is selected from normal blocks and then the furthest block to the selected block is found, which this far block is not paired with any other blocks. The similar operation must be done for all normal and rough blocks.
As mentioned before, 5-MSB bits is calculated for all 16 × 16 blocks and these bits will be used for tamper detection in all blocks and tamper recovery in the smooth blocks. According to this fact that these 5-MSB bits of the average values are important in tamper localization, so the blocks must have a sufficient distance with their dependent blocks. For this purpose, a block distance structure is proposed, which is shown in Fig.4. Using this block distance structure, each block has a minimum and maximum distance with its dependent blocks. In this structure, four main areas (A, B, C and D) are considered for the original image, which each area have four dependent sub-areas.
For tamper localization, each random block of A1 subarea keeps 5-MSB bits from a random block of A4 sub-area and 5-MSB bits from a random block of A2 sub-area. The A4 and A2 are previous and next dependent sub-areas with A1 sub-area. The two-sided circular block dependency between blocks is shown in Fig.5. There is similar block dependency for other blocks in other sub-areas in B, C, and D areas. This dependency is more powerful than the pairwise block dependency and the one side circular block dependency, in the tamper localization phase.
This method needs to know the type of each block in the tamper detection and recovery phases. So, the smooth, normal and rough block types are shown by binary bits "01", "10" and "11", respectively. In this method, the block type of four corresponding blocks, in four related sub-areas, are concatenated and 8 bits are created for them. Then, four copies of these 8 bits are embedded in the random location of the four related sub-areas. These four copy of block types will be used in a voting mechanism in tamper detection and recovery phases. The 8 bits creation operation is shown in Fig. 6.
So far, each block (with smooth, normal or rough type) must maintain 10 bits for two side circular block dependency and 8 bits for determining the block's type. Furthermore, normal and rough blocks need to maintain 20 and 45 bits, respectively for their pairwise dependency. Thus, each smooth, normal and rough block needs to maintain 18, 38 and 63 bits, respectively.
The calculated bits for each block is embedded in first approximation sub-band (LL1) of Integer Wavelet Transform (IWT) of each block using Quantization Index Modulation (QIM) method that are shown in relations 2 -4. For embedding in each block, 18, 38 and 63 coefficients of LL1 sub-band are selected randomly for smooth, normal and rough blocks, respectively. In these relations C n is the selected coefficient for embedding andC n is the embedded coefficient. Also, w n is one of the watermark bits. After any tamper in the watermarked image, the proposed algorithm needs to find the tampered blocks and the type of all blocks and finally recovering the tampered blocks, which these phases are described in the following subsections.
C n = v 1 , if |Cn − v 1 | ≤ |Cn − v 2 |, v 2 , otherwise. (2) v 1 = 2S Cn 2S , if wn == 0, 2S Cn 2S + S, if wn == 1. (3) v 2 = v 1 + 2S.(4)
Tamper Detection
As mentioned in the previous section, 10 bits are embedded in each block for previous and next dependent blocks using two side circular block dependency. These 10 bits consists of 5-MSB bits for the average gray level value of the previous dependent block and 5-MSB bits for the average gray level value of next dependent block. Relation 5 is used in order to extract these 10 bits for each block from the coefficients of LL1 sub-band.
w n = 0, if round(C n S ) == even, 1, if round(C n S ) == odd.(5)
In this step, the status of each block is defined by healthful block, fully destroyed block or partially destroyed block. The status of a block is healthful if the 5-bits generated from the average gray level value of current block be extractable from at least one of the previous or next dependent blocks and also the 5-MSB bits generated from the average gray level value of the previous and next dependent blocks be extractable from the current block. The status of a block is fully destroyed if the 5-bits generated from the average gray level value of current block not be extractable from both of the previous and next dependent blocks and also the 5-MSB bits generated from the average gray level value of the previous and next dependent blocks not be extractable from the current block.
The status of a block is partially destroyed if the 5-MSB bits generated from the average gray level value of the previous and next dependent blocks not be extractable from the current block and also the 5-bits generated from the average gray level value of current block be extractable from at least one of the previous or next dependent blocks. The pairwise block dependency or the one-sided circular bock dependency in [17] are not able to distinguish the partially destroyed blocks. This type of distinguished blocks increases the accuracy of tamper localization and recovery algorithms. Making decision operations on the status of exemplary block B in Fig.7 , are shown in relations 6 -8. In these relations, the meaning of Gen B is 5-MSB bits generated from the average gray level value of current block B. Also, Ext B1 and Ext B2 are the extracted 5-MSB information bits from block B that are corresponding to the average gray level value of block A and C in embedding phase, respectively. Figure 6: Concatenation of block types ("01" for a smooth block, "10" for a normal block and "11" for a rough block) for corresponding blocks of four sub-areas.
Status(B) is Healthful if:
{(Gen A == Ext B1 ) (Gen C == Ext B2 )}(6)
Status(B) is Fully Destroyed if:
{(Gen A = Ext B1 ) & (Gen C = Ext B2 ) & (Gen B = Ext A2 ) & (Gen B = Ext C1 )}(7)
Status(B) is Partially Destroyed if:
{[(Gen A = Ext B1 ) & (Gen C = Ext B2 )] & [(Gen B == Ext A2 ) (Gen B == Ext C1 )]} (8) A B C GenB GenC GenA ExtB1 ExtB2 ExtA2 ExtA1
Make Decision On Blocks Type
As mentioned in subsection 3.1, for four corresponding blocks in four sub-areas A1, A2, A3 and A4, eight bits of types are concatenated and four copy of these eight bits are embedded in random locations of four sub-areas A1, A2, A3 and A4. Type detection of each block is done by voting on the four copy of eight bits. There are two states for voting. The first state is for partially or fully destroyed blocks and the second state is for healthful blocks. In the first state, the votes of other three blocks are gathered for current block type decision. In the second state, the vote of the current block and two other healthful blocks in other three sub-areas are used for current block type decision. So, these two states try to use just healthy blocks in making the decision on the block's type.
Algorithm 1: Information extraction and embedding algorithm
Input: Original image Output: Watermarked image 1 Dividing image into non-overlapping 16 × 16 blocks; 2 Calculation of standard deviation for each block and normalize all of them; 3 Categorization of blocks into smooth, normal or rough block types using two thresholds T h 1 and T h 2 on the normalized standard deviation values (Fig. 2); 4 Dividing the smooth, normal and rough blocks into 1, 4, and 9 sub-blocks, respectively (Fig. 3); 5 Extracting 5-MSB bits from the average gray value of each sub-block and making 20 and 45 bits for normal and rough blocks, respectively; 6 Extracting 5-MSB bits of the average gray level value for all 16 × 16 blocks; 7 Concatenation the 5-MSB bits of the previous and next dependent blocks for each block (Fig. 5); 8 Generating 8 bits from four corresponding blocks type for each block (Fig. 6); 9 Finding a far pair block for each normal and rough block with the same type; 10 making 18, 38 and 63 bits using the results of steps 5, 6 and 7 for each smooth, normal and rough block, respectively.; 11 Embedding the extracted bits for each block in the LL 1 sub-band of IWT transform of block using the QIM method in Eq. 2 -3.
Tamper Recovery
After detecting the type of the exact block and tampered blocks, there is need to extract the embedded information bits of smooth, normal and rough blocks. As seen from the previous subsections the information bits of smooth blocks are embedded in the previous and next dependent blocks using the two side circular block dependency. For a tampered smooth block, its corresponding information bits must be calculated from a not destroyed dependent block. Also, the paired block of each normal or rough block is in a random far location with the same block type. So, in this step, the paired block of each normal and rough block must be found and then the embedded information bits (20 and 45 bits for normal and rough blocks, respectively) must be extracted. As mentioned in the subsection 3.1, the information bits for each block are embedded in the LL1 sub-band of the corresponding block IWT transform and must be extracted using the relation 5.
So, in this step, 20 bits are calculated for four sub-blocks of each normal block and 45 bits are calculated for nine sub-blocks of each rough block. Every five bits of 20 or 45 bits are the 5-MSB bits of the average gray level value of a sub-block. So, the binary value "100" is appended to the LSB bits of these 5-MSB bits in order to form an eight 6 bits binary value and the decimal value of the eight bits binary value is used as the average gray level value of each sub-block.
Each smooth, normal and rough block is consist of 1, 4 and 9 decimal values, respectively. In this method, for recovery of tampered blocks, the cubic interpolation is applied to the calculated decimal values for sub-blocks. In order to increase the benefits of cubic interpolation, the contents of smooth and normal blocks are reformed to the form of the rough block as shown in Fig.8. The reformation and cubic interpolation are done on the whole of tampered image blocks. These operations combine the neighbor subblocks and blocks, which decrease the blocking effect in tampered regions.
Experimental Results
For evaluation of the proposed method twenty 512 × 512 standard gray images are used, which are selected from the USC-SIPI image database [25]. Some of the standard images of this database are shown in Fig. 9. For evaluation of robustness, the JPEG and JPEG2000 compressions and copy-move attacks are applied to the watermarked image.
Peak Signal to Noise Ratio (PSNR) is used for evaluation of the visual quality of the watermarked image and the recovered image. This criterion compares pixel by pixel similarity between the original image, watermarked image and recovered image and is defined as Eq. 9.
PSNR(f, f w ) = 10 log 10 max ∀(m,n) f 2 (m, n)
1 N f ∀(m,n) (f w (m, n) − f (m, n)) 2 .
(9) In Eq. 9, f (m, n) is the original image, f w (m, n) is the watermarked (or recovered) image and N f is the number of pixels in image.
Another visual quality measure that is used in this paper is Structural Similarity (SSIM), which the structure of the image is considered in it. The values of SSIM measure is (e) (f ) (g) Figure 11: Tamper detection steps on the Lena standard image. ÙŚFig.11a is the watermarked image with PSNR=35.20 dB. Fig.11b, shows the tampeded and JPEG compressed image with QF=50. Fig.11c, shows the three detected block types (black is the smooth block, gray is the fully destroyed block, white is the rough block). Fig.11d, shows the conversion of partially destroyed block to fully destroyed block if they have continuty. Fig.11e, shows the blocks that do not have any role in the continuing tamper detection process. Fig.11f, shows the continuous partially destroyed blocks and fully destroyed blocks independently. Fig.11f, shows the tamper deteted blocks after filling the hole and contour blocks (FR=0% and FA=3.2%).
Another visual quality measure that is used in this paper is Structural Similarity (SSIM), which the structure of the image is considered in it. The values of SSIM measure is in the range [0,1], which 0 is the minimum similarity and 1 is the maximum similarity.
Dividing the original image into 16 × 16 blocks and categorization of them into smooth, normal and rough blocks according to the standard deviation of each block is shown in Fig. 10. Two thresholds T h 1 and T h 2 are set to 0. Fig. 13 shows the summary of the tamper detection steps for some of the standard images. In these examples, the size of the tampered region is considered 160 × 160 (equal to 100 blocks) and the JPEG compression power that is applied to the watermarked image is QF=50%. In Fig.11c, the three detected block types (black is the smooth block, gray is the fully destroyed block, white is the rough block) are shown. The Fig.11d, shows the conversion of the partially destroyed block to fully destroyed block if they have continuity. This continuity between the partially destroyed block and the fully destroyed blocks can be either directly or through some intermediate partially destroyed blocks. The Fig.11e, shows the blocks that do not have any role in the continuing of tamper detection process. Separation of partially destroyed blocks and fully destroyed blocks has enabled us to remove some not tampered blocks in the later steps of tamper detection. This step does not exist in the method [17] because in that paper is not any difference between partially destroyed block and fully destroyed block. , Figure 12: Tamper detection steps on the House standard image. ÙŚFig.12a is the watermarked image with PSNR=34.38 dB. Fig.12b, shows the tampeded and JPEG compressed image with QF=50. Fig.12c, shows the three detected block types (black is the smooth block, gray is the fully destroyed block, white is the rough block). Fig.12d, shows the conversion of partially destroyed block to fully destroyed block if they have continuty. Fig.12e, shows the blocks that do not have any role in the continuing tamper detection process. Fig.12f, shows the continuous partially destroyed blocks and fully destroyed blocks independently. Fig.12f, shows the tamper deteted blocks after filling the hole and contour blocks (FR=0% and FA=3.7%).
The Fig. 11f, shows the combined continuous partially destroyed blocks and fully destroyed blocks independently. And finally, the Fig. 11f shows the tamper detected blocks after filling the hole and contour blocks. A 3 × 3 neighborhood kernel is used for the filling operation. In the Fig. 11f the healthfull blocks (black color) who have more than two destroyed blocks (white color), are considered as destroyed blocks. The average value of tamper localization power, for the Figs. 11, 12 and 13, in terms of false rejection (FR) and false acceptance (FA) is 1.4% and 3.2%, respectively. These measures are reported near to zero in method [17].
Recovery results for Lena image are shown in Figs. 14 and 15. Lena image is a good standard image that is composed of three regions (smooth, normal and rough). As shown in these figures, the smooth region (such as the body) is recovered with lower quality and the rough region (such as edges) is recovered with higher quality. The reason for this difference in the quality of regions is the difference in the amount of extracted and used information bits for the blocks of each region. The visual comparison between the proposed method and methods [16,17,18,20] is shown in Fig. 16. In these comparisons, the same region of the image is tampered and recovered. Also, the JPEG compression ratio is equal in the all of them. Due to the Fig. 4 and the distance between the dependent blocks, the proposed method is able to detect and recover a square 256 × 256 region of the image in the best case.
In Tables. 1 and 2, a 256 × 256 region on the top-left side of the watermarked images is tampered and recovered in presence of a different ratio of JPEG compression and JPEG2000 compression, respectively. The quality factor (QF) of JPEG compression in these experiments are 100, 95, 90, 85, 80 and 75. Also, the compression ratio (CR) of JPEG2000 compression is represented by 1, 2, 3, 4, 5 and 6, which CR=2 implies that the output image size is half of the input image size or less. As seen from these tables the average PSNR value for the recovered region is 25.06 dB for JPEG compression with QF=80% and also is 24.88 dB for JPEG2000 compression with CR=5, which these results are high PSNR values. Applying the cubic interpolation on the reformed sub-blocks values, which is shown in Fig.8, makes the proposed algorithm to be efficient in recovery of tampered regions in comparison with the state-of-the-art methods.
Also, the PSNR value after tamper detection and recovery on the 70 miscellaneous standard image from the USC-SIPI image database [25] is calculated. In this experiment, a random square portion (100 × 100) is filled with black gray level and then is detected and recovered. Also, this experiment is done in presence of JPEG compression with QF=80%. Also, The PSNR Value for the recovered portion of all 70 images is calculated. The mean and variance values of the PSNR are 24.27 dB and 27.35. Some image results are near 15 dB (not acceptable) because the proposed method is reliable in presence of JPEG compression with QF=85%-100%. The False Rejection and False Acceptance values of the proposed method are 2.5 and 2.8 percent in the average state.
Conclusions
In this paper, a new semi-fragile watermarking algorithm is proposed for tamper detection and recovery. Three main ideas that are proposed in this paper are region attention attitude, tamper localization method and the usage of cubic interpolation method in the recovery phase for solving the blocking problem. Region attention is defined as the extracting fewer information bits for the smooth region of image and more information bits for normal and rough regions of the image. Also, in order to provide more imperceptibility, the information bits extracted for each type of blocks are embedded in a dependent far block with the same type. Two side circular block dependency is proposed for increasing the power of tamper localization and solving the weakness of the pairwise dependency and the one side circular block dependency.
In order to recover the tampered regions, the average values of sub-blocks are extracted and used by cubic interpolation. So, a good combination is provided between sub-blocks and blocks. The PSNR measure for watermarked image and recovered region are 35.06 dB and 25.06 dB, respectively. Also, localization power based on the false rejection (FR) and false acceptance (FA) measures are 1.2 % and 3.4 %, respectively in the average state.
Usage of halftone image as the extracted information bits for each region can be considered as a future work. Figure 13: Tamper detection results on some standard images. First column is the watermarked image. Second column is tampered image, which the size of tampered region is considered 160 × 160 and the JPEG compression power is considered QF=50%. Third column is the detection of three blocks status (Black is healthful, Gray is fully destroyed and White is partially destroyed). Fourth column is the tampered region. (c) (d) Figure 14: Fig.14a shows the watermarked image with PSNR 35.20 dB. Fig.14b shows the tampered region (176 × 176) and the JPEG compression power is QF=80%. Fig.14c is the recovered image with PSNR=31.12 dB. Fig.14d shows the magnified recovered region with PSNR=24.51 dB.
One of the most challenges in the usage of halftone image with the proposed framework is the adaptation and mixing the zero and one bits of halftone image in the neighbor blocks.
As seen from the visual results, the original image is divided into 16 non-overlapping blocks and each smooth, normal and rough block is divided into 1, 4 and 9 fix sub-blocks, respectively. So, as another future work, the quadtree or Q-tree structure can be considered for dividing each high texture block into more sub-blocks in front of fix sub-blocking. Figure 15: Fig.14a shows the watermarked image with PSNR 35.20 dB. Fig.14b shows the tampered region (256 × 256) and the JPEG compression power is QF=80%. Fig.14c is the recovered image with PSNR=30.76 dB. Fig.14d shows the magnified recovered region with PSNR=25.43 dB.
(a) Proposed method (b) Method in [16] (c) Proposed method (d) Method in [17] (e) Proposed method (f ) Method in [20] (g) Proposed method (h) Method in [18] Figure 16: Visual comparison between the proposed method and methods [16,17,18,20]. All images except for the image in Fig.16f are in presence of JPEG compression with QF=80%. The Fig.16f is in presence of JPEG compression with QF=85%. In Figs.16a, 16b, 16e, 16f, 16g and 16h the tampered and recovered region are shown using red countour. Figs.16c and 16d are full recovery of image using the embedded information without any tampering.
Figure 1 :Figure 2 :
12Diagram of the information extraction and the embedding phase. Categorization of image blocks using standard deviation.Fig.2ais the original image andFig.2bis the categorized blocks. The Red blocks are smooth, the Green blocks are normal and the Blue blocks are rough.
Figure 3 :
3The smooth, normal and rough blocks are divided into 1, 4 and 9 sub-blocks, respectively.
Figure 5 :
5Two sided circular block dependency between subarea blocks of area A1.
Figure 7 :
7Making decision operations on the status of exemplar block B.
Figure 8 :
8Reforming of the smooth and normal blocks to the rough block form.
Figure 9 :
9Some of the standard images of USC-SIPI image database[25].
Figure 10 :
10categorization of blocks. (a) Original image. (b) Standard deviation values. (c) Categorization using thresholds T h1 = 0.1 and T h2 = 0.3 (black color for smooth, gray color for normal and white color for rough block). (d) Representation of each block on image (red color for smooth, green color for normal and blue color for rough block).
1 and 0.3, respectively, for categorization of the normalized standard deviation values. The normalized values in range [0,1), [0.1,0.3) and [0.3,1.0] are smooth, normal and rough blocks, respectively. The Figs. 11, 12 and 13 are examples of tamper detection operations. In order to enhance understanding, more steps of tamper detection steps are shown in Figs. 11 and 12. The
(
Algorithm 2 :
2Tamper detection and recovery algorithmInput: Tampered image Output: Recovered image 1 Dividing image into non-overlapping 16 × 16 blocks; Applying cubic interpolation on the values of image sub-blocks to become equal to the size of original image; 10 replacing the tampered blocks with the results of step 9.2 Extracting 8 bits that are related to the type of four
correspond blocks from LL 1 sub-band of IWT
transform using Eq. 5;
3 Applying a voting algorithm on the block types for
make decision on the type of image blocks
(subsection 3.3);
4 Extracting 10 bits that are related to the previous
and next dependent block using Eq. 5;
5 Making decision on the status of each block using
Eq. 6 -8;
6 Finding the coressponding far pair block for each
normal and rough block;
7 Extracting 20 and 45 bits from the corresponding far
pair block for each normal and rough block using
Eq. 5;
8 Converting the form of smooth and normal blocks to
the form of a rough block (Fig. 8);
9 in the range [0,1], which 0 is the minimum similarity and 1
is the maximum similarity.
Table 1 :
1Results of proposed method in presense of JPEG compression. Tampered region is a 256 × 256 region in the top-left side of the watermarked images.JPEG compression QF (Quality Factor)
Table 2 :
2Results of proposed method in presense of JPEG2000 compression. Tampered region is a 256 × 256 region in the top-left side of the watermarked images.JPEG2000 compression CR (Compression Ratio)
Table 3 :
3Comparison between the quality of proposed method and some related works.Method
PSNR
Watermarked Image
SSIM
Watermarked Image
PSNR
Recovered Region
Li 2014 [17]
36.00
0.81
21.00
Wang 2014 [16]
37.41
0.95
21.68
Korus 2015 [19]
36.00
-
21.00
Phadikar 2012 [20]
35.00
0.86
23.46
Proposed Method
35.06
0.88
25.06
a) PSNR=34.62 dB(b)
(c)
(d) FR=0%, FA=3.3%
(e) PSNR=35.20 dB
(f )
(g)
(h) FR=5%, FA=2.7%
(i) PSNR=34.16 dB
(j)
(k)
(l) FR=0%, FA=3.8%
(m) PSNR=36.71 dB
(n)
(o)
(p) FR=5%, FA=2.2%
(q) PSNR=35.16 dB
(r)
(s)
(t) FR=0%, FA=4%
I Cox, M Miller, J Bloom, J Fridrich, T Kalker, Digital watermarking and steganography. Morgan KaufmannI. Cox, M. Miller, J. Bloom, J. Fridrich, T. Kalker, Digital watermarking and steganography, Morgan Kaufmann, 2007.
T.-S Nguyen, C.-C Chang, X.-Q Yang, 10.1016/j.aeue.2016.05.003A reversible image authentication scheme based on fragile watermarking in discrete wavelet transform domain. 70T.-S. Nguyen, C.-C. Chang, X.-Q. Yang, A reversible image authentication scheme based on fragile watermarking in discrete wavelet transform domain, AEU-International J. Electron. Com- mun. 70 (2016) 1055âĂŞ1061. doi:10.1016/j.aeue.2016.05.003.
New fragile watermarking method for stereo image authentication with localization and recovery. M Yu, J Wang, G Jiang, Z Peng, F Shao, T Luo, AEU-International J. Electron. Commun. 69M. Yu, J. Wang, G. Jiang, Z. Peng, F. Shao, T. Luo, New fragile watermarking method for stereo image authentication with localization and recovery, AEU-International J. Electron. Commun. 69 (2015) 361âĂŞ370.
Binomial transform based fragile watermarking for image authentication. S K Ghosal, J K , J. Inf. Secur. Appl. 19S.K. Ghosal, J.K. Mandal, Binomial transform based fragile watermarking for image authentication, J. Inf. Secur. Appl. 19 (2014) 272âĂŞ281.
Chaos-based selfembedding fragile watermarking with flexible watermark payload. F Chen, H He, H.-M Tai, H Wang, Multimed. Tools Appl. 72F. Chen, H. He, H.-M. Tai, H. Wang, Chaos-based self- embedding fragile watermarking with flexible watermark payload, Multimed. Tools Appl. 72 (2014) 41âĂŞ56.
Fragile image watermarking scheme based on VQ index sharing and self-embedding. C Qin, P Ji, J Wang, C.-C Chang, Multimed. Tools Appl. 76C. Qin, P. Ji, J. Wang, C.-C. Chang, Fragile image watermark- ing scheme based on VQ index sharing and self-embedding, Multimed. Tools Appl. 76 (2017) 2267âĂŞ2287.
A new spread spectrum watermarking method using two levels DCT. A H Taherinia, M Jamzad, Int. J. Electron. Secur. Digit. Forensics. 3A.H. Taherinia, M. Jamzad, A new spread spectrum watermark- ing method using two levels DCT, Int. J. Electron. Secur. Digit. Forensics. 3 (2010) 1âĂŞ26.
Robust image watermarking based on ICA-DCT and noise augmentation technique. S H Soleymani, A H Taherinia, Comput. Knowl. Eng. (ICCKE), 2015 5th Int. Conf., IEEE. S.H. Soleymani, A.H. Taherinia, Robust image watermarking based on ICA-DCT and noise augmentation technique, in: Comput. Knowl. Eng. (ICCKE), 2015 5th Int. Conf., IEEE, 2015: pp. 18âĂŞ23.
Double expanding robust image watermarking based on Spread Spectrum technique and BCH coding. S H Soleymani, A H Taherinia, 10.1007/s11042-016-3734-2Multimed. Tools Appl. S.H. Soleymani, A.H. Taherinia, Double expanding robust image watermarking based on Spread Spectrum technique and BCH coding, Multimed. Tools Appl. (2016) 1âĂŞ19. doi:10.1007/s11042-016-3734-2.
Semi-fragile watermarking for image authentication with sensitive tamper localization in the wavelet domain. R O Preda, Measurement. 46R.O. Preda, Semi-fragile watermarking for image authentication with sensitive tamper localization in the wavelet domain, Measurement. 46 (2013) 367âĂŞ373.
A region-adaptive semi-fragile dual watermarking scheme. H Shi, M Li, C Guo, R Tan, Multimed. Tools Appl. 75H. Shi, M. Li, C. Guo, R. Tan, A region-adaptive semi-fragile dual watermarking scheme, Multimed. Tools Appl. 75 (2016) 465âĂŞ495.
A semi-fragile image watermarking algorithm with two-stage detection. Y Huo, H He, F Chen, Multimed. Tools Appl. 72Y. Huo, H. He, F. Chen, A semi-fragile image watermarking algorithm with two-stage detection, Multimed. Tools Appl. 72 (2014) 123âĂŞ149.
Digital image authentication and recovery: employing integer transform based information embedding and extraction. R Chamlawi, A Khan, 10.1016/j.ins.2010.08.039Inf. Sci. (Ny). 180R. Chamlawi, A. Khan, Digital image authentication and recovery: employing integer transform based information em- bedding and extraction, Inf. Sci. (Ny). 180 (2010) 4909âĂŞ4928. doi:10.1016/j.ins.2010.08.039.
Dual-purpose semi-fragile watermark: Authentication and recovery of digital images. R Ullah, A Khan, A S Malik, 10.1016/j.compeleceng.2013.04.024Comput. Electr. Eng. 39R. Ullah, A. Khan, A.S. Malik, Dual-purpose semi-fragile watermark: Authentication and recovery of digital im- ages, Comput. Electr. Eng. 39 (2013) 2019âĂŞ2030. doi:10.1016/j.compeleceng.2013.04.024.
A novel fast self-restoration semi-fragile watermarking algorithm for image content authentication resistant to JPEG compression. H Wang, A T S Ho, X Zhao, Int. Work. Digit. Watermarking. SpringerH. Wang, A.T.S. Ho, X. Zhao, A novel fast self-restoration semi-fragile watermarking algorithm for image content authen- tication resistant to JPEG compression, in: Int. Work. Digit. Watermarking, Springer, 2011: pp. 72âĂŞ85.
A novel image restoration scheme based on structured side information and its application to image watermarking, Signal Process. H Wang, A T S Ho, S Li, 10.1016/j.image.2014.05.001Image Commun. 29H. Wang, A.T.S. Ho, S. Li, A novel image restoration scheme based on structured side information and its application to image watermarking, Signal Process. Image Commun. 29 (2014) 773âĂŞ787. doi:10.1016/j.image.2014.05.001.
Semi-fragile self-recoverable watermarking algorithm based on wavelet group quantization and double authentication. C Li, A Zhang, Z Liu, L Liao, D Huang, 10.1007/s11042-014-2188-7Multimed. Tools Appl. 74C. Li, A. Zhang, Z. Liu, L. Liao, D. Huang, Semi-fragile self-recoverable watermarking algorithm based on wavelet group quantization and double authentication, Multimed. Tools Appl. 74 (2015) 10581âĂŞ10604. doi:10.1007/s11042-014-2188-7.
Semi-fragile watermarking for image tamper localization and self-recovery. Y Li, L Du, Secur. Pattern Anal. Cybern. (SPAC). IEEEY. Li, L. Du, Semi-fragile watermarking for image tamper localization and self-recovery, in: Secur. Pattern Anal. Cybern. (SPAC), 2014 Int. Conf., IEEE, 2014: pp. 328âĂŞ333.
Iterative filtering for semifragile self-recovery. P Korus, J Biaåćas, A Dziech, 10.1109/WIFS.2014.7084300IEEE Int. Work. Inf. Forensics Secur. IEEEP. Korus, J. BiaÅĆas, A. Dziech, Iterative filtering for semi- fragile self-recovery, in: 2014 IEEE Int. Work. Inf. Forensics Se- cur., IEEE, 2014: pp. 36âĂŞ41. doi:10.1109/WIFS.2014.7084300.
Novel wavelet-based QIM data hiding technique for tamper detection and correction of digital images. A Phadikar, S P Maity, M , 10.1016/j.jvcir.2012.01.005J. Vis. Commun. Image Represent. 23A. Phadikar, S.P. Maity, M. Mandal, Novel wavelet-based QIM data hiding technique for tamper detection and correction of digital images, J. Vis. Commun. Image Represent. 23 (2012) 454âĂŞ466. doi:10.1016/j.jvcir.2012.01.005.
Tamper detection and self-recovery scheme by DWT watermarking. O Benrhouma, H Hermassi, S Belghith, 10.1007/s11071-014-1777-3Nonlinear Dyn. 79O. Benrhouma, H. Hermassi, S. Belghith, Tamper detection and self-recovery scheme by DWT watermarking, Nonlinear Dyn. 79 (2015) 1817âĂŞ1833. doi:10.1007/s11071-014-1777-3.
A ROI-based semi-fragile watermarking for image tamper detection and recovery. S D Lin, J Lin, C Chen, Int. J. Innov. Comput. Inf. Control. 7S.D. Lin, J. Lin, C. Chen, A ROI-based semi-fragile water- marking for image tamper detection and recovery, Int. J. Innov. Comput. Inf. Control. 7 (2011) 6875âĂŞ6888.
Semi-fragile self-recoverable watermarking scheme for face image protection. C Li, R Yang, Z Liu, J Li, Z Guo, Comput. Electr. Eng. C. Li, R. Yang, Z. Liu, J. Li, Z. Guo, Semi-fragile self-recoverable watermarking scheme for face image protection, Comput. Electr. Eng. (2016).
C Cruz-Ramos, M Nakano-Miyatake, H Perez-Meana, R Reyes-Reyes, L Rosales-Roldan, 10.1007/s11042-014-2006-2Face region authentication and recovery system based on SPIHT and watermarking. 74C. Cruz-Ramos, M. Nakano-Miyatake, H. Perez-Meana, R. Reyes-Reyes, L. Rosales-Roldan, Face region authentication and recovery system based on SPIHT and watermarking, Multimed. Tools Appl. 74 (2014) 7685âĂŞ7709. doi:10.1007/s11042-014- 2006-2.
. The USC-SIPI Image Database. The USC-SIPI Image Database, (2016).
| [] |
[
"PARTIAL CLASSICALITY OF HILBERT MODULAR FORMS",
"PARTIAL CLASSICALITY OF HILBERT MODULAR FORMS"
] | [
"Chi-Yun Hsu "
] | [] | [] | Let F be a totally real field and p a rational prime unramified in F . We prove a partial classicality theorem for overconvergent Hilbert modular forms: when the slope is small compared to a subset of weights, an overconvergent form is partially classical. We use the method of analytic continuation.Date: May 31, 2022. | 10.1016/j.jnt.2022.04.007 | [
"https://arxiv.org/pdf/2109.00470v2.pdf"
] | 237,372,011 | 2109.00470 | bed757c1ce0959cb5a9fee55b7d7d6b3fe2acea9 |
PARTIAL CLASSICALITY OF HILBERT MODULAR FORMS
28 May 2022
Chi-Yun Hsu
PARTIAL CLASSICALITY OF HILBERT MODULAR FORMS
28 May 2022arXiv:2109.00470v2 [math.NT]
Let F be a totally real field and p a rational prime unramified in F . We prove a partial classicality theorem for overconvergent Hilbert modular forms: when the slope is small compared to a subset of weights, an overconvergent form is partially classical. We use the method of analytic continuation.Date: May 31, 2022.
Introduction
Coleman [Col96] proved that a p-adic overconvergent modular form of weight k ∈ Z must be classical if its slope, i.e., the p-adic valuation of the U p -eigenvalue, is less than k − 1. His proof involves analyzing the rigid cohomology of modular curves. On the other hand, Buzzard [Buz03] and Kassaei [Kas06] developed the alternate method of analytic continuation to prove classicality theorems. The key is to understand the dynamic of the U p Hecke operator.
Let F be a totally real field of degree g over Q. In the situation of Hilbert modular forms associated to F , many results about classicality are also known. Coleman's cohomological method was developed by Tian-Xiao [TX16] to prove a classicality theorem, assuming p is unramified in F . The method of analytic continuation was worked out first in the case when p splits completely in F by Sasaki [Sas10], then in the case when p is unramified by Kassaei [Kas16] and Pilloni-Stroh [PS17], and finally when p is allowed to be ramified by Bijakowski [Bij16].
Let Σ be the set of archimedean embeddings of F , which we identify with the set of p-adic embeddings of F through some fixed isomorphism C ∼ = Q p . For each prime p of F above p, denote by Σ p ⊆ Σ the subset of p-adic embeddings inducing p. Let e p be the ramification index, and f p the residue degree of p. Then the classicality theorem for overconvergent Hilbert modular forms proved by analytic continuation is as follows. ask in the flavor of Kisin's interpretation of Fontaine-Mazur conjecture: if f is overconvergent and ρ f | Gal F p is (I ∩ Σ p )-de Rham for all p | p, is f I-classical? For the organization of this paper: In Section 2, we define the degree function and partially classical overconvergent forms. In Section 3, we prove Theorem 2.
Notations. Fix a totally real field F of degree g over Q. Let Σ denote the set of archimedean places of F ; in particular #Σ = g. Fix a rational prime p which is unramified in F and (p) = p 1 · · · p r in F . For each prime p of F above p, let f p be the residue degree of p. Fix an isomorphism ι p : C ∼ − → Q p , and identify archimedean embeddings τ : F → C with p-adic embeddings ι p • τ : F → Q p . For each prime p of F above p, let Σ p ⊆ Σ be the subset of p-adic embeddings inducing p. Hence #Σ p = f p . Let L be a finite extension of Q p containing the image of all p-adic embeddings ι • τ of F . Since p is assumed to be unramified in F , we may also assume that L is an unramified extension of Q p . Let k L denote the residue field of L. Let δ F be the different ideal of F . • A is an abelian scheme of relative dimension g over an O L -scheme S,
• i : O F ֒→ End S (A) is a ring homomorphism. Here i is called a real multiplication on A, • λ : (P A , P + A ) → (c, c + )
is an isomorphism of O F -modules identifying the positive elements, and inducing an isomorphism
A⊗ O F c ∼ = A ∨ . Here P A = Hom O F (A, A ∨ ) sym is the projective O F -
module of rank 1 consisting of symmetric morphisms from A to its dual abelian scheme A ∨ , and P + A ⊆ P A is the cone of polarizations. Here λ is called a c-polarization of A,
• α : µ N ⊗ δ −1 F → A is a closed immersion of O F -group schemes.
Here α is called a Γ 1 (N )-level structure, and • H ⊆ A[p] is a finite flat O F -subgroup scheme of rank p g which is isotropic with respect to the µ-Weil pairing for some polarization µ ∈ P + A of degree prime to p. Let Cl(F ) + be the narrow class group of F , namely the quotient of the abelian group of fractional ideals of F by the subgroup of principal ideals generated by totally positive elements. Let {c i } be a set of representatives of Cl(F ) + . Define Y = c i Y c i , which is independent of the choice of the representatives {c i }. Denote by Y the completion of Y along its special fiber, and by Y the rigid generic fiber of the formal scheme Y. We also use this convention of letter styles for other schemes: when K/Q p is a finite extension and S is a scheme over O K , we denote by S the associated formal scheme and by S the rigid generic fiber of S.
Directional degrees.
We first recall the definition of the degree for a commutative finite flat group scheme. See [Far10] for more detailed studies of the concept.
Let S be a scheme and G a commutative finite flat group scheme over S. Let ω G be the sheaf of invariant differentials on G. Define δ G := Fitt 0 ω G as the 0-th Fitting ideal of ω G . This is an invertible ideal sheaf in O S . Now let K/Q p be a finite extension and S = Spec O K . Then the degree of G is defined as [Far10, Définition 4] the rational number deg G = deg ω G := val p (δ G ).
Writing
ω G = i O K /x i O K , then deg G = i val p (x i ). Equivalently, deg G = ℓ(ω G )/e K , where ℓ(ω G )
is the length of the O K -module ω G , and e K is the ramification index of K. Recall that the height ht G of G is such that |G| = p ht G . Hence G is étale if and only if deg G = 0, and G is multiplicative if and only if deg G = ht G.
More generally, let S be a scheme over O K . Each closed point s in the rigid analytic space S is defined over the ring of integer of a finite extension of K [BLR95, Section 8.3, Lemma 6]. Hence we obtain the degree function deg :
S → [0, ∞) ∩ Q s → deg G s . The inverse image of a (open, closed, or half-open) interval in [0, ∞)
is an admissible open of S. Moreover, when the interval is closed and its end points a ≤ b are rational numbers, then the inverse image is quasi-compact.
We record some properties of deg which we will constantly use for computation.
Lemma 2.1. [Far10, lemme 4] Let 0 → G ′ → G → G ′′ → 0
In particular, if
A is of dimension g, then deg A[p] = g.
When G has an O F -module structure, we can define directional degree functions on S. Instead of a general exposition, we only explain this for S = Y, the Hilbert modular variety over L. See also [PS17, Section 4.2] or [Kas16, Section 2.9]. Let (A univ , H univ ) be the universal abelian scheme over the Hilbert modular variety Y . Let ω H univ be the sheaf of invariant differentials of H univ , which is a
O F /pO F -module. Since p is unramified in F , Σ is in bijection with the embeddings O F /pO F ֒→ k L . We decompose ω H univ according to the embeddings O F /pO F ֒→ k L to obtain ω H univ = τ ∈Σ ω H univ ,τ .
For each τ ∈ Σ, define δ τ := Fitt 0 (ω H univ ,τ ), which is an invertible ideal sheaf in O Y .
Let y = (A, H) be a closed point of Y. Let K be the finite extension of L over which y is defined. Then we have the rational number deg ω H,τ . In addition, deg ω H,τ ∈ [0, 1]. Indeed, for each p | p, the subgroup scheme H[p] of H is a Raynaud group scheme over Spec O K , namely a
k p = O F /pO F -vector space scheme of dimension 1. For each tuple (d τ ) τ ∈Σp of elements of O K with val p (d τ ) ≤ 1, Raynaud associates a k p -vector space scheme of dimension 1 H (dτ ) := Spec O K [X τ , τ ∈ Σ p ]/(X p σ −1 •τ − d τ X τ ), where σ is the Frobenius automorphism of L over Q p lifting x → x p modulo p, and the k p -action on X τ is given by the character k × p → O × K induced by τ : F → L. Moreover, each k p -vector space scheme of dimension 1 over O K is isomorphic to some H (dτ ) [Ray74, THÉORÈME 1.4.1]. Since ω H (dτ ) ,τ = O K /d τ O K , we have deg ω H,τ = deg ω H (dτ ) ,τ = val p (d τ ) ∈ [0, 1].
Hence for each τ ∈ Σ, we can define the directional degree function
deg τ : Y → [0, 1] ∩ Q y = (A, H) → deg ω H,τ , as well as deg : Y → ([0, 1] ∩ Q) Σ y → (deg τ y) τ .
As before, the inverse image of deg τ (resp. deg) of a subset of [0, 1] (resp. [0, 1]) Σ ) defined by a finite number of affine inequalities is an admissible open of Y. Moreover, when the inequalities are all non-strict and the coefficients are all rational numbers, then the inverse image is quasi-compact.
x 12 = (0, 0)
x 2 = (1, 0) x ∅ = (1, 1) x 1 = (0, 1) Figure 1. F Σ when g = 2
Given I ⊆ Σ, we define
F I := τ ∈Σ F I,τ , where F I,τ = [0, 1], τ ∈ I [1, 1], τ ∈ I. Then F I is a closed |I|-dimensional hypercube in ([0, 1] ∩ Q) Σ = F Σ . We also define x I ∈ [0, 1] Σ to be the vertex x I,τ = 0, τ ∈ I 1, τ ∈ I.
Hence the vertices of F I are exactly the
x J 's with J ⊆ I. Denote by YF I the quasi-compact admissible open deg −1 F I of Y. Definition 2.3. Let p | p be a prime of F . For τ ∈ Σ p , define the twisted directional degreẽ deg τ : Y → [0, p fp − 1 p − 1 ] ∩ Q byd eg τ := fp−1 j=0 p fp−1−j deg σ j •τ = p fp−1 deg τ +p fp−2 deg σ•τ + · · · + deg σ f p −1 •τ .
Here σ is the Frobenius automorphism of the unramified extension L over Q p , lifting x → x p mod p. We also defined
eg : Y → ([0, p fp − 1 p − 1 ] ∩ Q) Σ y → (d eg τ y) τ .
We use the overhead tilde notation(·) to denote the image under the linear transformation
R Σ → R Σ (x τ ) τ → (x τ ) τ , wherex τ = fp−1 j=0 p fp−1−j x σ j •τ for τ ∈ Σ p .
In particular, if (x τ ) τ = deg y for some y ∈ Y, then (x τ ) τ =d eg y. For example,x I is the vertex ofF Σ given byx
I,τ = fp−1 j=0 p fp−1−j x I,σ j •τ for τ ∈ Σ p .
See Figures 1 and 2 for an example of F Σ andF Σ .
x 12 = (0, 0)x 2 = (p, 1)
x ∅ = (p + 1, p + 1)
x 1 = (1, p) Figure 2.F Σ when g = 2 2.3. Hilbert modular forms. Let ω = ω A univ be the sheaf of relative differentials of the universal abelian scheme over Y . The sheaf ω is an O F ⊗ Z O Y -module, locally free of rank 1. The O F -module structure on ω provides the decomposition with respect to embeddings τ : F → L ω = τ ∈Σ ω τ , where each ω τ is an O Y -module, locally free of rank 1. Given k = (k τ ) τ ∈Σ ∈ Z Σ , we define an invertible sheaf on Y ω k = τ ∈Σ ω kτ τ .
We use the same notation ω k for the invertible sheaf on Y coming from analytifying ω k . The space of Hilbert modular forms of level Γ 1 (N )∩Γ 0 (p) and weight k is defined to be H 0 (Y, ω k ). By GAGA and Koecher principle, it is the same as H 0 (Y, ω k ) [PS17, Proposition 5.1.2].
Definition 2.4. Let I ⊆ Σ. The space of I-classical overconvergent Hilbert modular forms of level Γ 1 (N ) ∩ Γ 0 (p) and weight k is
H 0, † (I, ω k ) := lim − → V H 0 (V, ω k ),
where V runs through strict neighborhoods of YF I in Y.
When I = ∅, I-classical simply means overconvergent, and when I = Σ, I-classical means classical. Whenever J ⊆ I, we have a map
H 0, † (I, ω k ) → H 0, † (J, ω k )
given by restriction. This is an injective map.
2.4. U p -operators. Let p | p be a prime of F above p and f p the residue degree of p.
Let Y (p) → Spec L be the moduli space whose S-points consist of (A, H, H 1 ), where (A, H) ∈ Y (S) and H 1 ⊆ A[p] is a finite flat isotropic O F -subgroup scheme of rank p fp and
H 1 = H[p]. We have the U p -correspondence of Y ⊗ O L L: Y (p) Y ⊗ O L L Y ⊗ O L L p 1 p 2
Here the projections are , which is a rigid analytic space over L. We have the induced U p -correspondence, p 1 and p 2 over (Y ⊗ L) an . Note that (Y ⊗ L) an contains Y. Let Y(p) := Y (p) an × (Y ⊗L) an ,p 1 Y. We then have the U p -correspondence, p 1 and p 2 over Y.
Given a subset U of Y, we then obtain a subset of Y
U p (U ) := p 2 p −1 1 (U ). Given two admissible opens U , V ⊆ Y such that U p (V) ⊆ U , we have U p : ω k (U ) → ω k (V) defined by (U p f )(A, H) = 1 p fp (A/H 1 ,H)∈Up(A,H) pr * f (A/H 1 ,H),
Partial classicality
The content of this section is to prove the following partial classicality theorem.
Theorem 3.1. Let f be an overconvergent Hilbert modular form of weight k. Let I ⊆ Σ. Assume that for all p | p, U p (f ) = a p f such that
val p (a p ) < inf τ ∈I∩Σp {k τ } − f p .(1)
Then f is I-classical.
Remark 3.2. In the case of I = Σ, this is a theorem of Kassaei [Kas16] or Pilloni-Stroh [PS17]. Although when I = Σ, Bijakowski [Bij16] proved a classicality theorem not assuming p is unramified, it is Kassaei's approach that is more suitable for partial classicality. Both use the idea of analytic continuation. Kassaei made efforts to analyze how U p -operators affect deg τ for all τ ∈ Σ p , but only when p is unramified. On the other hand, Bijakowski was able to use only deg H[p] to prove the classicality even when p is ramified. In the situation of partial classicality, the weight k τ with τ ∈ Σ in the slope condition is independent of each other, while the U p -operator intertwines all directional degrees inducing p, so we do need to understand the directional degrees. Throughout the section, we will assume that p is inert in F . To prove Theorem 3.1 for a general unramified p, we can apply the same argument to each prime p | p. For example, see [Sas10] and [PS17, Lemma 7.4.2]. Now we begin to prove Theorem 3.1 assuming p is inert in F ; in particular, f p = g. We will show that if U p (f ) = a p f such that val p (a p ) < inf τ ∈I k τ − g, then f is J-classical for all J ⊆ I, and hence f is I-classical. We do this by induction on |J|. For |J| = 0, it simply means that f is overconvergent, which is true by assumption. Assume that f is J-classical for all J I, say f is defined on a strict neighborhood of YF J = deg −1 F J . In particular, f is defined on a strict neighborhood of deg −1 x J for all J I.
Automatic analytic continuation.
In the subsection, with the assumption that the slope of f is finite (but not necessarily small), we can already show that f can be analytically continued to a large region in YF I .
Let I ⊆ Σ and ǫ > 0. Define
U I (ǫ) = {y ∈ Y :
τ ∈Id eg τ y ≥ τ ∈Ix I,τ + ǫ,d eg τ y ≥ p g−2 + · · · + 1 + ǫ, ∀τ ∈ I}.
See Figures 3 and 4 for examples of the image of U I (ǫ) under deg, and Figures 5 and 6 for examples of the image of U I (ǫ) underd eg. Because U I (ǫ) is defined by a finite number of affine inequalities withd eg τ (equivalently, with deg τ ), we know that U I (ǫ) is an admissible open of Y. Note that whenever ǫ ′ < ǫ, we have
U I (ǫ ′ ) ⊇ U I (ǫ).
Let f be an overconvergent Hilbert modular form of weight k. Assume that U p (f ) = a p f with val p (a p ) < ∞. Proof. First of all, note that U I (ǫ) is U p -stable because U p increases twisted directional degrees (Proposition 2.5(1)).
By Proposition 2.5(2), U p strictly increases τ ∈Σd eg τ except at points y ∈ Y such that deg y ∈ {0, 1} g , i.e., deg y = x J for some J ⊆ Σ. Suppose that y ∈ U I (ǫ) satisfies deg y = x J . We claim that J I. Indeed, for τ ∈ J,d eg τ y ≤ p g−2 + · · · + 1. Hence the second condition of U I (ǫ) deg τ y ≥ p g−2 + · · · + 1 + ǫ, ∀τ ∈ I says that τ / ∈ I implies τ ∈ J, i.e., J ⊆ I. The first condition of U I (ǫ)
V J = {y ∈ Y : deg τ y ≤ ǫ τ if τ ∈ J, deg τ y ≥ 1 − ǫ τ , if τ ∈ J},
for some rational ǫ τ > 0. On the other hand, let ǫ ′ τ < ǫ τ be a rational number, and define
V = y ∈ Y : deg τ y ≥ ǫ ′ τ if τ ∈ I, deg τ y ≤ 1 − ǫ ′ τ , if τ ∈ I; τ ∈Id eg τ y ≥ τ ∈Ix I,τ + ǫ,d eg τ y ≥ p g−2 + · · · + 1 + ǫ, ∀τ ∈ I .
Because V J 's and V are defined by a finite number of affine non-strict inequalities with rational coefficients, they are quasi-compact admissible opens of U I (ǫ). We hence have an admissible cover
U I (ǫ) = J I V J ∪ V.
Since V is disjoint from deg −1 x J for any J I from its definition, U p strictly increases τ ∈Σd eg τ on V. Using the Maximum Modulus Principle, the quasi-compactness of V implies that there is a positive lower bound for the increase of τ ∈Σd eg τ under U p on V. 3.2. Analytic continuation near vertices. In this subsection, we will make use of the small slope assumption (1) to extend f to a strict neighborhood of deg −1 x I .
Because U I (ǫ) is U p -stable, there exists M > 0 such that U M p V ⊆ J I V J . Since f is defined on J I V J ,
Let's first give an outline of the strategy. By (1), for any small enough ǫ > 0 we have
val p (a p ) ≤ inf τ ∈I k τ − g − ǫ τ ∈I k τ .(2)
Possibly making it smaller, we will first fix such a rational number ǫ. Then we will choose a rational number δ > 0 based on ǫ, and define a sequence of strict neighborhoods S I,0 (δ) ⊇ S I,1 (δ) ⊇ · · · of deg −1 x I . When δ ′ < δ we will show that S I,m (δ ′ ) S I,m (δ). We have extended f to U I (δ) by Lemma 3.3. Further applying some power of Up ap , we will be able to extend f to S I,0 (δ) \ S I,m (δ ′ ), named f m . We will also define F m on S I,m (δ). With the help of the estimates in Section 3.3, we will show that when m → ∞, f m and F m glue to define an extension of f on S I,0 (δ).
To begin, we prove the following lemma regarding the twisted directional degrees of points in the set U p (y), when y ∈ Y satisfies deg y = x I . The lemma will be used to decompose the U pcorrespondence Y(p) over S I,1 (δ) into the special part Y(p) sp and the non-special part Y(p) nsp , and so the U p -operator becomes U sp p + U nsp p . i. If y, y 1 ∈d eg −1x I for some I ⊆ Σ, theñ
deg τ H 2 = inf(d eg τ H,d eg τ H 1 ), for all τ ∈ Σ.
ii. There exists arbitrarily small positive rational number ǫ so that if |d eg τ (y) −x I,τ | ≤ ǫ and |d eg τ (y 1 ) −x I,τ | ≤ ǫ for some I ⊆ Σ, theñ
deg τ H 2 = inf(d eg τ H,d eg τ H 1 ), for all τ ∈ Σ.
In particular, y 2 ∈ U ∅ (ǫ).
Proof. For the proof of i., see [Kas16, Lemma 5.1.5 1.] The first statement of ii. follows from [Kas16, Lemma 5.1.5 2(a)].
The only statement remained to be proved is the one after "In particular". By assumption,
deg τ H 2 = inf(d eg τ H,d eg τ H 1 ) = d eg τ H if τ ∈ Ĩ deg τ H 1 if τ ∈ I andd eg τ y 2 = (p g−1 + · · · + 1) −d eg τ H 2 = (p g−1 + · · · + 1) −d eg τ H τ ∈ I (p g−1 + · · · + 1) −d eg τ H 1 τ ∈ I. ≥ (p g−1 + · · · + 1) −x I,τ − ǫ τ ∈ Ĩ x I,τ − ǫ τ ∈ I ≥ p g−1 − ǫ
If we further require that ǫ < 1 2 (p g−1 − p g−2 − · · · − 1), thend eg τ y 2 ≥ p g−2 + · · · + 1 + ǫ, i.e., y 2 ∈ U ∅ (ǫ).
Corollary 3.5. Let I ⊆ Σ and I = ∅. Let ǫ be a rational number as in Lemma 3.4 ii. such that ǫ < 1 2 (p g−1 − p g−2 − · · · − 1). Let y ∈ Y be such that |d eg τ (y) −x I,τ | ≤ ǫ for all τ ∈ Σ. Then there exists at most one point y 1 ∈ U p (y) such that |d eg τ (y 1 ) −x I,τ | ≤ ǫ for all τ ∈ Σ.
Proof. By the proof of Lemma 3.4 ii., if y 2 ∈ U p (y) and y 2 = y 1 , thend eg τ (y 2 ) ≥ p g−1 − ǫ for all τ ∈ Σ. Since I = ∅, we pick an arbitrary τ 0 ∈ I. Theñ deg τ 0 (y 2 ) − x I,τ 0 ≥ (p g−1 − ǫ) − (p g−2 + · · · + 1) > ǫ.
For any rational number δ > 0, consider the strict neighborhood of deg −1 x I : Proposition 3.7. Let f be an overconvergent Hilbert modular form of weight k. Let I ⊆ Σ. Suppose that f is defined on a strict neighborhood of deg −1 x J for all J I. Let ǫ be a small enough rational number as in Lemma 3.4 ii. such that ǫ < 1 2 (p g−1 − p g−2 − · · · − 1), and that val p (a p ) ≤ inf
S I,0 (δ) := y ∈ Y : τ ∈Id eg τ y ≤ τ ∈Ix I,τ + δ,d eg τ y ≥x I,τ − δ, ∀τ ∈ I ,τ ∈I k τ − g − ǫ τ ∈I k τ .
Let δ > 0 be a rational number so that S I,0 (δ) ⊆ {y ∈ Y : |d eg τ y −x I,τ | < ǫ} and S I,0 (δ) ⊆ {y ∈ Y : | deg τ y − x I,τ | < ǫ}. Then f can be extended to S I,0 (δ), which is a strict neighborhood of deg −1 x I .
Proof. By definition, S I,m−1 (δ) ⊇ S I,m (δ). In addition, U m p (S I,0 (δ) \ S I,m (δ)) ⊆ V I (δ). By Lemma 3.3, we can extend f to V I (δ) ⊆ U I (δ). Then we can further extend f by (
Up ap ) m f to (U p ) −m V I (δ) ⊇ S I,0 (δ) \ S I,m (δ).
Similarly, for any other rational number δ ′ < δ, we can extend f by (
Up ap ) m f to (U p ) −m V I (δ ′ ) ⊇ S I,0 (δ ′ ) \ S I,m (δ ′ )
. Because S I,0 (δ) \ S I,m (δ) and S I,0 (δ ′ ) \ S I,m (δ ′ ) form an admissible covering of S I,0 (δ) \ S I,m (δ ′ ), we can actually extend f to S I,0 (δ) \ S I,m (δ ′ ).
We denote by f m the extension of f to S I,0 (δ) \ S I,m (δ ′ ).
On the other hand, by Lemma 3.3, we can extend f to U ∅ (ǫ). Then
F m := m−1 j=0 ( 1 a p ) j+1 U nsp p (U sp p ) j f can be defined on (U sp p ) −(m−1) (U nsp p ) −1 (U ∅ (ǫ)) ⊇ S I,m (δ)
. Assume the norm estimates in Proposition 3.8 in the next subsection. By (2), we can choose a subsequence so that F m mod p m and f m mod p m glue as h m (only defined modulo p m ) under the admissible covering S I,0 (δ)\S I,m (δ ′ ) and S I,m (δ) of S I,0 (δ). We have h m ≡ f (mod p m ) on S I,0 (δ)\ S I,m (δ ′ ). By (3), we can further choose a subsequence so that h m+1 mod p m agrees with h m mod p m on S I,m+1 (δ). Hence h = lim m→∞ h m is defined on S I,0 (δ), and h = f on S I,0 (δ) \ m S I,m (δ ′ ). Hence h is the desired extension of f to S I,0 (δ).
3.3. Norm estimates. Assume that val p (a p ) ≤ inf τ ∈I k τ −g−ǫ τ ∈I k τ . Choose a rational number δ > 0 so that S I,0 (δ) ⊆ {y ∈ Y : |d eg τ y −x I,τ | < ǫ} and S I,0 (δ) ⊆ {y ∈ Y : | deg τ y − x I,τ | < ǫ}. Also let δ ′ < δ be another positive rational number.
Let f m defined on S I,0 (δ) and F m defined on S I,0 (δ) \ S I,m (δ ′ ) as in the previous section. The following proposition records the norm estimates used to glue f m and F m in the previous section.
Proposition 3.8.
(1) |F m | S I,m (δ) and |f m | S I,0 (δ)\S I,m (δ ′ ) are bounded.
(2) |F m − f m | S I,m (δ)\S I,m (δ ′ ) → 0.
(3) |F m+1 − F m | S I,m+1 (δ) → 0.
We need the following two lemmas to prove Proposition 3.8.
Lemma 3.9. Let V ⊆ S I,1 (δ) and h ∈ ω k (U sp p (V)). Then |U sp p (h)| V ≤ p g− τ ∈I kτ (1−ǫ) |h| U sp p (V) . In particular, if val p (a p ) < inf τ ∈I k τ − g − ǫ τ ∈I k τ , then
| U sp p a p h| V ≤ p −µ |h| U sp p (V)
for some small enough µ > 0.
Hence U is a strict neighborhood of YF I . We have shown in the proof of Lemma 3.3 that the condition of U implies that if y ∈ U is such that deg(y) = x J for some J ⊆ Σ, then J ⊆ I. Moreover, U is U p -stable because U p increases twisted directional degrees (Proposition 2.5(1)). For each J ⊆ I, let V J be a strict neighborhood of deg −1 x J on which f is defined, and we explicitly choose V J in the form V J = {y ∈ Y : deg τ y ≤ ǫ τ if τ ∈ J, deg τ y ≥ 1 − ǫ τ , if τ ∈ J}, for some rational ǫ τ > 0. Let ǫ ′ τ < ǫ τ be a rational number, and define the quasi-compact admissible open V = y ∈ Y : deg τ y ≥ ǫ ′ τ if τ ∈ I, deg τ y ≤ 1 − ǫ ′ τ , if τ ∈ I; deg τ y ≥ p g−2 + · · · + 1 + ǫ, ∀τ ∈ I .
We have an admissible cover U = J⊆I V J ∪ V.
Since V is disjoint from deg −1 x J for any J ⊆ I from its definition, U p strictly increases τ ∈Σd eg τ on V by Proposition 2.5(2). Using the Maximum Modulus Principle, the quasi-compactness of V implies that there is a positive lower bound for the increase of τ ∈Σd eg τ under U p on V. Because U is U p -stable, there exists M > 0 such that U M p V ⊆ J⊆I V J . Since f is defined on J⊆I V J , we may define f on V by ( Up ap ) M f . On the intersection ( J⊆I V J ) ∩ V, the definitions of f coincide since a p is the U p -eigenvalue of f . We can then define f on the whole U through the admissible cover U = J⊆I V J ∪ V.
2 .
2Partially classical overconvergent forms 2.1. Hilbert modular varieties. Let N ≥ 4 be an integer, and p ∤ N . Let c be a fractional ideal of F . Denote by c + ⊆ c the cone of totally positive elements, i.e., the elements in c which are positive under every embedding τ : F → R. Let Y c → Spec O L be the Hilbert modular scheme classifying (A, H) = (A/S, i, λ, α, H) where
be a short exact sequence of finite flat group schemes over S. Then deg G = deg G ′ + deg G ′′ . Lemma 2.2. [Far10, p. 2] Let λ : A → B be an isogeny of p-power degree between abelian schemes over S. Let G := ker λ. Let ω A/S and ω B/S be the sheaves of invariant differentials of A and B, respectively. Let λ * : ω B/S → ω A/S be the induced pullback map. Then deg G = val p (det λ * ).
the image of H under A → A/H 1 . Let Y (p) an be the rigid analytification of Y (p) [BLR95, Section 5.4, Corollary 5]
Proposition 2. 5 .
5Let y = (A, H) ∈ Y. Let p | p be a prime of F above p, and y ′ = (A/H 1 ,H) ∈ U p (y). Then (1)d eg τ (y ′ ) ≥d eg τ (y) for all τ ∈ Σ p , and (2) if τ ∈Σpd eg τ y ′ = τ ∈Σpd eg τ y, equivalently, τ ∈Σp deg τ y ′ = τ ∈Σp deg τ y, then deg τ y ∈ {0, 1} for all τ ∈ Σ p .
Figure 3 .
3deg U 1 (ǫ) when g = 2
Figure 4 . 2 Figure 5
425deg U Σ (ǫ) when g = .d eg U 1 (ǫ) when g = 2Figure 6.d eg U Σ (ǫ) when g = 2
Lemma 3 . 3 .
33Let I ⊆ Σ. Suppose that f is defined on a strict neighborhood of deg −1 x J =d eg −1x Jfor all J I. Then f can be extended to U I (ǫ) for any rational number ǫ > 0.
that J = I. For each J I, let V J be a strict neighborhood of deg −1 x J on which f is defined. Moreover we can choose V J in the form
we may define f on V by ( Up ap ) M f . On the intersection ( J I V J ) ∩ V, the definitions of f coincide since a p is the U p -eigenvalue of f . We can then define f on the whole U I (ǫ) through the admissible cover U I (ǫ) = J I V J ∪ V.
Lemma 3. 4 .
4Let y = (A, H) ∈ Y. Let y 1 = (A/H 1 ,H = A[p]/H 1 ) and y 2 = (A/H 2 ,H = A[p]/H 2 ) be in U p (y) and y 1 = y 2 .
which is a quasi-compact admissible open. Recall from Section 2.4 that the U p -correspondence is given byp 1 : Y(p) → Y, (A, H, H 1 ) → (A, H) and p 2 : Y(p) → Y, (A, H, H 1 ) → (A/H 1 ,H). Define S I,1 (δ) := p 1 (p −1 1 S I,0 (δ) ∩ p −1 2 S I,0 (δ)), which is a quasi-compact admissible open of Y because it is the pushforward of a quasi-compact admissible open by the finite étale morphism p 1 . Note that S I,1 (δ) = {y ∈ S I,0 (δ) : ∃y 1 ∈ U p (y) also in S I,0 (δ)}, so S I,1 (δ) is called the special locus of order 1 in S I,0 (δ).
where pr : A → A/H 1 is the natural projection.Werecord the dynamic of U p with respect to the (twisted) directional degrees. See [Kas16, Proposition 5.1.4, 5.1.14] or [PS17, Proposition 4.4.1, 4.4.2].
Let ǫ be a small enough rational number as in Lemma 3.4 ii. such that ǫ < 1 2 (p g−1 −p g−2 −· · ·−1), and that the small slope condition (2) is satisfied. Note that the S I,0 (δ)'s contain a fundamental system of strict neighborhoods of deg −1 x I . Hence we choose a rational number δ > 0 so that S I,0 (δ) ⊆ {y ∈ Y : |d eg τ y −x I,τ | < ǫ} and S I,0 (δ) ⊆ {y ∈ Y : | deg τ y − x I,τ | < ǫ}. With this choice of δ, we see by Corollary 3.5 that the y 1 in the definition of S I,1 (δ) is unique.Hence we have a correspondence Y(p) sp := p −1 1 S I,0 (δ) ∩ p −1 2 S I,0 (δ) ⊆ Y(p) Y(p) spwhere p sp i is the restriction of p i to Y(p) sp , and p sp 1 is an isomorphism. Then as before in Section 2.4, for any subset U ⊆ S I,1 (δ), let U sp Then S I,0 (δ)∪V I (δ) is U p -stable because U p increases twisted directional degrees (Proposition 2.5(1)). Hence we have U p (S I,0 (δ) \ S I,1 (δ)) ⊆ V I (δ). Note that V I (δ) ⊆ U I (δ), and the latter was defined in Section 3.1.Lemma 3.6. Let δ ′ < δ be two positive rational numbers. Then S I,1 (δ) is a strict neighborhood of S I,1 (δ ′ ).Proof. Because S I,0 (δ) is defined by inequalities of twisted directional degrees, when δ ′ < δ are two positive rational numbers, then S I,0 (δ) is a strict neighborhood of S I,0 (δ ′ ). By definition,Since p 1 is finite étale, pushforward by p 1 preserves quasicompact admissible opens, and hence S I,1 (δ) is a strict neighborhood of S I,1 (δ ′ ).As explained above, for any admissible open, which is quasi-compact because S I,0 (δ) is. Lemma 3.6 says that if δ ′ < δ are two positive rational numbers, then S I,1 (δ) and S I,0 (δ) \ S I,1 (δ ′ ) form an admissible covering of S I,0 (δ). Then S I,m (δ) and S I,0 (δ) \ S I,m (δ ′ ) also form an admissible covering of S I,0 (δ). Now we are ready to prove analytic continuation near vertices.Proof. Recall that U p is defined byLet y ∈ V and y 1 ∈ U sp p (y). ThenBy assumption, y 1 ∈ V ⊆ S I,0 (δ), i.e., τ ∈Id eg τ y 1 ≤ τ ∈Ix I + δ,d eg τ y 1 ≥x I − δ. Hence by our choice of δ, we have. Proof. Recall that we have fixed δ ′ < δ, and f m is defined on S I,0 (δ) \ S I,m (δ ′ ). In particular,(1) Because f is defined on the quasi-compact open V I (δ), |f | V I (δ) is bounded. Since U p is a compact operator, |f 1 | S I,0 (δ)\S I,1 (δ) ≤ | Up ap f | V I (δ) is also bounded. Similarly, |f 1 | S I,0 (δ ′ )\S I,1 (δ ′ ) is bounded, and hence |f 1 | S I,0 (δ)\S I,1 (δ ′ ) is bounded.We will show that |f m | S I,0 (δ)\S I,m (δ ′ ) ≤ sup(|f 1 | S I,0 (δ)\S I,1 (δ ′ ) , |F 1 | S I,m (δ) ) for all m ≥ 1. As for |F m | S I,m (δ) , by Lemma 3.9,= |F 1 | S I,1 (δ) .(2) By Lemma 3.10 and Lemma 3.9,→ 0 as m → ∞.(3) By Lemma 3.9,3.4. Finishing the proof of Theorem 3.1. Following the paragraph just before Section 3.1, we assume that the overconvergent form f is defined on a strict neighborhood of deg −1 x J for all J I.We also assume that f satisfies the small slope condition (1). Let ǫ be a small enough rational number as in Lemma 3.4 ii. such that ǫ < 1 2 (p g−1 − p g−2 − · · · − 1), and thatBy Proposition 3.7 we can extend f to a strict neighborhood S I,0 (δ) of deg −1 x I for any small enough rational number δ > 0. Note that the vertices in F I are exactly the x J 's with J ⊆ I, so we have extended f to a strict neighborhood of the inverse image of deg of all the vertices of F I . We will show that f can be extended to a strict neighborhood U of YF I , again using the argument in Lemma 3.3 that U p strictly increases the sum of twisted directional degrees when the deg is not one of the vertices of [0, 1] g .Define a quasi-compact admissible open U = {y ∈ Y :d eg τ y ≥ p g−2 + · · · + 1 + ǫ, ∀τ ∈ I}.Recall that YF I = {y ∈ Y : deg τ y = 1, ∀τ ∈ I}. If y ∈ YF I , then for τ ∈ I, deg τ y ≥ p g−1 > p g−2 + · · · + 1 + ǫ.
Stéphane Bijakowski, MR 3581175Classicité de formes modulaires de Hilbert. Stéphane Bijakowski, Classicité de formes modulaires de Hilbert, Astérisque (2016), no. 382, 49-71. MR 3581175
Formal and rigid geometry. III. The relative maximum principle. Siegfried Bosch, Werner Lütkebohmert, Michel Raynaud, MR 1329445Math. Ann. 3021Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Formal and rigid geometry. III. The relative maximum principle, Math. Ann. 302 (1995), no. 1, 1-29. MR 1329445
Conjectures de classicité sur les formes de hilbert surconvergentes de pente finie. Christophe Breuil, Christophe Breuil, Conjectures de classicité sur les formes de hilbert surconvergentes de pente finie, https://www.imo.universite-paris-saclay.fr/ breuil/PUBLICATIONS/classicHilbert.pdf, March 2010.
Parabolic eigenvarieties via overconvergent cohomology. Daniel Barrera Salazar, Chris Williams, 961-995. MR 4311626Math. Z. 2991-2Daniel Barrera Salazar and Chris Williams, Parabolic eigenvarieties via overconvergent cohomology, Math. Z. 299 (2021), no. 1-2, 961-995. MR 4311626
Analytic continuation of overconvergent eigenforms. Kevin Buzzard, J. Amer. Math. Soc. 161MRKevin Buzzard, Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc. 16 (2003), no. 1, 29-55. MR 1937198
Classical and overconvergent modular forms. Robert F Coleman, MR 1369416Invent. Math. 1241-3Robert F. Coleman, Classical and overconvergent modular forms, Invent. Math. 124 (1996), no. 1-3, 215- 241. MR 1369416
L-invariants, partially de Rham families, and local-global compatibility. Yiwen Ding, MR 3711132Ann. Inst. Fourier (Grenoble). 4Yiwen Ding, L-invariants, partially de Rham families, and local-global compatibility, Ann. Inst. Fourier (Grenoble) 67 (2017), no. 4, 1457-1519. MR 3711132
La filtration de Harder-Narasimhan des schémas en groupes finis et plats. Laurent Fargues, MR 2673421J. Reine Angew. Math. 645Laurent Fargues, La filtration de Harder-Narasimhan des schémas en groupes finis et plats, J. Reine Angew. Math. 645 (2010), 1-39. MR 2673421
A gluing lemma and overconvergent modular forms. L Payman, Kassaei, MR 2219265Duke Math. J. 1323Payman L. Kassaei, A gluing lemma and overconvergent modular forms, Duke Math. J. 132 (2006), no. 3, 509-529. MR 2219265
Analytic continuation of overconvergent Hilbert modular forms. MR 3581174Astérisque. 382, Analytic continuation of overconvergent Hilbert modular forms, Astérisque (2016), no. 382, 1-48. MR 3581174
Surconvergence et classicité: le cas Hilbert. Vincent Pilloni, Benoît Stroh, MR 3733761J. Ramanujan Math. Soc. 324Vincent Pilloni and Benoît Stroh, Surconvergence et classicité: le cas Hilbert, J. Ramanujan Math. Soc. 32 (2017), no. 4, 355-396. MR 3733761
Michel Raynaud, MR 419467Schémas en groupes de type. 102Michel Raynaud, Schémas en groupes de type (p, . . . , p), Bull. Soc. Math. France 102 (1974), 241-280. MR 419467
Analytic continuation of overconvergent Hilbert eigenforms in the totally split case. Shu Sasaki, 541-560. MR 2644926Compos. Math. 1463Shu Sasaki, Analytic continuation of overconvergent Hilbert eigenforms in the totally split case, Compos. Math. 146 (2010), no. 3, 541-560. MR 2644926
Yichao Tian, Liang Xiao, MR 3581176p-adic cohomology and classicality of overconvergent Hilbert modular forms. Yichao Tian and Liang Xiao, p-adic cohomology and classicality of overconvergent Hilbert modular forms, Astérisque (2016), no. 382, 73-162. MR 3581176
| [] |
[
"Some cut-generating functions for second-order conic sets",
"Some cut-generating functions for second-order conic sets"
] | [
"Asteroide Santana \nISyE\nGeorgia Institute of Technology\n765 Ferst Drive NW30332-0205AtlantaGAUSA\n",
"Santanu S Dey \nISyE\nGeorgia Institute of Technology\n765 Ferst Drive NW30332-0205AtlantaGAUSA\n"
] | [
"ISyE\nGeorgia Institute of Technology\n765 Ferst Drive NW30332-0205AtlantaGAUSA",
"ISyE\nGeorgia Institute of Technology\n765 Ferst Drive NW30332-0205AtlantaGAUSA"
] | [] | In this paper, we study cut generating functions for conic sets. Our first main result shows that if the conic set is bounded, then cut generating functions for integer linear programs can easily be adapted to give the integer hull of the conic integer program. Then we introduce a new class of cut generating functions which are non-decreasing with respect to second-order cone. We show that, under some minor technical conditions, these functions together with integer linear programming-based functions are sufficient to yield the integer hull of intersections of conic sections in R 2 . | 10.1016/j.disopt.2016.11.001 | [
"https://arxiv.org/pdf/1606.00385v3.pdf"
] | 195,394 | 1606.00385 | 00525b91a81dd91cd806f3e3a1ff277804a424e1 |
Some cut-generating functions for second-order conic sets
11 Nov 2016
Asteroide Santana
ISyE
Georgia Institute of Technology
765 Ferst Drive NW30332-0205AtlantaGAUSA
Santanu S Dey
ISyE
Georgia Institute of Technology
765 Ferst Drive NW30332-0205AtlantaGAUSA
Some cut-generating functions for second-order conic sets
11 Nov 2016arXiv:1606.00385v3 [math.OC]integer conic programminginteger hull of conic setcut generating functionsubadditive functionsecond-order cone
In this paper, we study cut generating functions for conic sets. Our first main result shows that if the conic set is bounded, then cut generating functions for integer linear programs can easily be adapted to give the integer hull of the conic integer program. Then we introduce a new class of cut generating functions which are non-decreasing with respect to second-order cone. We show that, under some minor technical conditions, these functions together with integer linear programming-based functions are sufficient to yield the integer hull of intersections of conic sections in R 2 .
Then it is straightforward to see that the inequality
n j=1 f (A j )x j ≥ f (b),(2)
is valid for the conic integer program (1), where A j is the j-th column of A. We denote the set of functions satisfying (1.), (2.) and (3.) above as F K . In the paper [2], it was shown that, assuming a technical 'discrete Slater' condition holds, the closure of the convex hull of the set of integer feasible solutions to (1) is described by inequalities of the form (2) obtained from F K . This result from [2] generalizes result on subadditive duality of linear integer programs [3,4,5,6], that is inequalities (2) give the convex hull of (1) when K = R m + and the constraint matrix A is rational. Also see [7,8] for related models and results. In the case where K = R m + and assuming A is rational, a lot more is known about the subset of functions from F R m + that are sufficient to describe the convex hull of integer solutions (also called as the integer hull). For example, these functions have a constructive characterization using the Chvátal-Gomory procedure [9], it is sufficient to consider functions that are applied to every 2 n subset of constraints at a time (see [10], Theorem 16.5), or for a fixed A there is a finite list of functions independent of b that describes the integer hull [6].
The main goal of this paper is to similarly better understand structural properties of subsets of functions from F K that are sufficient to produce the integer hull of the underlying conic representable set {x ∈ R n | Ax K b}.
Main results
We will refer to the dual cone of a cone K as K * which we remind the reader is the set K * := {y ∈ R m | y ⊤ x ≥ 0 ∀x ∈ K}. Given a positive integer m, we denote the set {1, . . . , m} by [m]. And given a subset X of R n we denote its integer hull by X I .
Bounded sets
Given a regular cone K we call as linear composition the set of functions f obtained as follows: Let the vectors w 1 , w 2 , . . . , w p ∈ K * and the function f : R m → R be given by
f (v) = g((w 1 ) ⊤ v, (w 2 ) ⊤ v, · · · , (w p ) ⊤ v),(3)
where g ∈ F R p + satisfies g(u) = −g(−u) for all u ∈ R p . It is straightforward to see that linear composition functions belong to F K and also satisfy f (v) = − f (−v) for all v ∈ R m , which implies that f generates valid inequalities of the form (2) even when the variables are not required to be non-negative. Our first result describes a class of conic sets for which linear composition functions are sufficient to produce the convex hull. Theorem 1. Let K ⊆ R m be a regular cone. Consider the conic set T = {x ∈ R n | Ax K b}, where A ∈ R m×n and b ∈ R m . Assume T has nonempty interior. Let π ⊤ x ≥ π 0 be a valid inequality for T I where π ∈ Z n is non-zero. Assume B := {x ∈ T | π ⊤ x ≤ π 0 } is nonempty and bounded. Then, for some natural number p ≤ 2 n , there exist vectors y 1 , y 2 , . . . , y p ∈ K * such that π ⊤ x ≥ π 0 is a valid inequality for the integer hull of the polyhedron Q = {x ∈ R n | (y i ) ⊤ Ax ≥ (y i ) ⊤ b, i ∈ [p]}, where (y i ) ⊤ A is rational for all i ∈ [p].
We highlight here that particular care was taken in Theorem 1 to ensure that the outer approximating polyhedron has rational constraints.
Since a valid inequality for Q I can be obtained using a subadditive function g ∈ F R p + that satisfies g(u) = −g(−u) for all u ∈ R p [11] (note that the constraints matrix defining Q is rational), Theorem 1 implies that if a cut separates a bounded set from T , then it can be obtained using exactly one function (3) with p ≤ 2 n . Geometrically, Theorem 1 can be interpreted as the fact that if the set of points separated is bounded, then the cut can be obtained using a rational polyhedral outer approximation.
We obtain the following corollary immediately: If the set {x ∈ R n | Ax K b} is compact and has non-empty interior, then it is sufficient to restrict attention to linear composition functions to obtain the convex hull. A proof of Theorem 1 is presented in Section 3.
New family of cut-generating functions
In the previous section we stated that any valid inequality for the integer hull of a bounded conic set can be obtained using linear composition functions. So what happens when the underlying set is not bounded? Consider the simple unbounded set
T ′ = {(x 1 , x 2 ) ∈ R 2 + | x 1 x 2 ≥ 1}, which is one branch of a hyperbola 1 . This set is conic representable, that is T ′ = {x ∈ R 2 + | Ax K b}, where K is the second-order cone L 3 and A = 0 0 1 −1 1 1 , b = −2 0 0 .
(4) (We use the notation L m := x ∈ R m | x 2 1 + x 2 2 + · · · + x 2 m−1 ≤ x m to represent the second-order cone in R m .) The integer hull of T ′ is given by the following two inequalities:
x 1 ≥ 1, x 2 ≥ 1.(5)
It is straightforward to verify that the inequalities (5) are not valid for any polyhedral outer approximation of T ′ . Indeed any polyhedral outer approximation of T ′ contains integer points not belonging to T ′ (see Proposition 3). Therefore, applying the cut-generating recipe (3) a finite number of times (that is considering integer hulls of a finite number of polyhedral outer approximations of T ′ ) does not yield x 1 ≥ 1. However, we note here that we can use linear composition (3) to obtain a cut of the form x 1 + x 2 /k ≥ 1 where k ∈ Z + and k ≥ 1. Clearly
k∈Z + ,k≥1 x ∈ R 2 | x 1 + x 2 /k ≥ 1 = {x ∈ R 2 | x 1 ≥ 1}.
However, it would be much nicer if we could directly obtain x 1 ≥ 1 without resorting to obtaining it as an implication of an infinite sequence of cuts. Many papers [12,13,14,15,16,17,18,19,20] have explored various families of subadditive functions for linear integer programs. Our second result, in the same spirit, is a parametrized family of functions that belongs to F K , where K is the second-order cone L m . The formal result is as follows:
Theorem 2. Let j ∈ [m − 1]. Define Γ j := {γ ∈ R m | γ m ≥ m−1 i=1 |γ i |, γ m > |γ j |}. Suppose γ ∈ Γ j ∪ interior (L m ). Consider the real-valued function f γ : R m → R defined as: f γ (v) = γ ⊤ v + 1 if v j 0 and γ ⊤ v ∈ Z, γ ⊤ v otherwise.(6)
Then, f γ ∈ F L m . To see an example of use of f γ , consider j = 1 and γ = (0, 0.5, 0.5). Then applying the resulting function f γ to the columns of (4) we obtain the inequality x 1 ≥ 1.
Note that the validity of the first inequality in (5) can be explained via the disjunction x 1 ≤ 0 ∨ x 1 ≥ 1. Therefore, some of the cuts generated using (6) can be viewed as split disjunctive cuts. Significant research has gone into describing split disjunctive cuts (newer implied conic constraints) for conic sections [21,22,23,24,25,26,27]. However, to the best of our knowledge, there is no family of subadditive functions in F L m which have been described in closed form previously.
It is instructive to compare cuts obtained using (6) with two well-known approaches for generating cuts for the integer hull of second-order conic sets [28,29]. Note that the CG cuts described in [28] are a special case 2 of cuts generated via linear composition (3). Therefore as discussed above, the CG cuts described in [28] cannot generate (5) directly. The conic MIR procedure described in [29] begins with first generating an extended formulation which applied to T ′ would be of the form:
t 0 ≤ x 1 + x 2 t 1 ≥ 2 t 2 ≥ |x 1 − x 2 | t 0 ≥ | |t| | 2 x 1 , x 2 ∈ Z + , t ∈ R 3 + . Then, cuts for the set {(x, t 2 ) ∈ Z 2 + × R | t 2 ≥ |x 1 − x 2 |} are considered.
However, this set is integral in this case and therefore no cuts are obtained. Thus, the conic MIR procedure does not generate the inequalities (5).
Remark 1.
The function f γ defined in (6) is piecewise linear, and it is therefore tempting to think it may also belong to F R m + . However it is straightforward to check that f γ is not necessarily non-decreasing with respect to R 3 + . Let j = 1 and γ = (0, ρ, ρ) where ρ is a positive scalar. Then
f γ (v 1 , v 2 , v 3 ) = ρ(v 2 + v 3 ) + 1 if v 1 0 and ρ(v 2 + v 3 ) ∈ Z, ⌈ρ(v 2 + v 3 )⌉ otherwise.
Consider the vectors u = (0, 0, 1/ρ) and v = (−1, 0, 1/ρ).
Then u ≥ R 3 + v, whereas f γ (u) = 1 < 2 = f γ (v).
A proof of Theorem 2 is presented in Section 4.
Cuts for integer conic sets in R 2
As mentioned earlier, the family of functions (6) yields the inequalities (5). Indeed, we are able to verify a more general result in R 2 . To explain this result, we will need the following results:
Lemma 1. Let G be one branch of a hyperbola in R 2 . Then G can be represented as G = {x ∈ R 2 | Ax L 3 b}, where A ∈ R 3×2 is such that A 11 , A 12 = 0. Moreover, the asymptotes of G have equations (A 21 + A 31 )x 1 + (A 22 + A 32 )x 2 = b 3 + b 2 (7) (−A 21 + A 31 )x 1 + (−A 22 + A 32 )x 2 = b 3 − b 2 .(8)
In order to generate cuts for G in Lemma 1 using functions (6) we first require the variables to be non-negative. Therefore, let us write G as
A 1 x + 1 − A 1 x − 1 + A 2 x + 1 − A 2 x − 1 L 3 b (9) x + 1 , x − 1 , x + 2 , x − 2 ≥ 0 (10) x j = x + j − x − j j ∈ {1, 2}.(11)
Assuming that the asymptotes of G are rational, we may assume that the coefficients in (7) and (8) are integers and then let τ = gcd(A 21 + A 31 , A 22 + A 32 ). Let j = 1 and γ = (0, 1/τ, 1/τ). Then we apply the function f γ to obtain the following cut for (9), (10):
(A 21 + A 31 ) τ x + 1 − (A 21 + A 31 ) τ x − 1 + (A 22 + A 32 ) τ x + 2 − (A 22 + A 32 ) τ x − 2 ≥ f γ (b).(12)
Now, using (11) and observing that the coefficient of (12), j = 1, 2, we can project the inequality (12) to the space of the original x variables. The resulting cut is parallel to the asymptote (7). We can do a similar calculation to obtain a cut parallel to the other asymptote (8). We state all this concisely in the next proposition.
x + j is the negative of the coefficient of x − j inProposition 1. Let G = {x ∈ R 2 | Ax L 3
+ b} be one branch of a hyperbola with rational asymptotes, where A ∈ R 3×2 and A 11 , A 12 = 0. Then the following inequalities are valid for G I :
(u j ) ⊤ A 1 x 1 + (u j ) ⊤ A 2 x 2 ≥ τ j f γ j (b),(13)where u 1 = (0, 1, 1), u 2 = (0, −1, 1), τ j = gcd((u j ) ⊤ A 1 , (u j ) ⊤ A 2 ) and γ j := u j /τ j , j = 1, 2.
We are now ready to state the main result of this section. 1. If W ∩Z 2 = ∅, then this fact can be certified with the application of at most two inequalities generated from (3) or (13)
Theorem 3. Let W = i∈[m] W i , where W i = {x ∈ R 2 | A i x L m i b i }, A i ∈ R m i ×2 , b i ∈ R m i and L m i is the second-order cone in R m i . Assume W; 2. Assume interior(W) ∩ Z 2 ∅. If π ⊤ x ≥ π 0 defines a face of W I where π ∈ Z 2 is non-zero,
then this inequality can be obtained with application of exactly one function (3) or it is one of the inequalities (13).
Proof of Lemmma 1 and Theorem 3 are presented in Section 5.
Cutting-planes separating bounded set of points
In this section, we prove Theorem 1. We begin by stating three well-known lemmas.
Lemma 2.
Let K ∈ R n be a closed cone and let K * denote its dual. Then interior (
K * ) = {y ∈ R n | y ⊤ x > 0 ∀x ∈ K \ {0}}.
Hereafter, we will denote the recession cone of a set C by rec.cone(C) and the dual of rec.cone(C) by rec.cone * (C).
Lemma 3.
Let C ⊆ R n be a nonempty closed convex set. Then the following statements hold:
(i) for every c ∈ interior (rec.cone * (C)) the problem inf{c ⊤ x | x ∈ C} is bounded. (ii) for every c rec.cone * (C) the problem inf{c ⊤ x | x ∈ C} is unbounded.
Lemma 4 (Conic strong duality [30]). Let K ⊆ R m be a regular cone. Consider the conic set
T = {x ∈ R n | Ax K b}, where A ∈ R m×n and b ∈ R m . Assume interior T ∅. If c ∈ R n is such that inf{c ⊤ x | x ∈ T } is bounded, then there exists y ∈ K * such that y ⊤ A = c ⊤ and y ⊤ b = inf{c ⊤ x | x ∈ T }.
The next lemma states that under some conditions it is possible to separate a point from a set using a rational separating hyperplane.
Lemma 5. Let C ⊆ R n be a closed convex set. Assume interior (rec.cone * (C)) ∅. Let z C.
Then there exist π ∈ Q n , π 0, and π 0 ∈ R such that π ⊤ z < π 0 ≤ π ⊤ x for all x ∈ C.
Proof. The standard separation theorem ensures that there exist w ∈ R n , w 0, and w 0 ∈ R such that w ⊤ z < w 0 ≤ w ⊤ x for all x ∈ C. As interior (rec.cone * (C)) ∅ there exist w 1 , w 2 , . . . , w n+1 ∈ interior (rec.cone * (C)) affinely independent. For every
i ∈ [n + 1] let w i 0 = inf{(w i ) ⊤ x | x ∈ C}. In view of Lemma 3 we have that w i 0 is finite for all i ∈ [n + 1]. Since w 0 − w ⊤ z > 0 and z is fixed, we can chose ε i > 0, i ∈ [n + 1], such that | n+1 i=1 ε i (w i ) ⊤ z − n+1 i=1 ε i w i 0 | < w 0 − w ⊤ z.(14)
Moreover, since w 1 , w 2 , . . . , w n+1 are affinity independent, the cone generated by these vectors is full dimensional. Thus, the scalars
ε i > 0, i ∈ [n+1], can be chosen such that π := w+ n+1 i=1 ε i w i ∈ Q n . Now observe that π ⊤ z < w 0 + n+1 i=1 ε i w i 0 ≤ inf{w ⊤ x | x ∈ C} + n+1 i=1 inf{(ε i w i ) ⊤ x | x ∈ C} ≤ inf{(w ⊤ + n+1 i=1 ε i w i ) ⊤ x | x ∈ C} ≤ π ⊤ x ∀x ∈ C,
where the first strict inequality follows from (14). Therefore,
π ⊤ z < π 0 ≤ π ⊤ x for all x ∈ C, where π 0 := w 0 + n+1 i=1 ε i w i 0 .
The next result will imply Theorem 1.
Proposition 2. Let T be the set as in the statement of Lemma 4. Consider the set B
:= {x ∈ T | π ⊤ x ≤ π 0 }, where π ∈ Z n is non-zero.
Then B is bounded if and only if π ∈ interior(rec.cone * (T )), in which case for some natural number p ′ , there exist vectors y 1 , y 2 . . . , y p ′ ∈ K * such that the polyhedron
P = {x ∈ R n | π ⊤ x ≤ π 0 , (y i ) ⊤ Ax ≥ (y i ) ⊤ b, i ∈ [p ′ ]} contains B and P I = B I , where (y i ) ⊤ A is rational for all i ∈ [p ′ ].
Proof. Assume B is bounded. We claim that d ⊤ π > 0, for all d ∈ rec.cone(T ) \ {0}. Indeed, if d ∈ rec.cone(T ) is such that d ⊤ π ≤ 0, then d ∈ rec.cone(B), which implies that d = 0 since B is bounded. Now, in view of Lemma 2, the claim implies that π ∈ interior (rec.cone * (T )).
Assume π ∈ interior (rec.cone * (T )). As π ∈ Z n , let {v 1 , v 2 , . . . , v n−1 } ⊆ Q n be an orthogonal basis of the linear subspace orthogonal to π. Since π ∈ interior (rec.cone * (T )), there exists a positive constant ε such that w i := π + εv i and w i+n−1 := π − εv i belong to interior (rec.cone * (T )) for all i ∈ [n − 1]. As we may assume that ε is rational, we obtain that w i is rational for all i ∈ [2n − 2]. It follows from Lemma 3 and Lemma 4 that for all i ∈ [2n − 2] there exists
y i ∈ K * such that (y i ) ⊤ Ax ≥ (y i ) ⊤ b is a valid inequality for T , where (y i ) ⊤ A = w i ∈ Q n .
Since π ∈ interior (rec.cone * (T )), Lemma 3 and Lemma 4 also imply that there exists
y 2n−1 ∈ K * such that (y 2n−1 ) ⊤ Ax ≥ (y 2n−1 ) ⊤ b is a valid inequality for T , where (y 2n−1 ) ⊤ A = π ⊤ ∈ Q n . Now, let P 1 = {x ∈ R n | π ⊤ x ≤ π 0 , (y i ) ⊤ Ax ≥ (y i ) ⊤ b, i ∈ [2n − 1]}.
By our choice of w i and using the fact that (y 2n−1 ) ⊤ b ≤ π ⊤ x ≤ π 0 for all x ∈ P 1 (if π 0 ≤ (y 2n−1 ) ⊤ b, then P 1 = ∅), it is easy to verify that P 1 is bounded. Since P 1 contains B, we obtain that B is also bounded.
If (P 1 ) I = B I , then we are done by setting P to P 1 , in which case p ′ = 2n − 1. Otherwise, as P 1 is bounded, there is only a finite number of integer points z ∈ P 1 \ B. For each one of these points z, we construct a rational valid inequality w 0 ≤ w ⊤ x for T that is guaranteed by Lemma 5 that separates z from B, that is w ⊤ z < w 0 . It remains to show that this inequality can be obtained 'via dual multipliers': This is straightforward by again examining the conic program inf{w ⊤ x | x ∈ T } and applying Lemma 4. 7
Proof. of Theorem 1 Let π ⊤ x ≥ π 0 be a valid inequality for T I , where π ∈ Z n is non-zero. Suppose B = {x ∈ T | π ⊤ x ≤ π 0 } is nonempty and bounded. Then, by Proposition 2, using dual multipliers y 0 , y 1 , . . . , y p ′ ∈ K * , and letting
P = {x ∈ R n | π ⊤ x ≤ π 0 , (y i ) ⊤ Ax ≥ (y i ) ⊤ b, i ∈ [p ′ ]},
we have that (i) P ⊇ B and (ii) P ∩ Z n = B ∩ Z n . Note that interior(B) ∩ Z n = ∅ and the only integer points in B are those that satisfy π ⊤ x = π 0 . Now using an argument similar to Corollary 16.5a [10], there is a subset of 2 n inequalities defining P together with π T x < π 0 such that the resulting set contains no integer points. WLOG
{x ∈ R n | π ⊤ x ≤ π 0 , (y i ) ⊤ Ax ≥ (y i ) ⊤ b, i ∈ [p]} is lattice-free, where p ≤ 2 n , i.e., π ⊤ x ≥ π 0 is a valid inequality for the integer hull of Q = {x ∈ R n | (y i ) ⊤ Ax ≥ (y i ) ⊤ b, i ∈ [p]} where (y i ) ⊤ A ∈ Q n for i ∈ [p].
Remark 2.
If T ∩ Z n ∅, then using the same argument as in the proof of Corollary 16.6 [10] (also see [31]), the bound of 2 n in Theorem 1 can be improved to 2 n − 1.
The next proposition illustrates that if the set B in the statement of Theorem 1 is not bounded, then the result may not hold.
Proposition 3. Let T
′ := {(x ∈ R 2 + | x 1 x 2 ≥ 1}.
Every polyhedral outer approximation of T ′ contains points of the form (0, k) (and similarly points of form (k, 0)) for k sufficiently large natural number.
Proof. Suppose {x ∈ R 2 | α i 1 x 1 + α i 2 x 2 ≥ β i , i ∈ [q]
}, is a polyhedral outer approximation of T ′ where q is some natural number. Since the recession cone of this polyhedron contains the recession cone of T ′ , that is R 2 + , we have that α i 1 , α i 2 ≥ 0. We will prove that there exist points of the form (0, k) belonging to this outer approximation by showing that for all i ∈ [q] there exists a k i such that (
α i ) ⊤ (0, t) ≥ β i for all t ∈ [k i , ∞) ∩ Z. If α i 2 = 0, then β i ≤ 0 (since α i 1 /k + α i 2 k ≥ β i for all k ∈ R + ). Therefore k i = 0. If α i 2 > 0, then k i = β i /α i 2 .
A family of cut-generating functions in F L m and its properties
In this section, we show that f γ defined in (6) belongs to F K . Clearly f γ satisfies property (3.) in the definition of F K , that is f γ (0) = 0. In Proposition 4 and 5 we prove that f γ also satisfies properties (1.) and (2.).
Proposition 4. The function f γ defined in (6) is subadditive.
Proof. Let u, v ∈ R m . If at least one of these vectors fits in the first clause of (6), then we have
f γ (u + v) ≤ γ ⊤ (u + v) + 1 ≤ γ ⊤ u + γ ⊤ v + 1 ≤ f γ (u) + f γ (v).
Now, suppose that neither u nor v satisfies the first clause. If u + v does not fit in the first clause, then we are done because ⌈·⌉ is a subadditive function. Assume u + v satisfies the first clause, that is
u j + v j 0, γ ⊤ (u + v) = γ ⊤ u + γ ⊤ v ∈ Z.(15)
In this case, u j and v j cannot be simultaneously zero, say u j 0. Then
γ ⊤ u Z,(16)
because u does not satisfies the first clause. It follows from (15) and (16) that
γ ⊤ v Z.(17)
Finally, (15), (16), (17) together imply
f γ (u) + f γ (v) = γ ⊤ u + γ ⊤ v = γ ⊤ u + γ ⊤ v + 1 = f γ (u + v),
where the second inequality follows from the fact that γ ⊤ u + γ ⊤ v ∈ Z.
Lemma 6.
Let w ∈ L m and j ∈ [m − 1]. Let Γ j be the set as in the statement of Theorem 2. If γ ∈ L m , then γ ⊤ w ≥ 0. If, in addition, γ ∈ Γ j ∪ interior (L m ) and w j 0, then γ ⊤ w > 0.
Proof. We have that γ ∈ L m . Therefore, since w L m 0 and L m is a self-dual cone, we conclude that γ ⊤ w ≥ 0. Now, assume w j 0. If either γ or w is in the interior of L m , then it follows directly from Lemma 2 that γ ⊤ w > 0. Assume γ, w interior (L m ). Then
w m = w 2 1 + w 2 2 + · · · + w 2 m−1 (18) γ m = γ 2 1 + γ 2 2 + · · · + γ 2 m−1 .(19)
Two observations follows: (i) as w j 0, equation (18)
γ i w m + i∈[m−1]: γ i <0 −γ i w m > i∈[m−1]: γ i ≥0 γ i (−w i ) + i∈[m−1]: γ i <0 (−γ i )w i ⇒ i∈[m−1] |γ i |w m > − i∈[m−1] γ i w i ⇒ γ m w m > − i∈[m−1] γ i w i ,
where the last implication follows from the fact that γ m ≥ m−1 i=1 |γ i | and w m ≥ 0. The result follows from this last inequality. (6) is non-decreasing with respect to L m .
Proposition 5. The function f γ defined in
Proof. Let u, v ∈ R m . Suppose u L m v. By applying Lemma 6 to w = u − v we conclude that
γ ⊤ u ≥ γ ⊤ v,(20)
where the inequality (20) holds strictly whenever u j − v j 0. Now, we use these facts to prove that f γ (v) ≤ f γ (u). If u fits in the first clause of (6), then
f γ (v) ≤ γ ⊤ v + 1 ≤ γ ⊤ u + 1 = f γ (u),
where the second inequality follows from (20). Assume u does not satisfies the first clause. If v does not fit in the first clause, then the result follows directly from (20) and the fact that ⌈·⌉ is non-decreasing. Suppose v satisfies the first clause, that is v j 0 and γ ⊤ v ∈ Z. In this case, if u j = 0, then u j − v j 0 and hence (20) holds strictly. Therefore, we conclude that f γ (v) = γ ⊤ v + 1 ≤ ⌈γ ⊤ u⌉ = f γ (u). On the other hand, if u j 0, then γ ⊤ u Z (since u does not satisfy the first clause), and using (20) we obtain γ ⊤ v < ⌈γ ⊤ u⌉ and hence f γ (v) = γ ⊤ v + 1 ≤ ⌈γ ⊤ u⌉ = f γ (u), which completes the proof.
Application of cut-generating functions in R 2
In this section, we will prove Theorem 3. We begin with proofs of two technical lemmas.
Lemma 7. Let W i = {x ∈ R 2 | A i x L m i b i } be a parabola, where A i ∈ R m i ×2 , b i ∈ R m i and L m i is the second-order cone in R m i . If π ∈ rec.cone * (W i ) \ interior (rec.cone * (W i )), π 0, then the problem inf{π ⊤ x | x ∈ W i } is unbounded.
Proof. Up to a rotation, any parabola in R 2 can be written as {(x, y) ∈ R 2 | y ≥ ρ(x − x 0 ) 2 + y 0 }, where ρ > 0. In this case, the recession cone of the parabola is a vertical line. As π ∈ rec.cone * (W i ) \ interior (rec.cone * (W i )) we must have π 2 = 0, in which case π 1 0 and the problem is clearly unbounded.
Lemma 8.
Let W be the set as in the statement of Theorem 3. Assume, in addition, that W is unbounded. Let π 0 be such that π interior (rec.cone * (W)). If the problem
α := inf{π ⊤ x | x ∈ W} (21) is bounded, then there exists i 0 ∈ [m] such that α = inf{π ⊤ x | W i 0 }.(22)
Moreover,
W i 0 = {x ∈ R 2 | A i 0 x L m i 0 b i 0 } is either: (i) a half-space defined by π ⊤ x ≥ α;
or (ii) one branch of a hyperbola whose one of the asymptotes is orthogonal to π.
Proof. Since the primal problem (21) is bounded and strictly feasible, we have that its dual
sup{ m i=1 (b i ) ⊤ y i | m i=1 (y i ) ⊤ A i = π ⊤ , y i ∈ L * m i ∀i ∈ [m]}(23)
is solvable [30]. We will show that (23) admits an optimal solution for which y i = 0 for all i ∈ [m] except for one particular i 0 ∈ [m].
Since (21) is bounded, it follows from Lemma 3 that π ∈ rec.cone * (W). On the other hand, by assumption π is not in the interior of that cone. Therefore, using Lemma 2 we conclude that there exists a non-zero vector d 0 ∈ rec.cone(W) such that π ⊤ d 0 = 0. Then any feasible solution (y 1 , y 2 , · · · , y m ) of (23) satisfies
0 = π ⊤ d 0 = m i=1 (y i ) ⊤ A i d 0 .
Moreover, each term in this summation is non-negative since
A i d 0 L m i 0 (recall d 0 ∈ rec.cone(W)) and y i ∈ L * m i , for all i ∈ [m]. As a result, we have (y i ) ⊤ A i d 0 = 0 ∀i ∈ [m].
As d 0 is a non-zero vector in R 2 , we conclude that for each i ∈ [m] there must exist a scalar λ i such that
(y i ) ⊤ A i = λ i π ⊤ .(24)
We claim that λ i ≥ 0 for all i ∈ [m]. To prove the claim, all we need to show is that (y i ) ⊤ A i and π are in the same half-space. By assumption π ∈ rec.cone * (W). Since rec.cone * (W) is contained in a half-space (otherwise we would have rec.cone(W) = {0} which contradicts the 10 fact that W is unbounded), it is enough to prove that (y i ) ⊤ A i ∈ rec.cone * (W). To see why this is true, note that for all d ∈ rec.cone(
W i ) we have A i d L m i 0, which implies (y i ) ⊤ A i d ≥ 0. Thus, (y i ) ⊤ A i ∈ rec.cone * (W i ) ⊆ rec.cone * (W)
, where the last containment follows from the fact that rec.cone(W i ) ⊇ rec.cone(W). Now, suppose (y 1 , y 2 , · · · , y m ) is an optimal solution of the dual problem (23). If λ i = 0, then we must have (b i ) ⊤ y i = 0, because (b i ) ⊤ y i > 0 would imply the dual problem to be unbounded and (b i ) ⊤ y i < 0 would imply that the current solution is not optimal. Hence we have that if λ i = 0, then we can set y i = 0 without altering the objective value. On the other hand, since π 0, (24) combined with the equality in (23) imply that the λ's add up to 1. Thus, we cannot have
λ i = 0 for all i ∈ [m]. Suppose λ i , λ j > 0 for some i, j ∈ [m], i j. We claim that (b i ) ⊤ y i = (λ i /λ j )(b j ) ⊤ y j .
Without loss of generality, assume by contradiction that
(b i ) ⊤ y i < (λ i /λ j )(b j ) ⊤ y j . Then, since λ i + λ j ≤ 1 we obtain (b i ) ⊤ y i + (b j ) ⊤ y j < λ i λ j (b j ) ⊤ y j + (b j ) ⊤ y j ≤ 1 λ j (b j ) ⊤ y j .
In this case, we could set λ i = 0, λ j = 1 and y i = 0 to obtain a new feasible solution with objective value strictly larger. But this contradicts the fact that y is an optimal solution. Thus, the claim holds and by setting λ i = 0, λ j = 1 and y i = 0 we obtain a new feasible solution with the same objective value, and hence optimal. In this case, we set i 0 = j. Consider now the primal-dual pair
β := inf{π ⊤ x | A i 0 x L m i 0 b i 0 },(25)sup{(b i 0 ) ⊤ y i 0 | (y i 0 ) ⊤ A i 0 = π ⊤ , y i 0 ∈ L * m i 0 }.(26)
Let x * be an ε-optimal solution to the original primal (21), that is x * ∈ W and π ⊤ x * ≤ α + ε.
Clearly, x * is feasible for (25). Note now that the dual solution constructed above for (23), when restricted to the y i 0 component is a feasible solution to (26) with objective value α. Thus, we have α ≤ β ≤ π ⊤ x * ≤ α + ε, where the first inequality follows from weak duality to the primal-dual pair (25)(26) and the second inequality follows from fisibility of x * to (25). By taking the limit as ε goes to zero, we obtain (22). To prove the second part of the lemma, we first observe that rec.cone * (W i 0 ) ⊆ rec.cone * (W). If π rec.cone * (W i 0 ), then (22) would be unbounded by Lemma 3. As π interior (rec.cone * (W)), we have that π interior (rec.cone * (W i 0 )). Hence, π ∈ rec.cone * (W i 0 ) \ interior (rec.cone * (W i 0 )). Now, W i 0 cannot define an ellipse because then W ⊆ W i 0 would be bounded. Since π ∈ rec.cone * (W i 0 ) \ interior(rec.cone * (W i 0 )), if W i 0 was a parabola, then problem (22) would be unbounded in view of Lemma 7. Therefore, only two possibilities remain: (i) W i 0 is defined by a linear inequality, say µ ⊤ x ≥ µ 0 . In this case µ must be a multiple of π, otherwise problem (22) would be unbounded. Thus, we may assume π = µ and then µ 0 = α. (ii) W i 0 is one branch of a hyperbola. In this case, rec.cone(W i 0 ) is defined by the asymptotes of the hyperbola. As π ∈ rec.cone * (W i 0 ) \ interior(rec.cone * (W i 0 )), π must be orthogonal to one of the asymptotes.
Next we prove Lemma 1 that was stated in Section 2.3.
Proof. of Lemma 1 Any conic section (parabola, ellipse, hyperbola) in R 2 is a curve defined by a quadratic equation of the form
1 2 x ⊤ Qx + d ⊤ x + s = 0,(27)
where s is a scalar, d ∈ R 2 and Q = VDV ⊤ . In this factorization, V ∈ R 2×2 is orthonormal and
D = λ 1 0 0 λ 2 ,
where λ 1 , λ 2 are the eigenvalues of Q. In particular, the curve defined by (27) is a hyperbola if and only if one of these eigenvalues is positive and the other is negative. After changing variables y := V ′ x and completing squares, equation (27) can be written in exactly one of the following forms
[β 1 (y 1 − α 1 )] 2 − [β 2 (y 2 − α 2 )] 2 = ±η 2 ,(28)
where η and α i , β i , for i = 1, 2, are constants depending on the coefficients of (27). In what follows, we assume that the coefficient of η 2 is positive. If it was negative, then we could multiply (28) by −1 and all we will do next would be analogous. Under this assumption, one branch of the hyperbola is given by
G + := {y ∈ R 2 | (η) 2 + [β 2 (y 2 − α 2 )] 2 ≤ [β 1 (y 1 − α 1 )] 2 , β 1 (y 1 − α 1 ) ≥ 0} = {y ∈ R 2 | η 2 + [β 2 (y 2 − α 2 )] 2 ≤ β 1 (y 1 − α 1 )} = {y ∈ R 2 | (η, β 2 (y 2 − α 2 ), β 1 (y 1 − α 1 )) ∈ L 3 } = {y ∈ R 2 | 0 0 0 β 2 β 1 0 y 1 y 2 L 3 −η β 2 α 2 β 1 α 1 }.
Then, going back to the space of the original variables we obtain
G + = {x ∈ R 2 | 0 0 β 2 v 12 β 2 v 22 β 1 v 11 β 1 v 21 x 1 x 2 L 3 −η β 2 α 2 β 1 α 1 },
where v i j are the entries of the matrix V. The other branch of the hyperbola is given by
G − := {y ∈ R 2 | (η) 2 + [β 2 (y 2 − α 2 )] 2 ≤ [β 1 (y 1 − α 1 )] 2 , β 1 (y 1 − α 1 ) ≤ 0}.
After the change of variablesỹ := −y we obtain
G − = {ỹ ∈ R 2 | (η) 2 + [β 2 (−ỹ 2 − α 2 )] 2 ≤ [β 1 (−ỹ 1 − α 1 )] 2 , β 1 (−ỹ 1 − α 1 ) ≤ 0} = {ỹ ∈ R 2 | (η) 2 + [β 2 (ỹ 2 + α 2 )] 2 ≤ [β 1 (ỹ 1 + α 1 )] 2 , β 1 (ỹ 1 + α 1 ) ≥ 0} = {ỹ ∈ R 2 | (η, β 2 (ỹ 2 + α 2 ), β 1 (ỹ 1 + α 1 )) ∈ L 3 } = {ỹ ∈ R 2 | 0 0 0 β 2 β 1 0 ỹ 1 y 2 L 3 −η −β 2 α 2 −β 1 α 1 }.
Going back to the space of the original variables we obtain
G − = {x ∈ R 2 | 0 0 −β 2 v 12 −β 2 v 22 −β 1 v 11 −β 1 v 21 x 1 x 2 L 3 −η −β 2 α 2 −β 1 α 1 }.
It follows from (28) that the asymptotes of G + have equations
β 1 y 1 + β 2 y 2 = β 1 α 1 + β 2 α 2 , β 1 y 1 − β 2 y 2 = β 1 α 1 − β 2 α 2 .
In the space of x variables they become
(β 1 v 11 + β 2 v 12 )x 1 + (β 1 v 21 + β 2 v 22 )x 2 = β 1 α 1 + β 2 α 2 ,(29)(β 1 v 11 − β 2 v 12 )x 1 + (β 1 v 21 − β 2 v 22 )x 2 = β 1 α 1 − β 2 α 2 .
The asymptotes of G − are obtained in a similar way.
Lemma 9.
Let G be one branch of a non-degenerate hyperbola in R 2 . Let π ⊤ x ≥ π 0 be a face of G I such that π ∈ Z 2 is non-zero and orthogonal to one of the asymptotes. Then π ⊤ x ≥ π 0 is one of the inequalities (13).
Proof. Using the same notation adopted in the proof of Lemma 1 above, we assume G = G + . If G = G − , then the proof is analogous. Note that G is contained in the set
H := {x ∈ R 2 | (β 1 v 11 + β 2 v 12 )x 1 + (β 1 v 21 + β 2 v 22 )x 2 ≥ β 1 α 1 + β 2 α 2 , (β 1 v 11 − β 2 v 12 )x 1 + (β 1 v 21 − β 2 v 22 )x 2 ≥ β 1 α 1 − β 2 α 2 }.
Assume π is orthogonal to the asymptote (29). The proof of the case in which π is orthogonal to the second asymptote is similar. Since π ∈ Z 2 is non-zero, we may assume that the coefficients of x 1 and x 2 in (29) are integers. Let
τ := gcd{β 1 v 11 + β 2 v 12 , β 1 v 21 + β 2 v 22 }.
Since the hyperbola is non-degenerate, the line
(β 1 v 11 + β 2 v 12 )x 1 + (β 1 v 21 + β 2 v 22 )x 2 = β 1 α 1 + β 2 α 2
does not intersect G. However, for all ε > 0, the equation
β 1 v 11 + β 2 v 12 τ x 1 + β 1 v 21 + β 2 v 22 τ x 2 = β 1 α 1 + β 2 α 2 τ + ε(30)
intersects G along a ray. Moreover, (30) has integral solutions if and only if the right-hand-side is integral. Therefore, if (β 1 α 1 + β 2 α 2 )/τ ∈ Z, then the inequality
β 1 v 11 + β 2 v 12 τ x 1 + β 1 v 21 + β 2 v 22 τ x 2 ≥ β 1 α 1 + β 2 α 2 τ + 1(31)
13 is a face of G I , and hence it is equivalent to π ⊤ x ≥ π 0 . On the other hand, if (β 1 α 1 + β 2 α 2 )/τ Z, then
β 1 v 11 + β 2 v 12 τ x 1 + β 1 v 21 + β 2 v 22 τ x 2 ≥ β 1 α 1 + β 2 α 2 τ(32)
is a face of G I , and hence it is equivalent to π ⊤ x ≥ π 0 . Observe now that (31) and (32) are one of the inequalities (13) in view of Proposition 1.
Next we use Lemma 8 and Lemma 9 above to proof Theorem 3.
Proof. of Theorem 3 First, we observe that if W is bounded, then the result follows directly from Theorem 1. Suppose W is unbounded. We have two cases: Case 1: W ∩ Z 2 = ∅. In this case, there exist π = (π 1 , π 2 ) with π 1 , π 2 integer relatively prime and a integer π 0 such that [32,33]
W ⊆ {x ∈ R 2 | π 0 ≤ π ⊤ x ≤ π 0 + 1}.(33)
We will show that the cut π ⊤ x ≥ π 0 + 1 can be obtained using subadditive functions (3) or using one of the inequalities (13). Analogous proof holds for the cut π ⊤ x ≤ π 0 . A consequence of W being between these two lines is that rec.cone(W) is orthogonal to π and, therefore, π interior (rec.cone * (W)) in view of Lemma 2. Then, by Lemma 8,
α := inf{π ⊤ x | W i 0 } = inf{π ⊤ x | x ∈ W},
for some i 0 ∈ [m], where there are only two possibilities for
W i 0 = {x ∈ R 2 | A i 0 x L m i 0 b i 0 }: (i) W i 0 is the half-space π ⊤ x ≥ α: In this case, since A i 0 x L m i 0 b i 0 is non-redundant,
we have that the line π ⊤ x = α intersects W. Note that π 0 ≤ α in view of (33). Since W is unbounded and its recession cone is orthogonal to π, if α = π 0 , then W would contain a integer point from the line π ⊤ x = π 0 . Therefore, α > π 0 in which case π ⊤ x ≥ ⌈α⌉ = π 0 + 1 is a valid inequality for W I and this cut can be obtained using a subadditive function (3).
(ii) W i 0 is a hyperbola whose one of the asymptotes is orthogonal to π: Without loss of generality, we may assume that the asymptote orthogonal to π has equation π ⊤ x = α. Let
β = α + 1 if α ∈ Z ⌈α⌉ if α Z.(34)
Since the hyperbola is non-degenerate, we have that π ⊤ x ≥ β is a valid inequality for (W i 0 ) I . Moreover, π ⊤ x = β contains a ray of W i 0 since β > α. Then, since π 1 and π 2 are relatively prime and β ∈ Z, we have that π ⊤ x ≥ β is, in addition, a face of (W i 0 ) I . Now, it follows from Lemma 9 that this face is one of the inequalities (13). Finally, note that π 0 ≤ α < π 0 + 1. Thus, we have that β = π 0 + 1. Case 2: interior(W) ∩ Z 2 ∅. By assumption, the components of π are integers and, without loss of generality, we may also assume they are relatively prime. We now have three cases.
1. π rec.cone * (W): In this case, by Lemma 3, we have that inf{π ⊤ x | x ∈ W} is unbounded. Since we assume that interior(W) ∩ Z 2 ∅, we obtain that inf{π ⊤ x | x ∈ W ∩ Z 2 } is unbounded [34], which contradicts the fact that π ⊤ x ≥ π 0 is a valid inequality for W I . 2. π ∈ interior(rec.cone * (W)) : In this case, {x ∈ W | π ⊤ x ≤ π 0 } is bounded in view of Proposition 2. Therefore, it follows from Theorem 1 that the valid inequality π ⊤ x ≥ π 0 can be obtained using functions (3). 14 3. π ∈ rec.cone * (W) \ interior(rec.cone * (W)): Since interior(W) ∩ Z 2 ∅ and inf{π ⊤ x | x ∈ W ∩ Z 2 } is bounded, we have that α := inf{π ⊤ x | x ∈ W} is bounded [34]. Then, by Lemma 8, α = inf{π ⊤ x | W i 0 }, for some i 0 ∈ [m], where there are only two possibilities for
W i 0 = {x ∈ R 2 | A i 0 x L m i 0 b i 0 }: (i) W i 0 is the half-space π ⊤ x ≥ α: Since A i 0 x L m i 0 b i 0 is non-redundant,
we have that the line π ⊤ x = α intersects W. Thus, π ⊤ x ≥ ⌈α⌉ is a valid inequality for W I and this cut can be obtained using a subadditive function (3). Now, we only need to show that ⌈α⌉ = π 0 . It is enough to show that the line π ⊤ x = ⌈α⌉ intersects W ∩ Z 2 . Note that the line π ⊤ x = ⌈α⌉ intersects W (otherwise we would have W ⊆ {x ∈ R 2 | π ⊤ x < ⌈α⌉} which contradicts the fact that W ∩ Z 2 ∅ since π ⊤ x ≥ ⌈α⌉ is valid inequality for W I ). Thus, {x ∈ W | π ⊤ x = ⌈α⌉} ∅. Moreover, since π ∈ rec.cone * (W) \ interior(rec.cone * (W)), there exists a non-zero vector d ∈ rec.cone(W) such that π ⊤ d = 0. Therefore, d is in the recession cone of {x ∈ W | π ⊤ x = ⌈α⌉}. Hence, π ⊤ x = ⌈α⌉ contains a ray of W. Thus, π ⊤ x = ⌈α⌉ contains an integer point of W since π 1 and π 2 are relatively prime. (ii) W i 0 is a hyperbola one of whose asymptotes is orthogonal to π: As in Case 1 (ii), we can show that π ⊤ x ≥ β is a face of W i 0 , where β is defined in (34). Moreover, by Lemma 9, π ⊤ x ≥ β is one of the inequalities (13). Now, only remains to show that β = π 0 . It is enough to show that π ⊤ x = β intersects W ∩ Z 2 . Clearly, π ⊤ x ≥ β is a valid inequality for W I ⊆ W i 0 . Since α < β, we have that the line π ⊤ x = β intersects W (otherwise we would have W ⊆ {x ∈ R 2 | π ⊤ x < β} which contradicts the fact that W ∩ Z 2 ∅). Therefore, as in the case (i) above, we can prove that π ⊤ x = β contains a ray of W. Thus, π ⊤ x = β contains an integer point of W since π 1 and π 2 are relatively prime and β ∈ Z.
Figure 1 :
1Slice at x 3 = 1 of the second-order cone L 3 and Γ 1 .
implies that for all i ∈ [m − 1] such that i j we have w m > |w i |; (ii) since γ m > |γ j |, equation (19) implies that γ i 0 for some i ∈ [m−1] such that i j. Now, for all i ∈ [m − 1]such that γ i ≥ 0, we multiply w m > −w i by γ i and, for all i ∈ [m − 1] such that γ i < 0, we multiply w m > w i by −γ i . In view of observations (i) and (ii), at least one of the resulting inequalities remains strict. Then adding them all we obtaini∈[m−1]: γ i ≥0
In this paper, we refer to the curve, as well as the convex region delimited by this curve, as the branch of a hyperbola. Same for parabolas and ellipses.
More precisely, in[28] the variables are assumed to be non-negative, in which case we can drop the requirement of g satisfying g(u) = −g(−u) in the definition of linear composition.
AcknowledgmentsThe authors thank to two anonymous referees for their helpful corrections and suggestions.
Cut-generating functions and S-free sets. M Conforti, G Cornuéjols, A Daniilidis, C Lemaréchal, J Malick, 10.1287/moor.2014.0670Mathematics of Operations Research. 402M. Conforti, G. Cornuéjols, A. Daniilidis, C. Lemaréchal, J. Malick, Cut-generating functions and S-free sets, Mathematics of Operations Research 40 (2) (2015) 276-391. doi:10.1287/moor.2014.0670. URL http://dx.doi.org/10.1287/moor.2014.0670
A strong dual for conic mixed-integer programs. D A Morán, R , S S Dey, J P Vielma, 10.1137/110840868doi:10.1137/110840868SIAM Journal on Optimization. 223D. A. Morán R., S. S. Dey, J. P. Vielma, A strong dual for conic mixed-integer programs, SIAM Jour- nal on Optimization 22 (3) (2012) 1136-1150. arXiv:http://dx.doi.org/10.1137/110840868 , doi:10.1137/110840868. URL http://dx.doi.org/10.1137/110840868
Minimal inequalities. R G Jeroslow, 10.1007/BF01588222Mathematical Programming. 171R. G. Jeroslow, Minimal inequalities, Mathematical Programming 17 (1) (1979) 1-15. doi:10.1007/BF01588222 . URL http://dx.doi.org/10.1007/BF01588222
E L Johnson, 10.1016/B978-0-12-358350-5.50009-9Cyclic groups, cutting planes, shortest paths. T. Hu, S. M. RobinsonAcademic PressMathematical ProgrammingE. L. Johnson, Cyclic groups, cutting planes, shortest paths, in: T. Hu, S. M. Robin- son (Eds.), Mathematical Programming, Academic Press, 1973, pp. 185 - 211. doi:http://dx.doi.org/10.1016/B978-0-12-358350-5.50009-9. URL http://www.sciencedirect.com/science/article/pii/B9780123583505500099
E L Johnson, 10.1016/S0167-5060(08)70345-5Discrete Optimization II Proceedings of the Advanced Research Institute on Discrete Optimization and Systems Applications of the Systems Science Panel of NATO and of the Discrete Optimization Symposium co-sponsored by IBM Canada and SIAM Banff. E. J. P.L. Hammer, B. KorteElsevier5Aha. and VancouverE. L. Johnson, On the group problem and a subadditive approach to integer programming, in: E. J. P.L. Hammer, B. Korte (Eds.), Discrete Optimization II Proceedings of the Advanced Research Institute on Discrete Optimization and Systems Applications of the Systems Science Panel of NATO and of the Discrete Optimization Symposium co-sponsored by IBM Canada and SIAM Banff, Aha. and Vancouver, Vol. 5 of Annals of Discrete Mathematics, Elsevier, 1979, pp. 97 -112. doi:http://dx.doi.org/10.1016/S0167-5060(08)70345-5. URL http://www.sciencedirect.com/science/article/pii/S0167506008703455
The b-hull of an integer program. L Wolsey, 10.1016/0166-218X(81)90016-0Discrete Applied Mathematics. 33L. Wolsey, The b-hull of an integer program, Discrete Applied Mathematics 3 (3) (1981) 193 -201. doi:http://dx.doi.org/10.1016/0166-218X(81)90016-0. URL http://www.sciencedirect.com/science/article/pii/0166218X81900160
On minimal valid inequalities for mixed integer conic programs. F Kılınç-Karzan, 10.1287/moor.2015.0737Mathematics of Operations Research. 412F. Kılınç-Karzan, On minimal valid inequalities for mixed integer conic programs, Mathematics of Operations Research 41 (2) (2016) 477-510. doi:10.1287/moor.2015.0737 .
On sublinear inequalities for mixed integer conic programs. F Kılınç-Karzan, D E Steffy, 10.1007/s10107-015-0968-0Mathematical Programming. 1591F. Kılınç-Karzan, D. E. Steffy, On sublinear inequalities for mixed integer conic programs, Mathematical Program- ming 159 (1) (2016) 585-605. doi:10.1007/s10107-015-0968-0. URL http://dx.doi.org/10.1007/s10107-015-0968-0
The value function of an integer program. C E Blair, R G Jeroslow, 10.1007/BF01583794Mathematical Programming. 231C. E. Blair, R. G. Jeroslow, The value function of an integer program, Mathematical Programming 23 (1) (1982) 237-273. doi:10.1007/BF01583794 . URL http://dx.doi.org/10.1007/BF01583794
Theory of linear and integer programming, Wiley-Interscience series in discrete mathematics and optimization. A Schrijver, WileyA. Schrijver, Theory of linear and integer programming, Wiley-Interscience series in discrete mathematics and optimization, Wiley, 1999.
A recursive procedure to generate all cuts for 0-1 mixed integer programs. G L Nemhauser, L A Wolsey, 10.1007/BF01585752Mathematical Programming. 463G. L. Nemhauser, L. A. Wolsey, A recursive procedure to generate all cuts for 0-1 mixed integer programs, Math- ematical Programming 46 (3) (1990) 379-390. doi:10.1007/BF01585752. URL http://dx.doi.org/10.1007/BF01585752
Some continuous functions related to corner polyhedra, part I. R E Gomory, E L Johnson, Mathematical Programming. 3R. E. Gomory, E. L. Johnson, Some continuous functions related to corner polyhedra, part I, Mathematical Pro- gramming 3 (1972) 23-85.
T-space and cutting planes. R E Gomory, E L Johnson, Mathematical Programming. 96R. E. Gomory, E. L. Johnson, T-space and cutting planes, Mathematical Programming 96 (2003) 341-375.
Valid inequalities based on simple mixed integer set. S Dash, O Günlük, Mathematical Programming. 106S. Dash, O. Günlük, Valid inequalities based on simple mixed integer set, Mathematical Programming 106 (2006) 29-53.
Facets of the two-dimensional infinite group problems. S S Dey, J.-P P Richard, Mathematics of Operations Research. 33S. S. Dey, J.-P. P. Richard, Facets of the two-dimensional infinite group problems, Mathematics of Operations Research 33 (2008) 140-166.
Some relations between facets of low-and high-dimensional group problems. S S Dey, J.-P P Richard, Mathematical Programming. 123S. S. Dey, J.-P. P. Richard, Some relations between facets of low-and high-dimensional group problems, Mathe- matical Programming 123 (2010) 285-313.
Valid inequalities for mips and group polyhedra from approximate liftings. J P Richard, Y Li, L A Miller, 10.1007/s10107-007-0190-9Mathematical Programming. 1182J. P. Richard, Y. Li, L. A. Miller, Valid inequalities for mips and group polyhedra from approximate liftings, Math- ematical Programming 118 (2) (2009) 253-277. doi:10.1007/s10107-007-0190-9 . URL http://dx.doi.org/10.1007/s10107-007-0190-9
50 Years of Integer Programming 1958-2008 -From the Early Years to the State-of-the-Art. J P Richard, S S Dey, ; M Jünger, T M Liebling, D Naddef, G L Nemhauser, W R Pulleyblank, G Reinelt, G Rinaldi, 10.1007/978-3-540-68279-0_19L. A. WolseySpringerThe group-theoretic approach in mixed integer programmingJ. P. Richard, S. S. Dey, The group-theoretic approach in mixed integer programming, in: M. Jünger, T. M. Liebling, D. Naddef, G. L. Nemhauser, W. R. Pulleyblank, G. Reinelt, G. Rinaldi, L. A. Wolsey (Eds.), 50 Years of Integer Programming 1958-2008 -From the Early Years to the State-of-the-Art, Springer, 2010, pp. 727-801. doi:10.1007/978-3-540-68279-0_19. URL http://dx.doi.org/10.1007/978-3-540-68279-0_19
A Basu, R Hildebrand, M Köppe, Light on the infinite group relaxation i: foundations and taxonomy. A. Basu, R. Hildebrand, M. Köppe, Light on the infinite group relaxation i: foundations and taxonomy, 4OR 14 (1) (2016) 1-40.
An electronic compendium of extreme functions for the gomory-johnson infinite group problem. M Köppe, Y Zhou, arXiv:1411.5121M. Köppe, Y. Zhou, An electronic compendium of extreme functions for the gomory-johnson infinite group prob- lem, arXiv:1411.5121.
The split closure of a strictly convex body. D Dadush, S S Dey, J P Vielma, 10.1016/j.orl.2011.02.002Operations Research Letters. 392D. Dadush, S. S. Dey, J. P. Vielma, The split closure of a strictly convex body, Operations Research Letters 39 (2) (2011) 121-126. doi:10.1016/j.orl.2011.02.002. URL http://dx.doi.org/10.1016/j.orl.2011.02.002
On families of quadratic surfaces having fixed intersections with two hyperplanes. P Belotti, J C Góez, I Pólik, T K Ralphs, T Terlaky, 10.1016/j.dam.2013.05.017Discrete Applied Mathematics. 161P. Belotti, J. C. Góez, I. Pólik, T. K. Ralphs, T. Terlaky, On families of quadratic surfaces having fixed intersections with two hyperplanes, Discrete Applied Mathematics 161 (16-17) (2013) 2778-2793. doi:10.1016/j.dam.2013.05.017. URL http://dx.doi.org/10.1016/j.dam.2013.05.017
Split cuts and extended formulations for mixed integer conic quadratic programming. S Modaresi, M R Kilinç, J P Vielma, 10.1016/j.orl.2014.10.006Operations Research Letters. 431S. Modaresi, M. R. Kilinç, J. P. Vielma, Split cuts and extended formulations for mixed integer conic quadratic programming, Operations Research Letters 43 (1) (2015) 10-15. doi:10.1016/j.orl.2014.10.006. URL http://dx.doi.org/10.1016/j.orl.2014.10.006
. S Modaresi, J P Vielma, S. Modaresi, J. P. Vielma, Convex hull of two quadratic or a conic quadratic and a quadratic inequality, http://www.optimization-online.org/DBHTML/2014/11/4641.html.
Two-term disjunctions on the second-order cone. F Kilinç-Karzan, S Yildiz, 10.1007/s10107-015-0903-4Mathematical Programming. 1541-2F. Kilinç-Karzan, S. Yildiz, Two-term disjunctions on the second-order cone, Mathematical Programming 154 (1- 2) (2015) 463-491. doi:10.1007/s10107-015-0903-4 . URL http://dx.doi.org/10.1007/s10107-015-0903-4
Disjunctive cuts for cross-sections of the second-order cone. S Yildiz, G Cornuéjols, 10.1016/j.orl.2015.06.001Operations Research Letter. 434S. Yildiz, G. Cornuéjols, Disjunctive cuts for cross-sections of the second-order cone, Operations Research Letter 43 (4) (2015) 432-437. doi:10.1016/j.orl.2015.06.001. URL http://dx.doi.org/10.1016/j.orl.2015.06.001
How to convexify the intersection of a second order cone and a nonconvex quadratic. S Burer, F Kılınç-Karzan, 10.1007/s10107-016-1045-zMathematical Programming. S. Burer, F. Kılınç-Karzan, How to convexify the intersection of a second order cone and a nonconvex quadratic, Mathematical Programming (2016) 1-37doi:10.1007/s10107-016-1045-z. URL http://dx.doi.org/10.1007/s10107-016-1045-z
Cuts for mixed 0-1 conic programming. M T Ezik, G Iyengar, 10.1007/s10107-005-0578-3Mathematical Programming. 1041M. T. Ç ezik, G. Iyengar, Cuts for mixed 0-1 conic programming, Mathematical Programming 104 (1) (2005) 179- 202. doi:10.1007/s10107-005-0578-3. URL http://dx.doi.org/10.1007/s10107-005-0578-3
Conic mixed-integer rounding cuts. A Atamtürk, V Narayanan, 10.1007/s10107-008-0239-4Mathematical Programming. 1221A. Atamtürk, V. Narayanan, Conic mixed-integer rounding cuts, Mathematical Programming 122 (1) (2008) 1-20. doi:10.1007/s10107-008-0239-4. URL http://dx.doi.org/10.1007/s10107-008-0239-4
A Ben-Tal, A Nemirovski, http:/epubs.siam.org/doi/abs/10.1137/1.9780898718829doi:10.1137/1.9780898718829Lectures on Modern Convex Optimization, Society for Industrial and Applied Mathematics. A. Ben-Tal, A. Nemirovski, Lectures on Modern Convex Optimization, Society for Industrial and Ap- plied Mathematics, 2001. arXiv:http://epubs.siam.org/doi/pdf/10.1137/1.9780898718829 , doi:10.1137/1.9780898718829. URL http://epubs.siam.org/doi/abs/10.1137/1.9780898718829
Binding constraints and helly numbers. A J Hoffman, 10.1111/j.1749-6632.1979.tb32803.xAnnals of the New York Academy of Sciences. 3191A. J. Hoffman, Binding constraints and helly numbers, Annals of the New York Academy of Sciences 319 (1) (1979) 284-288. doi:10.1111/j.1749-6632.1979.tb32803.x. URL http://dx.doi.org/10.1111/j.1749-6632.1979.tb32803.x
Two row mixed-integer cuts via lifting. S S Dey, L A Wolsey, 10.1007/s10107-010-0362-xMathematical Programming. 1241S. S. Dey, L. A. Wolsey, Two row mixed-integer cuts via lifting, Mathematical Programming 124 (1) (2010) 143- 174. doi:10.1007/s10107-010-0362-x. URL http://dx.doi.org/10.1007/s10107-010-0362-x
Maximal lattice-free convex sets in linear subspaces. A Basu, M Conforti, G Cornuéjols, G Zambelli, 10.1287/moor.1100.0461Mathematics of Operations Research. 353A. Basu, M. Conforti, G. Cornuéjols, G. Zambelli, Maximal lattice-free convex sets in linear subspaces, Mathe- matics of Operations Research 35 (3) (2010) 704-720. doi:10.1287/moor.1100.0461 . URL http://dx.doi.org/10.1287/moor.1100.0461
Some properties of convex hulls of integer points contained in general convex sets. S S Dey, D A Morán, R , 10.1007/s10107-012-0538-7Mathematical Programming. 1411S. S. Dey, D. A. Morán R., Some properties of convex hulls of integer points contained in general convex sets, Mathematical Programming 141 (1) (2013) 507-526. doi:10.1007/s10107-012-0538-7. URL http://dx.doi.org/10.1007/s10107-012-0538-7
| [] |
[
"Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case",
"Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case"
] | [
"Luc Molinet ",
"Stéphane Vento "
] | [] | [] | We complete the known results on the Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in H −1 (R) with a solution-map that is analytic from, as soon as s < −1, in the sense that the flow-map u 0 → u(t) cannot be continuous from H s (R) to even D ′ (R) at any fixed t > 0 small enough. As far as we know, this is the first result of this type for a dispersive-dissipative equation. The framework we develop here should be useful to prove similar results for other dispersivedissipative models. | null | [
"https://arxiv.org/pdf/0911.5256v2.pdf"
] | 16,517,764 | 0911.5256 | 1a1fb6ced883f3e70a186b867c74633434da91fb |
Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case
31 Dec 2009
Luc Molinet
Stéphane Vento
Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the real line case
31 Dec 2009
We complete the known results on the Cauchy problem in Sobolev spaces for the KdV-Burgers equation by proving that this equation is well-posed in H −1 (R) with a solution-map that is analytic from, as soon as s < −1, in the sense that the flow-map u 0 → u(t) cannot be continuous from H s (R) to even D ′ (R) at any fixed t > 0 small enough. As far as we know, this is the first result of this type for a dispersive-dissipative equation. The framework we develop here should be useful to prove similar results for other dispersivedissipative models.
Introduction and main results
The aim of this paper is to establish positive and negative optimal results on the local Cauchy problem in Sobolev spaces for the Korteweg-de Vries-Burgers (KdV-B) equation posed on the real line :
u t + u xxx − u xx + uu x = 0 (1.1)
where u = u(t, x) is a real valued function. This equation has been derived as an asymptotic model for the propagation of weakly nonlinear dispersive long waves in some physical contexts when dissipative effects occur (see [17]). It thus seems natural to compare the well-posedness results on the Cauchy problem for the KdV-B equation with the ones for the Korteweg-de-Vries (KdV) equation
u t + u xxx + uu x = 0 (1.2)
that correspond to the case when dissipative effects are negligible and for the dissipative Burgers (dB) equation
u t − u xx + uu x = 0 (1.3)
that corresponds to the case when dissipative effect are dominant.
To make this comparison more transparent it is convenient to define different notions of well-posedness (and consequently ill-posedness) related to the smoothness of the flow-map (see in the same spirit [13], [8]).
Throughout this paper we shall say that a Cauchy problem is (locally) C 0 -well-posed in some normed function space X if, for any initial data u 0 ∈ X, there exist a radius R > 0, a time T > 0 and a unique solution u, belonging to some space-time function space continuously embedded in C([0, T ]; X), such that for any t ∈ [0, T ] the map u 0 → u(t) is continuous from the ball of X centered at u 0 with radius R into X. If the map u 0 → u(t) is of class C k , k ∈ N ∪ {∞}, (resp. analytic) we will say that the Cauchy is C k -well-posed (resp. analytically well-posed). Finally a Cauchy problem will be said to be C k -ill-posed, k ∈ N ∪ {∞}, if it is not C k -well-posed.
For the KdV equation on the line the situation is as follows: it is analytically well-posed in H −3/4 (R) (cf. [14] and [11] for the limit case) and C 3 -ill-posed below this index 1 (cf. [4]). On the other hand the results for the dissipative Burgers equation are much clear. Indeed this equation is known to be analytically well-posed in H s (R) for s ≥ −1/2 (cf [7] and [1] for the limit case) and C 0 ill-posed in H s for s < −1/2 (cf. [7] ). At this stage it is interesting to notice that the critical Sobolev exponents obtained by scaling considerations are respectively −3/2 for the KdV equation and −1/2 for the dissipative Burgers equation. Hence for the KdV equation there is an important gap between this critical exponent and the best exponent obtained for well-posedness. Now, concerning the KdV-Burgers equation, Molinet and Ribaud [16] proved that this equation is analytically well-posed in H s (R) as soon as s > −1. They also established that the index −1 is critical for the C 2well-posedness. The surprising part of this result was that, according to the above results, the C ∞ critical index s ∞ c (KdV B) = −1 was lower that the one of the KdV equation s ∞ c (KdV ) = −3/4 and also lower than the C ∞ index s ∞ c (dB) = −1/2 of the dissipative Burgers equation. In this paper we want in some sense to complete this study by proving that the KdV-Burgers equation is analytically well-posed in H −1 (R) and C 0 -ill-posed in H s (R) for s < −1 in the sense that the flow-map defined on H −1 (R) is not continuous for the topology inducted by H s , s < −1, with values even in D ′ (R). It is worth emphasizing that the critical index s 0 c = −1 is still far away from the critical index s c = −3/2 given by the scaling symmetry of the KdV equation. We believe that this result strongly suggest that the KdV equation should also be C 0 -ill-posed in H s (R) for s < −1.
To reach the critical Sobolev space H −1 (R) we adapt the refinement of Bourgain's spaces that appeared in [20] and [19] to the framework developed in [16]. One of the main difficulty is related to the choice of the extension for negative times of the Duhamel operator (see the discussion in the beginning of Section 4). The approach we develop here to overcome this difficulty should be useful to prove optimal results for other dispersivedissipative models. The ill-posedness result is due to a high to low frequency cascade phenomena that was first observed in [2] for a quadratic Schrödinger equation..
At this stage it is worth noticing that, using the integrability theory, it was recently proved in [13] that the flow-map of KdV equation can be uniquely continuously extended in H −1 (T). Therefore, on the torus, KdV is C 0 -well-posed in H −1 if one takes as uniqueness class, the class of strong limit in C([0, T ]; H −1 (T)) of smooth solutions. In the present work we use in a crucial way the global Kato smoothing effect that does not hold on the torus. However, in a forthcoming paper ( [15]) we will show how one can modify the approach developed here to prove that the same results hold on the torus, i.e. analytic well-posedness in H −1 (T) and C 0 -ill-posedness in H s (T) for s < −1. In view of the result of Kappeler and Topalov for KdV it thus appears that, at least on the torus, even if the dissipation part of the KdV-Burgers equation (it is important to notice that the dissipative term −u xx is of lower order than the dispersive one u xxx ) allows to lower the C ∞ critical index with respect to the KdV equation, it does not permit to improve the C 0 critical index .
Our results can be summarized as follows: Acknowlegements: L.M. was partially supported by the ANR project "Equa-Disp".
∈ H −1 (R) there exist T = T (u 0 ) > 0 and R = R(u 0 ) > 0 such that the solution-map u 0 → u is analytic from the ball centered at u 0 with radius R of H −1 (R) into C([0, T ]; H −1 (R)
Ill-posedness
The ill-posedness result can be viewed as an application of a general result proved in [2]. Roughly speaking this general ill-posedness result requires the two following ingredients:
1. The equation is analytically well-posed until some index s ∞ c with a solution-map that is also analytic.
2. Below this index one iteration of the Picard scheme is not continuous.
The discontinuity should be driven by high frequency interactions that blow up in frequencies of order least or equal to one.
The first ingredient is given by Theorem 1.1 whereas the second one has been derived in [16] where the discontinuity of the second iteration of the Picard scheme in H s (R) and H s (T) for s < −1 is established. However, due to the nature of the equation, our result is a little better than the one given by the general theory developed in [2]. Indeed, we will be able to prove the discontinuity of the flow-map u 0 → u(t) for any fixed t > 0 less than some T > 0 and not only of the solution-map u 0 → u. Therefore for sake of completeness we will prove the result with hand here.
Let us first recall the counter-example constructed in [16] that we renormalize here in H −1 (R). We define the sequence of initial data {φ N } N ≥1 byφ
N = N −1 χ I N (ξ) + χ I N (−ξ) ,(2.1)
where I N = [N, N + 2] andφ N denotes the space Fourier transform of φ N . Note that φ N H −1 (R) ∼ 1 and φ N → 0 in H s (R) for s < −1. This sequence yields a counter-example to the continuity of the second iteration of the Picard Scheme in H s (R), s < −1, that is given by
A 2 (t, h, h) = t 0 S(t − t ′ )∂ x [S(t ′ )h] 2 dt ′
where S is the semi-group associated to the linear part of (1.1) (see (3.2)). Indeed, computing the space Fourier transform we get
F x (A 2 (t, φ N , φ N ))(ξ) = R e −tξ 2 e itξ 3φ N (ξ 1 )φ N (ξ − ξ 1 ) (iξ) t 0 e −(ξ 2 1 +(ξ−ξ 1 ) 2 −ξ 2 )t ′ e i(ξ 3 1 +(ξ−ξ 1 ) 3 −ξ 3 )t ′ dt ′ dξ 1 = (iξ) e itξ 3 e −tξ 2 Rφ N (ξ 1 )φ N (ξ − ξ 1 ) e −(ξ 2 1 +(ξ−ξ 1 ) 2 −ξ 2 )t e i3ξξ 1 (ξ−ξ 1 )t − 1 −2ξ 1 (ξ − ξ 1 ) + i3ξξ 1 (ξ − ξ 1 ) dξ 1 , so that A 2 (t, φ N , φ N ) 2 H s ≥ 1/2 −1/2 (1 + |ξ| 2 ) s |F x (A 2 (t, φ N , φ N ))(ξ)| 2 dξ = N 4 1/2 −1/2 (1 + |ξ| 2 ) s |ξ| 2 K ξ e −(ξ 2 1 +(ξ−ξ 1 ) 2 )t e i3ξξ 1 (ξ−ξ 1 )t − e −ξ 2 t −2ξ 1 (ξ − ξ 1 ) + i3ξξ 1 (ξ − ξ 1 ) dξ 1 2 dξ , where K ξ = {ξ 1 / ξ − ξ 1 ∈ I N , ξ 1 ∈ −I N } ∪ {ξ 1 / ξ 1 ∈ I N , ξ − ξ 1 ∈ −I N } .
Note that for any ξ ∈ [−1/2, 1/2], one has mes(K ξ ) ≥ 1 and
3ξξ 1 (ξ − ξ 1 ) ∼ N 2 2ξ 1 (ξ − ξ 1 ) ∼ N 2 , ∀ξ 1 ∈ K ξ .
Therefore, fixing 0 < t < 1 we have
Re (e −(ξ 2 1 +(ξ−ξ 1 ) 2 )t e i3ξξ 1 (ξ−ξ 1 )t − e −ξ 2 t ) ≤ −e −t/4 + e −2(N +2) 2 t ,
which leads for N = N (t) > 0 large enough to
K ξ e −(ξ 2 1 +(ξ−ξ 1 ) 2 )t e i3ξξ 1 (ξ−ξ 1 )t − e −ξ 2 t −2ξ 1 (ξ − ξ 1 ) + i3ξξ 1 (ξ − ξ 1 ) dξ 1 ≥ C e −t/4 N 2 and thus A 2 (t, φ N , φ N ) 2 H s ≥ Ce −t/4 ≥ C 0 (2.2)
for some positive constant C 0 > 0. Since φ N → 0 in H s (R), for s < −1, this ensures that, for any fixed t > 0, the map u 0 → A 2 (t, u 0 , u 0 ) is not continuous at the origin from H s (R) into D ′ (R). Now, we will use that A 2 (t, φ N , φ N ) is of order at least one in H s (R) to prove that somehow A 2 (t, εφ N , εφ N ) is the main contribution to u(t, εφ N ) in H s (R) as soon as s < −1, ε > 0 is small and N is large enough. The discontinuity of u 0 → u(t) will then follow from the one of u 0 → A 2 (t, u 0 , u 0 ). According to Theorem 1.1 there exist T > 0 and ε 0 > 0 such that for any |ε| ≤ ε 0 , any h H −1 (R) ≤ 1 and 0 ≤ t ≤ T ,
u(t, εh) = εS(t)h + +∞ k=2 ε k A k (t, h k ) where h k := (h, . . . , h), h k → A k (t, h k ) is a k-linear continuous map from H −1 (R) k into C([0, T ]; H −1 (R)
) and the series converges absolutely in C([0, T ]; H −1 (R)).
In particular,
u(t, εφ N ) − ε 2 A 2 (t, φ N , φ N ) = εS(t)φ N + +∞ k=3 ε k A k (t, φ k N ) .
On the other hand,
S(t)φ N H s (R) ≤ φ N H s (R) ∼ N 1+s and ∞ k=3 ε k A k (t, φ k N ) H −1 ≤ ε ε 0 3 ∞ k=3 ε k 0 A k (t, φ N ) H −1 ≤ Cε 3 .
Hence, for s < −1,
sup t∈[0,T ] u(t, εφ N ) − ε 2 A 2 (t, φ N , φ N ) H s (R) ≤ Cε 3 + O(N 1+s ) .
In view of (2.2) this ensures that, fixing 0 < t < 1 and taking ε small enough and N large enough, ε 2 A 2 (t, φ N , φ N ) is a "good" approximation of u(t, εφ N ). In particular, taking ε ≤ C 0 C −1 /4 we get
u(t, εφ N ) H s (R) ≥ C 0 ε 2 /2 + O(N 1+s ) .
Since u(t, 0) ≡ 0 and φ N → 0 in H s (R) for s < −1 this leads to the discontinuity of the flow-map at the origin by letting N tend to infinity. It is worth noticing that since φ N ⇀ 0 in H −1 (R) we also get that u 0 → u(t, u 0 ) is discontinuous from H −1 (R) equipped with its weak topology with values even in D ′ (R).
Resolution space
In this section we introduce a few notation and we define our functional framework.
For A, B > 0, A B means that there exists c > 0 such that A ≤ cB. When c is a small constant we use A ≪ B. We write A ∼ B to denote the statement that A B A. For u = u(t, x) ∈ S ′ (R 2 )
, we denote by u (or F x u) its Fourier transform in space, and u (or Fu) the space-time Fourier transform of u. We consider the usual Lebesgue spaces L p , L p x L q t and abbreviate L p x L p t as L p . Let us define the Japanese bracket x = (1+|x| 2 ) 1/2 so that the standard non-homogeneous Sobolev spaces are endowed with the
norm f H s = ∇ s f L 2 .
We also need a Littlewood-Paley analysis. Let η ∈ C ∞ 0 (R) be such that
η ≥ 0, supp η ⊂ [−2, 2], η ≡ 1 on [−1, 1]. We define next ϕ(ξ) = η(ξ)−η(2ξ).
Any summations over capitalized variables such as N, L are presumed to be dyadic, i.e. these variables range over numbers of the form 2 ℓ , ℓ ∈ Z. We set ϕ N (ξ) = ϕ(ξ/N ) and define the operator
P N by F(P N u) = ϕ N u. We introduce ψ L (τ, ξ) = ϕ L (τ − ξ 3 ) and for any u ∈ S ′ (R 2 ), F x (P N u(t))(ξ) = ϕ N (ξ)û(t, ξ), F(Q L u)(τ, ξ) = ψ L (τ, ξ)ũ(τ, ξ).
Roughly speaking, the operator P N localizes in the annulus {|ξ| ∼ N } whereas Q L localizes in the region {|τ − ξ 3 | ∼ L}.
Furthermore we define more general projection
P N = N 1 N P N 1 , Q ≫L = L 1 ≫L Q L 1 etc.
Let e −t∂xxx be the propagator associated to the Airy equation and define the two parameters linear operator W by
F x (W (t, t ′ )φ)(ξ) = exp(itξ 3 − |t ′ |ξ 2 )φ(ξ), t ∈ R. (3.1)
The operator W : t → W (t, t) is clearly an extension to R of the linear semi-group S(·) associated with (1.1) that is given by
F x (S(t)φ)(ξ) = exp(itξ 3 − tξ 2 )φ(ξ), t ∈ R + . (3.2)
We will mainly work on the integral formulation of (1.1):
u(t) = S(t)u 0 − 1 2 t 0 S(t − t ′ )∂ x u 2 (t ′ )dt ′ , t ∈ R + . (3.3)
Actually, to prove the local existence result, we will apply a fixed point argument to the following extension of (3.3) (See Section 4 for some explanations on this choice).
u(t) = η(t) W (t)u 0 − 1 2 χ R + (t) t 0 W (t − t ′ , t − t ′ )∂ x u 2 (t ′ )dt ′ − 1 2 χ R − (t) t 0 W (t − t ′ , t + t ′ )∂ x u 2 (t ′ )dt ′ . (3.4) If u solves (3.4) then u is a solution of (3.3) on [0, T ], T < 1.
In [16], the authors performed the iteration process in the space X s,b equipped with the norm
u X s,b = i(τ − ξ 3 ) + ξ 2 b ξ s u L 2
which take advantage of the mixed dispersive-dissipative part of the equation. In order to handle the endpoint index s = −1 without encountering logarithmic divergence, we will rather work in its Besov version X s,b,q (with q = 1) defined as the weak closure of the test functions that are uniformly bounded by the norm
u X s,b,q = N L N sq L + N 2 bq P N Q L u q L 2 xt 2/q 1/2 .
This Besov refinement, which usually provides suitable controls for nonlinear terms, is not sufficient here to get the desired bound especially in the highhigh regime, where the nonlinearity interacts two components of the solution u with the same high frequency. To handle these divergences, inspired by [19], we introduce,
for b ∈ { 1 2 , − 1 2 }, the space Y s,b endowed with the norm u Y s,b = N [ N s F −1 [(i(τ − ξ 3 ) + ξ 2 + 1) b+1/2 ϕ N u] L 1 t L 2 x ] 2 1/2 , so that u Y −1, 1 2 ∼ N [ N −1 (∂ t + ∂ xxx − ∂ xx + I)P N u L 1 t L 2 x ] 2 1/2 .
Next we form the resolution space S s = X s, 1 2 ,1 +Y s, 1 2 , and the "nonlinear space" N s = X s,− 1 2 ,1 + Y s,− 1 2 in the usual way:
u X+Y = inf{ u 1 X + u 2 Y : u 1 ∈ X, u 2 ∈ Y, u = u 1 + u 2 }.
In the rest of this section, we study some basic properties of the function space S −1 .
Lemma 3.1. For any φ ∈ L 2 , L [L 1/2 Q L (e −t∂xxx φ) L 2 ] 2 1/2 φ L 2 .
Proof. From Plancherel theorem, we have
L [L 1/2 Q L (e −t∂xxx φ) L 2 ] 2 1/2 ∼ |τ − ξ 3 | 1/2 F(e −t∂xxx φ) L 2 .
Moreover if we set η T (t) = η(t/T ) for T > 0, then
F(η T (t)e −t∂xxx φ)(τ, ξ) = η T (τ − ξ 3 ) φ(ξ).
Thus we obtain with the changes of variables τ − ξ 3 → τ ′ and T τ ′ → σ that
|τ − ξ 3 | 1/2 F(η T (t)e −t∂xxx φ) L 2 φ L 2 |τ ′ | 1/2 T η(T τ ′ ) L 2 τ ′ φ L 2 .
Taking the limit T → ∞, this completes the proof.
Lemma 3.2. 1. For each dyadic N , we have (∂ t + ∂ xxx )P N u L 1 t L 2 x P N u Y 0, 1 2 . (3.5) 2. For all u ∈ S −1 , u L 2 xt u S −1 . (3.6) 3. For all u ∈ S 0 , L [L 1/2 Q L u L 2 ] 2 1/2 u S 0 . (3.7)
Proof.
1. From the definition of Y 0, 1 2 , the right-hand side of (3.5) can be rewritten as
P N u Y 0, 1 2 = (∂ t + ∂ xxx − ∂ xx + I)P N u L 1 t L 2 x .
Thus, by the triangle inequality, we reduce to show (3.5) with ∂ t +∂ xxx replaced by I − ∂ xx . Using Plancherel theorem as well as Young and Hölder inequalities, we get
(I − ∂ xx )P N u L 1 t L 2 x F −1 t ξ 2 + 1 i(τ − ξ 3 ) + ξ 2 + 1 (i(τ − ξ 3 ) + ξ 2 + 1)ϕ N u L 1 t L 2 ξ .
In the sequel, it will be convenient to write
ϕ N for ϕ N/2 + ϕ N + ϕ 2N .
With this slight abuse of notation, we obtain
(I − ∂ xx )P N u L 1 t L 2 x F −1 t ϕ N (ξ)(ξ 2 + 1) i(τ − ξ 3 ) + ξ 2 + 1 L 1 t L ∞ ξ (∂ t + ∂ xxx − ∂ xx + I)P N u L 1 t L 2 x .
On the other hand, a direct computation yields
F −1 t ϕ N (ξ)(ξ 2 + 1) i(τ − ξ 3 ) + ξ 2 + 1 = Cϕ N (ξ)(1 + ξ 2 )e −t(1+ξ 2 ) χ R + (t) so that F −1 t ϕ N (ξ)(ξ 2 + 1) i(τ − ξ 3 ) + ξ 2 + 1 L 1 t L ∞ ξ N 2 e −t N 2 χ R + (t) L 1 t 1,
and the claim follows.
2. We show that for any fixed dyadic N , we have
P N u L 2 P N u S −1 . (3.8)
Estimate (3.6) then follows after square-summing. Observe that (3.8) follows immediately from the estimate N −1 L + N 2 1/2 1 if the right-hand side is replaced by P N u X −1, 1 2 ,1 , so it suffices to prove (3.8) with P N u Y −1, 1 2 in the right-hand side. But applying again Young and Hölder's inequalities, this is easily verified:
P N u L 2 = F −1 t 1 i(τ − ξ 3 ) + ξ 2 + 1 (i(τ − ξ 3 ) + ξ 2 + 1)ϕ N u L 2 tξ F −1 t ϕ N (ξ) i(τ − ξ 3 ) + ξ 2 + 1 L 2 t L ∞ x P N u Y 0, 1 2 e −t N 2 χ R + (t) L 2 t P N u Y 0, 1 2 N −1 P N u Y 0, 1 2 P N u Y −1, 1 2 .
First it is clear from definitions that
L [L 1/2 Q L u L 2 ] 2 1/2 u X 0, 1 2 ,1 . Setting now v = (∂ t + ∂ xxx )u,
we see that u can be rewritten as
u(t) = e −t∂xxx u(0) + t 0 e −(t−t ′ )∂xxx v(t ′ )dt ′ . By virtue of Lemma 3.1, we have L [L 1/2 Q L e −t∂xxx u(0) L 2 ] 2 1/2 u(0) L 2 u L ∞ t L 2 x .
Moreover, we get as previously
u L ∞ t L 2 x F −1 t 1 i(τ − ξ 3 ) + ξ 2 + 1 L ∞ tξ u Y 0, 1 2 u Y 0, 1 2 . (3.9)
Thanks to estimate (3.5), it remains to show that
L L 1/2 Q L t 0 e −(t−t ′ )∂xxx v(t ′ )dt ′ L 2 2 1/2 v L 1 t L 2 x . (3.10)
In order to prove this, we split the integral
t 0 = t −∞ − 0 −∞ . By Lemma 3.1, the contribution with integrand on (−∞, 0) is bounded by 0 −∞ e t ′ ∂xxx v(t ′ )dt ′ L 2 x v L 1 t L 2 x .
For the last term, we reduce by Minkowski to show that
L [L 1/2 Q L (χ t>t ′ e −(t−t ′ )∂xxx v(t ′ )) L 2 tx ] 2 1/2 v(t ′ ) L 2 x .
This can be proved by a time-restriction argument. Indeed, for any T > 0, we have
L [L 1/2 Q L (η T (t)χ t>t ′ e −(t−t ′ )∂xxx v(t ′ )) L 2 ] 2 1/2 |τ | 1/2 v(t ′ )F t (η T (t)χ t>t ′ )(τ ) L 2 v(t ′ ) L 2 |τ | 1/2 F t (η(t)χ tT >t ′ ) L 2 v(t ′ ) L 2 .
We conclude by passing to the limit T → ∞.
Now we state a general and classical result which ensures that our resolution space is well compatible with dispersive properties of the Airy equation. Actually, it is a direct consequence of Lemma 4.1 in [19] together with the fact that the resolution space S 0 used by Tao to solve 4-KdV contains our space S 0 thanks to estimate (3.5) Lemma 3.3 (Extension lemma). Let Z be a Banach space of functions on R × R with the property that
g(t)u(t, x) Z g L ∞ t u(t, x) Z
holds for any u ∈ Z and g ∈ L ∞ t (R). Let T be a spacial linear operator for which one has the estimate
T (e −t∂xxx P N φ) Z P N φ L 2
for some dyadic N and for all φ. Then one has the embedding
T (P N u) Z P N u S 0 .
Combined with the unitary of the Airy group in L 2 and the sharp Kato smoothing effect
∂ x e −t∂xxx φ L ∞ x L 2 t φ L 2 , ∀φ ∈ L 2 ,(3.11)
we deduce the following result.
Corollary 3.1. For any u, we have 1 u L ∞ t H −1 x u S −1 , (3.12) P N u L ∞ x L 2 t N −1 P N u S 0 ,(3.
13)
provided the right-hand side is finite. In particular, S −1 ֒→ L ∞ t H −1 .
Linear estimates
In this section we prove linear estimates related to the operator W as well as to the extension of the Duhamel operator introduced in (3.4). At this this stage let us give some explanations on our choice of this extension. Let us keep in mind that this extension has to be compatible with linear estimates in both norms X s,1/2,1 and Y s,1/2 . First, since X s,1/2,1 is a Besov in time space we are not allowed to simply multiply the Duhamel term by χ R + (t). Second, in order to prove the desired linear estimate in Y s,1/2 the strategy is to use that the Duhamel term satisfies a forced KdV-Burgers equation. Unfortunately, it turns out that the extension introduced in [16], that makes the calculus simple, does not satisfy such PDE for negative time. The new extension that we introduce in this work has the properties to satisfy some forced PDE related to KdV-Burgers for negative times (see (4.14)) and to be compatible with linear estimates in X s,1/2,1 . However the proof is now a little more complicated even if it follows the same lines than the one of Propositions 2.3 in [16], see also Proposition 4.4, [12].
The following lemma is a dyadic version of Proposition 2.1 in [16].
Proposition 4.1. For all φ ∈ H −1 (R), we have η(t)W (t)φ S −1 φ H −1 . (4.1)
Proof. We bound the left-hand side in (4.1) by the X −1, 1 2 ,1 -norm of η(t)W (t)φ. After square-summing in N , we may reduce to prove
L L + N 2 1/2 P N Q L (η(t)W (t)φ) L 2 xt P N φ L 2 (4.2)
for each dyadic N . Using Plancherel, we obtain
L L + N 2 1/2 P N Q L (η(t)W (t)φ) L 2 xt L L + N 2 1/2 ϕ N (ξ)ϕ L (τ ) φ(ξ)F t (η(t)e −|t|ξ 2 )(τ ) L 2 τ ξ P N φ L 2 L L + N 2 1/2 ϕ N (ξ)P L (η(t)e −|t|ξ 2 ) L ∞ ξ L 2 t .
Hence it remains to show that
L L + N 2 1/2 ϕ N (ξ)P L (η(t)e −|t|ξ 2 ) L ∞ ξ L 2 t 1. (4.3)
We split the summand into L ≤ N 2 and L ≥ N 2 . In the former case, we get by Bernstein
L≤ N 2 L + N 2 1/2 ϕ N (ξ)P L (η(t)e −|t|ξ 2 ) L ∞ ξ L 2 t L≤ N 2 N L 1/2 sup |ξ|∼N η(t)e −|t|ξ 2 L 1 t
Also, one can bound η(t)e −|t|ξ 2 L 1 either by η L 1 or by e −|t|ξ 2
L 1 t ∼ |ξ| −2 . It follows that L≤ N 2 L + N 2 1/2 ϕ N (ξ)P L (η(t)e −|t|ξ 2 ) L ∞ ξ L 2 t N 2 min(1, N −2 ) 1.
Now we deal with the case L ≥ N 2 . A standard paraproduct rearrangement allows us to write
P L (η(t)e −|t|ξ 2 ) = P L M L (P M η(t)P M e −|t|ξ 2 + P M η(t)P M e −|t|ξ 2 = P L (I) + P L (II).
Using the Schur's test, the term P L (I) is directly bounded by
L≥ N 2 L + N 2 1/2 ϕ N P L (I) L ∞ ξ L 2 t L L 1/2 M L ϕ N P M η(t) L ∞ ξ L 2 t ϕ N P M e −|t|ξ 2 L ∞ ξt M M 1/2 P M η L 2 t 1.
Similarly for P L (II), we have
L≥ N 2 L + N 2 1/2 ϕ N P L (II) L ∞ ξ L 2 t L L 1/2 M L ϕ N P M η(t) L ∞ ξt ϕ N P M e −|t|ξ 2 L ∞ ξ L 2 t M M 1/2 ϕ N P M e −|t|ξ 2 L ∞ ξ L 2 t .
Moreover, it is not too hard to check that if |ξ| ∼ N , then P M e −|t|ξ 2
L 2 t P M e −|t|N 2 L 2 t , thus L≥ N 2 L + N 2 1/2 ϕ N P L (II) L ∞ ξ L 2 t M M 1/2 P M e −|t|N 2 L 2 t 1,
where we used the fact that the Besov spaceḂ 1/2 2,1 has a scaling invariance and e −|t| ∈Ḃ 1/2 2,1 .
Lemma 4.1. For w ∈ S(R 2 ), consider k ξ defined on R by k ξ (t) = η(t)ϕ N (ξ) R e itτ e (t−|t|)ξ 2 − e −|t|ξ 2 iτ + ξ 2 w(τ )dτ.
Then, for all ξ ∈ R, it holds
L L + N 2 1/2 P L k ξ L 2 t L L + N 2 −1/2 ϕ L (τ )ϕ N (ξ) w L 2 τ .
Proof. Following [16], we rewrite k ξ as
k ξ (t) = η(t)e (t−|t|)ξ 2 |τ |≤1 e itτ − 1 iτ + ξ 2 w N (τ )dτ + η(t) |τ |≤1 e (t−|t|)ξ 2 − e −|t|ξ 2 iτ + ξ 2 w N (τ )dτ + η(t)e (t−|t|)ξ 2 |τ |≥1 e itτ iτ + ξ 2 w N (τ )dτ − η(t) |τ |≥1 e −|t|ξ 2 iτ + ξ 2 w N (τ )dτ = I + II + III − IV where w N is defined by F x (w N )(ξ) = ϕ N (ξ)F x (w)(ξ).
Contribution of IV . Clearly we have
P L (IV ) L 2 t P L (η(t)e −|t|ξ 2 ) L 2 t |τ |≥1 | w N (τ )| iτ + ξ 2 dτ.
On the other hand, by Cauchy-Schwarz in τ ,
|τ |≥1 | w N (τ )| iτ + ξ 2 dτ L L+N 2 −1 ϕ L w N L 1 τ L L+N 2 −1/2 ϕ L w N L 2 τ ,
which combined with (4.3) yields the desired bound.
Contribution of II. By Cauchy-Schwarz inequality,
P L (II) L 2 t P L (η(t)(e (t−|t|)ξ 2 − e −|t|ξ| 2 )) L 2 t × | w N (τ )| 2 iτ + ξ 2 dτ 1/2 |τ |≤1 iτ + ξ 2 |iτ + ξ 2 | 2 dτ 1/2 P L (η(t)(e (t−|t|)ξ 2 − e −|t|ξ| 2 )) L 2 t × N −2 N L L + N 2 −1/2 ϕ L w N L 2 τ . (4.4)
Hence we need to estimate
L L + N 2 1/2 P L (η(t)(e (t−|t|)ξ 2 − e −|t|ξ| 2 )) L 2 t L L + N 2 1/2 ( P L (η(t)e (t−|t|)ξ 2 ) L 2 t + P L (η(t)e −|t|ξ 2 ) L 2 t ).
The second term in the right-hand side is bounded by 1 thanks to estimate(4.3). Denote θ(t) = η(t)e (t−|t|)ξ 2 . It is not too hard to check that one integration by parts yields |θ(τ )| 1 |τ | whereas two integrations by parts give us |θ(τ )| ξ 2 |τ | 2 . We thus infer that
L L + N 2 1/2 ϕ Lθ L 2 τ L≤1 N L 1/2 θ L 1 t + 1≤L≤ N 2 N L 1/2 + L≥ N 2 L 1/2 N 2 L 3/2 N . (4.5)
This provides the result for N ≥ 1. In the case N ≤ 1, we use a Taylor expansion and obtain
P L (η(t)(e (t−|t|)ξ 2 − 1 + 1 − e −|t|ξ 2 ) L 2 t n≥1 |ξ| 2n n! P L (|t| n η(t)) L 2 t + 2 n P L (t n η(t)χ R − (t)) L 2 t .
According to the Sobolev embedding H 1 ֒→ B 1/2 2,1 as well as the estimate
χ R − f H 1 f H 1 provided f (0) = 0, we deduce L L + N 2 1/2 P L (η(t)(e (t−|t|)ξ 2 − e −|t|ξ 2 )) L 2 t ξ 2 n≥1 1 n! ( |t| n η(t) B 1/2 2,1 + 2 n t n η(t)χ R − (t) B 1/2 2,1 ) N 2 n≥1 2 n n! |t| n η(t) H 1 t N 2 .
Gathering this and (4.4) we conclude that
L L + N 2 1/2 P L (II) L 2 t L L + N 2 −1/2 ϕ L w N L 2 τ .
Contribution of I. Since I can be rewritten as
I = η(t)e (t−|t|)ξ 2 |τ |≤1 n≥1 (itτ ) n n! w N (τ ) iτ + ξ 2 dτ, we have P L (I) L 2 t n≥1 1 n! P L (t n θ(t)) L 2 t |τ |≤1 |τ | n |iτ + ξ 2 | | w N (τ )|dτ.
Using Cauchy-Schwarz we get, for n ≥ 1,
|τ |≤1 |τ | n |iτ + ξ 2 | | w N (τ )dτ | w N (τ )| 2 iτ + ξ 2 dτ 1/2 |τ |≤1 |τ | 2 iτ + ξ 2 |iτ + ξ 2 | 2 dτ 1/2 N −1 L L + N 2 −1/2 ϕ L w N L 2 τ .
Thus we see that it suffices to show that (see above the contribution of II for the definition of θ)
L L + N 2 1/2 n≥1 1 n! P L (t n θ(t)) L 2 t N .
But again we have |F t (t n θ(t))| 2 n min( 1 |τ | , ξ 2 τ 2 ) and arguing as in (4.5), we get
L L + N 2 1/2 n≥1 1 n! P L (t n θ(t)) L 2 t n≥1 N 2 n n! N .
Contribution of III. Settingĝ(τ ) = w N (τ ) iτ +ξ 2 χ |τ |≥1 , we have to prove
L L + N 2 1/2 P L (θg) L 2 t L L + N 2 1/2 P L g L 2 t . (4.6)
Using the paraproduct decomposition, we have
P L (θg) = P L M L (P M θP ∼M g + P ∼M θP M g) = P L (III 1 ) + P L (III 2 )
and we estimate the contributions of these two terms separately.
Contribution of III 1 . The sum over L ≥ N 2 is estimated in the following way:
L≥ N 2 L + N 2 1/2 P L (III 1 ) L 2 t L≥ N 2 L 1/2 M L P M θ L ∞ t P M g L 2 t M M 1/2 P M g L 2 t .
Now we deal with the case where L N 2 . Ifθ is localized in an annulus {|τ | ∼ M }, we get from Bernstein inequality that
L≤ N 2 L + N 2 1/2 M L P L (P M θP M g) L 2 t M N L M L 1/2 P M θP M g L 1 t M N M 1/2 P M θ L 2 t P M g L 2 M N P M g L 2 t , (4.7)
where we used the estimate P M θ L
L + N 2 1/2 M ∼L P L (P ≪M θP M g) L 2 t L N P ≪L θ L ∞ t P L g L 2 t ,
which is acceptable.
Contribution of III 2 . Consider the case L ≥ N 2 . Since |θ|
ξ 2 τ 2 , we have P L (P M θP M g) L 2 t ϕ Mθ L 1 τ P M g L 2 t N 2 M g L 2 t .
It follows that
L≥ N 2 L + N 2 1/2 P L (III 2 ) L 2 t M N 2 M 1/2 N 2 M g L 2 t N g L 2 t .
It remains to establish the bound in the case L ≤ N 2 . We may assume thatĝ is supported in a ball {|τ | ≪ M } since the other case has already been treated (cf. estimate (4.7)). Therefore, M ∼ L and L≤ N 2
L + N 2 1/2 M L P L (P M θP ≪M g) L 2 t L N P L θP ≪L g L 2 t L N P L θ L 2 t M ≪L P M g L ∞ t L N L −1/2 M ≪L M 1/2 P M g L 2 t M N P M g L 2 t .
The proof of Lemma 4.1 is complete.
Lf (t, x) = η(t) χ R + (t) t 0 W (t − t ′ , t − t ′ )f (t ′ )dt ′ +χ R − (t) t 0 W (t − t ′ , t + t ′ )f (t ′ )dt ′ . (4.8) If f ∈ N −1 , then Lf S −1 f N −1 . (4.9)
Proof. It suffices to show that
Lf X −1, 1 2 ,1 f X −1,− 1 2 ,−1 (4.10)
and Lf
Y −1, 1 2 f Y −1,− 1 2 . (4.11)
Taking the x-Fourier transform, we get
Lf (t, x) = U (t) χ R+ (t) η(t) R e ixξ t 0 e −|t−t ′ |ξ 2 F x (U (−t ′ )f (t ′ ))(ξ) dt ′ dξ + χ R− (t) η(t) R e ixξ t 0 e −|t+t ′ |ξ 2 F x (U (−t ′ )f (t ′ ))(ξ) dt ′ dξ = U (t) η(t) R e ixξ t 0 e −|t|ξ 2 e t ′ ξ 2 F x (U (−t ′ )f (t ′ ))(ξ) dt ′ dξ .
Setting w(t ′ ) = U (−t ′ )f (t ′ ), and using the time Fourier transform, we infer that
Lf (t, x) = U (t) η(t) R 2 e ixξ e itτ e (t−|t|)ξ 2 − e −|t|ξ 2 iτ + ξ 2w (τ, ξ)dτ dξ .
Estimate (4.10) follows then easily from Lemma 4.1. Now we turn to estimate (4.11). After square summing, it suffices to prove that for any dyadic N ,
(∂ t + ∂ xxx − ∂ xx + I)P N Lf L 1 t L 2 x P N f L 1 t L 2 x .
(4.12)
In view of the expression of L it suffices to prove (4.12) separately for χ R + Lf and χ R − Lf First, a straightforward calculation leads to
(∂ t + ∂ xxx − ∂ xx + I)(χ R + Lf (t)) = η(t)χ R + (t)f (t) + (η ′ (t) + η(t))χ R + (t) t 0 W (t − t ′ , t − t ′ )f (t ′ )dt ′ .
Computing the L 1 t L 2 x norm, we get
(∂ t + ∂ xxx − ∂ xx + I)P N (χ R + Lf ) L 1 t L 2 x f L 1 t L 2 x + η ′ + η L 1 t sup t ∞ 0 e i(t−t ′ )ξ 3 e −(t−t ′ )ξ 2 f (t ′ ) L 2 ξ dt ′ ,
and estimate (4.12) follows. Now, let us tackle the proof for χ R − Lf . We have to work a little more since clearly Lf does not satisfy the same equation for negative times. Actually, one can check that and thus
(∂ t + ∂ xxx + ∂ xx + I)(χ R − Lf (t)) = η(t)χ R − (t)W (0, 2t)f (t) + (η ′ (t) + η(t))χ R − (t) t 0 W (t − t ′ , t + t ′ )f (t ′ )dt ′ .(∂ t + ∂ xxx − ∂ xx + I)(χ R − Lf (t)) = −2∂ xx (χ R − Lf (t)) + η(t)χ R − (t)W (2t, 0)f (t) + (η ′ (t) + η(t))χ R − (t) t 0 W (t − t ′ , t + t ′ )f (t ′ )dt ′ .
(4.14)
Setting w := P N (χ R − Lf (t)) and
g := η(t)χ R − (t)W (2t, 0)f (t) + (η ′ (t) + η(t))χ R − (t) t 0 W (t − t ′ , t + t ′ )f (t ′ )dt ′
we first note as above that
g L 1 t L 2 x f L 1 t L 2 x . (4.15)
Now, according to (4.13), w satisfies
w t − w xxx + w xx + w = g
Taking the L 2 x -scalar product with w and using Cauchy-Schwarz yield
1 2 d dt w 2 L 2 x − w x 2 L 2 x + w 2 L 2 x ≥ − g L 2 x w L 2 x . (4.16)
By the frequencies localization of w and Bernstein inequality, w x L 2 x ≥ 1 2 N w L 2 x . Therefore, for t > 0, such that w(t) L 2 x = 0, we can divide (4.16) by w(t) L 2 x to get
N 2 w(t) L 2 x d dt w(t) L 2 x + w(t) L 2 x + g(t) L 2 x (4.17)
On the other hand, for t > 0, the smoothness and non negativity of t → w(t) L 2 x forces d dt w(t) 2
L 2
x = 0 as soon as w(t) L 2 x = 0. This ensures that (4.17) is actually valid for all t > 0. Therefore integrating (4.17) on ]0, t[ we infer that
w xx L 1 t L 2 x ∼ N 2 w L 1 t L 2 x w L ∞ t L 2 x + w L 1 t L 2 x + g L 1 t L 2 x . Since obviously, w L 1 t L 2 x + w L ∞ t L 2 x sup t ∞ 0 e i(t−t ′ )ξ 3 e −|t+t ′ |ξ 2 P N f (t ′ ) L 2 ξ dt ′ P N f L 1 t L 2 x it follows that w xx L 1 t L 2 x P N f L 1 t L 2
x which concludes the proof together with (4.14) and (4.15).
Bilinear estimate
In this section we provide a proof of the following crucial bilinear estimate.
Proposition 5.1. For all u, v ∈ S −1 , we have ∂ x (uv) N −1 u S −1 v S −1 . (5.1)
First we remark that because of the L 2 ξ structure of the spaces involved in our analysis we have the following localization property
f S −1 ∼ N P N f 2 S −1 1/2 and f N −1 ∼ N P N f 2 N −1 1/2 .
Performing a dyadic decomposition for u, v we thus obtain
∂ x (uv) N −1 ∼ N N 1 ,N 2 P N ∂ x (P N 1 uP N 2 v) 2 N −1 1/2 . (5.2)
We can now reduce the number of case to analyze by noting that the righthand side vanishes unless one of the following cases holds:
• (high-low interaction) N ∼ N 2 and N 1 N ,
• • (low-high interaction) N ∼ N 1 and N 2 N ,
• (high-high interaction) N ≪ N 1 ∼ N 2 . The former two cases are symmetric. In the first case, we can rewrite the right-hand side of (5.2) as
∂ x (uv) N −1 ∼ N P N ∂ x (P N uP N v) 2 N −1 1/2 ,
and it suffices to prove the high-low estimate
P N ∂ x (P N uP N v) N −1 u S −1 P N v S −1 (HL)
for any dyadic N . If we consider now the third case, we easily get
∂ x (uv) N −1 N 1 P ≪N 1 ∂ x (P N 1 uP N 1 v) N −1 ,
and it suffices to prove for any N 1 the high-high estimate
P ≪N 1 ∂ x (P N 1 uP N 1 v) N −1 P N 1 u S −1 P N 1 v S −1 (HH)
since the claim follows then from Cauchy-Schwarz.
Proof of (HL)
We decompose the bilinear term as
P N ∂ x (P N uP N v) = N 1 N L,L 1 ,L 2 P N Q L ∂ x (P N 1 Q L 1 uP N Q L 2 v).
Using the well-known resonance relation
ξ 3 1 + ξ 3 2 + ξ 3 3 = 3ξ 1 ξ 2 ξ 3 whenever ξ 1 + ξ 2 + ξ 3 = 0,(5.3)
we see that non-trivial interactions only happen when
L max ∼ max(N 2 N 1 , L med ) (5.4)
where L max ≥ L med ≥ L min holds for L, L 1 , L 2 .
First we consider the easiest case N 1 1. We take advantage of the Y −1,− 1 2 part of N −1 as well as Hölder and Bernstein inequalities to obtain
N 1 1 P N ∂ x (P N 1 uP N v) Y −1,− 1 2 N 1 1 N −1 N P N (P N 1 uP N v) L 1 t L 2 x N 1 1 P N 1 u L 2 t L ∞ x P N v L 2 N 1 1 N 1/2 1 P N 1 u L 2 P N v L 2 u S −1 v S −1
where we used (3.6) in the last estimate. One can now assume we have large space frequencies, i.e. N N 1 1.
Case L max = L
In light of (5.4), we are in the region L N 2 N 1 . From the definition of X −1, 1 2 ,1 we have
1 N 1 N L N 2 N 1 P N Q L ∂ x (P N 1 uP N v) X −1,− 1 2 ,1 1 N 1 N L N 2 N 1 N −1 L −1/2 N P N Q L (P N 1 uP N v) L 2 .
Then, estimates (3.6) and (3.12) lead to the bound
1 N 1 N N −1 N −1/2 1 P N 1 u L ∞ t L 2 x P N v L 2 t L ∞ x 1 N 1 N N 1/2 1 N −1/2 P N 1 u L ∞ t H −1 x P N v L 2 u S −1 P N v S −1 .
Case
L max = L 1
Here we must have either L 1 ∼ N 2 N 1 or L 1 ∼ L med . Note that the second case has been treated in Subsection 5.1.1 when L med = L and we reduce to
L 1 ∼ L 2 .
The contribution for the former case can be estimated as follows:
1 N 1 N L 1 ∼N 2 N 1 P N Q L ∂ x (P N 1 Q L 1 uP N v) Y −1,− 1 2 1 N 1 N P N 1 Q N 2 N 1 uP N v L 1 t L 2 x 1 N 1 N N 1/2 1 P N 1 Q N 2 N 1 u L 2 P N v L 2 .
Now we can exploit the smoothing relation L 1 ∼ N 2 N 1 and obtain
N 1/2 1 P N 1 Q N 2 N 1 u L 2 N −1 N 1 (N −1 1 L 1/2 1 P N 1 Q L 1 u L 2 ), (5.5)
which combined with (3.6), (3.7) and Cauchy-Schwarz in N 1 yields the desired bound. It remains to treat the case L 1 ∼ L 2 N 2 N 1 where we can use both on L 1 and L 2 the smoothing relation. Arguing as before we get
1 N 1 N L 1 ∼L 2 N 2 N 1 P N ∂ x (P N 1 Q L 1 uP N Q L 2 v) Y −1,− 1 2 1 N 1 N L 1 N 2 N 1 N 1/2 1 P N 1 Q L 1 u L 2 P N Q L 1 v L 2 .
In this regime, (5.5) is still valid if we replace Q N 2 N 1 by Q L 1 . Applied on u and v, this provides the bound
1 N 1 N N 1/2 1 N −1 L 1 (N −1 1 L 1/2 1 P N 1 Q L 1 u L 2 ) 2 1/2 L 1 (N −1 L 1/2 1 P N Q L 1 v L 2 ) 2 1/2 1 N 1 N N 1/2 1 N −1 P N 1 u S −1 P N v S −1 ,
which is acceptable (with about N −1/2 of spare).
Case
L max = L 2
By (5.4), it suffices to consider the case L 2 ∼ N 2 N 1 . With a similar argument we get
N 1 N L 2 ∼N 2 N 1 P N ∂ x (P N 1 uP N Q L 2 v) Y −1,− 1 2 N 1 N N 1/2 1 P N 1 u L 2 P N Q N 2 N 1 v L 2 N 1 P N 1 u 2 L 2 1/2 N 1 (N 1/2 1 P N Q N 2 N 1 v L 2 ) 2 1/2 u S −1 L 2 (N −1 L 1/2 2 P N Q L 2 v L 2 ) 2 1/2 ,
which achieves the proof of (HL).
Proof of (HH)
Performing the decomposition
P ≪N 1 ∂ x (P N 1 uP N 1 v) = N ≪N 1 L,L 1 ,L 2 P N Q L ∂ x (P N 1 Q L 1 uP N 1 Q L 2 v),
we see from (5.3) that we may restrict ourself to the region where
L max ∼ max(N 2 1 N, L med ). (5.6)
Moreover, we may assume by symmetry that L 1 ≥ L 2 . Low frequencies N 1 are easily handled:
N 1 P N ∂ x (P N 1 uP N 1 v) Y −1,− 1 2 N 1 N −1 N P N (P N 1 uP N 1 v) L 1 t L 2 x N 1 P N 1 u L 2 P N 1 v L 2 P N 1 u S −1 P N 1 v S −1 .
Therefore it is sufficient to consider N 1 ≫ N 1.
Case L max = L
In this region one has L N 2 1 N . Let us assume L 1 N 2 1 N 1−ε for some ε > 0 so that we wish to bound
1 N ≪N 1 P N Q N 2 1 N ∂ x (P N 1 Q N 2 1 N 1−ε uP N 1 v) X −1,− 1 2 ,1 . (5.7)
Using the triangle inequality we reduce to estimate
1 N ≪N 1 L N 2 1 N L 1 N 2 1 N 1−ε L −1/2 P N 1 Q L 1 uP N 1 v L 2 .
In order to get a suitable control for this term, we apply the Kato smoothing effect (3.13) together with estimate (3.6) to get
P N 1 Q L 1 uP N 1 v L 2 P N 1 Q L 1 u L 2 x L ∞ t P N 1 v L ∞ x L 2 t L 1/2 1 P N 1 u S −1 P N 1 v S −1 .
Therefore it remains to establish
1 N ≪N 1 L N 2 1 N L 1 N 2 1 N 1−ε L −1/2 L 1/2 1 1,
but this is easily verified by Schur's test for any ε > 0. The situation where L 2 N 2 1 N 1−ε is identical to the previous case and we suppose now L 1 , L 2 N 2 1 N 1−ε . Estimating the N −1 -norm by the Y −1,− 1 2 -norm, and using the Hölder and Bernstein inequalities we see that the contribution in this case is bounded by
1 N ≪N 1 P N (P N 1 Q N 2 1 N 1−ε uP N 1 Q N 2 1 N 1−ε v L 1 t L 2 x 1 N ≪N 1 N 1/2 P N 1 Q N 2 1 N 1−ε u L 2 P N 1 Q N 2 1 N 1−ε v L 2 . (5.8)
On the other hand the resonance relation and (3.7) yield
N 1/2 P N 1 Q N 2 1 N 1−ε u L 2 N ε/2 L 1 [N −1 1 L 1/2 1 P N 1 Q L 1 u L 2 ] 2 1/2 N ε/2 P N 1 u S −1 ,
and similarly for v. Inserting this into (5.8) we deduce (5.8)
N 1 N −1/2+ε P N 1 u S −1 P N 1 v S −1 ,
which is acceptable for ε < 1/2.
Case
L max = L 1
First we consider the region L 1 ∼ N 2 1 N and we want to estimate
N ≪N 1 P N ∂ x (P N 1 Q N 2 1 N uP N 1 v) Y −1,− 1 2 N [N 1/2 P N 1 Q N 2 1 N u L 2 P N 1 v L 2 ] 2 1/2
where we took care of not using the triangle inequality in order to keep the ℓ 2 -norm in N . The term P N 1 u L 2 can be handled with help of (3.6), while the change of variable N ∼ L 1 N −2 1 for fixed N 1 leads to the bound
L 1 [N −1 1 L 1/2 1 P N 1 Q L 1 u L 2 ] 2 1/2 P N 1 v S −1 P N 1 u S −1 P N 1 v S −1 .
Finally in the case L 1 ∼ L 2 N 2 1 N , arguing as in Subsection 5.1.2, we get
1 N ≪N 1 L 1 ∼L 2 ≫N 2 1 N P N ∂ x (P N 1 Q L 1 uP N 1 Q L 2 v) Y −1,− 1 2 1 N ≪N 1 L 1 ≫N 2 1 N N 1/2 P N 1 Q L 1 u L 2 P N 1 Q L 1 v L 2 N 1 N −1/2 L 1 (N −1 1 L 1/2 1 P N 1 Q L 1 u L 2 ) 2 1/2 L 1 (N −1 1 L 1/2 1 P N 1 Q L 1 v L 2 ) 2 1/2 ,
which is acceptable (with about N −1/2 of spare).
Well-posedness
In this section, we prove the well-posedness result. Using a standard fixed point procedure, it is clear that the bilinear estimate (5.1) allows us to show local well-posedness but for small initial data only. This is because H −1 appears as a critical space for KdV-Burgers and thus we can't get the desired contraction factor in our estimates. In order to remove the size restriction on the data, we need to change the metric on our resolution space. For β ≥ 1, let us define the following norm on S −1 ,
u Z β = inf u=u 1 +u 2 u 1 ∈S −1 ,u 2 ∈S 0 u 1 S −1 + 1 β u 2 S 0 .
Note that this norm is equivalent to · S −1 . Now we will need the following modification of Proposition 5.1. This new proposition means that as soon as we assume more regularity on u we can get a contractive factor for small times in the bilinear estimate.
Proposition 6.1. There exists ν > 0 such that for all (u, v) ∈ S 0 × S −1 , with compact support (in time) in [−T, T ], it holds ∂ x (uv) N −1 T ν u S 0 v S −1 .F −1 t,x f (τ, ξ) τ − ξ 3 θ L 2 t,x T µ f L 2,2 t,x . (6.2)
According to (3.7) this ensures, in particular, that for any w ∈ S 0 with compact support in [−T, T ] it holds
w L 2 t H 3/4 w X 0,3/8,2 T µ( 1 8 ) w X 0,1/2,2 T µ( 1 8 ) w S 0 . (6.3)
It is pretty clear that the interactions between high frequencies of u and high or low frequencies of v can be treated by following the proof of Proposition 5.1 and using (6.3). The region that seems the most dangerous is the one of interactions between low frequencies of u and high frequencies of v, that is the region of (HL) in the proof of Proposition 5.1. But actually this region can also be easily treated. For instance in the case 5.1.1 it suffices to notice
that 1 N 1 N L N 2 N 1 P N Q L ∂ x (P N 1 uP N v) X −1,− 1 2 ,1 1 N 1 N L N 2 N 1 N −1 L −1/2 N P N Q L (P N 1 uP N v) L 2 . 1 N 1 N N −1 N −1/2 1 P N 1 u L 2 t L ∞ x P N v L ∞ t L 2 x 1 N 1 N N −1/2 1 P N 1 u L 2 t H 1/2 x P N v L ∞ t H −1 x T µ( 1 8 ) u S 0 v S −1
and in the case 5.1.2 it suffices to replace (5.5) by simply
N 1/2 1 P N 1 Q N 2 N 1 u L 2 N −1/4 1 P N 1 u L 2 t H 3/4 x N −1/4 1 T µ( 1 8 ) u S 0 .
The other cases can be handle in a similar way.
We are now in position to prove that the application
F T φ : u → η(t) W (t)φ − 1 2 L∂ x (η T u) 2 ,
where L is defined in (4.8), is contractive on a ball of Z β for a suitable β > 0 and T > 0 small enough. Assuming this for a while, the local part of Theorem 1.1 follows by using standard arguments. Note that the uniqueness will hold in the restriction spaces S −1 τ endowed with the norm
u S −1 τ := inf v∈S −1 { v S −1 , v ≡ u on [0, τ ]} .
Finally, to see that the solution u can be extended for all positive times and belongs to C(R * + ; H ∞ ) it suffices to notice that, according to (3.6), u ∈ S −1 τ ֒→ L 2 (]0, τ [×R) . Therefore, for any 0 < τ ′ < τ there exists t 0 ∈]0, τ ′ [, such that u(t 0 ) belongs to L 2 (R) . Since according to [16], (1.1) is globally well-posed in L 2 (R) with a solution belonging to C(R * + ; H ∞ (R)), the conclusion follows.
In order to prove that F T φ is contractive, the first step is to establish the following result.
L∂ x (uv) Z β u Z β v Z β . (6.4)
Assume for the moment that (6.4) holds and let u 0 ∈ H −1 and α > 0. Split the data u 0 into low and high frequencies: u 0 = P N u 0 + P ≫N u 0 for a dyadic number N . Taking N = N (α) large enough, it is obvious to check that P ≫N u 0 H −1 ≤ α. Hence, according to (4.1), η(·)W (·)P ≫N u 0 Z β α.
Using now the S 0 -part of Z β , we control the low frequencies as follows:
η(·)W (·)P N u 0 S 0 1 β P N u 0 L 2 N β u 0 H −1 .
Thus we get η(·)W (·)P N u 0 Z β α for β N u 0 H −1 α .
Since α can be chosen as small as needed, we conclude with (6.4) that F T φ is contractive on a ball of Z β of radius R ∼ α as soon as β N u 0 H −1 /α and T = T (β).
Proof of Proposition 6.2. By definition on the function space Z β , there exist u 1 , v 1 ∈ S −1 and u 2 , v 2 ∈ S 0 such that u = u 1 + v 1 , v = v 1 + v 2 and
u 1 S −1 + 1 β u 2 S 0 ≤2 u Z β , v 1 S −1 + 1 β v 2 S 0 ≤2 v Z β .
Thus one can decompose the left-hand side of (6.4) as
L∂ x (uv) Z β L∂ x (u 1 v 1 ) S −1 + L∂ x (u 1 v 2 + u 2 v 1 ) S −1 + L∂ x (u 2 v 2 ) S −1 = I + II + III.
From the estimates (4.9) and (5.1) we get
I ∂ x (u 1 v 1 ) N −1 u 1 S −1 v 1 S −1 u Z β v Z β .
On the other hand, we obtain from (6.1) that
III T ν u 2 S 0 v 2 S 0 β 2 T ν u Z β v Z β . and II T ν ( u 1 S −1 v 2 S 0 + u 2 S 0 v 1 S −1 ) βT ν u Z β v Z β .
We thus get
L∂ x (uv) Z β (1 + (β + β 2 )T ν ) u Z β v Z β .
This ensures that (6.4) holds for T ∼ β −2/ν ≤ 1.
.
Ifθ is localized in a ball {|τ | ≪ M }, then we must have M ∼ L and thus L≤ N 2
Proposition 4. 2 .
2Let L : f → Lf denote the linear operator
( 6 . 1 )
61Proof. It suffices to slightly modify the proof of Proposition 5.1 to make use of the following result that can be found in [[10], Lemma 3.1] (see also [[16], Lemma 3.6]): For any θ > 0, there exists µ = µ(θ) > 0 such that for any smooth function f with compact support in time in [−T, T ],
Proposition 6 . 2 .
62For any β ≥ 1 there exists 0 < T = T (β) < 1 such that for any u, v ∈ Z β with compact support in [−T, T ] we have
Theorem 1.1. The Cauchy problem associated to (1.1) is locally analytically well-posed in H −1 (R). Moreover, at every point u 0
See also[5] where it is proven that the solution-map is even not uniformly continuous on bounded sets below this index
Note that(3.12) can also be deduced from estimate(3.9).
The initial-value problem for the generalized Burgers' equation. D Bekiranov, Diff. Int. Eq. 96D.Bekiranov, The initial-value problem for the generalized Burgers' equation, Diff. Int. Eq., 9 (6) (1996), pp. 1253-1265.
Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation. I Bejenaru, T Tao, J. Funct. Anal. 2331I. Bejenaru and T. Tao, Sharp well-posedness and ill-posedness results for a quadratic non-linear Schrödinger equation. J. Funct. Anal. 233 (2006), no. 1, 228-259.
Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations II. The KdV equation. J Bourgain, GAFA. 3J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations II. The KdV equation, GAFA, 3 (1993), pp. 209-262.
Periodic Korteveg de Vries equation with measures as initial data. J Bourgain, Sel. Math. New. Ser. 3J. Bourgain, Periodic Korteveg de Vries equation with measures as ini- tial data , Sel. Math. New. Ser. 3 (1993), pp. 115-159.
Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. M Christ, J Colliander, T Tao, Amer. J. Math. 1256M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modula- tion, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003), no. 6, 1235-1293.
Sharp global well-posedness results for perdiodic and non-periodic KdV and modified KdV on R and T. J Colliander, M Keel, G Staffilani, H Takaoka, T Tao, J. Amer. Math. Soc. 16J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness results for perdiodic and non-periodic KdV and modified KdV on R and T, J. Amer. Math. Soc. 16 (2003), pp. 705-749.
Nonuniqueness and uniqueness in the initial-value problem for Burger's equation. D B Dix, SIAM J. Math. Anal. 273D.B. Dix, Nonuniqueness and uniqueness in the initial-value problem for Burger's equation, SIAM J. Math. Anal., 27 (3) (1996), pp. 708-724.
Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem. P Gérard, International Congress of Mathematicians. IIIEur. Math. Soc.P. Gérard, Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem, International Congress of Mathematicians. Vol. III, 157-182, Eur. Math. Soc., Zürich, 2006.
Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), in Séminaire Bourbaki 796. J Ginibre, Astérique. 237J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), in Séminaire Bourbaki 796, Astérique 237, 1995, 163-187.
On the Cauchy problem for the Zakharov system. J Ginibre, Y Tsutsumi, G Velo, J. Funct. Analysis. 133J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Analysis, 133 (1995), pp. 50-68.
Global Well-posedness of Korteweg-de Vries equation in H −3/4 (R) , to appear. Z Guo, J. Math. Pures Appl. Z. Guo, Global Well-posedness of Korteweg-de Vries equation in H −3/4 (R) , to appear J. Math. Pures Appl.
Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation. Z Guo, B Wang, J. Diff. Eq. 24610Z. Guo, B. Wang, Global well-posedness and inviscid limit for the Korteweg-de Vries-Burgers equation, J. Diff. Eq. 246 (2009), no. 10, 3864-3901.
T Kappeler, P Topalov ; T, R ) , Duke Math, Global wellposedness of KdV in H −1. 135T. Kappeler and P. Topalov, Global wellposedness of KdV in H −1 (T, R), Duke Math. J. 135 (2006), no. 2, 327-360.
A bilinear estimate with applications to the KdV equation. C E Kenig, G Ponce, L Vega, J. Amer. Math. Soc. 9C. E. Kenig, G. Ponce, and L. Vega, A bilinear estimate with applica- tions to the KdV equation, J. Amer. Math. Soc., 9 (1996), pp. 573-603.
Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case. L Molinet, S Vento, to be completedL. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case (to be completed).
On the low regularity of the Korteweg-de Vries-Burgers equation. L Molinet, F Ribaud, I.M.R.N. 37L. Molinet and F. Ribaud, On the low regularity of the Korteweg-de Vries-Burgers equation, I.M.R.N. 37, (2002), pp. 1979-2005.
Damping of solitary waves. E Ott, N Sudan, Phys. Fluids. 136E. Ott and N. Sudan, Damping of solitary waves, Phys. Fluids, 13(6) (1970), pp. 1432-1434.
Multilinear weighted convolution of L 2 -functions, and applications to nonlinear dispersive equations. T Tao, Amer. J. Math. 1235T. Tao, Multilinear weighted convolution of L 2 -functions, and appli- cations to nonlinear dispersive equations. Amer. J. Math. 123 (2001), no.5, 839-908.
Scattering for the quartic generalised Korteweg-de Vries equation. T Tao, J. Diff. Eq. 2322T. Tao, Scattering for the quartic generalised Korteweg-de Vries equa- tion J. Diff. Eq. 232 (2007), no. 2, 623-651.
On global existence and scattering for the wave maps equation. D Tataru, Amer. J. Math. 1231D. Tataru, On global existence and scattering for the wave maps equa- tion, Amer. J. Math. 123 (1) (2001), 37-77.
. Luc Molinet, Luc Molinet,
Université François Rabelais Tours, Fédération Denis Poisson-CNRS. Physique Laboratoire De Mathématiques, Théorique, Parc Grandmont, 37200 ToursFRANCE. [email protected] Stéphane VentoLaboratoire de Mathématiques et Physique Théorique, Université François Rabelais Tours, Fédération Denis Poisson-CNRS, Parc Grandmont, 37200 Tours, FRANCE. [email protected] Stéphane Vento
. L A G A Institut Galilée, Université Paris 13L.A.G.A., Institut Galilée, Université Paris 13,
. France Villetaneuse, Villetaneuse, FRANCE. [email protected]
| [] |
[
"X-ray source population study of the starburst galaxy M 83 with XMM-Newton ⋆,⋆⋆,⋆⋆⋆",
"X-ray source population study of the starburst galaxy M 83 with XMM-Newton ⋆,⋆⋆,⋆⋆⋆"
] | [
"L Ducci [email protected] \nInstitut für Astronomie und Astrophysik\nEberhard Karls Universität\nSand 172076TübingenGermany\n",
"M Sasaki \nInstitut für Astronomie und Astrophysik\nEberhard Karls Universität\nSand 172076TübingenGermany\n",
"F Haberl \nMax-Planck-Institut für Extraterrestrische Physik\n85741Giessenbachstraße, GarchingGermany\n",
"W Pietsch \nMax-Planck-Institut für Extraterrestrische Physik\n85741Giessenbachstraße, GarchingGermany\n"
] | [
"Institut für Astronomie und Astrophysik\nEberhard Karls Universität\nSand 172076TübingenGermany",
"Institut für Astronomie und Astrophysik\nEberhard Karls Universität\nSand 172076TübingenGermany",
"Max-Planck-Institut für Extraterrestrische Physik\n85741Giessenbachstraße, GarchingGermany",
"Max-Planck-Institut für Extraterrestrische Physik\n85741Giessenbachstraße, GarchingGermany"
] | [] | Aims. We present the results obtained from the analysis of three XMM-Newton observations of M 83. The aims of the paper are studying the X-ray source populations in M 83 and calculating the X-ray luminosity functions of X-ray binaries for different regions of the galaxy. Methods. We detected 189 sources in the XMM-Newton field of view in the energy range of 0.2 − 12 keV. We constrained their nature by means of spectral analysis, hardness ratios, studies of the X-ray variability, and cross-correlations with catalogues in X-ray, optical, infrared, and radio wavelengths. Results. We identified and classified 12 background objects, five foreground stars, two X-ray binaries, one supernova remnant candidate, one super-soft source candidate and one ultra-luminous X-ray source. Among these sources, we classified for the first time three active galactic nuclei (AGN) candidates. We derived X-ray luminosity functions of the X-ray sources in M 83 in the 2 − 10 keV energy range, within and outside the D 25 ellipse, correcting the total X-ray luminosity function for incompleteness and subtracting the AGN contribution. The X-ray luminosity function inside the D 25 ellipse is consistent with that previously observed by Chandra.The Kolmogorov-Smirnov test shows that the X-ray luminosity function of the outer disc and the AGN luminosity distribution are uncorrelated with a probability of ∼ 99.3%. We also found that the X-ray sources detected outside the D 25 ellipse and the uniform spatial distribution of AGNs are spatially uncorrelated with a significance of 99.5%. We interpret these results as an indication that part of the observed X-ray sources are X-ray binaries in the outer disc of M 83. | 10.1051/0004-6361/201321035 | [
"https://arxiv.org/pdf/1303.4306v1.pdf"
] | 55,157,117 | 1303.4306 | 7f8111b18ebaf534c3020d4def6126d7dbaae19a |
X-ray source population study of the starburst galaxy M 83 with XMM-Newton ⋆,⋆⋆,⋆⋆⋆
18 Mar 2013 February 6, 2014
L Ducci [email protected]
Institut für Astronomie und Astrophysik
Eberhard Karls Universität
Sand 172076TübingenGermany
M Sasaki
Institut für Astronomie und Astrophysik
Eberhard Karls Universität
Sand 172076TübingenGermany
F Haberl
Max-Planck-Institut für Extraterrestrische Physik
85741Giessenbachstraße, GarchingGermany
W Pietsch
Max-Planck-Institut für Extraterrestrische Physik
85741Giessenbachstraße, GarchingGermany
X-ray source population study of the starburst galaxy M 83 with XMM-Newton ⋆,⋆⋆,⋆⋆⋆
18 Mar 2013 February 6, 2014arXiv:1303.4306v1 [astro-ph.HE] Astronomy & Astrophysics manuscript no. lducci_m83 c ESO 2014galaxies: individual; M 83 − X-rays: galaxies
Aims. We present the results obtained from the analysis of three XMM-Newton observations of M 83. The aims of the paper are studying the X-ray source populations in M 83 and calculating the X-ray luminosity functions of X-ray binaries for different regions of the galaxy. Methods. We detected 189 sources in the XMM-Newton field of view in the energy range of 0.2 − 12 keV. We constrained their nature by means of spectral analysis, hardness ratios, studies of the X-ray variability, and cross-correlations with catalogues in X-ray, optical, infrared, and radio wavelengths. Results. We identified and classified 12 background objects, five foreground stars, two X-ray binaries, one supernova remnant candidate, one super-soft source candidate and one ultra-luminous X-ray source. Among these sources, we classified for the first time three active galactic nuclei (AGN) candidates. We derived X-ray luminosity functions of the X-ray sources in M 83 in the 2 − 10 keV energy range, within and outside the D 25 ellipse, correcting the total X-ray luminosity function for incompleteness and subtracting the AGN contribution. The X-ray luminosity function inside the D 25 ellipse is consistent with that previously observed by Chandra.The Kolmogorov-Smirnov test shows that the X-ray luminosity function of the outer disc and the AGN luminosity distribution are uncorrelated with a probability of ∼ 99.3%. We also found that the X-ray sources detected outside the D 25 ellipse and the uniform spatial distribution of AGNs are spatially uncorrelated with a significance of 99.5%. We interpret these results as an indication that part of the observed X-ray sources are X-ray binaries in the outer disc of M 83.
Introduction
M 83 (NGC 5236) is a grand-design barred spiral galaxy (SAB(s)c; de Vaucouleurs et al. 1992) located at 4.5 ± 0.3 Mpc from the Milky Way (Thim et al. 2003). M 83 is oriented nearly face-on (i = 24 • ; Rogstad et al. 1974) and shows a galactic disc spanning 12.9 ′ × 11.5 ′ (17 kpc × 15.2 kpc; Wofford et al. 2011).
M83 is experiencing a starburst activity with a present-day star formation rate (SFR) of 3 − 4 M ⊙ yr −1 in three regions: the nuclear region (galactocentric distance d 300 pc; Harris et al. 2001), the inner disc (300 pc d 7.5 kpc), and the outer disc (7.5 kpc d 20 kpc; Dong et al. 2008). Ultraviolet (UV) images of M 83 obtained with the Galaxy Evolution Explorer (GALEX) satellite revealed a population of young stars ( 400 Myr) in the outer disc of M 83 (Thilker et al. 2005). Although this would indicate recent star-forming activity, using Spitzer and GALEX data, Dong et al. (2008) discovered that the star formation in the outer disc started at least 1 Gyr ago. These results are confirmed by the study of AGB stars of Davidge (2010). Bigiel et al. (2010) compared the HI data from the National Radio Astronomy Observatory (NRAO) Very Large Array (VLA) and far-ultraviolet (FUV) data from GALEX in the outer disc of M 83, and discovered that the star formation traced by the FUV emission and HI are spatially correlated out to almost four optical radii. Bigiel et al. (2010) also found that the star formation rate in the outer disc (∼ 0.01 M ⊙ yr −1 ; Bresolin et al. 2009) implies that the star formation activity is not completely consuming the HI reservoir, which will be available as fuel for star formation in the inner disc.
M 83 was observed in the X-ray bands by Einstein in 1979-1981(Trinchieri et al. 1985, Ginga in 1987 (Ohashi et al. 1990), ROSAT in 1992-1994(Ehle et al. 1998Immler et al. 1999), ASCA in 1994 (Okada et al. 1997), and Chandra in 2000 (Soria & Wu 2002;Soria & Wu 2003, SW03 hereafter). SW03 identified 127 discrete sources near the centre of M 83 (8.3 ′ × 8.3 ′ ) and resolved for the first time the nuclear region in X-rays. The diffuse X-ray emission of M 83 has been studied by Owen & Warwick (2009) with an XMM-Newton observation performed in January 27, 2003 (obsid 0110910201). They obtained a good fit to the spectrum assuming a two-temperature thermal model, which is typical of the diffuse emission in normal and starburst galaxies. They also found that the soft X-ray emission mainly overlaps with the inner spiral arm, and shows a strong correlation with the distribution of UV emission, indicative of a correlation between X-ray emission and recent star formation.
The recent high star formation activity experienced by the nucleus and the spiral arms of M 83 provided an unusually large number of supernova remnants (SNRs). In fact, the optical sur- vey performed at the Cerro Tololo Inter-American Observatory in Chile by Blair & Long (2004) identified 71 sources as SNR candidates, the Hubble Space Telescope (HST) observations of the nuclear region of M 83 (Dopita et al. 2010) provided the identification of 60 SNR candidates, and the Magellan I survey 271 SNR candidates ).
In a normal galaxy such as M 83, X-ray binaries (XRBs) are the most prominent class of X-ray sources. XRBs show Xray luminosities ranging from ∼ 10 32 erg s −1 to the Eddington luminosity, and sometimes they can exceed this limit (see e.g. White & Carpenter 1978). They are composed of a compact object (a neutron star or a black hole) and a companion star, which can be a main-sequence, giant, or supergiant star, and in some cases a white dwarf (e.g. van Paradijs 1998). The strong X-ray emission is produced by the accretion of matter from the companion star onto the compact object. XRBs are usually divided into two classes: low mass X-ray binaries (LMXBs), and high mass X-ray binaries (HMXBs). The companion stars of LMXBs have masses lower than ∼ 1 M ⊙ . The lifetime of an LMXB is determined by the nuclear evolution time-scale of the companion star to 10 8 − 10 9 yr (e.g. Tauris & van den Heuvel 2006), and their number is correlated to the total stellar mass of a galaxy (Gilfanov 2004). The companion star of LMXBs usually tranfers mass by Roche-lobe overflow, and the compact object accretes from an accretion disc (e.g. van Paradijs 1998). The donors in HMXBs have masses 8 M ⊙ , and their typical lifetime does not exceed 10 6 − 10 7 yr. Therefore, the presence of HMXBs in a particular region of a galaxy is associated with a relatively recent star formation event (e.g. Fabbiano 2006). The X-ray emission from HMXBs is usually explained with the accretion of a fraction of the stellar wind ejected by the donor star onto the compact object, or through mass transfer via Roche-lobe overflow (see e.g. Treves et al. 1988 and references therein). As a first approximation, two standard models are commonly used to describe the X-ray spectra of XRBs in nearby galaxies: an absorbed disc-blackbody model, with temperatures ranging from ∼ 0.5 to ∼ 1 keV (e.g. Makishima et al. 1986), or an absorbed powerlaw model. X-ray spectra of LMXBs below 10 keV are described by absorbed powerlaw with photon indices 1 − 3. HMXBs usually show harder X-ray spectra in the energy range 1 − 10 keV, with photon indices 1 − 2 and a high intrinsic absorption (White et al. 1995). Within each of these classes, the properties of the X-ray spectra can also depend on the type of the accreting compact object. Accreting black holes can show states of high luminosity (e.g. Jones 1977), with very soft spectra, with slopes steeper than those shown by accreting neutron stars (see e.g. White & Marshall 1984). Given the wide variety of spectral shapes shown by XRBs, they can be confused with background AGNs, whose X-ray spectra have roughly a powerlaw shape, with indices ranging from 1.6 to 2.5 (see e.g. Walter & Fink 1993;Vignali et al. 1999;Turner et al. 1991).
In this paper we report the results obtained from a study of the X-ray source populations of M 83, using three XMM-Newton observations covering both the inner and outer disc regions. The higher spatial resolution and sensitivity of XMM-Newton compared to the previous observations of ROSAT and Einstein allowed an increase of the number of detected sources in M 83. While the spatial coverage of the Chandra observation was limited to a region located at the centre of M 83 with a size of 8.3 ′ × 8.3 ′ (the ACIS S3 field of view), the XMM-Newton obser-Article number, page 2 of 30 vations allowed us to obtain a complete coverage of M 83, and to study also the outer parts of the galaxy, which in total provided us with a more representative sample of X-ray sources in M 83.
The paper is organised as follows: in Sect. 2 we describe the data reduction and analysis of XMM-Newton observations. In Sect. 3 we show the astrometrical corrections that have been applied. In Sect. 4 we present the techniques adopted to classify the X-ray sources (X-ray variability, spectral analysis, and hardness ratios). In Sect. 5 we describe the properties and classification of the detected sources. In Sect. 6 we derive the X-ray luminosity functions (XLFs) of X-ray binaries within and outside the D 25 ellipse, after correcting them for incompleteness and subtracting the AGN contribution, and we discuss our results. We examine in detail the properties of the sources that have been identified and classified in this work in Appendix A.
Reduction and data analysis
We analysed the public archival XMM-Newton data of M 83 (PIs: M. Watson, K.D. Kuntz). Table 1 lists the three observations that we analysed, one pointing at the centre of the galaxy (obs. 1) and two in the south, which covered the outer arms with a young population of stars discovered with GALEX. The data analysis was performed using the XMM-Newton Science Analysis System (SAS) 12.0. For each pointing we produced PN, MOS1, and MOS2 event files. We excluded times of high background due to soft proton flares as follows. For each observation and instrument, we created background lightcurves (with sources removed) in the 7−15 keV energy band. Good time intervals (GTIs) were determined by selecting count rates lower than 8 cts ks −1 arcmin −2 and 2.5 cts ks −1 arcmin −2 for PN and MOS, respectively.
For each observation, data were divided into five energy bands:
-R1: 0.2-0.5 keV; -R2: 0.5-1 keV; -R3: 1-2 keV; -R4: 2-4.5 keV; -R5: 4.5-12 keV.
For the PN data we used single-pixel events (PATTERN=0) in the first energy band and for the other energy bands single-and double-pixel events (PATTERN≤4) were selected. For the MOS data, single-pixel to quadruple-pixel events (PATTERN≤12) were used for all five bands.
We ran the source detection procedure separately for images of each observation, and simultaneously for five energy bands and three instruments with the SAS task edetect_chain. The source detection consists of three steps. The first step provides a list of source positions used to create the background maps. We adopted a minimum-detection likelihood 1 of 7 to obtain this list of sources. After removing the sources, a two-dimensional spline with 20 nodes was fitted to the exposure-corrected image. In the second step the background maps are used to improve the detection sensitivity and hence to create a new source list, assuming a minimum-detection likelihood of 4. In the last step, a maximum-likelihood point-spread function (PSF) fit to the source count distribution is performed simultaneously in all energy bands and each EPIC instrument, from the input list of source positions obtained in the previous step (a description of Table 2. Count rate to energy conversion factors for thin and medium filters of the EPIC instruments in the energy ranges R1-R5, assuming an absorbed powerlaw with a photon index of 1.7 and the Galactic foreground absorption 3.69 × 10 20 cm −2 in the direction of M 83. this algorithm is given by Cruddace et al. 1988). For each observation we generated the final source list adopting a lower threshold for the maximum-detection likelihood of 6, which corresponds to a detection probability of ∼ 99.75%. The source detection gives several parameters for each source, such as the coordinates, count rates, and likelihood of detection (see Table B.1 in the appendix B). As mentioned above, 20 nodes (more than the default 16) for the background spline map were used to better follow the central diffuse emission and to minimise false detections. We removed the remaining false detections due to diffuse emission structures by visual inspection. Fig. 1 shows the combined PN, MOS1, and MOS2 threecolour mosaic image obtained from the three observations. The numbers of the detected sources are overplotted on the image. The red, green, and blue colours represent the 0.2 − 1 keV, 1 − 2 keV, and 2 − 4.5 keV energy bands.
Astrometrical corrections
Corrections between XMM-Newton observations
We calculated the RA and Dec offsets of the three XMM-Newton observations using position of the sources detected in at least two observations. Sources were considered as detections in at least two different observations if their position was closer than 3× the combined statistical positional errors. We calculated the offsets of observations 2 and 3 with respect to the reference observation 1 as the weighted mean of RA and Dec of all sources, then recalculated all X-ray positions correcting for the shifts relative to the observation 1.
Correcting the position of the detected sources using X-ray and optical observations
We also applied the cross-correlation procedure described above to determine the systematic errors in the X-ray positions of the XMM-Newton observations by calculating the offsets in the Xray positions of the XMM-Newton sources with respect to the X-ray sources observed by SW03 with Chandra. The offset between the XMM-Newton and Chandra positions (given as the weighted mean of RA and Dec in arcsec) is ∆RA= −1.22 ± 0.16, ∆Dec= −0.72 ± 0.16. We point out that SW03 corrected the Chandra positions using only the position of the infrared nucleus of M 83 deduced from HST/WFPC2 observations. Therefore, to obtain more accurate positions from possible optical counterparts, we cross-correlated the XMM-Newton list of sources with the optical catalogue of the United States Naval Observatory USNO-B1 (Monet et al. 2003). For this calculation we were interested in associations between X-ray sources and foreground stars. As discussed in Sect. 5.1, we classified five sources as foreground star candidates. The offset between the (Buccheri et al. 1983). No statistically significant variability from the analysed sources was detected.
Long-term variability
To study the long-term time variability of sources observable at least in two different observations, we calculated the average flux (or the 3σ upper limit in case of non-detection) at the source position in each observation. We considered fluxes in the 0.2 − 4.5 keV energy band because, as Pietsch et al. (2004) noted, the band 4.5 − 12 keV has a lower sensitivity and is contaminated by hard background. We calculated the fluxes with the energy conversion factors (ECFs) reported in Table 2. Then, we searched for variable sources by comparing their fluxes (or upper limits) in different observations. We measured the X-ray variability of each source by its variability factor V f = F max /F min , where F max and F min are the maximum and minimum (or upperlimit) fluxes. To estimate the significance of the variability between different observations, we calculated the significance parameter S = (F max − F min )/ σ 2 max + σ 2 min , where σ max and σ min are the errors of the maximum and minimum flux (Primini et al. 1993).
We also studied the X-ray variability considering the Chandra observation of M 83. We converted the Chandra counts Article number, page 4 of 30 Table 3. Variability factors (V f ) with errors of sources observed in at least two XMM-Newton observations and in XMM-Newton and Chandra observations. Table also includes maximum fluxes and errors in the energy range 0.2 − 4.5 keV in units erg cm −2 s −1 , and the significance of the difference S . XMM-Newton: (2.00 ± 0.44) × 10 −14 1.9 0.2 2.0 (0.3 − 8 keV) of SW03 to 0.2 − 4.5 keV fluxes with the conversion factor calculated by SW03 and the distance of M 83 (d = 4.5 Mpc) assumed in this work. The conversion factor CF = 8 × 10 37 /300 erg s −1 counts −1 was calculated by SW03 assuming an absorbed powerlaw spectrum with Γ = 1.7, N H = 10 21 cm −2 , and a distance of 3.7 Mpc. For each Chandra source, we obtained the flux in the energy range 0.2 − 4.5 keV correcting the luminosity L 0.3−8 keV = counts ×CF by the absorption column density, the galaxy distance, and the energy range. The results are reported in Table 3. Fig. 2 shows the variability factor plotted versus the maximum detected flux and the hardness ratio (R2-R1)/(R1+R2) (see section 4.3) for each source. The left column shows the variability factors calculated for sources observed in at least two XMM-Newton observations. The right column shows the variability factors calculated for sources observed with Chandra and in at least one XMM-Newton observation.
Source flux max. V f error V f S Source flux max. V f error V f S 2 (3.
Applying a variability significance threshold of S = 3, we found 35 variable sources. Like XRBs and AGNs, SSSs can show high variability, and because of their soft spectrum (see section 5.6), they can be distinguished from the other sources: in Fig. 2 (lower panels), SSSs candidates should appear on the lefthand side, while XRBs (characterized by a much harder spectrum) are expected to appear on the right-hand side.
Spectral analysis
We extracted the X-ray spectra of sources with 300 counts in the energy range 0.2 − 12 keV. For each source, we fitted all three EPIC spectra simultaneously with different models: powerlaw, disc-blackbody, thermal plasma model (APEC Smith et al. 2001), and blackbody, using XSPEC (ver. 12.7.0, Arnaud 1996). For the absorption we used the PHABS model.
A good fit with one of the above-mentioned spectral models can be used to classify the sources into one of the following classes of sources: -X-ray binaries; -supernova remnants; -super-soft sources.
In total, we fitted the spectra of 12 sources (see section A). For sources that are not bright enough for spectral modelling, we only calculated their hardness ratios, as described in section 4.3.
Hardness-ratio diagrams
We used the hardness-ratio diagrams to separate different classes of sources according to their X-ray properties. They are especially helpful for sources that are too faint, for which spectral fitting is not possible. For each source, we computed four hardness ratios, defined as
HR i = R i+1 − R i R i+1 + R i for i = 1, ... , 4,(1)
where R i are the net source counts in five energy bands. To obtain the best statistics we combined the hardness-ratios of all three instruments. When a source was detected in more than one observation, we considered the observation with the highest number of counts. Some sources can exhibit different spectral states (which can be correlated with the X-ray flux), resulting in hardness-ratio changes between different observations (see e.g. Done et al. Article number, page 6 of 30 2007). Therefore, for some of these sources we only considered a state by adopting the highest number of counts when determining the hardness ratio. This approach allowed us to obtain the hardness ratios with small uncertainties for bright sources in their bright states. However, one has to be aware that if a source changes its state, the hardness-ratio may change as well. For fainter sources (with hardness ratio uncertainties 0.2), the hardness ratios are not sensitive to changes of the state of the source within uncertainties. The hardness ratios calculated for each source are reported in Table B.1. Fig. 3 shows the hardness ratios of sources with errors smaller than 0.3, detected in the field of view of M 83. We plotted sources classified as XRBs, SNRs, SSSs, ultra-luminous Xray sources (ULXs), foreground stars, and background objects (see section 5) with different symbols. On the same plot we also overlaid grids of hardness ratios calculated for different spectral models: three absorbed powerlaws with photon-index Γ = 1, 2, 3 (XRBs in hard state), two absorbed disc-blackbody models with temperatures at the inner disc radius of kT in = 0.5 and 1 keV (XRBs in soft state), four thermal plasma models APEC with temperatures kT apec = 0.2, 0.5, 1, 1.5 keV (SNRs), and two blackbody models with temperatures kT bb = 50 and 100 eV (SSSs, see section 5.6). The column densities range from N H = 10 20 cm −2 to N H = 10 24 cm −2 .
Source classification
We cross-correlated the list of sources observed with XMM-Newton with existing catalogues. For this purpose we used Xray (Trinchieri et al. 1985;Ehle et al. 1998;Immler et al. 1999;SW03;Di Stefano & Kong 2003), optical (Blair & Long 2004;Dopita et al. 2010;Jones et al. 2004;Rumstay & Kaufman 1983; USNO-B1, Monet et al. 2003), radio (Maddox et al. 2006;Cowan et al. 1994;Condon et al. 1998), and infrared (2MASS, Skrutskie et al. 2006) catalogues.
We considered two sources as associated to each other if their positions were closer than the 3× combined statistical errors. The optical counterparts of several X-ray sources cannot be determined uniquely. In such cases we assumed as counterpart the brightest optical object within the error circle. The cross-correlations are reported in Table B.2 in appendix B.
We used the previous classifications in X-rays and other wavelengths and the methods of classification described in sections 4.1 (X-ray variability), 4.2 (spectral analysis), and 4.3 (hardness ratios), to identify and classify sources as background objects, foreground stars, XRBs, SNRs, SSSs, and ULXs.
In this section we describe the observational properties for each class of sources and define the classification criteria.
Foreground stars
X-ray observations of nearby galaxies are contaminated by foreground stars, which have X-ray luminosities ranging from ∼ 10 26 to ∼ 10 30 erg s −1 for stars of spectral type F to M, and ∼ 10 29 to ∼ 10 34 erg s −1 for stars of spectral types O and B (Vaiana et al. 1981;Rosner et al. 1985). Stars of spectral classes F to M emit X-rays because of the intense magnetic fields that form a corona, in which the plasma is heated to temperatures of about ∼ 10 6 − 10 8 K (e.g. Güdel 2002). A mechanism proposed to explain the X-ray emission from stars of spectral types O-B is the formation of shocks in the coronal regions due to the instability of the wind-driven mechanism (see Puls et al. 2008 and references therein). In A-type stars, none of the above mechanisms for X-ray emission can operate efficiently. Therefore, A-type stars are expected to be weak Xray sources (Schröder & Schmitt 2007) and only very few have been observed in X-rays (see e.g. Robrade & Schmitt 2010;Schröder et al. 2008).
The X-ray spectra of foreground stars are relatively soft and can be described by models of optically thin plasma in collisional equilibrium (e.g. Raymond & Smith 1977) with temperatures ranging from 10 6 to 10 7 K. A common method to distinguish stars from other X-ray sources is comparing the X-ray-tooptical flux ratio, as suggested by Maccacaro et al. (1988):
log 10 ( f x / f opt ) = log 10 ( f x ) + m 2.5 + 5.37 ,(2)
where m is the visual magnitude m v . In the USNO-B1 catalogue the red and blue magnitudes are given, thus we assumed m v ≈ (m red + m blue )/2. We used the blue magnitude m blue as magnitude m when the red magnitude was not available.
For each X-ray source with an optical counterpart, we distinguished foreground stars from other sources by plotting X-rayto-optical flux ratios over the hardness ratios HR 2 and HR 3 (Fig. 5). The X-ray-to-optical flux ratios and the hardness ratios differ significantly between different classes of sources.
The soft X-ray flux of early-type stars (OB type) scales with f x ≈ 10 −7 f opt (Kudritzki & Puls 2000 and references therein), while the ratio f x / f opt of late-type stars (F to M) usually ranges from 10 −6 to 10 −1 (e.g. Krautter et al. 1999). In contrast, sources such as SNRs, SSSs, and XRBs radiate mainly in X-rays.
We also used optical and near-infared magnitudes and colours to classify foreground stars (Figs. 4 and 6). Fig. 4 is the colour-colour diagram for XMM-Newton sources with optical (USNO-B1) and infrared (2MASS) counterparts. Lines show the expected (B−R) and (J−K) colours for main-sequence, giant, and supergiant stars belonging to the Milky Way. We obtained Article number, page 7 of 30 these lines using intrinsic colours calculated by Johnson (1966). Stars located at the Galactic latitude of M 83 (b ≈ 32 • ) have on average a colour excess per kiloparsec of E(B − V) = 0.05 ± 0.05 mag kpc −1 (Gottlieb & Upson 1969). Therefore, the colour excesses E(J − K) and E(B − R) are negligible compared to the optical and infrared magnitude uncertainties (Schild 1977).
Figs. 4 and 6 allow to separate foreground stars from other classes of sources. Foreground stars are brighter in R than background objects or members of M 83, and sources with J − K 1.0 and B − R 2.0 are most likely foreground stars.
From previous considerations, we classified foreground stars when these conditions were met:
-log( f x / f opt ) −1;
Article number, page 8 of 30 -HR 2 0.3;
-HR 3 −0.4; -J − K 1.0; -B − R 2.0.
The five sources classified as foreground star candidates are reported in Table 4. A detailed discussion of the identification and classification of foreground stars is provided in sections A.1 and A.2.
Background objects
The identification of AGNs, normal galaxies, and galaxy clusters is based on SIMBAD and NED correlations, and is confirmed if there is an optical counterpart in the 2nd Digitized Sky Survey (DSS2) image. New classifications are based on the radio counterpart and hardness ratio HR 2 ≥ −0.4 (Pietsch et al. 2004).
We identified nine sources as background galaxies and AGNs (sources No. 7,17,31,65,83,89,144,148,158, see Table 5). We found radio counterparts of the sources No. 20, 37, 189 and classified them as AGN candidates for the first time (see section A.3). Based on the log N − log S calculated by Cappelluti et al. (2009) (see section 6.2), about 40 observed sources (with a 2 − 10 keV flux F x > 10 −14 erg cm −2 s −1 ) are expected to be background objects in each XMM-Newton observation of Table 1. From a comparison with other works (e.g. Misanovic et al. 2006), we expect a large difference between the predicted number of background objects from background surveys and the number of identified/classified background objects in an XMM-Newton observation. This difference is due to the difficulty in classifying sources which, because of their distance, are too faint (and therefore provide little information) to be classified with the methods at our disposal.
Nuclear sources
We detected two bright sources in the nuclear region of M 83 with the source detection procedure: sources No. 92 and No. 95. They are separated by ∼ 6.3 ′′ and are the brightest sources detected with XMM-Newton in M 83 (F No. 90
= [1.03±0.25]×10 −12 erg cm −2 s −1 ; F No. 93 = [2.59±0
.15]×10 −13 erg cm −2 s −1 ; 0.2−12 keV, assuming an absorbed powerlaw spectrum with index 1.8 and a foreground Galactic absorption of N H = 3.69×10 20 cm −2 ). The two nuclear sources coincide with ∼ 18 Chandra sources and the bright diffuse emission of the starburst nucleus, not resolved by XMM-Newton because of its high PSF, which causes source confusion in crowded regions, such as the nuclear region of M 83.
X-ray binaries
We classified sources as XRBs if the X-ray spectra or hardness ratios were compatible with the typical spectra of XRBs and we detected a flux periodicity. We identified two X-ray binaries (Nos. 81 and 120), previously classified by SW03 using Chandra observations (section A.4).
Supernova remnants
We assume that the X-ray spectra of SNRs are well described by the thermal plasma model APEC , with temperatures ranging from 0.2 to 1.5 keV. At this distance we are unable to resolve an SNR or to verify a more detailed spectral model assuming, e.g., a non-equilibrium ionisation.
We classified an X-ray source as SNR if HR 1 > 0.1, HR 2 < −0.4, the source was not a foreground star, and did not show a significant variability (Pietsch et al. 2004).
We identified the source No. 79 as source [SW03] 27, classified as a young SNR candidate by SW03 (section A.5).
SN1957D Long et al. (2012) reported the first detection of SN1957D in X-rays with Chandra. The source shows a luminosity of 1.7 × 10 37 erg cm −2 s −1 (d = 4.61 Mpc, Saha et al. 2006;0.3 − 8 keV), and the spectrum is well modelled with an absorbed powerlaw with an index ∼ 1.4, a foreground Galactic absorption of N H = 4 × 10 20 cm −2 and an intrinsic column density of N H = 2 × 10 22 cm −2 . We did not detect SN1957D in the XMM-Newton observations. In observation 1 the source is located near to the centre of the field of view, and in the other two observations the source is located at the edge of the field of view. Assuming the spectral parameters found by Long et al. (2012), we calculated a 3σ upper-limit in observation 1 of ∼ 2.4 × 10 −14 erg cm −2 s −1 (0.2 − 12 keV), corresponding to a luminosity of ∼ 5.8 × 10 37 erg s −1 , well above the luminosity detected by Long et al. (2012).
Article number, page 9 of 30 5.6. Super-soft sources Super-soft sources are a class of sources that are believed to be binary systems containing a white dwarf. The white dwarf accretes matter from a Roche-lobe-filling companion at high rates (Ṁ acc ∼ 10 −7 M ⊙ yr −1 ), which leads to quasi-steady nuclear burning on its surface (see e.g. van den Heuvel et al. 1992). SSSs show soft spectra with blackbody temperatures of 15 − 150 eV and X-ray luminosities ranging from ∼ 10 35 erg s −1 to 10 38 erg s −1 (Di Stefano & Kong 2003;Kahabka & van den Heuvel 1997). An additional harder component, due to interactions of the radiation with matter near to the white dwarf or wind interactions can be observed (Di Stefano & Kong 2003). Moreover, SSSs are often observed as transient X-ray sources (see Greiner 2000). Other classes of sources with soft spectra can be confused with SSSs. For example, some X-ray pulsars observed outside the beam of the pulsed radiation can show a soft (∼ 30 eV) component (Hughes 1994;Di Stefano & Kong 2003). Moreover, stripped cores of giant stars can be classified as SSSs (Di Stefano et al. 2001).
As our classification criteria, we assumed blackbody temperatures of kT bb ≤ 100 eV (in agreement with the selection procedure proposed by Di Stefano & Kong 2003) and hardness ratios that do not overlap with those of other classes of sources. These criteria are an HR 1 0 and HR 2 − EHR 2 < −0.9. We classified a source as SSS only if both criteria are fulfilled.
We identified source No. 91 as source M83-50, classified as an SSS candidate by Di Stefano & Kong (2003) using Chandra observations (section A.6).
Hard sources
Hard sources show hard X-ray spectra (or hard HRs, see Table 5 in Pietsch et al. 2004). Using their spectral properties and hardness ratios, we classified five hard sources (Nos. 16, 61, 103, 126, and 153; see section A.8.1) and we identified 11 hard sources (Nos. 60,80,92,97,99,106,107,108,114,116,129; see section A.8.2).
Ultra-luminous X-ray sources
ULXs are pointlike non-nuclear sources with X-ray luminosities in excess of the Eddington limit (L Edd ≃ 10 39 erg s −1 ) for a stellar mass black-hole (see e.g. Feng & Soria 2011). They are usually located in active star-forming environments (Miller & Colbert 2004), and their nature is still unclear; recent studies indicate that ULXs are a heterogeneous sample of objects (e.g. Gladstone 2011).
Several models have been proposed to explain the high Xray luminosity of ULXs, but there are three models that are often used for this class of sources. The first model requires that ULXs are intermediate-mass black-hole systems (IMBHs) with masses M ∼ 10 2 − 10 4 M ⊙ , accreting at sub-Eddington rates (e.g. Colbert & Mushotzky 1999). The other models assume that ULXs are stellar-mass black holes (with masses M 100 M ⊙ ) in a super-Eddington accretion regime (Poutanen et al. 2007) or with beamed radiation (see e.g. King 2009).
We identified ULX No. 133, discovered by Trinchieri et al. (1985) with Einstein (source H2), and previously observed in Xrays with ROSAT by Ehle et al. (1998) and Immler et al. (1999) (see section A.7).
X-ray luminosity functions
For each observation, we calculated the XLFs in the energy range 2 − 10 keV excluding the softer bands to reduce the effect of incompleteness of the observed source sample due to absorption. Moreover, from an XLF calculated in this energy band, it is possible to easily subtract the contribution of the log N −log S of the AGNs, which was calculated from several surveys performed by XMM-Newton and Chandra (see section 6.2).
We considered for XLFs only sources with a detection likelihood greater than 6 in the energy range 2 − 12 keV. For each source, we converted the count rates to the 2 − 10 keV fluxes using the ECFs of Table 2 for the energy bands R4 and R5.
We excluded the region inside a circle centred on the nuclear region of M 83 with radius R = 26 ′′ from the XLF calculation, where the large PSF of EPIC in a crowded region causes source confusion effects (see section 5.3). Since we were interested in obtaining XLFs of XRBs, we also excluded the sources previously classified as SNRs, SSSs, ULXs, and foreground stars (section 5). For each observation, we calculated the XLFs of sources detected within two regions of M 83: the inner disc inside the D 25 ellipse, and the outer disc outside the D 25 ellipse.
XLFs corrected for incompleteness
The sensitivity of the EPIC instruments depends on the exposure, background, and PSF, which are not uniform across the FOV. Indeed, the exposure time is relatively high at the centre of the FOV and decreases with increasing off-axis angle (vignetting effect). The background, modelled by the task esplinemap, decreases with increasing angular distance from the nuclear region of M 83 (due to the diffuse emission in the disc of M 83), and the optical properties of the X-ray telescope introduce a degradation of the PSF with increasing off-axis angle. Therefore, the sensitivity also varies across the observed area, allowing the detection of the brightest sources across the entire observed area, whereas the effective area for the detection of faint sources is smaller. This effect leads to an underestimation of the number of sources observed at the faintest flux levels.
We corrected the XLFs by taking into account the incompleteness effect described above by calculating the sky coverage Article number, page 10 of 30 function, which is the effective area covered by the observation as a function of flux. For each observation, we first created the combined sensitivity maps of PN, MOS1, and MOS2 with the SAS task esensmap, which requires as input files the exposure maps, the background images, and the detection masks created by the source detection procedure. We used the sensitivity maps to calculate the sky coverage function for each observation (Fig. 7). The cumulative XLF corrected for incompleteness is given by
N(> F x ) = A tot N s i=1 1 Ω(F i ) ,(3)
where N(> F x ) is the number of sources with a flux higher than F x , weighted by the fraction of the surveyed area Ω(F i )/A tot over which sources with flux F i can be detected; A tot is the total area of the sky observed by EPIC, Ω(F i ) is the sky coverage (Fig. 7), and N s is the total number of the detected sources. Therefore, with equation (3), every source is weighted with a factor correcting for incompleteness at its flux. The variance of the source number counts is defined as
σ 2 = N s i=1 1 Ω i 2 .
(4)
AGN-corrected XLFs
The XLFs obtained in section 6.1 consist of sources belonging to M 83 (XRBs) and AGNs. We subtracted the AGN contribution using the AGN XLF of Cappelluti et al. 2009, who derived the XLFs from the 2 deg 2 of the XMM-COSMOS survey (Scoville et al. 2007). These authors found that the XLF of AGNs in the energy range 2 − 10 keV is described by a broken powerlaw:
dN dF = AF −α 1 F > F b BF −α 2 F ≤ F b ,(5)
where A = BF α 1 −α 2 b is the normalisation, α 1 = 2.46 ± 0.08, α 2 = 1.55±0.18, F b = (1.05±0.16)×10 −14 erg cm −2 s −1 , and A = 413. Fig. 8 shows the XLFs of sources detected within the D 25 ellipse and outside, calculated for each XMM-Newton observation. Blue lines are the observed XLFs, and black lines are the XLFs corrected for incompleteness. Solid green lines are the AGN XLFs of equation (5) with relative uncertainties (dashed green lines). Solid red lines show the XLFs corrected for incompleteness and AGN-subtracted, and dashed red lines are the 90% confidence errors, obtained from equation (4) and the 90% confidence errors of the AGN distribution.
Vertical black lines in the right column of Fig. 8 show the level at which the survey is 90% complete (see section 6.3.2), defined as the flux at which
N s i=1 N(F i )/ N s i=1 A tot /Ω(F i ) = 0.9 .
Fit
We fitted the differential XLFs corrected for incompleteness and AGN-subtracted with a powerlaw:
A(F) = kF α ,(6)
where k is the normalisation and α the powerlaw index. We also Table 6. Best-fitting parameters of the differential XLFs of observations 1, 2, and 3, calculated for sources within and outside the D 25 ellipse. For each observation, the best-fitting parameters were obtained using the total XLF corrected for incompleteness and AGN-subtracted.
< R 25 Obs. 1 Obs. 2 Obs. 3 powerlaw a α −1.0 ± 0.3 −1.8 ± 0.4 −1.6 ± 1.2 k 10.4 +1.8 −1.6
11.2 +5.9 −3.9
21.4 +38.9
−13.8 (15) 11.74 (5) 9.20 (5) broken (11) 4.41 (5) 8.68 (9) Notes: a : see equation (6); b : see equation (7); fitted the differential XLFs with a broken powerlaw:
χ 2 (d.o.f.) 23.48powerlaw b α 1 −3.0 +0.9 −0.2 −2.9 +0.8 0.2 α 2 −1.1 +0.1 −0.5 −1.1 +0.1 −0.4 k 20.9 +13.0 −8.0 37.2 +48.0 −20.9 F b (10 −14 erg cm −2 s −1 ) 5.6 +1.0 −0.4 6.5 +0.8 −0.7 χ 2 (d.o.f.) 21.01 (13) 9.17 (3) > R 25 Obs. 1 Obs. 2 Obs. 3 powerlaw a α −1.9 ± 0.5 −3.3 ± 1.1 −1.2 ± 0.4 k 13.4 +4.7 −3.5 8.7 +3.5 −2.5 7.1 +2.0 −1.6 χ 2 (d.o.f.) 11.62A(F) = kF α 2 −α 1 b F α 1 F > F b kF α 2 F ≤ F b ,(7)
where F b is the break point. The resulting parameters obtained from the fit are reported in Table 6.
Inner disc
From Chandra observation, SW03 calculated the XLFs of sources located in the inner region (distance < 60 ′′ from the nucleus) and outer region (60 ′′ < d < R 25 ) of the optical disc. They found that the inner region sources have a powerlaw luminosity distribution with an differential index of −1.7, while the luminosity distribution of the outer region sources shows a lack of bright sources above ∼ 10 38 erg s −1 . These authors modelled the XLF of these sources with a broken powerlaw with a break around ∼ 10 38 erg s −1 and differential indices of −1.6 and −2.6. They explained the XLF of the inner region sources in terms of current starburst activity, while the XLF of the outer region may result from an older population of disc sources mixing with a younger population.
We recall that we cannot study the innermost region because of poor spatial resolution of XMM-Newton compared to Chandra. We compared the best-fitting parameters of the XLF of the outer region sources (60 ′′ < d < R 25 ) obtained by SW03 with those obtained from the XMM-Newton analysis (Table 6). In particular, we considered the broken powerlaw fit of sources detected in observation 1. Only during this observation was the whole optical disc of M 83 observed. We found that the indices Article number, page 11 of 30 α 1 , α 2 and the break F b of equation 7 agree within the uncertainties with the parameters found by SW03. Grimm et al. (2003) studied the XLFs of a sample of galaxies and found the probable existence of a universal HMXB XLF (in the luminosity range ∼ 4 × 10 36 − 10 40 erg s −1 ), described by a powerlaw with differential slope of −1.6. They found that the number of HMXBs with L x > 2 × 10 38 erg s −1 in a star-forming galaxy is directly proportional to the SFR, and proposed that the number and the total X-ray luminosity of HMXBs can be used to measure the star formation rate of a galaxy. Based on a much larger sample of galaxies, Mineo et al. (2012) found that the properties of populations of HMXBs and their relation with the SFR agree with those obtained by Grimm et al. (2003). We estimated the SFR in the optical disc of M 83 using the Article number, page 12 of 30 N HMXBs −SFR relation of Mineo et al. (2012):
N(> 10 38 erg s −1 ) = 3.22 × SFR (M ⊙ yr −1 ) .(8)
We assumed that the XLF we used for this calculation provides a good approximation of the HXMB XLF in M 83. The contribution of LMXBs to the XLF is negligible for a starburst galaxy such as M 83 when L x 10 38 erg s −1 (Grimm et al. 2003). Moreover, the contribution of LMXBs to the XLF is minimized by excluding the nuclear region of the galaxy, from which a strong contribution to the total number of LMXBs is expected. Using the XLF of sources detected in observation 1 within the D 25 ellipse, from equation 8 we found an SFR ≈ 3.1 M ⊙ yr −1 , in agreement with the SFR estimates obtained from observations in other wavelengths (see e.g. Boissier et al. 2005;Dong et al. 2008;Grimm et al. 2003 and references therein).
Outer disc
The XLFs of the outer disc (d > R 25 ) show an excess of sources (with respect to the expected number of AGNs) in the luminosity range ∼ 10 37 to ∼ 2 × 10 38 erg s −1 (Fig. 8).
We are interested in calculating the probability of the luminosity distribution of the observed sources to be consistent with the luminosity distribution of equation (5) which represents the AGN distribution. Therefore, we compared for each observation the luminosity distribution of the sources detected in the outer disc (d > R 25 ) that was not corrected for incompleteness (see section 6) with a distribution of simulated sources over the EPIC FOV obtained from a uniform spatial distribution of sources with a luminosity distribution given by equation (5), filtered to exclude sources with a flux below the detection threshold calculated at the position of each source in the sensitivity map. The Kolmogorov-Smirnov test applied to these source samples showed that the probabilities that the luminosity distributions of the observed sources are consistent with the luminosity distribution of AGNs (equation 5) are almost zero, being 0.04% in observation 1, 0.7% in observation 2, and 0.6% in observation 3.
To quantify the probability that the set of X-ray sources located outside the D 25 ellipse are AGNs (which are expected to be uniformly distributed across the sky) or XRBs (whose distribution should not be uniform, because the position of XRBs should correlate with the arms extending out of the optical disc), we performed a two-dimensional Kolmogorov-Smirnov test (Fasano & Franceschini 1987;Peacock 1983). This test is based on the statistic δ, which in the unidimensional Kolmogorov-Smirnov test represents the largest difference between two cumulative distributions. We applied this test to two data samples:
1. all X-ray sources detected in observation 1 that are located outside the D 25 ellipse. The number of these sources is N 1 = 39; 2. a distribution of simulated sources in the EPIC FOV of observation 1, obtained from a uniform spatial distribution of sources (which represents the uniform spatial distribution of AGNs) modified to take into account the incompleteness effect described in section 6.1. We obtained this spatial distribution of sources as follows. We first generated a uniform spatial distribution of sources with fluxes given by the XLF of AGNs described in section 6.2. Then, we selected sources with flux higher than that corresponding to the position of each source in the sensitivity map. We additionally selected sources with luminosity > 10 37 erg s −1 in the energy range 2 − 10 keV that are located outside the D 25 ellipse. With this method, we generated a sample of N 2 = 10 4 coordinate pairs (RA, Dec) of sources (see Fig. 9).
From the number of data points N 1 and N 2 of the two data sets, the significance level was calculated from the probability distribution of the quantity
Z n ≡ δ √ n,(9)
where n = N 1 N 2 /(N 1 + N 2 ). The analytical formula for calculating of the probability that the two data samples come from the same distribution is accurate enough for large data sets with n > 80 (Fasano & Franceschini 1987). Since in our case n ≈ 39, we needed to use Monte Carlo simulations. We generated many synthetic data samples simulating the uniformly distributed AGNs with the same method previously used to calculate sample 2; each of the synthetic data samples has the same number of sources as the observed data set 1 (N 1 = 39). For each data set we applied the 2D Kolmogorov-Smirnov test by comparing the synthetic data set with the set of 10 4 sources distributed across the EPIC FOV previously described, then we calculated the quantity Z n using equation (9). The probability of the observed Z n is given by the fraction of the times the simulated Z n are larger than the observed Z n . Applying this statistical method to our data, we found a probability of 99.5% that the observed sample 1 and the simulated homogeneously distributed sample 2 are significantly different, which suggests a non-uniform distribution of the observed X-ray sources and therefore a possible correlation between the positions of these sources and the extended arms of M 83.
The incompleteness correction given by equation (3) is based on the hypothesis that sources are uniformly distributed. However, we have demonstrated that the X-ray sources located outside the D 25 ellipse have a non-uniform distribution, hence the associated XLFs corrected for incompleteness of Fig. 8 (right column) are not reliable at low luminosities. Therefore we only considered the part of the XLFs with luminosities higher than the level at which the survey is 90% complete (to the right of the vertical black lines in Fig. 8). We found that the 90% complete XLFs of observations 1 and 3 (for which we have enough data points to find a good fit) are well fitted with a powerlaw with differential slopes α = −2.2 ± 0.5 (observation 1), and α = −1.7 ± 0.4 (observation 3), which are consistent with each other within errors. These are also consistent with the AGN slope of Cappelluti et al. (2009).
Assuming that the spatial distribution of AGNs and their number density are not subject to strong fluctuations on small angular scales corresponding to different directions in the M 83 field, the observed excess of sources (with respect to the AGN distribution) in the luminosity range ∼ 10 37 to ∼ 2 × 10 38 erg s −1 (Fig. 8) can probably be ascribed to a population of XRBs located in the outer disc of M 83. The recent star-forming activity discovered by GALEX in this region indicates that a large portion of the observed X-ray sources are HMXBs. However, the observed XLF slope is steeper than the slope of the universal HMXB XLF inferred by Grimm et al. (2003). A possible explanation for the difference between the two slopes could be that the observed XLFs are the result of a mix of XRB populations formed after starbursts of different ages. An alternative explanation is that the mass distribution of the population of stars in the low-density regions of the outer disc of M 83 is described by a truncated initial mass function (IMF), whose existence was proposed to explain the production of fewer high-mass stars (compared to the standard IMF) in low-density environments (see e.g. Krumholz & McKee 2008;Meurer et al. 2009). The universality of the IMF is still a matter of debate (Bastian et al. 2010); in this context, a recent Subaru Hα observation of the outer disc of M 83 revealed O stars even in small clusters (M 10 3 M ⊙ ), which supports the hypothesis that the IMF is not truncated in low-density environments (Koda et al. 2012).
Summary
We presented an analysis of three XMM-Newton observations of M 83. We performed the source detection procedure separately for images of each observation, and we obtained a catalogue containing 189 sources.
Based on cross-correlations with other catalogues we identified counterparts for 103 sources, 12 of which were identified or classified as background objects and 5 as foreground stars (one as candidate CV). We performed spectral analysis of the sources with the largest number of counts, as well as studies of the X-ray variability and the hardness ratio diagrams. The spectral analysis of ULX No. 133 in observations 2 and 3 showed good fits with the standard IMBHs model as well as with accreting stellarmass black-hole model, in agreement with the results obtained by Stobbart et al. (2006) from observation 1.
In Sect. 6 we presented the XLFs of sources in the 2 − 10 keV energy band, within and outside the D 25 ellipse. We corrected the XLFs for incompleteness and subtracted the contribution of background AGNs from the total XLF to obtain the XLFs of XRBs. The XLF of the optical disc is well fitted with a powerlaw or a broken powerlaw, while the XLF of the outer disc is well fitted with a simple powerlaw. The broken powerlaw fit parameters agree (within the uncertainties) with the parameters found by SW03 with Chandra. From the XMM-Newton XLF, we obtained an SFR ≈ 3.1 M ⊙ yr −1 in the optical disc of M 83, which agree with previous estimates obtained in other wavelengths.
The XLFs of these sources show an excess of sources (compared to the AGNs distribution) in the luminosity range ∼ 10 37 to ∼ 2 × 10 38 erg s −1 . The application of the Kolmogorov-Smirnov test to the X-ray sources detected outside the D 25 ellipse allowed us to find that this population of sources is significantly different from the population of background AGNs, which is supposed to have a homogeneous distribution. These results led us to suggest that a part of the X-ray sources observed outside the D 25 ellipse belongs to the outer disc of M 83. The 90% complete XLFs of the outer disc are well fitted with a simple powerlaw with differential slope α = −2.2 ± 0.5 (observation 1), and α = −1.7 ± 0.4 (observation 3) steeper than the universal HMXB XLF discovered by Grimm et al. (2003). We proposed as a possible origin for the steep slope of the observed XLF that the observed XLFs are the result of a mix of XRB populations of different ages, or, as an alternative explanation, that the IMF in the low-density regions of the outer disc of M 83 is truncated, as previously suggested by e.g. Krumholz & McKee (2008) and Meurer et al. (2009) to explain the low production of high-mass stars in lowdensity environments. Additional X-rays and UV observations of the outer disc of M 83, analysed with most effective methods such as the one used by Bodaghee et al. (2012) to measure the spatial cross-correlation of HMXBs and OB star-forming complexes in the Milky-Way, will be fundamental to confirm our hypothesis. (Figs. 4, 6), and their optical-to-X-ray ratios as a function of the hardness ratios (Fig. 5). Although the hardness ratio criterion HR 3 −0.4 of source No. 174 is not fulfilled, we classified this source as a foreground star because of the large uncertainty of the hardness ratio (see Fig. 5).
Appendix A: Classification and identification of the XMM-Newton sources
Source No. 24 has optical and infrared counterparts and log 10 ( f x / f opt ) < −1, but violates the hardness ratio HR 2 criterion (see Fig. 5). The optical counterpart is bright (m B,No. 24 = 14.1), and the B − R and J − K colours are consistent with those of foreground stars (Figs. 4 and 6), thus this source most likely belongs to the Milky Way. It has been detected in observations 1 and 2 in all three EPIC cameras. In all cases, source No. 24 shows hard HR 2 (Fig. 5, left panel), inconsistent with the expected X-ray spectra of foreground stars. The properties of the optical companion and the hard X-ray spectra may indicate a cataclysmicvariable nature for this source. This class of sources can show short-and long-term time variability, therefore we produced the X-ray lightcurve in the energy range 0.5 − 4.5 keV to give more evidence for this identification. However, the resulting X-ray lightcurve (with a bin-time of 2000 s) shows neither short-nor long-term variability.
Appendix A.2: Sources that are not foreground stars Sources No. 12, 137, 164, and 189 coincide with ROSAT sources H2, H31, H34 and H36. They were classified by Immler et al. (1999) as foreground stars based on positional coincidences with optical sources of the APM Northern Sky Catalogue (Irwin et al. 1994).
We found possible optical counterparts in the USNO-B1 catalogue for source No. 164 (USNO−B1 0601 − 0299090) and source No. 12 (USNO−B1 0602 − 0301227). However, their X-ray-to-optical flux ratios (equation 2) are log( f x / f opt ) ≈ 0.10 and 0.11 respectively ( f opt of both sources was calculated using visual magnitude), hence the foreground star classification for these sources is ruled out. The refined positions of sources Nos. 137 and 189 obtained with XMM-Newton, allowed us to exclude their association with the optical counterparts proposed by Immler et al. (1999). Source No. 189 can be associated with a new optical counterpart, USNO−B1 0600 − 0300832, which is ∼ 3 orders of magnitude fainter than the previous one (USNO−B1 0600 − 0300831). However, the new X-ray-to-optical flux ratio is log( f x / f opt ) ≈ 0.68 ( f opt was calculated using visual magnitude), too high for a foreground star (see Sect. A.3). Hardness ratios of sources No. 164 and 137 are consistent with a powerlaw or diskblackbody spectrum. Therefore, the spectra of these sources are too hard to be classified as foreground stars.
Appendix A.3: Background objects
We found radio counterparts of the sources No. 20, 37, and 189 and classified them as AGN candidates for the first time.
Source No. 20 is located outside the D 25 ellipse (D 25 = 11.5 ′ ; Tully 1988) at ∼ 0.41 • from the centre of the galaxy. It coincides with the radio source NVSS J133618−301459. We detected this source with XMM-Newton in observations 2 and 3 in the outer disc of M 83. Source No. 20 shows a significant long-term variability (Table 3), and the hardness ratios are roughly consistent with a spectrum described by an APEC model with a temperature of kT apec ∼ 0.5 keV (HR 2 = −0.2±0.1; HR 3 = −0.81±0.11). Therefore, source No. 20 can be classified as an AGN candidate (with a soft spectral component) or an SNR candidate. The distance of this source from the nuclear region of M 83 of ∼ 32 kpc rather indicates that source No. 20 does not belong to the galaxy, therefore it is more likely an AGN than an SNR candidate.
Sources No. 37 and 189 coincide with the radio sources NVSS J133630−301651 and NVSS J133805−295748, respectively. Source No. 189 was previously classified as a foreground star by Immler et al. (1999) (see Sect. A.2). We detected these sources with XMM-Newton in observation 3. Their hardness ratios are consistent with a spectrum described with a powerlaw or disc-blackbody model (No. 37: HR 2 = 0.62 ± 0.12; HR 3 = −0.37 ± 0.13; No. 189: HR 2 = 0.07 ± 0.10; HR 3 = −0.25 ± 0.12). Therefore, they can be classified as AGN candidates.
Appendix A.4: X-ray binaries
Source No. 81 coincides with the Chandra source [SW03] 33, classified as an accreting X-ray pulsar, with a hard spectrum (Γ ≈ 1.7) and a spin period of 174.9 s.
We observed source No. 81 in all XMM-Newton observations. The hardness ratios are consistent with an absorbed powerlaw spectrum, and this source shows a significant long-term X-ray variability (V f = 2.5, S = 3.0, Table 3). We applied a Fourier transform periodicity search and a Z 2 n analysis (section 4.1), which did not reveal any significant periodicity. We calculated the upper-limit on the pulsed fraction (defined as the semi-amplitude of the sinusoidal modulation divided by the mean count rate) using the procedure described by Vaughan et al. (1994). The upper limit on the pulsed fraction obtained from the combined PN and MOS events of observation 1 is 16% at the 99% confidence level. This upper limit is marginally compatible with the pulsed fraction of (50 ± 15)% of source [SW03] 33.
Source No. 120 corresponds to the X-ray source [SW03] 113. Using the spectral properties and the 201.5 s periodicity detected with Chandra, SW03 classified source [SW03] 113 as an XRB in a soft state.
We observed source No. 120 with XMM-Newton in observations 1 and 3. The hardness ratios of this source are consistent with an absorbed powerlaw spectrum with N H ∼ 5 × 10 21 cm −2 and Γ ∼ 1.5. Similarly to source No. 81, a Fourier transform periodicity search and a Z 2 n analysis did not reveal any significant periodicity. At the 99% confidence level, the upper limit on the pulsed-fraction of source No. 120 derived from the MOS events is 49%. This upper limit is compatible with the (50±19)% pulsed fraction of [SW03] 113. the Chandra source [SW03] 27. The Chandra spectrum shows emission lines, suggesting the possibility of emission from optically thin thermal plasma, and has been fitted by SW03 with an absorbed powerlaw with Γ ∼ 1.4 and N H ∼ 7×10 20 cm −2 . SW03 classified this source as a young SNR candidate. Another possible explanation for the hard powerlaw spectrum with superposition of emission lines of [SW03] 27 is that the source is an XRB surrounded by a photoionised nebula (SW03). However, XRBs showing these spectral properties usually have a higher absorbing column density than that of [SW03] 27 (see e.g. Sako et al. 1999).
The XMM-Newton hardness ratios of source No. 79 below 2 keV are consistent with an APEC model with temperature kT apec 1.5 keV, while at higher energies the hardness ratios are consistent with a powerlaw with photon index ∼ 2. The spectral shape of source No. 79 derived from XMM-Newton hardnessratio diagrams agrees with the X-ray spectrum of [SW03] 27 presented by SW03 (see Figure 6 in SW03) and can be interpreted as an SNR exhibiting both a thin-thermal emission (below ∼ 2 keV) and an additional hard component, which dominates at energies above ∼ 2 keV. Also, source No. 79 does not show any significant long-term variability (see Table 3).
Appendix A.6: Super-soft source candidates
Source No. 91
coincides with Einstein source 3 (Trinchieri et al. 1985) and Chandra source [SW03] 55 classified by Di Stefano & Kong (2003) as an SSS candidate (source M83-50 in Di Stefano & Kong 2003).
Di Stefano & Kong (2003) fitted the X-ray spectrum of M83-50 with an absorbed blackbody with a temperature of kT bb = 66 +13 −24 eV, a column density of N H = 2.4 +7.4 −2.4 × 10 20 cm −2 , and a luminosity of L x = 2.8 × 10 37 erg s −1 (0.3 − 7 keV, d = 4.5 Mpc).
We detected source No. 91 in observation 1, where the hardness ratios are consistent with a blackbody spectrum (with column density in the range ≈ 10 20 − 10 21 cm −2 ) and marginally compatible with an APEC spectrum with temperature in the range ≈ 0.2 − 0.5 keV (Fig. A.1). Source No. 91 has a 0.2 − 4.5 keV luminosity of L x = (2.2 ± 0.2) × 10 37 erg s −1 and does not show any significant variability compared to the Chandra observation.
Appendix A.7: Ultra-luminous X-ray sources Two ULXs have been discovered in M 83: H2 (Trinchieri et al. 1985), and a transient ULX discovered with Chandra on 23 December 2010 with a luminosity of L x ∼ 4 × 10 39 erg s −1 (0.3 − 10 keV) by Soria et al. (2010), and classified as an accretionpowered black hole with mass M BH ≈ 40 − 100 M ⊙ (Soria et al. 2012). This ULX has not been detected in the XMM-Newton data. Soria et al. (2012) measured an upper limit to the X-ray luminosity of ∼ 10 37 erg s −1 (0.3 − 10 keV) from the three XMM-Newton observations.
Source No. 133
We observed the ULX as source No. 133 in all XMM-Newton observations. Ehle et al. (1998) and Immler et al. (1999) found a faint extended optical source within the error circle of the ROSAT source position. Roberts et al. (2008) used HST images in three Advanced Camera for Survey (ACS) filters to find the counterparts to six ULXs in different galaxies. For the ULX in M 83, they compared the optical position with the X-ray position from a Chandra High Resolution Camera for Imaging (HRC-I) observation. They detected a counterpart to the ULX with magnitudes B = 25.66 ± 0.13, V = 25.36 ± 0.17. They also noticed that the ULX is located at ∼ 5 ′′ from the centre of a background galaxy, and although the latter is outside the error circle, Roberts et al. (2008) did not completely rule out a possible association between the ULX and the background galaxy. Stobbart et al. (2006) reported the XMM-Newton spectral analysis of source No. 133 during observation 1. They found that the X-ray spectrum is well fitted with a cool disc-blackbody (kT in ∼ 0.2 keV) plus a powerlaw (Γ ∼ 2.5), or with a cool blackbody (kT bb ∼ 0.2 keV) plus a warm disc-blackbody (kT in ∼ 1.1 keV). The first spectral model is the standard IMBH model, where the low disc temperature is due to a black hole with mass of ∼ 1000 M ⊙ , while the origin of the powerlaw component is still not clear (see Roberts et al. 2005). Instead, the spectral parameters obtained with the second spectral model suggest that No. 133 is a stellar-mass black hole accreting close to the Eddington limit. In this model, the cool blackbody component represents the optically thick wind from the stellar-mass black-hole accreting at or above the Eddington limit, while the high temperature of the disc follows the standard trend L x ∝ T 4 shown by the Galactic stellar-mass black-hole binaries.
We analysed all XMM-Newton observations of the ULX No. 133 and fitted the PN, MOS1 and MOS2 spectra simultaneously with a model assuming an IMBH (phabs*[diskbb + powerlaw] in XSPEC), and a model assuming a stellar-mass BH (phabs*[bbody + diskbb]). We used two absorption components: the Galactic absorption column density (N H = 3.69 × 10 20 cm −2 ) and the absorption within M 83 plus the intrinsic column density of the ULX. In all fits we obtained a good fit with both spectral models with the resulting spectral parameters in agreement with those obtained by Stobbart et al. (2006) from observation 1. However, the spectral parameters in observation 3 are only poorly constrained due to the poor statistics (only MOS1 and MOS2 data were available for this observation). Therefore, we fitted the spectrum of observation 3 with a single component model and found that an absorbed powerlaw can adequately fit the data (Fig. A.2 Left panel shows the fit with an absorbed cool disc-blackbody plus hard powerlaw, while the right panel shows the fit with an absorbed cool blackbody plus a warm disc-blackbody (see Table A.1). Immler et al. (1999). This source is located outside the optical disc of M 83, and its position overlaps with the outer disc of M 83 observed by GALEX (e.g. Thilker et al. 2005).
We detected source No. 16 in all XMM-Newton observations, but only in observation 1 was it bright enough to allow spectral analysis. The spectrum can be well fitted with an absorbed powerlaw with Γ = 2.6 +0.3 −0.3 , compatible with that of an XRB or an AGN (see Table A.2). Source No. 16 shows a significant long-term variability (S = 9.5) with a variability factor of V f = 10.6 ± 0.3 (Table 3). It also shows a significant variability A&A-lducci_m83, Online Material p 19 within observation 1, with a variability factor of V f = 6.6 ± 4.5 and significance S = 4.0.
Source No. 61 is in the field of view of XMM-Newton during observation 1, where it shows an X-ray luminosity of L x ≈ 4 × 10 38 erg s −1 (see Table A.2). It has not been previously detected in X-ray, optical, radio, infrared, or UV. The X-ray spectrum is well fitted with an absorbed powerlaw with Γ = 2.4 +0.3 −0.3 or a discblackbody model with temperature kT in = 0.82 +0.13 −0.11 keV (Table A.2). Source No. 61 shows a significant long-term variability (S = 8.6) with a variability factor of V f = 4.3 ± 0.1 (Table 3).
Source No. 103 is located at a distance of ∼ 6 ′′ from a radio source (6 in Cowan et al. 1994, 36 in Maddox et al. 2006, and at 1.6 ′′ from the Chandra source [SW03] 84, which shows hardness ratios compatible with a powerlaw or a disc-blackbody spectrum.
We detected source No. 103 only in the XMM-Newton observation 2, with a flux of (2.23 +3.26 −1.34 ) × 10 −13 erg cm −2 s −1 (0.2 − 12 keV). The X-ray spectrum is well fitted with an absorbed powerlaw with Γ = 1.8 +0.4 −0.4 (Table A.2). We did not detect source No. 103 in observations 1 and 3, thus we calculated the flux upper-limits and we found a significant (S = 7.2) long-term variability, with a variability factor of V f = 12.78 ± 0.12 (Table 3, Fig. 2).
Source No. 126 coincides with X-ray source 30 (Ehle et al. 1998) discovered with ROSAT. Source No. 126 also crosscorrelates with the optical counterpart USNO-B1 0599 − 0300335, but the ratio log 10 (F x /F opt ) does not match the criteria previously specified to classify foreground stars. Source No. 126 is located outside the optical disc of M 83, and its position overlaps with an extended arm of the galaxy.
We observed source No. 126 in all XMM-Newton observations. The X-ray spectra extracted from each observation can be well fitted with an absorbed powerlaw with Γ ≈ 1.8 and the flux is consistent with that measured by Ehle et al. (1998) (Table A.2).
Source No. 153 is detected in all XMM-Newton observations, and has not been previously detected in X-rays, optical, radio, infrared, or UV bands. It is located in the extended arms observed by GALEX, ≈ 10 ′ away from the nuclear region of M 83.
The spectra extracted from each observation can be well fitted with an absorbed powerlaw with Γ ≈ 1.5, suggesting an XRB nature for this source (see Table A.2).
Appendix A.8.2: Identifications Source No. 60 correlates with the Chandra source [SW03] 5 SW03 suggested that this source is an XRB candidate.
We observed source No. 60 in all XMM-Newton observations. The source shows a significant long-term variability (V f = 2.4, S = 3.8, Table 3) with respect to the Chandra observation. X-ray colours of No. 60 are consistent with a powerlaw or discblackbody spectrum, in agreement with the spectral analysis of SW03.
Source No. 80 correlates with the Chandra source [SW03] 31. From the spectral properties, SW03 suggested that [SW03] 31 is an XRB candidate.
We observed source No. 80 with XMM-Newton in observation 1. The hardness ratios are consistent with a powerlaw or disc-blackbody spectrum with column density of ∼ 10 21 cm −2 Source No. 92 coincides with the Chandra source [SW03] 60. SW03 suggested that No. 92 is a XRB candidate because of its hard spectrum (Γ ∼ 1.6).
We observed source No. 92 with XMM-Newton in observation 1. The hardness ratios are consistent with a spectrum described by an absorbed powerlaw model with Γ ∼ 2. Source No. 92 also shows a high long-term variability by a factor of V f = 2.7, with a variability significance of S = 4.3 (see Table 3).
Source No. 97 coincides with the Chandra source [SW03] 72 and with a ROSAT source (source 7 in Ehle et al. 1998 andsource H20 in Immler et al. 1999).
We observed source No. 97 in all XMM-Newton observations. The spectra extracted from each observation can be well fitted with an absorbed powerlaw or a disc-blackbody model (Table A.3), with spectral parameters in agreement with the spectral analysis of SW03. Source No. 97 shows a significant longterm variability between XMM-Newton and Chandra observations (V f = 2.8 ± 0.1, S = 6.6; Table 3). Within observation 1 we found a variability of V f = 6.4 ± 2.7 with a significance of S = 4.8.
Source No. 99 coincides with the Chandra source [SW03] 73, and it is associated with the radio source MCK 34 (Maddox et al. 2006). located in a HII region (RK 137, Rumstay & Kaufman 1983). From a spectral study, SW03 proposed that [SW03] 73 is more likely an XRB than a young SNR.
We observed source No. 99 with XMM-Newton in observations 2 and 3. The source shows a significant variability (S = 5.0), with a variability factor of V f = 15.3 (Table 3). and the hardness ratios are consistent with an absorbed powerlaw or disk-blackbody spectrum.
Source No. 106 corresponds to the X-ray source H25 observed by Immler et al. (1999) in a ROSAT observation and the Chandra source [SW03] 85.
We observed source No. 106 in all the observations. During observation 1 the source was bright enough to allow spectral analysis. The spectrum can be well fitted with an absorbed powerlaw (see Table A.3), with spectral parameters in agreement with those previously obtained by SW03. Ehle et al. (1998) (source 9) and Immler et al. (1999) (source H26) in ROSAT (PSPC and HRI) observations. Immler et al. (1999) found that H26 coincides with a compact radio source (source 8 in Cowan et al. 1994), and with a giant HII region (Rumstay & Kaufman 1983). Hence, they classified this source as an SNR candidate. Moreover, also the observation of Hα and Hβ emission anti-coincident with HI emission (Tilanus & Allen 1993) supports the SNR hypothesis. Source No. 107 was also observed in 2000 April 29 by Chandra (source [SW03] 86). From a spectral analysis, SW03 proposed that No. 110 is more likely an XRB (BH candidate) than an SNR. A&A-lducci_m83, Online Material p 20 No. 97,106,107,108,114,and 129. We fitted the spectra with an absorbed powerlaw. Γ is the powerlaw photon-index, F x is the absorbed flux in the energy range 0.2 − 12 keV, L x is the X-ray luminosity in the same energy range of F x (errors at 90% confidence level). 2.0 +1.9 −0.9 × 10 −5 0.867 (18) 6.7 +20.9 −5.1 × 10 −14 3.1 +5.4 −1.4 × 10 38 1 PN,MOS1
Source No. 107 was detected by
We detected source No. 107 in all XMM-Newton observations with a luminosity of ∼ 7 × 10 38 erg s −1 . In observations 1 and 3 the source was bright enough to allow spectral analysis. The spectra can be well fitted with an absorbed powerlaw or a disc-blackbody (see Table A.3). The obtained spectral parameters are consistent with those previously found by SW03 with Chandra. Source No. 107 shows a significant long-term variability between XMM-Newton observations (V f = 1.97 ± 0.12, S = 5.1).
Source No. 108 was first detected in X-rays by Trinchieri et al. (1985) (source 4) with the Einstein satellite and by Ehle et al. (1998) (source 8) and Immler et al. (1999) (source H27) with ROSAT. It also coincides with the Chandra source [SW03] 88.
We observed source No. 108 in all XMM-Newton observations. During observation 1 source No. 108 was in the centre of the field of view, providing enough statistic to allow spectral analysis. We extracted the PN and MOS1 spectra (the position of source No. 108 was in a gap of MOS2) and we found that an absorbed powerlaw or an absorbed disc-blackbody provide acceptable fits (Table A.3), with spectral parameters consistents with those obtained by SW03. Source No. 108 shows a significant long-term X-ray variability (V f = 1.4±0.1, S = 3.3 Table 3), and during observation 1 we found a variability of V f = 4.2±1.5, with a significance of S = 4.4.
Source No. 114
coincides with the Chandra source [SW03] 104.
We observed source No. 114 in all XMM-Newton observations. During observation 1 source No. 114 was in the centre of the field of view, providing enough statistics to allow a spectral analysis. The spectrum is well fitted with an absorbed powerlaw with spectral parameters consistent with those found by SW03 with Chandra (see Table A.3). Source No. 114 also shows a significant long-term variability (V f = 2.4 ± 0.2, S = 4.1).
Source
No . We observed source No. 116 with XMM-Newton in observation 1, where it shows a significant X-ray variability (V f = 20.8, S = 6.2, Table 3) compared to the Chandra observation, and the X-ray hardness ratios are consistent with a hard spectrum. These properties indicate that source No. 116 is most likely an XRB. We overlaid the 3σ error circles of source No. 116,[SW03] 105, and BL53 on the emission line images Hα and SII obtained from the public Wide Field Camera 3 (WFC3) observation of 2009-08-20 (Fig. A.3). Hα and SII images are used in extragalactic searches of SNRs because their optical spectra show high [SII]:Hα ratios compared to the spectra of normal HII regions (see e.g. Blair & Long 2004 Ehle et al. 1998, source H29 in the catalogue of Immler et al. 1999).
A&A-lducci_m83, Online Material p 21
We observed source No. 129 during observation 1, where it was in the XMM-Newton field of view. The spectrum is well fitted with an absorbed powerlaw with spectral parameters consistent with those found by SW03 with Chandra (see Table A.3). We did not detect source No. 129 in observations 2 and 3, thus we calculated the flux upper limits and we found a significant (S 6.4) long-term variability with a variability factor of V f = 3.94 ± 0.11 (Table 3).
Appendix B: X-ray source catalogue of the XMM-Newton EPIC M 83 observation (Only in the electronic version) M 83 sources detected by XMM-Newton cross-correlated with optical and radio counterparts and X-ray sources detected with ROSAT (Ehle et al. 1998; Immler et al. 1999), Einstein (Trinchieri et al. 1985), and Chandra [SW2003]. For each source the identification proposed by the respective authors and our classification are given. Uncertain classifications are given in brackets. For each counterpart, we assigned a "grade" A, B, or C if the positions of the X-ray source and its counterpart are closer than 1×, 2×, 3× combined statistical errors, respectively.
Fig. 1 .
1Combined PN, MOS1, and MOS2 three-colour mosaic image of M 83. The crowded central region is shown in higher resolution. The white circle is the D 25 ellipse (diameter= 11.5 ′ ; Tully 1988).
Fig. 3 .
3Hardness-ratio diagrams of sources with error-bars smaller than 0.3. Black squares are sources classified as XRBs (section 5.4), orange diamonds are SNRs (section 5.5), violet plus signs are SSSs (section 5.6), green crosses are ULXs (section 5.8), cyan stars are foreground stars (section 5.1), red triangles are background sources (section 5.2), and blue circles are sources not classified. The lines are the hardness ratios calculated for different spectral models and column densities, as described in section 4.3.
Fig. 4 .
4Colour-colour diagram of XMM-Newton sources with optical (USNO-B1) and infrared (2MASS) counterparts. Sources located below the black dashed line are very likely foreground stars.
Fig. 5 .
5Flux ratio log( f x / f opt ) over hardness ratios HR 2 and HR 3 .
Fig. 6 .
6Colour-magnitude diagrams of XMM-Newton sources correlating with sources in the USNO-B1 (left panel) and 2MASS (right panel) catalogues.
Fig. 7 .
7Sky coverage as a function of the X-ray flux (2 − 10 keV) for the region inside the D 25 ellipse (observation 1), calculated excluding the region within the circle centered on the nuclear region of M 83 with radius R = 26 ′′ .
Fig. 8 .
8Cumulative XLFs in the 2 − 10 keV energy band. Blue lines correspond to the XLFs without the contribution of SNRs, SSSs, ULX, and foreground stars, not corrected for incompleteness. Black lines are the XLFs corrected for incompleteness. Solid green lines are the AGN XLFs ofCappelluti et al. (2009), and dashed green lines are the 90% confidence errors. Solid red lines are the XLFs corrected for incompleteness and AGN-subtracted, and the dashed red lines are the resulting uncertainties.
Fig. 9 .
9Sample of 10 4 simulated sources, distributed over the EPIC field of view of observation 1 and located outside the D 25 ellipse.
Appendix A.5: Supernova remnant candidates Source No. 79 The position of this source corresponds to the position of the ROSAT source H15 (Immler et al. 1999) and A&A-lducci_m83, Online Material p 17 Fig. A.1. Hardness-ratio diagram of source No. 91 observed with XMM-Newton. Thick lines are different spectral models as function of the N H , thin lines are different column densities N H (from left to right: 10 20 , 10 21 , 10 22 cm −2 ) as a function of the spectral parameters.
Fig
. A.2. EPIC counts spectra, together with residuals in units of standard deviations for source No. 133 detected in the observation 2.
Fig
. A.3. Emission line HST/WFC3 images of the region surrounding source No. 116. Left panel: WFC3 image with the narrowband filter F657N, corresponding to Hα line emission. Right panel: WFC3 image with the narrowband filter F673N, corresponding to SII line emission. The radii of the circles of Chandra (SW105) and XMM-Newton (116) sources give the 3σ accuracy of the position of the sources. The circle labelled BL53 gives the position of the opitcal SNR candidate.
Table 1 .
1XMM-Newton observations of M 83. The exposure times after the screening for high background are given in units of ks. Mode: EFF=extended full frame imaging mode; FF=full frame imaging mode.Fig. 2. Left panels: variability factor as a function of the maximum flux (upper panel) and hardness ratio HR 1 (bottom panel) based on XMM-Newton observations. Right panels: variability factor as a function of the maximum flux (upper panel) and hardness ratio HR 1 (bottom panel) based on XMM-Newton and Chandra observations. The lower limits of the variability factors are marked as arrows.X-ray positions and optical positions corrected for proper motion (given as the weighted mean of RA and Dec in arcsec) is ∆RA= −2.02 ± 0.43, ∆Dec= −0.44 ± 0.43. The measured offset in RA agrees with the expected precision of the XMM-Newton Attitude Measurement System(Guainazzi 2012). We used these systematic offsets to correct the position of all detected sources.Obs. ID.
Date
Pointing direction
EPIC PN
EPIC MOS1
EPIC MOS2
Mode
RA
Dec
filter
T exp
filter
T exp
filter
T exp
PN MOS
1 0110910201 2003-01-27 13:37:05.16 -29:51:46.1
thin
21.2 medium 24.6 medium 24.6 EFF
FF
2 0503230101 2008-01-16 13:37:01.09 -30:03:49.9 medium 15.4 medium 19.0 medium 19.0 EFF
FF
3 0552080101 2008-08-16 13:36:50.87 -30:03:55.2 medium 25.0 medium 28.8 medium 28.8 EFF
FF
Table 4 .
4M83 X-ray sources and their associated candidate sources in our Galaxy.No.
RA
Dec
USNO-B1 B mag. R mag.
(J2000)
(J2000)
21 13 36 18.73 -30 01 38.1 0599-0299962 18.0
15.5
24 13 36 19.95 -29 51 08.3 0601-0298625 14.1
13.2
143 13 37 27.29 -29 55 45.5 0600-0300561 18.2
16.4
174 13 37 44.79 -30 07 49.2 0598-0301638 12.9
11.2
182 13 37 57.77 -30 01 40.6 0599-0300696 15.1
13.9
Table 5 .
5X-ray sources identified and classified as galaxies or AGNs and their counterparts or previous X-ray classifications.No.
RA
Dec
Name
(J2000)
(J2000)
(SIMBAD)
Identifications:
7 13 36 04.66 -30 08 30.8
QSO B1333−298
17 13 36 15.42 -29 57 58.2
[I1999] 5 3
31 13 36 28.13 -29 42 27.9 2MASS 13362821−2942266
65 13 36 45.78 -29 59 13.0 6dFGS gJ133645.8−295913
83 13 36 58.26 -29 51 04.3
[MCK2006] 28 1
89 13 36 59.68 -30 00 58.8
[BRK2009] 7 2
144 13 37 27.46 -30 02 28.3 6dFGS gJ133727.5−300228
148 13 37 29.36 -29 50 27.4 6dFGS gJ133729.5−295028
158 13 37 32.94 -29 51 01.2
ESO 444−85
New classifications:
20 13 36 18.21 -30 15 00.5
NVSS J133618−301459
37 13 36 30.53 -30 16 57.0
NVSS J133630−301651
189 13 38 05.57 -29 57 45.4
NVSS J133805−295748
Notes:
1 : Maddox et al. (2006);
2 : Bresolin et al. (2009);
3 : Immler et al. (1999).
Table A . 1 .
A1Best-fitting parameters of the X-ray spectra of source No. 133 (errors at 90% confidence level).obs. 2
obs. 3
model
powerlaw + diskbb bbody + diskbb
powerlaw
N H (10 22 cm −2 )
0.30 +0.12
−0.09
≤ 0.03
0.12 ± 0.05
Γ or kT bb (keV)
2.64 +0.19
−0.18
0.30 +0.02
−0.04
2.6 +0.3
−0.2
norm.
4.4 +1.0
−0.4 × 10 −4
3.6 +0.8
−0.9 × 10 −6
1.4 +0.3
−0.3 × 10 −4
kT in (keV)
0.09 +0.02
−0.03
1.4 +0.3
−0.2
norm.
4.8 +47.4
−4.4 × 10 3
7.8 +9.7
−3.1 × 10 −3
χ 2
ν (d.o.f.)
0.946 (184)
0.994 (184)
0.918 (62)
F x (0.2 − 12 keV, erg cm −2 s −1 )
8.9 +57.5
−3.7 × 10 −13
8.5 +24.5
−5.2 × 10 −13 3.7 +2.2
−1.4 × 10 −13
L x (d = 4.5 Mpc, erg s −1 )
1.3 +20.4
−0.6 × 10 40
2.2 +6.3
−1.3 × 10 39
2.2 +0.9
−0.5 × 10 39
Table A . 2 .
A2Best-fitting parameters of sourcesNo. 16, 61, 103, 126, 153. We fitted the spectra with an absorbed powerlaw. Γ is the powerlaw photon-index, F x is the absorbed flux in the energy range 0.2 − 12 keV, L x is the X-ray luminosity in the same energy range of F x (errors at 90% confidence level).Source
Parameters
Analysed data
N H
Γ
norm.χ 2
ν (d.o.f.)
F x
L x
obs.
instrument
(10 21 cm −2 )
(erg cm −2 s −1 )
(erg s −1 )
16
0.8 +0.4
−0.3
2.6 +0.3
−0.3
4.1 +0.9
−0.7 × 10 −5 0.928 (38) 1.4 +0.8
−0.5 × 10 −13 6.3 +2.7
−1.3 × 10 38
1
PN,MOS1,MOS2
61
2.01 +0.75
−0.65
2.4 +0.3
−0.3
2.9 +0.9
−0.6 × 10 −5 1.04 (36)
8.0 +6.8
−3.7 × 10 −14 4.3 +1.4
−0.6 × 10 38
1
PN,MOS1,MOS2
103
0.7 +1.0
−0.7
1.8 +0.4
−0.4
3.7 +1.8
−1.2 × 10 −5 0.876 (19) 2.2 +3.3
−1.3 × 10 −13 6.6 +6.7
−2.7 × 10 38
1
PN,MOS2
126
0.01 +0.56
−0.01
1.8 +0.4
−0.2
8.8 +2.7
−1.2 × 10 −6 0.743 (17) 6.3 +3.1
−2.9 × 10 −14 1.5 +0.8
−0.4 × 10 38
2
PN,MOS1,MOS2
153
0 < N H ≤ 1.5 1.4 +0.7
−0.3
7.6 +5.8
−1.6 × 10 −6 1.027 (15) 8.8 +13.7
−6.2 × 10 −14 2.1 +3.3
−1.2 × 10 38
1
PN,MOS1,MOS2
Appendix A.8: Hard sources
Appendix A.8.1: New classifications
Source No. 16 coincides with the ROSAT source H3 discov-
ered by
Table A .
A3. Best-fitting parameters of sources
). Fig. A.3 shows that the shell of the optical SNR is located only in the error circles of BL53 and [SW03] 105, indicating that source No. 116 and [SW03] 105 cannot be the same source. Therefore, source No. 116 is more likely a transient source not associated with BL53.Source No. 129
coincides with the Chandra source
[SW03] 121 and with a ROSAT source (source 12 in the
catalogue of
Table B
BA&A-lducci_m83, Online Material p 24A&A-lducci_m83, Online Material p 25.1. (Only in the electronic version) M 83 sources detected by XMM-Newton.
No.
RA(2000)
DEC(2000)
pos. err.
rate (obs1)
rate (obs2)
rate (obs3)
likelihood
HR1
HR2
HR3
HR4
(obs1)
(obs2)
(obs3)
1
13 35 48.83
-29 56 39.5
1.57 ′′
0.012 ± 0.002
33.2
−0.07 ± 0.21
−0.06 ± 0.23
−0.43 ± 0.34
−0.58 ± 1.11
2
13 35 49.82
-30 00 38.4
2.94 ′′
0.016 ± 0.004
21.2
0.06 ± 0.31
0.27 ± 0.25
−0.60 ± 0.32
0.50 ± 0.44
3
13 35 52.30
-30 02 59.8
1.72 ′′
0.019 ± 0.004
0.0048 ± 0.0008
28.2
57.0
1.00 ± 0.08
0.06 ± 0.17
−0.64 ± 0.20
−1.00 ± 0.76
4
13 35 55.46
-30 13 23.3
1.96 ′′
0.013 ± 0.004
16.2
0.22 ± 0.28
−0.79 ± 0.31
0.64 ± 0.50
0.41 ± 0.40
5
13 35 59.23
-29 59 05.5
2.19 ′′
0.010 ± 0.003
6.7
1.00 ± 0.25
0.13 ± 0.41
0.15 ± 0.37
0.39 ± 0.31
6
13 36 03.03
-30 03 21.3
1.37 ′′
0.0067 ± 0.0013
25.5
0.45 ± 0.31
0.36 ± 0.19
−0.48 ± 0.20
−0.50 ± 0.61
7
13 36 04.66
-30 08 30.8
0.77 ′′
0.036 ± 0.004
0.080 ± 0.004
158.6
1247.9
0.52 ± 0.06
0.17 ± 0.05
−0.27 ± 0.05
−0.40 ± 0.10
8
13 36 05.29
-29 52 52.8
2.13 ′′
0.006 ± 0.002
9.1
0.28 ± 0.31
−0.28 ± 0.27
−0.32 ± 0.48
−0.15 ± 1.08
9
13 36 05.95
-29 52 00.8
1.30 ′′
0.012 ± 0.002
30.3
0.11 ± 0.22
0.10 ± 0.20
−0.28 ± 0.23
−0.59 ± 0.61
10
13 36 06.77
-30 06 51.7
1.60 ′′
0.0078 ± 0.0018
13.8
1.00 ± 0.56
0.44 ± 0.21
−0.62 ± 0.22
−1.00 ± 1.04
11
13 36 10.66
-29 54 03.1
1.65 ′′
0.0097 ± 0.0029
14.0
0.27 ± 0.29
−0.10 ± 0.29
−0.47 ± 0.42
0.25 ± 0.80
12
13 36 11.72
-29 43 36.6
0.47 ′′
0.058 ± 0.004
414.4
0.17 ± 0.08
−0.12 ± 0.08
−0.48 ± 0.11
−0.26 ± 0.31
13
13 36 12.43
-30 04 24.1
1.10 ′′
0.0064 ± 0.0012
23.0
0.51 ± 0.21
−0.50 ± 0.22
−0.29 ± 0.43
0.64 ± 0.26
14
13 36 12.71
-29 58 37.8
1.74 ′′
0.015 ± 0.004
0.0069 ± 0.0014
9.3
16.1
0.66 ± 0.31
0.18 ± 0.21
−0.45 ± 0.22
0.02 ± 0.47
15
13 36 13.25
-29 59 40.4
1.26 ′′
0.013 ± 0.002
0.015 ± 0.002
0.0182 ± 0.0018
35.6
55.8
138.5
0.12 ± 0.13
−0.03 ± 0.12
−0.35 ± 0.14
0.21 ± 0.20
16
13 36 13.88
-29 56 13.4
0.32 ′′
0.105 ± 0.004
0.010 ± 0.002
0.0043 ± 0.0009
1522.4
15.0
21.7
0.24 ± 0.05
−0.09 ± 0.05
−0.58 ± 0.05
−0.93 ± 0.14
17
13 36 15.42
-29 57 58.2
0.64 ′′
0.049 ± 0.005
0.033 ± 0.003
0.028 ± 0.002
243.6
171.8
253.2
0.44 ± 0.11
−0.08 ± 0.10
−0.45 ± 0.11
−0.19 ± 0.25
18
13 36 17.09
-30 13 55.4
1.76 ′′
0.0018 ± 0.0006
9.2
0.38 ± 0.42
0.17 ± 0.31
−1.00 ± 0.16
19
13 36 18.13
-29 40 13.0
1.71 ′′
0.008 ± 0.003
6.2
−1.00 ± 0.58
1.00 ± 0.33
0.40 ± 0.32
−0.02 ± 0.42
20
13 36 18.21
-30 15 00.5
0.47 ′′
0.053 ± 0.005
0.0062 ± 0.0009
302.7
89.2
0.08 ± 0.09
−0.20 ± 0.10
−0.81 ± 0.11
−0.06 ± 0.83
21
13 36 18.73
-30 01 38.1
0.56 ′′
0.023 ± 0.003
0.015 ± 0.002
0.0190 ± 0.0017
138.5
71.1
222.4
0.42 ± 0.09
−0.36 ± 0.09
−0.82 ± 0.13
0.42 ± 0.43
22
13 36 18.96
-30 06 12.6
1.87 ′′
0.0033 ± 0.0009
7.0
0.41 ± 0.67
0.52 ± 0.31
−0.15 ± 0.28
−0.17 ± 0.44
23
13 36 19.84
-30 05 18.3
1.17 ′′
0.0047 ± 0.0009
15.9
0.28 ± 0.42
0.37 ± 0.24
−0.11 ± 0.22
−0.44 ± 0.32
24
13 36 19.94
-29 51 08.4
1.09 ′′
0.0101 ± 0.0016
0.008 ± 0.003
40.8
8.4
0.03 ± 0.35
0.55 ± 0.19
−0.45 ± 0.17
0.11 ± 0.31
25
13 36 19.96
-29 41 11.6
1.15 ′′
0.014 ± 0.003
41.7
0.34 ± 0.20
−0.06 ± 0.20
−0.30 ± 0.25
−0.70 ± 0.68
26
13 36 24.05
-30 14 56.6
0.89 ′′
0.019 ± 0.003
59.9
0.70 ± 0.15
−0.20 ± 0.15
−0.25 ± 0.21
−1.00 ± 0.55
27
13 36 24.17
-29 54 00.6
1.20 ′′
0.0132 ± 0.0019
0.010 ± 0.002
0.0154 ± 0.0019
51.9
8.9
71.7
0.47 ± 0.22
0.31 ± 0.14
−0.39 ± 0.14
0.03 ± 0.25
28
13 36 26.53
-30 06 31.8
1.56 ′′
0.0047 ± 0.0009
19.5
0.90 ± 0.31
0.50 ± 0.19
−0.43 ± 0.19
−0.15 ± 0.42
29
13 36 26.62
-29 53 09.4
1.65 ′′
0.0049 ± 0.0012
9.3
0.79 ± 0.63
0.08 ± 0.30
0.18 ± 0.26
−0.94 ± 0.40
30
13 36 26.64
-29 55 35.6
0.90 ′′
0.0108 ± 0.0014
0.0082 ± 0.0015
59.3
34.3
0.02 ± 0.19
0.28 ± 0.16
−0.44 ± 0.16
−0.83 ± 0.46
31
13 36 28.13
-29 42 27.9
1.61 ′′
0.010 ± 0.002
9.7
0.25 ± 0.39
−0.20 ± 0.39
−0.09 ± 0.48
0.73 ± 0.17
32
13 36 28.58
-29 57 16.3
1.58 ′′
0.0062 ± 0.0017
6.9
0.06 ± 1.11
0.74 ± 0.37
0.07 ± 0.27
−0.01 ± 0.36
33
13 36 28.94
-29 55 39.4
0.83 ′′
0.0152 ± 0.0016
0.0136 ± 0.0021
0.0121 ± 0.0015
112.4
45.5
62.2
0.59 ± 0.15
0.09 ± 0.12
−0.27 ± 0.13
−0.42 ± 0.27
34
13 36 28.95
-29 51 22.9
0.57 ′′
0.025 ± 0.004
0.022 ± 0.004
0.0070 ± 0.0012
197.3
52.9
66.3
0.96 ± 0.40
0.41 ± 0.44
0.58 ± 0.16
−0.04 ± 0.16
35
13 36 29.63
-29 42 47.3
1.18 ′′
0.0097 ± 0.0017
32.1
0.68 ± 0.19
−0.20 ± 0.18
−0.57 ± 0.24
−0.14 ± 0.72
36
13 36 30.28
-29 41 56.9
0.99 ′′
0.0110 ± 0.0018
31.5
0.47 ± 0.23
0.09 ± 0.18
−0.37 ± 0.21
−0.51 ± 0.49
37
13 36 30.53
-30 16 57.0
0.88 ′′
0.0095 ± 0.0013
112.1
0.48 ± 0.29
0.62 ± 0.12
−0.37 ± 0.13
−0.51 ± 0.33
38
13 36 30.80
-29 45 51.7
1.23 ′′
0.0078 ± 0.0016
18.2
0.37 ± 0.70
0.26 ± 0.53
0.50 ± 0.24
0.16 ± 0.20
39
13 36 31.10
-29 49 24.9
1.48 ′′
0.0051 ± 0.0011
12.1
0.40 ± 0.23
−0.45 ± 0.24
−0.29 ± 0.44
0.20 ± 0.56
40
13 36 31.64
-29 57 11.4
1.19 ′′
0.0072 ± 0.0014
0.0061 ± 0.0011
18.3
26.4
0.16 ± 0.40
0.60 ± 0.19
−0.34 ± 0.19
−0.08 ± 0.35
41
13 36 33.45
-30 00 18.2
1.10 ′′
0.0073 ± 0.0013
0.0036 ± 0.0016
38.6
10.2
1.00 ± 0.86
0.80 ± 0.16
−0.34 ± 0.16
−0.40 ± 0.34
42
13 36 33.77
-29 45 22.5
1.57 ′′
0.0042 ± 0.0012
6.2
−0.43 ± 0.62
0.65 ± 0.42
−0.23 ± 0.31
−0.04 ± 0.50
43
13 36 34.92
-30 09 01.2
0.99 ′′
0.0128 ± 0.0019
0.0037 ± 0.0008
64.2
17.9
0.03 ± 0.21
0.33 ± 0.16
−0.39 ± 0.17
−0.37 ± 0.49
44
13 36 35.07
-30 11 07.4
1.58 ′′
0.0028 ± 0.0008
8.2
−0.34 ± 0.52
0.56 ± 0.38
−0.62 ± 0.24
0.52 ± 0.37
Continued on next page
A&A-lducci_m83, Online Material p 23
Continued from previous page
No.
RA(2000)
DEC(2000)
pos. err.
rate (obs1)
rate (obs2)
rate (obs3)
likelihood
HR1
HR2
HR3
HR4
(obs1)
(obs2)
(obs3)
45
13 36 35.82
-29 41 17.0
1.44 ′′
0.0050 ± 0.0015
8.1
0.23 ± 0.43
0.27 ± 0.33
−0.39 ± 0.31
−0.32 ± 0.95
46
13 36 35.83
-29 51 20.2
0.77 ′′
0.0098 ± 0.0012
71.2
0.06 ± 0.20
0.24 ± 0.16
−0.16 ± 0.15
−0.78 ± 0.27
47
13 36 36.02
-30 03 37.7
1.49 ′′
0.0030 ± 0.0006
10.6
0.18 ± 0.29
0.09 ± 0.25
−0.72 ± 0.26
−0.75 ± 1.08
48
13 36 36.76
-29 50 18.1
0.98 ′′
0.0090 ± 0.0018
39.8
0.61 ± 0.24
0.22 ± 0.20
−0.83 ± 0.11
0.53 ± 0.41
49
13 36 38.46
-29 55 51.5
1.48 ′′
0.0061 ± 0.0011
0.0057 ± 0.0011
28.5
28.4
1.00 ± 0.55
0.53 ± 0.17
−0.39 ± 0.17
−0.68 ± 0.44
50
13 36 38.95
-29 47 43.3
0.43 ′′
0.0263 ± 0.0018
384.0
0.35 ± 0.10
0.22 ± 0.07
−0.46 ± 0.08
−0.42 ± 0.19
51
13 36 39.18
-30 04 07.8
1.38 ′′
0.009 ± 0.002
0.0069 ± 0.0011
0.0131 ± 0.0011
22.1
46.6
258.2
0.59 ± 0.17
0.36 ± 0.09
−0.31 ± 0.09
−0.50 ± 0.15
52
13 36 39.42
-30 10 08.6
1.80 ′′
0.004 ± 0.0010
9.8
−0.49 ± 0.41
0.34 ± 0.51
0.18 ± 0.35
−1.00 ± 0.58
53
13 36 40.14
-29 59 52.6
0.82 ′′
0.0154 ± 0.0017
0.0130 ± 0.0015
0.0134 ± 0.0012
92.7
103.6
188.2
0.22 ± 0.12
0.03 ± 0.11
−0.29 ± 0.11
−0.39 ± 0.22
54
13 36 40.52
-30 05 47.0
1.34 ′′
0.0029 ± 0.0007
10.6
0.45 ± 0.35
−0.17 ± 0.28
−0.56 ± 0.46
0.63 ± 0.40
55
13 36 40.73
-29 51 09.1
1.33 ′′
0.0121 ± 0.0025
0.015 ± 0.002
12.2
29.4
0.89 ± 0.18
0.24 ± 0.17
−0.44 ± 0.17
0.06 ± 0.34
56
13 36 41.41
-30 13 26.8
1.38 ′′
0.010 ± 0.002
0.009 ± 0.003
22.2
22.2
0.19 ± 0.27
−0.05 ± 0.25
0.00 ± 0.27
−0.37 ± 0.52
57
13 36 41.79
-30 11 17.4
1.64 ′′
0.0032 ± 0.0008
9.1
1.00 ± 0.26
−0.03 ± 0.26
−0.09 ± 0.34
58
13 36 42.24
-30 03 31.4
1.40 ′′
0.0020 ± 0.0006
6.5
1.00 ± 1.35
0.63 ± 0.31
−0.41 ± 0.29
−0.49 ± 0.67
59
13 36 42.49
-30 09 34.5
1.40 ′′
0.0030 ± 0.0007
9.4
0.74 ± 0.26
0.00 ± 0.26
−0.73 ± 0.25
0.55 ± 0.43
60
13 36 43.44
-29 51 06.5
0.50 ′′
0.03 ± 1.58
0.034 ± 0.004
0.0382 ± 0.0031
306.0
94.7
272.8
0.72 ± 0.92
0.89 ± 0.10
−0.04 ± 0.11
−0.69 ± 7.21
61
13 36 44.16
-29 48 41.8
0.28 ′′
0.051 ± 0.002
1211.9
0.49 ± 0.06
0.18 ± 0.05
−0.46 ± 0.05
−0.97 ± 0.07
62
13 36 45.26
-30 00 38.0
1.10 ′′
0.0113 ± 0.0016
0.0124 ± 0.0014
0.0094 ± 0.0009
50.7
112.3
125.3
0.39 ± 0.11
−0.26 ± 0.11
−0.83 ± 0.16
−0.46 ± 0.97
63
13 36 45.34
-30 14 41.4
1.61 ′′
0.0060 ± 0.0013
16.4
0.15 ± 0.26
−0.12 ± 0.26
−0.16 ± 0.31
−0.43 ± 0.77
64
13 36 45.56
-30 03 27.5
1.12 ′′
0.021 ± 0.003
0.0116 ± 0.0013
0.0135 ± 0.0010
81.5
93.6
256.9
0.44 ± 0.10
−0.05 ± 0.09
−0.30 ± 0.10
−0.54 ± 0.16
65
13 36 45.78
-29 59 13.0
0.26 ′′
0.114 ± 0.004
0.048 ± 0.002
0.0189 ± 0.0015
2376.9
878.9
314.8
0.59 ± 0.04
0.11 ± 0.04
−0.60 ± 0.04
−0.33 ± 0.11
66
13 36 45.94
-30 00 00.2
1.69 ′′
0.0043 ± 0.0013
0.0046 ± 0.0007
7.9
30.4
0.55 ± 0.21
−0.11 ± 0.18
−0.33 ± 0.22
−1.00 ± 0.29
67
13 36 47.76
-30 02 44.4
1.37 ′′
0.004 ± 0.001
0.0023 ± 0.0006
14.2
6.2
0.15 ± 0.30
−0.10 ± 0.31
−0.15 ± 0.34
−0.84 ± 0.53
68
13 36 47.79
-29 46 47.2
1.50 ′′
0.0040 ± 0.0008
16.3
1.00 ± 0.44
0.24 ± 0.26
−0.06 ± 0.21
−1.00 ± 0.32
69
13 36 49.07
-29 52 58.7
0.50 ′′
0.032 ± 0.002
0.009 ± 0.002
271.8
9.0
0.53 ± 0.11
0.20 ± 0.08
−0.68 ± 0.06
−0.76 ± 0.18
70
13 36 50.59
-30 14 35.1
1.45 ′′
0.0061 ± 0.0014
13.1
0.16 ± 0.44
−0.20 ± 0.52
0.51 ± 0.36
0.24 ± 0.25
71
13 36 51.15
-29 41 54.7
1.84 ′′
0.0058 ± 0.0014
6.8
0.34 ± 0.63
0.47 ± 0.31
−0.13 ± 0.26
−0.08 ± 0.37
72
13 36 51.39
-30 18 01.7
2.03 ′′
0.0022 ± 0.0008
6.2
0.53 ± 0.62
0.51 ± 0.33
−0.47 ± 0.46
0.36 ± 0.60
73
13 36 51.59
-29 53 34.5
1.12 ′′
0.0041 ± 0.0012
0.0089 ± 0.0020
11.2
8.2
1.00 ± 0.70
0.10 ± 0.44
−0.17 ± 0.26
−0.12 ± 0.31
74
13 36 52.38
-29 51 43.6
1.72 ′′
0.0079 ± 0.0016
0.010 ± 0.002
7.6
6.3
0.22 ± 0.21
−0.66 ± 0.20
−0.96 ± 0.49
0.86 ± 1.75
75
13 36 53.43
-30 08 40.3
1.49 ′′
0.005 ± 0.001
0.005 ± 0.002
18.3
36.9
0.93 ± 1.68
−0.43 ± 0.70
0.77 ± 0.28
−0.71 ± 0.40
76
13 36 53.53
-29 55 59.1
1.58 ′′
0.0039 ± 0.0012
7.7
0.82 ± 0.18
−0.57 ± 0.31
−1.00 ± 0.40
1.00 ± 0.35
77
13 36 53.78
-29 48 49.5
1.63 ′′
0.0038 ± 0.0009
6.2
0.15 ± 0.23
−0.68 ± 0.23
−1.00 ± 0.88
1.00 ± 13.45
78
13 36 54.86
-30 09 46.7
1.63 ′′
0.0034 ± 0.0008
9.6
0.65 ± 0.37
−0.02 ± 0.27
−0.10 ± 0.29
−0.20 ± 0.45
79
13 36 55.43
-29 55 09.0
0.69 ′′
0.015 ± 0.002
0.024 ± 0.002
0.0221 ± 0.0019
105.5
146.5
192.6
0.11 ± 0.22
0.51 ± 0.12
−0.08 ± 0.09
−0.33 ± 0.15
80
13 36 56.56
-29 49 12.2
0.55 ′′
0.0232 ± 0.0018
254.3
0.65 ± 0.16
0.28 ± 0.11
−0.20 ± 0.08
−0.62 ± 0.11
81
13 36 57.22
-29 53 38.3
0.36 ′′
0.0322 ± 0.0019
0.024 ± 0.003
0.0203 ± 0.0024
392.4
97.4
54.1
0.50 ± 0.11
0.21 ± 0.08
−0.33 ± 0.06
−0.33 ± 0.10
82
13 36 57.29
-29 47 28.1
1.13 ′′
0.0041 ± 0.0008
20.6
−0.79 ± 1.60
0.97 ± 0.23
0.02 ± 0.19
−0.45 ± 0.28
83
13 36 58.26
-29 51 04.3
0.86 ′′
0.0053 ± 0.0009
30.0
1.00 ± 0.23
0.24 ± 0.20
−0.31 ± 0.17
84
13 36 58.26
-29 48 32.8
0.78 ′′
0.0115 ± 0.0013
90.9
0.85 ± 0.51
0.54 ± 0.18
−0.14 ± 0.11
−0.31 ± 0.13
85
13 36 58.67
-29 43 35.7
0.85 ′′
0.0162 ± 0.0016
144.7
0.40 ± 0.13
−0.10 ± 0.11
−0.30 ± 0.14
−0.08 ± 0.22
86
13 36 58.84
-30 05 18.1
1.82 ′′
0.007 ± 0.002
0.007 ± 0.001
0.0070 ± 0.0008
13.5
64.6
82.5
0.78 ± 0.16
0.21 ± 0.14
−0.31 ± 0.14
−0.18 ± 0.21
87
13 36 59.35
-29 49 58.4
0.24 ′′
0.075 ± 0.003
0.025 ± 0.003
0.057 ± 0.004
1750.7
197.4
266.3
0.52 ± 0.06
0.17 ± 0.04
−0.12 ± 0.04
−0.53 ± 0.05
88
13 36 59.51
-29 54 13.9
0.97 ′′
0.0057 ± 0.0009
32.1
0.62 ± 4.08
0.95 ± 0.25
−0.12 ± 0.15
−0.48 ± 0.20
89
13 36 59.68
-30 00 58.8
1.60 ′′
0.0051 ± 0.0012
0.0040 ± 0.0009
0.008 ± 0.001
15.5
13.0
81.4
0.99 ± 0.29
0.59 ± 0.13
−0.18 ± 0.13
−0.75 ± 0.25
Continued on next page
Continued from previous page
No.
RA(2000)
DEC(2000)
pos. err.
rate (obs1)
rate (obs2)
rate (obs3)
likelihood
HR1
HR2
HR3
HR4
(obs1)
(obs2)
(obs3)
90
13 37 00.35
-29 51 57.8
0.16 ′′
0.881 ± 0.009
0.959 ± 0.016
1.173 ± 0.014
7696.9
0.62 ± 0.01
−0.21 ± 0.01
−0.62 ± 0.01
−0.57 ± 0.03
91
13 37 00.47
-29 50 52.3
1.23 ′′
0.0051 ± 0.0011
7.8
−0.41 ± 0.24
−1.00 ± 0.30
1.00 ± 22.70
0.44 ± 3.24
92
13 37 00.51
-29 53 18.7
0.99 ′′
0.0089 ± 0.0018
11.5
0.06 ± 0.56
0.60 ± 0.30
−0.29 ± 0.21
−0.29 ± 0.26
93
13 37 00.83
-29 51 59.9
0.19 ′′
0.243 ± 0.006
1121.0
0.55 ± 0.03
−0.41 ± 0.03
−0.51 ± 0.04
−0.63 ± 0.07
94
13 37 00.97
-30 16 43.0
1.07 ′′
0.014 ± 0.002
41.8
0.28 ± 0.21
0.08 ± 0.17
−0.44 ± 0.19
0.35 ± 0.28
95
13 37 01.12
-29 52 46.2
1.07 ′′
0.004 ± 0.002
15.0
−1.00 ± 1.60
1.00 ± 1.27
0.21 ± 0.67
0.01 ± 0.70
96
13 37 01.16
-30 00 36.2
1.00 ′′
0.0040 ± 0.0009
0.0036 ± 0.0008
23.1
12.5
0.98 ± 0.48
−0.66 ± 0.49
0.73 ± 0.39
0.25 ± 0.22
97
13 37 01.36
-29 53 25.0
0.23 ′′
0.078 ± 0.003
0.044 ± 0.003
0.083 ± 0.004
1708.1
282.9
853.5
0.97 ± 0.07
0.65 ± 0.04
−0.30 ± 0.03
−0.78 ± 0.04
98
13 37 01.49
-29 47 42.7
0.38 ′′
0.0235 ± 0.0015
462.1
0.87 ± 0.11
0.53 ± 0.07
−0.19 ± 0.07
−0.56 ± 0.10
99
13 37 01.75
-29 51 26.4
1.47 ′′
0.074 ± 0.012
0.036 ± 0.004
39.6
84.0
0.67 ± 0.15
0.12 ± 0.12
−0.27 ± 0.10
−0.94 ± 0.16
100
13 37 01.99
-29 55 17.5
0.58 ′′
0.0211 ± 0.0017
0.014 ± 0.003
0.023 ± 0.002
215.7
38.9
153.3
0.41 ± 0.16
0.21 ± 0.12
−0.08 ± 0.10
−0.31 ± 0.12
101
13 37 02.10
-30 12 28.3
1.72 ′′
0.0068 ± 0.0015
0.0060 ± 0.0011
12.0
29.3
0.91 ± 0.21
0.44 ± 0.21
−0.20 ± 0.19
−0.16 ± 0.35
102
13 37 02.25
-29 44 26.8
1.35 ′′
0.0075 ± 0.0015
16.7
−0.09 ± 0.55
0.35 ± 0.42
0.42 ± 0.25
0.13 ± 0.22
103
13 37 02.72
-29 52 25.5
0.46 ′′
0.065 ± 0.005
234.4
0.46 ± 0.13
0.19 ± 0.10
−0.24 ± 0.08
−0.51 ± 0.15
104
13 37 02.79
-29 57 36.6
1.59 ′′
0.0032 ± 0.0008
0.004 ± 0.001
7.0
8.7
−0.64 ± 0.40
0.73 ± 0.31
−0.27 ± 0.29
−0.63 ± 0.52
105
13 37 03.69
-30 06 31.0
0.95 ′′
0.0085 ± 0.0015
0.0054 ± 0.0008
54.7
47.7
0.69 ± 1.10
0.81 ± 0.14
−0.24 ± 0.18
−0.40 ± 0.27
106
13 37 03.80
-29 49 29.9
0.40 ′′
0.0288 ± 0.0018
0.0055 ± 0.0014
0.013 ± 0.003
354.0
17.0
21.4
0.26 ± 0.10
0.03 ± 0.08
−0.25 ± 0.07
−0.63 ± 0.11
107
13 37 04.21
-29 54 03.1
0.30 ′′
0.045 ± 0.002
0.054 ± 0.004
0.087 ± 0.004
853.6
350.7
1165.7
0.69 ± 0.07
0.43 ± 0.04
−0.46 ± 0.04
−0.78 ± 0.09
108
13 37 04.32
-29 51 21.0
0.20 ′′
0.089 ± 0.003
0.016 ± 0.003
0.123 ± 0.005
1854.7
47.2
852.1
0.60 ± 0.05
0.35 ± 0.04
−0.43 ± 0.03
−0.88 ± 0.04
109
13 37 05.12
-29 52 26.5
0.46 ′′
0.0224 ± 0.0016
0.0058 ± 0.0017
220.1
8.6
0.52 ± 0.14
0.31 ± 0.09
−0.25 ± 0.07
−0.71 ± 0.10
110
13 37 05.49
-29 57 56.5
1.35 ′′
0.0039 ± 0.0009
0.0053 ± 0.0011
0.0058 ± 0.0010
10.4
23.4
21.6
0.97 ± 0.25
0.19 ± 0.23
−0.18 ± 0.23
−0.67 ± 0.47
111
13 37 06.30
-29 51 03.2
1.04 ′′
0.031 ± 0.005
108.3
−0.01 ± 1.06
0.85 ± 0.28
0.16 ± 0.17
−0.56 ± 0.20
112
13 37 06.32
-30 17 24.5
2.88 ′′
0.007 ± 0.002
0.010 ± 0.002
6.1
18.6
0.34 ± 0.25
−0.13 ± 0.24
0.09 ± 0.26
−0.46 ± 0.43
113
13 37 06.45
-30 13 40.4
1.99 ′′
0.009 ± 0.003
8.6
1.00 ± 3.86
0.90 ± 0.31
−0.03 ± 0.29
−0.36 ± 0.54
114
13 37 06.99
-29 51 02.1
0.40 ′′
0.0291 ± 0.0017
0.016 ± 0.002
0.033 ± 0.004
436.0
118.4
96.7
0.98 ± 0.15
0.68 ± 0.09
−0.14 ± 0.06
−0.44 ± 0.07
115
13 37 07.09
-29 46 49.5
0.87 ′′
0.006 ± 0.001
45.1
0.14 ± 0.34
0.55 ± 0.18
−0.37 ± 0.17
−0.40 ± 0.30
116
13 37 07.23
-29 51 32.9
0.52 ′′
0.0179 ± 0.0014
218.6
1.00 ± 0.26
0.72 ± 0.13
−0.11 ± 0.08
−0.49 ± 0.09
117
13 37 08.13
-30 03 05.0
1.11 ′′
0.0046 ± 0.0009
0.0039 ± 0.0010
20.6
12.7
0.32 ± 0.32
0.14 ± 0.24
−0.04 ± 0.23
−0.53 ± 0.38
118
13 37 08.26
-29 53 36.1
2.90 ′′
0.020 ± 0.004
0.008 ± 0.002
34.0
11.4
0.85 ± 0.57
0.65 ± 0.34
−0.09 ± 0.19
−0.77 ± 0.15
119
13 37 10.65
-30 11 18.9
1.11 ′′
0.0136 ± 0.0017
0.011 ± 0.003
90.2
21.1
0.03 ± 0.14
−0.19 ± 0.15
−0.61 ± 0.20
−0.91 ± 0.82
120
13 37 12.54
-29 51 53.2
0.78 ′′
0.011 ± 0.003
0.008 ± 0.002
44.4
11.7
0.97 ± 0.49
−0.25 ± 0.37
−0.25 ± 0.36
0.35 ± 0.28
121
13 37 12.60
-30 09 01.0
0.74 ′′
0.0147 ± .0015
0.0177 ± 0.0016
125.8
178.7
0.24 ± 0.15
0.38 ± 0.10
−0.42 ± 0.10
−0.03 ± 0.19
122
13 37 12.66
-29 43 10.0
1.02 ′′
0.0147 ± 0.0017
87.0
0.15 ± 0.16
0.13 ± 0.14
−0.14 ± 0.15
−0.29 ± 0.25
123
13 37 12.80
-30 05 33.0
1.51 ′′
0.0102 ± 0.0020
0.0135 ± 0.0013
0.0076 ± 0.0009
23.6
172.3
94.8
0.27 ± 0.19
0.35 ± 0.11
−0.23 ± 0.11
−0.60 ± 0.20
124
13 37 12.91
-29 45 08.9
1.14 ′′
0.0071 ± 0.0011
33.7
−0.03 ± 0.22
0.12 ± 0.21
−0.37 ± 0.22
−0.10 ± 0.40
125
13 37 14.65
-29 54 28.8
1.05 ′′
0.0051 ± 0.0009
25.9
0.80 ± 0.47
0.56 ± 0.22
−0.48 ± 0.16
−0.24 ± 0.31
126
13 37 15.83
-30 02 56.3
0.62 ′′
0.028 ± 0.002
0.0339 ± 0.0019
0.0240 ± 0.0017
259.7
597.8
334.1
0.22 ± 0.08
−0.03 ± 0.07
−0.37 ± 0.08
−0.52 ± 0.14
127
13 37 16.06
-29 56 55.5
1.24 ′′
0.004 ± 0.001
0.0057 ± 0.0014
14.0
8.7
0.98 ± 0.71
0.70 ± 0.20
−0.84 ± 0.17
0.28 ± 0.82
128
13 37 16.22
-29 41 56.2
1.37 ′′
0.0072 ± 0.0014
19.3
0.73 ± 0.42
0.41 ± 0.21
−0.50 ± 0.20
−0.09 ± 0.44
129
13 37 16.27
-29 49 38.3
0.36 ′′
0.0295 ± 0.0017
595.3
0.48 ± 0.14
0.49 ± 0.07
−0.17 ± 0.06
−0.53 ± 0.09
130
13 37 17.22
-29 51 53.1
0.88 ′′
0.0093 ± 0.0011
0.008 ± 0.002
65.1
6.0
0.80 ± 0.12
−0.38 ± 0.11
−1.00 ± 0.17
1.00 ± 1.88
131
13 37 19.23
-29 57 09.6
0.74 ′′
0.0115 ± 0.0013
0.0071 ± 0.0016
0.0063 ± 0.0013
103.8
16.0
12.9
0.63 ± 0.19
0.27 ± 0.13
−0.21 ± 0.12
−0.33 ± 0.21
132
13 37 19.54
-30 04 29.2
1.08 ′′
0.014 ± 0.002
0.0176 ± 0.0015
0.0172 ± 0.0014
56.3
199.3
244.4
0.30 ± 0.11
0.04 ± 0.09
−0.40 ± 0.10
−0.28 ± 0.23
133
13 37 19.73
-29 53 48.1
0.12 ′′
0.294 ± 0.004
0.473 ± 0.009
0.261 ± 0.007
13526.54
8642.0
4230.5
0.37 ± 0.02
0.04 ± 0.02
−0.46 ± 0.02
−0.73 ± 0.03
134
13 37 19.98
-30 09 03.7
1.90 ′′
0.0045 ± 0.0014
0.0073 ± 0.0012
6.3
31.7
0.38 ± 0.22
−0.29 ± 0.21
0.28 ± 0.21
−0.33 ± 0.29
Continued on next page
Continued from previous page
No.
RA(2000)
DEC(2000)
pos. err.
rate (obs1)
rate (obs2)
rate (obs3)
likelihood
HR1
HR2
HR3
HR4
(obs1)
(obs2)
(obs3)
135
13 37 22.41
-29 40 31.3
1.07 ′′
0.014 ± 0.002
79.2
0.44 ± 0.22
0.22 ± 0.17
−0.40 ± 0.17
−0.68 ± 0.37
136
13 37 22.46
-30 08 23.7
1.34 ′′
0.0046 ± 0.0013
0.0036 ± 0.0011
13.3
10.1
0.65 ± 0.99
0.14 ± 0.51
0.54 ± 0.26
−0.27 ± 0.34
137
13 37 24.66
-29 58 56.3
0.65 ′′
0.0168 ± 0.0016
0.0073 ± 0.0012
0.0099 ± 0.0013
156.0
33.9
53.0
0.53 ± 0.14
0.19 ± 0.11
−0.35 ± 0.11
−0.54 ± 0.27
138
13 37 25.36
-30 01 59.9
1.53 ′′
0.0045 ± 0.0011
0.0058 ± 0.0011
11.4
24.4
0.59 ± 0.31
0.36 ± 0.19
−0.28 ± 0.20
−0.43 ± 0.44
139
13 37 25.47
-29 54 33.6
0.99 ′′
0.006 ± 0.001
27.4
0.23 ± 0.25
0.17 ± 0.20
−0.50 ± 0.20
−0.28 ± 0.48
140
13 37 26.20
-30 00 30.2
0.84 ′′
0.0114 ± 0.0015
0.0154 ± 0.0018
0.0157 ± 0.0015
78.2
92.3
146.3
0.76 ± 0.13
0.25 ± 0.10
−0.33 ± 0.11
−0.45 ± 0.25
141
13 37 26.36
-29 48 33.0
1.24 ′′
0.0057 ± 0.0009
34.6
0.61 ± 0.20
0.18 ± 0.17
−0.71 ± 0.16
0.11 ± 0.51
142
13 37 26.68
-30 01 47.4
1.50 ′′
0.0066 ± 0.0014
14.7
0.35 ± 0.43
0.46 ± 0.23
−0.30 ± 0.23
−0.50 ± 0.50
143
13 37 27.24
-29 55 47.6
1.33 ′′
0.0112 ± 0.0026
0.0110 ± 0.0018
31.6
50.7
0.47 ± 0.18
−0.05 ± 0.16
−0.99 ± 0.17
0.95 ± 1.08
144
13 37 27.46
-30 02 28.3
1.80 ′′
0.0061 ± 0.0015
0.0037 ± 0.0009
11.3
10.6
0.60 ± 0.24
−0.32 ± 0.22
−0.93 ± 0.53
0.50 ± 3.29
145
13 37 27.47
-30 13 56.1
1.17 ′′
0.014 ± 0.002
0.009 ± 0.005
35.6
16.0
0.50 ± 0.25
0.14 ± 0.18
−0.42 ± 0.20
0.07 ± 0.38
146
13 37 28.34
-29 54 25.3
1.55 ′′
0.0048 ± 0.0012
12.8
0.82 ± 0.32
0.02 ± 0.26
0.03 ± 0.26
−0.75 ± 0.62
147
13 37 28.78
-29 49 43.1
1.20 ′′
0.005 ± 0.001
26.7
0.13 ± 0.60
0.61 ± 0.25
0.01 ± 0.20
−0.12 ± 0.27
148
13 37 29.36
-29 50 27.4
1.35 ′′
0.0039 ± 0.0009
8.2
0.01 ± 0.29
−0.04 ± 0.29
−0.37 ± 0.35
0.07 ± 0.53
149
13 37 29.48
-29 50 08.5
1.00 ′′
0.0093 ± 0.0012
57.8
0.25 ± 0.18
0.07 ± 0.15
−0.29 ± 0.16
−0.32 ± 0.34
150
13 37 29.51
-30 01 47.6
1.44 ′′
0.0042 ± 0.0012
0.0064 ± 0.0013
0.0095 ± 0.0017
6.8
14.7
29.0
0.76 ± 0.32
0.39 ± 0.18
−0.24 ± 0.19
0.17 ± 0.27
151
13 37 29.52
-30 04 16.6
1.06 ′′
0.0106 ± 0.0019
38.9
0.83 ± 0.16
−0.02 ± 0.17
−0.88 ± 0.16
0.53 ± 0.63
152
13 37 29.91
-29 48 27.6
1.40 ′′
0.0064 ± 0.0017
26.1
0.13 ± 1.22
0.81 ± 0.32
−0.64 ± 0.21
−0.22 ± 0.64
153
13 37 30.47
-29 59 37.5
0.58 ′′
0.0253 ± 0.0020
0.0247 ± 0.0022
0.0189 ± 0.0018
229.3
233.8
145.0
0.61 ± 0.11
0.07 ± 0.10
−0.40 ± 0.11
−0.27 ± 0.23
154
13 37 30.97
-29 42 34.4
0.49 ′′
0.072 ± 0.004
728.3
0.94 ± 0.05
0.31 ± 0.07
−0.27 ± 0.06
−0.26 ± 0.11
155
13 37 31.17
-29 51 56.8
0.47 ′′
0.029 ± 0.002
0.0033 ± 0.0011
375.3
11.4
0.52 ± 0.13
0.32 ± 0.09
−0.22 ± 0.08
−0.24 ± 0.13
156
13 37 31.98
-30 05 58.8
1.18 ′′
0.0064 ± 0.0013
25.0
0.47 ± 0.23
−0.09 ± 0.20
−0.30 ± 0.25
−0.81 ± 1.10
157
13 37 32.39
-30 10 19.9
1.18 ′′
0.013 ± 0.002
0.0196 ± 0.0021
41.9
126.6
0.44 ± 0.13
−0.05 ± 0.12
−0.10 ± 0.13
−0.72 ± 0.34
158
13 37 32.94
-29 51 01.2
1.20 ′′
0.0059 ± 0.0012
19.8
0.78 ± 0.22
−0.10 ± 0.23
−0.13 ± 0.25
−0.20 ± 0.41
159
13 37 33.01
-30 06 40.6
1.75 ′′
0.0051 ± 0.0013
0.0080 ± 0.0016
9.9
14.9
0.14 ± 0.38
0.48 ± 0.22
−0.26 ± 0.21
−0.06 ± 0.40
160
13 37 33.32
-29 55 15.6
1.27 ′′
0.004 ± 0.001
0.052 ± 0.003
18.5
416.3
0.64 ± 0.10
0.29 ± 0.07
−0.25 ± 0.07
−0.23 ± 0.13
161
13 37 33.35
-29 57 02.6
0.84 ′′
0.0114 ± 0.0014
0.0099 ± 0.0019
0.0078 ± 0.0015
84.5
39.0
25.3
0.26 ± 0.17
0.06 ± 0.14
−0.20 ± 0.15
−0.76 ± 0.40
162
13 37 33.80
-29 59 58.8
1.08 ′′
0.0087 ± 0.0015
0.004 ± 0.001
35.6
10.1
0.57 ± 0.45
0.58 ± 0.21
−0.12 ± 0.17
−0.49 ± 0.30
163
13 37 36.42
-30 10 52.1
1.28 ′′
0.0075 ± 0.0018
0.0097 ± 0.0021
15.1
21.1
0.83 ± 0.23
0.26 ± 0.22
−0.30 ± 0.22
−0.22 ± 0.53
164
13 37 36.72
-29 48 18.3
0.51 ′′
0.0191 ± 0.0017
195.4
0.29 ± 0.11
−0.07 ± 0.10
−0.33 ± 0.12
−0.41 ± 0.27
165
13 37 39.04
-30 03 32.5
1.29 ′′
0.0061 ± 0.0013
19.6
0.27 ± 0.48
0.59 ± 0.21
−0.45 ± 0.20
−1.00 ± 0.51
166
13 37 39.27
-29 43 21.5
1.09 ′′
0.0129 ± 0.0018
47.6
0.70 ± 0.19
0.16 ± 0.16
−0.22 ± 0.16
−0.41 ± 0.33
167
13 37 40.17
-30 03 16.7
1.38 ′′
0.0059 ± 0.0014
12.5
0.67 ± 0.27
−0.10 ± 0.24
−0.52 ± 0.31
0.19 ± 0.58
168
13 37 40.30
-29 51 23.9
1.42 ′′
0.0050 ± 0.0011
11.6
0.25 ± 0.23
−0.25 ± 0.24
−0.50 ± 0.36
−0.09 ± 0.97
169
13 37 41.38
-30 06 04.3
1.34 ′′
0.010 ± 0.002
0.0081 ± 0.0015
20.4
23.5
0.80 ± 0.31
0.19 ± 0.19
−0.32 ± 0.21
−0.29 ± 0.41
170
13 37 42.47
-29 51 36.9
1.19 ′′
0.0060 ± 0.0012
22.9
0.76 ± 0.29
0.23 ± 0.19
−0.43 ± 0.23
−0.38 ± 0.61
171
13 37 42.88
-30 05 17.3
0.48 ′′
0.041 ± 0.003
0.030 ± 0.002
392.5
221.6
0.24 ± 0.09
−0.03 ± 0.09
−0.21 ± 0.10
−0.67 ± 0.16
172
13 37 43.32
-30 06 01.4
0.84 ′′
0.019 ± 0.002
0.018 ± 0.002
90.5
102.2
0.15 ± 0.14
0.01 ± 0.13
−0.40 ± 0.15
−0.33 ± 0.40
173
13 37 44.45
-29 53 06.4
0.55 ′′
0.029 ± 0.002
364.4
0.20 ± 0.09
0.09 ± 0.08
−0.40 ± 0.09
−0.48 ± 0.24
174
13 37 44.72
-30 07 47.3
1.72 ′′
0.0063 ± 0.0015
0.0074 ± 0.0018
10.5
10.7
0.81 ± 0.19
−0.17 ± 0.23
−1.00 ± 0.32
1.00 ± 0.33
175
13 37 49.64
-29 55 40.9
0.92 ′′
0.0124 ± 0.0019
0.0031 ± 0.0009
0.007 ± 0.002
53.8
13.1
12.7
0.79 ± 0.24
0.38 ± 0.18
−0.20 ± 0.16
−0.60 ± 0.29
176
13 37 49.97
-29 52 17.4
0.75 ′′
0.0206 ± 0.0019
171.3
0.34 ± 0.13
0.11 ± 0.10
−0.33 ± 0.12
−0.66 ± 0.28
177
13 37 50.30
-29 56 43.3
0.76 ′′
0.0126 ± 0.0014
0.0061 ± 0.0014
0.024 ± 0.005
139.6
23.3
35.9
0.59 ± 0.16
0.22 ± 0.13
−0.31 ± 0.14
−0.18 ± 0.24
178
13 37 51.98
-29 48 27.4
1.74 ′′
0.0073 ± 0.0016
12.5
−0.22 ± 0.37
0.32 ± 0.33
0.05 ± 0.25
−0.20 ± 0.38
179
13 37 52.98
-29 44 06.6
1.29 ′′
0.0022 ± 0.0007
8.8
0.20 ± 0.47
0.10 ± 0.42
−0.41 ± 0.45
0.48 ± 0.44
Continued on next page
Table B .2.
B
The detection likelihood L is defined by the relationship L = − ln(p), where p is the probability that a Poissonian fluctuation in the background is detected as a spurious source.
Article number, page 5 of 30
Article number, page 13 of 30
Acknowledgements. We thank the referee Eric M. Schlegel for constructive comments, which helped to improve the manuscript. This research is funded by the Deutsche Forschungsgemeinschaft through the Emmy Noether Research Grant SA 2131/1. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This publication has made use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has made use of SAOImage DS9, developed by Smithsonian Astrophysical Observatory.Continued from previous page No.RA(2000)DEC(2000)
K A Arnaud, Astronomical Society of the Pacific Conference Series. G. H. Jacoby & J. Barnes10117Astronomical Data Analysis Software and Systems VArnaud, K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Ja- coby & J. Barnes, 17
. N Bastian, K R Covey, M R Meyer, ARA&A. 48339Bastian, N., Covey, K. R., & Meyer, M. R. 2010, ARA&A, 48, 339
. F Bigiel, A Leroy, M Seibert, ApJ. 72031Bigiel, F., Leroy, A., Seibert, M., et al. 2010, ApJ, 720, L31
. W P Blair, K S Long, ApJS. 155101Blair, W. P. & Long, K. S. 2004, ApJS, 155, 101
. W P Blair, P F Winkler, K S Long, ApJS. 2038Blair, W. P., Winkler, P. F., & Long, K. S. 2012, ApJS, 203, 8
. A Bodaghee, J A Tomsick, J Rodriguez, J B James, ApJ. 744108Bodaghee, A., Tomsick, J. A., Rodriguez, J., & James, J. B. 2012, ApJ, 744, 108
. S Boissier, A Gil De Paz, B F Madore, ApJ. 61983Boissier, S., Gil de Paz, A., Madore, B. F., et al. 2005, ApJ, 619, L83
. F Bresolin, E Ryan-Weber, R C Kennicutt, Q Goddard, ApJ. 695580Bresolin, F., Ryan-Weber, E., Kennicutt, R. C., & Goddard, Q. 2009, ApJ, 695, 580
. R Buccheri, K Bennett, G F Bignami, A&A. 128245Buccheri, R., Bennett, K., Bignami, G. F., et al. 1983, A&A, 128, 245
. N Cappelluti, M Brusa, G Hasinger, A&A. 497635Cappelluti, N., Brusa, M., Hasinger, G., et al. 2009, A&A, 497, 635
. E J M Colbert, R F Mushotzky, ApJ. 51989Colbert, E. J. M. & Mushotzky, R. F. 1999, ApJ, 519, 89
. J J Condon, W D Cotton, E W Greisen, AJ. 1151693Condon, J. J., Cotton, W. D., Greisen, E. W., et al. 1998, AJ, 115, 1693
. J J Cowan, D A Roberts, D Branch, ApJ. 434128Cowan, J. J., Roberts, D. A., & Branch, D. 1994, ApJ, 434, 128
R G Cruddace, G R Hasinger, J H Schmitt, European Southern Observatory Conference and Workshop Proceedings. F. Murtagh, A. Heck, & P. Benvenuti28European Southern Observatory Conference and Workshop ProceedingsCruddace, R. G., Hasinger, G. R., & Schmitt, J. H. 1988, in European South- ern Observatory Conference and Workshop Proceedings, Vol. 28, Euro- pean Southern Observatory Conference and Workshop Proceedings, ed. F. Murtagh, A. Heck, & P. Benvenuti, 177-182
. T J Davidge, ApJ. 7181428Davidge, T. J. 2010, ApJ, 718, 1428
VizieR Online Data Catalog. G De Vaucouleurs, A De Vaucouleurs, H G CorwinJr, 71370de Vaucouleurs, G., de Vaucouleurs, A., Corwin, Jr., H. G., et al. 1992, VizieR Online Data Catalog, 7137, 0
. Di Stefano, R Greiner, J Murray, S Garcia, M , ApJ. 55137Di Stefano, R., Greiner, J., Murray, S., & Garcia, M. 2001, ApJ, 551, L37
. Di Stefano, R Kong, A K H , ApJ. 592884Di Stefano, R. & Kong, A. K. H. 2003, ApJ, 592, 884
. C Done, M Gierliński, A Kubota, A&A Rev. 151Done, C., Gierliński, M., & Kubota, A. 2007, A&A Rev., 15, 1
. H Dong, D Calzetti, M Regan, AJ. 136479Dong, H., Calzetti, D., Regan, M., et al. 2008, AJ, 136, 479
. M A Dopita, W P Blair, K S Long, ApJ. 710964Dopita, M. A., Blair, W. P., Long, K. S., et al. 2010, ApJ, 710, 964
. M Ehle, W Pietsch, R Beck, U Klein, A&A. 32939Ehle, M., Pietsch, W., Beck, R., & Klein, U. 1998, A&A, 329, 39
. G Fabbiano, ARA&A. 44323Fabbiano, G. 2006, ARA&A, 44, 323
. G Fasano, A Franceschini, MNRAS. 225155Fasano, G. & Franceschini, A. 1987, MNRAS, 225, 155
. H Feng, R Soria, New A Rev. 55166Feng, H. & Soria, R. 2011, New A Rev., 55, 166
. M Gilfanov, MNRAS. 349146Gilfanov, M. 2004, MNRAS, 349, 146
. J C Gladstone, D M Gottlieb, Upson, W L Ii, ApJ. 157611Gladstone, J. C. 2011, ArXiv e-prints Gottlieb, D. M. & Upson, II, W. L. 1969, ApJ, 157, 611
Article number, page 14 of 30. Article number, page 14 of 30
X-ray source population study of the galaxy M 83 with XMM-Newton Greiner. L Ducci, New A. 5137J.L. Ducci et al.: X-ray source population study of the galaxy M 83 with XMM-Newton Greiner, J. 2000, New A, 5, 137
. H.-J Grimm, M Gilfanov, R Sunyaev, MNRAS. 339793Grimm, H.-J., Gilfanov, M., & Sunyaev, R. 2003, MNRAS, 339, 793
XMM-Newton Calibration Technical Note. M Guainazzi, XMM-SOC- CAL-TN-0018Guainazzi, M. 2012, XMM-Newton Calibration Technical Note; XMM-SOC- CAL-TN-0018
. M Güdel, Royal Society of London Philosophical Transactions Series A. 360Güdel, M. 2002, Royal Society of London Philosophical Transactions Series A, 360, 1935
. J Harris, D Calzetti, Gallagher, J S Iii, C J Conselice, D A Smith, AJ. 1223046Harris, J., Calzetti, D., Gallagher, III, J. S., Conselice, C. J., & Smith, D. A. 2001, AJ, 122, 3046
. J P Hughes, ApJ. 42725Hughes, J. P. 1994, ApJ, 427, L25
. S Immler, A Vogler, M Ehle, W Pietsch, A&A. 352415Immler, S., Vogler, A., Ehle, M., & Pietsch, W. 1999, A&A, 352, 415
. M Irwin, S Maddox, R Mcmahon, IEEE Spectrum. 214Irwin, M., Maddox, S., & McMahon, R. 1994, IEEE Spectrum, 2, 14
. H L Johnson, ARA&A. 4193Johnson, H. L. 1966, ARA&A, 4, 193
. C Jones, ApJ. 214856Jones, C. 1977, ApJ, 214, 856
. D H Jones, W Saunders, M Colless, MNRAS. 355747Jones, D. H., Saunders, W., Colless, M., et al. 2004, MNRAS, 355, 747
. P Kahabka, Van Den, E P J Heuvel, ARA&A. 3569Kahabka, P. & van den Heuvel, E. P. J. 1997, ARA&A, 35, 69
. A R King, MNRAS. 39341King, A. R. 2009, MNRAS, 393, L41
. J Koda, M Yagi, S Boissier, ApJ. 74920Koda, J., Yagi, M., Boissier, S., et al. 2012, ApJ, 749, 20
. J Krautter, F.-J Zickgraf, I Appenzeller, A&A. 350743Krautter, J., Zickgraf, F.-J., Appenzeller, I., et al. 1999, A&A, 350, 743
. M R Krumholz, C F Mckee, Nature. 4511082Krumholz, M. R. & McKee, C. F. 2008, Nature, 451, 1082
. R.-P Kudritzki, J Puls, ARA&A. 38613Kudritzki, R.-P. & Puls, J. 2000, ARA&A, 38, 613
. K S Long, W P Blair, L E H Godfrey, ApJ. 75618Long, K. S., Blair, W. P., Godfrey, L. E. H., et al. 2012, ApJ, 756, 18
. T Maccacaro, I M Gioia, A Wolter, G Zamorani, J T Stocke, ApJ. 326680Maccacaro, T., Gioia, I. M., Wolter, A., Zamorani, G., & Stocke, J. T. 1988, ApJ, 326, 680
. L A Maddox, J J Cowan, R E Kilgard, AJ. 132310Maddox, L. A., Cowan, J. J., Kilgard, R. E., et al. 2006, AJ, 132, 310
. K Makishima, Y Maejima, K Mitsuda, ApJ. 308635Makishima, K., Maejima, Y., Mitsuda, K., et al. 1986, ApJ, 308, 635
. G R Meurer, O I Wong, J H Kim, ApJ. 695765Meurer, G. R., Wong, O. I., Kim, J. H., et al. 2009, ApJ, 695, 765
. M C Miller, E J M Colbert, International Journal of Modern Physics D. 131Miller, M. C. & Colbert, E. J. M. 2004, International Journal of Modern Physics D, 13, 1
. S Mineo, M Gilfanov, R Sunyaev, MNRAS. 4192095Mineo, S., Gilfanov, M., & Sunyaev, R. 2012, MNRAS, 419, 2095
. Z Misanovic, W Pietsch, F Haberl, A&A. 4481247Misanovic, Z., Pietsch, W., Haberl, F., et al. 2006, A&A, 448, 1247
. D G Monet, S E Levine, B Canzian, AJ. 125984Monet, D. G., Levine, S. E., Canzian, B., et al. 2003, AJ, 125, 984
. T Ohashi, K Makishima, T Tsuru, ApJ. 365180Ohashi, T., Makishima, K., Tsuru, T., et al. 1990, ApJ, 365, 180
. K Okada, K Mitsuda, T Dotani, PASJ. 49653Okada, K., Mitsuda, K., & Dotani, T. 1997, PASJ, 49, 653
. R A Owen, R S Warwick, MNRAS. 3941741Owen, R. A. & Warwick, R. S. 2009, MNRAS, 394, 1741
. J A Peacock, MNRAS. 202615Peacock, J. A. 1983, MNRAS, 202, 615
. W Pietsch, Z Misanovic, F Haberl, A&A. 42611Pietsch, W., Misanovic, Z., Haberl, F., et al. 2004, A&A, 426, 11
. J Poutanen, G Lipunova, S Fabrika, A G Butkevich, P Abolmasov, MNRAS. 3771187Poutanen, J., Lipunova, G., Fabrika, S., Butkevich, A. G., & Abolmasov, P. 2007, MNRAS, 377, 1187
. F A Primini, W Forman, C Jones, ApJ. 410615Primini, F. A., Forman, W., & Jones, C. 1993, ApJ, 410, 615
. J Puls, J S Vink, F Najarro, A&A Rev. 16209Puls, J., Vink, J. S., & Najarro, F. 2008, A&A Rev., 16, 209
. J C Raymond, B W Smith, ApJS. 35419Raymond, J. C. & Smith, B. W. 1977, ApJS, 35, 419
. T P Roberts, A J Levan, M R Goad, MNRAS. 38773Roberts, T. P., Levan, A. J., & Goad, M. R. 2008, MNRAS, 387, 73
. T P Roberts, R S Warwick, M J Ward, M R Goad, L P Jenkins, MNRAS. 3571363Roberts, T. P., Warwick, R. S., Ward, M. J., Goad, M. R., & Jenkins, L. P. 2005, MNRAS, 357, 1363
. J Robrade, J H M M Schmitt, A&A. 51638Robrade, J. & Schmitt, J. H. M. M. 2010, A&A, 516, A38
. D H Rogstad, I A Lockhart, M C H Wright, ApJ. 193309Rogstad, D. H., Lockhart, I. A., & Wright, M. C. H. 1974, ApJ, 193, 309
. R Rosner, L Golub, G S Vaiana, ARA&A. 23413Rosner, R., Golub, L., & Vaiana, G. S. 1985, ARA&A, 23, 413
. K S Rumstay, M Kaufman, ApJ. 274611Rumstay, K. S. & Kaufman, M. 1983, ApJ, 274, 611
. A Saha, F Thim, G A Tammann, B Reindl, A Sandage, ApJS. 165108Saha, A., Thim, F., Tammann, G. A., Reindl, B., & Sandage, A. 2006, ApJS, 165, 108
. M Sako, D A Liedahl, S M Kahn, F Paerels, ApJ. 525921Sako, M., Liedahl, D. A., Kahn, S. M., & Paerels, F. 1999, ApJ, 525, 921
. R E Schild, AJ. 82337Schild, R. E. 1977, AJ, 82, 337
. C Schröder, S Hubrig, J H M M Schmitt, A&A. 484479Schröder, C., Hubrig, S., & Schmitt, J. H. M. M. 2008, A&A, 484, 479
. C Schröder, J H M M Schmitt, A&A. 475677Schröder, C. & Schmitt, J. H. M. M. 2007, A&A, 475, 677
. N Scoville, H Aussel, M Brusa, ApJS. 1721Scoville, N., Aussel, H., Brusa, M., et al. 2007, ApJS, 172, 1
. M F Skrutskie, R M Cutri, R Stiening, AJ. 1311163Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163
. R K Smith, N S Brickhouse, D A Liedahl, J C Raymond, ApJ. 55691Smith, R. K., Brickhouse, N. S., Liedahl, D. A., & Raymond, J. C. 2001, ApJ, 556, L91
. R Soria, K D Kuntz, P F Winkler, Soria, R., Kuntz, K. D., Winkler, P. F., et al. 2012, ArXiv e-prints
R Soria, K S Long, L Bianchi, The Astronomer's Telegram. 30921Soria, R., Long, K. S., Bianchi, L., et al. 2010, The Astronomer's Telegram, 3092, 1
. R Soria, K Wu, A&A. 38499Soria, R. & Wu, K. 2002, A&A, 384, 99
. R Soria, K Wu, A&A. 41053Soria, R. & Wu, K. 2003, A&A, 410, 53
. A.-M Stobbart, T P Roberts, J Wilms, MNRAS. 368397Stobbart, A.-M., Roberts, T. P., & Wilms, J. 2006, MNRAS, 368, 397
Formation and evolution of compact stellar X-ray sources. T M Tauris, Van Den, E P J Heuvel, W H G Lewin, M Van Der Klis, Tauris, T. M. & van den Heuvel, E. P. J. 2006, Formation and evolution of com- pact stellar X-ray sources, ed. Lewin, W. H. G. and van der Klis, M., 623-665
. D A Thilker, L Bianchi, S Boissier, ApJ. 61979Thilker, D. A., Bianchi, L., Boissier, S., et al. 2005, ApJ, 619, L79
. F Thim, G A Tammann, A Saha, ApJ. 590256Thim, F., Tammann, G. A., Saha, A., et al. 2003, ApJ, 590, 256
. R P J Tilanus, R J Allen, A&A. 274707Tilanus, R. P. J. & Allen, R. J. 1993, A&A, 274, 707
. A Treves, L Maraschi, M Abramowicz, PASP. 100427Treves, A., Maraschi, L., & Abramowicz, M. 1988, PASP, 100, 427
. G Trinchieri, G Fabbiano, G G C Palumbo, ApJ. 29096Trinchieri, G., Fabbiano, G., & Palumbo, G. G. C. 1985, ApJ, 290, 96
R B Tully, Nearby galaxies catalog. Tully, R. B. 1988, Nearby galaxies catalog
. T J Turner, K A Weaver, R F Mushotzky, S S Holt, G M Madejski, ApJ. 38185Turner, T. J., Weaver, K. A., Mushotzky, R. F., Holt, S. S., & Madejski, G. M. 1991, ApJ, 381, 85
. G S Vaiana, J P Cassinelli, G Fabbiano, ApJ. 24597A&AVaiana, G. S., Cassinelli, J. P., Fabbiano, G., et al. 1981, ApJ, 245, 163 van den Heuvel, E. P. J., Bhattacharya, D., Nomoto, K., & Rappaport, S. A. 1992, A&A, 262, 97
J Van Paradijs, The Many Faces of Neutron Stars. R. Buccheri, J. van Paradijs, & A. Alpar515279NATO ASIC Procvan Paradijs, J. 1998, in NATO ASIC Proc. 515: The Many Faces of Neutron Stars., ed. R. Buccheri, J. van Paradijs, & A. Alpar, 279
. B A Vaughan, M Van Der Klis, K S Wood, ApJ. 435362Vaughan, B. A., van der Klis, M., Wood, K. S., et al. 1994, ApJ, 435, 362
. C Vignali, A Comastri, M Cappi, ApJ. 516582Vignali, C., Comastri, A., Cappi, M., et al. 1999, ApJ, 516, 582
. R Walter, H H Fink, A&A. 274105Walter, R. & Fink, H. H. 1993, A&A, 274, 105
. N E White, G F Carpenter, MNRAS. 18311White, N. E. & Carpenter, G. F. 1978, MNRAS, 183, 11P
. N E White, F E Marshall, ApJ. 281354White, N. E. & Marshall, F. E. 1984, ApJ, 281, 354
. N E White, F Nagase, A N Parmar, X-ray Binaries, 1White, N. E., Nagase, F., & Parmar, A. N. 1995, X-ray Binaries, 1
. A Wofford, C Leitherer, R Chandar, ApJ. 727100Wofford, A., Leitherer, C., & Chandar, R. 2011, ApJ, 727, 100
Article number, page 15 of 30. Article number, page 15 of 30
A&a-Lducci_M83, E1998] 19 (B) AGN 18 19 20 NVSS J133618-301459 (B) (AGN) 21 0599-0299962 (C) 13361884-3001381 (C) [E1998] 20 (B) fg. star 22 23 24 0601-0298625 (C) 13362007-2951058 (C) (CV) 25 26 0597-0300800 (C) 27 28 0598-0301061 (A) 29 0601-0298670 (A) 30 [E1998] 22 (BOnline Material p. AGN16C). C) [E1998] 22 (B)A&A-lducci_m83, Online Material p 16 0598-0300943 (C) 13360646-3006475 (C) 11 12 0602-0301227 (C) [I1999] 2 star (B), [E1998] 17 (A) 13 14 [I1999] 4 (C) 15 0600-0300086 (C) 16 0600-0300087 (C) [I1999] 3 (B), [E1998] 18 (B) hard source 17 0600-0300099 (C) [I1999] 5 gal. (B), [E1998] 19 (B) AGN 18 19 20 NVSS J133618-301459 (B) (AGN) 21 0599-0299962 (C) 13361884-3001381 (C) [E1998] 20 (B) fg. star 22 23 24 0601-0298625 (C) 13362007-2951058 (C) (CV) 25 26 0597-0300800 (C) 27 28 0598-0301061 (A) 29 0601-0298670 (A) 30 [E1998] 22 (B) 31 0602-0301338 (C) 13362821-2942266 (B) GAL 32 33 0600-0300208 (C) [E1998] 22 (B) 34 0601-0298683 (C) 13362901-2951232 (B) 35 36 37 NVSS J133630-301651 (C) (AGN)
A&a-Lducci_M83, Online Material p. 29A&A-lducci_m83, Online Material p 29
ULX 134 135 0603-0300266 (A) 136 137 [I1999] 31 star (B), [E1998] 31 (B) 0300561 (A) 13372725-2955475 (A) [I1999] 32 star (B) fg. star 144 0599-0300410 (A) 13372747-3002283 (A) 6dFGS gJ133727.5-300228, gal (A) GAL 145 0597-0301232 (A) 146 147 0601-0298991USNO B1 2MASS optical radio X-ray class. 114 [SW2003] 104 xrb cand. (C) hard source 115 116 [BL2004] 53, snr cand. (C) [SW2003] 105 (C), [DK03] 88 sss cand. (C) hard source 117. 118SW2003] 113 xrb (B) XRB. I1999] 29 (B), [E1998] 12 (B) hard source 130 [SW2003] 122 (A) 131 132 133 [I1999] 30 ULX (C), [E1998] 13 (C), [T1985] 2 (A). B) 13372947-2950283 (B) 6dFGS gJ133729.5-295028, gal (BContinued from previous page No. USNO B1 2MASS optical radio X-ray class. 114 [SW2003] 104 xrb cand. (C) hard source 115 116 [BL2004] 53, snr cand. (C) [SW2003] 105 (C), [DK03] 88 sss cand. (C) hard source 117 118 0601-0298889 (C) 13370870-2953313 (C) 119 120 [SW2003] 113 xrb (B) XRB 121 122 0602-0301643 (B) 123 0599-0300322 (B) 124 125 [SW2003] 119 (B) 126 0599-0300335 (C) [E1998] 30 (C) hard source 127 128 129 [SW2003] 121 xrb cand. (C), [I1999] 29 (B), [E1998] 12 (B) hard source 130 [SW2003] 122 (A) 131 132 133 [I1999] 30 ULX (C), [E1998] 13 (C), [T1985] 2 (A) ULX 134 135 0603-0300266 (A) 136 137 [I1999] 31 star (B), [E1998] 31 (B) 0300561 (A) 13372725-2955475 (A) [I1999] 32 star (B) fg. star 144 0599-0300410 (A) 13372747-3002283 (A) 6dFGS gJ133727.5-300228, gal (A) GAL 145 0597-0301232 (A) 146 147 0601-0298991 (C) 148 0601-0298995 (B) 13372947-2950283 (B) 6dFGS gJ133729.5-295028, gal (B)
. E1998] 32 (A) 155 [I1999] 33 (B), [E1998] 33 (B) 156 157 158 0601-0299030 (C) 13373327-2951007 (C) ESO 444-85gal (C) GAL. 159C). C) [I1999] 34 (A), [E1998] 34 (A) 165 [E1998] 35 (B). C) [I1999] 35 (B), [E1998] 36 (A) 167 [E1998] 35 (C). A) 169 170 [E1998] 37 (A) 171 172 0598-0301628 (C) [E1998] 38 (B) Continued on next page[E1998] 32 (A) 155 [I1999] 33 (B), [E1998] 33 (B) 156 157 158 0601-0299030 (C) 13373327-2951007 (C) ESO 444-85, gal (C) GAL 159 0598-0301544 (A) 160 0600-0300590 (C) 13373342-2955182 (C) 161 162 163 0598-0301583 (C) 164 0601-0299090 (C) [I1999] 34 (A), [E1998] 34 (A) 165 [E1998] 35 (B) 166 0602-0301832 (C) [I1999] 35 (B), [E1998] 36 (A) 167 [E1998] 35 (C) 168 0601-0299123 (B) 13374032-2951233 (A) 169 170 [E1998] 37 (A) 171 172 0598-0301628 (C) [E1998] 38 (B) Continued on next page
| [] |
[
"SOBOLEV, BESOV AND TRIEBEL-LIZORKIN SPACES ON QUANTUM TORI",
"SOBOLEV, BESOV AND TRIEBEL-LIZORKIN SPACES ON QUANTUM TORI"
] | [
"Xiao Xiong ",
"ANDQuanhua Xu ",
"Zhi Yin "
] | [] | [] | This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative d-torus T d θ (with θ a skew symmetric real d × d-matrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincaré type inequality for Sobolev spaces. We also show that the Sobolev space W k ∞ (T d θ ) coincides with the Lipschitz space of order k, already studied by Weaver in the case k = 1. We establish the embedding inequalities of all these spaces, including the Besov and Sobolev embedding theorems. We obtain Littlewood-Paley type characterizations for Besov and Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms of the Poisson, heat semigroups and differences. Some of them are new even in the commutative case, for instance, our Poisson semigroup characterizations improve the classical ones. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Brézis -Mironescu and Maz'ya-Shaposhnikova on the limits of Besov norms. The same characterization implies that the Besov space B α ∞,∞ (T d θ ) for α > 0 is the quantum analogue of the usual Zygmund class of order α. We investigate the interpolation of all these spaces, in particular, determine explicitly the K-functional of the couple (Lp(T d θ ), W k p (T d θ )), which is the quantum analogue of a classical result due to Johnen and Scherer. Finally, we show that the completely bounded Fourier multipliers on all these spaces do not depend on the matrix θ, so coincide with those on the corresponding spaces on the usual d-torus. We also give a quite simple description of (completely) bounded Fourier multipliers on the Besov spaces in terms of their behavior on the Lp-components in the Littlewood-Paley decomposition. | 10.1090/memo/1203 | [
"https://arxiv.org/pdf/1507.01789v2.pdf"
] | 118,907,531 | 1507.01789 | 801026a4e60905dac3545a105f0619169afc3f6a |
SOBOLEV, BESOV AND TRIEBEL-LIZORKIN SPACES ON QUANTUM TORI
7 Jul 2015
Xiao Xiong
ANDQuanhua Xu
Zhi Yin
SOBOLEV, BESOV AND TRIEBEL-LIZORKIN SPACES ON QUANTUM TORI
7 Jul 2015Quantum torinoncommutative Lp-spacesBessel and Riesz potentials(potential) Sobolev spacesBesov spacesTriebel-Lizorkin spacesHardy spacescharacterizationsPoisson and heat semigroupsembedding inequalitiesinterpolation(completely) bounded Fourier multipliers
This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative d-torus T d θ (with θ a skew symmetric real d × d-matrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincaré type inequality for Sobolev spaces. We also show that the Sobolev space W k ∞ (T d θ ) coincides with the Lipschitz space of order k, already studied by Weaver in the case k = 1. We establish the embedding inequalities of all these spaces, including the Besov and Sobolev embedding theorems. We obtain Littlewood-Paley type characterizations for Besov and Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms of the Poisson, heat semigroups and differences. Some of them are new even in the commutative case, for instance, our Poisson semigroup characterizations improve the classical ones. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Brézis -Mironescu and Maz'ya-Shaposhnikova on the limits of Besov norms. The same characterization implies that the Besov space B α ∞,∞ (T d θ ) for α > 0 is the quantum analogue of the usual Zygmund class of order α. We investigate the interpolation of all these spaces, in particular, determine explicitly the K-functional of the couple (Lp(T d θ ), W k p (T d θ )), which is the quantum analogue of a classical result due to Johnen and Scherer. Finally, we show that the completely bounded Fourier multipliers on all these spaces do not depend on the matrix θ, so coincide with those on the corresponding spaces on the usual d-torus. We also give a quite simple description of (completely) bounded Fourier multipliers on the Besov spaces in terms of their behavior on the Lp-components in the Littlewood-Paley decomposition.
Contents
Chapter 0. Chapter 0. Introduction This paper is the second part of our project about analysis on quantum tori. The previous one [17] studies several subjects of harmonic analysis on these objects, including maximal inequalities, mean and pointwise convergences of Fourier series, completely bounded Fourier multipliers on L pspaces and the theory of Hardy spaces. It was directly inspired by the current line of investigation on noncommutative harmonic analysis. As pointed out there, very little had been done about the analytic aspect of quantum tori before [17]; this situation is in strong contrast with their geometry on which there exists a considerably long list of publications. Presumably, this deficiency is due to numerous difficulties one may encounter when dealing with noncommutative L p -spaces, since these spaces come up unavoidably if one wishes to do analysis. [17] was made possible by the recent developments on noncommutative martingale/ergodic inequalities and the Littlewood-Paley-Stein theory for quantum Markovian semigroups, which had been achieved thanks to the efforts of many researchers; see, for instance, [56,31,36,37,61,62,52], and [32,44,45,33,34].
This second part intends to study Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori. In the classical setting, these spaces are fundamental for many branches of mathematics such as harmonic analysis, PDE, functional analysis and approximation theory. Our references for the classical theory are [1,42,49,53,72,73]. However, they have never been investigated so far in the quantum setting, except two special cases to our best knowledge. Firstly, Sobolev spaces with the L 2 -norm were studied by Spera [64] in view of applications to the Yang-Mills theory for quantum tori [65] (see also [26,39,58,63] for related works). On the other hand, inspired by Connes' noncommutative geometry [18], or more precisely, the part on noncommutative metric spaces, Weaver [77,78] developed the Lipschitz classes of order α for 0 < α ≤ 1 on quantum tori. The fact that only these two cases have been studied so far illustrates once more the above mentioned difficulties related to noncommutativity.
Among these difficulties, a specific one is to be emphasized: it is notably relevant to this paper, and is the lack of a noncommutative analogue of the usual pointwise maximal function. However, maximal function techniques play a paramount role in the classical theory of Besov and Triebel-Lizorkin spaces (as well as in the theory of Hardy spaces). They are no longer available in the quantum setting, which forces us to invent new tools, like in the previously quoted works on noncommutative martingale inequalities and the quantum Littlewood-Paley-Stein theory where the same difficulty already appeared.
One powerful tool used in [17] is the transference method. It consists in transferring problems on quantum tori to the corresponding ones in the case of operator-valued functions on the usual tori, in order to use existing results in the latter case or adapt classical arguments. This method is efficient for several problems studied in [17], including the maximal inequalities and Hardy spaces. It is still useful for some parts of the present work; for instance, Besov spaces can be investigated through the classical vector-valued Besov spaces by means of transference, the relevant Banach spaces being the noncommutative L p -spaces on a quantum torus. However, it becomes inefficient for others. For example, the Sobolev or Besov embedding inequalities cannot be proved by transference. On the other hand, if one wishes to study Triebel-Lizorkin spaces on quantum tori via transference, one should first develop the theory of operator-valued Triebel-Lizorkin spaces on the classical tori. The latter is as hard as the former. Contrary to [17] , the transference method will play a very limited role in the present paper. Instead, we will use Fourier multipliers in a crucial way, this approach is of interest in its own right. We thus develop an intrinsic differential analysis on quantum tori, without frequently referring to the usual tori via transference as in [17]. This is a major advantage of the present methods over those of [17]. We hope that the study carried out here would open new perspectives of applications and motivate more future research works on quantum tori or in similar circumstances. In fact, one of our main objectives of developing analysis on quantum tori is to gain more insights on the geometrical structures of these objects, so ultimately to return back to their differential geometry.
To describe the content of the paper, we need some definitions and notation (see the respective sections below for more details). Let d ≥ 2 and θ = (θ kj ) be a real skew-symmetric d × d-matrix. The d-dimensional noncommutative torus A θ is the universal C*-algebra generated by d unitary operators U 1 , . . . , U d satisfying the following commutation relation U k U j = e 2πiθ kj U j U k , 1 ≤ j, k ≤ d.
Let U = (U 1 , · · · , U d ). For m = (m 1 , · · · , m d ) ∈ Z d , set
U m = U m1 1 · · · U m d d .
A polynomial in U is a finite sum:
x = m∈Z d α m U m , α m ∈ C.
For such a polynomial x, we define τ (x) = α 0 . Then τ extends to a faithful tracial state on A θ . Let T d θ be the w*-closure of A θ in the GNS representation of τ . This is our d-dimensional quantum torus. It is to be viewed as a deformation of the usual d-torus T d , or more precisely, of the commutative algebra L ∞ (T d ). The noncommutative L p -spaces associated to (T d θ , τ ) are denoted by L p (T d θ ). The Fourier transform of an element x ∈ L 1 (T d θ ) is defined by
x(m) = τ (U m ) * x , m ∈ Z d .
The formal Fourier series of x is
x ∼ m∈Z d
x(m)U m .
The differential structure of T d θ is modeled on that of T d . Let
S(T d θ ) = m∈Z d a m U m : {a m } m∈Z d rapidly decreasing .
This is the deformation of the space of infinitely differentiable functions on T d ; it is the Schwartz class of T d θ . Like in the commutative case, S(T d θ ) carries a natural locally convex topology. Its topological dual S ′ (T d θ ) is the space of distributions on T d θ . The partial derivations on S(T d θ ) are determined by ∂ j (U j ) = 2πiU j and ∂ j (U k ) = 0, k = j, 1 ≤ j, k ≤ d. Given m = (m 1 , . . . , m d ) ∈ N d 0 (N d 0 denoting the set of nonnegative integers), the associated partial derivation D m is defined to be ∂ m1 1 · · · ∂ m d d . The order of D m is |m| 1 = m 1 + · · · + m d . Let ∆ = ∂ 2 1 + · · · + ∂ 2 d be the Laplacian. By duality, the derivations and Fourier transform transfer to S ′ (T d θ ) too. Fix a Schwartz function ϕ on R d satisfying the usual Littlewood-Paley decomposition property expressed in (3.1). For each k ≥ 0 let ϕ k be the function whose Fourier transform is equal to ϕ(2 −k ·). For a distribution x on T d θ , define
ϕ k * x = m∈Z d ϕ(2 −k m) x(m)U m .
So x → ϕ k * x is the Fourier multiplier with symbol ϕ(2 −k ·).
We can now define the four families of function spaces on T d θ to be studied . Let 1 ≤ p, q ≤ ∞ and k ∈ N, α ∈ R, and let J α be the Bessel potential of order α: J α = (1 − (2π) −2 ∆) α 2 .
• Sobolev spaces:
W k p (T d θ ) = x ∈ S ′ (T d θ ) : D m x ∈ L p (T d θ ) for each m ∈ N d 0 with |m| 1 ≤ k .
• Potential or fractoinal Sobolev spaces:
H α p (T d θ ) = x ∈ S ′ (T d θ ) : J α x ∈ L p (T d θ ) . • Besov spaces: B α p,q (T d θ ) = x ∈ S ′ (T d θ ) : | x(0)| q + k≥0 2 qkα ϕ k * x q p 1 q < ∞ .
• Triebel-Lizorkin spaces for p < ∞ :
F α,c p (T d θ ) = x ∈ S ′ (T d θ ) : | x(0)| 2 + k≥0 2 2kα | ϕ k * x| 2 1 2 p < ∞ .
Equipped with their natural norms, all these spaces become Banach spaces. Now we can describe the main results proved in this paper by classifying them into five families.
Basic properties. A common basic property of potential Sobolev, Besov and Triebel-Lizorkin spaces is a reduction theorem by the Bessel potential. For example, J β is an isomorphism from B α p,q (T d θ ) onto B α−β p,q (T d θ ) for all 1 ≤ p, q ≤ ∞ and α, β ∈ R; this is the so-called lifting or reduction theorem. Specifically to Triebel-Lizorkin spaces, J α establishes an isomorphism between F α,c p (T d θ ) and the Hardy space H c p (T d θ ) for any 1 ≤ p < ∞. As a consequence, we deduce that the potential Sobolev space H α p (T d θ ) admits a Littlewood-Paley type characterization for 1 < p < ∞. Another type of reduction for Besov and Triebel-Lizorkin spaces is that for any positive integer k, x ∈ F α,c p (T d θ ) (resp. B α p,q (T d θ )) iff all its partial derivatives of order k belong to F α−k,c p (T d θ ) (resp. B α−k p,q (T d θ )). Concerning Sobolev spaces, we obtain a Poincaré type inequality: For any x ∈ W 1 p (T d θ ) with 1 ≤ p ≤ ∞, we have
x − x(0) p ∇x p .
Our proof of this inequality greatly differs with standard arguments for such results in the commutative case.
We also show that W k ∞ (T d θ ) is the analogue for T d θ of the classical Lipschitz class of order k. For u ∈ R d , define ∆ u x = π z (x) − x, where z = (e 2πiu1 , · · · , e 2πiu d ) and π z is the automorphism of T d θ determined by U j → z j U j for 1 ≤ j ≤ d. Then for a positive integer k, ∆ k u is the kth difference operator on T d θ associated to u. Note that ∆ k u is also the Fourier multiplier with symbol e k u , where e u (ξ) = e 2πiu·ξ − 1. The kth order modulus of L p -smoothness of an x ∈ L p (T d θ ) is defined to be
ω k p (x, ε) = sup 0<|u|≤ε ∆ k u x p .
We then prove that for any 1 ≤ p ≤ ∞ and k ∈ N,
sup ε>0 ω k p (x, ε) ε k ≈ m∈N d 0 , |m|1=k D m x p .
In particular, we recover Weaver's results [77,78] on the Lipschitz class on T d θ when p = ∞ and k = 1.
Embedding. The second family of results concern the embedding of the preceding spaces. A typical one is the analogue of the classical Sobolev embedding inequality for W k p (T d θ ): If 1 < p < q < ∞ such that 1 q = 1 p − k d , then W k p (T d θ ) ⊂ L q (T d θ ) continuously. Similar embedding inequalities hold for the other spaces too. Combined with real interpolation, the embedding inequality of B α p,q (T d θ ) yields the above Sobolev embedding. Our proofs of these embedding inequalities are based on Varopolous' celebrated semigroup approach [75] to the Littlewood-Sobolev theory, which has also been developed by Junge and Mei [34] in the noncommutative setting for the study of BMO spaces on quantum Markovian semigroups. Thus the characterization of Besov spaces by Poisson or heat semigroup described below is essential for the proof of our embedding inequalities.
We also establish compact embedding theorems. For instance, the previously mentioned Sobolev embedding becomes a compact one W k p (T d θ ) ֒→ L q * (T d θ ) for any q * with 1 ≤ q * < q. Characterizations. The third family of results are various characterizations of Besov and Triebel-Lizorkin spaces. This is the most difficult and technical part of the paper. In the classical case, all existing proofs of these characterizations that we know use maximal function techniques in a crucial way. As pointed out earlier, these techniques are no longer available. Instead, we use frequently Fourier multipliers. We would like to emphasize that our results are better than those in literature even in the commutative case. Let us illustrate this by stating the characterization of Besov spaces in terms of the circular Poisson semigroup.
Given a distribution x on T d θ and k ∈ Z, let P r (x) = m∈Z d
x(m)r |m| U m and J k r P r (x) =
m∈Z d C m,k x(m)r |m|−k U m , 0 ≤ r < 1 ,
where | · | denotes the Euclidean norm of R d and C m,k = |m| · · · (|m| − k + 1) if k ≥ 0 and C m,k = 1 (|m| + 1) · · · (|m| − k)
if k < 0.
Note that J k r is the kth derivation operator relative to r if k ≥ 0, and the (−k)th integration operator if k < 0. Then our characterization asserts that for 1 ≤ p, q ≤ ∞ and α ∈ R, k ∈ Z with k > α,
x B α p,q ≈ max |m|<k | x(m)| q + 1 0 (1 − r) (k−α)q J k r P r (x k ) q p dr 1 − r 1 q , where x k = x − |m|<k x(m)U m .
The use of the integration operator (corresponding to negative k) in the above statement is completely new even in the case θ = 0 (the commutative case). This is very natural, and consistent with the fact that smaller α is, lower smoothness the elements of B α p,q (T d θ ) have. This is also consistent with the previously mentioned lifting theorem. A similar result holds for Triebel-Lizorkin spaces too. But its proof is much subtler. For the latter spaces, another improvement of our characterization over the classical one lies on the assumption on k: in the classical case, k is required to be greater than d + max(α, 0), while we only need to assume k > α.
The classical characterization of Besov spaces by differences is also extended to the quantum setting. This result resembles the previous one in terms of the derivations of the Poisson semigroup. For 1 ≤ p, q ≤ ∞ and α ∈ R, k ∈ N with 0 < α < k, let
x B α,ω p,q = 1 0 ε −αq ω k p (x, ε) q dε ε 1 q .
Then x ∈ B α p,q (T d θ ) iff x B α,ω p,q < ∞. The difference characterization of Besov spaces shows that B α ∞,∞ (T d θ ) is the quantum analogue of the classical Zygmund class. In particular, for 0 < α < 1, B α ∞,∞ (T d θ ) is the Hölder class of order α, already studied by Weaver [78].
In the commutative case, the limit behavior of the quantity x B α,ω p,q as α → k or α → 0 are object of a recent series of publications. This line of research was initiated by Bourgain, Brézis and Mironescu [13,14] who considered the case α → 1 (k = 1). Their work was later simplified and extended by Maz'ya and Shaposhnikova [41]. Here, we obtain the following analogue for T d θ of their results: For 1 ≤ p ≤ ∞, 1 ≤ q < ∞ and 0 < α < k with k ∈ N,
lim α→k (k − α) 1 q x B α,ω p,q ≈ q − 1 q m∈N d 0 , |m|1=k D m x p , lim α→0 α 1 q x B α,ω p,q ≈ q − 1 q x p
with relevant constants depending only on d and k.
Interpolation. Our fourth family of results deal with interpolation. Like in the usual case, the interpolation of Besov spaces is quite simple, and that of Triebel-Lizorkin spaces can be easily reduced to the corresponding problem of Hardy spaces. Thus the really hard task here concerns the interpolation of Sobolev spaces for which we have obtained only partial results. The most interesting couple is W k 1 (T d θ ), W k ∞ (T d θ ) . Recall that the complex interpolation problem of this couple remains always unsolved even in the commutative case (a well-known longstanding open problem which is explicitly posed by P. Jones in [27, p. 173]), while its real interpolation spaces were completely determined by DeVore and Scherer [21]. We do not know, unfortunately, how to prove the quantum analogue of DeVore and Scherer's theorem.
However, we are able to extend to the quantum tori the K-functional formula of the couple L p (R d ), W k p (R d ) obtained by Johnen and Scherer [30]. This result reads as follows:
K(x, ε k ; L p (T d θ ), W k p (T d θ )) ≈ ε k | x(0)| + ω k p (x, ε), 0 < ε ≤ 1. As a consequence, we determine the real interpolation spaces of L p (T d θ ), W k p (T d θ ) , which are Besov spaces.
The real interpolation of L p (R d ), W k p (R d ) is closely related to the limit behavior of Besov norms described previously. We show that it implies the optimal order (relative to α) of the best constant in the embedding of B α p,p (T d θ ) into L q (T d θ ) for 1 q = 1 p − α d and 0 < α < 1, which is the quantum analogue of a result of Bourgain, Brézis and Mironescu. On the other hand, the latter result is equivalent to the Sobolev embedding W 1
p (T d θ ) ⊂ L q (T d θ ) for 1 q = 1 p − 1 d . Multipliers.
The last family of results of the paper describe Fourier multipliers on the preceding spaces. Like in the L p case treated in [17], we are mainly concerned with completely bounded Fourier multipliers. All spaces in consideration carry a natural operator space structure in Pisier's sense. We show that the completely bounded Fourier multipliers on W k p (T d θ ) are independent of θ, so they coincide with those on the usual Sobolev space W k p (T d ). This is the Sobolev analogue of the corresponding result for L p proved in [17]. The main tool is Neuwirth-Ricard's transference between Fourier multipliers and Schur multipliers in [48]. A similar result holds for the Triebel-Lizorkin spaces too.
The situation for Besov spaces is very satisfactory since it is well known that Fourier multipliers behave much better on Besov spaces than on L p -spaces (in the commutative case). We prove that a function φ on Z d is a (completely) bounded Fourier multiplier on B α p,q (T d θ ) iff the φϕ(2 −k ·)'s are (completely) bounded Fourier multipliers on L p (T d θ ) uniformly in k ≥ 0. Consequently, the Fourier multipliers on B α p,q (T d θ ) are completely determined by the Fourier multipliers on L p (T d θ ) associated to their components in the Littlewood-Paley decomposition. So the completely bounded multipliers on B α p,q (T d θ ) depend solely on p. In the case of p = 1, a multiplier is bounded on B α 1,q (T d θ ) iff it is completely bounded iff it is the Fourier transform of an element of B 0 1,∞ (T d ). Using a classical example of Stein-Zygmund [69], we show that there exists a φ which is a completely bounded Fourier multiplier on B α p,q (T d θ ) for all p but bounded on L p (T d θ ) for no p = 2. We will frequently use the notation A B, which is an inequality up to a constant: A ≤ c B for some constant c > 0. The relevant constants in all such inequalities may depend on the dimension d, the test function ϕ or ψ, etc. but never on the functions f or distributions x in consideration. The main results of this paper have been announced in [81].
Chapter 1. Preliminaries
This chapter collects the necessary preliminaries for the whole paper. The first two sections present the definitions and some basic facts about noncommutative L p -spaces and quantum tori which are the central objects of the paper. The third one contains some results on Fourier multipliers that will play a paramount role in the whole paper. The last section gives the definitions and some fundamental results on operator-valued Hardy spaces on the usual and quantum tori. This section will be needed only starting from chapter 4 on Triebel-Lizorkin spaces.
Noncommutative L p -spaces
Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ and S + M be the set of all positive elements x in M with τ (s(x)) < ∞, where s(x) denotes the support of x, i.e., the smallest projection e such that exe = x. Let S M be the linear span of S + M . Then every x ∈ S M has finite trace, and S M is a w*-dense * -subalgebra of M.
Let 0 < p < ∞. For any x ∈ S M , the operator |x| p belongs to S + M (recalling |x| = (x * x) 1 2 ). We define
x p = τ (|x| p ) 1 p . One can check that · p is a norm on S M . The completion of (S M , · p ) is denoted by L p (M), which is the usual noncommutative L p -space associated to (M, τ ). For convenience, we set L ∞ (M) = M equipped with the operator norm · M . The norm of L p (M) will be often denoted simply by · p . But if different L p -spaces appear in a same context, we will sometimes precise their norms in order to avoid possible ambiguity. The reader is referred to [57] and [82] for more information on noncommutative L p -spaces.
The elements of L p (M) can be described as closed densely defined operators on H (H being the Hilbert space on which M acts). A closed densely defined operator x on H is said to be affiliated with M if ux = xu for any unitary u in the commutant M ′ of M. An operator x affiliated with M is said to be measurable with respect to (M, τ ) (or simply measurable) if for any δ > 0 there exists a projection e ∈ B(H) such that
e(H) ⊂ Dom(x) and τ (e ⊥ ) ≤ δ,
where Dom(x) defines the domain of x. We denote by L 0 (M, τ ), or simply L 0 (M) the family of all measurable operators. For such an operator x, we define
λ s (x) = τ (e ⊥ s (|x|)), s > 0 where e ⊥ s (x) = 1 (s,∞) (x)
is the spectrum projection of x corresponding to the interval (s, ∞), and µ t (x) = inf{s > 0 : λ s (x) < t}, t > 0.
The function s → λ s (x) is called the distribution function of x and the µ t (x) the generalized singular numbers of x. Similarly to the classical case, for 0 < p < ∞, 0 < q ≤ ∞, the noncommutative Lorentz space L p,q (M) is defined to be the collection of all measurable operators x such that
x p,q = ∞ 0 (t 1 p µ t (x)) q dt t 1 q < ∞. Clearly, L p,p (M) = L p (M). The space L p,∞ (M) is usually called a weak L p -space, 0 < p < ∞, and x p,∞ = sup s>0 sλ s (x) 1 p .
Like the classical L p -spaces, noncommutative L p -spaces behave well with respect to interpolation. Our reference for interpolation theory is [8]. Let 1 ≤ p 0 < p 1 ≤ ∞, 1 ≤ q ≤ ∞ and 0 < η < 1. Then
(1.1) L p0 (M), L p1 (M) η = L p (M) and L p0 (M), L p1 (M) η,q = L p,q (M), where 1 p = 1−η p0 + η p1
. Now we introduce noncommutative Hilbert space-valued L p -spaces L p (M; H c ) and L p (M; H r ), which are studied at length in [32]. Let H be a Hilbert space and v a norm one element of H. Let p v be the orthogonal projection onto the one-dimensional subspace generated by v. Then define the following row and column noncommutative L p -spaces:
L p (M; H r ) = (p v ⊗ 1 M )L p (B(H)⊗M), L p (M; H c ) = L p (B(H)⊗M)(p v ⊗ 1 M ),
where the tensor product B(H)⊗M is equipped with the tensor trace while B(H) is equipped with the usual trace. For f ∈ L p (M; H c ),
f Lp(M;H c ) = (f * f ) 1 2 Lp(M) .
A similar formula holds for the row space by passing to adjoints:
f ∈ L p (M; H r ) iff f * ∈ L p (M; H c ), and f Lp(M;H r ) = f * Lp(M;H c )
. It is clear that L p (M; H c ) and L p (M; H r ) are 1-complemented subspaces of L p (B(H)⊗M) for any p. Thus they also form interpolation scales with respect to both complex and real interpolation methods: Let 1 ≤ p 0 , p 1 ≤ ∞ and 0 < η < 1. Then
L p0 (M; H c ), L p1 (M; H c ) η = L p (M; H c ), L p0 (M; H c ), L p1 (M; H c ) η,p = L p (M; H c ), (1.2) where 1 p = 1−η p0 + η p1 .
The same formulas hold for row spaces too.
Quantum tori
Let d ≥ 2 and θ = (θ kj ) be a real skew symmetric d × d-matrix. The associated d-dimensional noncommutative torus A θ is the universal C * -algebra generated by d unitary operators U 1 , . . . , U d satisfying the following commutation relation
(1.3) U k U j = e 2πiθ kj U j U k , j, k = 1, . . . , d.
We will use standard notation from multiple Fourier series.
Let U = (U 1 , · · · , U d ). For m = (m 1 , · · · , m d ) ∈ Z d we define U m = U m1 1 · · · U m d d . A polynomial in U is a finite sum x = m∈Z d α m U m with α m ∈ C,
that is, α m = 0 for all but finite indices m ∈ Z d . The involution algebra P θ of all such polynomials is dense in A θ . For any polynomial x as above we define
τ (x) = α 0 ,
where 0 = (0, · · · , 0). Then, τ extends to a faithful tracial state on A θ . Let T d θ be the w * -closure of A θ in the GNS representation of τ . This is our d-dimensional quantum torus. The state τ extends to a normal faithful tracial state on T d θ that will be denoted again by τ . Recall that the von Neumann algebra T d θ is hyperfinite. Any x ∈ L 1 (T d θ ) admits a formal Fourier series:
x ∼ m∈Z d x(m)U m , where x(m) = τ ((U m ) * x), m ∈ Z d
are the Fourier coefficients of x. The operator x is, of course, uniquely determined by its Fourier series.
We introduced in [17] a transference method to overcome the full noncommutativity of quantum tori and use methods of operator-valued harmonic analysis. Let T d be the usual d-torus equipped with normalized Haar measure dz. Let N θ = L ∞ (T d )⊗T d θ , equipped with the tensor trace ν = dz ⊗ τ . It is well known that for every 0 < p < ∞,
L p (N θ , ν) ∼ = L p (T d ; L p (T d θ )).
The space on the right-hand side is the space of Bochner p-integrable functions from T d to L p (T d θ ). In general, for any Banach space X and any measure space (Ω, µ), we use L p (Ω; X) to denote the space of Bochner p-integrable functions from Ω to X. For each z ∈ T d , define π z to be the isomorphism of T d θ determined by
(1.4) π z (U m ) = z m U m = z m1 1 · · · z m d d U m1 1 · · · U m d d .
Since τ (π z (x)) = τ (x) for any x ∈ T d θ , π z preserves the trace τ. Thus for every 0 < p < ∞,
(1.5) π z (x) p = x p , ∀x ∈ L p (T d θ )
. Now we state the transference method as follows (see [17]). Lemma 1.1. For any x ∈ L p (T d θ ), the function x : z → π z (x) is continuous from T d to L p (T d θ ) (with respect to the w*-topology for p = ∞). If x ∈ A θ , it is continuous from T d to A θ .
Corollary 1.2. (i) Let 0 < p ≤ ∞. If x ∈ L p (T d θ )
, then x ∈ L p (N θ ) and x| p = x p , that is, x → x is an isometric embedding from L p (T d θ ) into L p (N θ ). Moreover, this map is also an isomorphism from A θ into C(T d ; A θ ).
(ii) Let T d θ = { x : x ∈ T d θ }.
Then T d θ is a von Neumann subalgebra of N θ and the associated conditional expectation is given by
E(f )(z) = π z T d π w f (w) dw , z ∈ T d , f ∈ N θ .
Moreover, E extends to a contractive projection from
L p (N θ ) onto L p ( T d θ ) for 1 ≤ p ≤ ∞. (iii) L p (T d θ ) is isometric to L p ( T d θ ) for every 0 < p ≤ ∞.
Fourier multipliers
Fourier multipliers will be the most important tool for the whole work. Now we present some known results on them for later use. Given a function φ :
Z d → C, let M φ denote the associated Fourier multiplier on T d , namely, M φ f (m) = φ(m) f (m) for any trigonometric polynomial f on T d . We call φ a multiplier on L p (T d ) if M φ extends to a bounded map on L p (T d ).
Fourier multipliers on T d θ are defined exactly in the same way, we still use the same symbol M φ to denote the corresponding multiplier on T d θ too. Note that the isomorphism π z defined in (1.4) is the Fourier multiplier associated to the function φ given by φ(m) = z m .
It is natural to ask if the boundedness of M φ on L p (T d ) is equivalent to that on L p (T d θ ). This is still an open problem. However, it is proved in [17] that the answer is affirmative if "boundedness" is replaced by "complete boundedness", a notion from operator space theory for which we refer to [23] and [55]. All noncommutative L p -spaces are equipped with their natural operator space structure introduced by Pisier [54,55].
We will use the following fundamental property of completely bounded (c.b. for short) maps due to Pisier [54]. Let E and F be operator spaces. Then a linear map T :
E → F is c.b. iff Id Sp ⊗ T : S p [E] → S p [F ] is bounded for some 1 ≤ p ≤ ∞. In this case, T cb = Id Sp ⊗ T : S p [E] → S p [F ] .
Here S p [E] denotes the E-valued Schatten p-class. In particular, if E = C, S p [C] = S p is the noncommutative L p -space associated to B(ℓ 2 ), equipped with the usual trace. Applying this criterion to the special case where E = F = L p (M), we see that a map T on L p (M) is c.b. iff Id Sp ⊗ T : L p (B(ℓ 2 )⊗M) → L p (B(ℓ 2 )⊗M) is bounded. The readers unfamiliar with operator space theory can take this property as the definition of c.b. maps between L p -spaces.
Thus φ is a c.b. multiplier on
L p (T d θ ) if M φ is c.b. on L p (T d θ ), or equivalently, if Id Sp ⊗ M φ is bounded on L p (B(ℓ 2 )⊗T d θ ). Let M(L p (T d θ )) (resp. M cb (L p (T d θ )
)) denote the space of Fourier multipliers (resp. c.b. Fourier multipliers) on L p (T d θ ), equipped with the natural norm. When θ = 0, we recover the (c.b.) Fourier multipliers on the usual d-torus T d . The corresponding multiplier spaces are denoted by M(L p (T d )) and M cb (L p (T d )). Note that in the latter case (θ = 0),
L p (B(ℓ 2 )⊗T d θ ) = L p (T d ; S p ), thus φ is a c.b. multiplier on L p (T d ) iff M φ extends to a bounded map on L p (T d ; S p ).
The following result is taken from [17].
Lemma 1.3. Let 1 ≤ p ≤ ∞. Then M cb (L p (T d θ )) = M cb (L p (T d )) with equal norms. Remark 1.4. Note that M cb (L 1 (T d )) = M cb (L ∞ (T d ))
coincides with the space of the Fourier transforms of bounded measures on T d , and M cb (L 2 (T d )) with the space of bounded functions on Z d .
The most efficient criterion for Fourier multipliers on L p (T d ) for 1 < p < ∞ is Mikhlin's condition. Although it can be formulated in the periodic case, it is more convenient to state this condition in the case of R d . On the other hand, the Fourier multipliers on T d used later will be the restrictions to Z d of continuous Fourier multipliers on R d . As usual, for m = (m 1 , · · · , m d ) ∈ N d 0 (recalling that N 0 denotes the set of nonnegative integers), we set
D m = ∂ m1 1 · · · ∂ m d d ,
where ∂ k denotes the kth partial derivation on R d . Also put |m| 1 = m 1 + · · · + m d . The Euclidean norm of R d is denoted by | · |: |ξ| = ξ 2 1 + · · · + ξ 2 d .
Definition 1.5. A function φ : R d → C is called a Mikhlin multiplier if it is d-times differentiable on R d \ {0}
and satisfies the following condition
φ M = sup |ξ| |m|1 |D m φ(ξ)| : ξ ∈ R d \ {0}, m ∈ N d 0 , |m| 1 ≤ d < ∞.
Note that the usual Mikhlin condition requires only partial derivatives up to order [ d 2 ] + 1 (see, for instance, [25,section II.6] or [66,Theorem 4.3.2]). Our requirement above up to order d is imposed by the boundedness of these multipliers on UMD spaces. We refer to section 4.1 for the usual Mikhlin condition and more multiplier results on T d θ . It is a classical result that every Mikhlin multiplier is a Fourier multiplier on L p (R d ) for 1 < p < ∞ (cf. [25,section II.6] or [66,Theorem 4.3.2]), so its restriction φ Z d is a Fourier multiplier on L p (T d ) too. It is, however, less classical that such a multiplier is also c.b. on L p (R d ) or L p (T d ). This follows from a general result on UMD spaces. Recall that a Banach space X is called a UMD space if the X-valued martingale differences are unconditional in L p (Ω; X) for any 1 < p < ∞ and any probability space (Ω, P ). This is equivalent to the requirement that the Hilbert transform be bounded on L p (R d ; X) for 1 < p < ∞. Any noncommutative L p -space with 1 < p < ∞ is a UMD space. We refer to [10,15,16] for more information.
Before proceeding further, we make a convention used throughout the paper: Convention. To simplify the notational system, we will use the same derivation symbols for R d and T d . Thus for a multi-index m ∈ N d 0 , D m = ∂ m1 1 · · · ∂ m d d , introduced previously, will also denote the partial derivation of order m on T d . Similarly, ∆ = ∂ 2 1 + · · · + ∂ 2 d will denote the Laplacian on both R d and T d . In the same spirit, for a function φ on R d , we will call φ rather than φ Z d a Fourier multiplier on L p (T d ) or L p (T d θ ). This should not cause any ambiguity in concrete contexts. Considered as a map on L p (T d ) or L p (T d θ ), M φ will be often denoted by f → φ * f or x → φ * x. Note that the notation φ * f coincides with the usual convolution when φ is good enough. Indeed, let φ be the 1-periodization of the inverse Fourier transform of φ whenever it exists in a reasonable sense:
φ(s) = m∈Z d F −1 (φ)(s + m), s ∈ R d .
Viewed as a function on T d , φ admits the following Fourier series:
φ(z) = m∈Z d φ(m)z m .
Thus for any trigonometric polynomial f ,
φ * f (z) = T d φ(zw −1 )f (w)dw, z ∈ T d .
The following lemma is proved in [43,84] (see also [12] for the one-dimensional case).
Lemma 1.6. Let X be a UMD space and 1 < p < ∞. Let φ be a Mikhlin multiplier. Then φ is a Fourier multiplier on L p (T d ; X). Moreover, its norm is controlled by φ M , p and the UMD constant of X.
Since S p is a UMD space for 1 < p < ∞, combining Lemmas 1.3 and 1.6 and Remark 1.4, we obtain the following result.
Lemma 1.7. Let φ be a function on R d . (i) If F −1 (φ) is integrable on R d , then φ is a c.b. Fourier multiplier on L p (T d θ ) for 1 ≤ p ≤ ∞. Moreover, its c.b. norm is not greater than F −1 (φ) 1 .
(ii) If φ is a Mikhlin multiplier, then φ is a c.b. Fourier multiplier on L p (T d θ ) for 1 < p < ∞. Moreover, its c.b. norm is controlled by φ M and p.
Hardy spaces
We now present some preliminaries on operator-valued Hardy spaces on T d and Hardy spaces on T d θ . Motivated by the developments of noncommutative martingale inequalities in [56,36] and quantum Markovian semigroups in [32], Mei [44] developed the theory of operator-valued Hardy spaces on R d . More recently, Mei's work was extended to the torus case in [17] with the objective of developing the Hardy space theory in the quantum torus setting. We now recall the definitions and results that will be needed later. Throughout this section, M will denote a von Neumann algebra equipped with a normal faithful tracial state τ and N = L ∞ (T d )⊗M with the tensor trace. In our future applications, M will be T d θ . A cube of T d is a product Q = I 1 × · · · × I d , where each I j is an interval (= arc) of T. As in the Euclidean case, we use |Q| to denote the normalized volume (= measure) of Q. The whole T d is now a cube too (of volume 1).
We will often identify T d with the unit cube I d = [0, 1) d via (e 2πis1 , · · · , e 2πis d ) ↔ (s 1 , · · · , s d ). Under this identification, the addition in I d is the usual addition modulo 1 coordinatewise; an interval of I is either a subinterval of I or a union [b, 1] ∪ [0, a] with 0 < a < b < 1, the latter union being the interval [b − 1, a] of I (modulo 1). So the cubes of I d are exactly those of T d . Accordingly, functions on T d and I d are identified too; they are considered as 1-periodic functions on R d . Thus
N = L ∞ (T d )⊗M = L ∞ (I d )⊗M.
We define BMO c (T d , M) to be the space of all f ∈ L 2 (N ) such that
f BMO c = max f T d M , sup Q⊂T d cube 1 |Q| Q f (z) − 1 |Q| Q f (w)dw 2 dz 1 2 M < ∞. The row BMO r (T d , M) consists of all f such that f * ∈ BMO c (T d , M), equipped with f BMO r = f * BMO c .
Finally, we define mixture space BMO(T d , M) as the intersection of the column and row BMO spaces:
BMO(T d , M) = BMO c (T d , M) ∩ BMO r (T d , M)
,
equipped with f BMO = max( f BMO c , f BMO r ).
As in the Euclidean case, these spaces can be characterized by the circular Poisson semigroup. Let P r denote the circular Poisson kernel of T d :
(1.6) P r (z) = m∈Z d r |m| z m , z ∈ T d , 0 ≤ r < 1.
The Poisson integral of f ∈ L 1 (N ) is
P r (f )(z) = T d P r (zw −1 )f (w)dw = m∈Z d f (m)r |m| z m .
Here f denotes, of course, the Fourier transform of f :
f (m) = T d f (z) z −m dz.
It is proved in [17] that
(1.7) sup Q⊂T d cube 1 |Q| Q f (z) − 1 |Q| Q f (w)dw 2 dz M ≈ sup 0≤r<1 P r (|f − P r (f )| 2 ) N
with relevant constants depending only on d. Thus
f BMO c ≈ max f (0) M , sup 0≤r<1 P r (|f − P r (f )| 2 ) 1 2 N .
Now we turn to the operator-valued Hardy spaces on T d which are defined by the Littlewood-Paley functions associated to the circular Poisson kernel. For f ∈ L 1 (N ) define
s c (f )(z) = 1 0 ∂ r P r (f )(z) 2 (1 − r)dr 1 2 , z ∈ T d . For 1 ≤ p < ∞, let H c p (T d , M) = {f ∈ L 1 (N ) : f H c p < ∞}, where f H c p = f (0) Lp(M) + s c (f ) Lp(N ) .
The row Hardy space H r p (T d , M) is defined to be the space of all f such that f * ∈ H c p (T d , M), equipped with the natural norm. Then we define
H p (T d , M) = H c p (T d , M) + H r p (T d , M) if 1 ≤ p < 2, H c p (T d , M) ∩ H r p (T d , M) if 2 ≤ p < ∞, equipped
with the sum and intersection norms, respectively:
f Hp = inf g H c p + h H r p : f = g + h if 1 ≤ p < 2, max f H c p , f H r p if 2 ≤ p < ∞.
The following is the main results of [17, Section 8]
Lemma 1.8. (i) Let 1 < p < ∞. Then H p (T d , M) = L p (N ) with equivalent norms. (ii) The dual space of H c 1 (T d , M) coincides isomorphically with BMO c (T d , M). (iii) Let 1 < p < ∞. Then (BMO c (T d , M), H c 1 (T d , M)) 1 p = H c p (T d , M) (BMO c (T d , M), H c 1 (T d , M)) 1 p ,p = H c p (T d , M)
. Similar statements hold for the row and mixture spaces too.
By transference, the previous results can be transferred to the quantum torus case. The Poisson integral of an element
x in L 1 (T d θ ) is defined by P r (x) = m∈Z d x(m)r |m| U m , 0 ≤ r < 1.
Its associated Littlewood-Paley g-function is
s c (x) = 1 0 ∂ r P r (x) 2 (1 − r)dr 1 2 . For 1 ≤ p < ∞ let x H c p = | x(0)| + s c (x) Lp(T d θ ) . The column Hardy space H c p (T d θ ) is then defined to be H c p (T d
The following lemma is the main result of [80]. We will need it essentially in the case of p = 1.
Lemma 1.10. Let 1 ≤ p < ∞, and let ψ be a Schwartz function that does not vanish in {ξ
: 1 ≤ |ξ| < 2}. Then x ∈ H c p (T d θ ) iff s c ψ (x) ∈ L p (T d θ )
. In this case, we have
x H c p ≈ | x(0)| + s c ψ (x) Lp(T d θ )
, where the equivalence constants depend only on d, p and ψ.
Chapter 2. Sobolev spaces
This chapter starts with a brief introduction to distributions on quantum tori. We then pass to the definitions of Sobolev spaces on T d θ and give some fundamental properties of them. Two families of Sobolev spaces are studied: W k p (T d θ ) and the fractional Sobolev spaces H α p (T d θ ). We prove a Poincaré type inequality for W k p (T d θ ) for any 1 ≤ p ≤ ∞. Our approach to this inequality seems very different from existing proofs for such an inequality in the classical case. We show that W k ∞ (T d θ ) coincides with the Lipschitz class of order k, studied by Weaver [77,78]. We conclude the chapter with a section on the link between the quantum Sobolev spaces and the vector-valued Sobolev spaces on the usual d-torus T d .
Distributions on quantum tori
In this section we give an outline of the distribution theory on quantum tori. Let
S(T d θ ) = m∈Z d a m U m : {a m } m∈Z d rapidly decreasing .
This is a w*-dense * -subalgebra of T d θ and contains all polynomials. We simply write S(T d 0 ) = S(T d ), the algebras of infinitely differentiable functions on T d . Thus for a general θ, S(T d θ ) should be viewed as a noncommutative deformation of S(T d ). We will need the differential structure on S(T d θ ), which is similar to that on S(T d ). According to our convention made in section 1.3 and in order to lighten the notational system, we will use the same derivation notation on T d θ as on T d . For every 1 ≤ j ≤ d, define the following derivations, which are operators on S(T d θ ): ∂ j (U j ) = 2πi U j and ∂ j (U k ) = 0 for k = j.
These operators ∂ j commute with the adjoint operation * , and play the role of the partial derivations in the classical analysis on the usual d-torus. Given m = (m 1 , . . . , m d ) ∈ N d 0 , the associated partial derivation D m is ∂ m1 1 · · · ∂ m d d . We also use ∆ to denote the Laplacian: ∆ = ∂ 2 1 + · · · + ∂ 2 d . The elementary fact expressed in the following remark will be frequently used later on.
Restricted to L 2 (T d θ ), the partial derivation ∂ j is a densely defined closed (unbounded) operator whose adjoint is equal to −∂ j . This is an immediate consequence of the following obvious fact (cf. [63]):
Lemma 2.1. If x, y ∈ S(T d θ ), then τ (∂ j (x)y) = −τ (x∂ j (y)) for j = 1, · · · , d. Thus ∆ = −(∂ * 1 ∂ 1 + · · · + ∂ * d ∂ d ), so −∆ is a positive operator on L 2 (T d θ ) with spectrum equal to {4π 2 |m| 2 : m ∈ Z d }.
Remark 2.2. Given u ∈ R d let e u be the function on R d defined by e u (ξ) = e 2πiu·ξ , where u · ξ denotes the inner product of R d . The Fourier multiplier on T d θ associated to e u coincides with π z in (1.4) with z = (e 2πiu1 , · · · e 2πiu d ). This Fourier multiplier will play an important role in the sequel. By analogy with the classical case, we will call it the translation by u and denote it by T u : T u (x) = π z (x) for any x ∈ S(T d θ ). Then it is clear that
(2.1) ∂ ∂u j T u (x) = T u (∂ j x) , so ∂ ∂u j T u (x) u=0 = ∂ j x.
Following the classical setting as in [22], we now endow S(T d θ ) with an appropriate topology. For each k ∈ N 0 define a norm p k on S(T d θ ) by
p k (x) = sup 0≤|m|1≤k D m x ∞ .
The sequence {p k } k≥0 induces a locally convex topology on S(T d θ ). This topology is metrizable by the following distance: Definition 2.6. Let 1 ≤ p ≤ ∞, k ∈ N and α ∈ R.
(i) The Sobolev space of order k on T d θ is defined to be
W k p (T d θ ) = x ∈ S ′ (T d θ ) : D m x ∈ L p (T d θ ) for each m ∈ N d 0 with |m| 1 ≤ k , equipped with the norm x W k p = 0≤|m|1≤k D m x p p 1 p .
(ii) The potential (or fractional) Sobolev space of order α is defined to be
H α p (T d θ ) = x ∈ S ′ (T d θ ) : J α x ∈ L p (T d θ ) , equipped with the norm x H α p = J α x p .
In the above definition of x W k p , we have followed the usual convention for p = ∞ that the right-hand side is replaced by the corresponding supremum. This convention will be always made in the sequel. We collect some basic properties of these spaces in the following:
Proposition 2.7. Let 1 ≤ p ≤ ∞, k ∈ N and α ∈ R. (i) W k p (T d θ ) and H α p (T d θ ) are Banach spaces. (ii) The polynomial subalgebra P θ of T d θ is dense in W k p (T d θ ) and H α p (T d θ ) for 1 ≤ p < ∞. Conse- quently, S(T d θ ) is dense in W k p (T d θ ) and H α p (T d θ ). (iii) For any β ∈ R, J β is an isometry from H α p (T d θ ) onto H α−β p (T d θ ). In particular, J α is an isometry from H α p (T d θ ) onto L p (T d θ ). (iv) H α p (T d θ ) ⊂ H β p (T d θ ) continuously whenever β < α. Proof. (iii) is obvious. It implies (i) for H α p (T d θ ). (i) It suffices to show that W k p (T d θ ) is complete. Assume that {x n } ⊂ W k p (T d θ )
is a Cauchy sequence. Then for every |m| 1 ≤ k, {D m x n } is a Cauchy sequence in L p (T d θ ), so D m x n → y m in L p (T d θ ). Particularly, {D m x n } n converges to y m in S ′ (T d θ ). On the other hand, since x n → y 0 in L p (T d θ ), for every x ∈ S(T d θ ) we have τ (x n D m x) → τ (y 0 D m x). Thus {D m x n } n converges to D m y 0 in S ′ (T d θ ). Consequently, D m y 0 = y m for |m| 1 ≤ k. Hence, y 0 ∈ W k p (T d θ ) and x n → y 0 in W k p (T d θ ). (ii) Consider the square Fejér mean
F N (x) = m∈Z d , maxj |mj|≤N 1 − |m 1 | N + 1 · · · 1 − |m d | N + 1 x(m)U m .
By [17,Proposition 3.1]
, lim N →∞ F N (x) = x in L p (T d θ ). On the other hand, F N commutes with D m : F N (D m x) = D m F N (x). We then deduce that lim N →∞ F N (x) = x in W k p (T d θ ) for every x ∈ W k p (T d θ ). Thus P θ is dense in W k p (T d θ )
. On the other hand, F N and J α commute; so by (iii), the density of P θ in L p (T d θ ) implies its density in H α p (T d θ ). (iv) It is well known that if γ < 0, the inverse Fourier transform of J γ is an integrable function on R d (see [66,Proposition V.3]). Thus, Lemma 1.7 implies that J β−α is a bounded map on L p (T d θ ) with norm majorized by F −1 (J β−α ) L1(R d ) . This is the desired assertion.
The following shows that the potential Sobolev spaces can be also characterized by the Riesz potential.
Theorem 2.8. Let 1 ≤ p ≤ ∞. Then x H α p ≈ | x(0)| p + I α (x − x(0)) p p 1 p ,
where the equivalence constants depend only on α and d.
Proof. By changing α to −α, we can assume α > 0. It suffices to show I α x p ≈ J α x p for x(0) = 0. By [66,Lemma V.3.2], Iα Jα is the Fourier transform of a bounded measure on R d , which, together with Lemma 1.7, yields I α x p J α x p .
To show the converse inequality, let η be an infinite differentiable function on R d such that η(ξ) = 0 for |ξ| ≤ 1 2 and η(ξ) = 1 for |ξ| ≥ 1, and let φ = J α I −α η. Then the Fourier multiplier with symbol φI α coincides with J α on the subspace of distributions on T d θ with vanishing Fourier coefficients at the origin. Thus we are reduced to proving F −1 (φ) ∈ L 1 (R d ). To that end, first observe that for any m ∈ N d 0 ,
D m φ(ξ) 1 |ξ| |m|1+2 .
Consider first the case d ≥ 3. Choose positive integers ℓ and k such that d 2 − 2 < ℓ < d 2 and k > d 2 . Then by the Cauchy-Schwarz inequality and the Plancherel theorem,
|s|<1 |F −1 φ(s)|ds 2 ≤ |s|<1 |s| −2ℓ ds |s|<1 |s| 2ℓ |F −1 φ(s)| 2 ds m∈N d 0 ,|m|1=ℓ R d |D m φ(ξ)| 2 dξ |ξ|≥ 1 2 1 |ξ| 2(ℓ+2) dξ 1.
On the other hand,
|s|≥1 |F −1 φ(s)|ds 2 ≤ |s|≥1 |s| −2k ds |s|≥1 |s| 2k |F −1 φ(s)| 2 ds m∈N d 0 ,|m|1=k R d |D m φ(ξ)| 2 dξ 1. Thus F −1 (φ) is integrable for d ≥ 3.
If d ≤ 2, the second part above remains valid, while the first one should be modified since the required positive integer ℓ does not exist for d ≤ 2. We will consider d = 2 and d = 1 separately. For d = 2, choosing 0 < ε < 1 2 , we have
|s|<1 |F −1 φ(s)|ds ≤ |s|<1 |s| −2ε ds 1 2 R d |s| 2ε |F −1 φ(s)| 2 ds 1 2 I ε φ 2 .
Writing I ε = I ε−1 I 1 and using the classical Hardy-Littlewood-Sobolev inequality (see [66, Theorem V.1]) and the Riesz transform, we obtain
I ε φ 2 I 1 φ q ≈ ∇φ q |ξ|≥ 1 2 1 |ξ| 3q dξ 1 q 1 , where 1 q = 1 − ε 2 (so 1 < q < ∞).
Thus we are done in the case d = 2. It remains to deal with the one-dimensional case. Write
φ(ξ) = (1 + ξ −2 ) α 2 η(ξ) = η(ξ) + ρ(ξ)η(ξ), ξ ∈ R \ {0},
where ρ(ξ) = O(ξ −2 ) as |ξ| → ∞. Since η − 1 is infinitely differentiable and supported by [−1, 1], its inverse Fourier transform is integrable. So η is the Fourier transform of a finite measure on R.
On the other hand, as ρη ∈ L 1 (R d ), F −1 (ρη) is a bounded continuous function, so it is integrable on [−1, 1]. On the other hand, by the second part of the preceding argument for d ≥ 3, we see that F −1 (ρη) is integrable outside [−1, 1] too, whence F −1 (ρη) ∈ L 1 (R d ). We thus deduce that φ is the Fourier transform of a finite measure on R. Hence the assertion is completely proved.
Theorem 2.9. Let 1 < p < ∞. Then H k p (T d θ ) = W k p (T d θ ) with equivalent norms.
Proof. This proof is based on Fourier multipliers by virtue of Lemma 1.7. For any m ∈ N d 0 with |m| 1 ≤ k, the function φ, defined by φ(ξ) = (2πi) |m|1 ξ m (1 + |ξ| 2 ) − k 2 , is clearly a Mikhlin multiplier. Then for any x ∈ S ′ (T d θ ),
D m x p = M φ (J k x) p J k x p , whence x W k p x H k p .
To prove the converse inequality, choose an infinite differentiable function χ on R such that χ = 0 on {ξ : |ξ| ≤ 4 −1 } and χ = 1 on {ξ : |ξ| ≥ 2 −1 }. Let
φ(ξ) = (1 + |ξ| 2 ) k 2 1 + χ(ξ 1 )|ξ 1 | k + · · · + χ(ξ d )|ξ d | k and φ j (ξ) = χ(ξ j )|ξ j | k (2πi ξ j ) k , 1 ≤ j ≤ d.
These are Mikhlin multipliers too, and
J k x = M φ (x + M φ1 ∂ k 1 x + · · · + M φ d ∂ k d x)
. It then follows that
x H k p x p p + d j=1 ∂ k j x p p 1 p ≤ x W k p .
The assertion is thus proved.
Remark 2.10. Incidentally, the above proof shows that if 1 < p < ∞, then
x W k p ≈ x p p + d j=1 ∂ k j x p p 1 p
with relevant constants depending only on p and d.
However, if one allows the above sum to run over all partial derivations of order k, then p can be equal to 1 or ∞. Namely, for any 1 ≤ p ≤ ∞,
x W k p ≈ x p p + m∈N d 0 , |m|1=k D m x p p 1 p
with relevant constants depending only on d. This equivalence can be proved by standard arguments (see Lemma 2.15 below and its proof). In fact, we have a nicer result, a Poincaré-type inequality:
x p d j=1 ∂ j x p
for any x ∈ W 1 p (T d θ ) with x(0) = 0. So x p can be removed from the right-hand side of the above equivalence. This inequality will be proved in the next section.
We conclude this section with an easy description of the dual space of W k p (T d θ ). Let N = N (d, k) = m∈N d 0 , 0≤|m|1≤k 1 and
L N p = N j=1 L p (T d θ ) equipped with the norm x L N p = N j=1 x j p p 1 p . The map x → (D m x) 0≤|m|1≤k establishes an isometry from W k p (T d θ ) into L N p . Therefore, the dual of W k p (T d θ ) with 1 ≤ p < ∞ is identified with a quotient of L N p ′ , where p ′ is the conjugate index of p. More precisely, for every ℓ ∈ (W k p (T d θ )) * there exists an element y = (y m ) m∈N d 0 , 0≤|m|1≤k ∈ L N p ′ such that (2.2) ℓ(x) = 0≤|m|1≤k τ (y m D m x), ∀x ∈ W k p (T d θ ),
and ℓ (W k p ) * = inf y L N p ′ , the infimum running over all y ∈ L N p ′ as above. (W k p (T d θ )) * can be described as a space of distributions. Indeed, let ℓ ∈ (W k p (T d θ )) * and y ∈ L N p ′ be a representative of ℓ as in (2.2). Define ℓ y ∈ S ′ (T d θ ) by
(2.3) ℓ y = 0≤|m|1≤k (−1) |m|1 D m y m .
Then
ℓ y (x) = 0≤|m|1≤k τ (y m D m x) = ℓ(x), x ∈ S(T d θ ).
So ℓ is an extension of ℓ y ; moreover,
ℓ (W k p ) * = min{ y L N p ′ : ℓ extends ℓ y given by (2.3)}. Conversely, suppose ℓ is an element of S ′ (T d θ ) of the above form ℓ y for some y ∈ L N p ′ . Then by the density of S(T d θ ) in W k p (T d θ )
, ℓ extends uniquely to a continuous functional on W k p (T d θ ). Thus we have proved the following
Proposition 2.11. Let 1 ≤ p < ∞ and W −k p ′ (T d θ )
be the space of those distributions ℓ which admit a representative ℓ y as above, equipped with the norm inf{ y L N p ′ : y as in
(2.3)}. Then (W k p (T d θ )) * is isometric to W −k p ′ (T d θ )
. Note that the duality problem for the potential Sobolev spaces is trivial. Since J α is an isometry
between H α p (T d θ ) and L p (T d θ ), we see that for 1 ≤ p < ∞ and α ∈ R, the dual space of H α p (T d θ ) coincides with H −α p ′ (T d θ ) isometrically.
A Poincaré-type inequality
For x ∈ W k p (T d θ ) let |x| W k p = m∈N d 0 , |m|1=k D m x p p 1 p . Theorem 2.12. Let 1 ≤ p ≤ ∞. Then for any x ∈ W 1 p (T d θ ), x − x(0) p |x| W 1 p . More generally, if k ∈ N and x ∈ W k p (T d θ ) with x(0) = 0, then |x| W j p |x| W k p , ∀ 0 ≤ j < k. Consequently, | x(0)| + |x| W k p is an equivalent norm on W k p (T d θ )
. The proof given below is quite different from standard approaches to the Poincaré inequality. We will divide it into several lemmas, each of which might be interesting in its own right. We start with the following definition which will be frequently used later. Note that the function e u and the translation operator T u have been defined in Remark 2.1.
Definition 2.13. Given u ∈ R d let d u = e u − 1. The Fourier multiplier on T d θ defined by d u is called the difference operator by u and denoted by ∆ u .
Remark 2.14. Note that e u is the Fourier transform of the Dirac measure δ u at u. Thus T u is an isometry and ∆ u is of norm 2 on L p (T d θ ) for any 1 ≤ p ≤ ∞.
Lemma 2.15. Let 1 ≤ p ≤ ∞, and j, k ∈ N with j < k. Then for any x ∈ W k p (T d θ ), |x| W j p x 1− j k p |x| j k W k p . Proof. Fix x ∈ W k p (T d θ ) with x(0) = 0. For any u, ξ ∈ R d we have d u (ξ) − ∂ ∂r d ru (ξ) r=0 = 1 0 ∂ ∂r d ru (ξ) − ∂ ∂r d ru (ξ) r=0 dr = 1 0 r 0 ∂ 2 ∂s 2 d su (ξ) ds dr. Since ∂ ∂r d ru (ξ) = e ru (ξ)(2πiu · ξ)
and ∂ 2 ∂s 2 d su (ξ) = e su (ξ)(2πiu · ξ) 2 , letting u = te j with t > 0 and e j the jth canonical basic vector of R d , we deduce
∆ u x − t∂ j x = 1 0 r 0 T su (t 2 ∂ 2 j x)ds dr. Thus t ∂ j x p ≤ ∆ u x p + t 2 1 0 r 0 T su (∂ 2 j x) p ds dr ≤ 2 x p + t 2 2 ∂ 2 j x p .
Dividing by t and taking the infimum over all t > 0, we get
(2.4) ∂ j x p ≤ 2 x p ∂ 2 j x p .
This gives the assertion for the case j = 1 and k = 2. An iteration argument yields the general case.
Lemma 2.16. Let j ∈ {1, · · · , d} and x ∈ W 2 p (T d θ ) such that m j = 0 whenever x(m) = 0 for m ∈ Z d . Then x p ≤ c ∂ 2 j x p , where c is a universal constant. More generally, for any x ∈ W 2 p (T d θ ) with x(0) = 0 x p ≤ c d j=1 ∂ 2 j x p . Proof. Assume j = 1. Define φ : Z → R by φ(m 1 ) = 1 m 2 1 for m 1 ∈ Z \ {0} and φ(m 1 ) = 0 for m 1 = 0.
We also view φ as a function on Z d , independent of (m 2 , · · · , m d ). Then the inequality to prove amounts to showing that φ is a bounded Fourier multiplier on L p (T d θ ) for any 1 ≤ p ≤ ∞. This is easy. Indeed, let ψ : R → R be the 2π-periodic even function determined by
ψ(s) = (π − s) 2 2 − π 2 6 for s ∈ [0, π).
Then ψ = φ and ψ L1(T) = 2π 2 9 √ 3 .
Thus by Lemma 1.3, φ is a bounded Fourier multiplier on L p (T d θ ) with norm 2π 2 9 √ 3 , which proves the first inequality of the lemma.
The second one is an immediate consequence of the first. Indeed, let E U1,··· ,U d−1 be the trace preserving conditional expectation from T d θ onto the subalgebra generated by (U 1 , · · · , U d−1 ). Let
x ′ = E U1,··· ,U d−1 (x) and x d = x − x ′ . Then m d = 0 whenever x d (m) = 0 for m ∈ Z d . Thus x d p ≤ c ∂ 2 d x d p = c ∂ 2 d x p . Since x ′ depends only on (U 1 , · · · , U d−1 )
, an induction argument then yields the desired inequality.
Lemma 2.17. The sequence {|x| W k p } k≥1 is increasing, up to constants. More precisely, there exists a constant c d,k such that |x| W k p ≤ c d,k |x| W k+1 p , ∀k ≥ 1.
Proof. The proof is done easily by induction with the help of the previous two lemmas. Indeed, we have (assuming
x(0) = 0) |x| W 1 p x p |x| W 2 p |x| W 2 p .
Thus the assertion is proved for k = 1. Then induction gives the general case.
Proof of Theorem 2.12. By the preceding lemma, it remains to show x p |x| W 1 p for any x ∈ W 1 p (T d θ ) with x(0) = 0. By approximation, we can assume that x is a polynomial. We proceed by induction on the dimension d. Consider first the case d = 1. Then
x = m1∈Z\{0} x(m 1 )U m1 1 . Define y = m1∈Z\{0} 1 2πim 1 x(m 1 )U m1 1 .
Then
∂ 1 y = x and ∂ 2 1 y = ∂ 1 x. Thus Lemma 2.16 implies x p ∂ 1 x p . Now consider a polynomial x in (U 1 , · · · , U d ). As in the proof of Lemma 2.16, let x ′ = E U1,··· ,U d−1 (x) and x d = x − x ′ . The induction hypothesis implies x ′ p |x ′ | W 1 p = d−1 j=1 E U1,··· ,U d−1 (∂ j x) p p 1 p ≤ |x| W 1 p ,
where we have used the commutation between E U1,··· ,U d−1 and the partial derivations.
To handle the term x d , recalling that m d = 0 whenever x d (m) = 0 for m ∈ Z d , we introduce
y d = m∈Z d 1 2πim d x d (m)U m .
Then ∂ d y d = x d . So by (2.4) and Lemma 2.16,
x d p y d p ∂ 2 d y d p ∂ 2 d y d p = ∂ d x d p .
Thus we are done, so the theorem is proved.
Lipschitz classes
In this section we show that W k ∞ (T d θ ) is the quantum analogue of the classical Lipschitz class of order k. We will use the translation and difference operators introduced in Remark 2.1 and Definition 2.13. Note that for any positive integer k, T k u = T ku and ∆ k u is the kth difference operator by u ∈ R d . Definition 2.18. Let k be a positive integer and x ∈ L p (T d θ ) with 1 ≤ p ≤ ∞. The kth order modulus of L p -smoothness of x is defined by
ω k p (x, ε) = sup 0<|u|≤ε ∆ k u x p . Remark 2.19. It is clear that ω k p (x, ε) ≤ 2 k x p and ω k p (x, ε) is nondecreasing in ε. On the other hand, ω 1 p (x, ε) is subadditive in ε; for k ≥ 2, ω k p (x, ε) is quasi subadditive in the sense that there exists a constant c = c k such that ω k p (x, ε + η) ≤ c (ω k p (x, ε) + ω k p (x, η)).
The following is the main result of this section. It shows that W k ∞ (T d θ ) is the Lipschitz class of order k. We thus recover a result of Weaver [77,78] for k = 1.
Theorem 2.20. For any x ∈ W k p (T d θ ), we have sup ε>0 ω k p (x, ε) ε k ≈ |x| W k p ,
where the equivalence constants depend only on d and k.
We require the following lemma for the proof.
Lemma 2.21. For any x ∈ L p (T d θ ), lim ε→0 ω k p (x, ε) ε k = sup 0<ε≤1 ω k p (x, ε) ε k .
Proof. The assertion for k = 1 is a common property of increasing and subadditive functions (in ε), and easy to check. Indeed, for any 0 < ε, δ < 1, choose n ∈ N such that nδ ≤ ε < (n + 1)δ.
Then
ω 1 p (x, ε) ε ≤ n + 1 n ω 1 p (x, δ) δ ,
whence the result for k = 1. The general case is treated in the same way. Instead of being subadditive, ω k p (x, ε) is quasi subadditive in the sense that ω k p (x, nε) ≤ n k ω k p (x, ε) for any n ∈ N. The latter follows immediately from
d k nu = n−1 j=0 e ju k d k u , so ∆ k nu = n−1 j=0 T ju k ∆ k u .
Thus the lemma is proved.
Proof of Theorem 2.20. If the assertion is proved for all p < ∞ with constants independent of p, the case p = ∞ will follow by letting p → ∞. So we will assume p < ∞.
We first consider the case k = 1 whose proof contains all main ideas. As in the proof of Lemma 2.15, for any u ∈ R d , we have
d u (ξ) = 1 0 ∂ ∂t d tu (ξ)dt = 1 0 e tu (ξ) (2πiu · ξ)dt, ξ ∈ R d .
In terms of Fourier multipliers, this yields
∆ u x = 1 0 T tu (u · ∇x)dt, where u · ∇x = u 1 ∂ 1 x + · · · + u d ∂ d x.
Since the translation T tu is isometric, it then follows that
(2.5) ∆ u x p ≤ |u| |∂ 1 x| 2 + · · · + |∂ d x| 2 1 2 p def = |u| ∇x p , whence lim ε→0 ω 1 p (x, ε) ε ≤ ∇x p .
To show the converse inequality, by the density of
P θ in W k p (T d θ ) (see Proposition 2.7), we can assume that x is a polynomial. Given u ∈ R d define φ on R d by φ(ξ) = d u (ξ) − ∂ ∂t d tu (ξ) t=0 .
Then the Fourier multiplier on T d θ associated to φ is
φ * x = ∆ u x − u · ∇x. Thus if |u| = ε, u · ∇x p ≤ ω p (x, ε) + sup |u|=ε ∆ u x − u · ∇x p . Since x is a polynomial, lim ε→0 sup |u|=ε ∆ u x − u · ∇x p ε = 0.
For u = (ε, 0, · · · , 0), we then deduce
∂ 1 x p ≤ lim ε→0 ω p (x, ε) ε .
Hence the desired assertion for k = 1 is proved. Now we consider the case k > 1. (2.5) can be easily iterated as follows:
∆ k u x p ≤ |u| d j=1 ∂ j ∆ k−1 u x p = |u| d j=1 ∆ k−1 u ∂ j x p ≤ |u| k |m|1=k ∆ m x p ≈ |u| k |x| W k p . So sup ε>0 ω k p (x, ε) ε k |x| W k p .
The converse inequality is proved similarly to the case k = 1. Let m ∈ N d 0 with |m| 1 = k. For each j with m j > 0, using the Taylor formula of the function d εej (recalling that (e 1 , · · · , e d ) denotes the canonical basis of R d ), we get
∆ mj εej x = ε mj ∂ mj j x + o(ε mj ) , which implies d j=0 ∆ mj εej x = ε k D m x + o(ε k ) as ε → 0.
Thus by the next lemma, we deduce
D m x p ≤ ε −k d j=0 ∆ mj εej x p + o(1) ε −k ω k p (x, ε) + o(1),
whence the desired converse inequality by letting ε → 0. So the theorem is proved modulo the next lemma.
Lemma 2.22. Let u 1 , · · · , u k ∈ R d . Then ∆ u1 · · · ∆ u k = D⊂{1,··· ,k} (−1) |D| T uD ∆ k uD ,
where the sum runs over all subsets of {1, · · · , k}, and where
u D = j∈D u j , u D = j∈D 1 j u j .
Consequently, for ε > 0 and
x ∈ L p (T d θ ), sup |u1|≤ε,··· ,|u k |≤ε ∆ u1 · · · ∆ u k x p ≈ ω k p (x, ε).
Proof. This is a well-known lemma in the classical setting (see [6,Lemma 5.4.11]). We outline its proof for the convenience of the reader. The above equality is equivalent to the corresponding one with ∆ u and T u replaced by d u and e u , respectively. Setting v = ℓ∈D ℓu ℓ and w = − ℓ∈D u ℓ ,
for each 0 ≤ j ≤ k, we have k ℓ=1 d (ℓ−j)u ℓ = D⊂{1,··· ,k} (−1) k−|D| ℓ∈D e (ℓ−j)u ℓ = D⊂{1,··· ,k} (−1) k−|D| e v (e w ) j .
The left hand side is nonzero only for j = 0. Multiply by (−1) k−j k j and sum over 0 ≤ j ≤ k; then replacing u ℓ by u ℓ ℓ gives the desired equality.
Remark 2.23. It might be interesting to note that in the commutative case, the proof of Theorem 2.20 shows
sup 0<ε≤1 ω p (x, ε) ε = lim ε→0 ω p (x, ε) ε = ∇x p .
So Lemma 2.21 is not needed in this case.
The link with the classical Sobolev spaces
The transference enables us to establish a strong link between the quantum Sobolev spaces defined previously and the vector-valued Sobolev spaces on T d . Note that the theory of vectorvalued Sobolev spaces is well-known and can be found in many books on the subject (see, for instance, [2]). Here we just recall some basic notions. In the sequel, X will always denote a (complex) Banach space.
Let S(T d ; X) be the space of X-valued infinitely differentiable functions on T d with the standard Fréchet topology, and S ′ (T d ; X) be the space of continuous linear maps from S(T d ) to X. All operations on S(T d ) such as derivations, convolution and Fourier transform transfer to S ′ (T d ; X) in the usual way. On the other hand,
L p (T d ; X) naturally embeds into S ′ (T d ; X) for 1 ≤ p ≤ ∞, where L p (T d ; X) stands for the space of strongly p-integrable functions from T d to X. Definition 2.24. Let 1 ≤ p ≤ ∞, k ∈ N and α ∈ R. (i) The X-valued Sobolev space of order k is W k p (T d ; X) = f ∈ S ′ (T d ; X) : D m f ∈ L p (T d ; X) for each m ∈ N d 0 with |m| 1 ≤ k equipped with the norm f W k p = 0≤|m|1≤k D m f p Lp(T d ;X) 1 p . (ii) The X-valued potential Sobolev space of order α is H α p (T d ; X) = f ∈ S ′ (T d ; X) : J α f ∈ L p (T d ; X) equipped with the norm f H α p = J α f Lp(T d ;X) . Remark 2.25.
There exists a parallel theory of vector-valued Sobolev spaces on R d . In fact, a majority of the literature on the subject is devoted to the case of R d which is simpler from the pointview of treatment. The corresponding spaces are W k p (R d ; X) and H α p (R d ; X). They are subspaces of S ′ (R d ; X). The latter is the space of X-valued distributions on R d , that is, the space of continuous linear maps from S(R d ) to X. We will sometimes use the space
S(R d ; X) of X-valued Schwartz functions on R d . We set W k p (R d ) = W k p (R d ; C) and H k p (R d ) = H k p (R d ; C)
. The properties of the Sobolev spaces on T d θ in the previous sections also hold for the present setting. For instance, the proof of Proposition 2.9 and Lemma 1.6 give the following well-known result:
Remark 2.26. Let X be a UMD space. Then W k p (T d ; X) = H k p (T d ; X) with equivalent norms for 1 < p < ∞.
Let us also mention that Theorem 2.12, the Poincaré inequality, transfers to the vector-valued case too. It seems that this result does not appear in literature but it should be known to experts. We record it explicitly here.
Theorem 2.27. Let X be a Banach space and 1 ≤ p ≤ ∞, k ∈ N. Then f (0) p X + m∈N d 0 , |m|1=k D m f p Lp(T d ;X) 1 p
is an equivalent norm on W k p (T d ; X) with relevant constants depending only on d and k. Now we use the transference method in Corollary 1.2. It is clear that the map x → x there commutes with ∂ j , that is, ∂ j x = ∂ j x (noting that the ∂ j on the left-hand side is the jth partial derivation on T d and that on the right-hand side is the one on T d θ ). On the other hand, the expectation in that corollary commutes with ∂ j too. We then deduce the following:
Proposition 2.28. Let 1 ≤ p ≤ ∞. The map x → x is an isometric embedding from W k p (T d θ ) and H α p (T d θ ) into W k p (T d ; L p (T d θ )
) and H α p (T d ; L p (T d θ )), respectively. Moreover, the ranges of these embeddings are 1-complemented in their respective spaces.
This result allows us to reduce many problems about W k p (T d θ ) to the corresponding ones about W k p (T d ; L p (T d θ )). For example, we could deduce the properties of W k p (T d θ ) in the preceding sections from those of W k p (T d ; L p (T d θ )). But we have chosen to work directly on T d θ for the following reasons. It is more desirable to develop an intrinsic quantum theory, so we work directly on T d θ whenever possible. On the other hand, the existing literature on vector-valued Sobolev spaces often concerns the case of R d , for instance, there exist few publications on periodic Besov or Triebel-Lizorkin spaces. In order to use existing results, we have to transfer them from R d to T d . However, although it is often easy, this transfer may not be obvious at all, which is the case for Hardy spaces treated in [17] and recalled in section 1.4. This difficulty will appear again later for Besov and Triebel-Lizorkin spaces.
Remark 2.29. The preceding discussion on vector-valued Sobolev spaces on T d can be also transferred to the quantum case. We have seen in section 1.3 that all noncommutative L p -spaces are equipped with their natural operator space structure. Thus W k p (T d θ ) and H α p (T d θ ) becomes operator spaces in the natural way. More generally, given an operator space E, following Pisier [54],
we define the E-valued noncommutative L p -space L p (T d θ ; E) (recalling that T d θ is an injective von Neumann algebra). Similarly, we define the E-valued distribution space S ′ (T d θ ; E) that(T d θ ; E) = W k p (T d θ ; E)
for any 1 < p < ∞ and any OUMD space E (OUMD is the operator space version of UMD; see [54]). Note that we recover
W k p (T d ; E) and H α p (T d ; E) if θ = 0 and if E is equipped with its minimal operator space structure.
Chapter 3. Besov spaces
We study Besov spaces on T d θ in this chapter. The first section presents the relevant definitions and some basic properties of these spaces. The second one shows a general characterization of them. This is the most technical part of the chapter. The formulation of our characterization is very similar to Triebel's classical theorem. Although modeled on Triebel's pattern, our proof is harder than his. The main difficulty is due to the unavailability in the noncommutative setting of the usual techniques involving maximal functions which play an important role in the study of the classical Besov and Triebel-Lizorkin spaces. Like for the Sobolev spaces in the previous chapter, Fourier multipliers are our main tool. We then concretize this general characterization by means of Poisson, heat kernels and differences. We would like to point out that when θ = 0 (the commutative case), these characterizations (except that by differences) improve the corresponding ones in the classical case. Using the characterization by differences, we extend a recent result of Bourgain, Brézis and Mironescu on the limits of Besov norms to the quantum setting. The chapter ends with a section on vector-valued Besov spaces on T d .
Definitions and basic properties
We will use Littlewood-Paley decompositions as in the classical theory. Let ϕ be a Schwartz function on R d such that
(3.1) supp ϕ ⊂ {ξ : 2 −1 ≤ |ξ| ≤ 2}, ϕ > 0 on {ξ : 2 −1 < |ξ| < 2}, k∈Z ϕ(2 −k ξ) = 1, ξ = 0. Note that if m ∈ Z d with m = 0, ϕ(2 −k m) = 0 for all k < 0, so k≥0 ϕ(2 −k m) = 1, m ∈ Z d \ {0} .
On the other hand, the support of the function ϕ(2 −k ·) is equal to {ξ :
2 k−1 ≤ |ξ| ≤ 2 k+1 }, thus supp ϕ(2 −k ·) ∩ supp ϕ(2 −j ·) = ∅ whenver |j − k| ≥ 2; consequently, (3.2) ϕ(2 −k ·) = ϕ(2 −k ·) k+1 j=k−1 ϕ(2 −j ·), k ≥ 0. Therefore, the sequence {ϕ(2 −k ·)} k≥0 is a Littlewood-Paley decomposition of T d , modulo constant functions. The Fourier multiplier on S ′ (T d θ ) with symbol ϕ(2 −k ·) is denoted by x → ϕ k * x: ϕ k * x = m∈Z d ϕ(2 −k m) x(m)U m .
As noted in section 1.3, the convolution here has the usual sense. Indeed, let ϕ k be the function whose Fourier transform is equal to ϕ(2 −k ·), and let ϕ k be its 1-periodization, that is,
ϕ k (s) = m∈Z d ϕ k (s + m).
Viewed as a function on T d by our convention, ϕ k admits the following Fourier series:
ϕ k (z) = m∈Z d ϕ(2 −k m)z m .
Thus for any distribution f on T d ,
ϕ k * f (z) = m∈Z d ϕ(2 −k m) f (m)z m .
We will maintain the above notation throughout the remainder of the paper.
We can now start our study of quantum Besov spaces.
Definition 3.1. Let 1 ≤ p, q ≤ ∞ and α ∈ R. The associated Besov space on T d θ is defined by B α p,q (T d θ ) = x ∈ S ′ (T d θ ) : x B α p,q < ∞ , where x B α p,q = | x(0)| q + k≥0 2 qkα ϕ k * x q p 1 q . Let B α p,c0 (T d θ ) be the subspace of B α p,∞ (T d θ )
consisting of all x such that 2 kα ϕ k * x p → 0 as k → ∞. Remark 3.2. The Besov spaces defined above are independent of the choice of the function ϕ, up to equivalent norms. More generally, let {ψ (k) } k≥0 be a sequence of Schwartz functions such that
supp ψ (k) ⊂ {ξ : 2 k−1 ≤ |ξ| ≤ 2 k+1 }, sup k≥0 F −1 (ψ (k) ) 1 < ∞, k≥0 ψ (k) (m) = 1, ∀ m ∈ Z d \ {0}.
Let ψ k = F −1 (ψ (k) ) and ψ k be the periodization of ψ k . Then
x B α p,q ≈ | x(0)| q + k≥0 2 qkα ψ k * x q p 1 q .
Let us justify this remark. By the discussion leading to (3.2), we have (with ϕ −1 = 0)
ψ k * x = k+1 j=k−1 ψ k * ϕ j * x.
By Lemma 1.7,
ψ k * x p k+1 j=k−1 ϕ j * x p .
It then follows that
| x(0)| q + k≥0 2 qkα ψ k * x q p 1 q | x(0)| q + k≥0 2 qkα ϕ k * x q p 1 q .
The reverse inequality is proved similarly.
Proposition 3.3. Let 1 ≤ p, q ≤ ∞ and α ∈ R. (i) B α p,q (T d θ ) is a Banach space. (ii) B α p,q (T d θ ) ⊂ B α p,r (T d θ ) for r > q and B α p,q (T d θ ) ⊂ B β p,r (T d θ ) for β < α and 1 ≤ r ≤ ∞. (iii) P θ is dense in B α p,q (T d θ ) and B α p,c0 (T d θ ) for 1 ≤ p ≤ ∞ and 1 ≤ q < ∞. (iv) The dual space of B α p,q (T d θ ) coincides isomorphically with B −α p ′ ,q ′ (T d θ ) for 1 ≤ p ≤ ∞ and 1 ≤ q < ∞, where p ′ denotes the conjugate index of p. Moreover, the dual space of B α p,c0 (T d θ ) coincides isomorphically with B −α p ′ ,1 (T d θ ). Proof. (i) To show the completeness of B α p,q (T d θ ), let {x n } be a Cauchy sequence in B α p,q (T d θ ). Then { x n (0)} converges to some y(0) in C, and for every k ≥ 0, { ϕ k * x n } converges to some y k in L p (T d θ ). Let y = y(0) + k≥0 y k . Since y k is supported in {m ∈ Z d : 2 k−1 < |m| < 2 k+1 }, the above series converges in S ′ (T d θ )
. On the other hand, by (3.2), as n → ∞, we have
ϕ k * x n = k+1 j=k−1 ϕ k * ϕ j * x n → k+1 j=k−1 ϕ k * y j = ϕ k * y. We then deduce that y ∈ B α p,q (T d θ ) and x n → y in B α p,q (T d θ ). (ii) is obvious. (iii) We only show the density of P θ in B α p,q (T d θ ) for finite q. For N ∈ N let x N = x(0) + N j=0 ϕ j * x. Then by (3.1), ϕ k * (x − x N ) = 0 for k ≤ N − 1, ϕ k * (x − x N ) = ϕ k * x for k > N + 1, and ϕ N * (x − x N ) = ϕ N * x − ϕ N * ϕ N * x, ϕ N +1 * (x − x N ) = ϕ N +1 * x − ϕ N +1 * ϕ N * x. We then deduce x − x N q B α p,q ≤ 2 k≥N 2 qkα ϕ k * x q p → 0 as N → ∞. (iv) In this part, we view B α p,q (T d θ ) as B α p,c0 (T d θ ) when q = ∞. Let y ∈ B −α p ′ ,q ′ (T d θ ). Define ℓ y (x) = τ (xy) for x ∈ P θ . Then |ℓ y (x)| = x(0) y(0) + k≥0 τ ϕ k * x k+1 j=k−1 ϕ j * y ≤ | x(0) y(0)| + k≥0 ϕ k * x p k+1 j=k−1 ϕ j * y p ′ x B α p,q y B −α p ′ ,q ′ . Thus by the density of P θ in B α p,q (T d θ ), ℓ y defines a continuous functional on B α p,q (T d θ )
. To prove the converse, we need a more notation. Given a Banach space X, let ℓ α q (X) be the weighted direct sum of (C, X, X, · · · ) in the ℓ q -sense, that is, this is the space of all sequences (a, x 0 , x 1 , · · · ) with a ∈ C and x k ∈ X, equipped with the norm
|a| q + k≥0 2 qkα x k q 1 q .
If q = ∞, we replace ℓ α q (X) by its subspace c α 0 (X) consisting of sequences (a, x 0 , x 1 , · · · ) such that
2 kα x k → 0 as k → ∞. Recall that the dual space of ℓ α q (X) is ℓ −α q ′ (X * ). By definition, B α p,q (T d θ ) embeds into ℓ α q (L p (T d θ )) via x → ( x(0), ϕ 0 * x, ϕ 1 * x, · · · )
. Now let ℓ be a continuous functional on B α p,q (T d θ ) for p < ∞. Then by the Hahn-Banach theorem, ℓ extends to a continuous functional on ℓ α q (L p (T d θ )) of unit norm, so there exists a unit element (b, y 0 , y 1 , · · · ) belonging to ℓ −α
q ′ (L p ′ (T d θ )) such that ℓ(x) = b x(0) + k≥0 τ (y k ϕ k * x), x ∈ B α p,q (T d θ ). Let y = b + k≥0 ( ϕ k−1 * y k + ϕ k * y k + ϕ k+1 * y k ) .
Then clearly y ∈ B −α p ′ ,q ′ (T d θ ) and ℓ = ℓ y when p < ∞. The same argument works for p = ∞ too. Indeed, for ℓ as above, there exists a unit element (b, y 0 ,
y 1 , · · · ) belonging to ℓ −α q ′ (L ∞ (T d θ ) * ) such that ℓ(x) = b x(0) + k≥0 y k , ϕ k * x , x ∈ B α p,q (T d θ ).
Let y be defined as above. Then y is still a distribution and
| y(0)| q ′ + k≥0 2 q ′ kα ϕ k * y q ′ L∞(T d θ ) * 1 q ′ < ∞ .
Since it is a polynomial, ϕ k * y belongs to L 1 (T d θ ); and we have
ϕ k * y L∞(T d θ ) * = ϕ k * y L1(T d θ ) . Thus we are done for p = ∞ too.
To proceed further, we require some elementary lemmas. Recall that J α (ξ) = (1 + |ξ| 2 ) α 2 and I α (ξ) = |ξ| α . Lemma 3.4. Let α ∈ R and k ∈ N 0 . Then
F −1 (J α ϕ k ) 1 2 αk and F −1 (I α ϕ k ) 1 2 αk .
where the constants depend only on ϕ, α and d.
Consequently, for x ∈ L p (T d θ ) with 1 ≤ p ≤ ∞, J α ( ϕ k * x) p 2 αk ϕ k * x p and I α ( ϕ k * x) p 2 αk ϕ k * x p .
Proof. The first part is well-known and easy to check. Indeed,
F −1 (J α ϕ k ) 1 = 2 αk F −1 ((4 −k + | · | 2 ) α 2 ϕ) 1 ;
the function (4 −k + | · | 2 ) α 2 ϕ is a Schwartz function supported by {ξ : 2 −1 ≤ |ξ| ≤ 2}, whose all partial derivatives, up to a fixed order, are bounded uniformly in k, so
sup k≥0 F −1 ((4 −k + | · | 2 ) α 2 ϕ) 1 < ∞. Similarly, F −1 (I α ϕ k ) 1 = 2 αk F −1 (I α ϕ) 1 .
Since ϕ k = ϕ k (ϕ k−1 + ϕ k + ϕ k+1 ), by Lemma 1.7, we obtain the second part.
Given a ∈ R + , we define D i,a (ξ) = (2πiξ i ) a for ξ ∈ R d , and D a i to be the associated Fourier multiplier on T d θ . We set D a = D 1,a1 · · · D d,a d and D a = D a1 1 · · · D a d d for any a = (a 1 , · · · , a d ) ∈ R d + . Note that if a is a positive integer, D a i = ∂ a i , so there does not exist any conflict of notation. The following lemma is well-known. We include a sketch of proof for the reader's convenience (see the proof of Remark 1 in Section 2.4.1 of [73]). Lemma 3.5. Let ρ be a compactly supported infinitely differentiable function on R d . Assume σ, β ∈ R + and a ∈ R d + such that σ > d 2 , β > σ − d 2 and |a| 1 > σ − d 2 . Then the functions I β ρ and D a ρ belong to H σ 2 (R d ).
Proof. If σ is a positive integer, the assertion clearly holds in view of H σ
2 (R d ) = W σ 2 (R d ). On the other hand, I β ρ ∈ L 2 (R d ) = H 0 2 (R d ) for β > − d 2 .
The general case follows by complex interpolation. Indeed, under the assumption on σ and β, we can choose σ 1 ∈ N, β 1 , β 0 ∈ R and η ∈ (0 , 1) such that
σ 1 > σ, β 1 > σ 1 − d 2 , β 0 > − d 2 , σ = ησ 1 , β = (1 − η)β 0 + ηβ 1 . For a complex number z in the strip {z ∈ C : 0 ≤ Re(z) ≤ 1} define F z (ξ) = e (z−η) 2 |ξ| β0(1−z)+β1z ρ(ξ). Then sup b∈R F ib L2 I β0 ρ L2 and sup b∈R F 1+ib H σ 1 2 I β1 ρ H σ 1 2 .
It thus follows that
I β ρ = F η ∈ (L 2 (R d ), H σ1 2 (R d )) η .
The second assertion is proved in the same way.
The usefulness of the previous lemma relies upon the following well-known fact.
Remark 3.6. Let σ > d 2 and f ∈ H σ 2 (R d ). Then F −1 (f ) 1 f H σ 2 .
The verification is extremely easy:
F −1 (f ) 1 = |s|≤1 F −1 (f )(s) ds + k≥0 2 k <|s|≤2 k+1 F −1 (f )(s) ds |s|≤1 F −1 (f )(s) 2 ds + k≥0 2 2kσ 2 k <|s|≤2 k+1 F −1 (f )(s) 2 ds 1 2 ≈ f H σ 2 .
The following is the so-called reduction (or lifting) theorem of Besov spaces.
Theorem 3.7. Let 1 ≤ p, q ≤ ∞, α ∈ R. (i) For any β ∈ R, both J β and I β are isomorphisms between B α p,q (T d θ ) and B α−β p,q (T d θ ). (ii) Let a ∈ R d + . If x ∈ B α p,q (T d θ ), then D a x ∈ B α−|a|1 p,q (T d θ ) and D a x B α−|a| 1 p,q x B α p,q . (iii) Let β > 0. Then x ∈ B α p,q (T d θ ) iff D β i x ∈ B α−β p,q (T d θ ) for all i = 1, · · · , d. Moreover, in this case, x B α p,q ≈ | x(0)| + d i=1 D β i x B α−β p,q . Proof. (i) Let x ∈ B s p,q (T d θ ) with x(0) = 0. Then by Lemma 3.4, J β x B α−β p,q = k≥0 2 k(α−β) J β ( ϕ k * x) p q 1 q k≥0 2 kα ϕ k * x p q 1 q = x B α p,q . Thus J β is bounded from B α p,q (T d θ ) to B α−β p,q (T d θ )
, its inverse, which is J −β , is bounded too. The case of I β is treated similarly.
(ii) By Lemma 3.5 and Remark 3.6, we have
F −1 (D a ϕ k ) 1 = 2 k|a|1 F −1 (D a ϕ) 1 2 k|a|1 .
Consequently, by Lemma 1.7,
ϕ k * D a x p 2 k|a|1 ϕ k * x p , ∀j ≥ 0, whence D a x B α−|a| 1 p,q x B α p,q . (iii) One implication is contained in (ii). To show the other, choose an infinitely differentiable function χ : R → R + such that χ(s) = 0 if |s| < 1 4 √ d and χ(s) = 1 if |s| ≥ 1 2 √ d . For i = 1, · · · , d, let χ i on R d be defined by χ i (ξ) = 1 χ(ξ 1 )|ξ 1 | β + · · · + χ(ξ d )|ξ d | β χ(ξ i )|ξ i | β (2πiξ i ) β
whenever the first denominator is positive, which is the case when |ξ| ≥ 2 −1 . Then for any k ≥ 0,
χ i ϕ k is a well-defined infinitely differentiable function on R d \ {ξ : ξ i = 0}. We have F −1 (χ i ϕ k ) 1 = 2 −kβ F −1 (ψϕ) 1 , where ψ(ξ) = 1 χ(2 k ξ 1 )|ξ 1 | β + · · · + χ(2 k ξ d )|ξ d | β χ(2 k ξ i )|ξ i | β (2πiξ i ) β . The function ψϕ is supported in {ξ : 2 −1 ≤ |ξ| ≤ 2}.
An inspection reveals that all its partial derivatives of order less than a fixed integer are bounded uniformly in k. It then follows that the
L 1 -norm of F −1 (ψϕ) is majorized by a constant independent of k, so F −1 (χ i ϕ k ) 1 2 −kβ ,
and by Lemma 1.7,
χ i * ϕ k * D β i x p 2 −kβ ϕ k * D β i x p . Since ϕ k = d i=1 χ i D i,β ϕ k , we deduce ϕ k * x 2 −kβ d i=1 ϕ k * D β i x p ,
which implies
x B α p,q | x(0)| + d i=1 D β i x B α−β p,q .
Thus (iii) is proved.
The following result relates the Besov and potential Sobolev spaces.
Theorem 3.8. Let 1 ≤ p ≤ ∞ and α ∈ R d . Then we have the following continuous inclusions:
B α p,min(p,2) (T d θ ) ⊂ H α p (T d θ ) ⊂ B α p,max(p,2) (T d θ ).
Proof. By Propositions 2.7 and 3.7, we can assume α = 0. In this case,
H a p (T d θ ) = L p (T d θ ). Let x be a distribution on T d θ with x(0) = 0. Since x = k≥0 ϕ k * x, we see that the inclusion B 0 p,1 (T d θ ) ⊂ L p (T d θ )
follows immediately from triangular inequality. On the other hand, the inequality
ϕ k * x p x p , k ≥ 0 yields the inclusion L p (T d θ ) ⊂ B 0 p,∞ (T d θ )
. Both inclusions can be improved in the range 1 < p < ∞. Let us consider only the case 2 ≤ p < ∞. Then the inclusion L p (T d θ ) ⊂ B 0 p,p (T d θ ) can be easily proved by interpolation. Indeed, the two spaces coincide isometrically when p = 2. The other extreme case p = ∞ has been already proved. We then deduce the case 2 < p < ∞ by complex interpolation and by embedding B 0
p,∞ (T d θ ) into ℓ ∞ (L p (T d θ )). The converse inclusion B 0 p,2 (T d θ ) ⊂ L p (T d θ ) is subtler.
To show it, we use Hardy spaces and the equality L p (T d θ ) = H p (T d θ ) (see Lemma 1.9). Then we must show
max( x H c p , x H r p ) x B 0 p,2 .
To this end, we appeal to Lemma 1.10. The function ψ there is now equal to ϕ. The associated square function of x is thus given by
s c ϕ (x) = k≥0 | ϕ k * x| 2 1 2 .
Recall the following well-known inequality
k≥0 |x k | 2 1 2 p ≤ k≥0 x k 2 p 1 2 for x k ∈ L p (T d θ ) and 2 ≤ p ≤ ∞.
Note that this inequality is proved simply by the triangular inequality in L p 2 (T d θ ). Thus
x H c p ≈ s c ϕ (x) p ≤ k≥0 ϕ k * x 2 p 1 2 = x B 0 p,2 .
Passing to adjoints, we get x H r p x B 0 p,2 . Therefore, the desired inequality follows.
A general characterization
In this and next sections we extend some characterizations of the classical Besov spaces to the quantum setting. Our treatment follows Triebel [73] rather closely.
We give a general characterization in this section. We have observed in the previous section that the definition of the Besov spaces is independent of the choice of ϕ satisfying (3.1). We now show that ϕ can be replaced by more general functions. To state the required conditions, we introduce an auxiliary Schwartz function h such that
(3.3) supp h ⊂ {ξ ∈ R d : |ξ| ≤ 4} and h = 1 on {ξ ∈ R d : |ξ| ≤ 2}. Let α 0 , α 1 ∈ R. Let ψ be an infinitely differentiable function on R d \ {0} satisfying the following conditions (3.4) |ψ| > 0 on {ξ : 2 −1 ≤ |ξ| ≤ 2}, F −1 (ψhI −α1 ) ∈ L 1 (R d ), sup j∈N0 2 −α0j F −1 (ψ(2 j ·)ϕ) 1 < ∞.
The first nonvanishing condition above on ψ is a Tauberian condition. The integrability of the inverse Fourier transforms can be reduced to a handier criterion by means of the potential Sobolev space H σ 2 (R d ) with σ > d 2 ; see Remark 3.6. We will use the same notation for ψ as for ϕ. In particular, ψ k is the inverse Fourier transform of ψ(2 −k ·) and ψ k is the Fourier multiplier on T d θ with symbol ψ(2 −k ·). It is to note that compared with [73, Theorem 2.5.1], we need not require α 1 > 0 in the following theorem. This will have interesting consequences in the next section.
Theorem 3.9. Let 1 ≤ p, q ≤ ∞ and α ∈ R. Assume α 0 < α < α 1 . Let ψ satisfy the above assumption. Then a distribution x on T d θ belongs to B α p,q (T d θ ) iff k≥0 2 kα ψ k * x p q 1 q < ∞.
If this is the case, then
(3.5) x B α p,q ≈ | x(0)| q + k≥0 2 kα ψ k * x p q 1 q
with relevant constants depending only on ϕ, ψ, α, α 0 , α 1 and d.
Proof. We will follow the pattern of the proof of [73,Theorem 2.4.1]. Given a function f on R d , we will use the notation that f (k) = f (2 −k ·) for k ≥ 0 and f (k) = 0 for k ≤ −1. Recall that f k is the inverse Fourier transform of f (k) and f k is the 1-periodization of f k :
f k (s) = m∈Z d f k (s + m).
In the following, we will fix a distribution x on T d θ . Without loss of generality, we assume x(0) = 0. We will denote the right-hand side of (3.5) by x B α,ψ p,q when it is finite. For clarity, we divide the proof into several steps.
Step 1. In the first two steps, we assume x ∈ B α p,q (T d θ ). Let K be a positive integer to be determined later in step 3. By (3.1), we have
ψ (j) = ∞ k=0 ψ (j) ϕ (k) = K k=−∞ ψ (j) ϕ (j+k) + ∞ k=K ψ (j) ϕ (j+k) on {ξ : |ξ| ≥ 1}. Then (3.6) ψ j * x = k≤K ψ j * ϕ j+k * x + k>K ψ j * ϕ j+k * x .
For the moment, we do not care about the convergence issue of the second series above, which is postponed to the last step. Let a j,k = 2 jα ψ j * ϕ j+k * x p . Then
(3.7) x B α,ψ p,q ≤ ∞ j=0 k≤K a j,k q 1 q + ∞ j=0 k>K a j,k q 1 q .
We will treat the two sums on the right-hand side separately. For the first one, by the support assumption on ϕ and h, for k ≤ K, we can write
ψ (j) (ξ)ϕ (j+k) (ξ) = 2 kα1 ψ (j) (ξ) |2 −j ξ| α1 h (j+K) (ξ)|2 −j−k ξ| α1 ϕ (j+k) (ξ) = 2 kα1 η (j) (ξ)ρ (j+k) (ξ),(3.8)
where η and ρ are defined by
η(ξ) = ψ(ξ) |ξ| α1 h (K) (ξ) and ρ(ξ) = |ξ| α1 ϕ(ξ). Note that F −1 (η) is integrable on R d . Indeed, write (3.9) η(ξ) = ψ(ξ) |ξ| α1 h(ξ) + ψ(ξ) |ξ| α1 (h (K) (ξ) − h(ξ)).
By (3.4), the inverse Fourier transform of the first function on the right-hand side is integrable. The second one is an infinitely differentiable function with compact support, so its inverse Fourier transform is also integrable with L 1 -norm controlled by a constant depending only on ψ, h, α 1 and K. Therefore, Lemma 1.7 implies that each η (j) is a Fourier multiplier on L p (T d θ ) for all 1 ≤ p ≤ ∞ with norm controlled by a constant c 1 , depending only ψ, h, α 1 and K. Therefore,
(3.10) a j,k ≤ c 1 2 jα+kα1 ρ j+k * x p = c 1 2 k(α1−α) 2 (j+k)α ρ j+k * x p .
Thus by triangular inequality and Lemma 3.4, we deduce ∞ j=0 k≤K
a j,k q 1 q ≤ c 1 k≤K 2 k(α1−α) ∞ j=−∞ 2 (j+k)α ρ j+k * x p q 1 q = c 1 k≤K 2 k(α1−α) ∞ j=0 2 jα 2 −jα1 I α1 ϕ j * x p q 1 q ≤ c ′ 1 x B α p,q , where c ′
1 depends only ψ, h, K, α and α 1 .
Step 2. The second sum on the right-hand side of (3.7) is treated similarly. Like in step 1 and by (3.2), we write
ψ (j) (ξ)ϕ (j+k) (ξ) = ψ (j) (ξ) |2 −j−k ξ| α0 (ϕ (j+k−1) + ϕ (j+k) + ϕ (j+k+1) )(ξ)|2 −j−k ξ| α0 ϕ (j+k) (ξ) = ψ(2 −j−k · 2 k ξ) |2 −j−k ξ| α0 H(2 −j−k ξ) ρ (j+k) (ξ),(3.11)
where H = ϕ (−1) + ϕ + ϕ (1) , and where ρ is now defined by ρ(ξ) = |ξ| α0 ϕ(ξ).
The L 1 -norm of the inverse Fourier transform of the function
ψ(2 −j−k · 2 k ξ) |2 −j−k ξ| α0 H(2 −j−k ξ)
is equal to F −1 (I −α0 Hψ(2 k ·)) 1 . Using Lemma 3.4, we see that the last norm is majorized by
F −1 (ψ(2 k ·)H) 1 F −1 (ψ(2 k ·)ϕ) 1 .
Then, using (3.4), for k > K we get
(3.12) a j,k ≤ c 2 2 k(α0−α) 2 (j+k)α ρ j+k * x p ,
where c 2 depends only on ϕ, α 0 and the supremum in (3.4). Thus as before, we get ∞ j=0 k>K a j,k
q 1 q ≤ c ′ 2 2 K(α0−α) 1 − 2 α0−α x B α p,q ,
which, together with the inequality obtained in step 1, yields
x B α,ψ p,q x B α p,q .
Step 3. Now we prove the inequality reverse to the previous one. We first assume that x is a polynomial. We write
(3.13) ϕ (j) = ϕ (j) h (j+K) = ϕ (j) ψ (j) h (j+K) ψ (j) .
The function ϕψ −1 is an infinitely differentiable function with compact support, so its inverse Fourier transform belongs to L 1 (R d ). Thus by Lemma 1.7,
ϕ j * x p ≤ c 3 h j+K * ψ j * x p , where c 3 = F −1 (ϕψ −1 ) 1 . Hence, x B α p,q ≤ c 3 ∞ j=0 2 jα h j+K * ψ j * x p q 1 q .
To handle the right-hand side, we let λ = 1 − h and write h (j+K) ψ (j) = ψ (j) − λ (j+K) ψ (j) . Then
∞ j=0 2 jα h j+K * ψ j * x p q 1 q ≤ x B α,ψ p,q + ∞ j=0 2 jα λ j+K * ψ j * x p q 1 q .
Thus it remains to deal with the last sum. We do this as in the previous steps with ψ replaced by λψ, by writing
λ (j+K) ψ (j) = ∞ k=−∞ λ (j+K) ψ (j) ϕ (j+k) .
The crucial point now is the fact that λ (j+K) ϕ (j+k) = 0 for all k ≤ K and all j. So
λ (j+K) ψ (j) = k>K λ (j+K) ψ (j) ϕ (j+k) ,
that is, only the second sum on the right-hand side of (3.7) survives now:
∞ j=0 (2 jα λ j+K * ψ j * x p ) q 1 q ≤ ∞ j=0 2 jα k>K λ j+K * ψ j * ϕ j+k * x p q 1 q .
Let us reexamine the argument of step 2 and formulate (3.11) with λ (K) ψ instead of ψ. We then arrive at majorizing the norm F −1 λ(2 k−K ·)ψ(2 k ·)ϕ 1 :
F −1 λ(2 k−K ·)ψ(2 k ·)ϕ 1 ≤ F −1 (λ) 1 F −1 ψ(2 k ·)ϕ 1 .
Keeping the notation of step 2 and as for (3.12), we get
λ j+K * ψ j * ϕ j+k * x p ≤ cc 2 2 k(α0−α) 2 (j+k)α ρ j+k * x p , where c = F −1 (λ) 1 . Thus ∞ j=0 2 αj k>K λ j+K * ψ j * ϕ j+k * x p q 1 q ≤ cc ′ 2 2 K(α0−α) 1 − 2 α0−α x B α p,q .
Combining the preceding inequalities, we obtain
x B α p,q ≤ c 3 x B α,ψ p,q + cc ′ 2 2 K(α0−α) 1 − 2 α0−α x B α p,q . Choosing K so that c c ′ 2 2 K(α0−α) 1 − 2 α0−α ≤ 1 2 ,
we finally deduce x B α p,q ≤ 2c 3 x B α,ψ p,q , which shows (3.5) in case x is a polynomial.
The general case can be easily reduced to this special one. Indeed, assume x B α,ψ p,q < ∞. Then using the Fejér means F N as in the proof of Proposition 2.7, we see that
F N (x) B α,ψ p,q ≤ x B α,ψ p,q .
Applying the above part already proved for polynomials yields
F N (x) B α p,q ≤ 2c 3 F N (x) B α,ψ p,q ≤ 2c 3 x B α,ψ p,q , However, it is easy to check that lim N →∞ F N (x) B α p,q = x B α p,q .
We thus deduce (3.5) for general x, modulo the convergence problem on the second series of (3.6).
Step 4. We now settle up the convergence issue left previously. Each term ψ j * ϕ j+k * x is a polynomial, so a distribution on T d θ . We must show that the series converges in S ′ (T d θ ). By (3.12), for any L > K, by the Hölder inequality (with q ′ the conjugate index of q), we get
2 jα L k=K+1 ψ j * ϕ j+k * x p ≤ c ′ 2 L k=K+1 2 k(α0−α) 2 (j+k)α ϕ j+k * x p ≤ c ′ 2 R K,L L k=K+1 2 (j+k)α ϕ j+k * x p q 1 q c ′ 2 R K,L x B α p,q , where R K,L = L k=K+1 2 q ′ k(α0−α) 1 q ′ .
Since α 0 < α, R K,L → 0 as K tends to ∞. Thus the series k>K ψ j * ϕ j+k * x converges in L p (T d θ ), so in S ′ (T d θ ) too. In the same way, we show that the series also converges in B α p,q (T d θ ). Hence, the proof of the theorem is complete.
Remark 3.10. The infinite differentiability of ψ can be substantially relaxed without changing the proof. Indeed, we have used this condition only once to insure that the inverse Fourier transform of the second term on the right-hand side of (3.9) is integrable. This integrability is guaranteed when ψ is continuously differentiable up to order [ d 2 ] + 1. The latter condition can be replaced by the following slightly weaker one: there exists σ > d 2 + 1 such that ψη ∈ H σ 2 (R d ) for any compactly supported infinite differentiable function η which vanishes in a neighborhood of the origin.
The following is the continuous version of Theorem 3.9. We will use similar notation for continuous parameters: given ε > 0, ψ ε denotes the function with Fourier transform ψ (ε) = ψ(ε·), and ψ ε denotes the Fourier multiplier on T d θ associated to ψ (ε) .
Theorem 3.11. Keep the assumption of the previous theorem. Then for any distribution x on T d θ ,
(3.14) x B α p,q ≈ | x(0)| q + 1 0 ε −qα ψ ε * x q p dε ε 1 q .
The above equivalence is understood in the sense that if one side is finite, so is the other, and the two are then equivalent with constants independent of x.
Proof. This proof is very similar to the previous one. Keeping the notation there, we will point out only the necessary changes. Let us first discretize the integral on the right-hand side of (3.14):
1 0 ε −α ψ ε * x p q dε ε ≈ ∞ j=0 2 jqα 2 −j 2 −j−1 ψ ε * x q p dε ε .
Now for j ≥ 0 and 2 −j−1 < ε ≤ 2 −j , we transfer (3.8) to the present setting:
ψ (ε) (ξ)ϕ (j+k) (ξ) = 2 α1k ψ(2 −j · 2 j εξ) |2 −j ξ| α1 h (j+K) (ξ) ρ (j+k) (ξ).
We then must estimate the L 1 -norm of the inverse Fourier transform of the function in the brackets. It is equal to
F −1 I −α1 ψ(2 j ε·)h (K) 1 = δ −α1 F −1 I −α1 ψh(δ2 −K ·) 1 , where δ = 2 −j ε −1 .
The last norm is estimated as follows:
F −1 I −α1 ψh(δ2 −K ·) 1 ≤ F −1 I −α1 ψh 1 + F −1 I −α1 ψ [h − h(δ2 −K ·)] 1 ≤ F −1 I −α1 ψh 1 + sup 1≤δ≤2 F −1 I −α1 ψ [h − h(δ2 −K ·)] 1 .
Note that the above supremum is finite since
I −α1 ψ[h− h(δ2 −K ·)]
is a compactly supported infinite differentiable function and its inverse Fourier transform depends continuously on δ. It follows that for k ≤ K and 2
−j−1 ≤ ε ≤ 2 −j (3.15) 2 jα ψ ε * ϕ j+k * x p 2 k(α1−α) 2 (j+k)α ρ j+k * x p ,
which is the analogue of (3.10). Thus, we get the continuous analogue of the final inequality of step 1 in the preceding proof. We can make similar modifications in step 2, and then show the second part. Hence, we have proved
1 0 ε −α ψ ε * x p q dε ε 1 q x B α p,q .
To show the converse inequality, we proceed as in step 3 above. By (3.4), there exists a constant a > 2 such that ψ > 0 on {ξ : a −1 ≤ |ξ| ≤ a}. Assume also a ≤ 2 √ 2. For j ≥ 0 let R j = (a −1 2 −j−1 , a2 −j+1 ]. The R j 's are disjoint subintervals of (0, 1]. Now we slightly modify (3.13) as follows:
(3.16) ϕ (j) = ϕ (j) h (j+K) = ϕ (j) ψ (ε) h (j+K) ψ (ε) , ε ∈ R j .
Then
F −1 ϕ (j) ψ (ε) 1 = F −1 ϕ(2 −j ε −1 ·) ψ 1 ≤ sup 2a −1 ≤δ≤a2 −1 F −1 ϕ (δ) ψ 1 < ∞.
Like in step 3, we deduce
x B α p,q ∞ j=0 2 jα Rj h j+K * ψ ε * x p q dε ε 1 q 1 0 ε −α ψ ε * x p q dε ε 1 q + ∞ j=0 2 jα Rj h j+K * ψ ε * x p q dε ε 1 q .
The remaining of the proof follows step 3 with necessary modifications as in the first part.
(T d θ ) iff lim ε→0 ψ ε * x p ε α = 0.
This easily follows from Theorem 3.11 for q = ∞. The same remark applies to the characterizations by the Poisson, heat semigroups and differences in the next two sections.
The characterizations by Poisson and heat semigroups
We now concretize the general characterization in the previous section to the case of Poisson and heat kernels. We begin with the Poisson case. Recall that P denotes the Poisson kernel of R d and
P ε (x) = P ε * x = m∈Z d e −2πε|m| x(m)U m .
So for any positive integer k, the kth derivation relative to ε is given by
∂ k ∂ε k P ε (x) = m∈Z d (−2π|m|) k e −2πε|m| x(m)U m .
The inverse of the kth derivation is the kth integration I k defined for x with x(0) = 0 by
I k ε P ε (x) = ∞ ε ∞ ε k · · · ∞ ε2 P ε1 (x)dε 1 · · · dε k−1 dε k = m∈Z d \{0} (2π|m|) −k e −2πε|m| x(m)U m .
In order to simplify the presentation, for any k ∈ Z, we define
J k ε = ∂ k ∂ε k for k ≥ 0 and J k ε = I −k ε for k < 0.
It is worth to point out that all concrete characterizations in this section in terms of integration operators are new even in the classical case. Also, compare the following theorem with [73, Section 2.6.4], in which k is assumed to be a positive integer in the Poisson characterization, and a nonnegative integer in the heat characterization.
Theorem 3.13. Let 1 ≤ p, q ≤ ∞, α ∈ R and k ∈ Z such that k > α. Then for any distribution x on T d θ , we have
x B α p,q ≈ | x(0)| q + 1 0 ε q(k−α) J k ε P ε (x) q p dε ε 1 q .
Proof. Recall that P = P 1 . Thanks to P(ξ) = e −2π|ξ| , we introduce the function ψ(ξ) = (−sgn(k)2π|ξ|) k e −2π|ξ| .
Then
ψ(εξ) = ε k J k ε e −2πε|ξ| = ε k J k ε P ε (ξ). It follows that for x ∈ B α p,q (T d θ ), ψ ε * x = ε k J k ε P ε * x = ε k J k ε P ε (x)
. Thus by Theorem 3.11, it remains to check that ψ satisfies (3.4) for some α 0 < α < α 1 . It is clear that the last condition there is verified for any α 0 . For the second one, choosing k = α 1 > α, we have I −α1 h ψ = (−sgn(k)2π) k h P. So
F −1 I k−α1 h ψ 1 ≤ (2π) k F −1 (h) 1 P 1 < ∞ .
The theorem is thus proved.
There exists an analogous characterization in terms of the heat kernel. Let W ε be the heat semigroup of R d :
W ε (s) = 1 (4πε) d 2 e − |s| 2 4ε .
As usual, let W ε be the periodization of W ε . Given a distribution x on T d θ let
W ε (x) = W ε * x = m∈Z d W( √ ε m) x(m)U m , where W = W 1 . Recall that W(ξ) = e −4π 2 |ξ| 2 .
Theorem 3.14. Let 1 ≤ p, q ≤ ∞, α ∈ R and k ∈ Z such that k > α 2 . Then for any distribution
x on T d θ , x B α p,q ≈ | x(0)| q + 1 0 ε q(k− α 2 ) J k ε W ε (x) q p dε ε 1 q .
Proof. This proof is similar to and simpler than the previous one. This time, the function ψ is defined by ψ(ξ) = (−sgn(k)4π 2 |ξ| 2 ) k e −4π 2 |ξ| 2 . Clearly, it satisfies (3.4) for 2k = α 1 > α and any α 0 < α. Thus Theorem 3.11 holds for this choice of ψ. Note that a simple change of variables shows that the integral in (3.14) is equal to
2 − 1 q 1 0 ε − αq 2 ψ √ ε * x q p dε ε 1 q .
Then using the identity ψ( √ ε ξ) = ε k J k ε W ε (ξ), we obtain the desired assertion. Now we wish to formulate Theorems 3.13 and 3.14 directly in terms of the circular Poisson and heat semigroups of T d . Recall that P r denote the circular Poisson kernel of T d introduced by (1.6) and the Poisson integral of a distribution x on T d θ is defined by
P r (x) = m∈Z d x(m)r |m| U m , 0 ≤ r < 1.
Accordingly, we introduce the circular heat kernel W of T d :
(3.17) W r (z) = m∈Z d r |m| 2 z m , z ∈ T d , 0 ≤ r < 1.
Then for x ∈ S ′ (T d θ ) we put W r (x) = m∈Z d
x(m)r |m| 2 U m , 0 ≤ r < 1.
As before, J k r denotes the kth derivation ∂ k ∂r k if k ≥ 0 and the (−k)th integration I −k r if k < 0:
J k r P r (x) = m∈Z d C m,k x(m)r |m|−k U m , where C m,k = |m| · · · (|m| − k + 1) if k ≥ 0 and C m,k = 1 (|m| + 1) · · · (|m| − k) if k < 0. J k r W r (x)
is defined similarly. Since |m| is not necessarily an integer, the coefficient C m,k may not vanish for |m| < k and k ≥ 2. In this case, the corresponding term in J k r P r (x) above cause a certain problem of integrability since r (|m|−k)q is integrable on (0, 1) only when (|m| − k)q > −1. This explains why we will remove all these terms from J k r P r (x) in the following theorem. However, this difficulty does not occur for the heat semigroup.
The following is new even in the classical case, that is, in the case of θ = 0.
Theorem 3.15. Let 1 ≤ p, q ≤ ∞, α ∈ R and k ∈ Z. Let x be a distribution on T d θ . (i) If k > α, then
x B α p,q ≈ max |m|<k | x(m)| q + 1 0 (1 − r) q(k−α) J k r P r (x k ) q p dr 1 − r 1 q , where x k = x − |m|<k x(m)U m . (ii) If k > α 2 , then x B α p,q ≈ max |m| 2 <k | x(m)| q + 1 0 (1 − r) q(k− α 2 ) J k r W r (x) q p dr 1 − r 1 q .
Proof. We consider only the case of the Poisson kernel. Fix x ∈ B α p,q (T d θ ) with x(0) = 0. We first claim that for any 0 < ε 0 < 1,
1 0 ε q(k−α) J k ε P ε (x) q p dε ε 1 q ≈ ε0 0 ε q(k−α) J k ε P ε (x) q p dε ε 1 q .
Indeed, since
J k ε P ε (x) = m∈Z d \{0} (−sgn(k)2π|m|) k e −2πε|m| x(m)U m , we have ε0 0 ε q(k−α) J k ε P ε (x) q p dε ε 1 q ≥ sup m∈Z d \{0} (2π|m|) k | x(m)| ε0 0 ε q(k−α) e −2πε|m|q dε ε 1 q sup m∈Z d \{0} |m| α | x(m)| .
On the other hand, by triangular inequality,
1 ε0 ε q(k−α) J k ε P ε (x) q p dε ε 1 q ≤ m∈Z d \{0} (2π|m|) k | x(m)| 1 ε0 ε q(k−α) e −2πε|m|q dε ε 1 q sup m∈Z d \{0} |m| α | x(m)| m∈Z d |m| k−α e −2πε0|m| sup m∈Z d \{0} |m| α | x(m)| .
We then deduce the claim. Similarly, we can show that for any 0 < r 0 < 1,
1 0 (1 − r) q(k−α) J k r P r (x k ) q p dr 1 − r 1 q ≈ 1 r0 (1 − r) q(k−α) J k r P r (x k ) q p dr 1 − r 1 q .
Now we use the relation r = e −2πε . If ε 0 > 0 is sufficiently small, then
1 − r ≈ ε for ε ∈ (0, ε 0 ). So ε0 0 ε q(k−α) J k ε P ε (x) q p dε ε 1 q ≈ sup 0<|m|<k | x(m)| q + 1 r0 (1 − r) q(k−α) J k r P r (x k ) q p dr 1 − r 1 q .
Combining this with Theorem 3.13, we get the desired assertion.
The characterization by differences
In this section we show the quantum analogue of the classical characterization of Besov spaces by differences. Recall that ω k p (x, ε) is the L p -modulus of smoothness of x introduced in Definition 2.18. The result of this section is the following: Theorem 3.16. Let 1 ≤ p, q ≤ ∞ and 0 < α < n with n ∈ N. Then for any distribution x on T d θ ,
(3.18) x B α p,q ≈ | x(0)| q + 1 0 ε −qα ω n p (x, ε) q dε ε 1 q .
Proof. We will derive the result from Theorem 3.11, or more precisely, from its proof. Since α > 0, we take α 0 = 0 and α 1 = n in that theorem. Recall that d u (ξ) = e 2πiu·ξ − 1. Then the last condition of (3.4) with ψ = d n u is satisfied uniformly in u since
F −1 (d u (2 j ·) n ϕ) 1 = ∆ n 2 j u F −1 (ϕ) 1 ≤ 2 n F −1 (ϕ) 1 .
We will use a variant of the second one (which is not necessarily verified). To this end, let us come back to (3.8) and rewrite it as follows:
ψ (j) (ξ)ϕ (j+k) (ξ) = 2 nk ψ (j) (ξ) (2 −j u · ξ) n h (j+K) (ξ)(2 −j−k u · ξ) n ϕ (j+k) (ξ) = 2 nk η (j) (ξ)ρ (j+k) (ξ),
where η and ρ are now defined by η(ξ) = ψ(ξ) (u · ξ) n h (K) (ξ) and ρ(ξ) = (u · ξ) n ϕ(ξ).
The second condition of (3.4) becomes the requirement that sup u∈R d ,|u|≤1
F −1 (η) 1 < ∞.
The crucial point here is that ψ(ξ) = d n u (ξ) = (u · ξ) n ζ(u · ξ), where ζ is an analytic function on R. This shows that the above supremum is finite.
However, the first condition of (3.4), the Tauberian condition is not verified for a single d n u . We will return back to this point later. For the moment, we just observe that the Tauberian condition has not been used in steps 1 and 2 of the proof of Theorem 3.9. Reexamining those two steps with ψ = d n u , we see that all estimates there can be made independent of u. For instance, (3.15) now becomes (with α 1 = n) 2 jα ∆ n εu ϕ j+k * x p 2 (α1−α)k 2 (j+k)α ρ j+k * x p , where the new function ρ is defined as above. Thus taking the supremum over all u with |u| ≤ 1, we get 2 jα ω n p (x, ε) 2 k(α1−α) 2 (j+k)α ρ (j+k) * x p . Therefore, by Lemma 3.4, we obtain
1 0 ε −qα ω n p (x, ε) q dε ε 1 q x B α p,q .
The reverse inequality requires necessarily a Tauberian-type condition. Although a single d n u does not satisfy it, a finite family of d n u 's does satisfy this condition, which we precise below. Choose a 1 2 -net {v ℓ } 1≤ℓ≤L of the unit sphere of R d . Let u ℓ = 4 −1 v ℓ and
Ω ℓ = ξ : 2 −1 ≤ |ξ| ≤ 2, ξ |ξ| − v ℓ ≤ 2 −1 .
Then the union of the Ω ℓ 's is equal to {ξ : 2 −1 ≤ |ξ| ≤ 2} and |d n u ℓ | > 0 on Ω ℓ . So the family {d n u ℓ } 1≤ℓ≤L satisfies the following Tauberian-type condition:
L ℓ=1 |d n u ℓ | > 0 on {ξ : 2 −1 ≤ |ξ| ≤ 2}.
Now we reexamine step 3 of the proof of Theorem 3.9. To adapt it to the present setting, by an appropriate partition of unity, we decompose ϕ into a sum of infinitely differentiable functions, ϕ = ϕ 1 + · · · + ϕ L such that supp ϕ ℓ ⊂ Ω ℓ . Accordingly, for every j ≥ 0, let
ϕ (j) = L ℓ=1 ϕ (j) ℓ .
Then we write the corresponding (3.16) with (ϕ ℓ , d n u ℓ ) in place of (ϕ, ψ) for every ℓ ∈ {1, · · · , L}. Arguing as in step 3 of the proof of Theorem 3.9, we get
x B α p,q | x(0)| q + L ℓ=1 1 0 ε −qα (d n u ℓ ) ε * x q p dε ε 1 q . Since (d n u ℓ ) ε * x = ∆ n εu ℓ x, we deduce x B α p,q | x(0)| q + 1 0 ε −qα sup 1≤ℓ≤L ∆ n εu ℓ x q p dε ε 1 q | x(0)| q + 1 0 ε −qα ω n p (x, ε) q dε ε 1 q .
Thus the theorem is proved.
As a byproduct, the preceding theorem implies that the right-hand side of (3.18) does not depend on n with n > α, up to equivalence. This fact admits a direct simple proof and is an immediate consequence of the following analogue of Marchaud's classical inequality which is of interest in its own right.
Proposition 3.17. For any positive integers n and N with n < N and for any ε > 0, we have
2 n−N ω N p (x, ε) ≤ ω n p (x, ε) ε n ∞ ε ω N p (x, δ) δ n dδ δ .
Proof. The argument below is standard. Using the identity ∆ N u = ∆ N −n u ∆ n u , we get ∆ N u (x) p ≤ 2 N −n ∆ n u (x) p , whence the lower estimate. The upper one is less obvious. By elementary calculations, for any u ∈ R d , we have In terms of Fourier multipliers, this means ∆ n 2u = 2 n ∆ n u + n j=0 n j
j−1 i=0 T iu ∆ n+1 u .
It then follows that ω n p (x, ε) ≤ n 2 ω n+1 p (x, ε) + 2 −n ω n p (x, 2ε). Iterating this inequality yields
ω n p (x, ε) ≤ n 2 J−1 j=1 ω n+1 p (x, 2 j ε) + 2 −Jn ω n p (x, 2 J ε),
from which we deduce the desired inequality for N = n + 1 as J → ∞. Another iteration argument then yields the general case.
Remark 3.18. Theorem 3.16 shows that if α > 0, B α ∞,∞ (T d θ ) coincides with the quantum analogue of the classical Zygmund class of order α. In particular, for 0 < α < 1, B α ∞,∞ (T d θ ) and B α ∞,c0 (T d θ ) are the Lipschitz and little Lipschitz classes of order α, already studied by Weaver [78]. Note that like in the classical setting, if k is a positive integer, B k ∞,∞ (T d θ ) is closely related to but different from the Lipschitz class W k ∞ (T d θ ) discussed in section 2.4.
Limits of Besov norms
In this section we consider the behavior of the right-hand side of (3.18) as α → n. The study of this behavior is the subject of several recent publications in the classical setting; see, for instance, [4,5,38,41,74]. It originated from [14] in which Bourgain, Brézis and Mironescu proved that for any 1 ≤ p < ∞ and any f ∈ C ∞ 0 (R d )
lim α→1 (1 − α) R d ×R d |f (s) − f (t)| p |s − t| αp+d ds dt 1 p = C p,d ∇f (t) p .
It is well known that
R d ×R d |f (s) − f (t)| p |s − t| αp+d ds dt 1 p ≈ ∞ 0 sup u∈R d ,|u|≤ε ∆ u f p p dε ε 1 p .
The right-hand side is the norm of f in the Besov space B 1 p,p (R d ). Higher order extensions have been obtained in [38,74].
The main result of the present section is the following quantum extension of these results. Let (3.19) x
B α,ω p,q = 1 0 ε −αq ω k p (x, ε) q dε ε 1 q .
Theorem 3.19. Let 1 ≤ p ≤ ∞, 1 ≤ q < ∞ and 0 < α < k with k an integer. Then for
x ∈ W k p (T d θ ), lim α→k (k − α) 1 q x B α,ω p,q ≈ q − 1 q |x| W k p
with relevant constants depending only on d and k.
Proof. The proof is easy by using the results of section 2.4. Let x ∈ W k p (T d θ ) with x(0) = 0. Let A denote the limit in Lemma 2.21. Then
1 0 ε −αq ω k p (x, ε) q dε ε ≤ A q 1 0 ε (k−α)q dε ε , whence lim sup α→k (k − α) 1 0 ε −αq ω p (x, ε) q dε ε ≤ A q q .
Conversely, for any η > 0, choose δ ∈ (0, 1) such that
ω k p (x, ε) ε k ≥ A − η, ∀ε ≤ δ. Then (k − α) 1 0 ε −αq ω k p (x, ε) q dε ε ≥ (A − η) q q δ (k−α)q , which implies lim inf α→k (k − α) 1 0 ε −αq ω k p (x, ε) q dε ε ≥ (A − η) q q .
Therefore,
lim α→k (k − α) 1 0 ε −αq ω k p (x, ε) q dε ε = 1 q lim ε→0 ω k p (x, ε) ε k .
So Theorem 2.20 implies the desired assertion.
Remark 3.20. We will determine later the behavior of x B α,ω p,q when α → 0; see Corollary 5.20 below.
The link with the classical Besov spaces
Like for the Sobolev spaces on T d θ , there exists a strong link between B α p,q (T d θ ) and the classical vector-valued Besov spaces on T d . Let us give a precise definition of the latter spaces. We maintain the assumption and notation on ϕ in section 3.1. In particular, f → ϕ k * f is the Fourier multiplier on T d associated to ϕ(2 −k ·):
ϕ k * f = m∈Z d ϕ(2 −k m) f (m)z m for any f ∈ S ′ (T d ; X). Here X is a Banach space. Definition 3.21. Let 1 ≤ p, q ≤ ∞ and α ∈ R. Define B α p,q (T d ; X) = f ∈ S ′ (T d ; X) : f B α p,q < ∞ , where f B α p,q = f (0) q X + k≥0 2 αkq ϕ k * f q Lp(T d ;X) 1 q .
These vector-valued Besov spaces have been largely studied in literature. Note that almost all publications concern only the case of R d , but the periodic theory is parallel (see, for instance, [24,70]; see also [3] for the vector-valued case). B α p,q (R d ; X) is defined in the same way with the necessary modifications among them the main difference concerns the term f (0) X above which is replaced by φ * f Lp(R d ;X) , where φ is the function whose Fourier transform is equal to
1 − k≥0 ϕ(2 −k ·).
All results proved in the previous sections remain valid in the present vector-valued setting with essentially the same proofs for any Banach space X, except Theorem 3.8 whose vector-valued version holds only if X is isomorphic to a Hilbert space. On the other hand, the duality assertion in Proposition 3.3 should be slightly modified by requiring that X * have the Radon-Nikodym property.
Let us state the vector-valued analogue of Theorem 3.15. As said before, it is new even in the scalar case. The circular Poisson and heat semigroups are extended to the present case too. For any f ∈ S ′ (T d ; X),
P r (f )(z) = m∈Z d r |m| f (m)z m and W r (f )(z) = m∈Z d r |m| 2 f (m)z m , z ∈ T d , 0 ≤ r < 1.
The operator J k r has the same meaning as before, for instance, in the Poisson case, we have
J k r P r (f ) = m∈Z d C m,k f (m)r |m|−k z m , where C m,k = |m| · · · (|m| − k + 1) if k ≥ 0 and C m,k = 1 (|m| + 1) · · · (|m| − k) if k < 0. Theorem 3.22. Let 1 ≤ p, q ≤ ∞, α ∈ R and k ∈ Z. Let X be a Banach space. (i) If k > α, then for any f ∈ B α p,q (T d ; X), f B α p,q ≈ sup |m|<k f (m) q X + 1 0 (1 − r) (k−α)q J k r P r (f k ) q Lp(T d ;X) dr 1 − r 1 q , where f k = f − |m|<k f (m)z m . (ii) If k > α 2 , then for any f ∈ B α p,q (T d ; X), f B α p,q ≈ sup |m| 2 <k f (m) q X + 1 0 (1 − r) (k− α 2 )q J k r W r (f ) q Lp(T d ;X) dr 1 − r 1 q .
The following transference result from T d θ to T d is clear. It can be used to prove a majority of the preceding results on T d θ , under the hypothesis that the corresponding results in the vector-valued case on T d are known.
Proposition 3.23. Let 1 ≤ p, q ≤ ∞ and α ∈ R. The map x → x in Corollary 1.2 is an isometric embedding from B α p,q (T d θ ) into B α p,q (T d ; L p (T d θ )
) with 1-complemented range. Remark 3.24. As a subspace of ℓ α q (L p (T d θ )) (see the proof of Proposition 3.3 for the definition of this space), B α p,q (T d θ ) can be equipped with a natural operator space structure in Pisier's sense [54]. Moreover, in the spirit of the preceding vector-valued case, we can also introduce the vectorvalued quantum Besov spaces. Given an operator space E, B α p,q (T d θ ; E) is defined exactly as in the scalar case; it is a subspace of ℓ α q (L p (T d θ ; E)). Then all results of this chapter are extended to this vector-valued case, except the duality in Proposition 3.3 and Theorem 3.15.
Chapter 4. Triebel-Lizorkin spaces
This chapter is devoted to the study of Triebel-Lizorkin spaces. These spaces are much subtler than Besov spaces even in the classical setting. Like Besov spaces, the classical Triebel-Lizorkin spaces F α p,q (R d ) have three parameters, p, q and α. The difference is that the ℓ q -norm is now taken before the L p -norm. Namely, f ∈ F α p,q (R d ) iff k≥0 2 kαq |ϕ k * f | q 1 q p is finite. Besides the usual subtlety just mentioned, more difficulties appear in the noncommutative setting. For instance, a first elementary one concerns the choice of the internal ℓ q -norm. It is a well-known fact in the noncommutative integration that the simple replacement of the usual absolute value by the modulus of operators does not give a norm except for q = 2. Alternately, one could use Pisier's definition of ℓ q -valued noncommutative L p -spaces in the category of operator spaces. However, we will not study the latter choice and will confine ourselves only to the case q = 2, by considering the column and row norms (and their mixture) for the internal ℓ 2 -norms. This choice is dictated by the theory of noncommutative Hardy spaces. In fact, we will show that the so-defined Triebel-Lizorkin spaces on T d θ are isomorphic to the Hardy spaces developed in [17]. Another difficulty is linked with the frequent use of maximal functions in the commutative case. These functions play a crucial role in the classical theory. However, the pointwise analogue of maximal functions is no longer available in the present setting, which makes our study harder than the classical case. We have already encountered this difficulty in the study of Besov spaces. It is much more substantial now. Instead, our development will rely heavily on the theory of Hardy spaces developed in [80] through a Fourier multiplier theorem that is proved in the first section. It is this multiplier theorem which clears the obstacles on our route. After the definitions and basic properties, we prove some characterizations of the quantum Triebel-Lizorkin spaces. Like in the Besov case, they are better than the classical ones even in the commutative case. We conclude the chapter with a short section on the operator-valued Triebel-Lizorkin spaces on T d (or R d ). These spaces are interesting in view of the theory of operator-valued Hardy spaces.
Throughout this chapter, we will use the notation introduced in the previous one. In particular, ϕ is a function satisfying (3.1), ϕ (k) = ϕ(2 −k ·) and ϕ k = ϕ (k) .
A multiplier theorem
The following multiplier result will play a crucial role in this chapter.
Theorem 4.1. Let σ ∈ R with σ > d 2 .
Assume that (φ j ) ≥0 and (ρ j ) ≥0 are two sequences of continuous functions on R d \ {0} such that
(4.1) supp(φ j ρ j ) ⊂ {ξ : 2 j−1 ≤ |ξ| ≤ 2 j+1 }, ∀j ≥ 0, sup j≥0 φ j (2 j ·)ϕ H σ 2 (R d ) < ∞.
(i) Let 1 < p < ∞. Then for any distribution x on T d θ ,
j≥0 2 2jα | φ j * ρ j * x| 2 1 2 Lp(T d θ ) sup j≥0 φ j (2 j ·)ϕ H σ 2 j≥0 2 2jα | ρ j * x| 2 1 2 Lp(T d θ ) ,
where the constant depends only on p, σ, d and ϕ. (ii) Assume, in addition, that ρ j = ρ(2 −j ·) for some Schwartz function with supp(ρ) = {ξ : 2 −1 ≤ |ξ| ≤ 2}. Then the above inequality also holds for p = 1 with relevant constant depending additionally on ρ.
The remainder of this section is devoted to the proof of the above theorem. As one can guess, the proof is based on the Calderón-Zygmund theory. We require several lemmas. The first one is an elementary inequality.
Lemma 4.2. Assume that f : R d → ℓ 2 and g : R d → C satisfy f ∈ H σ 2 (R d ; ℓ 2 ) and R d (1 + |s| 2 ) σ |F −1 (g)(s)|ds < ∞.
Then f g ∈ H σ 2 (R d ; ℓ 2 ) and
f g H σ 2 (R d ;ℓ2) ≤ f H σ 2 (R d ;ℓ2) R d (1 + |s| 2 ) σ |F −1 (g)(s)|ds.
Proof. The norm · below is that of ℓ 2 . By the Cauchy-Schwarz inequality, for s ∈ R d , we have
F −1 (f g)(s) 2 = F −1 (f ) * F −1 (g)(s) 2 ≤ F −1 (g) 1 R d F −1 (f )(s − t) 2 F −1 (g)(t) dt.
It then follows that
f g 2 H σ 2 (R d ;ℓ2) = R d (1 + |s| 2 ) σ F −1 (f g)(s) 2 ds ≤ F −1 (g) 1 R d (1 + |s| 2 ) σ R d F −1 (f )(s − t) 2 F −1 (g)(t) dt ds ≤ F −1 (g) 1 R d R d (1 + |s − t| 2 ) σ F −1 (f )(s − t) 2 ds (1 + |t| 2 ) σ |F −1 (g)(t)|dt ≤ f 2 H σ 2 (R d ;ℓ2) R d (1 + |t| 2 ) σ |F −1 (g)(t)|dt 2 .
Thus the assertion is proved.
The following lemma is a well-known result in harmonic analysis, which asserts that every Hörmander multiplier is a Calderón-Zygmund operator. Note that the usual Hörmander condition is expressed in terms of partial derivatives up to order [ d 2 ] + 1, while the condition below, in terms of the potential Sobolev space H σ 2 (R d ), is not commonly used (it is explicitly stated on page 263 of [67]). Combined with the previous lemma, the standard argument as described in [
(4.2) sup k∈Z φ(2 k ·)ϕ H σ 2 (R d ;ℓ2) < ∞. Let k = (k j ) j≥0 with k j = F −1 (φ j ).
Then k is a Calderón-Zygmund kernel with values in ℓ 2 , more precisely,
• k L∞(R d ;ℓ2) sup k∈Z φ(2 k ·)ϕ H σ 2 (R d ;ℓ2) ; • sup t∈R d |s|>2|t| k(s − t) − k(s) ℓ2 ds sup k∈Z φ(2 k ·)ϕ H σ 2 (R d ;ℓ2)
. The relevant constants depend only on ϕ, σ and d.
The above kernel k defines a Calderón-Zygmund operator on R d . But we will consider only the periodic case, so we need to periodize k:
k(s) = m∈Z d k(s + m).
By a slight abuse of notation, we use k j to denote the Calderón-Zygmund operator on T d associated to k j too:
k j (f )(s) = I d k j (s − t)f (t)dt, where we have identified T with I = [0, 1). k j is the Fourier multiplier on T d with symbol φ j : f → φ j * f . We have k = k Z d . If φ satisfies (4.2), then Lemma 4.3 implies (4.3) k ℓ∞(Z d ;ℓ2) < ∞, sup t∈I d {s∈I d :|s|>2|t|} k(s − t) − k(s) ℓ2 ds < ∞.
Now let M be a von Neumann algebra equipped with a normal faithful tracial state τ , and let N = L ∞ (T d )⊗M, equipped with the tensor trace. The following lemma should be known to experts; it is closely related to similar results of [29,46,51], notably to [35,Lemma 2.3]. Note that the sole difference between the following condition (4.4) and (4.2) is that the supremum in (4.2) runs over all integers while the one below is restricted to nonnegative integers.
(4.4) φ h σ 2 = sup k≥0 φ(2 k ·)ϕ H σ 2 (R d ;ℓ2) < ∞. Then for 1 < p < ∞ and any finite sequence (f j ) ⊂ L p (N ), j≥0 | φ j * f j | 2 1 2 p φ h σ 2 j≥0 |f j | 2 1 2 p .
The relevant constant depends only on p, ϕ, σ and d.
Proof. The argument below is standard. First, note that the Fourier multiplier on T d with symbol φ j does not depend on the values of φ j in the open unit ball of R d . So letting η be an infinite differentiable function on R d such that η(ξ) = 0 for |ξ| ≤ 1 2 and η(ξ) = 1 for |ξ| ≥ 1, we see that φ j and ηφ j induce the same Fourier multiplier on T d (restricted to distributions with vanishing Fourier coefficients at the origin). On the other hand, it is easy to see that (4.4) implies that the sequence (ηφ j ) j≥0 satisfies (4.2) with (ηφ j ) j≥0 in place of φ. Thus replacing φ j by ηφ j if necessary, we will assume that φ satisfies the stronger condition (4.2).
We will use the Calderón-Zygmund theory and consider k as a diagonal matrix with diagonal entries ( k j ) j≥1 . The Calderón-Zygmund operator associated to k is thus the convolution operator:
k(f )(s) = I d k(s − t)f (t)dt
for any finite sequence f = (f j ) (viewed as a column matrix). Then the assertion to prove amounts to the boundedness of k on L p (N ; ℓ c 2 ). We first show that k is bounded from L ∞ (N ; ℓ c 2 ) into BMO c (T d , B(ℓ 2 )⊗M). Let f be a finite sequence in L ∞ (N ; ℓ c 2 ), and let Q be a cube of I d whose center is denoted by c. We decompose f as f = g + h with g = f 1 Q , where Q = 2Q, the cube with center c and twice the side length of Q. Setting
a = I d \ Q k(c − t)f (t)dt, we have k(f )(s) − a = k(g)(s) + I d ( k(s − t) − k(c − t))h(t)dt. Thus 1 |Q| Q | k(f )(s) − a| 2 ds ≤ 2(A + B), where A = 1 |Q| Q | k(g)(s)| 2 ds, B = 1 |Q| Q I d ( k(s − t) − k(c − t))h(t)dt 2 ds.
The first term A is easy to estimate. Indeed, by (4.3) and the Plancherel formula,
|Q|A ≤ I d | k(g)(s)| 2 ds = m∈Z d k(m) g(m) 2 = m∈Z d g(m) * k(m) * k(m) g(m) ≤ m∈Z d k(m) 2 B(ℓ2) | g(m)| 2 ≤ k ℓ∞(Z d ;ℓ∞) I d |g(s)| 2 ds ≤ k ℓ∞(Z d ;ℓ2) I d |g(s)| 2 ds | Q| f 2 L∞(N ;ℓ c 2 ) , whence A B(ℓ2)⊗M f 2 L∞(N ;ℓ c
To estimate B, let h = (h j ). Then by (4.3), for any s ∈ Q we have
I d ( k(s − t) − k(c − t))h(t)dt 2 = j I d ( k j (s − t) − k j (c − t))h j (t)dt 2 j I d \ Q | k j (s − t) − k j (c − t)| |h j (t)| 2 dt I d \ Q k(s − t) − k(c − t) ℓ∞ j |h j (t)| 2 dt f 2 L∞(N ;ℓ c 2 ) I d \ Q k(s − t) − k(c − t) ℓ2 dt f 2 L∞(N ;ℓ c 2 ) . Thus B B(ℓ2)⊗M ≤ 1 |Q| Q I d ( k(s − t) − k(c − t))h(t)dt 2 B(ℓ2)⊗M ds = 1 |Q| Q I d ( k(s − t) − k(c − t))h(t)dt 2 B(ℓ2)⊗M ds f 2 L∞(N ;ℓ c 2 ) . Therefore, k is bounded from L ∞ (N ; ℓ c 2 ) into BMO c (T d , B(ℓ 2 )⊗M). We next show that k is bounded from L ∞ (N ; ℓ c 2 ) into BMO r (T d , B(ℓ 2 )⊗M
). Let f, Q and a be as above. Now we have to estimate
1 |Q| Q k(f )(s) − a * 2 ds B(ℓ2)⊗M .
We will use the same decomoposition f = g + h. Then
1 |Q| Q k(f )(s) − a * 2 ds ≤ 2(A ′ + B ′ ), where A ′ = 1 |Q| Q k(g)(s) * 2 ds, B ′ = 1 |Q| Q I d ( k(s − t) − k(c − t))h(t) * dt 2 ds.
The estimate of B ′ can be reduced to that of B before. Indeed,
B ′ B(ℓ2)⊗M ≤ 1 |Q| Q I d ( k(s − t) − k(c − t))h(t) * dt 2 B(ℓ2)⊗M ds = 1 |Q| Q I d ( k(s − t) − k(c − t))h(t)dt * 2 B(ℓ2)⊗M ds = 1 |Q| Q I d ( k(s − t) − k(c − t))h(t)dt 2 B(ℓ2)⊗M ds f 2 L∞(N ;ℓ c 2 )
. However, A ′ needs a different argument. Setting g = (g j ), we have
A ′ B(ℓ2)⊗M = sup 1 |Q| Q τ i,j k i (g i )(s) k j (g j )(s) * a * j a i ds ,
where the supremum runs over all a = (a i ) in the unit ball of ℓ 2 (L 2 (M)). Considering a i as a constant function on I d , we can write
a i k i (g i ) = k i (a i g i ). Thus Q τ i,j k i (g i )(s) k j (g j )(s) * a * j a i ds = Q i k i (a i g i )(s) 2 L2(M) ds.
So by the Plancherel formula,
Q i k i (a i g i )(s) 2 L2(M) ds ≤ I d i k i (a i g i )(s) 2 L2(M) ds = m∈Z d i k i (m) a i g i (m) 2 L2(M) .
On the other hand, by the Cauchy-Schwarz inequality, (4.3) and the Plancherel formula once more, we have
m∈Z d i k i (m) a i g i (m) 2 L2(M) ≤ m∈Z d k(m) 2 ℓ2 i τ (|a i g i (m)| 2 ) i τ a i m∈Z d g i (m) g i (m) * a * i = i τ a i I d g i (s)g i (s) * ds a * i = i τ a i Q f i (s)f i (s) * ds a * i ≤ | Q| i τ a i f i 2 L∞(N ) a * i |Q| f 2 L∞(N ;ℓ c 2 ) i τ (|a i | 2 ) ≤ |Q| f 2 L∞(N ;ℓ c 2 )
. Combining the above estimates, we get the desired estimate of A ′ :
A ′ B(ℓ2)⊗M f 2 L∞(N ;ℓ c 2 ) . Thus, k is bounded from L ∞ (N ; ℓ c 2 ) into BMO r (T d , B(ℓ 2 )⊗M), so is it from L ∞ (N ; ℓ c 2 ) into BMO(T d , B(ℓ 2 )⊗M).
It is clear that k is bounded from L 2 (N ; ℓ c 2 ) into L 2 (B(ℓ 2 )⊗N ). Hence, by interpolation via (1.2) and Lemma 1.8, k is bounded from L p (N ; ℓ c 2 ) into L p (B(ℓ 2 )⊗N ) for any 2 < p < ∞. This is the announced assertion for 2 ≤ p < ∞. The case 1 < p < 2 is obtained by duality.
Remark 4.5. In the commutative case, i.e., M = C, it is well known that the conclusion of the preceding lemma holds under the following weaker assumption on φ:
(4.5) sup k≥0 R d (1 + |s| 2 ) σ F −1 (φ(2 k ·)ϕ)(s) 2 ℓ∞ ds 1 2 < ∞.
Like at the beginning of the preceding proof, this assumption can be strengthened to
sup k∈Z R d (1 + |s| 2 ) σ F −1 (φ(2 k ·)ϕ)(s) 2 ℓ∞ ds 1 2 < ∞.
Then if we consider k = (k j ) j≥0 as a kernel with values in ℓ ∞ , Lemma 4.3 admits the following ℓ ∞ -analogue:
• k L∞(R d ;ℓ∞) < ∞; • sup t∈R d |s|>2|t| k(s − t) − k(s) ℓ∞ ds < ∞.
Transferring this to the periodic case, we have
• k ℓ∞(Z d ;ℓ∞) < ∞; • sup t∈I d {s∈I d :|s|>2|t|} k(s − t) − k(s) ℓ∞ ds < ∞.(T d , M), j≥0 | φ j * f | 2 1 2 Lp(N ) φ h σ 2 f H c p .
The relevant constant depends only on ϕ, σ and d.
Proof. Like in the proof of Lemma 4.4, we can assume, without loss of generality, that φ satisfies (4.2). We use again the Calderón-Zygmund theory. Now we view k = ( k j ) j≥0 as a column matrix and the associated Calderón-Zygmund operator k as defined on L p (N ):
k(f )(s) = I d k(s − t)f (t)dt.
Thus k maps functions to sequences of functions. We have to show that k is bounded from H c p (T d , M) to L p (N ; ℓ c 2 ) for 1 ≤ p ≤ 2. This is trivial for p = 2. So by Lemma 1.9 via interpolation, it suffices to consider the case p = 1. The argument below is based on the atomic decomposition of H c 1 (T d , M) obtained in [17] (see also [44]). Recall that an M c -atom is a function
a ∈ L 1 (M; L c 2 (T d )) such that • a is supported by a cube Q ⊂ T d ≈ I d ; • Q a(s)ds = 0; • τ Q |a(s)| 2 ds
1.
Let Q be the supporting cube of a. By translation invariance of the operator k, we can assume that Q is centered at the origin. Set Q = 2Q as before. Then However, by the Plancherel formula,
Q | k(a)(s)| 2 ds ≤ I d | k(a)(s)| 2 ds = m∈Z d | k(a)(m)| 2 = m∈Z d | k(m) a(m)| 2 ≤ k ℓ∞(Z d ;ℓ2) m∈Z d | a(m)| 2 = k ℓ∞(Z d ;ℓ2) Q |a(s)| 2 ds .
Therefore, by (4.3)
k(a)1 Q L1(N ;ℓ c 2 ) = τ Q | k(a)(s)|ds | Q| 1 2 τ Q |a(s)| 2 ds 1 2 1.
This is the desired estimate of the first term of the right-hand side of (4.6). For the second, since a is of vanishing mean, for every s ∈ Q we can write
k(a)(s) = Q [ k(s − t) − k(s)]a(t)dt.
Then by the Cauchy-Schwarz inequality via the operator convexity of the square function x → |x| 2 , we have
| k(a)(s)| 2 ≤ Q k(s − t) − k(s) ℓ2 dt · Q k(s − t) − k(s) ℓ2 |a(t)| 2 dt.
Thus by (4.3),
k(a)1 I d \ Q L1(N ;ℓ c 2 ) = τ I d \ Q | k(a)(s)|ds τ Q I d \ Q k(s − t) − k(s) ℓ2 ds dt 1 2 · Q I d \ Q k(s − t) − k(s) ℓ2 |a(t)| 2 ds dt 1 2 |Q| 1/2 τ Q |a(s)| 2 ds 1 2 1.
Hence the desired assertion is proved.
By transference, the previous lemmas imply the following. According to our convention used in the previous chapters, the map x → φ * x denotes the Fourier multiplier associated to φ on T d θ .
Lemma 4.7. Let φ = (φ j ) j satisfy (4.4). (i) For 1 < p < ∞ we have j≥0 | φ j * x j | 2 1 2 p φ h σ 2 j≥0 |x j | 2 1 2 p , x j ∈ L p (T d θ )
with relevant constant depending only on p, ϕ, σ and d.
(ii) For 1 ≤ p ≤ 2 we have j≥0 | φ j * x| 2 1 2 p φ h σ 2 x H c p , x ∈ H c p (T d θ )
with relevant constant depending only on ϕ, σ and d.
We are now ready to prove Theorem 4.1.
Proof of Theorem 4.1. Let ζ j = φ j (ϕ (j−1) + ϕ (j) + ϕ (j+1) ). By (3.2) and the support assumption on φ j ρ j , we have
φ j ρ j = ζ j ρ j , so φ j * ρ j * x = ζ j * ρ j * x
for any distribution x on T d θ . We claim that ζ = (ζ j ) j≥0 satisfies (4.4) in place of φ. Indeed, given k ∈ N 0 , by the support assumption on ϕ in (3.1), the sequence ζ(2 k ·)ϕ = (ζ j (2 k ·)ϕ) j≥0 has at most five nonzero terms of indices j such that k − 2 ≤ j ≤ k + 2. Thus
ζ(2 k ·)ϕ H σ 2 (R d ;ℓ2) ≤ k+2 j=k−2 ζ j (2 k ·)ϕ H σ 2 (R d ) .
However, by Lemma 4.2,
ζ j (2 k ·)ϕ H σ 2 (R d ) φ j (2 j ·)ϕ H σ 2 (R d ) , k − 2 ≤ j ≤ k + 2,
where the relevant constant depends only on d, σ and ϕ. Therefore, the second condition of (4.1) yields the claim. Now applying Lemma 4.7 (i) with ζ j instead of φ j and x j = 2 jα ϕ j * x, we prove part (i) of the theorem.
To show part (ii), we need the characterization of H c 1 (T d θ ) by discrete square functions stated in Lemma 1.10 with ψ = I −α ρ. Let x be a distribution on T d θ with x(0) = 0 such that j≥0 2 jα | ρ j * x| 2 1 2 1 < ∞.
Let y = I α (x). Then the discrete square function of y associated to ψ is given by
s c ψ (y) 2 = j≥0 | ψ j * y| 2 = j≥0 2 jα | ρ j * x| 2 .
So y ∈ H c 1 (T d θ ) and
y H c 1 ≈ j≥0 2 jα | ρ j * x| 2 1 2 1 .
We want to apply Lemma 4.7 (ii) to y but with a different multiplier in place of φ. To that end, let η j = 2 jα I −α φ j and η = (η j ) j≥0 . We claim that η satisfies (4.1) too. The support condition of (4.1) is obvious for η. To prove the second one, by (3.2), we write
η j (2 j ξ)ϕ(ξ) = |ξ| −α ϕ(ξ)φ j (2 j ξ) = |ξ| −α [ϕ(2 −1 ξ) + ϕ(ξ) + ϕ(2ξ)]ϕ(ξ)φ j (2 j ξ).
Since I −α (ϕ (−1) + ϕ + ϕ (1) ) is an infinitely differentiable function with compact support,
R d (1 + |s| 2 ) σ F −1 (I −α (ϕ (−1) + ϕ + ϕ (1) ))(s) ds < ∞.
Thus by Lemma 4.2,
η j (2 j ·)ϕ H s 2 (R d ) φ j (2 j ·)ϕ H s 2 (R d ) , whence the claim.
As in the first part of the proof, we define a new sequence ζ by setting ζ j = η j ρ j . Then the new sequence ζ satisfies (4.4) too and
sup k≥0 ζ j (2 k ·)ϕ H σ 2 (R d ) sup j≥0 η j (2 j ·)ϕ H σ 2 (R d ) sup j≥0 φ j (2 j ·)ϕ H σ 2 (R d ) .
On the other hand, we have 2 jα φ j * ρ j * x = ζ j * y .
Thus we can apply Lemma 4.7 (ii) to y with this new ζ instead of φ, and as before, we get
j≥0 2 2jα | φ j * ρ j * x| 2 1 2 1 = j≥0 | ζ j * y| 2 1 2 1 sup k≥0 ζ(2 k ·)ϕ H σ 2 (R d ;ℓ2) y H c p sup j≥0 φ j (2 j ·)ϕ H σ 2 (R d ) j≥0 2 jα | ρ j * x| 2 1 2 1 .
Hence the proof of the theorem is complete.
Definitions and basic properties
As said at the beginning of this chapter, we consider the Triebel-Lizorkin spaces on T d θ only for q = 2. In this case, there exist three different families of spaces according to the three choices of the internal ℓ 2 -norms. Definition 4.8. Let 1 ≤ p < ∞ and α ∈ R.
(i) The column Triebel-Lizorkin space F α,c
p (T d θ ) is defined by F α,c p (T d θ ) = x ∈ S ′ (T d θ ) : x F α,c p < ∞ , where x F α,c p = | x(0)| + k≥0 2 2kα | ϕ k * x| 2 1 2 p . (ii) The row space F α,r p (T d θ ) consists of all x such that x * ∈ F α,c p (T d θ ), equipped with the norm x F α,r p = x * F α,c p . (iii) The mixture space F α p (T d θ ) is defined to be F α p (T d θ ) = F α,c p (T d θ ) + F α,r p (T d θ ) if 1 ≤ p < 2, F α,c p (T d θ ) ∩ F α,r p (T d θ ) if 2 ≤ p < ∞, equipped with x F α p = inf y F α,c p + z F α,r p : x = y + z if 1 ≤ p < 2, max( x F α,c p , x F α,r p ) if 2 ≤ p < ∞.
In the sequel, we will concentrate our study only on the column Triebel-Lizorkin spaces. All results will admit the row and mixture analogues. The following shows that F α,c p (T d θ ) is independent of the choice of the function ϕ. Proposition 4.9. Let ψ be a Schwartz function satisfying the same condition (3.1) as ϕ. Let ψ k = ψ (k) = ψ(2 −k ·). Then
x F α,c p ≈ | x(0)| + k≥0 2 2kα | ψ k * x| 2 1 2 p .
Proof. Fix a distribution x on T d θ with x(0) = 0. By the support assumption on ψ (k) and (3.2), we have (with ϕ −1 = 0)
ψ k * x = 1 j=−1 ψ k * ϕ k+j * x.
Thus by Theorem 4.1,
k≥0 2 2kα | ψ k * x| 2 1 2 p ≤ 1 j=−1 k≥0 2 2kα | ψ k * ϕ k+j * x| 2 1 2 p k≥0 2 2kα | ϕ k * x| 2 1 2 p .
Changing the role of ϕ and ψ, we get the reverse inequality.
Proposition 4.10. Let 1 ≤ p < ∞ and α ∈ R.
(i) F α,c p (T d θ ) is a Banach space. (ii) F α,c p (T d θ ) ⊂ F β,c p (T d θ ) for β < α. (iii) P θ is dense in F α,c p (T d θ ) . (iv) F 0,c p (T d θ ) = H c p (T d θ ). (v) B α p, min(p,2) (T d θ ) ⊂ F α,c p (T d θ ) ⊂ B α p, max(p,2) (T d θ ).
Proof. (i) is proved as in the case of Besov spaces; see the corresponding proof of Proposition 3.3.
(ii) is obvious. To show (iii), we use the Fejér means as in the proof of Proposition 2.7. We need one more property of those means, that is, they are completely contractive. So they are also contractive on L p (B(ℓ 2 )⊗T d θ ), in particular, on the column subspace L p (T d θ ; ℓ c 2 ) too. We then deduce that F N is contractive on F α,c p (T d θ ) and lim N →∞ F N (x) = x for every x ∈ F α,c p (T d θ ). (iv) has been already observed during the proof of Theorem 4.1. Indeed, for any distribution x on T d θ , the square function associated to ϕ defined in Lemma 1.10 is given by
s c ϕ (x) = k≥0 | ϕ k * x| 2 1 2 . Thus x H c p ≈ x F 0,c p .
(v) follows from the following well-known property:
ℓ 2 (L p (T d θ )) ⊂ L p (T d θ ; ℓ c 2 ) ⊂ ℓ p (L p (T d θ )
) are contractive inclusions for 2 ≤ p ≤ ∞; both inclusions are reversed for 1 ≤ p ≤ 2. Note that the first inclusion is an immediate consequence of the triangular inequality of L p 2 (T d θ ), the second is proved by complex interpolation.
The following is the Triebel-Lizorkin analogue of Theorem 3.7. We keep the notation introduced before that theorem. (i) For any β ∈ R, both J β and I β are isomorphisms between F α,c p (T d θ ) and F α−β,c p (T d θ ). In particular, J α and I α are isomorphisms between F α,c p (T d θ ) and H c p (T d θ ).
(ii) Let a ∈ R d + . If x ∈ F α,c p (T d θ ), then D a x ∈ F α−|a|1,c p (T d θ ) and D a x F α−|a| 1 ,c p x F α,c p . (iii) Let β > 0. Then x ∈ F α,c p (T d θ ) iff D β i x ∈ F α−β,c p (T d θ )
for all i = 1, · · · , d. Moreover, in this case,
x F α,c p ≈ | x(0)| + d i=1 D β i x F α−β,c p . Proof. (i) Let x ∈ F α,c p (T d θ ) with x(0) = 0. By Theorem 4.1, J β x F α−β,c p = k≥0 2 2k(α−β) |J β * ϕ k * x| 2 1 2 p sup k≥0 2 −kβ J β (2 k ·)ϕ H σ 2 (R d ) k≥0 2 2kα | ϕ k * x| 2 1 2 p .
However, it is easy to see that all partial derivatives of the function 2 −kβ J β (2 k ·)ϕ, of order less than a fixed integer, are bounded uniformly in k. It then follows that
sup k≥0 2 −kβ J β (2 k ·)ϕ H σ 2 (R d ) < ∞. Thus J β x F α−β,c p x F α,c p . So J β is bounded from F α,c p (T d θ ) to F α−β,c p (T d θ ), its inverse, which is J −β , is bounded too. I β is handled similarly. If β = α, then F α−β,c p (T d θ ) = F 0,c p (T d θ ) = H c p (T d θ )
by Proposition 4.10 (iv). (ii) This proof is similar to the previous one by replacing J β by D a and using Lemma 3.5. (iii) One implication is contained in (ii). To show the other, we follow the proof of Theorem 3.7 (iii) and keep the notation there. Since
ϕ k = d i=1 χ i D i,β ϕ k , by Theorem 4.1, x F α,c p ≤ d i=1 k≥0 2 2kα | χ i * ϕ k * D β i x| 2 1 2 p d i=1 sup k≥0 2 kβ χ i (2 k ·)ϕ H σ 2 (R d ) k≥0 2 2k(α−β) | ϕ k * D β i x| 2 1 2 p .
However,
2 kβ χ i (2 k ·)ϕ H σ 2 (R d ) = ψϕ H σ 2 (R d ) , where ψ(ξ) = 1 χ(2 k ξ 1 )|ξ 1 | β + · · · + χ(2 k ξ d )|ξ d | β χ(2 k ξ i )|ξ i | β (2πiξ i ) β .
As all partial derivatives of ψϕ, of order less than a fixed integer, are bounded uniformly in k, the norm of ψϕ in H σ 2 (R d ) are controlled by a constant independent of k. We then deduce
x F α,c p d i=1 k≥0 2 2k(α−β) | ϕ k * D β i x| 2 1 2 p = d i=1 D β i x F α−β,c p .
The theorem is thus completely proved.
Corollary 4.12. Let 1 < p < ∞ and α ∈ R. Then F α p (T d θ ) = H α p (T d θ ) with equivalent norms. Proof. Since J α is an isomorphism from F α p (T d θ ) onto F 0 p (T d θ ), and from H α p (T d θ ) onto H 0 p (T d θ )
, it suffices to consider the case α = 0. But then H 0 p (T d θ ) = L p (T d θ ) by definition, and F 0 p (T d θ ) = H p (T d θ ) by Proposition 4.10. It remains to apply Lemma 1.9 to conclude F 0
p (T d θ ) = H 0 p (T d θ )
. We now discuss the duality of F α,c p (T d θ ). For this we need to define F α,c ∞ (T d θ ) that is excluded from the definition at the beginning of the present section. Let ℓ α 2 denote the Hilbert space of all complex sequences a = (a k ) k≥0 such that a = k≥0 2 2kα |a k | 2 1 2 < ∞.
Thus L p (T d θ ; ℓ α,c 2 ) is the column subspace of L p (B(ℓ α 2 )⊗T d θ ).
Definition 4.13. For α ∈ R we define F α,c ∞ (T d θ ) as the space of all distributions x on T d θ that admit a representation of the form
x = k≥0 ϕ k * x k with (x k ) k≥0 ∈ L ∞ (T d θ ; ℓ α,c 2 ),
and endow it with the norm
x F α,c ∞ = | x(0)| + inf k≥0 2 2kα | ϕ k * x k | 2 1 2 ∞ ,
where the infimum runs over all representations of x as above.
Proposition 4.14. Let 1 ≤ p < ∞ and α ∈ R. Then the dual space of F α,c
p (T d θ ) coincides isomorphically with F −α,c p ′ (T d θ )
. Proof. For simplicity, we will consider only distributions with vanishing Fourier coefficients at
m = 0. We view F α,c p (T d θ ) as an isometric subspace of L p (T d θ ; ℓ α,c 2 ) via x → ( ϕ k * x) k≥0 . Then the dual space of F α,c p (T d θ )
is identified with the following quotient of the latter:
G p ′ = y = k≥0 ϕ k * y k : (y k ) k≥0 ∈ L p ′ (T d θ ; ℓ −α,c 2 ) ,
equipped with the quotient norm
y = inf (y k ) L p ′ (T d θ ;ℓ −α,c 2 ) : y = k≥0 ϕ k * y k .
The duality bracket is given by x, y = τ (xy * ). If p = 1, then
G p ′ = F −α,c ∞ (T d θ ) by definition. It remains to show that G p ′ = F −α,c p ′ (T d θ ) for 1 < p < ∞. It is clear that F −α,c p ′ (T d θ ) ⊂ G p ′ ,
a contractive inclusion. Conversely, let y ∈ G p ′ and y = ϕ k * y k for some (y k ) k≥0 ∈ L p ′ (T d θ ; ℓ −α,c 2 ). Then ϕ k * y = ϕ k * ϕ k−1 * y k−1 + ϕ k * ϕ k * y k + ϕ k * ϕ k+1 * y k+1 .
Therefore, by Lemma 4.7,
k≥0 2 2kα | ϕ k * y| 2 1 2 p ′ ≤ 1 j=−1 k≥0 2 −2kα | ϕ k * ϕ k+j * y k+j | 2 1 2 p ′ k≥0 2 −2kα |y k | 2 1 2 p ′ . Thus y ∈ F −α,c p ′ (T d θ ) and y F −α,c p ′ y G p ′ . Remark 4.15. (i) The above proof shows that F α,c p (T d θ ) is a complemented subspace of L p (T d θ ; ℓ α,c 2 ) for 1 < p < ∞.
(ii) By duality, Propositions 4.9, 4.10 and Theorem 4.11 remain valid for p = ∞, except the density of P θ . In particular, F 0,c ∞ (T d θ ) = BMO c (T d θ ). We conclude this section with the following Fourier multiplier theorem, which is an immediate consequence of Theorem 4.1 for p < ∞. The case p = ∞ is obtained by duality. In the case of α = 0, this result is to be compared with Lemma 1.7 where more smoothness of φ is assumed.
sup k≥0 φ(2 k ·) ϕ H σ 2 (R d ) < ∞ for some σ > d 2 .
Then φ is a bounded Fourier multiplier on F α,c p (T d θ ) for all 1 ≤ p ≤ ∞ and α ∈ R. In particular, φ is a bounded Fourier multiplier on H c p (T d θ ) for 1 ≤ p < ∞ and on BMO c (T d θ ).
A general characterization
In this section we give a general characterization of Triebel-Lizorkin spaces on T d θ in the same spirit as that given in section 3.2 for Besov spaces.
Let α 0 , α 1 , σ ∈ R with σ > d 2 . Let h be a Schwartz function satisfying (3.3). Assume that ψ is an infinitely differentiable function on R d \ {0} such that
(4.7) |ψ| > 0 on {ξ : 2 −1 ≤ |ξ| ≤ 2}, R d (1 + |s| 2 ) σ F −1 (ψhI −α1 )(s) ds < ∞, sup k∈N0 2 −kα0 F −1 (ψ(2 k ·)ϕ) H σ 2 (R d ) < ∞.
Writing ϕ = ϕ(ϕ (−1) + ϕ + ϕ (1) ) and using Lemma 4.2, we have
F −1 (ψ(2 k ·)ϕ) H σ 2 (R d ) R d (1 + |s| 2 ) σ F −1 (ψ(2 k ·)ϕ)(s) ds.
So the third condition of (4.7) is weaker than the corresponding one assumed in [ Theorem 4.17. Let 1 ≤ p < ∞ and α ∈ R. Assume that α 0 < α < α 1 and ψ satisfies (4.7).
Then for any distribution x on T d θ , we have
(4.8) x F α,c p ≈ | x(0)| + k≥0 2 2kα | ψ k * x| 2 1 2 p .
The equivalence is understood in the sense that whenever one side is finite, so is the other, and the two are then equivalent with constants independent of x.
Proof. Although it resembles, in form, the proof of Theorem 3.9, the proof given below is harder and subtler than the Besov space case. The key new ingredient is Theorem 4.1. The main differences will already appear in the first part of the proof, which is an adaptation of step 1 of the proof of Theorem 3.9. In the following, we will fix x with x(0) = 0. By approximation, we can assume that x is a polynomial. We will denote the right-hand side of (4.8) by x F α,c p,ψ . Given a positive integer K, we write, as before
ψ (j) = ∞ k=0 ψ (j) ϕ (k) = K k=−∞ ψ (j) ϕ (j+k) + ∞ k=K ψ (j) ϕ (j+k) .
Then (4.9) x F α,c p,ψ ≤ I + II, where I = k≤K j 2 2jα | ψ j * ϕ j+k * x| 2 1 2 p ,
II = k>K j 2 2jα | ψ j * ϕ j+k * x| 2 1 2 p .
The estimate of the term I corresponds to step 1 of the proof of Theorem 3.9. We use again (3.8) with η and ρ defined there. Then applying Theorem 4.1 twice, we have
I = k≤K 2 k(α1−α) j 2 2(j+k)α | η j * ρ j+k * x| 2 1 2 p = k≤K 2 k(α1−α) j 2 2jα | η j−k * ρ j * x| 2 1 2 p k≤K 2 k(α1−α) η (−k) ϕ H σ 2 j 2 2jα | ρ j * x| 2 1 2 p I α1 ϕ H σ 2 k≤K 2 k(α1−α) η (−k) ϕ H σ 2 j 2 2jα | ϕ j * x| 2 1 2 p = I α1 ϕ H σ 2 k≤K 2 k(α1−α) η (−k) ϕ H σ 2 x F α,c p .
Being an infinitely differentiable function with compact support, I α1 ϕ belongs to H σ 2 (R d ), that is,
I α1 ϕ H σ 2 < ∞. Next, we must estimate η (−k) ϕ H σ 2 uniformly in k.
To that end, for s ∈ R d , using
F −1 (η (−k) ϕ)(s) 2 = R d F −1 (η)(t) * F −1 (ϕ)(s − 2 k t)dt 2 ≤ F −1 (η) 1 R d F −1 (η)(t) F −1 (ϕ)(s − 2 k t) 2 dt , for k ≤ K, we have η (−k) ϕ 2 H σ 2 = R d (1 + |s| 2 ) σ F −1 (η (−k) ϕ)(s) 2 ds ≤ F −1 (η) 1 R d (1 + |s| 2 ) σ R d F −1 (η)(t) F −1 (ϕ)(s − 2 k t) 2 dtds F −1 (η) 1 R d (1 + |2 k t| 2 ) σ F −1 (η)(t) R d (1 + |s − 2 k t| 2 ) σ F −1 (ϕ)(s − 2 k t) 2 dsdt ≤ 2 Kσ F −1 (η) 1 R d (1 + |t| 2 ) σ F −1 (η)(t) dt R d (1 + |s| 2 ) σ F −1 (ϕ)(s) 2 ds ≤ c ϕ,σ,K R d (1 + |t| 2 ) σ F −1 (η)(t) dt 2 .
In order to return back from η to ψ, write
η = I −α1 ψh + I −α1 ψ(h (K) − h).
Note that (4.10)
R d (1 + |t| 2 ) σ F −1 (I −α1 ψ(h (K) − h))(t) dt = c ψ,h,α1,σ,K < ∞ since I −α1 ψ(h K − h)
is an infinitely differentiable function with compact support. We then deduce
R d (1 + |t| 2 ) σ F −1 (η)(t) dt R d (1 + |t| 2 ) σ F −1 (I −α1 ψh)(t) dt.
The term on the right-hand side is the second condition of (4.7). Combining the preceding inequalities, we obtain
I R d (1 + |t| 2 ) σ F −1 (I −α1 ψh)(t) dt x F α,c p .
The second term II on the right-hand side of (4.9) is easier to estimate. Using (3.11), Theorem 4.1 and arguing as in the preceding part for the term I, we obtain
II I α0 ϕ H σ 2 k>K 2 −2kα I −α0 ψ(2 k ·)Hϕ H σ 2 x F α,c p k>K 2 −2kα I −α0 ψ(2 k ·)Hϕ H σ 2 x F α,c p ,
where H = ϕ(2 −1 ·) + ϕ + ϕ(2 ·). To treat the last Sobolev norm, noting that I −α0 H is an infinitely differentiable function with compact support, by Lemma 4.2, we have
I −α0 ψ(2 k ·)Hϕ H σ 2 ≤ ψ(2 k ·)ϕ H σ 2 R d (1 + |t| 2 ) σ F −1 (I −α0 H)(t) dt ψ(2 k ·)ϕ H σ 2 .
Therefore,
II sup k>K 2 −kα0 ψ(2 k ·)ϕ H σ 2 k>K 2 2k(α0−α) x F α,c p ≤ c sup k>K 2 −kα0 ψ(2 k ·)ϕ H σ 2 2 (α0−α)K 1 − 2 α0−α x F α,c p (4.11)
with some constant c independent of K. Putting this estimate together with that of I, we finally get
x F α,c p,ψ x F α,c p .
Now we show the reverse inequality by following step 3 of the proof of Theorem 3.9 (recalling that λ = 1 − h). By (3.13) and Theorem 4.1,
x F α,c p ψ −1 ϕ 2 H σ 2 ∞ j=0 2 2jα | h j+K * ψ j * x| 2 1 2 p ∞ j=0 2 2jα | h j+K * ψ j * x| 2 1 2 p ≤ x F α,c p,ψ + ∞ j=0 2 2jα | λ j+K * ψ j * x| 2 1 2 p .
Then combining the arguments in step 3 of the proof of Theorem 3.9 and (4.11) with λ (K) ψ in place of ψ, we deduce
∞ j=0 2 2jα | λ j+K * ψ j * x| 2 1 2 p ≤ c sup k>K 2 −kα0 λ(2 k−K ·)ψ(2 k ·)ϕ H σ 2 2 (α0−α)K 1 − 2 α0−α x F α,c p .
To remove λ(2 k−K ·) from the above Sobolev norm, by triangular inequality, we have
λ(2 k−K ·)ψ(2 k ·)ϕ H σ 2 ≤ ψ(2 k ·)ϕ H σ 2 + h(2 k−K ·)ψ(2 k ·)ϕ H σ 2 .
By the support assumption on h and ϕ, h(2 k−K ·)ϕ = 0 only for k ≤ K + 2, so the second term on the right hand side above matters only for k = K + 1 and k = K + 2. But for these two values of k, by Lemma 4.2, we have
h(2 k−K ·)ψ(2 k ·)ϕ H σ 2 ≤ c ′ ψ(2 k ·)ϕ H σ 2 ,
where c ′ depends only on h. Thus
λ(2 k−K ·)ψ(2 k ·)ϕ H σ 2 ≤ (1 + c ′ ) ψ(2 k ·)ϕ H σ 2 .
Putting together all estimates so far obtained, we deduce
x F α,c p ≤ x F α,c p,ψ + c (1 + c ′ ) sup k≥K 2 −kα0 ψ(2 k ·)ϕ H σ 2 2 (α0−α)K 1 − 2 α0−α x F α,c p .
So if K is chosen sufficiently large, we finally obtain
x F α,c p
x F α,c p,ψ , which finishes the proof of the theorem.
Remark 4.18. Note that we have used the infinite differentiability of ψ only to insure (4.10), which holds whenever ψ is continuously differentiable up to order [ 3d 2 ] + 1. More generally, we need only to assume that there exists σ > 3d 2 + 1 such that ψη ∈ H σ 2 (R d ) for any compactly supported infinite differentiable function η which vanishes in a neighborhood of the origin.
Like in the case of Besov spaces, Theorem 4.17 admits the following continuous version.
The difference is that this function ρ is not infinitely differentiable at the origin. However, the claim is true if σ 1 is an integer. Then by complex interpolation as in the proof of that lemma, we deduce the claim in the general case. Now choose σ such that d 2 < σ < 1 2 (σ 1 − d 2 ) and set η = σ 1 − 2σ. Then η > d 2 , and by the Cauchy-Schwarz inequality, we have
R d (1 + |s| 2 ) σ F −1 I k−α1 h P (s) ds ≤ R d (1 + |s| 2 ) 2σ+η F −1 I k−α1 h P (s) 2 ds 1 2 F −1 I k−α1 h P H σ 1 2 (R d )
. Therefore, the second condition of (4.7) is verified. This shows part (i) in the case k > d + α.
To deal with the remaining case k > α, we need the following: Lemma 4.22. Let 1 ≤ p < ∞ and k, ℓ ∈ Z such that ℓ > k > α. Then for any distribution x on T d θ with x(0) = 0,
1 0 ε 2(k−α) J k ε P ε (x) 2 dε ε 1 2 p ≈ 1 0 ε 2(ℓ−α) J ℓ ε P ε (x) 2 dε ε 1 2 p .
Proof. By induction, it suffices to consider the case ℓ = k + 1. We first show the lower estimate:
1 0 ε 2(k−α) J k ε P ε (x) 2 dε ε 1 2 p 1 0 ε 2(k+1−α) J ℓ ε P ε (x) 2 dε ε 1 2 p .
To that end, we use
J k ε P ε (x) = −sgn(k) ∞ ε J k+1 δ P δ (x)dδ.
Choose β ∈ (0, k−α). By the Cauchy-Schwarz inequality via the operator convexity of the function t → t 2 , we obtain
J k ε P ε (x) 2 ≤ ε −2β 2β ∞ ε δ 2(1+β) J k+1 δ P δ (x) 2 dδ δ .
It then follows that
1 0 ε 2(k−α) J k ε P ε (x) 2 dε ε ≤ 1 2β ∞ 0 δ 2(1+β) J k+1 δ P δ (x) 2 dδ δ δ 0 ε 2(k−α−β) dε ε = 1 4β(k − α − β) ∞ 0 δ 2(k+1−α) J k+1 δ P δ (x) 2 dδ δ .
Therefore,
1 0 ε 2(k−α) J k ε P ε (x) 2 dε ε 1 2 p ∞ 0 δ 2(k+1−α) J k+1 δ P δ (x) 2 dδ δ 1 2 p 1 0 δ 2(k+1−α) J k+1 δ P δ (x) 2 dδ δ 1 2 p , as desired.
The upper estimate is harder. This time, writing P ε1+ε2 = P ε1 * P ε2 , we have
δ k+1 J k+1 δ P δ δ=2ε = sgn(k)2 k+1 ε k+1 ∂ ∂ε P ε * J k ε P ε = sgn(k)2 k+1 ε k φ ε * J k ε P ε , where φ(ξ) = −2π|ξ| e −2π|ξ| . Thus 1 0 ε 2(k+1−α) J k+1 ε P ε (x) 2 dε ε 1 2 p = 1 2 0 δ k+1−α J k+1 δ P δ δ=2ε 2 dε ε 1 2 p = 2 k+1−α 1 2 0 ε 2(k−α) φ ε * J k ε P ε (x) 2 dε ε 1 2 p ≤ 2 k+1−α 1 0 ε 2(k−α) φ ε * J k ε P ε (x) 2 dε ε 1 2 p .
Now our task is to remove φ ε from the integrand on the right-hand side in the spirit of Theorem 4.1.
To that end, we will use a multiplier theorem analogous to Lemma 4.7. Let H = L 2 ((0, 1), dε ε ) and define the H-valued kernel k on R d by k(s) = φ ε (s) 0<ε<1 . It is a well-known elementary fact that this is a Calderón-Zygmund kernel, namely,
• k L∞(R d ;H) < ∞; • sup t∈R d |s|>2|t| k(s − t) − k(s) H ds < ∞.
Thus by Lemma 4.7 (i) (more exactly, following its proof), we obtain that the singular integral operator associated to k is bounded on L p (T d θ ; H c ) for any 1 < p < ∞; consequently,
(4.12) 1 0 ε 2(k−α) φ ε * J k ε P ε (x) 2 dε ε 1 2 p 1 0 ε 2(k−α) J k ε P ε (x) 2 dε ε 1 2 p , whence 1 0 ε 2(k+1−α) J k+1 ε P ε (x) 2 dε ε 1 2 p 1 0 ε 2(k−α) J k ε P ε (x) 2 dε ε 1 2 p .
Thus the lemma is proved for 1 < p < ∞. The case p = 1 necessitates a separate argument like Lemma 4.7. We will require a more characterization of H c 1 (T d θ ) which is a complement to Lemma 1.10. It is the following equivalence proved in [80]:
Let β > 0. Then for a distribution x on T d θ with x(0) = 0, we have
(4.13) x H c 1 ≈ 1 0 (I β P) ε * x 2 dε ε 1 2 1 .
Armed with this characterization, we can easily complete the proof of the lemma. Indeed,
( I k−α P) ε * (I α x) = (−sgn(k)2π) −k ε k−α J k ε P ε (x) . Thus by (4.13) with β = k − α, 1 0 ε 2(k−α) J k ε P ε (x) 2 dε ε 1 2 1 ≈ I α x H c 1 .
It then remains to apply Lemma 4.7 (ii) to I α x to conclude that (4.12) holds for p = 1 too, so the proof of the lemma is complete.
End of the proof of Theorem 4.20. The preceding lemma shows that the norm in the right-hand side of the equivalence in part (i) is independent of k with k > α. As (i) has been already proved to be true for k > d + α, we deduce the assertion in full generality.
We end this section with a Littlewood-Paley type characterizations of Sobolev spaces. The following is an immediate consequence of Corollary 4.12 and the characterizations proved previously in this chapter.
Proposition 4.23. Let ψ satisfy (4.7), k > α and 1 < p < ∞. Then for any distribution on T d θ ,
x H α p ≈ | x(0)|+ inf k≥0 2 2kα | ψ k * y| 2 1 2 p + k≥0 2 2kα |( ψ k * z) * | 2 1 2 p if 1 < p < 2, max k≥0 2 2kα | ψ k * x| 2 1 2 p , k≥0 2 2kα |( ψ k * x) * | 2 1 2 p if 2 ≤ p < ∞; and x H α p ≈ | x(0)|+ inf 1 0 ε 2(k−α) J k ε P ε (y) 2 dε ε 1 2 p + 1 0 ε 2(k−α) J k ε P ε (z) * 2 dε ε 1 2 p if 1 < p < 2, max 1 0 ε 2(k−α) J k ε P ε (x) 2 dε ε 1 2 p , 1 0 ε 2(k−α) J k ε P ε (x) * 2 dε ε 1 2 p if 2 ≤ p < ∞.
The above infima are taken above all decompositions x = y + z.
Operator-valued Triebel-Lizorkin spaces
Unlike Sobolev and Besov spaces, the study of vector-valued Triebel-Lizorkin spaces in the classical setting does not allow one to handle their counterparts in quantum tori by means of transference. Given a Banach space X, a straightforward way of defining the X-valued Triebel-Lizorkin spaces on T d is as follows: for 1 ≤ p < ∞, 1 ≤ q ≤ ∞ and α ∈ R, an X-valued distribution
f on T d belongs to F α p,q (T d ; X) if f F α p,q = f (0) X + k≥0 2 qkα ϕ k * f q X 1 q Lp(T d ) < ∞.
A majority of the classical results on Triebel-Lizorkin spaces can be proved to be true in this vectorvalued setting with essentially the same methods. Contrary to the Sobolev or Besov case, the space F α p,2 (T d ; L p (T d θ )) is very different from the previously studied space F α,c p,2 (T d θ ). This explains why the transference method is not efficient here.
However, there exists another way of defining F α p,2 (T d ; X). Let (r k ) be a Rademacher sequence, that is, an independent sequence of random variables on a probability space (Ω, P ), taking only two values ±1 with equal probability. We define F α p,rad (T d ; X) to be the space of all X-valued distributions f on T d such that
f F α p,rad = f (0) X + k≥0 r k 2 kα ϕ k * f Lp(Ω×T d ;X) < ∞.
It seems that these spaces F α p,rad (T d ; X) have never been studied so far in literature. They might be worth to be investigated. If X is a Banach lattice of finite concavity, then by the Khintchine inequality,
f F α p,rad ≈ f (0) X + k≥0 2 2kα | ϕ k * f | 2 1 2 Lp(T d ;X) .
This norm resembles, in form, more the previous one f F α p,2 . Moreover, in this case, one can also define a similar space by replacing the internal ℓ 2 -norm by any ℓ q -norm.
But what we are interested in here is the noncommutative case, where X is a noncommutative L p -space, say, X = L p (T d θ ). Then by the noncommutative Khintchine inequality [40], we can show that for 2 ≤ p < ∞ (assuming f (0) = 0),
f F α p,rad ≈ max k≥0 2 2kα | ϕ k * f | 2 1 2 p , k≥0 2 2kα |( ϕ k * f ) * | 2 1 2 p .
Here p is the norm of L p (T d ; L p (T d θ )). Thus the right hand-side is closely related to the norm of F α p (T d θ ) defined in section 4.2. In fact, if x → x denotes the transference map introduced in Corollary 1.2, then for 1 < p < ∞, we have
x F α p (T d θ ) ≈ x F α p,rad (T d ;Lp(T d θ )) .
This shows that if one wishes to treat Triebel-Lizorkin spaces on T d θ via transference, one should first investigate the spaces F α p,rad (T d ; L p (T d θ )). The latter ones are as hard to deal with as F α p (T d θ ). We would like to point out, at this stage, that the method we have developed in this chapter applies as well to F α p,rad (T d ; L p (T d θ )). In view of operator-valued Hardy spaces, we will call F α p,rad (T d ; L p (T d θ )) an operator-valued Triebel-Lizorkin space on T d . We can define similarly its column and row counterparts. We will give below an outline of these operator-valued Triebel-Lizorkin spaces in the light of the development made in the previous sections. A systematic study will be given elsewhere. In the remainder of this section, M will denote a finite von Neumann algebra M with a faithful normal racial state τ and N = L ∞ (T d )⊗M.
Definition 4.24. Let 1 ≤ p < ∞ and α ∈ R. The column operator-valued Triebel-Lizorkin space
F α,c p (T d , M) is defined to be F α,c p (T d , M) = f ∈ S ′ (T d ; L 1 (M)) : f F α,c p < ∞ , where f F α,c p = f (0) Lp(M) + k≥0 2 2kα | ϕ k * f | 2 1 2 Lp(N ) .
The main ingredient for the study of these spaces is still a multiplier result like Theorem 4.1 that is restated as follows:
Theorem 4.25. Assume that (φ j ) ≥0 and (ρ j ) ≥0 satisfy (4.1) with some σ > d 2 .
(i) Let 1 < p < ∞. Then for any f ∈ S ′ (T d ; L 1 (M)), j≥0 2 2jα | φ j * ρ j * f 2 | 1 2 Lp(N ) sup j≥0 φ j (2 j ·)ϕ H σ 2 j≥0 2 2jα | ρ j * f | 2 1 2 Lp(N ) .
(ii) If ρ j = ρ(2 −j ·) for some Schwartz function ρ with supp(ρ) = {ξ : 2 −1 ≤ |ξ| ≤ 2}. Then the above inequality holds for p = 1 too.
The proof of Theorem 4.1 already gives the above result. Armed with this multiplier theorem, we can check that all results proved in the previous sections admit operator-valued analogues with the same proofs. For instance, the dual space of F α,c 1 (T d , M) can be described as a space F −α,c ∞ (T d , M) analogous to the one defined in Definition 4.13. However, following the H 1 -BMO duality developed in the theory of operator-valued Hardy spaces in [80], we can show the following nicer characterization of the latter space in the style of Carleson measures:
Theorem 4.26. A distribution f ∈ S ′ (T d ; L 1 (M)) with f (0) = 0 belongs to F α,c ∞ (T d , M) iff sup Q 1 |Q| Q k≥log 2 (l(Q)) 2 2kα | ϕ k * f (s)| 2 ds M < ∞,
where the supremum runs over all cubes of T d , and where l(Q) denotes the side length of Q.
The characterizations of Triebel-Lizorkin spaces given in the previous two sections can be transferred to the present setting too. Let us formulate only the analogue of Theorem 4.21.
Theorem 4.27. Let 1 ≤ p < ∞, α ∈ R and k ∈ Z.
(i) If k > α, then for any f ∈ S ′ (T d ; L 1 (M)),
f F α,c p ≈ max |m|<k f (m) Lp(M) + 1 0 (1 − r) 2(k−α) J k r P r (f k ) 2 dr 1 − r 1 2 Lp(N ) , where f k = f − |m|<k f (m)U m .
(ii) If k > α 2 , then for any f ∈ S ′ (T d ; L 1 (M)),
f F α,c p ≈ max |m| 2 <k f (m) Lp(M) + 1 0 (1 − r) 2(k− α 2 ) J k r W r (f ) 2 dr 1 − r 1 2 Lp(N )
.
Chapter 5. Interpolation
Now we study the interpolation of the various spaces introduced in the preceding three chapters. We start with the interpolation of Besov and Sobolev spaces. Like in the classical case, the interpolation of Besov spaces on T d θ is very simple. However, the situation of (fractional) Sobolev spaces is much more delicate. Recall that the complex interpolation problem of the classical couple ) are also considered. The most important problem left unsolved in the first section is to transfer DeVore and Scherer's theorem on the real interpolation of (W k 1 (R d ), W k ∞ (R d )) to the quantum setting. The main result of the second section characterizes the K-functional of the couple (L p (T d θ ), W k p (T d θ )) by the L p -modulus of smoothness, thereby extending a theorem of Johnen and Scherer to the quantum tori. This result is closely related to the limit theorem of Besov spaces proved in section 3.5. The last short section contains some simple results on the interpolation of Triebel-Lizorkin spaces.
(W k 1 (R d ), W k ∞ (R d ))
Interpolation of Besov and Sobolev spaces
This section collects some results on the interpolation of Besov and Sobolev spaces. We start with the Besov spaces.
Proposition 5.1. Let 0 < η < 1. Assume that α, α 0 , α 1 ∈ R and p, p 0 , p 1 , q, q 0 , q 1 ∈ [1, ∞] satisfy the constraints given in the formulas below. We have
(i) B α0 p,q0 (T d θ ), B α1 p,q1 (T d θ ) η,q = B α p,q (T d θ ), α 0 = α 1 , α = (1 − η)α 0 + ηα 1 ; (ii) B α p,q0 (T d θ ), B α p,q1 (T d θ ) η,q = B α p,q (T d θ ), 1 q = 1 − η q 0 + η q 1 ; (iii) B α0 p0,q0 (T d θ ), B α1 p1,q1 (T d θ ) η,q = B α p,q (T d θ ), α = (1 − η)α 0 + ηα 1 , 1 p = 1 − η p 0 + η p 1 , 1 q = 1 − η q 0 + η q 1 , p = q; (iv) B α0 p0,q0 (T d θ ), B α1 p1,q1 (T d θ ) η = B α p,q (T d θ ), α = (1 − η)α 0 + ηα 1 , 1 p = 1 − η p 0 + η p 1 , 1 q = 1 − η q 0 + η q 1 , q < ∞.
Proof. We will use the embedding of B α p,q (T d θ ) into ℓ α q (L p (T d θ )). Recall that given a Banach space X, ℓ α q (X) denotes the weighted ℓ q -direct sum of (C, X, X, · · · ), equipped with the norm
(a, x 0 , x 1 , · · · ) = |a| q + k≥0 2 kqα x k q 1 q .
Then B α p,q (T d θ ) isometrically embeds into ℓ α q (L p (T d θ )) via the map I defined by Ix = ( x(0), ϕ 0 * x, ϕ 1 * x, · · · ). On the other hand, it is easy to check that the range of I is 1-complemented. Indeed, let P :
ℓ α q (L p (T d θ )) → B α p,q (T d θ ) be defined by (with ϕ k = 0 for k ≤ −1) P(a, x 0 , x 1 , · · · ) = a + k≥0 ( ϕ k−1 + ϕ k + ϕ k+1 ) * x k .
Then by (3.2), PIx = x for all x ∈ B α p,q (T d θ ). On the other hand, letting y = P(a, x 0 , x 1 , · · · ), we have
ϕ j * y = j+2 k=j−2 ϕ j * ( ϕ k−1 + ϕ k + ϕ k+1 ) * x k , j ≥ 0.
Thus we deduce that P is bounded with norm at most 15.
Therefore, the interpolation of the Besov spaces is reduced to that of the spaces ℓ α q (L p (T d θ )), which is well-known and is treated in [8,Section 5.6]. Let us recall the results needed here. For a Banach space X and an interpolation couple (X 0 , X 1 ) of Banach spaces, we have
• ℓ α0 q0 (X), ℓ α1 q1 (X) η,q = ℓ α q (X), α 0 = α 1 , α = (1 − η)α 0 + ηα 1 ; • ℓ α0 q0 (X 0 ), ℓ α1 q1 (X 1 ) η,q = ℓ α q (X 0 , X 1 ) η,q , α = (1 − η)α 0 + ηα 1 , 1 q = 1 − η q 0 + η q 1 ; • ℓ α0 q0 (X 0 ), ℓ α1 q1 (X 1 ) η = ℓ α q (X 0 , X 1 ) η , α = (1 − η)α 0 + ηα 1 , 1 q = 1 − η q 0 + η q 1 , q < ∞.
It is then clear that the interpolation formulas of the theorem follow from the above ones thanks to the complementation result proved previously.
Remark 5.2. If q = ∞, part (iv) holds for Calderón's second interpolation method, namely,
B α0 p0,∞ (T d θ ), B α1 p1,∞ (T d θ ) η = B α p,∞ (T d θ ), α = (1 − η)α 0 + ηα 1 , 1 p = 1 − η p 0 + η p 1 .
On the other hand, if one wishes to stay with the first complex interpolation method in the case
q = ∞, one should replace B α p,∞ (T d θ ) by B α p,c0 (T d θ ): B α0 p0,c0 (T d θ ), B α1 p1,c0 (T d θ ) η = B α p,c0 (T d θ ) .
Now we consider the potential Sobolev spaces. Since J α is an isometry between H α p (T d θ ) and L p (T d θ ) for all 1 ≤ p ≤ ∞, we get immediately the following Remark 5.3. Let 0 < η < 1, α ∈ R, 1 ≤ p 0 , p 1 ≤ ∞ and 1 p = 1−η p0 + η p1 . Then
H α p0 (T d θ ), H α p1 (T d θ ) η = H α p (T d θ ) and H α p0 (T d θ ), H α p1 (T d θ ) η,p = H α p (T d θ ) .
The interpolation problem of the couple H α0 p0 (T d θ ), H α1 p1 (T d θ ) for α 0 = α 1 is delicate. At the time of this writing, we cannot, unfortunately, solve it completely. To our knowledge, it seems that even in the commutative case, its interpolation spaces by real or complex interpolation method have not been determined in full generality. We will prove some partial results.
Proposition 5.4. Let 0 < η < 1, α 0 = α 1 ∈ R and 1 ≤ p, q ≤ ∞. Then H α0 p (T d θ ), H α1 p (T d θ ) η,q = B α p,q (T d θ ), α = (1 − η)α 0 + ηα 1 .
Proof. The assertion follows from Theorem 3.8, the reiteration theorem and Proposition 5.1 (i).
To treat the complex interpolation, we introduce the potential Hardy-Sobolev spaces.
Definition 5.5. For α ∈ R, define H α H1 (T d θ ) = x ∈ S ′ (T d θ ) : J α x ∈ H 1 (T d θ ) with x H α H 1 = J α x H1 .
We define H α BMO (T d θ ) similarly. Theorem 5.6. Let α 0 , α 1 ∈ R and 1 < p < ∞. Then
H α0 BMO (T d θ ), H α1 H1 (T d θ ) 1 p = H α p (T d θ ), α = (1 − 1 p )α 0 + α 1 p .
We require the following result which extends Lemma 1.7(ii):
Lemma 5.7. Let φ be a Mikhlin multiplier in the sense of Definition 1.5. Then φ is a Fourier multiplier on both H 1 (T d θ ) and BMO(T d θ ) with norms majorized by c d φ M . Proof. This is an immediate consequence of Lemma 4.7 (the sequence (φ j ) there becomes now the single function φ). Indeed, by that Lemma, φ is a bounded Fourier multiplier on H 1 (T d θ ), so by duality, it is bounded on BMO(T d θ ) too. We will use Bessel potentials of complex order. For z ∈ C, define J z (ξ) = (1 + |ξ| 2 ) z 2 and J z to be the associated Fourier multiplier. Proof. One easily checks that J it is a Mikhlin multiplier and J it M ≤ c d (1 + |t|) d . Thus, the assertion follows from the previous lemma.
Proof of Theorem 5.6. Let x ∈ H α p (T d θ ) with norm less than 1, that is, J α x ∈ L p (T d θ ) and J α x p < 1. By Lemma 1.9, and the definition of complex interpolation, there exists a continuous function f from the strip S = {z ∈ C : 0 ≤ Re(z) ≤ 1} to H 1 (T d θ ), analytic in the interior, such that
f ( 1 p ) = J α x, sup t∈R f (it) BMO ≤ c and sup t∈R f (1 + it) H1 ≤ c. Define (with η = 1 p ) F (z) = e (z−η) 2 J −(1−z)α0−zα1 f (z), z ∈ S.
Then for any t ∈ R, by the preceding lemma,
F (it) H α 0 BMO = e −t 2 +η 2 J it(α0−α1) f (it) BMO ≤ c ′ .
A similar estimate holds for the other extreme point H α1 H1 (T d θ ). Therefore,
x = F (η) ∈ H α0 BMO (T d θ ), H α1 H1 (T d θ ) η with norm ≤ c ′ . We have thus proved H α p (T d θ ) ⊂ H α0 BMO (T d θ ), H α1 H1 (T d θ ) η . Since the dual space of H 1 (T d θ ) is BMO(T d θ ), we have H α1 H1 (T d θ ) * = H −α1 BMO (T d θ )
. Thus dualizing the above inclusion (for appropriate α i and p), we get
H α0 BMO (T d θ ) , H −α1 BMO (T d θ ) * η ⊂ H α p (T d θ ) ,
where ( · · ) η denotes Calderón's second complex interpolation method. However, by [7] H α0
BMO (T d θ ) , H −α1 BMO (T d θ ) * η ⊂ H α0 BMO (T d θ ) , H −α1 BMO (T d θ ) * η isometrically. Since H α1 H1 (T d θ ) ⊂ H −α1 BMO (T d θ ) * isometrically, we finally deduce H α0 BMO (T d θ ) , H α1 H1 (T d θ ) η ⊂ H α p (T d θ )
, which concludes the proof of the theorem. Corollary 5.9. Let 0 < η < 1, α 0 , α 1 ∈ R and 1 < p 0 , p 1 < ∞. Then
H α0 p0 (T d θ ), H α1 p1 (T d θ ) η = H α p (T d θ ) , α = (1 − η)α 0 + ηα 1 , 1 p = 1 − η p 0 + η p 1 .
Proof. The preceding proof works equally for this corollary. Alternately, in the case p 0 = p 1 , the corollary immediately follows from the previous theorem by reiteration. Indeed, if p 0 = p 1 , then for any α 0 , α 1 ∈ R there exist β 0 , β 1 ∈ R such that
(1 − 1 p 0 )β 0 + 1 p 0 β 1 = α 0 and (1 − 1 p 1 )β 0 + 1 p 1 β 1 = α 1 .
Thus the previous theorem implies
H β0 BMO (T d θ ) , H β1 H1 (T d θ ) 1 p j = H αj pj (T d θ ), j = 0, 1.
The corollary then follows by the reiteration theorem.
It is likely that the above corollary still holds for all 1 ≤ p 0 , p 1 ≤ ∞:
Conjecture 5.10. Let α 0 , α 1 ∈ R and 1 < p < ∞. Then H α0 ∞ (T d θ ), H α1 1 (T d θ ) 1 p = H α p (T d θ ) , α = (1 − 1 p )α 0 + α 1 p .
By duality and Wolff's reiteration theorem [79], the conjecture is reduced to showing that for any 0 < η < 1 and 1 < p 0 < ∞,
H α0 p0 (T d θ ), H α1 1 (T d θ ) η = H α p (T d θ ) , α = (1 − η)α 0 + ηα 1 , 1 p = 1 − η p 0 + η 1 . Since H α1 H1 (T d θ ) ⊂ H α1 1 (T d θ ), Theorem 5.6 implies H α p (T d θ ) ⊂ H α0 p0 (T d θ ), H α1 1 (T d θ ) η .
So the conjecture is equivalent to the validity of the converse inclusion.
Remark 5.11. The proof of Theorem 5.6 shows that for α 0 , α 1 ∈ R and 0 < η < 1,
H α0 H1 (T d θ ), H α1 H1 (T d θ ) η = H α H1 (T d θ ), α = (1 − η)α 0 + ηα 1 .
We do not know if this equality remains true for the couple H α0
1 (T d θ ), H α1 1 (T d θ )
. We conclude this section with a discussion on the interpolation of W k p0 (T d θ ), W k p1 (T d θ ) . Here, the most interesting case is, of course, that where p 0 = ∞ and p 1 = 1. Recall that in the commutative case, the K-functional of W k ∞ (T d ), W k 1 (T d ) is determined by DeVore and Scherer [21]; however, determining the complex interpolation spaces of this couple is a longstanding open problem.
Note that if 1 < p 0 ,
p 1 < ∞, W k p0 (T d θ ), W k p1 (T d θ ) reduces to H k p0 (T d θ ), H k p1 (T d θ )
by virtue of Theorem 2.9. So in this case, the interpolation problem is solved by the preceding results on potential Sobolev spaces. This reduction is, unfortunately, impossible when one of p 0 and p 1 is equal to 1 or ∞. However, in the spirit of potential Hardy Sobolev spaces, it remains valid if we work with the Hardy Sobolev spaces W k BMO (T d θ ) and W k H1 (T d θ ) instead of W k ∞ (T d θ ) and W k 1 (T d θ ), respectively. Here, the Hardy Sobolev spaces are defined as they should be.
Using Lemma 5.7, we see that the proof of Theorem 2.9 remains valid for the Hardy Sobolev spaces too. Thus we have the following:
Lemma 5.12. For any k ∈ N, W k BMO (T d θ ) = H k BMO (T d θ ) and W k H1 (T d θ ) = H k H1 (T d θ ).
Theorem 5.13. Let k ∈ N and 1 < p < ∞. Then for X = W k
H1 (T d θ ) or X = W k 1 (T d θ ), W k BMO (T d θ ), X 1 p = W k p (T d θ ) = W k BMO (T d θ ), X 1 p ,p .
Consequently, for any 0 < η < 1 and 1 < p 0 < ∞,
W k p0 (T d θ ), W k 1 (T d θ ) η = W k p (T d θ ) = W k p0 (T d θ ), W k 1 (T d θ ) η,p , 1 p = 1 − η p 0 + η 1 .
Proof. The first part for X = W k H1 (T d θ ) follows immediately from Remark 5.3, Theorem 5.6 and Lemma 5.12. Then by the reiteration theorem, for any 1 < p < ∞ and 0 < η < 1, we get
W k BMO (T d θ ), W k p (T d θ ) η = W k q (T d θ ) and W k p (T d θ ), W k H1 (T d θ ) η = W k r (T d θ ), where 1 q = 1−η ∞ + η p and 1 r = 1−η p + η 1 .
On the other hand, by the continuous inclusion
H 1 (T d θ ) ⊂ L 1 (T d θ ), we have W k r (T d θ ) = W k p (T d θ ), W k H1 (T d θ ) η ⊂ W k p (T d θ ), W k 1 (T d θ ) η ⊂ W k r (T d θ )
, the last inclusion above being trivial. Thus
W k p (T d θ ), W k 1 (T d θ ) η = W k r (T d θ ).
Therefore, by Wolff's reiteration theorem [79], we deduce the first part for X = W k 1 (T d θ ). The second part follows from the first by the reiteration theorem. 5.2. The K-functional of (L p , W k p ) In this section we characterize the K-functional of the couple (L p (T d θ ), W k p (T d θ )) for any 1 ≤ p ≤ ∞ and k ∈ N. First, recall the definition of the K-functional. For an interpolation couple (X 0 , X 1 ) of Banach spaces, we define
K(x, ε; X 0 , X 1 ) = inf x 0 X0 + ε x 0 X1 : x = x 0 + x 1 , x 0 ∈ X 0 , x 1 ∈ X 1 for ε > 0 and x ∈ X 0 +X 1 . Since W k p (T d θ ) ⊂ L p (T d θ ) contractively, K(x, ε; L p (T d θ )
, W k p (T d θ )) = x p for ε ≥ 1; so only the case ε < 1 is nontrivial. The following result is the quantum analogue of Johnen-Scherer's theorem for Sobolev spaces on R d (see [30]; see also [6,Theorem 5.4.12]). Recall that ω k p (x, ε) denotes the kth order modulus of L p -smoothness of x introduced in section 3.4. Theorem 5.16. Let 1 ≤ p ≤ ∞ and k ∈ N. Then
K(x, ε k ; L p (T d θ ), W k p (T d θ )) ≈ ε k | x(0)| + ω k p (x, ε)
, 0 < ε ≤ 1 with relevant constants depending only on d and k.
Proof. We will adapt the proof of [6,Theorem 5.4.12]. Denote K(x, ε; L p (T d θ ), W k p (T d θ )) simply by K(x, ε). It suffices to consider the elements of L p (T d θ ) whose Fourier coefficients vanish at m = 0. Fix such an element x. Let x = y + z with y ∈ L p (T d θ ) and z ∈ W k p (T d θ ) (with vanishing Fourier coefficients at 0). Then by Theorem 2.20, ω k p (x, ε) ≤ ω k p (y, ε) + ω k p (z, ε) y p + ε k |z| W k p , which implies ω k p (x, ε) K(x, ε k ). The converse inequality is harder. We have to produce an appropriate decomposition of x. To this end, let I = [0, 1) d and define the required decomposition by y = (−1) k I · · · I ∆ k εu (x)du 1 · · · du k and z = x − y,
where u = u 1 + · · · + u k . Then
y p ≤ I · · · I ∆ k εu (x) p du 1 · · · du k ≤ ω k p (x, k √ d ε) ω k p (x, ε).
To handle z, using the formula
∆ k εu = k j=0 (−1) k−j k j T jεu ,
we rewrite z as
z = (−1) k+1 k j=1 (−1) k−j k j I · · · I T jεu (x)du 1 · · · du k .
All terms on the right-hand side are treated in the same way. Let us consider only the first one by setting
z 1 = I · · · I T εu (x)du 1 · · · du k .
Write each u i in the canonical basis of R d :
u i = d j=1 u i,j e j .
We compute ∂ 1 z 1 explicitly, as example, in the spirit of (2.1):
∂ 1 z 1 = 1 ε k i=1 I · · · I ∂ ∂u i,1 T εu (x)du 1 · · · du k .
Integrating the partial derivative on the right-hand side with respect to u i,1 yields:
1 0 ∂ ∂u i,1 T εu (x)du i,1 = ∆ ε(e1+u−ui,1e1) (x) = 1 0 ∆ ε(e1+u−ui,1e1) (x)du i,1 ,
where for the second equality, we have used the fact that ∆ ε(e1+u−ui,1e1) (x) is constant in u i,1 . Thus
∂ 1 z 1 = 1 ε k i=1 I · · · I ∆ ε(e1+u−ui1e1) (x)du 1 · · · du k .
To iterate this formula, we use multi-index notation. For n ∈ N let
[[k]] n = i = (i 1 , · · · , i n ) : 1 ≤ i ℓ ≤ k, all i ℓ 's are distinct .
Then for any m 1 ∈ N with m 1 ≤ k, we have
∂ m1 1 z 1 = ε −m1 i 1 ∈[[k]] m 1 I · · · I ∆ m1 εu i 1 (x)du 1 · · · du k ,
where u i 1 = e 1 + u − (u i 1 1 ,1 + · · · + u i 1 m 1 ,1 )e 1 . Iterating this procedure, for any m ∈ N d 0 with |m| 1 = k, we get
D m z 1 = ε −k i d ∈[[k]] m d · · · i 1 ∈[[k]] m 1 I k ∆ m d εui d · · · ∆ m1 εui 1 (x)du 1 · · · du k ,
where the u i j 's are defined by induction u i j = e j + u i j−1 − (u i j 1 ,j + · · · + u i j m j ,j )e j , j = 2, · · · , d. Thus we are in a position of appealing Lemma 2.22 to conclude that
D m z 1 p ε −k ω k p (x, ε), whence |z| W k p ε −k ω k p (x, ε). Therefore, K(x, ε k ) ω k p (x, ε).
Remark 5.17. The preceding proof shows a little bit more: for any x ∈ W k p (T d θ ) with x(0) = 0, ω k p (x, ε) ≈ y p + ε k |z| W k p : x = y + z, y(0) = z(0) = 0 , 0 < ε ≤ 1. In particular, this implies
x p ω k p (x, ε), which is the analogue for moduli of L p -continuity of the inequality in Theorem 2.12 (the Poincaré inequality). On the other hand, together with Lemma 2.22, the above inequality provides an alternate proof of Theorem 2.12.
The preceding theorem, together with Theorem 3.16 and the reiteration theorem, implies the following Corollary 5.18. Let 0 < η < 1, α > 0, k, k 0 , k 1 ∈ N and 1 ≤ p, q, q 1 ≤ ∞. Then
(i) L p (T d θ ), W k p (T d θ ) η,q = B ηk p,q (T d θ ); (ii) W k p (T d θ ), B α p,q1 (T d θ ) η,q = B β p,q (T d θ ) , k = α, β = (1 − η)k + ηα; (iii) W k0 p (T d θ ), W k1 p (T d θ ) η,q = B α p,q (T d θ ) , k 0 = k 1 , α = (1 − η)k 0 + ηk 1 .
We can also consider the complex interpolation of L p (T d θ ), W k p (T d θ ) . If 1 < p < ∞, this is reduced to that of L p (T d θ ), H k p (T d θ ) ; so by the result of the previous section, for any 0 < η < 1,
L p (T d θ ), W k p (T d θ ) η = H ηk p (T d θ ).
Problem 5.19. Does the above equality hold for p = 1? The problem is closely related to that in Remark 5.11.
We conclude this section with a remark on the link between Theorem 3.19 and Theorem 5.16. The former can be easily deduced from the latter, by using the following elementary fact (see [8] p. 40): for any couple (X 0 , X 1 ) of Banach spaces and x ∈ X 0 ∩ X 1
lim η→1 η(1 − η) 1 q x (X0, X1)η,q = q − 1 q x X1 , lim η→0 η(1 − η) 1 q x (X0, X1)η,q = q − 1 q x X0 .
Here the norm of (X 0 , X 1 ) η,q is that defined by the K-functional. Then Theorem 3.19 follows from Theorem 5.16. and the first limit above. This is the approach adopted in [38,47]. It also allows us to determine the other extreme case α = 0 in Theorem 3.19, which was done by Maz'ya and Shaposhnikova [41] in the commutative case. Let us record this result here.
Corollary 5.20. Let 1 ≤ p ≤ ∞ and 1 ≤ q < ∞. Then for x ∈ B α0 p,q (T d θ ) with x(0) = 0 for some α 0 > 0, lim
α→0 α 1 q x B α,ω p,q ≈ q − 1 q x p .
Interpolation of Triebel-Lizorkin spaces
This short section contains some simple results on the interpolation of Triebel-Lizorkin spaces. They are similar to those for potential Sobolev spaces presented in section 5.1. It is surprising, however, that the real interpolation spaces of F α,c p (T d θ ) for a fixed p do not depend on the column structure.
Proposition 5.21. Let 1 ≤ p, q ≤ ∞ and α 0 , α 1 ∈ R with α 0 = α 1 . Then Proof. The assertion is an immediate consequence of Proposition 4.10 (v) and Proposition 5.1 (i). Note, however, that Proposition 4.10 (v) is stated for p < ∞; but by duality via Proposition 4.14, it continues to hold for p = ∞.
F α0,c p (T d θ ), F α1,c p (T d θ ) η,q = B α p,q (T d θ ), α = (1 − η)α 0 + ηα 1 .
On the other hand, the interpolation of F α,c p (T d θ ) for a fixed α is reduced to that of Hardy spaces by virtue of Proposition 4.10 (iv) and Lemma 1.9.
Remark 5.22. Let α ∈ R and 1 < p < ∞. Then
F α,c ∞ (T d θ ), F α,c 1 (T d θ ) 1 p = F α,c p (T d θ ) = F α,c ∞ (T d θ ), F α,c 1 (T d θ ) 1 p ,p . Proposition 5.23. Let α 0 , α 1 ∈ R and 1 < p < ∞. Then F α0,c ∞ (T d θ ), F α1,c 1 (T d θ ) 1 p = F α,c p (T d θ ), α = (1 − 1 p )α 0 + α 1 p .
Proof. This proof is similar to that of Theorem 5.6. Let x be in the unit ball of F α,c p (T d θ ). Then by Proposition 4.10, J α (x) ∈ H c p (T d θ ). Thus by Lemma 1.9, there exists a continuous function f from the strip S = {z ∈ C : 0 ≤ Re(z) ≤ 1} to H c 1 (T d θ ), analytic in the interior, such that f ( 1 p ) = J α (x) and such that sup
t∈R f (it) BMO c ≤ c, sup t∈R f (1 + it) H c 1 ≤ c. Define F (z) = e (z− 1 p ) 2 J −(1−z)α0−zα1 f (z)
, z ∈ S. By Remark 4.15 and Lemma 5.8, for any t ∈ R,
F (it) F α,c ∞ ≈ e −t 2 + 1 p 2 J it(α0−α1) f (it) BMO c ≤ c ′ . Similarly, F (1 + it) F α,c 1 ≈ e −t 2 +(1− 1 p ) 2 J it(α0−α1) f (1 + it) H c 1 ≤ c ′ .
Therefore,
x = F ( 1 p ) ∈ F α0,c ∞ (T d θ ), F α1,c 1 (T d θ ) 1 p , whence F α,c p (T d θ ) ⊂ F α0,c ∞ (T d θ ), F α1,c 1 (T d θ ) 1 p .
The converse inclusion is obtained by duality.
Chapter 6. Embedding
We consider the embedding problem in this chapter. We begin with Besov spaces, then pass to Sobolev spaces. Our embedding theorem for Besov spaces is complete; however, the embedding problem of W 1 1 (T d θ ) is, unfortunately, left unsolved at the time of this writing. The last section deals with the compact embedding.
Embedding of Besov spaces
This section deals with the embedding of Besov spaces. We will follow the semigroup approach developed by Varopolous [75] (see also [20,76]). This approach can be adapted to the noncommutative setting, which has been done by Junge and Mei [33]. Here we can use either the circular Poisson or heat semigroup of T d θ , already considered in section 3.3. We choose to work with the latter. Recall that for
x ∈ S ′ (T d θ ), W r (x) = m∈Z d x(m)r |m| 2 U m , 0 ≤ r < 1.
The following elementary lemma will be crucial.
Lemma 6.1. Let 1 ≤ p ≤ p 1 ≤ ∞. Then (6.1) W r (x) p1 (1 − r) d 2 ( 1 p 1 − 1 p ) x p , x ∈ L p (T d θ )
, 0 ≤ r < 1. Proof. Consider first the case p = 1 and p 1 = ∞. Then
W r (x) ∞ ≤ m∈Z d r |m| 2 | x(m)| ≤ x 1 m∈Z d r |m| 2 = x 1 k≥0 r k |m| 2 =k 1 x 1 k≥0 (1 + k) d 2 r k ≈ (1 − r) − d 2 x 1 .
The general case easily follows from this special one by interpolation. Indeed, the inequality just proved means that W r is bounded from
L 1 (T d θ ) to L ∞ (T d θ ) with norm controlled by (1 − r) − d 2 .
On the other hand, W r is a contraction on L p (T d θ ) for 1 ≤ p ≤ ∞. Interpolating these two cases, we get (6.1) for 1 < p < p 1 = ∞. The remaining case p 1 < ∞ is treated similarly.
The following is the main theorem of this section.
Theorem 6.2. Assume that 1 ≤ p < p 1 ≤ ∞, 1 ≤ q ≤ q 1 ≤ ∞ and α, α 1 ∈ R such that α − d p = α 1 − d p1 .
Then we have the following continuous inclusion:
B α p,q (T d θ ) ⊂ B α1 p1,q1 (T d θ ) . Proof. Since B α1 p1,q (T d θ ) ⊂ B α1 p1,q1 (T d θ )
, it suffices to consider the case q = q 1 . On the other hand, by the lifting Theorem 3.7, we can assume max{α, α 1 } < 0, so that we can take k = 0 in Theorem 3.15. Thus, we are reduced to showing
1 0 (1 − r) − qα 1 2 W r (x) q p1 dr 1 − r 1 q 1 0 (1 − r) − qα 2 W r (x) q p dr 1 − r 1 q .
To this end, we write W r (x) = W √ r W √ r (x) and apply (6.1) to get
W r (x) p1 (1 − √ r) d 2 ( 1 p 1 − 1 p ) W √ r (x) p . Thus 1 0 (1 − r) − qα 1 2 W r (x) q p1 dr 1 − r 1 q 1 0 (1 − r) − qα 1 2 (1 − √ r) qd 2 ( 1 p 1 − 1 p ) W √ r (x) q p dr 1 − r 1 q = 1 0 (1 − r 2 ) − qα 1 2 (1 − r) qd 2 ( 1 p 1 − 1 p ) W r (x) q p 2rdr 1 − r 2 1 q 1 0 (1 − r) − qα 2 W r (x) q p dr 1 − r 1 q ,
as desired.
Corollary 6.3. Assume that 1 ≤ p < p 1 ≤ ∞, 1 ≤ q ≤ ∞ and α = d( 1 p − 1 p1 ). Then B α p,q (T d θ ) ⊂ L p1,q (T d θ ) if p 1 < ∞ and B α p,1 (T d θ ) ⊂ L ∞ (T d θ ) if p 1 = ∞ . Proof.
Applying the previous theorem to α 1 = 0 and q = q 1 = 1, and by Theorem 3.8, we get
B α p,1 (T d θ ) ⊂ B 0 p1,1 (T d θ ) ⊂ L p1 (T d θ )
. This gives the assertion in the case p 1 = ∞. For p 1 < ∞, we fix p and choose two appropriate values of α (which give the two corresponding values of p 1 ); then we interpolate the resulting embeddings as above by real interpolation; finally, using (1.1) and Proposition 5.1, we obtain the announced embedding for p 1 < ∞. The preceding corollary admits a self-improvement in terms of modulus of smoothness.
Corollary 6.4. Assume that 1 ≤ p < p 1 ≤ ∞, α = d( 1 p − 1 p1 ) and k ∈ N such that k > α.
Then
ω k p1 (x, ε) ε 0 δ −α ω k p (x, δ) dδ δ , 0 < ε ≤ 1.
Proof. Without loss of generality, assume x(0) = 0. Then by the preceding corollary and Theorem 3.16, we have
x p1 1 0 δ −α ω k p (x, δ) dδ δ .
Now let u ∈ R d with |u| ≤ ε. Noting that ω k p (∆ u (x), δ) ≤ 2 k min ω k p (x, ε), ω k p (x, δ) ≤ 2 k ω k p (x, min(ε, δ)), Remark 6.8. Part (ii) of the preceding theorem implies W d p (T d θ ) ⊂ L ∞ (T d θ ) for all p > 1. In the commutative case, representing a function as an indefinite integral of its derivatives, one easily checks that this embedding remains true for p = 1. However, we do not know how to prove it in the noncommutative case. A related question concerns the embedding W k p (T d θ ) ⊂ B α1 ∞,∞ (T d θ ) in the case of odd k which is not covered by the same part (ii).
The quantum analogue of the Gagliardo-Nirenberg inequality can be also proved easily by interpolation. Proposition 6.9. Let k ∈ N, 1 < r, p < ∞, 1 ≤ q < ∞ and β ∈ N d 0 with 0 < |β| 1 < k. If
η = |β| 1 k and 1 r = 1 − η q + η p , then for every x ∈ W k p (T d θ ) L q (T d θ ), D β x r x 1−η q |m|=k D m x p η .
Proof. This inequality immediately follows from Theorem 5.13 and the well-known relation between real and complex interpolations:
L q (T d θ ), W k p (T d θ ) η,1 ⊂ L q (T d θ ), W k p (T d θ ) η = W |β|1 r (T d θ ).
It then follows that
x W |β| 1 r x 1−η q x η W k p .
Applying this inequality to x − x(0) instead of x and using Theorem 2.12, we get the desired Gagliardo-Nirenberg inequality.
An alternate approach to Sobolev embedding. Note that the preceding proof of Theorem 6.6 is based on Theorem 6.2, which is, in its turn, proved by Varopolous' semigroup approach. Varopolous initially developed his method for the Sobolev embedding, which was transferred to the noncommutative setting by Junge and Mei [33]. Our argument for the embedding of Besov spaces has followed this route. Let us now give an alternate proof of Theorem 6.6 (i) by the same way. We state its main part as the following lemma that is of interest in its own right.
Lemma 6.10. Let 1 ≤ p < q < ∞ such that 1 q = 1 p − 1 d . Then
W 1 p (T d θ ) ⊂ L q,∞ (T d θ ).
Proof. We will use again the heat semigroup W r of T d θ . Recall that W r = W ε with r = e −4π 2 ε , where W ε is the periodization of the usual heat kernel W ε of R d (see section 3.3). It is more convenient to work with W ε . In the following, we assume x ∈ S(T d θ ) and
x(0) = 0. Let ∆ j = ∆ −1 ∂ j , 1 ≤ j ≤ d. Then ∆ −1 x = 4π 2 ∞ 0 W ε (x)dε and ∆ j x = 4π 2 ∞ 0 W ε (∂ j x) dε.
We claim that for any 1 ≤ p ≤ ∞
(6.2) W ε (∂ j x) p ε − 1 2 x p and W ε (∂ j x) ∞ ε − 1 2 ( d p +1) x p , ε > 0.
Indeed, in order to prove the first inequality, by the transference method, it suffices to show a similar one for the Banach space valued heat semigroup of the usual d-torus. The latter immediately follows from the following standard estimate on the heat kernel W ε of R d :
sup ε>0 ε 1 2 R d ∇W ε (s) ds < ∞.
The second inequality of (6.2) is proved in the same way as (6.1). First, for the case p = 1, we have (recalling that x(0) = 0)
W ε (∂ j x) ∞ ≤ 2π m∈Z d \{0} |m j |e −ε|m| 2 | x(m)| ≤ 2π x 1 m∈Z d \{0} |m j |e −ε|m| 2 e −ε (1 − e −ε ) − d+1 2 x 1 ε − d+1 2 x 1 .
Interpolating this with the first inequality for p = ∞, we get the second one in the general case. Now let ε > 0 and decompose ∆ j x into the following two parts:
y = 4π 2 ∞ ε W δ (∂ j x) dδ and z = 4π 2 ε 0 W δ (∂ j x) dδ.
Then by (6.2),
y ∞ x p ∞ ε δ − 1 2 ( d p +1) dδ ≈ ε − 1 2 ( d p −1) x p and z p x p ε 0 δ − 1 2 dδ ≈ ε 1 2 x p .
Thus for any t > 0, choosing ε such that ε − d 2p = t, we deduce
y ∞ + t z p t 1− p d x p = t η x p , where η = 1 − p d . It then follows that ∆ j x q,∞ ≈ ∆ j x (L∞(T d θ ), Lp(T d θ ))η,∞ x p . Since x = − d j=1 ∆ j ∂ j x,
we finally get
x q,∞ d j=1 ∆ j ∂ j x q,∞ d j=1 ∂ j x p = ∇x p .
Thus the lemma is proved.
Alternate proof of Theorem 6.6 (i). For 1 < p < d, choose p 0 , p 1 such that 1 < p 0 < p < p 1 < d. Let 1 qi = 1 pi − 1 d for i = 0, 1. Then by the previous lemma, W 1 pi (T d θ ) ⊂ L qi,∞ (T d θ ), i = 0, 1. Interpolating these two inclusions by real method, we obtain
W 1 p (T d θ ) ⊂ L q,p (T d θ )
. This is the embedding of Sobolev spaces in Theorem 6.6 (i) for k = 1. The case k > 1 immediately follows by iteration. Then using real interpolation, we deduce the embedding of potential Sobolev spaces.
Sobolev embedding for p = 1. Now we discuss the case p = 1 which is not covered by Theorem 6.6. The main problem concerns the following:
(6.3) W 1 1 (T d θ ) ⊂ L d d−1 (T d θ )
. At the time of this writing, we are unable, unfortunately, to prove it. However, Lemma 6.10 provides a weak substitute, namely,
(6.4) W 1 1 (T d θ ) ⊂ L d d−1 ,∞ (T d θ )
. In the classical case, one can rather easily deduce (6.3) from (6.4). Let us explain the idea coming from [76, page 58]. It was kindly pointed out to us by Marius Junge. Let f be a nice real function on T d with f (0) = 0. For any t ∈ R let f t be the indicator function of the subset {f > t}. Then f can be decomposed as an integral of the f t 's:
(6.5) f = +∞ −∞ f t dt.
By triangular inequality (with q = d d−1 ),
f q ≤ +∞ −∞ f t q dt.
However, f t q = f t q,∞ ∀ t ∈ R. Thus by (6.4) for θ = 0, f t q f t 1 + ∇f t 1 . It comes now the crucial point which is the following
(6.6) +∞ −∞ ∇f t 1 dt ∇f 1 .
In fact, the two sides are equal in view of Sard's theorem. We then get the strong embedding (6.3) in the case θ = 0. Note that this proof yields a stronger embedding:
(6.7) W 1 1 (T d ) ⊂ L d d−1 ,1 (T d ).
The above decomposition of f is not smooth in the sense that f t is not derivable even though f is nice. In his proof of Hardy's inequality in Sobolev spaces, Bourgain [11] discovered independently the same decomposition but using nicer functions f t (see also [60]). Using (6.7) and the Hausdorff-Young inequality, Bourgain derived the following Hardy type inequality (assuming d ≥ 3):
m∈Z d | f (m)| (1 + |m|) d−1 f W 1 1 (T d ) .
We have encountered difficulties in the attempt of extending this approach to the noncommutative case. Let us formulate the corresponding open problems explicitly as follows: Problem 6.11. Let d ≥ 2.
(i) Does one have the following embedding By the previous discussion, part (i) is reduced to a decomposition for operators in W 1 1 (T d θ ) of the form (6.5) and satisfying (6.6). One could be attempted to do this by transference by first considering operator-valued functions on R d . With this in mind, the following observation, due to Marius Junge, might be helpful.
Given an interval I = [s, t] ⊂ R and an element a ∈ L 1 (T d θ ), we have ∂(1 I ⊗ a) = δ s ⊗ a − δ t ⊗ a, where ∂ denotes the distribution derivative relative to R. Let L denote the norm of the dual space C 0 (R; A θ ) * , which contains L 1 (R; L 1 (T d θ )) isometrically. If f is a (nice) linear combination of 1 I ⊗ a's, then we have the desired decomposition of f . Indeed, assume f = n i=1 α i 1 Ii ⊗ e i , where α i ∈ R + and the 1 Ii ⊗ e i 's are pairwise disjoint projections of L ∞ (R)⊗T d θ . Let f t = 1 (t,∞) (f ). Then
f = ∞ 0 f t dt.
So for any q ≥ 1,
f q ≤ ∞ 0 f t q dt.
On the other hand, by writing explicitly f t for every t, one easily checks ∂f L = ∞ 0 ∂f t L dt.
By iteration, the above decomposition can be extended to higher dimensional case for all functions f of the form n i=1 α i 1 Ri ⊗ e i , where α i ∈ R + , R i 's are rectangles (with sides parallel to the axes) and 1 Ri ⊗ e i 's are pairwise disjoint projections of L ∞ (R d )⊗T d θ . The next idea would be to apply Lemma 6.10 to these special functions. Then two difficulties come up to us, even in the commutative case. The first is that these functions do not belong to W 1 1 ; this difficulty can be resolved quite easily by regularization. The second one, substantial, is the density of these functions, more precisely, of suitable regularizations of them, in W 1 1 . Uniform Besov embedding. We end this section with a discussion on the link between a certain uniform embedding of Besov spaces and the embedding of Sobolev spaces. Let 0 < α < 1, 1 ≤ p < ∞ with αp < d and 1 r = 1 p − α d . Then
(6.8) x p r ≤ c d,p α(1 − α) (d − αp) x p B α,ω p,p , x ∈ B α p,p (T d θ ) ,
where x B α,ω p,p is the Besov norm defined by (3.19). In the commutative case, this inequality is proved in [13] for α close to 1 and in [41] for general α. One can show that (6.8) is essentially equivalent to the embedding of W 1 p (T d θ ) into L q (T d θ ) (or L q,p (T d θ )) for d > p and 1 q = 1 p − 1 d . Indeed, assume(6.8). Then taking limit in both sides of (6.8) as α → 1, by Theorem 3.19, we get 1 q x B α,ω p,p . This implies a variant of (6.8) since L r,p (T d θ ) ⊂ L r (T d θ ). Since we have proved the embedding W 1 p (T d θ ) ⊂ L q (T d θ ) for p > 1, (6.8) holds for p > 1. Let us record this explicitly as follows: Proposition 6.12. Let 0 < α < 1, 1 < p < ∞ with αp < d and 1 r = 1 p − α d . Then
x q |x| W 1 p for all x ∈ W 1 p (T d θ ) with x(0) = 0. Conversely, if W 1 p (T d θ ) ⊂ L q (T d θ ), then L p (T d θ ), W 1 p (T d θ ) α,p ⊂ L p (T d θ ), L q (T d θ ) α,p .x r α(1 − α) 1 p x p B α,ω p,p , x ∈ B α p,p (T d θ ) with relevant constant independent of α.
In the case p = 1, Problem 6.11 (i) is equivalent to (6.8) for p = 1 and α close to 1.
Compact embedding
This section deals with the compact embedding. The case p = 2 for potential Sobolev spaces was solved by Spera [64]:
Lemma 6.13. The embedding H α1 2 (T d θ ) ֒→ H α2 2 (T d θ ) is compact for α 1 > α 2 ≥ 0. We will require the following real interpolation result on compact operators, due to Cwikel [19]. Lemma 6.14. Let (X 0 , X 1 ) and (Y 0 , Y 1 ) be two interpolation couples of Banach spaces, and let T : X j → Y j be a bounded linear operator, j = 0, 1. If T : X 0 → Y 0 is compact, then T : (X 0 , X 1 ) η,p → (Y 0 , Y 1 ) η,p is compact too for any 0 < η < 1 and 1 ≤ p ≤ ∞. Theorem 6.15. Assume that 1 ≤ p < p 1 ≤ ∞, 1 ≤ p * < p 1 , 1 ≤ q ≤ q 1 ≤ ∞ and α − d p = α 1 − d p1 . Then the embedding B α p,q (T d θ ) ֒→ B α1 p * ,q1 (T d θ ) is compact. Proof. Without loss of generality, we can assume q = q 1 . First consider the case p = 2. Choose t sufficiently close to q and 0 < η < 1 such that 1 q = 1 − η 2 + η t .
Then by Proposition 5.1, B α 2,q (T d θ ) = B α 2,2 (T d θ ), B α 2,t (T d θ ) η,q . By Lemma 6.13, B α 2,2 (T d θ ) ֒→ B α1 2,2 (T d θ ) is compact. On the other hand, by Theorem 6.2, B α 2,t (T d θ ) ֒→ B α1 p1,t (T d θ ) is continuous. So by Lemma 6.14, B α 2,q (T d θ ) ֒→ B α1 2,2 (T d θ ), B α1 p1,t (T d θ ) η,q is compact. However, by the proof of Proposition 5.1 and (1.1), we have B α1 2,2 (T d θ ), B α1 p1,t (T d θ ) η,q ⊂ ℓ α1 q (L 2 (T d θ ), L p1 (T d θ ) η,q = ℓ α1 q (L s,q (T d θ )), where s is determined by
1 s = 1 − η 2 + η p 1 = 1 p 1 + (1 − η)(α − α 1 ) d .
Note that η tends to 1 as t tends to q, so we can choose η such that s > p * . Then L s,q (T d θ ) ⊂ L p * (T d θ ). Thus the desired assertion for p = 2 follows. The case p = 2 but p > 1 is dealt with similarly. Let t and η be as above. Choose r < p (r close to p). Then
B α 2,2 (T d θ ), B α r,t (T d θ ) η,q ⊂ ℓ α q (L 2 (T d θ ), L r (T d θ ) η,q = ℓ α q (L p0,q (T d θ )), where p 0 is determined by 1 p 0 = 1 − η 2 + η r .
If η is sufficiently close to 1, then p 0 < p that we will assume. Thus L p (T d θ ) ⊂ L p0,q (T d θ ). It then follows that B α p,q (T d θ ) ⊂ B α 2,2 (T d θ ), B α r,t (T d θ ) η,q . The rest of the proof is almost the same as the case p = 2, so is omitted.
The remaining case p = 1 can be easily reduced to the previous one. Indeed, first embed B α p,q (T d θ ) into B α2 p2,q (T d θ ) for some α 2 ∈ (α, α 1 ) (α 2 close to α) and p 2 determined by α − d p = α 2 − d p2 . Then by the previous case, the embedding B α2 p2,q (T d θ ) ֒→ B α1 p * ,q1 (T d θ ) is compact, so we are done. Theorem 6.16. Let 1 < p < p 1 < ∞ and α, α 1 ∈ R.
(
i) If α − d p = α 1 − d p1 , then H α p (T d θ ) ֒→ H α1 p * (T d θ )
is compact for p * < p 1 . In particular, if additionally α = k and α 1 = k 1 are nonnegative integers, then W k
p (T d θ ) ֒→ W k1 p * (T d θ ) is compact. (ii) If p(α − α 1 ) > d and α * < α 1 = α − d p , then H α p (T d θ ) ֒→ B α * ∞,∞ (T d θ )
is compact. In particular, if additionally α = k ∈ N, then W k p (T d θ ) ֒→ B α * ∞,∞ (T d θ ) is compact. Proof. Based on the preceding theorem, this proof is similar to that of Theorem 6.6 and left to the reader.
Chapter 7. Fourier multiplier
This chapter deals with Fourier multipliers on Sobolev, Besov and Triebel-Lizorkin spaces on T d θ . The first section concerns the Sobolev spaces. Its main result is the analogue for W k p (T d θ ) of [17,Theorem 7.3] (see also Lemma 1.3) on c.b. Fourier multipliers on L p (T d θ ); so the space of c.b. Fourier multipliers on W k p (T d θ ) is independent of θ. The second section turns to Besov spaces on which Fourier multipliers behave better. We extend some classical results to the present setting. We show that the space of c.b. Fourier multipliers on B α p,q (T d θ ) does not depend θ (nor on q or α). We also prove that a function on Z d is a Fourier multiplier on B α 1,q (T d θ ) iff it is the Fourier transform of an element of B 0 1,∞ (T d ). The last section deals with Fourier multipliers on Triebel-Lizorkin spaces.
Fourier multipliers on Sobolev spaces
We now investigate Fourier multipliers on Sobolev spaces. We refer to [59,9] for the study of Fourier multipliers on the classical Sobolev spaces. If X is a Banach space of distributions on T d θ , we denote by M(X) the space of bounded Fourier multipliers on X; if X is further equipped with an operator space structure, M cb (X) is the space of c.b. Fourier multipliers on X. These spaces are endowed with their natural norms. Recall that the Sobolev spaces W k p (T d θ ), H α p (T d θ ) and the Besov B α p,q (T d θ ) are equipped with their natural operator space structures as defined in Remarks 2.29 and 3.24.
The aim of this section is to extend [17,Theorem 7.3] (see also Lemma 1.3) on c.b. Fourier multipliers on L p (T d θ ) to Sobolev spaces. Inspired by Neuwirth and Ricard's transference theorem [48], we will relate Fourier multipliers with Schur multipliers. Given a distribution x on T d θ , we write its matrix in the basis (U m ) m∈Z d : where Sφ is the Schur multiplier with symbolφ. According to the definition of W k p (T d θ ), for any matrix a = (a m,n ) m,n∈Z d and ℓ ∈ N d 0 define D ℓ a = (2πi(m − n)) ℓ a m,n m,n∈Z d .
If x is a distribution on T d θ , then clearly
M φ D ℓ x = Sφ D ℓ [x] .
We introduce the space S k p = a = (a m,n ) m,n∈Z d : D ℓ a ∈ S p (ℓ 2 (Z d )), ∀ ℓ ∈ N d 0 , 0 ≤ |ℓ| 1 ≤ k and endow it with the norm
a S k p = 0≤|ℓ|1≤k D ℓ a p Sp 1 p .
Then S k p is a closed subspace of the ℓ p -direct sum of L copies of S p (ℓ 2 (Z d )) with L = 0≤|ℓ|1≤k 1. The latter direct sum is equipped with its natural operator space structure, which induces an operator space structure on S k p too. If ψ = (ψ m,n ) m,n∈Z d is a complex matrix, its associated Schur multiplier S ψ on S k p is defined by S ψ a = (ψ m,n a m,n ) m,n∈Z d . Let M cb (S k p ) denote the space of all c.b. Schur multipliers on S k p , equipped with the natural norm. Now assume thatφ is a c.b. Schur multiplier on S k p , then it is also a c.b. Schur multiplier on S k p (ℓ
Fourier multipliers on Besov spaces
It is well known that in the classical setting, Fourier multipliers behave better on Besov spaces than on L p -spaces. We will see that this fact remains true in the quantum case. We maintain the notation introduced in section 3.1. In particular, ϕ is a function satisfying (3.1) and ϕ (k) (ξ) = Conversely, assume φ = f with f ∈ B 0 1,∞ (T d ). Let g ∈ B 0 1,∞ (T d ). Then ϕ k * M φ (g) 1 = ϕ k * f * g 1 = ϕ k * f * ( ϕ k−1 + ϕ k + ϕ k+1 ) * g 1 ≤ ϕ k * f 1 ( ϕ k−1 + ϕ k + ϕ k+1 ) * g 1 , which allows us to conclude the proof too.
≤ 3 f B 0 1,∞ g B 0 1,∞ . Thus M φ (g) ∈ B 0 1,∞ (T d ) and M φ (g) B 0 1,∞ ≤ 3 f B 0 1,∞ g B 0 1,
We have seen previously that every bounded (c.b.) Fourier multiplier on L p (T d θ ) is a bounded (c.b.) Fourier multiplier on B α p,q (T d θ ). Corollary 7.7 shows that the converse is false for p = 1. We now show that it also is false for any p = 2.
Proposition 7.8. There exists a Fourier multiplier φ which is c.b. on B α p,q (T d θ ) for any p, q and α but never belongs to M(L p (T d θ )) for any p = 2 and any θ. Proof. The example constructed by Stein and Zygmund [69] for a similar circumstance can be shown to work for our setting too. Their example is a distribution on T defined as follows:
µ(z) = ∞ n=2 1 log n (w n z) 2 n D n (w n z)
for some appropriate w n ∈ T, where D n (z) = n j=0 z j , z ∈ T.
Since D n L1(T) ≈ log n, we see that µ ∈ B 0 1,∞ (T). Considered as a distribution on T d , µ ∈ B 0 1,∞ (T d ) too. Thus by Corollaries 7.5 and 7.7, φ = µ belongs to M cb (B α p,p (T d θ )) for any p, q and α. However, Stein and Zygmund proved that φ is not a Fourier multiplier on L p (T) for any p = 2 if the w n 's are appropriately chosen. Consequently, φ cannot be a Fourier multiplier on L p (T d θ ) for any p = 2 and any θ since L p (T) isometrically embeds into L p (T d θ ) by an embedding that is also a c.b. Fourier multiplier.
We conclude this section with some comments on the vector-valued case. The proof of Theorem 7.4 works equally for vector-valued Besov spaces. Recall that for an operator space E, B α p,q (T d θ ; E) denotes the E-valued Besov space on T d θ (see Remark 3.24). with equivalence constants depending only on α.
If θ = 0, we go back to the classical vector-valued case. The above proposition explains the well-known fact mentioned at the beginning of this section that Fourier multipliers behave better on Besov spaces than on L p -spaces. To see this, it is more convenient to write the above right-hand side in a different form:
φϕ (k)
M(Lp(T d θ ;E)) = φ(2 k ·)ϕ M(Lp(T d θ ;E)) . Thus if φ is homogeneous, the above multiplier norm is independent of k, so φ is a Fourier multiplier on B α p,q (T d ; E) for any p, q, α and any Banach space E. In particular, the Riesz transform is bounded on B α p,q (T d ; E). The preceding characterization of Fourier multipliers is, of course, valid for R d in place of T d . Let us record this here:
Lemma 4 . 3 .
43Let φ = (φ j ) j≥0 be a sequence of continuous functions on R d \ {0}, viewed as a function from R d to ℓ 2 . Assume that
Lemma 4 . 4 .
44Let φ = (φ j ) j≥0 be a sequence of continuous functions on R d \ {0} such that
we need only to show that for any atom a k(a) L1(N ;ℓ c 2 )
L1(N ;ℓ c 2 ) ≤ k(a)1 Q L1(N ;ℓ c 2 ) + k(a)1 I d \ Q L1(N ;ℓ c 2 ). The operator convexity of the square function x → |x| 2 implies
Theorem 4 . 11 .
411Let 1 ≤ p < ∞ and α ∈ R.
Theorem 4 . 16 .
416Let φ be a continuous function on R d \ {0} such that
Lemma 5 . 8 .
58Let t ∈ R. Then J it is bounded on both H 1 (T d θ ) and BMO(T d θ ) with norms majorized by c d (1 + |t|) d .
Remark 5 . 14 .
514The second part of the previous theorem had been proved by Marius Junge by a different method; he reduced it to the corresponding problem on H 1 too. The main problem left open at this stage is the following: Problem 5.15. Does the second part of the previous theorem hold for p 0 = ∞?
Similar statements hold for the row and mixture Triebel-Lizorkin spaces.
Theorem 5.16 implies that L p (T d θ ), W 1 p (T d θ ) α,p ⊂ B α p,p (T d θ ) with relevant constant depending only on d, here B α p,p (T d θ ) being equipped with the norm B α,ω p,p . On the other hand, By a classical result of Holmstedt [28] on real interpolation of L p -spaces (which readily extends to the noncommutative case, as observed in [37, Lemma 3.7]),L p (T d θ ), L q (T d θ ) α,p ⊂ L r,p
.
[x] = xU n , U m m,n∈Z d = x(m − n)e inθ(m−n) t m,n∈Z d .Here k t denotes the transpose of k = (k 1 , . . . , k d ) andθ is the following d × d-matrix deduced from the skew symmetric matrix θ:θ Now let φ : Z d → C and M φ be the associated Fourier multiplier on T d θ . Setφ = φ m−n m,n∈Z d . Then(7.1) M φ x = φ m−n x(m − n)e inθ(m−n) t m,n∈Z d = Sφ([x]),
Considering g with values in S ∞ , we show that φ is c.b. too. Alternately, since M(L 1 (T d )) = M cb (L 1 (T d )), Theorem 7.4 yields M(B 0 1,∞ (T d )) = M cb (B 0 1,∞ (T d ))
Almost all previous results remain valid in this vector-valued setting since all Fourier multipliers used in their proofs are c.b. maps. For instance, Theorem 2.9 (or Remark 2.26) now becomes H k pconsists
of continuous linear maps from S(T d
θ ) to E. Then as in Definition 2.6, we define the correspond-
ing Sobolev spaces W k
p (T d
θ ; E) and H α
p (T d
θ ; E).
25, p. 211-214], [67, p. 245-247] or [71, p. 161-165] can be easily adapted to the present setting.
The last two properties of the kernel k are exactly what is needed for the estimates of A and B in the proof of Lemma 4.4, so the conclusion holds when M = C.However, we do not know whether
Lemma 4.4 remains true when (4.4) is weakened to (4.5).
Lemma 4.6. Let φ = (φ j ) j≥0 be a sequence of continuous functions on R d \ {0} satisfying (4.4).
Then for 1 ≤ p ≤ 2 and any f ∈ H c
p
). Indeed, this is a variant of Lemma 3.5 with a = k − α 1 and ρ = h P.
(T 2 θ ). We show in the same way that it is c.b. too. Since T is not a measure, it does not belong to M(L 1 (T 2 )).
Acknowledgements. We wish to thank Marius Junge for discussions on the embedding and interpolation of Sobolev spaces, Tao Mei for discussions on characterizations of Hardy spaces and Eric Ricard for discussions on Fourier multipliers and comments. We are also grateful to Fedor Sukochev for comments on a preliminary version of the paper. We acknowledge the financial supports of ANR-2011-BS01-008-01, NSFC grant(No. 11271292, 11301401 and 11431011).Theorem 4.19. Under the assumption of the previous theorem, for any distribution x on T d θ ,Proof. This proof is very similar to that of Theorem 4.17. The main idea is, of course, to discretize the continuous square function:We can further discretize the internal integrals on the right-hand side. Indeed, by approximation and assuming that x is a polynomial, each internal integral can be approximated uniformly by discrete sums. Then we follow the proof of Theorem 3.11 with necessary modifications as in the preceding proof. The only difference is that when Theorem 4.1 is applied, the L 1 -norm of the inverse Fourier transforms of the various functions in consideration there must be replaced by the two norms of these functions appearing in(4.7). We omit the details.Concrete characterizationsThis section concretizes the general characterization in the previous one in terms of the Poisson and heat kernels. We keep the notation introduced in section 3.3.The following result improves[73,Section 2.6.4] at two aspects even in the classical case: Firstly, in addition to derivation operators, it can also use integration operators (corresponding to negative k); secondly, [73, Section 2.6.4] requires k > d + max(α, 0) for the Poisson characterization while we only need k > α.Theorem 4.20. Let 1 ≤ p < ∞ and α ∈ R.(i) Let k ∈ Z such that k > α. Then for any distribution x on T d θ ,(ii) Let k ∈ Z such that k > α 2 . Then for any distribution x on T d θ ,The preceding theorem can be formulated directly in terms of the circular Poisson and heat semigroups of T d θ . The proof of the following result is similar to that of Theorem 3.15, and is left to the reader. Theorem 4.21. Let 1 ≤ p < ∞, α ∈ R and k ∈ Z.(i) If k > α, then for any distribution x on T d θ ,(ii) If k > α 2 , then for any any distribution x on T d θ ,The proof of Theorem 4.20. Similar to the Besov case, the proof of (ii) is done by choosing α 1 = 2k > α. But (i) is much subtler. We will first prove (i) under the stronger assumption that k > d + α, the remaining case being postponed. The proof in this case is similar to and a little bit harder than the proof of Theorem 3.13. Let again ψ(ξ) = (−sgn(k)2π|ξ|) k e −2π|ξ| . As in that proof, it remains to show that ψ satisfies the second condition of (4.7) for some α 1 > α and σ > d 2 . Since k > d + α, we can choose α 1 such that α < α 1 < k − d. We claim thatTaking the supremum over all u with |u| ≤ ε yields the desired inequality.Remark 6.5. We will discuss the optimal order of the best constant of the embedding in Corollary 6.3 at the end of the next section.Embedding of Sobolev spacesThis section is devoted to the embedding of Sobolev spaces. The following is our main theorem. Recall that B α1 ∞,∞ (T d θ ) in the second part below is the quantum analogue of the classical Zygmund class of order α 1 (seeRemark 3.18).continuously. In particular, if additionally α = k and α 1 = k 1 are nonnegative integers, thencontinuously. In particular, if additionally α = k ∈ N, and if either p > 1 or p = 1 and k is even, then. Thus we just deal with the potential spaces H α p (T d θ ). On the other hand, by the lifting property of potential Sobolev spaces, we can assume α 1 = 0. By Theorem 3.8 and Corollary 6.3, we have. Now choose 0 < η < 1 and two indices s 0 , s 1 with 1 < s 0 , s 1 < d α such thatThen interpolating the above inclusions with s j in place of p for j = 0, 1, using Remark 5.3 and (1.1), we get. (ii) By Theorems 3.8 and 6.2, we obtain. Thus the theorem is proved. Remark 6.7. The case pα = d with α 1 = 0 is excluded from the preceding theorem. In this case, it is easy to see that H α p (T d θ ) ⊂ L q (T d θ ) for any q < ∞. It is well known in the classical case that this embedding is false for q = ∞. Consider, for instance, the ball B = {s ∈ R d : |s| ≤1 4} and the function f defined by f (s) = log log(1 + 1 |s| ). Then f belongs to W d 1 (B) but is unbounded on B. Now extending f to a 1-periodic function onwith equal norms. Proof. This proof is an adaptation of that of[17,Theorem 7.3]. We start with an elementary observation.where (e m,n ) are the canonical matrix units of B(ℓ 2 (Z d )). So,[x] = a m,n e m,n m,n∈Z d , a matrix with entries in B(ℓ 2 (Z d )). Since V is unitary, we have) a S k p . Therefore,φ is a bounded Schur multiplier on S k p . Considering matrices a = (a m,n ) m,n∈Z d with entries in S p , we show in the same way that Mφ is c.b. on S k p , soφ is a c.b. Schur multiplier on. To show the converse direction, introducing the following Fölner sequence of Z d :we define two maps A N and B N as follows:) with (a m,n ) → (a m,n ) m,n∈ZN ; andHere B(ℓ|ZN | 2) is endowed with the normalized trace. Both A N , B N are unital, completely positive and trace preserving, so extend to complete contractions between the corresponding L p -spaces. Moreover, limIf we define S k p (ℓ |ZN | 2 ) as before for S k p just replacing S p (ℓ 2 (Z d )) by S p (ℓ |ZN | 2 ), we see that A N extends to a complete contraction from W kThe following is an immediate consequence of the preceding theorem.Corollary 7.5. (i) M(B α p,q (T d θ )) is independent of α and q, up to equivalent norms.). Similar statements hold for the spaces M cb (B α p,q (T d θ )). Theorem 7.4 and Lemma 1.3 imply the following:) and f be the distribution on T d such that f = φ. By Theorem 7.4 and Lemma 1.3, we have supRecall that the Fourier transform of ϕ k is ϕ (k) and ϕ k is the periodization of ϕ k . SoNoting that by(3.2), ϕ k * f = M φϕ (k) ( ϕ k−1 + ϕ k + ϕ k+1 ), we getProposition 7.10. Let E be a Banach space. Then for any φ :where ψ is defined byFourier multipliers on Triebel-Lizorkin spacesAs we have seen in the chapter on Triebel-Lizorkin spaces, Fourier multipliers on such spaces are subtler than those on Sobolev and Besov spaces. Similarly to the previous two sections, our target here is to show that the c.b. Fourier multipliers on F α,c p (T d θ ) are independent of θ. By definition, F α,c p (T d θ ) can be viewed as a subspace of the column space L p (T d θ ; ℓ α,c 2 ), the latter is the column subspace of L p (B(ℓ α )⊗T d θ ). Thus F α,c p (T d θ ) inherits the natural operator space structure of. Similarly, the row Triebel-Lizorkin space F α,r p (T d θ ) carries a natural operator space structure too. Finally, the mixture space F α p (T d θ ) is equipped with the sum or intersection operator space structure according to p < 2 or p ≥ 2. Note that according to this definition, even though it is a commutative function space, the space F α p (T d ) (corresponding to θ = 0) is endowed with three different operator space structures, the first two being defined by its embedding into L p (T d ; ℓ α,c 2 ) and L p (T d ; ℓ α,r 2 ), the third one being the mixture of these two. The resulting operator spaces are denoted by F α,c p (T d ) , F α,r p (T d ) and F α p (T d ), respectively. Similarly, we introduce operator space structures on the Hardy space H c p (T d θ ), its row and mixture versions too. The main result of this section is the following:with equivalent norms. Similar statements hold for the row and mixture spaces.We will show the theorem only in the case p < ∞. The proof presented below can be easily modified to work for p = ∞ too. Alternately, the case p = ∞ can be also obtained by duality from the case p = 1. Note, however, that this duality argument yields only the first equality of the theorem with equivalent norms for p = ∞.We adapt the proof of Theorem 7.1 to the present situation, by introducing the space S α,c p = a = (a m,n ) m,n∈Z d :and endow it with the normThen S α,c p is a closed subspace of the column subspace of S p (ℓ α 2 ⊗ℓ 2 (Z d )), which introduces a natural operator space structure on S α,c p . Let M cb (S α,c p ) denote the space of all c.b. Schur multipliers on S α,c p , equipped with the natural norm. Lemma 7.12. Let 1 ≤ p < ∞ and α ∈ R. Then M cb (F α,c p (T d θ )) = M cb (S α,c p ) with equal norms. Proof. This proof is similar to the one of Theorem 7.1; we point out the necessary changes. Keeping the notation there, we have for a = (a m,n ) m,n∈Z d ∈ S α,c p andSuppose that φ ∈ M cb (F α,c p (T d θ )). It then follows thatTherefore,φ is a bounded Schur multiplier on S α,c p . Considering matrices a = (a m,n ) m,n∈Z d with entries in B(ℓ 2 ), we show in the same way that Sφ is c.b. on S α,c p , soφ is a c.b. Schur multiplier on S α,c. To show the opposite inequality, we just note that the contractive and convergence properties of the maps A N and B N introduced in the proof of Theorem 7.1 also hold on the corresponding F α,c p (T d θ ) or S α,c p spaces. To see this, we take A N for example. Since it is c.b. between the)). Applying this to the elements of the form )), the latter space being the finite dimensional analogue of S α,c p . We then argue as in the proof of Theorem 7.1 to deduce the desired opposite inequality.Proof of Theorem 7.11. The first part is an immediate consequence of the previous lemma. For the second, we need the c.b. version of Theorem 4.11 (i), whose proof is already contained in section 4.1. To see this, we just note that, letting M = B(ℓ 2 (Z d ))⊗T In fact, using arguments similar to the proof of the preceding theorem, we can show that the above equality holds with equal norms.
Sobolev Spaces. R A Adams, Academic PressNew York, London, S. FranciscoR. A. Adams. Sobolev Spaces. Academic Press, New York, London, S. Francisco, 1975.
Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. H Amann, Math. Nachr. 186H. Amann. Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr. 186 (1997), 5-56.
Operator-valued Fourier multipliers on periodic Besov spaces and applications. W Arendt, S Bu, Proc. Edinburgh Math. Soc. 47W. Arendt and S. Bu. Operator-valued Fourier multipliers on periodic Besov spaces and applications. Proc. Edinburgh Math. Soc. 47 (2004), 15-33.
Embedding estimates and fractional smoothness. W Beckner, Int. Math. Res. Notices. 2014W. Beckner. Embedding estimates and fractional smoothness. Int. Math. Res. Notices. 2014 (2014), 390-417.
Functionals for multilinear fractional embedding. W Beckner, Acta Math. Sinica. 31W. Beckner. Functionals for multilinear fractional embedding. Acta Math. Sinica. 31 (2015), 1-28.
Interpolation of operators. C Bennett, R Sharpley, Pure and Applied Mathematics. 129C. Bennett and R. Sharpley. Interpolation of operators. Pure and Applied Mathematics, Volume 129, 1988.
On the relation between the two complex methods of interpolation. J Bergh, Indiana Univ. Math. J. 28J. Bergh. On the relation between the two complex methods of interpolation. Indiana Univ. Math. J. 28 (1979), 775-777.
Interpolation Spaces: An Introduction. J Bergh, J Löfström, SpringerBerlinJ. Bergh and J. Löfström. Interpolation Spaces: An Introduction. Springer, Berlin, 1976.
Nonmultipliers of the Sobolev Spaces W k,1 (R n ). A Bonami, S Poornima, J. Funct. Anal. 71A. Bonami and S. Poornima. Nonmultipliers of the Sobolev Spaces W k,1 (R n ). J. Funct. Anal. 71 (1987), 175-181.
Some remarks on Banach spaces in which martingale difference sequences are unconditional. J Bourgain, Ark. Mat. 21J. Bourgain. Some remarks on Banach spaces in which martingale difference sequences are unconditional. Ark. Mat. 21 (1983), 163-168.
A Hardy inequality in Sobolev spaces. J Bourgain, BrusselsVrije UniversityJ. Bourgain. A Hardy inequality in Sobolev spaces. Vrije University, Brussels, 1981.
Vector-valued singular integrals and the H 1 -BMO duality. J Bourgain, Probability theory and harmonic analysis. Cleveland, Ohio; New YorkDekker98J. Bourgain. Vector-valued singular integrals and the H 1 -BMO duality. In Probability theory and harmonic analysis (Cleveland, Ohio, 1983). Monogr. Textbooks Pure Appl. Math. Vol. 98, pages 1-19. Dekker, New York, 1986.
Limiting embedding theorems for Ws,p when s ↑ 1 and applications. J Bourgain, H Brézis, P Mironescu, J. Anal. Math. 87J. Bourgain, H. Brézis and P. Mironescu. Limiting embedding theorems for Ws,p when s ↑ 1 and applications. J. Anal. Math. 87 (2002), 37-75.
Another Look at Sobolev Spaces. J Bourgain, H Brézis, P Mironescu, Optimal Control and Partial Differential Equations. AmsterdamIOS PressJ. Bourgain, H. Brézis and P. Mironescu. Another Look at Sobolev Spaces. In Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, 439-466.
A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. D L Burkholder, Ann. Probab. 9D. L. Burkholder. A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9 (1981), 997-1011.
A geometric condition that implies the existence of certain singular integrals of Banachspace-valued functions. D L Burkholder, Conference on harmonic analysis in honor of Antoni Zygmund. II (Chicago, Ill; Wadsworth, Belmont, CAID. L. Burkholder. A geometric condition that implies the existence of certain singular integrals of Banach- space-valued functions. In Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981). Wadsworth Math. Ser., pages 270-286. Wadsworth, Belmont, CA, 1983.
Harmonic analysis on quantum tori. Z Chen, Q Xu, Z Yin, Commun. Math. Phys. 322Z. Chen, Q. Xu and Z. Yin. Harmonic analysis on quantum tori. Commun. Math. Phys. 322 (2013), 755-805.
Noncommutative Geometry. A Connes, Academic PressSan Diego, CAA. Connes. Noncommutative Geometry. Academic Press, San Diego, CA, 1994.
Real and complex interpolation and extrapolation of compact operators. M Cwikel, Duke Math. J. 65M. Cwikel. Real and complex interpolation and extrapolation of compact operators. Duke Math. J. 65 (1992), 333-343.
Heat kernels and spectral theory. E B Davies, Cambridge University PressCambridgeE. B. Davies. Heat kernels and spectral theory Cambridge University Press, Cambridge, 1989.
Interpolation of linear operators on Sobolev spaces. R A Devore, K Scherer, Ann. Math. 109R. A. DeVore and K. Scherer. Interpolation of linear operators on Sobolev spaces. Ann. Math. 109 (1979), 583-599.
Fourier series: A modern introduction. R E Edwards, Springer-Verlag2New YorkR. E. Edwards. Fourier series: A modern introduction, Vol. 2. Springer-Verlag, New York, 1982.
Operator Spaces. E Effros, Z J Ruan, Oxford University PressOxfordE. Effros and Z. J. Ruan. Operator Spaces. Oxford University Press, Oxford, 2000.
Tenperatures, Bessel potentials and Lipschitz spaces. M T Flett, Proc. London Math. Soc. 32M. T. Flett. Tenperatures, Bessel potentials and Lipschitz spaces. Proc. London Math. Soc. 32 (1971), 385-451.
Weighted norm inequalities and related topics. J García-Cuerva, J L Rubio De Francia, North-Holland Publishing CoAmsterdamJ. García-Cuerva, and J.L. Rubio de Francia. Weighted norm inequalities and related topics. North-Holland Publishing Co., Amsterdam, 1985.
Wiener's lemma for twisted convolution and Gabor frames. K Gröchenig, M Leinert, J. Amer. Math. Soc. 17K. Gröchenig and M. Leinert. Wiener's lemma for twisted convolution and Gabor frames. J. Amer. Math. Soc. 17 (2004), 1-18.
Linear and complex analysis: Problem Book 3, Part II. V P Havin, N K Nikolski, Lect. Notes in Math. 1574Springer-VerlagV. P. Havin, and N. K. Nikolski. Linear and complex analysis: Problem Book 3, Part II. Lect. Notes in Math. Vol. 1574, Springer-Verlag, Berlin 1994.
Interpolation of quasi-normed spaces. T Holmstedt, Math. Scand. 26T. Holmstedt. Interpolation of quasi-normed spaces. Math. Scand. 26 (1970), 177-199.
Caldroón-Zygmund operator associated to matrixvalued kernels. G Hong, L D López-Sánchez, J M Martell, J Parcet, Int. Math. Res. Not. 2014G. Hong, L. D. López-Sánchez, J. M. Martell, and J. Parcet. Caldroón-Zygmund operator associated to matrix- valued kernels. Int. Math. Res. Not. 2014 (2014), 1221-1252.
On the equivalence of the K-functional and moduli of continuity and some applications. H Johnen, K Scherer, Lect. Notes in Math. 571H. Johnen and K. Scherer. On the equivalence of the K-functional and moduli of continuity and some applica- tions. Lect. Notes in Math. 571 (1976), 119-140.
Doob's inequality for non-commutative martingales. M Junge, J. Reine Angew. Math. 549M. Junge. Doob's inequality for non-commutative martingales. J. Reine Angew. Math. 549 (2002), 149-190.
H ∞ -functional calculus and square functions on noncommutative L p -spaces. M Junge, C Le Merdy, Q Xu, Astérisque. 305138M. Junge, C. Le Merdy, and Q. Xu. H ∞ -functional calculus and square functions on noncommutative L p -spaces. Astérisque. 305 (2006), vi + 138 pp.
Noncommutative Riesz transforms -a probabilistic approach. M Junge, T Mei, Amer. J. Math. 132M. Junge and T. Mei. Noncommutative Riesz transforms -a probabilistic approach. Amer. J. Math. 132 (2010), 611-681.
BMO spaces associated with semigroups of operators. M Junge, T Mei, Math. Ann. 352M. Junge and T. Mei. BMO spaces associated with semigroups of operators. Math. Ann. 352 (2012), 691-743.
Smooth Fourier multipliers on group von Neumann algebras. M Junge, T Mei, J Parcet, GAFA. 24M. Junge and T. Mei and J. Parcet. Smooth Fourier multipliers on group von Neumann algebras. GAFA. 24 (2014), 1913-1980.
Noncommutative Burkholder/Rosenthal inequalities, I and II. M Junge, Q Xu, Israel J. Math. 31Ann. Probab.M. Junge and Q. Xu. Noncommutative Burkholder/Rosenthal inequalities, I and II. Ann. Probab. 31 (2003), 948-995 and Israel J. Math. 167 (2008), 227-282.
Noncommutative maximal ergodic theorems. M Junge, Q Xu, J. Amer. Math. Soc. 20M. Junge and Q. Xu. Noncommutative maximal ergodic theorems. J. Amer. Math. Soc. 20 (2007), 385-439.
Limits of higher-order Besov spaces and sharp reiteration theorems. G E Karadzhov, M Milman, J Xiao, J. Funct. Anal. 221G. E. Karadzhov, M. Milman, and J. Xiao. Limits of higher-order Besov spaces and sharp reiteration theorems. J. Funct. Anal. 221 (2005), 323-339.
On spectral invariance of non-commutative tori. F Luef, Contemp. Math. 414F. Luef. On spectral invariance of non-commutative tori. Contemp. Math. 414 (2006), 131-146.
Noncommutative Khintchine and Paley Inequalities. F Lust-Piquard, G Pisier, Ark. Mat. 29F. Lust-Piquard and G. Pisier. Noncommutative Khintchine and Paley Inequalities. Ark. Mat. 29 (1991), 241- 260.
On the Bourgain, Brézis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. V Maz'ya, T Shaposhnikova, J. Funct. Anal. 195V. Maz'ya and T. Shaposhnikova. On the Bourgain, Brézis, and Mironescu theorem concerning limiting em- beddings of fractional Sobolev spaces. J. Funct. Anal. 195 (2002), 230-238.
Sobolev spaces. V Maz'ya, Springer-VerlagBerlin HeidelbergV. Maz'ya. Sobolev spaces. Springer-Verlag Berlin Heidelberg, 1980.
On Fourier multiplier transformations of Banach-valued functions. T R Mcconnell, Trans. Amer. Math. Soc. 285T. R. McConnell. On Fourier multiplier transformations of Banach-valued functions. Trans. Amer. Math. Soc. 285 (1984), 739-757.
Operator valued Hardy spaces. T Mei, Memoir Amer. Math. Soc. 88164T. Mei. Operator valued Hardy spaces. Memoir Amer. Math. Soc. 881 (2007), vi+64 pp.
Tent spaces associated with semigroups of operators. T Mei, J. Funct. Anal. 255T. Mei. Tent spaces associated with semigroups of operators. J. Funct. Anal. 255 (2008), 3356-3406.
Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities. T Mei, J Parcet, Intern. Math. Res. Not. 9T. Mei and J. Parcet. Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequal- ities. Intern. Math. Res. Not. 9 (2009), 1433-1487.
Notes on limits of Sobolev spaces and the continuity of interpolation scales. M Milman, Trans. Amer. Math. Soc. 357M. Milman. Notes on limits of Sobolev spaces and the continuity of interpolation scales. Trans. Amer. Math. Soc. 357 (2005), 3425-3442.
Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group. S Neuwirth, Ricard, Canad. J. Math. 63S. Neuwirth andÉ. Ricard. Transfer of Fourier multipliers into Schur multipliers and sumsets in a discrete group. Canad. J. Math. 63 (2011), 1161-1187.
Nikol'skij. Approximation of functions of several variables and imbedding theorems. S M , Springer-VerlagBerlin HeidelbergS.M. Nikol'skij. Approximation of functions of several variables and imbedding theorems. Springer-Verlag Berlin Heidelberg, 1975.
A non-inequality for differential operators in the L 1 -norm. D Ornstein, Arch. Rational Mech. Anal. 11D. Ornstein. A non-inequality for differential operators in the L 1 -norm. Arch. Rational Mech. Anal. 11 (1962), 40-49.
Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory. J Parcet, J. Funct. Anal. 256J. Parcet. Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory. J. Funct. Anal. 256 (2009), 509-593.
Gundy's decomposition for non-commutative martingales and applications. J Parcet, N Randrianantoanina, Proc. London Math. Soc. 93J. Parcet and N. Randrianantoanina. Gundy's decomposition for non-commutative martingales and applications. Proc. London Math. Soc. 93 (2006), 227-252.
New thoughts on Besov spaces. J Peetre, DurhamDuke University Math. Series, Duke UniversityJ. Peetre. New thoughts on Besov spaces. Duke University Math. Series, Duke University, Durham, 1976.
Noncommutative vector-valued Lp spaces and completely p-summing maps. G Pisier, Astérisque. 247131G. Pisier. Noncommutative vector-valued Lp spaces and completely p-summing maps. Astérisque. 247 (1998), vi+131 pp.
Introduction to operator space theory. G Pisier, Cambridge University PressCambridgeG. Pisier. Introduction to operator space theory. Cambridge University Press, Cambridge, 2003.
Non-commutative martingale inequalities. G Pisier, Q Xu, Commun. Math. Phys. 189G. Pisier and Q. Xu. Non-commutative martingale inequalities. Commun. Math. Phys. 189 (1997), 667-698.
Noncommutative L p -spaces. G Pisier, Q Xu, Handbook of the geometry of Banach spaces. W. B. Johnson and J. LindenstraussNorth-Holland, Amsterdam2G. Pisier and Q. Xu. Noncommutative L p -spaces. Handbook of the geometry of Banach spaces Vol. 2, ed. W. B. Johnson and J. Lindenstrauss, 2003, 1459-1517, North-Holland, Amsterdam.
Analogues of the exponential map associated with complex structures on noncommutative twotori. A Polishchuk, Pacific J. Math. 226A. Polishchuk. Analogues of the exponential map associated with complex structures on noncommutative two- tori. Pacific J. Math. 226 (2006), 153-178.
Multipliers of Sobolev spaces. S Poornima, J. Funct. Anal. 45S. Poornima. Multipliers of Sobolev spaces. J. Funct. Anal. 45 (1982), 1-27.
An embedding theorem for the Sobolev space W 1,1. S Poornima, Bull. Sci. Math. 107S. Poornima. An embedding theorem for the Sobolev space W 1,1 . Bull. Sci. Math. 107 (1983), 253-259.
Noncommutative martingale transforms. N Randrianantoanina, J. Funct. Anal. 194N. Randrianantoanina. Noncommutative martingale transforms. J. Funct. Anal. 194 (2002), 181-212.
Conditional square functions for noncommutative martingales. N Randrianantoanina, Ann. Proba. 35N. Randrianantoanina. Conditional square functions for noncommutative martingales. Ann. Proba. 35 (2007), 1039-1070.
Noncommutative variations on Laplace's equation. J Rosenberg, Anal. PDE. 1J. Rosenberg. Noncommutative variations on Laplace's equation. Anal. PDE. 1 (2008), 95-114.
Sobolev theory for noncommutative tori. M Spera, Rend. Sem. Mat. Univ. Padova. 86M. Spera. Sobolev theory for noncommutative tori. Rend. Sem. Mat. Univ. Padova. 86 (1992), 143-156.
A symplectic approach to Yang-Mills theory for noncommutative tori. M Spera, Canad. J. Math. 44M. Spera. A symplectic approach to Yang-Mills theory for noncommutative tori. Canad. J. Math. 44 (1992), 368-387.
Singular integrals and differentiability properties of functions. E M Stein, Princeton Univ. PressPrincetonE. M. Stein. Singular integrals and differentiability properties of functions. Princeton Univ. Press, Princeton, 1970.
Harmonic analysis. E M Stein, Princeton University PressPrincetonE. M. Stein. Harmonic analysis. Princeton University Press, Princeton, 1993.
Introduction to Fourier analysis on Euclidean spaces. E M Stein, G Weiss, Princeton University PressPrincetonE. M. Stein and G. Weiss. Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, 1975.
Boundedness of translation invariant operators on Hölder Spaces and L p -spaces. E M Stein, A Zygmund, Ann. Math. 85E. M. Stein and A. Zygmund. Boundedness of translation invariant operators on Hölder Spaces and L p -spaces. Ann. Math. 85 (1967), 337-349.
On the theory of Lipschitz spaces of distributions on Euclidean n-spaces, I, II. M H Taibleson, J. Math. Mech. 13M. H. Taibleson. On the theory of Lipschitz spaces of distributions on Euclidean n-spaces, I, II. J. Math. Mech. 13 (1964), 407-479; 14 (1965), 821-839.
Interpolation theory, function spaces, differential operators. H Triebel, Amsterdam, North-HollandH. Triebel. Interpolation theory, function spaces, differential operators. Amsterdam, North-Holland 1978.
Theory of function spaces. H Triebel, BirkhäuserBaselH. Triebel. Theory of function spaces. Birkhäuser, Basel, 1983.
Theory of function spaces. H Triebel, II. Birkhäuser. H. Triebel. Theory of function spaces, II. Birkhäuser, Basel, 1992.
Limits of Besov norms. H Triebel, Arch. Math. 96H. Triebel. Limits of Besov norms. Arch. Math. 96 (2011), 169-175.
Hardy-Littlewood theory for semigroups. N Th, Varopoulos, J. Funct. Anal. 63N. Th. Varopoulos. Hardy-Littlewood theory for semigroups. J. Funct. Anal. 63 (1985), 240-260.
Analysis and geometry on groups. N Th, L Varopoulos, T Saloff-Coste, Coulhon, Cambridge University PressCambridgeN. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon. Analysis and geometry on groups. Cambridge University Press, Cambridge, 1992.
Lipschitz algebras and derivations of von Neumann algebras. N Weaver, J. Funct. Anal. 139N. Weaver. Lipschitz algebras and derivations of von Neumann algebras. J. Funct. Anal. 139 (1996), 261-300.
α-Lipschitz algebras on the noncommutative torus. N Weaver, J. Operator Theory. 39N. Weaver. α-Lipschitz algebras on the noncommutative torus. J. Operator Theory. 39 (1998), 123-138.
A Note on Interpolation Spaces, Harmonic Analysis. T Wolff, Lect. Notes in Math. 908T. Wolff. A Note on Interpolation Spaces, Harmonic Analysis. Lect. Notes in Math. 908 (1982), 199-204.
Characterizations of operator-valued Hardy spaces and applications. R Xia, X Xiong, Q Xu, PreprintR. Xia, X. Xiong and Q. Xu. Characterizations of operator-valued Hardy spaces and applications. Preprint 2015.
Function spaces on quantum tori. X Xiong, Q Xu, Z Yin, C. R. Acad. Sci. Paris, Ser. I. onlineX. Xiong, Q. Xu and Z. Yin. Function spaces on quantum tori. C. R. Acad. Sci. Paris, Ser. I. 2015 (online).
Noncommutative Lp-spaces and martingale inequalities. Book manuscript. Q Xu, Q. Xu. Noncommutative Lp-spaces and martingale inequalities. Book manuscript, 2007.
Interactions between quantum probability and operator space theory. Q Xu, Lect. Notes in Math. 1954Q. Xu. Interactions between quantum probability and operator space theory. Lect. Notes in Math. 1954 (2008), 117-159.
On vector-valued Fourier multiplier theorems. F Zimmermann, Studia Math. 93F. Zimmermann. On vector-valued Fourier multiplier theorems. Studia Math. 93 (1989), 201-222.
Besançon Cedex, France E-mail address: xiao.xiong@univ-fcomte. 430072Wuhan; Besançon Cedex, France; WuhanSchool of Mathematics and Statistics, Wuhan University, Wuhan 430072, China and Laboratoire de Mathématiques, Université de Franche-Comté ; fr School of Mathematics and Statistics, Wuhan University ; China and Laboratoire de Mathématiques, Université de Franche-Comté ; fr School of Mathematics and Statistics, Wuhan UniversityE-mail address: qxu@univ-fcomte. China E-mail address: [email protected] of Mathematics and Statistics, Wuhan University, Wuhan 430072, China and Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France E-mail address: [email protected] School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China and Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France E-mail address: [email protected] School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China E-mail address: [email protected]
| [] |
[
"ON THE CONDITIONAL DISTRIBUTIONS AND THE EFFICIENT SIMULATIONS OF EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS",
"ON THE CONDITIONAL DISTRIBUTIONS AND THE EFFICIENT SIMULATIONS OF EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS"
] | [
"Jingchen Liu \nColumbia University\nUniversity of Minnesota\n\n",
"Gongjun Xu \nColumbia University\nUniversity of Minnesota\n\n"
] | [
"Columbia University\nUniversity of Minnesota\n",
"Columbia University\nUniversity of Minnesota\n"
] | [
"The Annals of Applied Probability"
] | In this paper, we consider the extreme behavior of a Gaussian random field f (t) living on a compact set T . In particular, we are interested in tail events associated with the integral T e f (t) dt. We construct a (non-Gaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field f (given that T e f (t) dt exceeds a large value) in total variation. Based on this approximation, we show that the tail event of T e f (t) dt is asymptotically equivalent to the tail event of sup T γ(t) where γ(t) is a Gaussian process and it is an affine function of f (t) and its derivative field. In addition to the asymptotic description of the conditional field, we construct an efficient Monte Carlo estimator that runs in polynomial time of log b to compute the probability P ( T e f (t) dt > b) with a prescribed relative accuracy. . This reprint differs from the original in pagination and typographic detail. 1 2 J. LIU AND G. XU of measure technique. In addition to the asymptotic descriptions, we design efficient Monte Carlo estimators that run in polynomial time of log b for computing the tail probabilitieswith a prescribed relative accuracy.1.1. The literature. In the probability literature, the extreme behaviors of Gaussian random fields have been studied extensively. The results range from general bounds to sharp asymptotic approximations. An incomplete list of works includes[15,20,23,35,37,39,46,50,52]. A few lines of investigations on the supremum norm are given as follows. Assuming locally stationary structure, the double-sum method [49] provides the exact asymptotic approximation of sup T f (t) over a compact set T , which is allowed to grow as the threshold tends to infinity. For almost surely at least twice differentiable fields, the authors of [2, 5, 53] derive the analytic form of the expected Euler-Poincaré characteristics of the excursion set [χ(A b )] which serves as a good approximation of the tail probability of the supremum. The tube method [51] takes advantage of the Karhune-Loève expansion and Weyl's formula. A recent related work along this line is given by[48]. The Rice method[11][12][13]provides an implicit description of sup T f (t). Change of measure based rare-event simulations are studied in[3]. The discussions also go beyond the Gaussian fields. For instance, [36] discusses the situations of Gaussian process with random variances. See also[4]for discussions on non-Gaussian cases. The distribution of I(T ) is studied in the literature when f (t) is a Brownian motion[29,56]. Recently,[42,43]derive the asymptotic approximations of P (I(T ) > b) as b → ∞ for three times differentiable and homogeneous Gaussian random fields.Besides the tail probability approximations, rigorous analysis of the conditional distributions of stochastic processes given the occurrence of rare events is also an important topic. In the classic large deviations analysis for light-tailed stochastic systems, the sample path(s) that admits the highest probability (the most likely sample path) under the conditional distribution given the occurrence of a rare event is central to the entire analysis in terms of determining the appropriate exponential change of measure, developing approximations of the tail probabilities and designing efficient simulation algorithms; see, for instance, standard textbook[30]. For heavytailed systems, the conditional distributions and the most likely paths, which typically admit the so-called "one-big-jump" principle, are also intensively studied[8,9,17]. These results not only provide intuitive and qualitative descriptions of the conditional distribution, but also shed light on the design of rare-event simulation algorithms[16][17][18]-the best importance sampling EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS 3 | 10.1214/13-aap960 | [
"https://arxiv.org/pdf/1204.5546v3.pdf"
] | 119,165,183 | 1204.5546 | 4ea23cdd5c9d993134a01cdd4ab455e1ad43bb78 |
ON THE CONDITIONAL DISTRIBUTIONS AND THE EFFICIENT SIMULATIONS OF EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS
19 May 2014. 2014
Jingchen Liu
Columbia University
University of Minnesota
Gongjun Xu
Columbia University
University of Minnesota
ON THE CONDITIONAL DISTRIBUTIONS AND THE EFFICIENT SIMULATIONS OF EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS
The Annals of Applied Probability
24419 May 2014. 201410.1214/13-AAP960
In this paper, we consider the extreme behavior of a Gaussian random field f (t) living on a compact set T . In particular, we are interested in tail events associated with the integral T e f (t) dt. We construct a (non-Gaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field f (given that T e f (t) dt exceeds a large value) in total variation. Based on this approximation, we show that the tail event of T e f (t) dt is asymptotically equivalent to the tail event of sup T γ(t) where γ(t) is a Gaussian process and it is an affine function of f (t) and its derivative field. In addition to the asymptotic description of the conditional field, we construct an efficient Monte Carlo estimator that runs in polynomial time of log b to compute the probability P ( T e f (t) dt > b) with a prescribed relative accuracy. . This reprint differs from the original in pagination and typographic detail. 1 2 J. LIU AND G. XU of measure technique. In addition to the asymptotic descriptions, we design efficient Monte Carlo estimators that run in polynomial time of log b for computing the tail probabilitieswith a prescribed relative accuracy.1.1. The literature. In the probability literature, the extreme behaviors of Gaussian random fields have been studied extensively. The results range from general bounds to sharp asymptotic approximations. An incomplete list of works includes[15,20,23,35,37,39,46,50,52]. A few lines of investigations on the supremum norm are given as follows. Assuming locally stationary structure, the double-sum method [49] provides the exact asymptotic approximation of sup T f (t) over a compact set T , which is allowed to grow as the threshold tends to infinity. For almost surely at least twice differentiable fields, the authors of [2, 5, 53] derive the analytic form of the expected Euler-Poincaré characteristics of the excursion set [χ(A b )] which serves as a good approximation of the tail probability of the supremum. The tube method [51] takes advantage of the Karhune-Loève expansion and Weyl's formula. A recent related work along this line is given by[48]. The Rice method[11][12][13]provides an implicit description of sup T f (t). Change of measure based rare-event simulations are studied in[3]. The discussions also go beyond the Gaussian fields. For instance, [36] discusses the situations of Gaussian process with random variances. See also[4]for discussions on non-Gaussian cases. The distribution of I(T ) is studied in the literature when f (t) is a Brownian motion[29,56]. Recently,[42,43]derive the asymptotic approximations of P (I(T ) > b) as b → ∞ for three times differentiable and homogeneous Gaussian random fields.Besides the tail probability approximations, rigorous analysis of the conditional distributions of stochastic processes given the occurrence of rare events is also an important topic. In the classic large deviations analysis for light-tailed stochastic systems, the sample path(s) that admits the highest probability (the most likely sample path) under the conditional distribution given the occurrence of a rare event is central to the entire analysis in terms of determining the appropriate exponential change of measure, developing approximations of the tail probabilities and designing efficient simulation algorithms; see, for instance, standard textbook[30]. For heavytailed systems, the conditional distributions and the most likely paths, which typically admit the so-called "one-big-jump" principle, are also intensively studied[8,9,17]. These results not only provide intuitive and qualitative descriptions of the conditional distribution, but also shed light on the design of rare-event simulation algorithms[16][17][18]-the best importance sampling EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS 3
1. Introduction. Consider a Gaussian random field {f (t) : t ∈ T } living on a d-dimensional domain T ⊂ R d with zero mean and unit variance, that is, for every finite subset {t 1 , . . . , t n } ⊂ T , (f (t 1 ), . . . , f (t n )) is a mean zero multivariate Gaussian random vector. Let µ(t) be a (deterministic) function and σ ∈ (0, ∞) be a scale factor. Define
I(T )
T e σf (t)+µ(t) dt. (1.1) In this paper, we develop a precise asymptotic description of the conditional distribution of f given that I(T ) exceeds a large value b, that is, P (·|I(T ) > b). In particular, we provide a tractable total variation approximation (in the sample path space) for such conditional random fields based on a change estimator of the rare-event probability uses a change of measure corresponding to the interesting conditional distribution. In addition, the conditional distribution (or the conditional expectations) is also of practical interest. For instance, in risk management, the conditional expected loss given some rare/disastrous event is an important risk measure and stress test.
In the literature of Gaussian random fields, the exact Slepian models [conditional field given a local maximum or level crossing of f (t)] are studied intensively for twice differentiable fields. For instance, Leadbetter, Lindgren and Rootzén [38] give the Slepian model conditioning on an upcrossing of level u at time zero. Lindgren [40] treats conditioning on a local maximum of height u at time zero. The first rigorous treatment of Slepian models for nonstationary processes is given by Lindgren [41]. Grigoriu [34] extends the results of Leadbetter, Lindgren and Rootzén [38] for level crossings to the general nonstationary case. This work is followed up by Gadrich and Adler [32]. In the later analysis, we will set an asymptotic equivalence between the conditional distribution given {I(T ) > b} and that given the high excursion of the supremem of f . The latter can be characterized by the Slepain model.
1.2.
Contributions. In this paper, we pursue along this line for the extreme behaviors of Gaussian processes and begin to describe the conditional distribution of f given the occurrence of the event {I(T ) > b}. In particular, we provide both quantitative and qualitative descriptions of this conditional distribution. Furthermore, from a computational point of view, we construct a Monte Carlo estimator that takes a polynomial computational cost (in log b) to estimate v(b) for a prescribed relative accuracy.
Central to the analysis is the construction of a change of measure on the space C(T ) (continuous functions living on T ). The application of the change of measure ideas is common in the study of large deviations analysis for the light-tailed stochastic systems. However, it is not at all standard in the study of Gaussian random fields. The proposed change of measure is not of a classical exponential-tilting form. This measure has several features that are appealing both theoretically and computationally. First, we show that the change of measure denoted by Q approximates the conditional measure P (·|I(T ) > b) in total variation as b → ∞. Second, the measure Q is analytically tractable in the sense that the distribution of f under Q has a closed form representation and the Radon-Nikodym derivative dQ/dP takes the form of a d-dimensional integral. This tractability property has useful consequences. From a methodological point of view, the measure Q provides a very precise description of the mechanism that drives the rare event {I(T ) > b}. This result allows us to directly use the intuitive mechanism to provide functional probabilistic descriptions that emphasize the most important elements that are present in the interesting rare events. More technically, the analytical computations associated with the measure Q are easy (compared to the conditional measure), and the expectation E Q [·] is theoretically much more tractable than E[·|I(T ) > b]. Based on this result, we show that the tail event {I(T ) > b} is asymptotically equivalent to the tail event of sup T γ(t) where γ(t) is an affine function of f (t) and its derivative field ∂ 2 f (t) and γ(t) implicitly depends on b. Thus, one can further characterize the conditional measure by means of the results on the Slepian model mentioned earlier.
Another contribution of this paper lies in the numerical evaluation of v(b). The importance sampling algorithm associated with the proposed change of measure yields an efficient estimator for computing v(b). An important issue concerns the implementation of the Monte Carlo method. The processes considered in this paper are continuous while computers can only represent discrete objects. Inevitably, we will introduce a suitable discretization scheme and use discrete (random) objects to approximate the continuous processes. A naturally raised issue lies in the control of the approximation error relative to the probability v(b). We will perform careful analysis and report the overall computational complexity of the proposed Monte Carlo estimators.
A key requirement of the current analysis is the twice differentiability of f . Our change of measure is written explicitly in the form of f , ∂f and ∂ 2 f . A very interesting future study would be developing parallel results for nondifferentiable fields. The technical challenges are two-fold. First, there is lack of asymptotic analysis for the exponential integral of general nondifferentiable fields. To the author's best knowledge, the behavior of I(T ) for nondifferentiable processes is investigated only when f is a Brownian motion whose techniques cannot be extend to general cases [29,56]. In addition, there is a lack of descriptive tools (such as derivatives and the Palm model) for nondifferentiable processes. This also leads to difficulties in describing the Slepian model for level crossing. To the author's best knowledge, analytic description of Slepian models for excursion of sup T f (t) are available only for twice differentiable fields. Despite of the smoothness limitation, the current analysis has important applications the details of which will be presented in the following section.
The rest of this paper is organized as follows. Two applications of this work are given in Section 2. In Section 3, we present the main results including the change of measure, the approximation of P (·|I(T ) > b) and the efficient Monte Carlo estimator of v(b). Proofs of the theorems are given in Sections 4-7. A supplemental material [45] is provided including all the supporting lemmas.
2. Applications. The integral of exponential functions of Gaussian random fields plays an important role in many probability models. We present two such models for which the conditional distribution is of interest and the underlying random fields are differentiable. In order to build in spatial dependence structure and to account for overdispersion, the log-intensity is typically modeled as a Gaussian random field, that is, log λ(t) = f (t) + µ(t) and then E[N (A)|λ(·)] = A e f (t)+µ(t) dt, where µ(t) is the mean function, and f (t) is a zero-mean Gaussian process. For instance, Chan and Ledolter [22] consider the time series setting in which T is a one-dimensional interval, µ(t) is modeled as the observed covariate process and f (t) is an autoregressive process; see [21,[24][25][26]57] for more examples in high-dimensional domains.
For the purpose of illustration, we consider a very concrete case that the point process N (·) represents the spatial distribution of asthma cases over a geographical domain T . The latent intensity λ(t) [or equivalently f (t)] represents the unobserved (and appropriately transformed) metric of the pollution severity at location t. The mean function can be written as a linear combination of the observed covariates that may affect the pollution level, that is, µ(t) = β ⊤ x(t) is treated as a deterministic function. It is well understood that λ(t) is a smooth function of the spatial parameter t at the macro level as the atmosphere mixes well; see, for example, [1]. One natural question in epidemiology is the following: upon observing an unusually high number of asthma cases, what is their geographical distribution, that is, the conditional distribution of the point process N (·) given that N (T ) > b for some large b?
First of all, Liu and Xu [43] show that P (N (T ) > b) ∼ P (I(T ) > b) as b → ∞. Following the same derivations, it is not difficult to establish the following convergence:
P (·|N (T ) > b) − P (·|I(T ) > b) → 0 in total variation as b → ∞.
The total count N (T ) is a Poisson random variable with mean I(T ). Intuitively speaking, the tail of the integral is similar to a lognormal random variable and thus is heavy-tailed. Its overshoot over level b
is O p (b/ log b).
However, a Poisson random variable with mean I(T ) ∼ b has standard de-
viation √ b ≪ b/ log b.
Thus, a large number of N (T ) is mainly caused by a large value of I(T ). The symmetric difference of the two sets {N (T ) > b} and {I(T ) > b} vanishes, and the probability law of the entire system conditional upon observing that N (T ) > b is asymptotically the same as that given I(T ) > b. Therefore, the conditional distribution of N (·) given N (T ) > b is asymptotically another doubly-stochastic Poisson process whose intensity is λ(t) = e µ(t)+f (t) where f (t) follows the conditional distribution of P (f ∈ ·|I(T ) > b).
Based on the main results presented momentarily, a qualitative description of the conditional distribution of N (·) is as follows. Given N (T ) > b, the overshoot is of order
O p (b/ log b), that is, N (T ) = b + O p (b/ log b).
The loca-tions of the points are i.i.d. samples approximately following a d-dimensional multivariate Gaussian distribution with mean τ ∈ T and variance Σ/ log b where Σ depends on the spectral moments of f . The distribution of τ is uniform over T if µ(t) is a constant; if µ(t) is not constant, τ has a density l(t) presented in (3.13).
2.2. Financial application. The exponential integral can be considered as a generalization of the sum of dependent lognormal random variables that has been studied intensively from different aspects in the applied probability literature (see [7,10,14,27,28,31,33]). In portfolio risk analysis, consider a portfolio of n assets S 1 , . . . , S n . The asset prices are usually modeled as lognormal random variables. That is, let X i = log S i and (X 1 , . . . , X n ) follows a multivariate normal distribution. The total portfolio value S = n i=1 w i S i is the weighted sum of dependent log-normal random variables.
An important question is the behavior of this sum when the portfolio size becomes large and the assets are highly correlated. One may employ a latent space approach used in the literature of social networks. More specifically, we construct a Gaussian process {f (t) : t ∈ T } and map each asset i to a latent variable t i ∈ T , that is, log S i = f (t i ). Then the log-asset prices fall into a subset of the continuous Gaussian process. Furthermore, we construct a (deterministic) function w(t) so that w(t i ) = w i . Then, the unit share value of the portfolio is 1 [19,43] for detailed discussions on the random field representations of large portfolios.
n w i S i = 1 n w(t i )e f (t i ) . See
In the asymptotic regime that n → ∞ and the correlations among the asset prices become close to one, the subset {t i } becomes dense in T . Ultimately, we obtain the limit
1 n n i=1 w i S i → T w(t)e f (t) h(t) dt,
where h(t) is the limiting spatial distribution of {t i } in T . Let µ(t) = log w(t)+log h(t). Then the (limiting) unit share price is
I(T ) = T e f (t)+µ(t) dt.
The current study provides an asymptotic description of the performance of each asset given the occurrence of the tail event I(T ) > b. This is of great importance in the study of the so-called stress test that evaluates the impact of shocks on and the vulnerability of a system. For instance, consider that another investor holds a different portfolio that has a substantial overlap with the current one, or it has exactly the same collection of assets but with different weights. Thus, this second portfolio corresponds to a different mean function µ ′ (t). The stress test investigates the performance of this second portfolio on the condition that a rare event has occurred to the first, that is,
P T e f (t)+µ ′ (t) dt ∈ · T e f (t)+µ(t) dt > b . EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS 7
To characterize the above distribution, we need a precise description of the conditional measure P (f ∈ ·| T e f (t)+µ(t) dt > b).
Main results.
3.1. Problem setting and notation. Throughout this discussion, we consider a homogeneous Gaussian random field {f (t) : t ∈ T } living on a domain T ⊂ R d . Let the covariance function be
C(t − s) = Cov(f (t), f (s)).
We impose the following assumptions:
(C1) f is stationary with Ef (t) = 0 and Ef 2 (t) = 1.
(C2) f is almost surely at least two times differentiable with respect to t. (C3) T is a d-dimensional compact set of R d with piecewise smooth boundary.
(C4) The Hessian matrix of C(t) at the origin is standardized to be −I, where I is the d × d identity matrix. In addition, C(t) has the following expansion when t is close to 0
C(t) = 1 − 1 2 t ⊤ t + C 4 (t) + R C (t), (3.1) where C 4 (t)= 1 24 ijkl ∂ 4 ijkl C(0)t i t j t k t l and R C (t)=O(|t| 4+δ 0 ) for some δ 0 > 0. (C5) For each t ∈ R d , the function C(λt) is a nonincreasing function of λ ∈ R + .
(C6) The mean function µ(t) falls into either of the two cases:
(a) µ(t) ≡ 0; (b) the maximum of µ(t) is unique and is attained in the interior of T and µ(t + ε) − µ(t) = ε ⊤ ∂µ(t) + ε ⊤ ∆µ(t)ε + O(|ε| 2+δ 0 ) as ε → 0.
We define a set of notation constantly used in the later development and provide some basic calculations. Let P * b be the conditional measure given {I(T ) > b}, that is,
P * b (f (·) ∈ A) = P (f (·) ∈ A|I(T ) > b)
. Let "∂" denote the gradient and "∆" denote the Hessian matrix with respect to t. The notation "∂ 2 " is used to denote the vector of second derivatives. The difference between ∂ 2 f (t) and ∆f (t) is that ∆f (t) is a d × d symmetric matrix whose diagonal and upper triangle consist of elements of ∂ 2 f (t). Furthermore, let ∂ j f (t) be the partial derivative with respect to the jth element of t. Finally, we define the following vectors: Suppose 0 ∈ T . It is well known that (f (0), ∂ 2 f (0), ∂f (0), f (t)) is a multivariate Gaussian random vector with mean zero and covariance matrix (cf. Chapter 5.5 of [5])
µ 1 (t) = −(∂ 1 C(t), . . . , ∂ d C(t)), µ 2 (t) = (∂ 2 ii C(t), i = 1, . . . , d; ∂ 2 ij C(t), 1 µ 20 0 C(t) µ 02 µ 22 0 µ ⊤ 2 (t) 0 0 I µ ⊤ 1 (t) C(t) µ 2 (t) µ 1 (t) 1 ,
where the matrix µ 22 is a d(d + 1)/2-dimensional positive definite matrix and contains the 4th order spectral moments arranged in an appropriate order according to the order of elements in ∂ 2 f (0). Let h(x, y, z) be the density function of (f (t), ∂f (t), ∂ 2 f (t)) evaluated at (x, y, z). Then, simple calculation yields that
h(x, y, z) (3.3) = det(Γ) −1/2 (2π) (d+1)(d+2)/4 e −(1/2)[y ⊤ y+(x−µ 20 µ −1 22 z) 2 /(1−µ 20 µ −1 22 µ 02 )+z ⊤ µ −1 22 z] ,
where det(·) is the determinant of a matrix and Γ = 1 µ 20 µ 02 µ 22 .
We define u as a function of b such that
2π σ d/2 u −d/2 e σu = b. (3.4)
Note that the above equation generally has two solutions: one is approximately σ −1 log b, and the other is close to zero as b → ∞. We choose u to be the one close to σ −1 log b. For µ(t) and σ appearing in (1.1), we define
µ σ (t) = µ(t)/σ, u t = u − µ σ (t). (3.5)
Approximately, u t is the level that f (t) needs to reach so that I(T ) > b. Furthermore, we need the following spatially varying set:
A t = {f (·) ∈ C(T ) : α t > u t − ηu −1 t }, (3.6)
where η > 0 is a tuning parameter that will be eventually sent to zero as b → ∞ and α t is a function of f (t) and its derivative fields taking the form of
α t = f (t) + |∂f (t)| 2 2u t + 1 ⊤f ′′ t 2σu t + B t u t . (3.7)
In the above equation (3.7),f ′′ t is defined as [with the notation in (3.
2)]
f ′′ t = ∂ 2 f (t) − u t µ 02 . (3.8) EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS 9
The term B t is a deterministic function depending only on C(t), µ(t) and σ,
B t = 1 ⊤ ∂ 2 µ σ (t) + d × µ σ (t) 2σ + 1 8σ 2 i ∂ 4 iiii C(0) + |∂µ σ (t)| 2 , (3.9)
where d is the dimension of T , and
d(d−1)/2 ) ⊤ . Note that α t ≈ f (t). Thus on the set A t , f (t) ≈ α t > u t − O(u −1 ). Together with the fact that E[∂ 2 f (t)|f (t) = u t ] = u t µ 02 ,f ′′
t is the standardized second derivative of f given that f (t) = u t . In Section 3.2, we will show that the event {I(T ) > b} is approximately t∈T A t .
For notational convenience, we write a u = O(b u ) if there exists a constant c > 0 independent of everything such that a u ≤ cb u for all u > 1, and
a u = o(b u ) if a u /b u → 0 as u → ∞, and the convergence is uniform in other quantities. We write a u = Θ(b u ) if a u = O(b u ) and b u = O(a u ). In addition, we write a u ∼ b u if a u /b u → 1 as u → ∞.
Remark 1. Condition C1 assumes unit variance. We treat the standard deviation σ as an additional parameter and consider e µ(t)+σf (t) dt. Condition C2 implies that C(t) is at least 4 times differentiable and the first and third derivatives at the origin are all zero. Differentiability is a crucial assumption in this analysis. Condition C3 restricts the results to finite horizon. Condition C4 assumes the Hessian matrix is standardized to be −I, which is to simplify notation. For any Gaussian process g(t) with covariance function C g (t) and ∆C g (0) = −Σ and det(Σ) > 0, identity Hessian matrix can be obtained by an affine transformation by letting g(t) = f (Σ 1/2 t) and T e µ(t)+σg(t) dt = det(Σ −1/2 )
{s:Σ −1/2 s∈T } e µ(Σ −1/2 s)+σf (s) ds.
Condition C5 is imposed for technical reasons so that we are able to localize the integration. For condition C6, we assume that µ(t) either is a constant or attains its global maximum at one place. If µ(t) has multiple (finitely many) maxima, the techniques developed in this paper still apply, but the derivations will be more tedious. Therefore, we stick to the uni-mode case.
Remark 2. The setting in (1.2) incorporates the case in which the integral is with respect to other measures with smooth densities. Then, if ν(dt) = κ(t) dt, we will have that A e µ(t)+σf (t) ν(dt) = A e µ(t)+log κ(t)+σf (t) dt, which shows that the density can be absorbed by the mean function.
3.2.
Approximation of the conditional distribution. In this subsection, we propose a change of measure Q on the sample path space C(T ) that approximates P * b in total variation. Let P be the original measure. The measure Q is defined such that P and Q are mutually absolutely continuous. We define the measure Q under two different scenarios: µ(t) is not a constant and µ(t) ≡ 0. Note that the measure Q obviously will depend on b. To simplify the notation, we omit the index b in Q whenever there is no ambiguity.
The measure Q takes a mixture form of three measures, which are weighted by (1 − ρ 1 − ρ 2 ), ρ 1 and ρ 2 , respectively (a natural constraint is that ρ 1 , ρ 2 and 1 − ρ 1 − ρ 2 ∈ [0, 1]). We define Q through the Radon-Nikodym derivative
dQ dP = (1 − ρ 1 − ρ 2 ) T l(t) · LR(t) dt + ρ 1 T l(t) · LR 1 (t) dt (3.10) + ρ 2 T LR 2 (t) mes(T ) dt,
where ρ 1 , ρ 2 will be eventually sent to 0 as b goes to infinity at the rate (log log b) −1 , mes(T ) is the Lebesgue measure of T and
LR(t) = h 0,t (f (t), ∂f (t), ∂ 2 f (t)) h(f (t), ∂f (t), ∂ 2 f (t))
,
LR 1 (t) = h 1,t (f (t), ∂f (t), ∂ 2 f (t)) h(f (t), ∂f (t), ∂ 2 f (t)) , (3.11) LR 2 (t) = 1/ √ 2πe −(1/2)(f (t)−ut ) 2 (1/ √ 2π)e −(1/2)f (t) 2 .
The density h(f (t), ∂f (t),
∂ 2 f (t)) is defined in (3.3), l(t)
is a density function on T , h 0,t and h 1,t are two density functions. Before presenting the specific forms of l(t), h 0,t and h 1,t , we would like to provide an intuitive explanation of dQ/dP from a simulation point of view. One can generate f (t) under the measure Q via the following steps:
(1) Generate ı ∼ Bernoulli(ρ 2 ). (2) If ı = 1, then: (a) generate τ uniformly from the index set T , that is, τ ∼ Unif(T ); (b) given the realized τ , generate f (τ ) ∼ N (u τ , 1); (c) given (τ, f (τ )), simulate {f (t) : t = τ } from the original conditional distribution under P . (3) If ı = 0:
(a) simulate a random variable τ following the density function l(t);
(b) given the realized τ , simulate f (τ ) = x, ∂f (τ ) = y, ∂ 2 f (τ ) = z from density function h all (x, y, z) = 1 − ρ 1 − ρ 2 1 − ρ 2 h 0,τ (x, y, z) + ρ 1 1 − ρ 2 h 1,τ (x, y, z); (3.12) EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS 11 (c) given (τ, f (τ ), ∂f (τ ), ∂ 2 f (τ )), simulate {f (t) : t = τ } from the origi- nal conditional distribution under P .
Thus, τ is a random index at which we twist the distribution of f and its derivatives. The likelihood ratio at a specific location τ is given by LR(τ ), LR 1 (τ ) or LR 2 (τ ) depending on the mixture component. The distribution of the rest of the field {f (t) : t = τ } given (f (τ ), ∂f (τ ), ∂ 2 f (τ )) is the same as that under P . It is not hard to verify that the above simulation procedure is consistent with the Radon-Nikodym derivative in (3.10).
We now provide the specific forms of the functions defining Q. We first consider the situation when µ(t) = 0. By condition C6, µ(t) admits its unique maximum at t * = arg sup t∈T µ(t) in the interior of T . Furthermore, the Hes-
sian matrix ∆µ σ (t * ) is negative definite. The function l(t) is a density on T such that for t ∈ T l(t) = (1 + o(1)) det(−∆µ σ (t * )) 1/2 u t * 2π d/2 e (ut * /2)(t−t * ) ⊤ ∆µσ(t * )(t−t * ) , (3.13)
which is approximately a Gaussian density centered around t * . As l(t) is defined on a compact set t, the o(1) term goes to zero as b tends to infinity. It is introduced to correct for the integral of l(t) outside the region T that is exponentially small and does not affect the current analysis. The functions h 0,t and h 1,t are density functions on the vector space where (f (t), ∂f (t), ∂ 2 f (t)) lives, and they are defined as follows (we will explain the following complicated functions momentarily):
h 0,t (f (t), ∂f (t), ∂ 2 f (t)) = I At × H λ × u t × e −λut(f (t)+(1 ⊤f ′′ t /(2σut))+Bt/ut−ut) × e −|∂f (t)| 2 /2 × exp − 1 2 |µ 20 µ −1 22f ′′ t | 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22f ′′ t − µ 1/2 22 1 2σ 2 , h 1,t (f (t), ∂f (t), ∂ 2 f (t)) = I A c t × H λ 1 × u t × e λ 1 ut(f (t)+(1 ⊤f ′′ t /(2σut))+Bt/ut−ut) × e −|∂f (t)| 2 /2 × exp − 1 2 |µ 20 µ −1 22f ′′ t | 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22f ′′ t − µ 1/2 22 1 2σ 2 , where I is the indicator function, A t = {f (·) : f (t) + |∂f (t)| 2 2ut + 1 ⊤f ′′ t 2σut + Bt ut > u t − η/u t } is defined as in (3.6),f ′′ t is defined as in (3.8)
, λ < 1 is positive and it will be sent to 1 as b goes to infinity, λ 1 is a fixed positive constant (e.g., λ 1 = 1) and the normalizing constants are defined as
H λ = e −λη (1 − λ) d/2 λ (2π) d/2 × R d(d+1)/2 e −(1/2)[|µ 20 µ −1 22 z| 2 /(1−µ 20 µ −1 22 µ 02 )+|µ −1/2 22 z−µ 1/2 22 1/(2σ)| 2 ] dz −1 ,
(3.14)
H λ 1 = e λ 1 η (1 + λ 1 ) d/2 λ 1 (2π) d/2 × R d(d+1)/2 e −(1/2)[|µ 20 µ −1 22 z| 2 /(1−µ 20 µ −1 22 µ 02 )+|µ −1/2 22 z−µ 1/2 22 1/(2σ)| 2 ] dz −1 .
The constants H λ and H λ 1 ensure that h 0,t and h 1,t are properly normalized densities.
Understanding the measure Q. The measure Q is designed such that the distribution of f under the measure Q is approximately the conditional distribution of f given I(T ) > b. The two terms corresponding to the probabilities ρ 1 and ρ 2 are included to ensure the absolute continuity and to control the tail of the likelihood ratio. Thus, ρ 1 and ρ 2 will be sent to zero eventually.
We now provide an explanation of the leading term corresponding to the probability 1 − ρ 1 − ρ 2 . To understand h 0,t , we use the notation α t in (3.7) and rewrite the density function as
h 0,t (f (t), ∂f (t), ∂ 2 f (t)) ∝ I At exp{−λu t (α t − u t )} × exp − 1 − λ 2 |∂f (t)| 2 × exp − 1 2 |µ 20 µ −1 22f ′′ t | 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22f ′′ t − µ 1/2 22 1 2σ 2 ,
which factorizes into three pieces consisting of α t , ∂f (t) andf ′′ t , respectively. We consider the change of variables from (f (t), ∂f (t), ∂ 2 f (t)) to (α t , ∂f (t),f ′′ t ). Then, under the distribution h 0,t , the random vectors α t , ∂f (t) andf ′′ t are independent. Note that h 0,t is defined on the set A t = {α t > u t − ηu −1 t } where η will be send to zero eventually. Then, α t − u t is approximately an exponential random variable with rate λu t ; ∂f (t), andf ′′ t are two independent Gaussian random vectors. The density h 1,t has a similar interpretation. The only difference is that h 1,t is defined on the set {α t − u t < −ηu −1 t } and u t − α t follows approximately an exponential distribution. For the last piece corresponding to ρ 2 , the density is simply an exponential tilting of f (t).
Under the dominating mixture component, to generate an f (t) from Q, a random index τ is first sampled from T following density l(t), then (f (τ ), ∂f (τ ), ∂ 2 f (τ )) is sampled according to h 0,τ . This implies that the large value of the integral T e µ(t)+σf (t) dt is mostly caused by the fact that the field reaches a high level at τ ; more precisely, α τ reaches a high level of u τ (with an exponential overshoot of rate λu τ ). Therefore, the random index τ localizes the position where the field α t goes very high. The distribution of τ given as in (3.13) is very concentrated around t * . This suggests that the
maximum of α t [or f (t)] is attained within O p (u −1/2 ) distance from t * .
We now consider the case where µ(t) ≡ 0. We choose l(t) to be the uniform distribution over set T and have that dQ dP
= (1 − ρ 1 − ρ 2 ) T LR(t) mes(T ) dt + ρ 1 T LR 1 (t) mes(T ) dt (3.15) + ρ 2 T LR 2 (t) mes(T ) dt,
where mes(·) is the Lebesgue measure. The following theorem states that Q is a good approximation of P * b with appropriate choice of the tuning parameters.
Theorem 3. Consider a Gaussian random field {f (t) : t ∈ T } living on a domain T satisfying conditions C1-C6. If we choose the parameters defining the change of measure η = ρ 1 = ρ 2 = 1 − λ = (log log b) −1 , then we have the following approximation:
lim b→∞ sup A∈F |Q(A) − P * b (A)| = 0,
where F is the σ-field where the measures are defined.
Remark 4. Theorem 3 is the central result of this paper. We present its detailed proof. The technical developments of other theorems are all based on that of Theorem 3. Therefore, we only layout their key steps and the major differences from that of Theorem 3.
Remark 5. The measure Q in the limit of the above theorem obviously depends on the tuning parameters (η, ρ 1 , ρ 2 , and λ) and the level b. To simplify the notation, we omit the indices of those parameters when there is no ambiguity.
Remark 6. The measure corresponding to the last mixture component in (3.10), T LR 2 (t) mes(T ) dt, has been employed by [43] to develop approximations for v(b). We emphasize that the measure constructed in this paper is substantially different. In fact, the measure corresponding to LR 2 (t) does not appear in the main proof. We included it to control the tail of the likelihood ratio in one lemma.
To illustrate the application of the measure Q, we provide a further characterization of the conditional distribution P * b by presenting another approximation result which is easier to understand at an intuitive level. Let
γ u (t) = f (t) + 1 ⊤f ′′ t 2σu t + B t u t + µ σ (t), β u (T ) = sup t∈T γ u (t), (3.16)P b (f (·) ∈ A) = P (f (·) ∈ A|β u (T ) > u)
. The process γ u (t) is slightly different than α t . The following theorem states that the measure Q also approximates the distributionP b in total variation for b large.
Theorem 7. Consider a Gaussian random field {f (t) : t ∈ T } living on a domain T satisfying conditions C1-C6. With the same choice of tuning parameters as in Theorem 3, that is,
η = ρ 1 = ρ 2 = 1 − λ = (log log b) −1 , Q approximatesP b in total variation, that is, lim b→∞ sup A∈F |Q(A) −P b (A)| = 0.
3.3. Some implications of the theorems. The results of Theorems 3 and 7 provide both qualitative and quantitative descriptions of P * b . From a qualitative point of view, Theorems 3 and 7 suggest that
sup A∈F |P * b (A) −P b (A)| → 0 (3.17)
as b → ∞. Note that γ u (t) itself is a Gaussian process. Thus, the above convergence result connects the tail events of exponential integrals to those of the supremum of another Gaussian random field that is a linear combination of f and its derivative field. We set up this connection mainly because the distribution of Gaussian random fields conditional on level crossing (also known as the Slepian model) is very well studied for smooth processes [32]. For the purpose of illustration, we cite one result in Chapter 6.2 of [6] when γ u (t) is stationary and twice differentiable. Let covariance function of γ u (t) be C γ (t). Conditional on γ u (t) achieving a local maximum at location t * at level x, we have the following closed form representation of the conditional field:
γ u (t * + t) = xC γ (t) − W x β(t) + g(t), (3.18) where β(t) = 1 µ γ 20 µ γ 02 µ γ 22 −1 µ γ⊤ 2 (t),
µ γ ij 's are the spectral moments of C γ (t), W x is a d(d + 1)/2 dimensional random vector whose density can be explicitly written down and g(t) is a mean zero Gaussian process whose covariance function is also in a closed form; see [6] for the specific forms. If we set x > u → ∞, the local maximum is asymptotically the global maximum. Furthermore, thanks to stationarity, the distribution of t * is asymptotically uniform over T . The overshoot x − u is asymptotically an exponential random variable. Thus, the conditional field γ u (t) can be written down explicitly through representation (3.18), the overshoot distribution and the distribution of t * . Furthermore, the conditional distribution of f (t) can be implied by (3.16) and conditional normal calculations.
From a quantitative point of view, Theorem 3 implies that for any bounded function Ξ :
C(T ) → R the conditional expectation E[Ξ(f )|I(T ) > b] can be approximated by E Q [Ξ(f )], more precisely, E[Ξ(f )|I(T ) > b] − E Q [Ξ(f )] → 0 (3.19) as b → ∞. The expectation E Q [Ξ(f )]
is much easier to compute (both analytically and numerically) via the following identity: (3.20) Note that the inner expectation is under the measure P in that the condi-
E Q [Ξ(f )] = E Q [E[Ξ(f )|ı, τ, f (τ ), ∂f (τ ), ∂ 2 f (τ )]].tional distribution of f given (f (τ ), ∂f (τ ), ∂ 2 f (τ )) under Q is the same as that under P . Furthermore, conditional on (f (τ ), ∂f (τ ), ∂ 2 f (τ ))
, the process f (t) is also a Gaussian process and has the expansion
f (t) = f (τ ) + ∂f (τ ) ⊤ (t − τ ) + 1 2 (t − τ ) ⊤ ∆f (τ )(t − τ ) + o(|t − τ | 2 )
. These results provide sufficient tools to evaluate the conditional expectation
E[Ξ(f )|ı, τ, f (τ ), ∂f (τ ), ∂ 2 f (τ )].
Once the above expectation has been evaluated, we may proceed to the outer expectation in (3.20). Note that the inner expectation is a function of (ı, τ, f (τ ), ∂f (τ ), ∂ 2 f (τ )), the joint distribution of which is in a closed form. Thus, evaluating the outer expectation is usually an easier task. In fact, the proof of Theorem 3 is an exercise of the above strategy by considering that Ξ(f ) = (dP/dQ) 2 .
Remark 8. According to the detailed proof of Theorem 3, the approximation (3.19) is applicable to all the functions such that
sup b E[Ξ 2 (f )|I(T ) > b] < ∞.
To see that, we need to change the statement and the proof of Lemma 13 presented in Section 4.
3.4. Efficient rare-event simulation for I(T ). In the preceding subsection we constructed a change of measure that asymptotically approximates the conditional distribution of f given I(T ) > b. In this section, we construct an efficient importance sampling estimator based on this change of measure to compute v(b) as b → ∞. We evaluate the overall computation efficiency using a concept that has its root in the general theory of computation in both continuous and discrete settings [47,54]. In particular, completely analogous notions in the setting of complexity theory of continuous problems lead to the notion of tractability of a computational problem [55]. Definition 9. A Monte Carlo estimator is said to be a fully polynomial randomized approximation scheme (FPRAS) for estimating v(b) if, for some q 1 , q 2 and d > 0, it outputs an averaged estimator that is guaranteed to have at most ε > 0 relative error with confidence at least
1 − δ ∈ (0, 1) in O(ε −q 1 δ −q 2 | log v(b)| d ) function evaluations.
Equivalently, one needs to compute an estimator
Z b with complexity O(ε −q 1 δ −q 2 | log v(b)| d ) such that P (|Z b /v(b) − 1| > ε) < δ. (3.21)
In the literature of rare-event simulations, an estimator L b is said to be
strongly efficient in estimating v(b) if EL b = v(b) and sup b Var L b /v 2 (b) < ∞.
Suppose that a strongly efficient estimator L b has been obtained. Let {L
Z b = 1 n n j=1 L (j) b
has a relative mean squared error equal to
E(Z b /v(b) − 1) 2 = Var(L b ) × n −1/2 v(b) −1 . A simple consequence of Chebyshev's inequlity yields P (|Z b /v(b) − 1| ≥ ε) ≤ Var(L b ) ε 2 nv 2 (b) .
Thus, it suffices to simulate n = O(ε −2 δ −1 ) i.i.d. replicates of L b to achieve the accuracy in (3.21). The so-called importance sampling is based on the identity P (A) = E Q [I A dP/dQ]. The random variable I A dP/dQ is an unbiased estimator of P (A). It is well known that if one chooses Q(·) = P (·|A), then I A dP/dQ has zero variance. The measure Q created in the previous subsection is a good approximation of P * b , and thus it naturally leads an estimator for v(b) with small variance.
In addition to the variance control, another issue is that the random fields considered in this paper are continuous objects. A computer can only perform discrete simulations. Thus we must use a discrete object approximating the continuous field to implement the algorithms. The bias caused by the discretization must be well controlled relative to v(b). In addition, the complexity of generating one such discrete object should also be considered in order to control the overall computational complexity to achieve an FPRAS.
We create a regular lattice covering T . Define
G N,d = i 1 N , i 2 N , . . . , i d N : i 1 , . . . , i d ∈ Z .I M (T ) = M i=1 mes(T N (t i )) × e σX i +µ(t i ) . (3.23)
We have the following theorem to control the bias.
Theorem 10. Consider a Gaussian random field f satisfying conditions in Theorem 3. For any ε 0 > 0, there exists κ 0 such that for any ε
∈ (0, 1), if N ≥ κ 0 ε −1−ε 0 (log b) 2+ε 0 , then for b > 2 |v M (b) − v(b)| v(b) < ε.
We estimate v M (b) using a discrete version of the change of measure proposed in the previous section. The specific algorithm is given as follows:
(1) Generate a random indicator ı ∼ Bernoulli(ρ 2 ). If ı = 1, then:
(a) generate ι uniformly from {1, . . . , M }; (b) generate X ι ∼ N (u tι , 1); (c) given (t ι , X ι ), simulate the joint field (f (t), ∂f (t), ∂ 2 f (t)) on the lattice T N \ {t ι } from the original conditional distribution under P .
(2) If ı = 0: (a) if µ(t) is not constant, simulate a random index ι proportional to l(t ι ), that is, P (ι = i) = l(t i )/κ and κ = M i=1 l(t i ); if µ(t) ≡ 0, then ι is simulated uniformly over {1, . . . , M }; (b) given the realized ι, simulate f (t ι ) = X ι = x, ∂f (t ι ) = y, ∂ 2 f (t ι ) = z from density function h all (x, y, z) = 1 − ρ 1 − ρ 2 1 − ρ 2 h 0,tι (x, y, z) + ρ 1 1 − ρ 2 h 1,tι (x, y, z);
(c) given (t ι , f (t ι ), ∂f (t ι ), ∂ 2 f (t ι )), simulate the joint field (f (t), ∂f (t), ∂ 2 f (t)) on the lattice T N \ {t ι } from the original conditional distribution under P .
(3) Output
L b = I {I M (T )>b} 1 − ρ 1 − ρ 2 κ M i=1 l(t i ) LR(t i ) + ρ 1 κ M i=1 l(t i ) LR 1 (t i ) (3.24) + ρ 2 M i=1 LR 2 (t i ) M .
Let Q M be the measure induced by the above simulation scheme. Then it is not hard to verify thatL b = I {I M (T )>b} dP/dQ M , and thusL b is an unbiased estimator of v M (b). The next theorem states the strong efficiency of the above algorithm.
Theorem 11. Suppose f is a Gaussian random field satisfying conditions in Theorem 3. If N is chosen as in Theorem 10 and all the other parameters are chosen as in Theorem 3, then there exists some constant
κ 1 > 0 such that sup b>1 E Q ML 2 b v 2 M (b) ≤ κ 1 .
Let Z b be the average of n i.i.d. copies ofL b . According to the results in Theorem 10, we have that
Z b v(b) − 1 ≤ Z b v M (b) (v M (b)/v(b) − 1) + Z b v M (b) − 1 ≤ ε Z b v M (b) + Z b v M (b) − 1 .
The results of Theorem 11 indicate that
P (|Z b /v M (b) − 1| ≥ ε) ≤ κ 1 ε 2 n .
If we choose n = κ 1 ε −2 δ −1 , then
P (|Z b /v(b) − 1| ≥ 3ε) ≤ δ.
Thus, the accuracy level as in (3.21) has been achieved. Note that simulating oneL b consists of generating a multivariate Gaussian random vector of 4. Proof of Theorem 3. We use the following simple yet powerful lemma to prove Theorem 3.
dimension M × (d + 1)(d + 2)/2 = O(N d ) = O((log b) (2+ε 0 )d ε −(1+ε 0 )d )
Lemma 13. Let Q 0 and Q 1 be probability measures defined on the same σ-field F such that dQ 1 = r −1 dQ 0 for a positive random variable r. Suppose that for some ε > 0,
E Q 1 [r 2 ] = E Q 0 [r] ≤ 1 + ε. Then sup |X|≤1 |E Q 1 (X) − E Q 0 (X)| ≤ ε 1/2 . Proof. |E Q 1 (X) − E Q 0 (X)| = |E Q 1 [(1 − r)X]| ≤ E Q 1 |r − 1| ≤ [E Q 1 (r − 1) 2 ] 1/2 = (E Q 1 [r 2 ] − 1) 1/2 ≤ ε 1/2 .
We also need the following approximations for the tail probability v(b). This proposition is an extension of Theorem 3.4 and Corollary 3.5 in [43]. We layout the key steps of its proof in the supplemental material [45].
v(b) ∼ (2π) d/2 det(−∆µ σ (t * )) −1/2 G(t * ) · u d/2−1 exp − (u − µ σ (t * )) 2 2 ,
where u is as defined in (3.4), and G(t) is defined as
det(Γ) −1/2 (2π) (d+1)(d+2)/4 e 1 T µ 22 1/(8σ 2 )+Bt × R d(d+1)/2 exp − 1 2 |µ 20 µ −1 22 z| 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22 z − µ 1/2 22 1 2σ 2 dz. If µ(t) ≡ 0, G(t) is a constant denoted by G. Then v(b) ∼ mes(T )G · u d−1 e −u 2 /2 . 4.1. Case 1: µ(t)
is not a constant. To make the proof smooth, we arrange the statement of the rest supporting lemmas in the Appendix. We start the proof of Theorem 3 when µ(t) is not a constant. Note that
E Q dP * b dQ 2 = v(b) −2 E Q dP dQ 2 ; I(T ) > b .
Thanks to Lemma 13, we only need to show that for any ε > 0 there exists b 0 such that for all b > b 0
E Q dP dQ 2 ; I(T ) > b = E Q E Q ı,τ dP dQ 2 ; I(T ) > b ≤ (1 + ε)v(b) 2 ,
where we use the notation E Q ı,τ [·] = E Q [·|ı, τ ] to denote the conditional expectation given ı and τ . τ ∈ T is the random index described as in the simulation scheme admitting a density function l(t) if ı = 0 and mes −1 (T )I T (t) if ı = 1. Note that
E Q ı,τ dP dQ 2 ; I(T ) > b = E Q ı,τ E Q ı,τ dP dQ 2 ; I(T ) > b f (τ ), ∂f (τ ), ∂ 2 f (τ ) .
For the rest of the proof, we mostly focus on the conditional expectation
E Q ı,τ dP dQ 2 ; I(T ) > b f (τ ), ∂f (τ ), ∂ 2 f (τ ) .
The rest of the discussion is conditional on ı and τ . To simplify notation, for a given τ , we define
f * (t) = f (t) − u τ C(t − τ ).
On the set {I(T ) > b}, f (τ ) reaches a level u τ , and E[f (t)|f (τ ) = u τ ] = u τ C(t − τ ). Thus, f * (t) is the field with the conditional expectation removed. From now on, we work with this shifted field f * (t). Correspondingly, we have
∂f * (t) = ∂f (t) − u τ ∂C(t − τ ), ∂ 2 f * (t) = ∂ 2 f (t) − u τ ∂ 2 C(t − τ ).
We further define the following notation:
w = f * (τ ), y = ∂f * (τ ), z = ∂ 2 f * (τ ), z = ∆f * (τ ), y = ∂f * (τ ) + ∂µ σ (τ ),z = ∆f * (τ ) + µ σ (τ )I + ∆µ σ (τ ), (4.1) w t = f * (t), y t = ∂f * (t), z t = ∂ 2 f * (t),z t = ∂ 2 f * (t) − u t µ 02 .
Under the measure Q and a given τ , if ı = 0, (w, y, z) has density function
h * all (w, y, z) = 1 − ρ 1 − ρ 2 1 − ρ 2 h * 0,τ (w, y, z) + ρ 1 1 − ρ 2 h * 1,τ (w, y, z); (4.2)
if ı = 1, then (w, y, z) follows density h * τ (w, y, z). The forms of the densities can be derived from h 0,t , h 1,t and h. In particular, their expressions are given .6). In the next step, we will compute dQ/dP in the form of f * (t). Basically, we replace f (t) by f * (t) + u τ C(t − τ ), ∂f (t) by y t + u τ ∂C(t − τ ), ∂ 2 f (t) by
h * 0,τ (w, y, z) ∝ I Aτ × exp −λu τ w + 1 ⊤ z 2σu τ + B τ u τ − 1 2 |y| 2 × exp − 1 2 |µ 20 µ −1 22 z| 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22 z − µ 1/2 22 1 2σ 2 , h * 1,τ (w, y, z) ∝ I A c τ × exp λ 1 u τ w + 1 ⊤ z 2σu τ + B τ u τ − 1 2 |y| 2 × exp − 1 2 |µ 20 µ −1 22 z| 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22 z − µ 1/2 22 1 2σ 2 , h * τ (w, y, z) = h(w, y, z) = det(Γ) −(1/2) (2π) (d+1)(d+2)/4 × exp − 1 2 y ⊤ y + |w − µ 20 µ −1 22 z| 2 1 − µ 20 µ −1 22 µ 02 + z ⊤ µ −1 22 z , and A τ = {w + y ⊤ y 2uτ + 1 ⊤ z 2σuτ + Bτ uτ > −ηu −1 τ } is defined as in (3z t + u τ ∂ 2 C(t − τ ) andf ′′ t = ∂ 2 f (t) − u t µ 02 byz t + u τ ∂ 2 C(t − τ )
. For the likelihood ratio terms LR and LR 1 in (3.11), note that the |∂f (t)| 2 terms in h 0,t and h 1,t cancel with those in h(f (t), ∂f (t), ∂ 2 f (t)), that is,
LR(t) = I At · H λ · u t exp −λu t f (t) + 1 ⊤f ′′ t 2σu t + B t u t − u t − 1 2 |µ 20 µ −1 22f ′′ t | 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22f ′′ t − µ 1/2 22 1 2σ 2 det(Γ) −1/2 (2π) (d+1)(d+2)/4 × e −(1/2)[(f (t)−µ 20 µ −1 22 ∂ 2 f (t)) 2 /(1−µ 20 µ −1 22 µ 02 )+∂ 2 f (t) ⊤ µ −1 22 ∂ 2 f (t)] .
We insert the notation in (4.1) and obtain that
LR(t) = I At · u t H λ exp −λu t w t + u τ C(t − τ ) + 1 ⊤ (z t + µ 2 (t − τ )u τ ) 2σu t + B t u t − u t × exp − 1 2 |µ 20 µ −1 22 (z t + µ 2 (t − τ )u τ )| 2 1 − µ 20 µ −1 22 µ 02 (4.3) + µ −1/2 22 (z t + µ 2 (t − τ )u τ ) − µ 1/2 22 1 2σ 2 × h −1 x,z (w t + u τ C(t − τ ), z t + u τ ∂ 2 C(t − τ )), where h x,z (x, z) = det(Γ) −1/2 (2π) (d+1)(d+2)/4 e −(1/2)[(x−µ 20 µ −1 22 z) 2 /(1−µ 20 µ −1 22 µ 02 )+z ⊤ µ −1 22 z] , (4.4)
which is the function h(x, y, z) with the |y| 2 term removed. Similarly, we have that )). With the analytic forms (4.3) and (4.5), we proceed to the likelihood ratio in (3.10)
LR 1 (t) = I A c t · u t H λ 1 exp λ 1 u t w t + u τ C(t − τ ) + 1 ⊤ (z t + µ 2 (t − τ )u τ ) 2σu t + B t u t − u t × exp − 1 2 |µ 20 µ −1 22 (z t + µ 2 (t − τ )u τ )| 2 1 − µ 20 µ −1 22 µ 02 (4.5) + µ −1/2 22 (z t + µ 2 (t − τ )u τ ) − µ 1/2 22 1 2σ 2 × h −1 x,z (w t + u τ C(t − τ ), z t + u τ ∂ 2 C(t − τdQ dP = (1 − ρ 1 − ρ 2 )K + ρ 1 K 1 + ρ 2 K 2 , (4.6) where K = A * l(t) LR(t) dt, K 1 = (A * ) c l(t) LR 1 (t) dt, K 2 = T e −(1/2)u 2 t +utwt+utuτ C(t−τ ) mes(T ) dt.
The set A * [depending on the sample path f * (t)] is defined as
t : w t + C(t − τ )u τ + |y t + u τ · ∂C(t − τ )| 2 2u t + 1 ⊤ (z t + u τ µ 2 (t − τ )) 2σu t + B t u t > u t − η u t .
EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS
23
We may equivalently define A * = {t : f ∈ A t }. Note that LR(t) = 0 if f / ∈ A t . Thus, the integral K is on the set A * , and K 1 is on the complement of A * .
Based on the above results, we have that
E Q dP dQ 2 ; I(T ) > b ≤ E Q E Q ı,τ 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; I(T ) > b (4.7) ≤ E Q E Q ı,τ 1 [(1 − ρ 1 − ρ 2 )K] 2 ; I(T ) > b, A τ ≥ 0 + E Q E Q ı,τ 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; I(T ) > b, A τ < 0 , where A τ = w + y ⊤ y 2u τ + 1 ⊤ z 2σu τ + B τ u τ . (4.8)
Note that the term K 2 is not used in the main analysis. In fact, K 2 is only used in Lemma 17 for the purpose of localization that will be presented later. The rest of the analysis consists of three main parts.
Part 1. Conditional on (ı, τ, f * (τ ), ∂f * (τ ), ∂ 2 f * (τ )), we study the event (4.9) and write the occurrence of this event almost as a deterministic function of f * (τ ), ∂f * (τ ) and ∂ 2 f * (τ ), equivalently, (w, y, z).
E b = {I(T ) > b},
Part 2. Conditional on (ı, τ, f * (τ ), ∂f * (τ ), ∂ 2 f * (τ )), we express K and K 1 as functions of f * (τ ), ∂f * (τ ), ∂ 2 f * (τ ) with small correction terms.
Part 3. We combine the results from the first two parts and obtain an approximation of (4.7).
All the subsequent derivations are conditional on ı and τ .
Preliminary calculations.
To proceed, we provide the Taylor expansions for f * (t), C(t) and µ(t):
• Expansion of f * (t) given (f * (τ ), ∂f * (τ ),
∂ 2 f * (τ )). Let t − τ = ((t − τ ) 1 , . . . , (t − τ ) d ).
Conditional on (f * (τ ), ∂f * (τ ), ∂ 2 f * (τ )), we first expand the random function
f * (t) = E[f * (t)|f * (τ ), ∂f * (τ ), ∂ 2 f * (τ )] + g(t − τ ) = f * (τ ) + ∂f * (τ ) ⊤ (t − τ ) + 1 2 (t − τ ) ⊤ ∆f * (τ )(t − τ ) (4.10) + R f (t − τ ) + g(t − τ ), 24 J. LIU AND G. XU where R f (t − τ ) = O(|t| 2+δ 0 (|f * (τ )| + |∂f * (τ )| + |∂ 2 f * (τ )|))
is the remainder term of the Taylor expansion of E[f * (t)|f * (τ ), ∂f * (τ ),
∂ 2 f * (τ )]. g(t)
is a mean zero Gaussian random field such that Eg 2 (t) = O(|t| 4+δ 0 ) as t → 0. In addition, the distribution of g(t) is independent of ı, τ, f * (τ ), ∂f * (τ ) and ∂ 2 f * (τ ).
• Expansion of C(t):
C(t) = 1 − 1 2 t ⊤ t + C 4 (t) + R C (t), (4.11) where C 4 (t) = 1 24 ijkl ∂ 4 ijkl C(0)t i t j t k t l and R C (t) = O(|t| 4+δ 0 ). • Expansion of µ(t): µ(t) = µ(τ ) + ∂µ(τ ) ⊤ (t − τ ) + 1 2 (t − τ ) ⊤ ∆µ(τ )(t − τ ) + R µ (t − τ ), (4.12) where R µ (t − τ ) = O(|t − τ | 2+δ 0 ).
We write
R(t) = R f (t) + u τ R C (t) + R µ (t)/σ
to denote all the remainder terms.
Choose small constants ǫ and δ such that 0 < ǫ ≪ δ ≪ δ 0 . By writing
x ≪ y, we mean that x/y is chosen sufficiently small, but x/y does not change with b. Let
L = |τ − t * | < u −1/2+ǫ , |w| ≤ u 1/2+ǫ , |y| < u ǫ , |z| < u ǫ , (4.13) sup |t−τ |<u −1+δ |z t − z| < u −ǫ , sup |t−τ |<u −1+δ |g(t)| < u −1−δ .
By Lemma 17 whose proof uses the last component LR 2 (t), we have that
E Q dP dQ 2 ; E b , L c = o(1)v 2 (b).
Therefore we only need to consider the second moment on the set L, that is,
E Q dP dQ 2 ; E b , L ≤ E Q E Q ı,τ 1 [(1 − ρ 1 − ρ 2 )K] 2 ; E b , L, A τ > 0 (4.14) + E Q E Q ı,τ 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; E b , L, A τ < 0 ,
EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS
25
where K and K 1 are given as in (4.6). We will focus on the terms on the right-hand side of (4.14) in the subsequent derivations. Now, we start to carry out each part of the program. 4.2. Part 1. All the derivations in this part are conditional on specific values of ı, τ , f * (τ ), ∂f * (τ ) and ∂ 2 f * (τ ), equivalently, ı, τ , w, y and z. By definition,
I(T ) = T e σf * (t)+σuτ C(t−τ )+µ(t) dt.
We insert the expansions in (4.10), (4.11) and (4.12) into the expression of I(T ) and obtain that
I(T ) = t∈T exp σ w + y ⊤ (t − τ ) + 1 2 (t − τ ) ⊤ z(t − τ ) + R f (t − τ ) + g(t − τ ) (4.15) × exp (σu − µ(τ )) × 1 − 1 2 (t − τ ) ⊤ (t − τ ) + C 4 (t − τ ) + R C (t − τ ) × exp µ(τ ) + ∂µ(τ ) ⊤ (t − τ ) + 1 2 (t − τ ) ⊤ ∆µ(τ )(t − τ ) + R µ (t − τ ) dt,
where the first row corresponds to the expansion of w t = f * (t), and the second and third rows correspond to those of C(t) and µ(t), respectively. We write the exponent inside the integral in a quadratic form of (t − τ ) and obtain that
I(T ) = exp σu + σw + σ 2ỹ ⊤ (uI −z) −1ỹ × t∈T exp − σ 2 (t − τ − (uI −z) −1ỹ ) ⊤ (uI −z) (4.16) × (t − τ − (uI −z) −1ỹ ) × exp{σu τ C 4 (t − τ ) + σR(t − τ )} × exp{σg(t − τ )} dt,
whereỹ andz are defined as in (4.1). Let a(s) and b(s) be two generic positive functions. Then we have the representation of the following integral:
T a(s)b(s) ds = E[b(S)] T a(s) ds,
where S is a random variable taking values in T with density a(s)/ T a(t) dt.
Using this representation and the change of variable that s = (uI −z) 1/2 (t − τ ), we write the big integral in (4.16) as a product of expectations and a normalizing constant, and obtain that
I(T ) = det(uI −z) −1/2 exp σu + σw + σ 2ỹ ⊤ (uI −z) −1ỹ × (uI−z) −1/2 s+τ ∈T exp − σ 2 (s − (uI −z) −1/2ỹ ) ⊤ × (s − (uI −z) −1/2ỹ ) ds × E[e σuτ C 4 ((uI−z) −1/2 S)+σR((uI−z) −1/2 S) ] × E[e σg((uI−z) −1/2S ) ].
The two expectations in the above display are taken with respect to S and S given the process g(t). S is a random variable taking values in the set {s : (uI −z) −1/2 s + τ ∈ T } with density proportional to
e −(σ/2)(s−(uI−z) −1/2ỹ ) ⊤ (s−(uI−z) −1/2ỹ ) , (4.17)
andS is a random variable taking values in the set {s : (uI −z) −1/2 s + τ ∈ T } with density proportional to
e −(σ/2)(s−(uI−z) −1/2ỹ ) ⊤ (s−(uI−z) −1/2ỹ )+σuτ C 4 ((uI−z) −1/2 s)+σR((uI−z) −1/2 s) .
Together with the definition of u that ( 2π σ ) d/2 u −d/2 e σu = b, we obtain that I(T ) > b if and only if We take log on both sides, and plug in the result of Lemma 20 that handles the big expectation term in (4.18). Then inequality (4.18) is equivalent to
I(T ) = det(uI −z) −1/2 e σu+σw+(σ/2)ỹ ⊤ (uI−z) −1ỹ × (uI−z) −1/2 s+τ ∈T e −(σ/2)(s−(uI−z) −1/2ỹ ) ⊤ (s−(uI−z) −1/2ỹ ) dsw +ỹ ⊤ (uI −z) −1ỹ 2 − log det(I −z/u) 2σ + i ∂ 4 iiii C(0) 8σ 2 u (4.20) > ξ u uσ + o(|w| + |y| + |z| + 1) u 1+δ 0 /4 .
On the set L, we further simplify (4.20) using the following facts (see Lemma 21):
∂µ σ (τ ) = O(u −1/2+ǫ ), log det I −z u = − 1 u Tr(z) + o(u −1−δ 0 /4 ) = − 1 ⊤ (z + ∂ 2 µ σ (τ )) + d · µ σ (τ ) u + o(u −1−δ 0 /4 ),
where Tr is the trace of a matrix. Therefore, on the set L, (4.20) is equivalent to
w + y ⊤ y 2u + 1 ⊤ (z + ∂ 2 µ σ (τ )) + d · µ σ (τ ) 2σu + i ∂ 4 iiii C(0) 8σ 2 u > ξ u uσ + o(|w| + |y| + |z| + 1) u 1+δ 0 /4 ,
and further, equivalently (by replacing u with u τ ),
w + y ⊤ y 2u τ + 1 ⊤ (z + ∂ 2 µ σ (τ )) + d · µ σ (τ ) 2σu τ + i ∂ 4 iiii C(0) 8σ 2 u τ > ξ u uσ + o(|w| + |y| + |z| + 1) u 1+δ 0 /4 .
Using the notation defined as in (3.9) and (4.8), I(T ) > b is equivalent to
A τ + o(|w| + |y| + |z| + 1) u 1+δ 0 /4 > ξ u uσ ,
where A τ is defined as in (4.8). Furthermore, with ǫ ≪ δ 0 and on the set
L, o(|y| + |z|)/u −1−δ 0 /4 = o(u −1−δ 0 /8 ).
For the above inequality, we absorb o(wu −1−δ 0 /4 ) into A τ and rewrite it as
A τ > (1 + o(u −1−δ 0 /4 )) ξ u σu + o(u −1−δ 0 /8 ) .
Part 2.
In part 2, we first consider (1 − ρ 1 − ρ 2 )K in the first expectation of (4.7) (which is on the set {A τ ≥ 0}) and then (1 − ρ 1 − ρ 2 )K + ρ 1 K 1 in the second expectation of (4.7).
Part 2.1: The analysis of K when A τ ≥ 0. Similarly to part 1, all the derivations are conditional on (ı, τ, w, y, z). We now proceed to the second part of the proof. More precisely, we simplify the term K defined as in (4.6), and write it as a deterministic function of (w, y, z) with a small correction term. Recall that
K = A * l(t)u t H λ exp −λu t w t + u τ C(t − τ ) + 1 ⊤ (z t + µ 2 (t − τ )u τ ) 2σu t + B t u t − u t × exp − 1 2 |µ 20 µ −1 22 (z t + µ 2 (t − τ )u τ )| 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22 (z t + µ 2 (t − τ )u τ ) − µ 1/2 22 1 2σ 2 × h −1 x,z (w t + u τ C(t − τ ), z t + u τ ∂ 2 C(t − τ )) dt.
We plug in the forms of h x,z and l(t) that are defined in (4.4) and (3.13) and obtain that
K = (2π) (d+1)(d+2)/4−d/2 det(Γ) 1/2 · det(−∆µ σ (t * )) 1/2 u d/2 t * H λ × A * exp u t * · (t − t * ) ⊤ ∆µ σ (t * )(t − t * ) 2 × u t × exp −λu t w t + u τ C(t − τ ) + 1 ⊤ (z t + µ 2 (t − τ )u τ ) 2σu t + B t u t − u t × exp − 1 2 |µ 20 µ −1 22 (z t + µ 2 (t − τ )u τ )| 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22 (z t + µ 2 (t − τ )u τ ) − µ 1/2 22 1 2σ 2 × exp 1 2 (w t + u τ C(t − τ ) − µ 20 µ −1 22 (z t + µ 2 (t − τ )u τ )) 2 1 − µ 20 µ −1 22 µ 02 + (z t + µ 2 (t − τ )u τ ) ⊤ µ −1 22 (z t + µ 2 (t − τ )u τ ) dt.
For some δ ′ such that ǫ < δ ′ < δ, where ǫ, δ are the parameters we used to define L, we further restrict the integration region by defining
I 2 = A * ,|t−τ |<u −1+δ ′ exp u t * (t − t * ) ⊤ ∆µ σ (t * )(t − t * ) 2 × u t × exp −λu t w t + u τ C(t − τ ) + 1 ⊤ (z t + µ 2 (t − τ )u τ ) 2σu t + B t u t − u t × exp − 1 2 |µ 20 µ −1 22 (z t + µ 2 (t − τ )u τ )| 2 1 − µ 20 µ −1 22 µ 02 (4.21) + µ −1/2 22 (z t + µ 2 (t − τ )u τ ) − µ 1/2 22 1 2σ 2 × exp 1 2 (w t + u τ C(t − τ ) − µ 20 µ −1 22 (z t + µ 2 (t − τ )u τ )) 2 1 − µ 20 µ −1 22 µ 02 + (z t + µ 2 (t − τ )u τ ) ⊤ µ −1 22 (z t + µ 2 (t − τ )u τ ) dt. Thus K ≥ (2π) (d+1)(d+2)/4−d/2 det(Γ) 1/2 × det(−∆µ σ (t * )) 1/2 u d/2 t * H λ · I 2 .
For the rest of part 2.1, we focus on I 2 . With some tedious algebra, Lemma 22 writes I 2 in a more manageable form; that is, I 2 equals
A * ,|t−τ |<u −1+δ ′ exp u t * (t − t * ) ⊤ ∆µ σ (t * )(t − t * ) 2 + u 2 t 2 × u t × exp (1 − λ)u t [w t + u τ C(t − τ ) − u t ] + (1 − λ) 2σ 1 ⊤ (z t − µ 02 u t + µ 2 (t − τ )u τ ) − λB t − 1 ⊤ µ 22 1 8σ 2 (4.22) × exp{((w t + u τ C(t − τ ) − u t ) 2 − 2(w t + u τ C(t − τ ) − u t )µ 20 µ −1 22 (z t − µ 02 u t + µ 2 (t − τ )u τ )) /(2(1 − µ 20 µ −1 22 µ 02 ))} dt. Lemma 23 implies that {|t − τ | < u −1+δ ′ } ⊂ A * . Thus, on the set {A τ > 0}, we have A * ∩ {|t − τ | < u −1+δ ′ } = {|t − τ | < u −1+δ ′ }
and we can remove A * from the integration region of I 2 . In addition, on the set L and |t − τ | < u −1+δ ′ , we have that
u τ − u t C(t − τ ) = O(u −1+2δ ′ ), µ 2 (t − τ ) = µ 20 + O(|t − τ | 2 ), |u τ µ 2 (t − τ ) − u t µ 20 | = O(u −1+2δ ′ ), (u τ − u t C(t − τ ))|z t | = o(1).
We insert the above estimates to (4.22). Together with the fact that
exp u t * (t − t * ) ⊤ ∆µ σ (t * )(t − t * ) 2 + u 2 t 2 = (1 + o(1)) exp 1 2 u 2 t * ,
we have that
I 2 ∼ u × exp 1 2 u 2 t * − λB t * − 1 ⊤ µ 22 1 8σ 2 × |t−τ |<u −1+δ ′ exp (1 − λ) × u t [w t + u τ (C(t − τ ) − 1) + (µ σ (t) − µ σ (τ ))] + (1 − λ) 1 ⊤ z 2σ + w 2 t − 2w t µ 20 µ −1 22 z t + o(1)w t 2(1 − µ 20 µ −1 22 µ 02 )
dt.
Further, we have that
w 2 t − 2w t µ 20 µ −1 22 z t + o(1)w t = o(1) + u · w · O(u −1/2+δ ′ ). Let ζ u = O(u −1/2+δ ′ )
, and we simplify I 2 to
I 2 ∼ u × exp 1 2 u 2 t * − λB t * − 1 ⊤ µ 22 1 8σ 2 × |t−τ |<u −1+δ ′ exp (1 − λ)(u τ + ζ u )[ζ u w + w t + u τ (C(t − τ ) − 1) + (µ σ (t) − µ σ (τ ))] + (1 − λ) 1 ⊤ z 2σ dt.
In what follows, we insert the expansions in (4.10), (4.11) and (4.12) into the expression of I 2 and write the exponent as a quadratic function of t − τ , and we obtain that on the set L
I 2 ∼ u × exp 1 2 u 2 t * − λB t * − 1 ⊤ µ 22 1 8σ 2 × exp (1 − λ)(u τ + ζ u ) (1 + ζ u )w + 1 2ỹ ⊤ (uI −z) −1ỹ + 1 ⊤ z 2σu τ (4.23) × |t−τ |<u −1+δ ′ e −(1/2)(1−λ)(uτ +ζu)(t−τ −(uI−z) −1ỹ ) ⊤ (uI−z)(t−τ −(uI−z) −1ỹ ) × e (1−λ)(uτ +ζu)[uτ C 4 (t−τ )+R(t−τ )+g(t−τ )] dt,
where we recall thatỹ = y + ∂µ σ (τ ) andz = z + u σ (τ )I + ∆µ σ (τ ). This derivation is very similar to that from (4.15) to (4.16). In the last row of the above display, on the set L and |t − τ | < u −1+δ ′ ,
u 2 C 4 (t − τ ) + uR(t − τ ) = o(1).
Therefore, they can be ignored. We consider the change of variable that
s = (1 − λ) 1/2 (u τ + ζ u ) 1/2 (uI −z) 1/2 (t − τ )
and obtain that I 2 equals (with the terms C 4 and R removed)
I 2 ∼ (1 − λ) −d/2 u −d+1 exp 1 2 u 2 t * − λB t * − 1 ⊤ µ 22 1 8σ 2 × exp (1 − λ)(u τ + ζ u ) (1 + ζ u )w + 1 2ỹ ⊤ (uI −z) −1ỹ + 1 ⊤ z 2σu (4.24) × s∈Su e −(1/2)|s−(1−λ) 1/2 (uτ +ζu) 1/2 (uI−z) −1/2ỹ | 2 ds ×E[e (1−λ)(uτ +ζu)g((1−λ) −1/2 (uτ +ζu) −1/2 (uI−z) −1/2 S ′ ) ], where S u = {s : |(1 − λ) −1/2 (u τ + ζ u ) −1/2 (uI −z) −1/2 s| < u −1+δ ′ },
and S ′ is a random variable taking values on the set S u with density proportional to e −(1/2)|s−(1−λ) 1/2 (uτ +ζu) 1/2 (uI−z) −1/2ỹ | 2 .
We use κ to denote the last two terms of (4.24), that is,
κ = Su e −(1/2)|s−(1−λ) 1/2 (uτ +ζu) 1/2 (uI−z) −1/2ỹ | 2 ds (4.25) × E[e (1−λ)(uτ +ζu)g((1−λ) −1/2 (uτ +ζu) −1/2 (uI−z) −1/2 S ′ ) ].
It is helpful to keep in mind that κ is approximately (2π) d/2 . We insert κ back to the expression of I 2 . Together with the fact thatỹ ⊤ (uI −z) −1ỹ = |ỹ| 2 /u + o(u −1 ), we have
I 2 ∼ κ(1 − λ) −d/2 u −d+1 exp 1 2 u 2 t * − λB t * − 1 ⊤ µ 22 1 8σ 2 (4.26) × exp (1 − λ)(u τ + ζ u ) (1 + ζ u )w + |ỹ| 2 2u τ + 1 ⊤ z 2σu τ .
Thus, we have that on the set {A τ > 0},
K ≥ (2π) (d+1)(d+2)/4−d/2 det(Γ) 1/2 · det(−∆µ σ (t * )) 1/2 u d/2 t * H λ · I 2 = (κ + o(1))(2π) (d+1)(d+2)/4−d/2 det(Γ) 1/2 × det(−∆µ σ (t * )) 1/2 H λ · (1 − λ) −d/2 u −d/2+1 (4.27) × exp 1 2 u 2 t * − λB t * − 1 ⊤ µ 22 1 8σ 2 + (1 − λ)(u τ + ζ u ) (1 + ζ u )w + |ỹ| 2 2u τ + 1 ⊤ z 2σu τ .
We further insert the A τ defined in (4.8) into (4.27) and obtain that
K ≥ (κ + o(1))(2π) (d+1)(d+2)/4−d/2 det(Γ) 1/2 × det(−∆µ σ (t * )) 1/2 H λ · (1 − λ) −d/2 u −d/2+1 (4.28) × exp 1 2 u 2 t * − B t * − 1 ⊤ µ 22 1 8σ 2 + (1 − λ)u τ (1 + o(1))A τ + (1 − λ)ζ u · (|ỹ| 2 + |z|) .
Part 2.2: The analysis of dP/dQ when A τ < 0. In this part, we focus mostly on the K 1 term, whose handling is very similar to that of K. Therefore, we only list out the key steps. For some large constant M , let D). For the first situation, the term K 1 is dominating; for the second situation, the term K (more precisely I 2 ) is dominating.
D = {|t − τ − (uI −z) −1ỹ | < M u −1 }
To simplify K 1 , we write it as
K 1 = (2π) (d+1)(d+2)/4−d/2 det(Γ) 1/2 · det(−∆µ σ (t * )) 1/2 u d/2 t * H λ 1 × (A * ) c ∩D + · · · + (A * ) c ∩D c · · · (2π) (d+1)(d+2)/4−d/2 det(Γ) 1/2 · det(−∆µ σ (t * )) 1/2 u d/2 t * H λ 1 × [I 1,2 + I 1,3 ].
Note that the difference between K 1 and K is that the term "−λ" has been replaced by "λ 1 ." With exactly the same derivation for (4.22), we obtain that I 1,2 equals [by replacing "−λ" in (4.22) by "λ 1 "]
(A * ) c ∩D exp u t * (t − t * ) ⊤ ∆µ σ (t * )(t − t * ) 2 + 1 2 u 2 t × u t × exp (1 + λ 1 )u t [w t + u τ C(t − τ ) − u t ] EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS 33 + (1 + λ 1 ) 2σ 1 ⊤ (z t − µ 02 u t + µ 2 (t − τ )u τ ) + λ 1 B t − 1 ⊤ µ 22 1 8σ 2 × exp{((w t + u τ C(t − τ ) − u t ) 2 (4.29) − 2(w t + (u τ C(t − τ ) − u t )) × µ 20 µ −1 22 (z t − µ 02 u t + µ 2 (t − τ )u τ )) /(2(1 − µ 20 µ −1 22 µ 02 ))} dt.
With a very similar derivation as in part 2.1, in particular, the result in (4.23), we have that
I 1,2 ∼ u × exp 1 2 u 2 t * + λ 1 B t * − 1 ⊤ µ 22 1 8σ 2 × exp (1 + λ 1 )(u τ + ζ u ) (1 + ζ u )w + 1 2ỹ ⊤ (uI −z) −1ỹ + 1 ⊤ z 2σu × (A * ) c ∩D exp (1 + λ 1 )(u τ + ζ u ) − 1 2 (t − τ − (uI −z) −1ỹ ) ⊤ (uI −z) (4.30) × (t − τ − (uI −z) −1ỹ ) × exp{(1 + λ 1 )(u τ + ζ u )[u τ C 4 (t − τ ) + R(t − τ ) + g(t − τ )]} dt.
Furthermore, similarly to the results in (4.26), we have that
I 1,2 ∼ κ 1,2 (1 + λ 1 ) −d/2 u −d+1 e (1/2)u 2 t * +λ 1 Bt * −1 ⊤ µ 22 1/(8σ 2 ) (4.31) × e (1+λ 1 )(uτ +ζu)((1+ζu)w+(1/2)ỹ ⊤ (uI−z) −1ỹ +1 ⊤ z/(2σuτ )) ,
where κ 1,2 is defined as
κ 1,2 = t 1 (s)∈(A * ) c ∩D e −1/2|s−(1+λ 1 ) 1/2 (uτ +ζu) 1/2 (uI−z) −1/2ỹ | 2 ds ×E[e (1+λ 1 )(uτ +ζu)g((1+λ 1 ) −1/2 (uτ +ζu) −1/2 (uI−z) −1/2 S 1,2 ) ],
the change of variable t 1 (s) = τ + (1 + λ 1 ) −1/2 (u τ + ζ u ) −1/2 (uI −z) −1/2 s and S 1,2 is a random variable taking values in the set {s : t(s) ∈ (A * ) c ∩ D} with an appropriately chosen density function similarly as in (4.24). In summary, the only difference between I 1,2 and I 2 lies in that the multiplier −λ is replaced by λ 1 . We now proceed to providing a lower bound of (1 − ρ 1 − ρ 2 )K + ρ 1 K 1 . Note that max{mes((A * ) c ∩ D), mes(A * ∩ D)} ≥ 1 2 mes(D).
Therefore at least one of (A * ) c ∩ D and A * ∩ D is nonempty. If mes((A * ) c ∩ D) ≥ 1 2 mes(D), we have the bound
(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ≥ ρ 1 K 1 ≥ Θ(1)ρ 1 u d/2 I 1,2 .
Similarly, if mes(A * ∩ D) ≥ 1 2 mes(D), we have that
(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ≥ Θ(1)(1 − ρ 1 − ρ 2 )u d/2 I 2 .
We further split I 2 in part 2.1 into two parts:
I 2 = A * ∩D · · · dt + A * ∩D c · · · dt I 2,1 + I 2,2 . (4.32)
Similarly to the derivation of I 1,2 , we have that
I 2,1 ∼ κ 2,1 (1 − λ) −d/2 u −d+1 e (1/2)u 2 t * −λBt * −1 ⊤ µ 22 1/(8σ 2 ) × e (1−λ)(uτ +ζu)((1+ζu)w+|ỹ| 2 /(2uτ )+1 ⊤ z/(2σuτ )) , where κ 2,1 = t 2 (s)∈A * ∩D e −1/2|s−(1−λ) 1/2 (uτ +ζu) 1/2 (uI−z) −1/2ỹ | 2 ds (4.33) ×E[e (1−λ)(uτ +ζu)g((1−λ) −1/2 (uτ +ζu) −1/2 (uI−z) −1/2 S 2,1 ) ].
S 2,1 is a random variable taking values on the set {s : t 2 (s) ∈ A * ∩ D} with an appropriate density function similarly as in (4.24) and t 2 (s) = τ + (1 − λ) −1/2 × (u τ + ζ u ) −1/2 (uI −z) −1/2 s. Then combining the above results of I 1,2 and I 2,1 , we have that for the case in which A τ < 0
ρ 1 K 1 + (1 − ρ 1 − ρ 2 )K ≥ Θ(1)u d/2 [I C 1 ρ 1 I 1,2 + I C 2 (1 − ρ 1 − ρ 2 )I 2,1 ] ≥ Θ(1)u −d/2+1 e (1/2)u 2 t * × [I C 1 · ρ 1 κ 1,2 e (1+λ 1 )(uτ +ζu)((1+ζu)w+|ỹ| 2 /(2uτ )+1 ⊤ z/(2σuτ )) + I C 2 · (1 − ρ 1 − ρ 2 )(1 − λ) −d/2 κ 2,1 × e (1−λ)(uτ +ζu)((1+ζu)w+|ỹ| 2 /(2uτ )+1 ⊤ z/(2σuτ )) ], where C 1 = {f (·) : mes((A * ) c ∩ D) ≥ mes(A * ∩ D)} and C 2 = C c 1 .
We further insert A τ defined in (4.8). Note that on the set {A τ < 0}, (1 + λ 1 )A τ < (1 − λ)A τ and B t is bounded away from zero and infinity. Then
(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ≥ Θ(1)u −d/2+1 e (1/2)u 2
t * · e (1+λ 1 )(1+ζu)uτ Aτ +ζu·(|ỹ| 2 +|z|) (4.34)
× [I C 1 · ρ 1 κ 1,2 + I C 2 · (1 − ρ 1 − ρ 2 )(1 − λ) −d/2 κ 2,1 ].
Part 3. We now put together the results in parts 1 and 2 and obtain an approximation for (4.7). Recall that
E Q dP dQ 2 ; E b , L ≤ E Q 1 [(1 − ρ 1 − ρ 2 )K] 2 ; E b , L, A τ ≥ 0 (4.35) + E Q 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; E b , L, A τ < 0 .
We consider the two terms on the right-hand side of the above display one by one. We start with the first term
E Q 1 [(1 − ρ 1 − ρ 2 )K] 2 ; E b , L, A τ ≥ 0 = E Q 1 [(1 − ρ 1 − ρ 2 )K] 2 ; E b , L, A τ ≥ 0, ı = 0 (4.36) + E Q 1 [(1 − ρ 1 − ρ 2 )K] 2 ; E b , L, A τ ≥ 0, ı = 1 .
The index τ admits density l(t) when ı = 0 and τ is uniformly distributed over T if ı = 1.
Consider the first expectation in (4.36). Note that conditionally on τ and ı = 0, on the set L ∩ {A τ ≥ 0}, (w, y, z) follows density (1 − ρ 1 − ρ 2 )h * 0,τ (w, y, z)/(1 − ρ 2 ) defined as in (4.2). Thus, according to (4.28), we have that the conditional expectation (1))Aτ +o(|y| 2 /(2u)+1 ⊤ z/(2σu))) · γ u (uσA τ )
E Q 1 (1 − ρ 1 − ρ 2 ) 2 K 2 ; E b , L, A τ ≥ 0 ı = 0, τ ≤ (1 + o(1)) H −1 λ det(Γ) −1/2 det(−∆µ σ (t * )) −1/2 (2π) (d+1)(d+2)/4−d/2 × (1 − λ) d/2 u d/2−1 e −(1/2)u 2 t * +Bt * +1 ⊤ µ 22 1/(8σ 2 ) 2 (4.37) × Aτ >0,L e −2(1−λ)u((1+o× 1 − ρ 1 − ρ 2 1 − ρ 2 h * 0,τ (w, y, z) dw dy dz, where γ u (x) = E 1 (1 − ρ 1 − ρ 2 ) 2 κ 2 ; x > (1 + o(u −1−δ 0 /4 ))[ξ u + o(u −δ 0 /8 )] ı, τ, w, y, z ,
with the expectation taken with respect to the process g(t). We insert the analytic form of h * 0,τ (w, y, z) into (4.2) and obtain that Aτ >0,L e −2(1−λ)u((1+o(1))Aτ +o(|y| 2 /(2u)+1 ⊤ z/(2σu))) · γ u (uσA τ )
× 1 − ρ 1 − ρ 2 1 − ρ 2 h * 0,τ (w, y, z) dw dy dz = (1 − ρ 1 − ρ 2 )H λ · u τ 1 − ρ 2 × Aτ >0 γ u (uσA τ ) exp{−2(1 − λ + o(1))uA τ + o(|z| + |y| 2 )} (4.38) × exp −λu τ A τ − 1 2 |µ 20 µ −1 22 z| 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22 z − µ 1/2 22 1 2σ 2 − 1 − λ 2 y ⊤ y dA τ dy dz.
Thanks to the Borel-TIS inequality (Lemma 16), Lemma 19 and the definition of κ in (4.25), for x > 0, γ u (x) is bounded and as b → ∞,
E 1 κ 2 ; x > (1 + o(u −1−δ 0 /4 ))[ξ u + o(u −δ 0 /8 )] → (2π) −d .
Thus, by the dominated convergence theorem and with H λ defined as in (3.14), as u → ∞, we have that
(4.38) ∼ (2π) −d (1 − ρ 1 − ρ 2 )(1 − ρ 2 ) e −λη λ 2 − λ .
We insert it back to (4.37) and obtain that
E Q 1 (1 − ρ 1 − ρ 2 ) 2 K 2 ; E b , L, A τ ≥ 0 ı = 0, τ ≤ (1 + o(1)) (2π) −d (1 − ρ 1 − ρ 2 )(1 − ρ 2 ) e −λη λ 2 − λ (4.39) × H −1 λ det(Γ) −1/2 det(∆µ σ (t * )) −1/2 (2π) (d+1)(d+2)/4−d/2 × (1 − λ) d/2 u d/2−1 e −(1/2)u 2 t * +Bt * +1 ⊤ µ 22 1/(8σ 2 ) 2 .
EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS
37
Using the asymptotic approximation of v(b) given by Proposition 14, we obtain that
E Q 1 [(1 − ρ 1 − ρ 2 )K] 2 ; E b , L, A τ ≥ 0, ı = 0 (4.40) ≤ 1 + o(1) 1 − ρ 1 − ρ 2 e λη λ(2 − λ) v 2 (b).
We choose
ρ 1 = ρ 2 = η = 1 − λ = 1/ log log b ∼ 1/ log u.
Then, the right-hand side of the above inequality is bounded by (1 + ε)v 2 (b) for b sufficiently large. The handling of the second term of (4.36) is similar except that (w, y, z) follows density h * τ (w, y, z). Thus, we only mention the key steps. Note that
E Q 1 (1 − ρ 1 − ρ 2 ) 2 K 2 ; E b , L, A τ ≥ 0 ı = 1, τ = (1 + o(1)) H −1 λ det(Γ) −1/2 det(−∆µ σ (t * )) −1/2 (2π) (d+1)(d+2)/4−d/2 × (1 − λ) d/2 u d/2−1 e −(1/2)u 2 t * +Bt * +1 ⊤ µ 22 1/(8σ 2 ) 2 (4.41) × det(Γ) −1/2 (2π) (d+1)(d+2)/4 × Aτ ≥0,L γ u (uσA τ ) × exp −2(1 − λ)uA τ − 1 + o(1) 2 × y ⊤ y + |w − µ 20 µ −1 22 z| 2 1 − µ 20 µ −1 22 µ 02 + z ⊤ µ −1 22 z dA τ dy dz = O(1)(1 − λ) −1 u −1 · u d−2 e −u 2 t * .
According to the asymptotic form of v(b) and with ρ 2 = 1 − λ = 1/ log log b, we have that
E Q 1 [(1 − ρ 1 − ρ 2 )K] 2 ; E b , L, A τ ≥ 0, ı = 1 (4.42) = O(1)ρ 2 (1 − λ) −1 u −1 · u d−2 e −u 2 t * = o(1)v 2 (b).
Therefore, combining the results in (4.40) and (4.42), we have the first term in (4.35) is bounded by (1 + 2ε)v 2 (b). The last step is to show that the second term of (4.35) is of a smaller order of v 2 (b). First, we split the expectation
E Q 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; E b , L, A τ < 0 = E Q 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; E b , L, A τ < 0, ı = 1 (4.43) + E Q 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; E b , L, −η/u τ < A τ < 0, ı = 0 + E Q 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; E b , L, A τ ≤ −η/u τ , ı = 0 .
We study these three terms one by one. Let
γ 1,u (x) = E 1 [I C 1 · ρ 1 κ 1,2 + I C 2 · (1 − ρ 1 − ρ 2 )(1 − λ) −d/2 κ 2,1 ] 2 ; (4.44) x > (1 + o(u −1−δ 0 /4 ))[ξ u + o(u −δ 0 /8 )] ı, τ, w, y, z .
We start with the first expectation in (4.43). Plugging in the lower bound for (1 − ρ 1 − ρ 2 )K + ρ 1 K 1 derived in (4.34), we have
E Q 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; E b , L, A τ < 0 ı = 1, τ = O(1)u d−2 e −u 2 t * × Aτ <0 γ 1,u (uσA τ ) (4.45) × exp −2(1 + λ 1 )uA τ − 1 2 y ⊤ y + |w − µ 20 µ −1 22 z| 2 1 − µ 20 µ −1 22 µ 02 + z ⊤ µ −1 22 z dA τ dy dz.
We deal with the γ 1,u (uσA τ ) term in the above integration. On the set L, uσA τ > −u 3/2+ǫ . By Lemma 24, for −u 3/2+ǫ < x < 0, there exists a constant δ * > 0 such that
E 1 ρ 2 1 κ 2 1,2 ; x > (1 + o(u −1−δ 0 /4 ))[ξ u + o(u −δ 0 /8 )] ı, τ, w, y, z, C 1 = O(1)ρ −2 1 e u δ * x EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS 39 and E 1 (1 − ρ 1 − ρ 2 ) 2 κ 2 2,1 ; x > (1 + o(u −1−δ 0 /4 ))[ξ u + o(u −δ 0 /8 )] ı, τ, w, y, z, C 2 = O(1)(1 − ρ 1 − ρ 2 ) −2 (1 − λ) −d e u δ * x .
Therefore, the above approximations and the dominated convergence theorem imply that conditionally on L,
Aτ <0 γ 1,u (uσA τ )e −2(1+λ 1 )uAτ dA τ = O(1) · max{ρ −2 1 , (1 − λ) −2d } · u −1−δ * .
Thus, (4.45) equals
(4.45) = O(1) max{ρ −2 1 , (1 − λ) −2d } · u −1−δ * · u d−2 e −u 2 t * .
Taking expectation of the above equation with respect to ı and τ and choosing ρ 1 , ρ 2 and 1 − λ be (log log b) −1 , we have
E 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; E b , L, A τ < 0, ı = 1 = o(1)v 2 (b). (4.46)
For the second term in (4.43), with the same bound of γ 1,u , we have
E Q 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; E b , L, −η/u τ < A τ < 0 ı = 0, τ = O(1)u d−2 e −u 2 t * × u τ −η/uτ <Aτ <0 γ 1,u (uσA τ )e −2(1+λ 1 )uAτ e −λuτ Aτ × exp − 1 2 |µ 20 µ −1 22 z| 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22 z − µ 1/2 22 1 2σ 2 − 1 − λ 2 y ⊤ y dA τ dy dz = O(1) · max{ρ −2 1 , (1 − λ) −2d } · u −δ * · u d−2 e −u 2 t * = o(1)v 2 (b),
and similarly for the third term in (4.43),
E Q 1 [(1 − ρ 1 − ρ 2 )K + ρ 1 K 1 ] 2 ; E b , L, A τ ≤ −η/u τ ı = 0, τ = O(1)ρ 1 · u d−2 e −u 2 t * u τ × Aτ <−η/uτ γ 1,u (uσA τ ) × e −2(1+λ 1 )uAτ × exp λ 1 u τ A τ (4.47) − 1 2 |µ 20 µ −1 22 z| 2 1 − µ 20 µ −1 22 µ 02 + µ −1/2 22 z − µ 1/2 22 1 2σ 2 − 1 + λ 1 2 y ⊤ y dA τ dy dz = O(1)ρ 1 · max{ρ −2 1 , (1 − λ) −2d } · u −δ * · u d−2 e −u 2 t * = o(1)v 2 (b).
We put all the estimates in (4.40), (4.42), (4.46) and (4.47) back to (4.35). For any ε > 0, if we choose η = ρ 1 = ρ 2 = 1 − λ = 1/ log log b, then for b sufficiently large we have that
E Q dP dQ 2 ; E b , L ≤ (1 + 3ε)v 2 (b).
We complete the proof of Theorem 3 for the case that µ(t) = 0.
4.4. Case 2: Constant mean function. The proof when µ(t) ≡ 0 is very similar, except that we need to consider two situations: first, τ is not close to the boundary of T and otherwise. More precisely, for a given δ ′ > 0 small enough, we consider the case when τ ∈ {t : |t − τ | ≤ u −1/2+δ ′ } ⊂ T and otherwise.
For the first situation, τ is "far away" from the boundary of T , which is the important case, the derivation is same as that of the case where µ(t) is not a constant. For the case in which τ is within u −1/2+δ ′ distance from the boundary of T , the contribution of the boundary case is o (v 2 (b)). An intuitive interpretation is that the important region of the integral I(T ) might be cut off by the boundary of T . Therefore, in cases that τ is too close to the boundary, the tail I(T ) is not heavier than that of the interior case. The rigorous analysis is basically repeating the parts 1, 2 and 3 on a truncated region. Therefore, we omit the details.
5.
Proof of Theorem 7. The proof of Theorem 7 is analogous to that of Theorem 3. According to Lemma 18, we focus on the set (for some small ǫ 0 > 0)
L * = L ∩ sup |t−τ |>2u −1/2+ǫ g(t) − ǫ 0 u|t| 2 < 0 . (5.1)
A similar three-part procedure is applied here.
In part 1, using the transformation from f to the process f * , we have
β u (T ) = sup t∈T f (t) + 1 ⊤f ′′ t 2σu t + B t u t + µ σ (t) = sup t∈T f * (t) + u τ C(t − τ ) + 1 ⊤ (z t − u t µ 02 + u τ µ 2 (t − τ )) 2σu t + B t u t + µ σ (t) .
We insert the expansions in (4.10), (4.11) and (4.12) into the expression of β u (T ) and obtain that β u (T ) equals
sup t∈T w + y ⊤ (t − τ ) + 1 2 (t − τ ) ⊤ z(t − τ ) + R f (t − τ ) + g(t − τ ) + u τ 1 − 1 2 (t − τ ) ⊤ (t − τ ) + C 4 (t − τ ) + R C (t − τ ) + µ σ (τ ) + ∂µ σ (τ ) ⊤ (t − τ ) + 1 2 (t − τ ) ⊤ ∆µ σ (τ )(t − τ ) + σ −1 R µ (t − τ ) + 1 ⊤ (z t − u t µ 02 + u τ µ 2 (t − τ )) 2σu t + B t u t = sup t∈T u + w + 1 2ỹ ⊤ (uI −z) −1ỹ − 1 2 (t − τ − (uI −z) −1ỹ ) ⊤ (uI −z)(t − τ − (uI −z) −1ỹ ) + u τ C 4 (t − τ ) + R(t − τ ) + g(t − τ ) + 1 ⊤ (z t − u t µ 02 + u τ µ 2 (t − τ )) 2σu t + B t u t .
Note that the above display is approximately a quadratic function of t − τ and is maximized approximately at t − τ = (uI −z) −1ỹ . In addition, on the set L * , we have that |τ − t * | < 2u −1/2+ǫ and thusỹ = y + O(u −1/2+ǫ ). Therefore, on the set L * , we have the following approximation of β u (T ):
A τ + inf |t−τ |<2u −1/2+ǫ g(t) ≤ β u (T ) − u + u −1−δ 0 /4 o(|w| + |y| + |z|) ≤ A τ + sup |t−τ |<2u −1/2+ǫ g(t).
Thus, we obtain the same representation as in part 1 in the proof of Theorem 3.
Since we use the same change of measure, the analysis of the likelihood ratio is exactly the same as part 2 of Theorem 3. For part 3, we compute the second moment of dP/dQ on the set {β u (T ) > u}. This is also identical to the proof of Theorem 3. Thus, with the same choice of tuning parameters, we have that
E Q dP dQ 2 ; β u (T ) > u ≤ (1 + ε)v 2 (b).
Additionally, Lemma 18 provides an approximation that P (β u (T ) > u) ∼ v(b). Thus, we use Lemma 13 (presented at the beginning of Section 4) and complete the proof.
6. Proof of Theorem 10. For the bias control, we need the following result [44].
Proposition 15.
Suppose that conditions C1-C6 are satisfied. Let F ′ (x) be the probability density function of log I(T ) = log T e σf (t)+µ(t) dt. Then the following approximation holds as x → ∞:
F ′ (x) ∼ σ −2 x · v(e x ).
Thus, for any small ε, P (b < I(T ) < b(1 + ε/ log b)|I(T ) > b) = Θ(ε). Let L ε = sup t∈T |∂f (t)| ≤ 2(1 − u −2 log ε)u .
Note that ∂f (t) is a d-dimensional Gaussian process. Using Borel-TIS lemma, we obtain that The last step is due to the result of Proposition 15 and further (6.1). Thus, it is sufficient to choose N = O(ε −1−ε 0 u 2+ε 0 ) so that the above probability is bounded by εv(b). The bound of P (I(T ) < b, I M (T ) > b, L ε ) is completely analogous.
P (L c ε ) = o(1)ε · v(b).
7. Proof of Theorem 11. The proof of Theorem 11 is similar to that of Theorem 3. Therefore, we only lay out the key steps. The only difference is that we replace the integral by a finite sum over T N . Recall that the proof of Theorem 3 consists of three parts: first, we write the event {I M (T ) > b} as a function of (w, y, z) (with an ignorable correction term); second, we write the likelihood ratio as a function of (w, y, z) (with an ignorable correction term); third, we integrate the likelihood ratio with respect to (ı, τ, w, y, z). For the current proof, we also have three similar parts. Under the discretization setup, we have
dQ M dP = 1 − ρ 1 − ρ 2 κ M i=1 l(t i ) LR(t i ) + ρ 1 κ M i=1 l(t i ) LR 1 (t i ) + ρ 2 M i=1 1 M LR 2 (t i ),
which is a discrete approximation of dQ/dP . In the proof of Theorem 3, after taking all the terms not consisting of t out of the integral [such as that in (4.23)], the discrete sum is essentially approximating the following integral:
|t−τ |<u −1+δ ′ e −((1−λ)(uτ +ζu)/2)(t−τ −(uI−z) −1ỹ ) ⊤ (uI−z)(t−τ −(uI−z) −1ỹ ) dt.
The above integral concentrates on a region of size O(u −1 ). Given that we choose N > u 2 , the discretized likelihood ratio in dQ M /dP approximate dQ/dP up to a constant in the sense that dQ M dP = Θ(1) dQ dP . (7.1) Part 3. With the results of parts 1 and 2, the analysis of part 3 is completely analogous to part 3 in the proof of Theorem 3. Thus, we conclude that
E Q M (L 2 b ) ≤ κ 1 v(b) 2 ,
where the constant κ 1 depends on the Θ(1) in (7.1).
APPENDIX: THE LEMMAS
In this section, we state all the lemmas used in the previous sections. To facilitate reading, we move several lengthy proofs ( Lemmas 17,18,20,22,23 and 24) to the supplemental materials [45], as those proofs are not particularly related to the proof of the theorems and mostly involve tedious elementary algebra.
The first lemma is known as the Borel-TIS lemma, which was proved independently by [20,23].
Lemma 16 (Borel-TIS). Let f (t), t ∈ U , U is a parameter set, be a mean zero Gaussian random field. f is almost surely bounded on U . Then E(sup U f (t)) < ∞, and In addition, we have the approximation P (β u (T ) > u) ∼ v(b).
Lemma 19. Let ξ u be as defined in (4.19), then there exist small constants δ * , λ ′ , λ ′′ > 0 such that for all x > 0 and sufficiently large u P (|ξ u | > x) ≤ e −λ ′ u δ * x 2 + e −λ ′′ u 2 .
Proof. For δ < δ 0 /10, we split the expectation into two parts {|S| ≤ u δ } and {|S| > u δ , τ + (uI − z) −1/2S ∈ T }. Note that |S| ≤ κu δ and g(t) is a mean zero Gaussian random field with Var(g(t)) = O(|t| 4+δ 0 ). A direct application of the Borel-TIS inequality (Lemma 16) yields the result of this lemma. If |y| ≤ u 1/2+ǫ and |z| ≤ u 1/2+ǫ and ǫ ≪ δ 0 , then log{Ee σ(u−µσ (τ ))C 4 ((uI−z) −(1/2) S)+σR((uI−z) −(1/2) S) }
= 1 8σu i ∂ 4 iiii C(0) + o(|w| + |y| + |z| + 1) u 1+δ 0 /4 ,
where the expectation is taken with respect to S. Proof. The result is immediate by noting that det(I −u −1 z) = d i=1 (1− λ i /u), and Tr(z) = d i=1 λ i .
Lemma 22.
On the set L, I 2 defined as in (4.21) can be written as
A * ,|t−τ |<u −1+δ ′ exp u t * (t − t * ) ⊤ ∆µ σ (t * )(t − t * ) 2 + u 2 t 2 × u t × exp (1 − λ)u t [w t + u τ C(t − τ ) − u t ] + (1 − λ) 2σ 1 ⊤ (z t − µ 02 u t + µ 2 (t − τ )u τ ) − λB t − 1 ⊤ µ 22 1 8σ 2 × exp{((w t + u τ C(t − τ ) − u t ) 2 − 2(w t + u τ C(t − τ ) − u t )
× µ 20 µ −1 22 (z t − µ 02 u t + µ 2 (t − τ )u τ )) /(2(1 − µ 20 µ −1 22 µ 02 ))} dt.
Spatial point process. In spatial point process modeling, let λ(t) be the intensity of a Poisson point process on T , denoted by {N (A) : A ⊂ T }.
= 1, . . . , d − 1, j = i + 1, . . . , d), µ ⊤ 02 = µ 20 = µ 2 (0).
b
: j = 1, . . . , n} be i.i.d. copies of L b . The averaged estimator
For
each t = (t 1 , . . . , t d ) ∈ G N,d , define T N (t) = {(s 1 , . . . , s d ) ∈ T : s j ∈ (t j − 1/N, t j ] for j = 1, . . . , d} that is, the 1 N -cube intersected with T and cornered at t. Furthermore, let T N = {t ∈ G N,d : T N (t) = ∅}. (3.22) Since T is compact, T N is a finite set. We enumerate the elements in T N = {t 1 , . . . , t M }, where M = O(N d ). We further define X = (X 1 , . . . , X M ) ⊤ (f (t 1 ), . . . , f (t M )) ⊤ and use v M (b) = P (I M (T ) > b) as an approximation of v(b) where
. The complexity of generating such a vector is at the most O(N 3 ). Thus the over-all complexity is O(ε −2−(3+3ε 0 )d δ −1 (log b) (6+3ε 0 )d ). The proposed estimator in (3.24) is a FPRAS.Remark 12. The proposed algorithm can also be used to compute conditional expectations via the representation E[Ξ(f )|I(T ) > b] = E[Ξ(f ); I(T ) > b]/v(b), where E[Ξ(f ); I(T ) > b] can be estimated by Ξ(f ) dP/dQ M and v(b) can be estimated by I {I(T )>b} dP/dQ M .
Proposition 14 .
14Consider a Gaussian random field {f (t) : t ∈ T } living on a domain T satisfying conditions C1-C6. If µ(t) has one unique maximum in T denoted by t * , then
u
×E[e σuτ C 4 ((uI−z) −1/2 S)+σR((uI−z) −1/2 S) ] · e −u −d/2 e σu ,where ξ u = −u log{E exp[σg((uI −z) −1/2S )]}.(4.19)
that is, the dominating region of the integral. We split the set D = (A * ∩ D) ∪ ((A * ) c ∩ D). There are two situations: mes((A * ) c ∩ D) > mes(A * ∩ D) and mes((A * ) c ∩ D) ≤ mes(A * ∩
the log-normal distribution, the overshoot of I(T ) is Θ(b/ log b). Note that |v M (b) − v(b)| ≤ P (I(T ) > b, I M (T ) < b) + P (I(T ) < b, I M (T ) > b).
Therefore, it is sufficient to control P (I(T ) > b, I M (T ) < b, L ε ) and P (I(T) < b, I M (T ) > b, L ε ).By the definition of I M in (3.23), there exists a constant c 1 > 0 such that∆ = |I(T ) − I M (T )| ≤ M i=1 T N (t i ) e σf (t)+µ(t) dt − mes(T N (t i )) · e σf (t i )+µ(t i ) ≤ c 1 min{I M (T ), I(T )} · sup t∈T |∂f (t)|/N.EXPONENTIAL INTEGRALS OF GAUSSIAN RANDOM FIELDS43Then we have, on the set L ε , ∆ ≤ 2c 1 min{I M (T ), I(T )}(1 − u −2 log ε)u/N , which implies thatP (I(T ) > b, I M (T ) < b, L ε ) ≤ P (b < I(T ) < b(1 + 2(1 − u −2 log ε)
Part 1 .
1For the first step in the proof of Theorem 3, we write I(T ) > b if and only if A τ + o(|w|+|y|+|z|+1) u 1+δ 0 /4 > u −1 σ −1 ξ u . With the current discretization size, as proved in Theorem 10, log I(T ) − log I M (T ) = o(u −1−ε 0 /2 ).Thus, we reach the same result thatI M (T ) > b if A τ + o(|w|+|y|+|z|) u 1+δ 0 /4 + o(u −1−ε 0 /2 ) > u −1 σ −1 ξ u .Part 2. Consider the likelihood ratiodQ dP = T (1 − ρ 1 − ρ 2 )l(t) LR(t) + ρ 1 l(t) LR 1 (t) + ρ 2 mes(T ) LR 2 (t) dt.
f
(t) − E max t∈U f (t) ≥ b ≤ e −b 2 /(2σ 2 U ) , where σ 2 U = max t∈U Var[f (t)].Lemma 17. Conditionally on the set L as defined in (4.13), we have thatE Q dP dQ 2 ; I(T ) > b, L c = o(1)v 2 (b).Lemma 18. On the set L * as defined in (5.1), we have that for k = 1 and 2E Q dP dQ k ; β u (T ) > u, L c * = o(1)P (β u (T ) > u) k .
Lemma 20 .
20Let S be a random variable taking values in {s : (uI −z) −1/2 s+ τ ∈ T } with density proportional to e −(σ/2)(s−(uI−z) −1/2ỹ ) ⊤ (s−(uI−z) −1/2ỹ ) .
Lemma 21 .
21log(det(I − u −1 z)) = −u −1 Tr(z) + 1 2 u −2 I 2 (z) + o(u −2 ), where Tr is the trace of a matrix, I 2 (z) = d i=1 λ 2 i , and λ i 's are the eigenvalues of z.
J. LIU AND G. XU
SUPPLEMENTARY MATERIAL
Distribution of the maximum in air pollution fields. S Aberg, P Guttorp, Environmetrics. 192420185Aberg, S. and Guttorp, P. (2008). Distribution of the maximum in air pollution fields. Environmetrics 19 183-208. MR2420185
The Geometry of Random Fields. R J Adler, 611857Wiley, ChichesterAdler, R. J. (1981). The Geometry of Random Fields. Wiley, Chichester. MR0611857
Efficient Monte Carlo for high excursions of Gaussian random fields. R J Adler, J H Blanchet, J Liu, Ann. Appl. Probab. 222977989Adler, R. J., Blanchet, J. H. and Liu, J. (2012). Efficient Monte Carlo for high excursions of Gaussian random fields. Ann. Appl. Probab. 22 1167-1214. MR2977989
High level excursion set geometry for non-Gaussian infinitely divisible random fields. R J Adler, G Samorodnitsky, J E Taylor, Ann. Probab. 41Adler, R. J., Samorodnitsky, G. and Taylor, J. E. (2013). High level excursion set geometry for non-Gaussian infinitely divisible random fields. Ann. Probab. 41 134-169. MR3059195
Random Fields and Geometry. R J Adler, J E Taylor, SpringerNew York. MR2319516Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York. MR2319516
R J Adler, J E Taylor, K J Worsley, Applications of random fields and geometry: Foundations and case studies. PreprintAdler, R. J., Taylor, J. E. and Worsley, K. J. (2009). Applications of random fields and geometry: Foundations and case studies. Preprint.
Portfolio selection in a lognormal securities market. S M Ahsan, Zeitschrift für Nationalokonomie-Journal of Economics. 38Ahsan, S. M. (1978). Portfolio selection in a lognormal securities market. Zeitschrift für Nationalokonomie-Journal of Economics 38 105-118.
Ruin Probabilities. S Asmussen, Advanced Series on Statistical Science & Applied Probability 2. World Scientific. 1794582Asmussen, S. (2000). Ruin Probabilities. Advanced Series on Statistical Science & Applied Probability 2. World Scientific, River Edge, NJ. MR1794582
Large deviations results for subexponential tails, with applications to insurance risk. S Asmussen, C Klüppelberg, Stochastic Process. Appl. 641419495Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexpo- nential tails, with applications to insurance risk. Stochastic Process. Appl. 64 103-125. MR1419495
Asymptotics of sums of lognormal random variables with Gaussian copula. S Asmussen, L Rojas-Nandayapa, Statist. Probab. Lett. 78Asmussen, S. and Rojas-Nandayapa, L. (2008). Asymptotics of sums of lognormal random variables with Gaussian copula. Statist. Probab. Lett. 78 2709-2714. MR2465111
On the distribution of the maximum of a Gaussian field with d parameters. J.-M Azaïs, M Wschebor, Ann. Appl. Probab. 152115043Azaïs, J.-M. and Wschebor, M. (2005). On the distribution of the maximum of a Gaussian field with d parameters. Ann. Appl. Probab. 15 254-278. MR2115043
A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail. J.-M Azaïs, M Wschebor, Stochastic Process. Appl. 118Azaïs, J.-M. and Wschebor, M. (2008). A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail. Stochastic Process. Appl. 118 1190-1218. MR2428714
Level Sets and Extrema of Random Processes and Fields. J.-M Azaïs, M Wschebor, Wiley2478201Hoboken, NJAzaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Pro- cesses and Fields. Wiley, Hoboken, NJ. MR2478201
Value-at-risk-based risk management: Optimal policies and asset prices. S Basak, A Shapiro, Review of Financial Studies. 14Basak, S. and Shapiro, A. (2001). Value-at-risk-based risk management: Optimal policies and asset prices. Review of Financial Studies 14 371-405.
An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments. S M Berman, J. Appl. Probab. 22789369Berman, S. M. (1985). An asymptotic formula for the distribution of the maximum of a Gaussian process with stationary increments. J. Appl. Probab. 22 454-460. MR0789369
Efficient importance sampling in ruin problems for multidimensional regularly varying random walks. J Blanchet, J Liu, J. Appl. Probab. 472668490Blanchet, J. and Liu, J. (2010). Efficient importance sampling in ruin problems for multidimensional regularly varying random walks. J. Appl. Probab. 47 301-322. MR2668490
Efficient simulation and conditional functional limit theorems for ruinous heavy-tailed random walks. J Blanchet, J Liu, Stochastic Process. Appl. 1222931349Blanchet, J. and Liu, J. (2012). Efficient simulation and conditional functional limit theorems for ruinous heavy-tailed random walks. Stochastic Process. Appl. 122 2994-3031. MR2931349
State-dependent importance sampling for regularly varying random walks. J H Blanchet, J Liu, Adv. in Appl. Probab. 40Blanchet, J. H. and Liu, J. (2008). State-dependent importance sampling for reg- ularly varying random walks. Adv. in Appl. Probab. 40 1104-1128. MR2488534
Monte Carlo for large credit portfolios with potentially high correlations. J H Blanchet, J Liu, X Yang, Proceedings of the 2010 Winter Simulation Conference. the 2010 Winter Simulation ConferenceBaltimore, MDBlanchet, J. H., Liu, J. and Yang, X. (2010). Monte Carlo for large credit port- folios with potentially high correlations. In Proceedings of the 2010 Winter Sim- ulation Conference, Baltimore, MD.
The Brunn-Minkowski inequality in Gauss space. C Borell, Invent. Math. 30Borell, C. (1975). The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 207-216. MR0399402
Time-series regression for counts-An investigation into the relationship between sudden-infant-death-syndrome and environmentaltemperature. M J Campbell, J. Roy. Statist. Soc. Ser. A. 157Campbell, M. J. (1994). Time-series regression for counts-An investigation into the relationship between sudden-infant-death-syndrome and environmental- temperature. J. Roy. Statist. Soc. Ser. A 157 191-208.
. K S Chan, J Ledolter, Chan, K. S. and Ledolter, J. (1995).
Monte Carlo EM estimation for time series models involving counts. J. Amer. Statist. Assoc. 901325132Monte Carlo EM estimation for time series models involving counts. J. Amer. Statist. Assoc. 90 242-252. MR1325132
Norms of Gaussian sample functions. B S Cirel'son, I A Ibragimov, V N Sudakov, Proceedings of the Third Japan-USSR Symposium on Probability Theory. the Third Japan-USSR Symposium on Probability TheoryTashkent; BerlinSpringer550458556Cirel'son, B. S., Ibragimov, I. A. and Sudakov, V. N. (1976). Norms of Gaussian sample functions. In Proceedings of the Third Japan-USSR Symposium on Prob- ability Theory (Tashkent, 1975). Lecture Notes in Math. 550 20-41. Springer, Berlin. MR0458556
Some statistical methods connected with series of events. D R Cox, J. R. Stat. Soc. Ser. B Stat. Methodol. 1792301Cox, D. R. (1955). Some statistical methods connected with series of events. J. R. Stat. Soc. Ser. B Stat. Methodol. 17 129-157; discussion, 157-164. MR0092301
Point Processes. D R Cox, V Isham, Chapman & Hall598033LondonCox, D. R. and Isham, V. (1980). Point Processes. Chapman & Hall, London. MR0598033
On autocorrelation in a Poisson regression model. R A Davis, W T M Dunsmuir, Y Wang, Biometrika. 871789805Davis, R. A., Dunsmuir, W. T. M. and Wang, Y. (2000). On autocorrelation in a Poisson regression model. Biometrika 87 491-505. MR1789805
Derivatives and Internal Models. H P Deutsch, Basingstoke, UK3rd ed. Palgrave MacmillanDeutsch, H. P. (2004). Derivatives and Internal Models, 3rd ed. Palgrave Macmil- lan, Basingstoke, UK.
An overview of value at risk. D Duffie, J Pan, The Journal of Derivatives. 4Duffie, D. and Pan, J. (1997). An overview of value at risk. The Journal of Deriva- tives 4 7-49.
The integral of geometric Brownian motion. D Dufresne, Adv. in Appl. Probab. 33Dufresne, D. (2001). The integral of geometric Brownian motion. Adv. in Appl. Probab. 33 223-241. MR1825324
A Weak Convergence Approach to the Theory of Large Deviations. P Dupuis, R S Ellis, WileyNew York. MR1431744Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York. MR1431744
On sums of conditionally independent subexponential random variables. S Foss, A Richards, Math. Oper. Res. 35Foss, S. and Richards, A. (2010). On sums of conditionally independent subexpo- nential random variables. Math. Oper. Res. 35 102-119. MR2676758
Slepian models for nonstationary Gaussian processes. T Gadrich, R J Adler, J. Appl. Probab. 301206355Gadrich, T. and Adler, R. J. (1993). Slepian models for nonstationary Gaussian processes. J. Appl. Probab. 30 98-111. MR1206355
Variance reduction techniques for estimating value-at-risk. P Glasserman, P Heidelberger, P Shahabuddin, Management Science. 46Glasserman, P., Heidelberger, P. and Shahabuddin, P. (2000). Variance reduc- tion techniques for estimating value-at-risk. Management Science 46 1349-1364.
Reliability of Daniels systems subject to quasistatic and dynamic non-stationary Gaussian load processes. M Grigoriu, Prob. Eng. Mech. 4Grigoriu, M. (1989). Reliability of Daniels systems subject to quasistatic and dy- namic non-stationary Gaussian load processes. Prob. Eng. Mech. 4 128-134.
Extreme values and high boundary crossings of locally stationary Gaussian processes. J Hüsler, MR1062062Ann. Probab. 18Hüsler, J. (1990). Extreme values and high boundary crossings of locally stationary Gaussian processes. Ann. Probab. 18 1141-1158. MR1062062
Extremes of Gaussian processes with random variance. J Hüsler, V Piterbarg, Y Zhang, Electron. J. Probab. 16Hüsler, J., Piterbarg, V. and Zhang, Y. (2011). Extremes of Gaussian processes with random variance. Electron. J. Probab. 16 1254-1280. MR2827458
On the supremum of a Gaussian process. H J Landau, L A Shepp, Sankhyā Ser. A. 32286167Landau, H. J. and Shepp, L. A. (1970). On the supremum of a Gaussian process. Sankhyā Ser. A 32 369-378. MR0286167
Extremes and Related Properties of Random Sequences and Processes. M R Leadbetter, G Lindgren, H Rootzén, SpringerNew York. MR0691492Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York. MR0691492
M Ledoux, M Talagrand, Probability in Banach Spaces: Isoperimetry and Processes. BerlinSpringer1102015Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperime- try and Processes. Ergebnisse der Mathematik und Ihrer Grenzgebiete (3) 23. Springer, Berlin. MR1102015
Some properties of a normal process near a local maximum. G Lindgren, Ann. Math. Statist. 41272051Lindgren, G. (1970). Some properties of a normal process near a local maximum. Ann. Math. Statist. 41 1870-1883. MR0272051
Prediction of level crossings for normal processes containing deterministic components. G Lindgren, Adv. in Appl. Probab. 11517553Lindgren, G. (1979). Prediction of level crossings for normal processes containing deterministic components. Adv. in Appl. Probab. 11 93-117. MR0517553
J Liu, Tail approximations of integrals of Gaussian random fields. 402962087Liu, J. (2012). Tail approximations of integrals of Gaussian random fields. Ann. Probab. 40 1069-1104. MR2962087
Some asymptotic results of Gaussian random fields with varying mean functions and the associated processes. J Liu, G Xu, Ann. Statist. 403014307Liu, J. and Xu, G. (2012). Some asymptotic results of Gaussian random fields with varying mean functions and the associated processes. Ann. Statist. 40 262-293. MR3014307
On the density functions of integrals of Gaussian random fields. J Liu, G Xu, Adv. in Appl. Probab. 453102457Liu, J. and Xu, G. (2013). On the density functions of integrals of Gaussian random fields. Adv. in Appl. Probab. 45 398-424. MR3102457
On the conditional distributions and the efficient simulations of exponential integrals of gaussian random fields. J Liu, G Xu, 10.1214/13-AAP960SUPPSupplement toLiu, J. and Xu, G. (2014). Supplement to "On the conditional distributions and the efficient simulations of exponential integrals of gaussian random fields." DOI:10.1214/13-AAP960SUPP.
Continuity of Gaussian processes. M B Marcus, L A Shepp, Trans. Amer. Math. Soc. 151264749Marcus, M. B. and Shepp, L. A. (1970). Continuity of Gaussian processes. Trans. Amer. Math. Soc. 151 377-391. MR0264749
Probability and Computing: Randomized Algorithms and Probabilistic Analysis. M Mitzenmacher, E Upfal, Cambridge Univ. Press2144605CambridgeMitzenmacher, M. and Upfal, E. (2005). Probability and Computing: Random- ized Algorithms and Probabilistic Analysis. Cambridge Univ. Press, Cambridge. MR2144605
The distribution of maxima of approximately Gaussian random fields. Y Nardi, D O Siegmund, B Yakir, Ann. Statist. 362418661Nardi, Y., Siegmund, D. O. and Yakir, B. (2008). The distribution of maxima of approximately Gaussian random fields. Ann. Statist. 36 1375-1403. MR2418661
Asymptotic Methods in the Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs 148. V I Piterbarg, Amer. Math. SocProvidence, RI. MR1361884Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. Translations of Mathematical Monographs 148. Amer. Math. Soc., Providence, RI. MR1361884
Extremal properties of half spaces for spherically invariant measures. V N Sudakov, B S Tsirelson, Zap. Nauchn. Sem. LOMI. 45Sudakov, V. N. and Tsirelson, B. S. (1974). Extremal properties of half spaces for spherically invariant measures. Zap. Nauchn. Sem. LOMI 45 75-82.
Tail probabilities of the maxima of Gaussian random fields. J Sun, Ann. Probab. 21Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. Ann. Probab. 21 34-71. MR1207215
Majorizing measures: The generic chaining. M Talagrand, Ann. Probab. 24Talagrand, M. (1996). Majorizing measures: The generic chaining. Ann. Probab. 24 1049-1103. MR1411488
Validity of the expected Euler characteristic heuristic. J Taylor, A Takemura, R J Adler, Ann. Probab. 332150192Taylor, J., Takemura, A. and Adler, R. J. (2005). Validity of the expected Euler characteristic heuristic. Ann. Probab. 33 1362-1396. MR2150192
Informationbased Complexity. J F Traub, G W Wasilkowski, H Woźniakowski, Academic Press958691Boston, MATraub, J. F., Wasilkowski, G. W. and Woźniakowski, H. (1988). Information- based Complexity. Academic Press, Boston, MA. MR0958691
Computational complexity of continuous problems. H Woźniakowski, Nonlinear Dynamics, Chaotic and Complex Systems. Zakopane; CambridgeCambridge Univ. Press1481726Woźniakowski, H. (1997). Computational complexity of continuous problems. In Nonlinear Dynamics, Chaotic and Complex Systems (Zakopane, 1995) 283-295. Cambridge Univ. Press, Cambridge. MR1481726
On some exponential functionals of Brownian motion. M Yor, Adv. in Appl. Probab. 24Yor, M. (1992). On some exponential functionals of Brownian motion. Adv. in Appl. Probab. 24 509-531. MR1174378
A regression model for time series of counts. S L Zeger, Biometrika. 75995107Zeger, S. L. (1988). A regression model for time series of counts. Biometrika 75 621-629. MR0995107
USA E-mail: [email protected] School of Statistics University of Minnesota 367 Ford Hall, 224 Church St. Amsterdam Avenue New York, New York; SE Minneapolis, Minnesota 55455 USA E-mailDepartment of Statistics Columbia [email protected] of Statistics Columbia University 1255 Amsterdam Avenue New York, New York 10027 USA E-mail: [email protected] School of Statistics University of Minnesota 367 Ford Hall, 224 Church St. SE Minneapolis, Minnesota 55455 USA E-mail: [email protected]
| [] |
[
"Robust Student's t based Stochastic Cubature Filter for Nonlinear Systems with Heavy-tailed Process and Measurement Noises",
"Robust Student's t based Stochastic Cubature Filter for Nonlinear Systems with Heavy-tailed Process and Measurement Noises"
] | [
"Yulong Huang ",
"Senior Member, IEEEYonggang Zhang "
] | [] | [] | In this paper, a new robust Student's t based stochastic cubature filter (RSTSCF) is proposed for nonlinear state-space model with heavy-tailed process and measurement noises. The heart of the RSTSCF is a stochastic Student's t spherical radial cubature rule (SSTSRCR), which is derived based on the thirddegree unbiased spherical rule and the proposed third-degree unbiased radial rule. The existing stochastic integration rule is a special case of the proposed SSTSRCR when the degrees of freedom parameter tends to infinity. The proposed filter is applied to a manoeuvring bearings-only tracking example, where an agile target is tracked and the bearing is observed in clutter. Simulation results show that the proposed RSTSCF can achieve higher estimation accuracy than the existing Gaussian approximate filter, Gaussian sum filter, Huber-based nonlinear Kalman filter, maximum correntropy criterion based Kalman filter, and robust Student's t based nonlinear filters, and is computationally much more efficient than the existing particle filter. | 10.1109/access.2017.2700428 | [
"https://arxiv.org/pdf/1704.00040v1.pdf"
] | 24,914,510 | 1704.00040 | 035aa952dc9f069d4edd9656dd1462398e8e2a46 |
Robust Student's t based Stochastic Cubature Filter for Nonlinear Systems with Heavy-tailed Process and Measurement Noises
31 Mar 2017
Yulong Huang
Senior Member, IEEEYonggang Zhang
Robust Student's t based Stochastic Cubature Filter for Nonlinear Systems with Heavy-tailed Process and Measurement Noises
31 Mar 20171Index Terms-Nonlinear filterheavy-tailed noiseStudent's t distributionStudent's t weighted integraloutliernonlinear system
In this paper, a new robust Student's t based stochastic cubature filter (RSTSCF) is proposed for nonlinear state-space model with heavy-tailed process and measurement noises. The heart of the RSTSCF is a stochastic Student's t spherical radial cubature rule (SSTSRCR), which is derived based on the thirddegree unbiased spherical rule and the proposed third-degree unbiased radial rule. The existing stochastic integration rule is a special case of the proposed SSTSRCR when the degrees of freedom parameter tends to infinity. The proposed filter is applied to a manoeuvring bearings-only tracking example, where an agile target is tracked and the bearing is observed in clutter. Simulation results show that the proposed RSTSCF can achieve higher estimation accuracy than the existing Gaussian approximate filter, Gaussian sum filter, Huber-based nonlinear Kalman filter, maximum correntropy criterion based Kalman filter, and robust Student's t based nonlinear filters, and is computationally much more efficient than the existing particle filter.
I. INTRODUCTION
N ONLINEAR filtering has been playing an important role in many applications, such as target tracking, detection, signal processing, communication and navigation. Under the Bayesian estimation framework, the nonlinear filtering problem is addressed by calculating the posterior probability density function (PDF) recursively based on the nonlinear state-space model. Unfortunately, there is not a closed form solution for posterior PDF for nonlinear state-space model since a closed PDF for nonlinear mapping doesn't exist [1]. As a result, there is not an optimal solution for nonlinear filtering problem, and an approximate approach is necessary to obtain a suboptimal solution. In general, the posterior PDF is approximated as Gaussian by assuming the jointly predicted PDF of the state and measurement vectors is Gaussian, and the resultant Gaussian approximate (GA) filter can provide tradeoffs between estimation accuracy and computational complexity [2], [3]. Up to present, many variants of the GA filter have been proposed using different Gaussian weighted integral rules [3]- [9]. However, in some engineering applications, such as tracking an agile target that is observed in clutter, the heavytailed process noise may be induced by severe manoeuvering and the heavy-tailed measurement noise may be induced by measurement outliers from unreliable sensors [10], [11], [12]. The performance of the GA filters may degrade for such engineering applications with heavy-tailed noises since they all model the process and measurement noises as Gaussian distributions so that they are sensitive to heavy-tailed non-Gaussian noises [11].
Particle filter (PF) is a common method to address non-Gaussian noises, in which the posterior PDF is approximated as a set of random samples with associated weights based on sequential Monte Carlo sampling technique [13]. The PF can model the process and measurement noises as arbitrary distributions, such as the Student's t distributions for heavy-tailed non-Gaussian noises [14], [15]. However, the PF suffers from substantial computational complexities in high-dimensional problems because the number of particles increases exponentially with the dimensionality of the state [16]. Gaussian sum filter (GSF) is an alternative method to handle heavy-tailed non-Gaussian noises, where the heavytailed process and measurement noises are modelled as a finite sum of Gaussian distributions, and the posterior distribution is then approximated as a weighted sum of Gaussian distributions by running a bank of GA filters [17]- [19]. However, for the GSF, it is very difficult to model the heavy-tailed process and measurement noises accurately using finite Gaussian distributions since the heavy-tailed non-Gaussian noises are induced by the unknown manoeuvering or outliers, which may degrade the estimation performance of the GSF.
To solve the filtering problem of nonlinear state-space model with heavy-tailed non-Gaussian noises, the Huber-based nonlinear Kalman filter (HNKF) has been proposed by minimising a Huber cost function that is a combined l 1 and l 2 norm [20]. A larger number of variants of the HNKF have been derived based on a linearized or statistical linearized method, such as the Huber-based extended Kalman filter [21], the Huber-based divided difference filter [22], the Huber-based unscented Kalman filter [23], the nonlinear regression Huber Kalman filter [24] and the adaptively robust unscented Kalman filter (ARUKF) [25]. However, the influence function of the HNKFs don't redescend, which may deteriorate the estimation performance of the HNKFs [12]. The maximum correntropy criterion based Kalman filter (MCCKF) has been proposed by maximising the correntropy of the predicted error and residual [26]- [29]. However, there is a lack of theoretical basis to develop the estimation error covariance matrix of the MCCKF, which may degrade the estimation accuracy [12].
A reasonable approach to improve the estimation performance is utilizing a Student's t distribution to model the heavy-tailed non-Gaussian noise. The Student's t distribution is a generalized Gaussian distribution but has heavier tails than the Gaussian distribution, which makes it more suitable for modelling the heavy-tailed non-Gaussian noise. A general framework of the robust Student's t based nonlinear filter (RSTNF) has been proposed, in which the jointly predicted PDF of the state and measurement vectors is assumed to be Student's t, and the posterior PDF is then approximated as Student's t [30]. The heart of the RSTNF is how to calculate the Student's t weighted integral, and the estimation accuracy of the associated RSTNF is determined by the employed numerical integral technique. Many variants of the RSTNF have been derived based on different numerical integral methods, such as the robust Student's t based extended filter (RSTEF) using the first-order linearization [10], the robust Student's t based unscented filter (RSTUF) using the unscented transform (UT) [30], [31], and the robust Student's t based cubature filter (RSTCF) using the third-degree Student's t spherical radial cubature rule (STSRCR) [32]. However, the existing Student's t integral rules can only capture the third-degree or fifth-degree information of the Taylor series expansion for nonlinear approximation, which may result in limited estimation accuracy. Although the Monte Carlo approach can be used to calculate the Student's t weighted integral, it has low accuracy and slow convergence when the integrand is not approximately constant and the number of random samples is finite [33]. Therefore, there is a great demand to develop more accurate numerical integral approach for the Student's t weighted integral to further improve the estimation accuracy of the existing RSTNFs.
In this paper, the Student's t weighted integral is decomposed into the spherical integral and the radial integral based on the spherical-radial transformation. A new stochastic STSRCR (SSTSRCR) is derived based on the third-degree unbiased spherical rule (USR) and the proposed third-degree unbiased radial rule (URR), from which a new robust Student's t based stochastic cubature filter (RSTSCF) is obtained. The existing stochastic integration rule (SIR) [8] is a special case of the proposed SSTSRCR when the degrees of freedom (dof) parameter tends to infinity. The proposed SSTSRCR can achieve better approximation to the Student's t weighted integral as compared with existing Student's t integral rules. As a result, the proposed RSTSCF has higher estimation accuracy than the existing RSTNFs. The proposed filter and existing filters are applied to a manoeuvring bearings-only tracking example, where an agile target is tracked and the bearing is observed in clutter. Simulation results show that the proposed RSTSCF can achieve higher estimation accuracy than the existing GA filter, GSF, HNKF, MCCKF and RSTNFs, and is computationally much more efficient than the existing PF.
The remainder of this paper is organized as follows. In Section II, a general frame of the RSTNF is reviewed. In Section III, a new SSTSRCR is derived based on the proposed third-degree URR, from which a new RSTSCF is obtained. Also, the relationship between the proposed SSTSRCR and the existing SIR is revealed in Section III. In Section IV, the proposed filter is applied to a manoeuvring bearings-only tracking example and simulation results are given. Concluding remarks are drawn in Section V.
II. PROBLEM STATEMENT
Consider the following discrete-time nonlinear stochastic system as represented by the state-space model [30] x
k = f k−1 (x k−1 ) + w k−1 (process equation) (1) z k = h k (x k ) + v k (measurement equation),(2)
where k is the discrete time index, x k ∈ R n is the state vector, z k ∈ R m is the measurement vector, and f k−1 (·) and h k (·) are known process and measurement functions respectively. w k ∈ R n and v k ∈ R m are heavy-tailed process and measurement noise vectors respectively, which are induced by process and measurement outliers, and their distributions are modelled as Student's t distributions, i.e.,
p(w k ) = St(w k ; 0, Q k , ν 1 ) (3) p(v k ) = St(v k ; 0, R k , ν 2 ),(4)
where St(·; µ, Σ, v) denotes the Student's t PDF with mean vector µ, scale matrix Σ, and dof parameter v, Q k and ν 1 are the scale matrix and dof parameter of process noise respectively, and R k and ν 2 are the scale matrix and dof parameter of measurement noise respectively. The initial state vector x 0 is also assumed to have a Student's t distribution with mean vectorx 0|0 , scale matrix P 0|0 , and dof parameter ν 3 , and x 0 , w k and v k are assumed to be mutually uncorrelated. To achieve the filtering estimation, a general framework of RSTNF is derived for the nonlinear system formulated in equations (1)-(4), where the jointly predicted PDF of the state and measurement vectors is assumed as Student's t, then the posterior PDF of the state vector can be approximated as Student's t [30]. The time update and measurement update of the recursive RSTNF are given as follows:
Time updatê
x k|k−1 = R n f k−1 (x k−1 )St(x k−1 ;x k−1|k−1 , P k−1|k−1 , ν 3 ) dx k−1 (5) P k|k−1 = ν 3 − 2 ν 3 R n f k−1 (x k−1 )f T k−1 (x k−1 )St(x k−1 ; x k−1|k−1 , P k−1|k−1 , ν 3 )dx k−1 − ν 3 − 2 ν 3x k|k−1x T k|k−1 + ν 1 (ν 3 − 2) (ν 1 − 2)ν 3 Q k−1 ,(6)
where (·) T denotes the transpose operation,x k|k−1 and P k|k−1 are respectively the mean vector and scale matrix of the one-step predicted PDF p(
x k |Z k−1 ), Z k−1 = {z j } k−1 j=1
is the set of k − 1 measurement vectors, and ν 3 denotes the dof parameter of the filtering PDF. Measurement update
∆ k = (z k −ẑ k|k−1 ) T (P zz k|k−1 ) −1 (z k −ẑ k|k−1 ) (7) K k = P xz k|k−1 (P zz k|k−1 ) −1 (8) x k|k =x k|k−1 + K k (z k −ẑ k|k−1 ) (9) P k|k = (ν 3 − 2)(ν 3 + ∆ 2 k ) ν 3 (ν 3 + m − 2) (P k|k−1 − K k P zz k|k−1 K T k ),(10)
where (·) −1 denotes the inverse operation,x k|k and P k|k are respectively the mean vector and scale matrix of the filtering PDF p(x k |Z k ),ẑ k|k−1 and P zz k|k−1 are respectively the mean vector and scale matrix of the likelihood PDF p(z k | Z k−1 ), and P xz k|k−1 is the cross scale matrix of state and measurement vectors, which are given bŷ
z k|k−1 = R n h k (x k )St(x k ;x k|k−1 , P k|k−1 , ν 3 )dx k (11) P zz k|k−1 = ν 3 − 2 ν 3 R n h k (x k )h T k (x k )St(x k ;x k|k−1 , P k|k−1 , ν 3 )dx k − ν 3 − 2 ν 3ẑ k|k−1ẑ T k|k−1 + ν 2 (ν 3 − 2) (ν 2 − 2)ν 3 R k (12) P xz k|k−1 = ν 3 − 2 ν 3 R n x k h T k (x k )St(x k ;x k|k−1 , P k|k−1 , ν 3 ) dx k − ν 3 − 2 ν 3x k|k−1ẑ T k|k−1 .(13)
The recursive RSTNF is composed of the analytical computations in equations (7)-(10) and the Student's t weighted integrals in equations (5)- (6) and (11)- (13). The key problem in the design of the RSTNF is calculating the nonlinear Student's t weighted integrals formulated in equations (5)- (6) and (11)- (13), whose integrands are all of the form nonlinear function×Student's t PDF. Therefore, the numerical integral technique is required to implement the RSTNF, which determines the estimation accuracy of associated RSTNF. Next, to further improve the estimation accuracy of existing RSTNFs, a new SSTSRCR will be proposed, based on which a new RSTSCF can be obtained.
III. MAIN RESULTS
A. Spherical-radial transformation
The Student's t weighted integrals involved in the RSTNF can be written as the general form as follows
I[g] = R n g(x)St(x; µ, Σ, ν)dx,(14)
where the Student's t PDF is given by
St(x; µ, Σ, ν) = Γ( ν+n 2 ) Γ( ν 2 ) 1 |νπΣ| × 1 + 1 ν (x − µ) T Σ −1 (x − µ) − ν+n 2 ,(15)
where Γ(·) and |·| denote the Gamma function and determinant operation respectively. To derive the SSTSRCR, the Student's t weighted integral in equation (14) requires to be transformed into a spherical-radial integral form. A change of variable is utilized as follows
x = µ + √ νΣy,(16)
where
√ Σ is the square-root of scale matrix Σ satisfying Σ = √ Σ √ Σ T .
Substituting equation (16) in equations (14)- (15) and using the identity | √ νΣ| = |νΣ| yields
I[g] = R n l(y)(1 + y T y) − ν+n 2 dy,(17)
where l(y) is given by
l(y) = Γ( ν+n 2 ) Γ( ν 2 )π n 2 g(µ + √ νΣy).(18)
Define y = rs with s T s = 1, then equation (17) can be rewritten as [34]
I[g] = +∞ 0 Un l(rs)r n−1 [1 + (rs) T (rs)] − ν+n 2 dσ(s)dr = +∞ 0 Un l(rs)r n−1 (1 + r 2 ) − ν+n 2 dσ(s)dr,(19)
where s = [s 1 , s 2 , · · · , s n ] T , U n = {s ∈ R n : s 2 1 + s 2 2 + · · · + s 2 n = 1}, and σ(s) is the spherical surface measure or an area element on U n .
According to equation (19), the Student's t weighted integral in equation (14) can be decomposed into the radial integral
I[g] = +∞ 0 S(r)r n−1 (1 + r 2 ) − ν+n 2 dr,(20)
and the spherical integral
S(r) = Un l(rs)dσ(s).(21)
Next, a new third-degree SSTSRCR will be derived, in which the spherical and the radial integrals are respectively calculated by the third-degree USR (Section III. B below) and the third-degree URR (Section III. C below). Before deriving the third-degree SSTSRCR, the unbiased integral rule is firstly defined as follows.
Definition 1. The integral rule g(x)p(x)dx ≈ N l=1 w l g(x l )
is unbiased if and only if [33]
g(x)p(x)dx = E N l=1 w l g(x l ) ,(22)
where x l and w l are respectively cubature points and corresponding weights, and E[·] denotes the expectation operation.
B. Unbiased spherical rule
The Stewart's method is employed to construct the thirddegree USR. If Q is a random orthogonal matrix drawn with a Haar distribution from the set of all matrices in the orthogonal group, the third-degree USR can be constructed as [33], [35]
S 3 u (r) ≈ A n 2n n i=1 [l(−rQe i ) + l(rQe i )] ,(23)
where A n = 2π
n 2 Γ( n 2 )
is the surface area of the unit sphere, and e i denotes the i-th column of an n × n unit matrix. To produce a random orthogonal matrix Q, a n × n matrix U of standard norm variables is first generated, then the required random orthogonal matrix Q is obtained based on the QR factorization, i.e., U = QR [35].
Next, a new third-degree URR will be proposed for the radial integral in equation (20).
C. Unbiased radial rule
Generally, the monomials S(r) = 1, S(r) = r, S(r) = r 2 , and S(r) = r 3 need to be matched to derive the third-degree URR. However, only monomials S(r) = 1 and S(r) = r 2 need to be matched for the third-degree URR since the USR and the resultant STSRCR are fully symmetry. Thus, two points {r 1 , ω r,1 } and {r 2 , ω r,2 } are sufficient to design the third-degree URR, where one point is used to match monomials S(r) = 1 and S(r) = r 2 and the other point is employed to retain unbiasedness. That is to say, the thirddegree URR can be written as
+∞ 0 S(r)r n−1 (1 + r 2 ) − ν+n 2 dr ≈ ω r,1 S(r 1 ) + ω r,2 S(r 2 ),(24)
where {r 1 , ω r,1 } and {r 2 , ω r,2 } satisfy the following equations
ω r,1 r 0 1 + ω r,2 r 0 2 = +∞ 0 r 0 r n−1 (1 + r 2 ) − ν+n 2 dr (25) ω r,1 r 2 1 + ω r,2 r 2 2 = +∞ 0 r 2 r n−1 (1 + r 2 ) − ν+n 2 dr(26)p(r 2 ) = 2r n+1 2 (1 + r 2 2 ) − ν+n 2 /B( n + 2 2 , ν − 2 2 ),(28)
where B(·, ·) denotes the beta function, then the third-degree URR is given by
I[g] ≈ 1 2 B( n 2 , ν 2 ) 1 − n (ν − 2)r 2 2 S(0) + n (ν − 2)r 2 2 S(r2) .(29)
Proof. Firstly, a general integral +∞ 0 r l r n−1 (1 + r 2 ) − ν+n 2 dr is calculated to obtain the right-hand parts in equations (25)- (26). Making a change of variable via t = r 2 results in
+∞ 0 r l r n−1 (1 + r 2 ) − ν+n 2 dr = 1 2 B( n + l 2 , ν − l 2 ),(30)
where B(·, ·) denotes the beta function. Substituting equation (30) in equations (25)-(26), we have
ω r,1 + ω r,2 = 1 2 B( n 2 , ν 2 )(31)ω r,1 r 2 1 + ω r,2 r 2 2 = 1 2 B( n + 2 2 , ν − 2 2 ).(32)
Utilizing the identities Γ(a + 1) = aΓ(a) and B(a, b) = Γ(a)Γ(b) Γ(a+b) in equation (32) yields
ω r,1 r 2 1 + ω r,2 r 2 2 = n 2(ν − 2) B( n 2 , ν 2 ).(33)
Employing r 1 = 0 in equation (33) yields
ω r,2 = n 2(ν − 2)r 2 2 B( n 2 , ν 2 ).(34)
Substituting equation (34) in equation (31) results in
ω r,1 = 1 2 B( n 2 , ν 2 ) 1 − n (ν − 2)r 2 2 .(35)
Utilizing r 1 = 0 and equations (34)- (35), the expectation of the third-degree radial rule with respect to p(r 2 ) is written as
E [ωr,1S(r1) + ωr,2S(r2)] = 1 2 B( n 2 , ν 2 )E 1 − n (ν − 2)r 2 2 × S(0) + 1 2 B( n 2 , ν 2 )E n (ν − 2)r 2 2 S(r2) .(36)
Using equations (28) and (30), we have (39), the third-degree URR can be formulated as equation (29).
E 1 − n (ν − 2)r 2 2 = +∞ 0 2r n+1 2 (1 + r 2 2 ) − ν+n 2 B( n+2 2 , ν−2 2 ) dr 2 − n (ν − 2) +∞ 0 2r n−1 2 (1 + r 2 2 ) − ν+n 2 B( n+2 2 , ν−2 2 ) dr 2 = 0 (37) E n (ν − 2)r 2 2 S(r2) = n (ν − 2) +∞ 0 2r n−1 2 (1 + r 2 2 ) − ν+n 2 B( n+2 2 , ν−2 2 ) × S(r2)dr2 = 2 B( n 2 , ν 2 ) +∞ 0 S(r)r n−1 (1 + r 2 ) − ν+n 2 dr.(38)
It is very difficult to directly generate random samples from p(r 2 ) since p(r 2 ) is not a special PDF. To solve this problem, Theorem 2 is presented as follows. , then random variable τ obeys the Beta distribution, i.e.,
p(τ ) = Beta(τ ; n + 2 2 , ν − 2 2 ) = τ n+2 2 −1 (1 − τ ) ν−2 2 −1 B( n+2 2 , ν−2 2 ) ,(40)
where Beta(·; α, β) denotes the beta PDF with parameters α and β.
Proof. Since τ = , r 2 is formulated as
r 2 = c(τ ) = τ 1 − τ τ ∈ [0, 1).(41)
Employing the transformation theorem and equation (41), the PDF of random variable τ is given by
p(τ ) = p r2 (c(τ ))c ′ (τ ),(42)
where p r2 (·) denotes the PDF of r 2 and c ′ (τ ) denotes the derivative of c(τ ) with respect to τ given by
c ′ (τ ) = 0.5τ − 1 2 (1 − τ ) − 3 2 .(43)
Substituting equations (28), (41) and (43) in equation (42) obtains
p(τ ) = τ n+2 2 −1 (1 − τ ) ν−2 2 −1 /B( n + 2 2 , ν − 2 2 ),(44)
which proves the theorem.
D. Stochastic STSRCR
A theorem is first presented to derive the unbiased STSRCR.
Theorem 3. If the spherical and radial rules are unbiased, then the resultant STSRCR is also unbiased.
Proof. If the spherical and radial rules are given by
S(r) ≈ Ns i=1 w s,i l(rs i ); I[g] ≈ Nr j=1 w r,j S(r j ),(45)
then the STSRCR can be formulated as
I[g] ≈ Nr j=1 Ns i=1 w r,j w s,i l(r j s i ),(46)
where s i and w s,i are respectively cubature points and weights of the spherical rule, and r j and w r,j are respectively quadrature points and weights of the radial rule.
Since the spherical and radial rules are unbiased, we obtain
S(r) = E Ns i=1 w s,i l(rs i ) ; I[g] = E Nr j=1 w r,j S(r j ) .(47)
Using equation (47) yields
I[g] = E Nr j=1 w r,j E Ns i=1 w s,i l(r j s i ) .(48)
Since the set {s i , w s,i } Ns i=1 is independent of the set {r j , w r,j } Nr j=1 , we have
I[g] = E Nr j=1 Ns i=1 w r,j w s,i l(r j s i ) ,(49)
which proves the theorem.
Using Theorems 1-3 obtains Inputs: µ, Σ, ν, g(·), n, N .
I[g] = E 1 − n (ν − 2)r 2 2 g(µ) + 1 2(ν − 2)r 2 2 × n i=1 g(µ − r 2 √ νΣQe i ) + g(µ + r 2 √ νΣQe i ) . (50)
Initialization: I 3 s [g] = 0.
for l = 1 : N 2. Generate a n × n matrix U l of standard norm variables.
3. Obtain the required random orthogonal matrix Q l using the QR factorization: U l = Q l R l .
4. Draw the random variable τ l from a Beta distribution:
τ l ∼ Beta( n+2 2 , ν−2 2 ).
5. Calculate the random quadrature point r 2,l :
r 2,l = τ l 1−τ l . 6. Update I 3 s
[g] at current iteration:
I 3 s [g] = I 3 s [g] + 1 N 1 − n (ν−2)r 2 2,l g(µ) + 1 2(ν−2)r 2 2,l × n i=1 g(µ − r 2,l √ νΣQ l e i ) + g(µ + r 2,l √ νΣQ l e i ) .
end for
Outputs: I[g] ≈ I 3 s [g].
By employing the Monte Carlo approach, the right-hand parts of equation (50) can be approximated as
I 3 s [g] = 1 N N l=1 1 − n (ν − 2)r 2 2,l g(µ) + 1 2(ν − 2)r 2 2,l × n i=1 g(µ − r 2,l √ νΣQ l e i ) + g(µ + r 2,l √ νΣQ l e i ) ,(51)
where N denotes the number of random samples, and Q l is a random orthogonal matrix, and r 2,l is drawn randomly from p(r 2 ). The form I 3 s [g] denotes the proposed thirddegree SSTSRCR, and the implementation pseudocode of the proposed SSTSRCR is shown in Table I
I 3 s [g] = I[g].(52)
Thus, the proposed SSTSRCR provides asymptotically exact integral evaluations when N tends to infinity. A new RSTSCF can be obtained by employing the proposed SST-SRCR to calculate the Student's t weighted integrals involved in the RSTNF, and the implementation pseudocode for one time step of the proposed RSTSCF is shown in Table II, where SSTSRCR(·) denotes the proposed SSTSRCR algorithm. The proposed SSTSRCR can achieve better approximation to the Student's t weighted integral as compared with existing Student's t integral rules. As a result, the proposed RSTSCF has higher estimation accuracy than the existing RSTNFs.
Inputs:x k−1|k−1 , P k−1|k−1 , z k , Q k−1 , R k , ν 1 , ν 2 , ν 3 , f k−1 (·),
h k (·), n, N .
Time update:
1.x k|k−1 = SSTSRCR(x k−1|k−1 , P k−1|k−1 , ν 3 , f k−1 (·), n, N ). 2. P k|k−1 = ν 3 −2 ν 3 SSTSRCR(x k−1|k−1 , P k−1|k−1 , ν 3 , f k−1 (·)f T k−1 (·), n, N ) − ν 3 −2 ν 3x k|k−1x T k|k−1 + ν 1 (ν 3 −2) (ν 1 −2)ν 3 Q k−1 .
Measurement update:
3.ẑ k|k−1 = SSTSRCR(x k|k−1 , P k|k−1 , ν 3 , h k (·), n, N ). 4. P zz k|k−1 = ν 3 −2 ν 3 SSTSRCR(x k|k−1 , P k|k−1 , ν 3 , h k (·)h T k (·), n, N ) − ν 3 −2 ν 3ẑ k|k−1ẑ T k|k−1 + ν 2 (ν 3 −2) (ν 2 −2)ν 3 R k . 5. P xz k|k−1 = ν 3 −2 ν 3 SSTSRCR(x k|k−1 , P k|k−1 , ν 3 , x k h k (·), n, N ) − ν 3 −2 ν 3x k|k−1ẑ T k|k−1 . 6. ∆ k = (z k −ẑ k|k−1 ) T (P zz k|k−1 ) −1 (z k −ẑ k|k−1 ). 7. K k = P xz k|k−1 (P zz k|k−1 ) −1 . 8.x k|k =x k|k−1 + K k (z k −ẑ k|k−1 ). 9. P k|k = (ν 3 −2)(ν 3 +∆ 2 k ) ν 3 (ν 3 +m−2) (P k|k−1 − K k P zz k|k−1 K T k ).
Outputs:x k|k , P k|k . Remark 1. The Monte Carlo approach can be also used to calculate the Student's t weighted integral, and it provides asymptotically exact integral evaluations when the number of random samples tends to infinity. However, it has low accuracy and slow convergence when the integrand is not approximately constant and the number of random samples is finite [33]. Fortunately, the proposed SSTSRCR is at least exact up to third-degree polynomials for any number of random samples, and it can capture more and more higher-degree moment information as the number of random samples increases.
E. Relationship between the proposed SSTSRCR and the existing SIR [8]
Theorem 4. The proposed SSTSRCR will degrade to the existing SIR when the dof parameter ν → +∞, i.e.
lim ν→+∞ I 3 s [g] = 1 N N l=1 1 − n ρ 2 l g(µ) + 1 2ρ 2 l × n i=1 g(µ − ρ l √ ΣQ l e i ) + g(µ + ρ l √ ΣQ l e i ) ,(53)
where the right-hand side of the equation (53) is the SIR for the Gaussian weighted integral, and ρ l is drawn randomly from p(ρ l ) that is given by
p(ρ l ) ∝ ρ n+1 l e − ρ 2 l 2 .(54)
Proof. Make a change of variable as follows
r 2,l = c(ρ l ) = ρ l √ ν − 2 , s.t. ν → +∞.(55)
Substituting equation (55) in equation (51) results in
I 3 s [g] = 1 N N l=1 1 − n ρ 2 l g(µ) + 1 2ρ 2 l n i=1 g(µ − ρ l ν ν − 2 ΣQ l e i ) + g(µ + ρ l ν ν − 2 ΣQ l e i ) .(56)
Taking the limit of equation (56) when the dof parameter ν → +∞, we can obtain equation (53).
Using the transformation theorem and equation (55), the PDF of random variable ρ l is given by
p(ρ l ) = p r2 (c(ρ l ))c ′ (ρ l ),(57)
where p r2 (·) denotes the PDF of r 2 , and c ′ (ρ l ) denotes the derivative of c(ρ l ) with respect to ρ l given by
c ′ (ρ l ) = 1 √ ν − 2 .(58)
Substituting equations (28), (55) and (58) in equation (57), we obtain
p(ρ l ) = 2ρ n+1 l lim ν→+∞ 1 (ν − 2) n+2 2 B( n+2 2 , ν−2 2 ) × lim ν→+∞ 1 + ρ 2 l ν − 2 − ν+n 2 .(59)
Utilizing the identity B(a, b) = Γ(a)Γ(b) Γ(a+b) , the first limit in equation (59) can be formulated as
lim ν→+∞ 1 (ν − 2) n+2 2 B( n+2 2 , ν−2 2 ) = 2 − n+2 2 Γ( n+2 2 ) × lim ν→+∞ Γ( ν−2 2 + n+2 2 ) Γ( ν−2 2 )( ν−2 2 ) n+2 2 .(60)
Using the property of Gamma function lim t→+∞ Γ(t+α) Γ(t)t α = 1 in equation (60) gives
lim ν→+∞ 1 (ν − 2) n+2 2 B( n+2 2 , ν−2 2 ) = 2 − n+2 2 Γ( n+2 2 )
.
The second limit in equation (59) can be reformulated as
lim ν→+∞ 1 + ρ 2 l ν − 2 − ν+n 2 = lim ν→+∞ s(ν) d(ν) = lim ν→+∞ s(ν) lim ν→+∞ d(ν) ,(62)
where the functions s(ν) and d(ν) are given by 7 Using the identity lim
s(ν) = 1 + ρ 2 l ν − 2 ν−2 ρ 2 l (63) d(ν) = − ρ 2 l (ν + n) 2(ν − 2) .(64)
Substituting equations (61) and (65) in (59), we can obtain (54), which proves the theorem.
Considering that the Student's t PDF turns into the Gaussian PDF as the dof parameter ν → +∞, we obtain
lim ν→+∞ I[g]= R n g(x) lim ν→+∞ St(x; µ, Σ, ν)dx = R n g(x)N(x; µ, Σ)dx.(66)
According to the Theorem 4 and equation (66), we can conclude that the proposed SSTSRCR with ν → +∞ can be utilized to calculate the Gaussian weighted integral. Thus, the proposed SSTSRCR is a generalized SIR, which can calculate not only the Gaussian weighted integral but also the Student's t weighted integral.
IV. SIMULATION STUDY
In this simulation, the superior performance of the proposed RSTSCF as compared with existing filters is shown in the problem of manoeuvring bearing-only tracking observed in clutter. The target moves according to the continuous white noise acceleration motion model [8] x
k = Fx k−1 + Gw k−1 ,(67)
where x k = [x k y kẋkẏk ], and x k , y k ,ẋ k andẏ k denote the cartesian coordinates and corresponding velocities respectively; F and G denote respectively the state transition matrix and noise matrix given by
F = I 2 ∆tI 2 0 I 2 G = Γ 0 2×1 0 2×1 Γ ,(68)
where ∆t = 1min is the sampling interval, and I 2 is the two dimensional identity matrix, and 0 2×1 is the two dimensional zero vector, and Γ = [0.5∆t 2 ∆t] T . The target is observed by an angle sensor installed in a manoeuvring platform. If the platform is located at (x p k , y p k ) at time k, then the measurement model is given by
z k = tan −1 ( y k − y p k x k − x p k ) + v k ,(69)
where z k is the angle between the target and the platform at time k. Outlier corrupted process and measurement noises are generated according to [10], [12], [30] w k ∼ N (0, Σ w ) w.p. 0.95 N (0, 100Σ w ) w.p. 0.05
(70) v k ∼ N (0, Σ v ) w.p. 0.95 N (0, 50Σ v ) w.p. 0.05 ,(71)
where w.p. denotes "with probability", the nominal process noise covariance matrix is Σ w = 10 −6 I 2 km 2 /min 2 , and the nominal measurement noise variance is Σ v = (0.02rad) 2 . Process and measurement noises, which are generated according to equations (70)-(71), have heavier tails. In our simulation scenario, the initial positions of the target and the platform are respectively (3km, 3km) and (0km, 0km). The target moves at a constant speed of 180 knots (1 knot is 1.852km/h) with a course of −135.4 • . The platform moves at a constant speed of 50 knots with a initial course of −80 • , and the course reaches 146 • at time k = 15min by executing a manoeuvre [8]. The initial estimation error covariance matrix is P 0|0 = diag[16km 2 16km 2 , 4km 2 /min 2 , 4km 2 /min 2 ], and the initial state estimatex 0|0 is chosen randomly from N(x 0 , P 0|0 ), where x 0 denotes the initial true state.
In this simulation, the stochastic integration filter (SIF) [8], the ARUKF with free parameter κ = 0 [25], the MCCKF with kernel size σ = 5 [29], the RSTEF [10], the 3rd-degree RSTUF with free parameter κ = 3 − n [30], the 3rd-degree RSTCF [32], the fifth-degree RSTUF [31], the robust Student's t based Monte Carlo filter (RSTMCF), the Gaussian sumcubature Kalman filter (GSCKF) [18], the PF [13], and the proposed RSTSCF are tested. In the RSTMCF, the Student's t weighted integral is calculated using the conventional Monte Carlo approach with 10000 random samples. In the GSCKF, the process and measurement noises are modelled as p(w k ) =
5 i=1 α i N (w k ; 0, λ i Σ w ) and p(v k ) = 5 i=1 α i N (v k ; 0, λ i Σ v ),
where the weights α 1 = 0.8 and α 2 = α 3 = α 4 = α 5 = 0.05, and the scale parameters λ 1 = 1, λ 2 = 50, λ 3 = 100, λ 4 = 500 and λ 5 = 1000. Moreover, to prevent the computational complexity of the GSCKF increasing exponentially as the time, the posterior distribution is approximated as a weighted sum of five Gaussian terms with the highest weights. In the PF, the process and measurement noises are modelled as Student's t distributions, and the number of particle is chosen as 10000.
In the existing RSTEF, 3rd-degree RSTUF, 3rd-degree RSTCF, fifth-degree RSTUF, RSTMCF, PF and the proposed RSTSCF, the dof parameters are all chosen as ν 1 = ν 2 = ν 3 = 5 and the scale matrices are all set as Q k = Σ w and R k = Σ v . In the SIF and the proposed RSTSCF, the number of random samples is selected as N = 100. The proposed filter and existing filters are coded with MATLAB and the simulations are run on a computer with Intel Core i7-3770 CPU at 3.40 GHz.
To compare the performances of the proposed filter and existing filters, the RMSEs and the averaged RMSEs (ARMSEs) of the position and velocity are chosen as performance metric. The RMSE and ARMSE in position are respectively defined as The RMSEs and ARMSEs of position and velocity from the proposed filter and existing filters are respectively shown in Fig. 1-Fig. 2 and Table III. The implementation times of the proposed filter and existing filters in single step run are given in Table III. Note that the existing ARUKF and RSTEF diverge in the simulation, as shown in Fig. 1-Fig. 2 and Table III.
RMSE pos (k) = 1 M M s=1 (x s k −x s k ) 2 + (y s k −ŷ s k ) 2 (72) ARMSE pos = 1 M T T k=1 M s=1 (x s k −x s k ) 2 + (y s k −ŷ s k ) 2 ,(73)
It is seen from Fig. 1-Fig. 2 and Table III that the RMSEs and ARMSEs of the proposed RSTSCF are smaller than the existing SIF, ARUKF, MCCKF, RSTEF, 3rd-degree RSTUF, 3rd-degree RSTCF, 5th-degree RSTUF, RSTMCF, GSCKF but larger than the existing PF. Furthermore, it can be also seen from Table III Fig. 2: RMSEs of the velocity from the proposed filter and existing filters RSTSCF are greater than the existing SIF, ARUKF, MCCKF, RSTEF, 3rd-degree RSTUF, 3rd-degree RSTCF, 5th-degree RSTUF but significantly smaller than the existing RSTMCF, GSCKF and PF. Therefore, the proposed RSTSCF has better estimation accuracy than the existing SIF, ARUKF, MCCKF, RSTNFs and GSCKF, and is computationally much more efficient than the existing PF.
V. CONCLUSION
In this paper, a new SSTSRCR was derived based on the third-degree USR and the proposed third-degree URR, from which a new RSTSCF was obtained. The existing SIR is a special case of the proposed SSTSRCR when the dof parameter tends to infinity. Simulation results for a manoeuvring bearings-only tracking example illustrated that the proposed RSTSCF can achieve higher estimation accuracy than the existing GA filter, GSF, HNKF, MCCKF and RSTNFs, and is computationally much more efficient than the existing PF.
This work was supported by the National Natural Science Foundation of China under Grant Nos. 61371173 and 61633008 and the Natural Science Foundation of Heilongjiang Province Grant No. F2016008. Corresponding author is Y. G. Zhang. Y. L. Huang and Y. G. Zhang are with the Department of Automation, Harbin Engineering University, Harbin 150001, China (e-mail: [email protected]; [email protected]).
+∞ 0 S
0(r)r n−1 (1+r 2 ) − ν+n 2 dr = E [ω r,1 S(r 1 ) + ω r,2 S(r 2 )] .(27) Since there are three equations and four variables in equations (25)-(27), there is one free variable. In order to get the STSRCR with the minimum number of points, r 1 is chosen as the free variable and set to zero. Theorem 1. If r 1 = 0 and the PDF of random variable r 2 is
Substituting equations (37)-(38) in equation (36) yields E [ω r,1 S(r 1 ) + ω r,2 S(r 2 )] = +∞ 0 S(r)r n−1 (1+r 2 ) − ν+n 2 dr. (39) With r 1 = 0, equations (34)-(35) and
Theorem 2 .
2If random variable τ =
r 2 ∈ [0, +∞), random variable τ ∈ [0, 1). According to τ =
where M = 1000 denotes the number of Monte Carlo runs, and T = 100min denotes the simulation time, and (x s k , y s k ) and
Fig. 1 :
1RMSEs of the position from the proposed filter and existing filters (x s k ,ŷ s k ) respectively denote the true and estimated positions at the s-th Monte Carlo run. Similar to the RMSE and ARMSE in position, we can also formulate the RMSE and ARMSE in velocity.
TABLE I :
IThe implementation pseudocode of the proposed SSTSRCR.
TABLE II :
IIThe implementation pseudocode for one time step of the proposed RSTSCF.
TABLE III :
IIIARMSEs and implementation times of the proposed filter and existing filters.Filters
ARMSEpos (km)
ARMSE vel (km/min)
Time (ms)
SIF
22.85
0.83
45.46
ARUKF
2.57 × 10 25
2.41 × 10 24
0.41
MCCKF
26.86
0.88
0.13
RSTEF
3.51 × 10 6
6.51 × 10 4
0.09
3rd RSTUF
88.86
2.43
0.53
3rd RSTCF
29.90
1.12
0.50
5th RSTUF
16.29
0.65
1.0
RSTMCF
53.65
1.65
193.7
GSCKF
19.23
0.58
54.6
PF
7.89
0.30
773.0
RSTSCF
12.08
0.56
48.2
10
20
30
40
50
60
70
80
90
100
20
40
60
80
100
120
140
Time (min)
RMSE
pos
(km)
SIF
ARUKF
MCCKF
RSTEF
3rd−degree RSTUF
3rd−degree RSTCF
5th−degree RSTUF
RSTMCF
GSCKF
PF
RSTSCF
that the implementation time of the proposed10
20
30
40
50
60
70
80
90
100
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time (min)
RMSE
vel
(km/min)
SIF
ARUKF
MCCKF
RSTEF
3rd−degree RSTUF
3rd−degree RSTCF
5th−degree RSTUF
RSTMCF
GSCKF
PF
RSTSCF
B D O Anderson, J B Moore, Optimal filtering. Englewood Cliffs, NJPrentice HallB. D. O. Anderson and J. B. Moore, Optimal filtering, Englewood Cliffs, NJ: Prentice Hall, 1979.
Gaussian filter for nonlinear systems with correlated noises at the same epoch. Y L Huang, Y G Zhang, X X Wang, L Zhao, Automatica. 6010Y. L. Huang, Y. G. Zhang, X. X. Wang, and L. Zhao, "Gaussian filter for nonlinear systems with correlated noises at the same epoch," Automatica, vol. 60, no. 10, pp. 122-126, oct. 2015.
Cubature Kalman filter. I Arasaratnam, S Haykin, IEEE Trans. Autom. Control. 546I. Arasaratnam and S. Haykin, "Cubature Kalman filter," IEEE Trans. Autom. Control, vol. 54, no. 6, pp. 1254−1269, June 2009.
Interpolatory cubature Kalman filters. Y G Zhang, Y L Huang, N Li, L Zhao, IET Control Theory & Appl. 911Y. G. Zhang, Y. L. Huang, N. Li, and L. Zhao. "Interpolatory cubature Kalman filters," IET Control Theory & Appl., vol. 9, no. 11, pp. 1731- 1739, Sep. 2015.
Gaussian filters for nonlinear filtering problems. K Ito, K Xiong, IEEE Trans. Autom. Control. 455K. Ito and K. Xiong, "Gaussian filters for nonlinear filtering problems," IEEE Trans. Autom. Control, vol. 45, no. 5, pp. 910−927, May 2000.
Unscented filtering and nonlinear estimation. S J Julier, J K Uhlman, Proc. IEEE. IEEE92S. J. Julier and J. K. Uhlman, "Unscented filtering and nonlinear estimation," Proc. IEEE, vol. 92, no. 3, pp. 401−422, Mar. 2004.
High-degree cubature Kalman filter. B Jia, M Xin, Y Cheng, Automatica. 492B. Jia, M. Xin, and Y. Cheng, "High-degree cubature Kalman filter," Automatica, vol. 49, no. 2, pp. 510−518, Feb. 2013.
Stochastic integration filter. J Duník, O Straka, M Šimandl, IEEE Trans. Automat. Control. 586J. Duník, O. Straka, and M.Šimandl, "Stochastic integration filter," IEEE Trans. Automat. Control, vol. 58, no. 6, pp. 1561-1566, June 2013.
Embedded cubature Kalman filter with adaptive setting of free parameter. Y G Zhang, Y L Huang, N Li, L Zhao, Signal Process. 1149Y. G. Zhang, Y. L. Huang, N. Li, and L. Zhao. "Embedded cubature Kalman filter with adaptive setting of free parameter," Signal Process., vol. 114, no. 9, pp. 112-116, Sep. 2015.
A Student's t filter for heavytailed process and measurement noise. M Roth, E Özkan, F Gustafsson, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). M. Roth, E.Özkan, and F. Gustafsson, "A Student's t filter for heavy- tailed process and measurement noise," in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 2013, pp. 5770-5774.
A robust Gaussian approximate fixed-interval smoother for nonlinear systems with heavytailed process and measurement noises. Y L Huang, Y G Zhang, N Li, J Chambers, IEEE Signal Proc. Let. 234Y. L. Huang, Y. G. Zhang, N. Li, and J. Chambers, "A robust Gaussian approximate fixed-interval smoother for nonlinear systems with heavy- tailed process and measurement noises," IEEE Signal Proc. Let., vol. 23, no. 4, pp. 468-472, Apr. 2016.
A novel robust Student's t based Kalman filter. Y L Huang, Y G Zhang, Z M Wu, N Li, J Chambers, Trans. Aero. Elec. Sys. to appear in IEEEY. L. Huang, Y. G. Zhang, Z. M. Wu, N. Li, and J. Chambers, "A novel robust Student's t based Kalman filter," to appear in IEEE Trans. Aero. Elec. Sys., 2017.
A tutorial on particle filters for on-line non-linear/non-Gaussian Bayesian tracking. S Arulampalam, S Maskell, N Gordon, T Clapp, IEEE Trans. Signal Process. 502S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, "A tutorial on particle filters for on-line non-linear/non-Gaussian Bayesian tracking," IEEE Trans. Signal Process., vol. 50, no. 2, pp. 174-189, Feb. 2002.
Approximate non-Gaussian Bayesian estimation and modal consistency. N Gordon, A Smith, J. R. Stat. Soc. B. 554N. Gordon and A. Smith, "Approximate non-Gaussian Bayesian esti- mation and modal consistency," J. R. Stat. Soc. B., vol. 55, no. 4, pp. 913-918, Apr. 1993.
A t-distribution based particle filter for target tracking. S J Li, H Wang, T Y Chai, American Control Conference. S. J. Li, H. Wang, and T. Y. Chai, "A t-distribution based particle filter for target tracking," in American Control Conference, June 2006, pp. 2191-2196.
Student mixture filter for robust, realtime visual tracking. J Loxam, T Drummond, Proceedings of 10th European Conference on Computer Vision: Part III. 10th European Conference on Computer Vision: Part IIIJ. Loxam and T. Drummond, "Student mixture filter for robust, real- time visual tracking," in Proceedings of 10th European Conference on Computer Vision: Part III, 2008.
Non-linear Bayesian estimation using Gaussian sum approximation. D L Alspach, H Sorenson, IEEE Trans. Autom. Control. 174D. L. Alspach and H. Sorenson, "Non-linear Bayesian estimation using Gaussian sum approximation," IEEE Trans. Autom. Control, vol. 17, no. 4, pp. 439-448, Apr. 1972.
Discrete-time nonlinear filtering algorithms using Gauss-Hermite quadrature. I Arasaratnam, S Haykin, R Elliott, Proc. IEEE. IEEE95I. Arasaratnam, S. Haykin, and R. Elliott, "Discrete-time nonlinear filtering algorithms using Gauss-Hermite quadrature," Proc. IEEE, vol. 95, no. 5, pp. 953−977, May 2007.
MMSE-based filtering in presence of non-Gaussian system and measurement noise. I Bilik, J Tabrikian, IEEE Trans. Aero. Elec. Sys. 463I. Bilik and J. Tabrikian, "MMSE-based filtering in presence of non- Gaussian system and measurement noise," IEEE Trans. Aero. Elec. Sys., vol. 46, no. 3, pp. 1153-1170, July 2010.
Huber-based unscented fltering and its application to vision-based relative navigation. X G Wang, N G Cui, J F Guo, IET Radar, Sonar, and Navigation. 41X. G. Wang, N. G. Cui, and J. F. Guo, "Huber-based unscented fltering and its application to vision-based relative navigation," IET Radar, Sonar, and Navigation, vol. 4, no. 1, pp. 134-141, Jan. 2010.
Robust regression-based EKF for tracking underwater targets. F El-Hawary, Y Jing, IEEE J. Oceanic Eng. 201F. El-Hawary and Y. Jing, "Robust regression-based EKF for tracking underwater targets, IEEE J. Oceanic Eng., vol. 20, no. 1, pp. 31-41, Jan. 1995.
Huber-based divided difference filtering. C D Karlgaard, H Schaub, J. Guid., Control, Dyn. 303C. D. Karlgaard, and H. Schaub, "Huber-based divided difference filtering," J. Guid., Control, Dyn., vol. 30, no. 3, pp. 885-891, Apr. 2007.
Multiple outliers suppression derivative-free filter based on unscented transformation. L B Chang, B Q Hu, G B Chang, A Li, J. Guid., Control, Dyn. 356L. B. Chang, B. Q. Hu, G. B. Chang, and A. Li, "Multiple outliers suppression derivative-free filter based on unscented transformation," J. Guid., Control, Dyn., vol. 35, no. 6, pp. 1902-1906, June 2012.
Nonlinear regression HuberCKalman filtering and fixed-interval smoothing. C D Karlgaard, J. Guid., Control, Dyn. 382C. D. Karlgaard, Nonlinear regression HuberCKalman filtering and fixed-interval smoothing, J. Guid., Control, Dyn., vol. 38, no. 2, pp. 322-330, Feb. 2015.
Adaptively robust unscented Kalman filter for tracking a maneuvering vehicle. Y D Wang, S M Sun, L Li, J. Guid., Control, Dyn. 375Y. D. Wang, S. M. Sun, and L. Li, "Adaptively robust unscented Kalman filter for tracking a maneuvering vehicle," J. Guid., Control, Dyn., vol. 37, no. 5, pp. 1696-1701, May 2014.
Maximum correntropy estimation is a smoothed MAP estimation. B Chen, J C Principe, IEEE Signal Proc. Let. 198B. Chen and J. C. Principe, "Maximum correntropy estimation is a smoothed MAP estimation," IEEE Signal Proc. Let., vol. 19, no. 8, pp. 491-494, Aug. 2012.
Convergence of a fixed-point algorithm under maximum correntropy criterion. B Chen, J Wang, H Zhao, N Zheng, J C Principe, IEEE Signal Proc. Let. 2210B. Chen, J. Wang, H. Zhao, N. Zheng, and J. C. Principe, "Convergence of a fixed-point algorithm under maximum correntropy criterion," IEEE Signal Proc. Let., vol. 22, no. 10, pp. 1723-1727, Oct. 2015.
Robust information filter based on maximum correntropy criterion. Y D Wang, W Zheng, S M Sun, L Li, J. Guid., Control, Dyn. 395Y. D. Wang, W. Zheng, S. M. Sun, and L. Li, "Robust information filter based on maximum correntropy criterion," J. Guid., Control, Dyn., vol. 39, no. 5, pp. 1126-1131, May 2016.
Kalman filtering based on the maximum correntropy criterion in the presence of non-Gaussian noise. R Izanloo, S A Fakoorian, H S Yazdi, D Simon, Proceedings of 50th Annual Conference on Information Science and Systems. 50th Annual Conference on Information Science and SystemsR. Izanloo, S. A. Fakoorian, H. S. Yazdi, and D. Simon, "Kalman filtering based on the maximum correntropy criterion in the presence of non-Gaussian noise," in Proceedings of 50th Annual Conference on Information Science and Systems, 2016.
Robust Student's t based nonlinear filter and smoother. Y L Huang, Y G Zhang, N Li, J Chambers, IEEE Trans. Aero. Elec. Sys. 525Y. L. Huang, Y. G. Zhang, N. Li, and J. Chambers, "Robust Student's t based nonlinear filter and smoother," IEEE Trans. Aero. Elec. Sys., vol. 52, no. 5, pp. 2586-2596, Oct. 2016.
Sigma-point filtering for nonlinear systems with non-additive heavy-tailed noise. F Tronarp, R Hostettler, S Särkkä, 19th International Conference on Information Fusion (FUSION). F. Tronarp, R. Hostettler, and S. Särkkä, "Sigma-point filtering for nonlinear systems with non-additive heavy-tailed noise," in 19th Inter- national Conference on Information Fusion (FUSION), July 2016, pp. 1859-1866.
A robust Student's t based cubature filter. Y L Huang, Y G Zhang, N Li, S M Naqvi, J Chambers, 19th International Conference on Information Fusion (FUSION). Y. L. Huang, Y. G. Zhang, N. Li, S. M. Naqvi, and J. Chambers, "A robust Student's t based cubature filter," in 19th International Conference on Information Fusion (FUSION), July 2016, pp. 9-16.
Stochastic integration rules for infinite regions. A Genz, J Monahan, SIAM J. Sci. Comput. 192A. Genz and J. Monahan, "Stochastic integration rules for infinite regions," SIAM J. Sci. Comput., vol. 19, no. 2, pp. 426-439, Feb. 1998.
Approximate calculation of multiple integrals. A H Stroud, Prentice HallEnglewood Cliffs, NJA. H. Stroud, Approximate calculation of multiple integrals, Englewood Cliffs, NJ: Prentice Hall, 1971.
The efficient generation of random orthogonal matrices with an application to condition estimation. G W Stewart, SIAM J. Numer. Anal. 173G. W. Stewart, "The efficient generation of random orthogonal matrices with an application to condition estimation," SIAM J. Numer. Anal., vol. 17, no. 3, pp. 403-409, Mar. 1980.
| [] |
[
"Quantum Interference Transport in two-dimensional Semi-Dirac Semimetals",
"Quantum Interference Transport in two-dimensional Semi-Dirac Semimetals"
] | [
"Shihao Bi [email protected]†[email protected]‡[email protected]§[email protected]:2109.09109v2[cond-mat.mes-hall] ",
"Yiting Deng ",
"Yan He ",
"Peng Li ",
"\nCollege of Physics\nKey Laboratory of High Energy Density Physics and Technology of Ministry of Education\nSichuan University\n610064ChengduPeople's Republic of China\n",
"\nSichuan University\n610064ChengduPeople's Republic of China\n"
] | [
"College of Physics\nKey Laboratory of High Energy Density Physics and Technology of Ministry of Education\nSichuan University\n610064ChengduPeople's Republic of China",
"Sichuan University\n610064ChengduPeople's Republic of China"
] | [] | Semi-Dirac semimetals have received enthusiastic research both theoretically and experimentally in the recent years. Due to the anisotropic dispersion, its physical properties are highly direction-dependent. In this work we employ the Feynman diagrammatic perturbation theory to study the transport properties in quantum diffusive regime. The magneto-conductivity with quantum interference corrections is derived, which demonstrate the weak localization effect in the semi-Dirac semimetal. Furthermore, the origin of anomalous Hall conductivity is also clarified, where both the intrinsic and side-jump contributions vanish and only the skew-scattering gives rise to non-zero transverse conductivity. The conductance fluctuations in both mesoscopic and quantum diffusive regimes are investigated in detail. Our work provides theoretical predictions for transport experiments, which can be examined by conductivity measurements at sufficiently low temperature. | null | [
"https://arxiv.org/pdf/2109.09109v2.pdf"
] | 237,572,193 | 2109.09109 | 2fcbc4ba32daa67816ac6f133402d5c607fe623c |
Quantum Interference Transport in two-dimensional Semi-Dirac Semimetals
(Dated: March 8, 2022) 7 Mar 2022
Shihao Bi [email protected]†[email protected]‡[email protected]§[email protected]:2109.09109v2[cond-mat.mes-hall]
Yiting Deng
Yan He
Peng Li
College of Physics
Key Laboratory of High Energy Density Physics and Technology of Ministry of Education
Sichuan University
610064ChengduPeople's Republic of China
Sichuan University
610064ChengduPeople's Republic of China
Quantum Interference Transport in two-dimensional Semi-Dirac Semimetals
(Dated: March 8, 2022) 7 Mar 2022* Electronic address:
Semi-Dirac semimetals have received enthusiastic research both theoretically and experimentally in the recent years. Due to the anisotropic dispersion, its physical properties are highly direction-dependent. In this work we employ the Feynman diagrammatic perturbation theory to study the transport properties in quantum diffusive regime. The magneto-conductivity with quantum interference corrections is derived, which demonstrate the weak localization effect in the semi-Dirac semimetal. Furthermore, the origin of anomalous Hall conductivity is also clarified, where both the intrinsic and side-jump contributions vanish and only the skew-scattering gives rise to non-zero transverse conductivity. The conductance fluctuations in both mesoscopic and quantum diffusive regimes are investigated in detail. Our work provides theoretical predictions for transport experiments, which can be examined by conductivity measurements at sufficiently low temperature.
I. Introduction
Semi-Dirac semimetals are exotic phases of matter hosting quasi-particles with linear and quadratic dispersion relations in different directions, which gives rise to the coalesce of Dirac fermions and ordinary non-relativistic fermions. Such peculiar dispersion have caused wide research interests both theoretically and experimentally. A lot of candidate materials such as transition metal oxides multilayer nano-structures [1][2][3], deformed Graphene [4,5], black phosphorus under high pressure [6][7][8] and Silicene oxide [9], etc., have been proposed to realize this dispersion in the past few years. In the meantime, its physical properties have been intensively explored, such as Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [10,11] between doped magnetic moments, novel magnetic field dependence of Landau levels [4,5,12] obtained from Wentzel-Kramers-Brillouin (WKB) approximation, quantum tunneling behavior [13][14][15], transport properties [16][17][18][19][20][21], non-Fermi liquid emergent from long-range Coulomb interaction [22][23][24] and its interplay with various types of disorder [25] in the framework of 2+1 dimensional quantum electrodynamics (QED 3 ), Floquet dynamics [21,26] under the illumination of polarized off-resonant light, and so on. Among all the physical properties mentioned above, the transport measurements are most straightforward to perform. However, relatively less attention has been paid to the transport behaviors of doped semi-Dirac semimetals in the magnetic field.
In realistic materials, impurities may have significant influence on the physical properties and sometimes are of vital importance. For examples, the impurity effects play an important role in Kondo effect, high temperature superconductors, quantum Hall insulators, and Anderson localization, etc. When considering the transport properties, it is well-known that the impurity scattering provides the key mechanism of electron momentum relaxation and leads to finite electrical conductivities. At low temperature, when the mean free path e is much less than the system size and phase coherence length φ (quantum diffusive regime), electrons can maintain their phase coherence even after being scattered for many times. In this quantum diffusive regime, the quantum interference between time-reversed scattering loops can give rise to weak localization or anti-localization correction to the conductivity [27][28][29][30][31]. Moreover, impurity scattering can also generate the anomalous Hall conductivity [32][33][34][35] other than the intrinsic contribution from the occupied bands.
Motivated by the theoretical and experimental progress on the transport properties of topological materials in recent years [36][37][38][39][40][41][42][43], we present the study of transport properties of doped semi-Dirac semimetals with quantum interference correction. We will consider the most general form of the impurity scattering mechanism potential. Following the conventional procedure in quantum transport theory, the dominate contributions come from the quantum states near the Fermi surface. We calculate the anisotropic longitudinal conductivities by the linear response theory, in which the effects of quantum interference is considered by including the Cooperon contributions.
With finite magnetic field, the magneto-conductivities are obtained by summing over the Landau levels bounded by the mean free path e and phase coherence length φ . It is also found that the skew-scattering induced by impurity potential can result in non-zero extrinsic anomalous Hall conductivity. Based on the same formalism, we also investigate the conductance fluctuations of semi-Dirac semimetal in both mesoscopic and quantum diffusive regimes. This paper is organized as follows. The model Hamiltonian and impurity potential are given in Sec. II. Some notations and physical quantities, such as relaxation times and density of states (DOS) are defined for later convenience. In Sec. III we present the diagrammatic perturbation calculation of longitudinal conductivities, which demonstrates the weak localization effect. Next, to understand the suppression of the quantum interference correction by magnetic field, we present the magneto-conductivity formulas with semi-classical quantized Landau levels in Sec. IV.
After that we investigate the anomalous Hall conductivity from side-jump and skew-scattering mechanisms in Sec. V. In Sec. VI, we study the universal conductance fluctuation (UCF) of the semi-Dirac semimetals in mesoscopic regime. Finally we draw some conclusions of our main results and make some remarks on possible extensions to our work in Sec. VII. For convenience, we set e =h = 1 throughout the whole paper.
II. Model Hamiltonian
The Hamiltonian of two dimensional semi-Dirac semimetal is given by
H(k) = λ k x σ x + k 2 y σ y , (II.1)
Here σ x,y,z is the Pauli matrices acting on the pseudospin space, such as the orbital or sublattice degree of freedom, λ is the effective velocity in the x direction with the dimension of inverse length, and k = (k x , k y ) is the two dimensional wave vector. The anisotropic dispersion relation is relativistic in the x direction, and parabolic in the y direction. Its dispersion relation is ε(k) = λ 2 k 2 x + k 4 y . The Hamiltonian Eq. (II.1) violates the time-reversal symmetry but possesses the chiral symmetry, and thus belongs to the AIII (unitary) class [44].
The impurity potential we considered includes both elastic and pseudospin scattering:
U(r) = ∑ α U α (r) = N α ∑ i,α u i,α σ α δ (r − R i ) , (II.2)
where α runs over 0, x, y, and z. σ 0 is the 2 × 2 identity matrix. The potential U 0 (r) will cause elastic the scattering on the same pseudospin states, and U x,y (r) will cause the scattering between the different pseudospin states. U z (r) is for the energy splitting between the two pseudospin states.
We will assume the scattering of different mechanisms are uncorrelated. R i s are the locations of the N α randomly distributed impurities, and u i,α is the energy fluctuation at R i of α-type scattering mechanism with zero mean value and variance u 2 α . Therefore, the average over the impurity configuration of the potential is U(r) imp = 0, and the potential correlation is U(r)U(r ) imp = ∑ α n α u 2 α δ (r − r ). Here the impurity average is
f (r) imp = N ∏ i=1 dR i S f (r), (II.3)
S is the sample area and n α is the impurity concentration of α-type.
The Bloch wavefunction for the conduction band is
|u(k) = 1 √ 2 1 ζ e iφ , ζ = sgn k x , (II.4)
with tan φ = k 2 y /λ k x . The momentum can be parametrized as k x = ε cos φ /λ and k y = |ε sin φ | sgn sin φ near the Fermi surface. The sgn(x) is the sign function. Then we can define the velocity operator v α k = dε dk α in the x and y direction, respectively.
v x k =λ cos φ , v y k =2 |ε sin φ | sin φ . (II.5)
The density of states (DOS) near the Fermi surface is ρ(ε) = K(1/2) √ 2λ π 2 ε 1/2 , obtained after the integration of the angular variable dΩ = K(x) is the complete elliptic integral of the first kind, and K(1/2) ≈ 1.854. We see that the DOS has a square-root dependence of the Fermi energy, which is distinguished from the linear dependence for Graphene or constant for two-dimensional electron gas. Here and throughout the text we assume that the Fermi energy is positive and lies across the conduction band. Finally, the relaxation time near the Fermi surface can be obtained by the Fermi golden rule
Σ BA (k, iω l ) = k = ∑ k |U kk | 2 G k , iω l (a) V a k Λ = v a k + V a k Λ (b) Γ k α k β q − k β q − k α = k α k β q − k α q − k β + Γ k α k µ k β q − k α q − k µ q − k β (d) V a k V a q−k k q − k q − k k Λ Λ Γ V a k V a q−k k q − k q − k k Λ Λ Γ V a k V a q−k k q − k q − k k Λ Λ Γ (c) *τ −1 =2π ∑ k U kk 2 imp δ (ε k − ε F ) =2πρ(ε F ) dΩ ∑ α n α u 2 α A α,kk 2 =πρ(ε F ) ∑ α n α u 2 α (II.7)
Here U kk is the Born scattering amplitude between two momenta,
U kk = d 2 r S ∑ i,α u i,α A α,kk δ (r − R i ) e −i(k−k )·r (II.8)
and A α,kk = u(k)| σ α |u(k ) . We find that for α-type scattering the relaxation time τ −1 α = πρ(ε F )n α u 2 α , thus we have τ −1 = ∑ α τ −1 α . In the whole paper, we will set τ x = τ y as they both denote the relaxation time of scattering between different pseudospin states.
III. Diagrammatic Perturbation Approach to the Conductivity
To compute the electrical conductivity we employ the Feynman diagrammatic perturbation method based on the Matsubara Green's function. The necessary diagrams are listed in Fig. 1.
The Matsubara Green's function under the Born approximation of the disorder scattering has the form
G(k, iω m ) = 1 iω m − ε k + i/(2τ ) sgn ω m (III.1)
where ω m = (2m + 1) π/β is the fermionic Matsubara frequency, and β is the inverse temperature.
After the calculation we will make analytic continuation iν n → ν + 0 + to obtain the zero temperature conductivity.
A. Drude-Boltzmann Conductivity
The Drude conductivity is given by the simple bubble diagram with bare velocity vertex as
follows σ µ µ (q, iν n ) = 1 β ν n ∑ m d 2 k (2π) 2 v µ k G(k, iω m )v µ k+q G(k + q, iω m + iν n ) (III.2)
here q is the momentum transferred, ν n is the bosonic Matsubara frequency and v µ k is the velocity operator defined in last section. In the long wavelength limit q → 0, the zero temperature Drude conductivities contributed from the electrons near the Fermi surface are
σ xx (0) = 2 3 ρ(ε F )λ 2 τ , σ yy (0) = 6π 5K(1/2) 2 ρ(ε F ) |ε F | τ (III.3)
According to the Einstein relation σ µ µ = D µ ρ(ε F ), we find that the bare diffusion constants are
D x = 2λ 2 τ /3, D y = 6π 5K(1/2) 2 |ε F | τ (III.4)
In order to find results consistent with Boltzmann equation, one has to take into account the ladder diagram shown in Fig. 1 (b), then the velocity operator is re-normalized as
V µ k = v µ k + ∑ k U kk 2 imp G(k, iω m )V µ k G(k + q, iω m + iν n ).
(III.5)
If we assume that the trial solution is V µ k = η µ v µ k , then we can determine that the vertex correction coefficient η µ at zero temperature is
1 − η −1 µ = 2πρ(ε F )τ dΩdΩ v µ k v µ k ∑ α n α u 2 α A α,kk 2 dΩv µ k v µ k . (III.6)
After performing the integral, we find that the vertex correction coefficients are
η −1 x =1 − 2 3 τ τ 0 − τ τ z , η −1 y =1 − 5π 48 τ τ 0 − τ τ z .
(III.7)
And the Boltzmann conductivity can be obtained with one dressed velocity operator inserted in the bubble diagram as
σ µ µ (q, iν n ) = 1 β ν n ∑ m d 2 k (2π) 2 V µ k G(k, iω m )v µ k+q G(k + q, iω m + iν n ) (III.8)
Then it is easy to see that the conductivities are given by
σ xx (0) = 2 3 ρ(ε F )λ 2 τ η x , σ yy (0) = 6π 5K(1/2) 2 ρ(ε F ) |ε F | τ η y (III.9)
The corresponding corrected diffusion constants are D µ = η µ D µ . It is known that the semiclassical conductivity will not response to the weak magnetic field. To reveal the influence of magnetic field on the conductivity, we should consider the quantum interference between the timereversal paths, which correspond to the maximally crossed diagrams in Fig. 1 (c,d).
B. Quantum Interference Correction
The quantum interference correction is given by the maximally crossed diagram with one bare and two dressed Hikami boxes. The conductivity with one bare Hikami box contribution is
σ 0 µ µ (q, iν n ) = 1 β ν n ∑ m d 2 k (2π) 2 ∑ q V µ k V µ q−k G (k, iω m ) G (q − k, iω m ) × Γ(q, iν n )G (q − k, iω m + iν n ) G (k, iω m + iν n ) (III.10)
At T = 0, one can arrive at the following result
σ 0 µ µ (0) = − 2D µ ρ(ε F )τ 2 η 2 x ∑ q Γ(q) (III.11)
with Cooperon vertex function Γ(q) to be determined later.
The two dressed Hikami boxes can be written as follows
σ R µ µ (q, iν n ) = 1 β ν n ∑ m d 2 k (2π) 2 d 2 k (2π) 2 ∑ q U k k U q−k q−k imp V µ k V µ q−k G (k, iω m ) G k , iω m × G (q − k, iω m ) G q − k , iω m Γ(q, iν n )G q − k , iω m + iν n G (k, iω m + iν n ) σ A µ µ (q, iν n ) = 1 β ν n ∑ m d 2 k (2π) 2 d 2 k (2π) 2 ∑ q U k k U q−k q−k imp V µ k V µ q−k G (k, iω m ) G q − k , iω m × Γ(q, iν n )G (k, iω m + iν n ) G k , iω m + iν n G (q − k, iω m + iν n ) G q − k , iω m + iν n (III.12)
Despite the the above complicated expressions, we find that the two dressed Hikami boxes contribute equally and are proportional to the bare Hikami box contributions. The proportional factors can be computed to give relatively simple results as follows
ξ x = σ R/A xx (0) σ 0 xx (0) = πρ(ε F )τ dΩdΩ v x k v x −k U k k U −k −k imp dΩv x k v x k = − 1 3 τ τ 0 − τ τ z − 2 τ τ x ξ y = σ R/A yy (0) σ 0 yy (0) = πρ(ε F )τ dΩdΩ v y k v y −k U k k U −k −k imp dΩv y k v y k = − 5π 96 τ τ 0 + τ τ z (III.13)
Put all the above results together, we find that the quantum interference correction to the conductivity is
σ qi µ µ = −2D µ ρ(ε F )τ η 2 µ 1 + 2ξ µ ∑ q Γ(q) (III.14)
In order to find the final results of conductivity, we will compute the Cooperon vertex function Γ(q) in the next sub-section.
C. Bethe-Salpeter Equation for the Cooperon
We start from the bare Cooperon vertex function which is given by
Γ 0 kk = U k k U −k −k imp = 1 2πρτ S τ τ 0 − τ τ z + τ τ 0 + τ τ z cos φ − φ − 2 τ τ x cos φ + φ (III.15)
It is more convenient to rewrite Γ 0 as the following form
Γ 0 kk = 1 2πρτ S ∑ ab z ab e i(aφ −bφ ) (III.16)
where a, b ∈ {−1, 0, 1}. The non-zero components of coefficients z ab are
z 00 = τ τ 0 − τ τ z z 11 = z 11 = 1 2 τ τ 0 + τ τ z z 11 = z 11 = − τ τ x (III.17)
Here we denote 1 = −1. Inspired by this, we make an ansatz of the full Cooperon vertex function with the same function form
Γ kk = 1 2πρτ S ∑ ab Z ab e i(aφ −bφ ) (III.18)
Here the coefficients Z ab are yet to be determined.
The Bethe-Salpeter equation for the Cooperon vertex function is
Γ k α k β =Γ 0 k α k β + ∑ k µ Γ 0 k α k µ G k µ , iω m G q − k µ , iω m + iν n Γ k µ k β (III.19)
Plug in Eq.(III.16) and (III.18) and performing the of the momentum summation of the electron propagator, we find that the Bethe-Salpeter equation transfers to an equation Z ab
Z ab =z ab + ∑ cd z ac dΩ µ 1 + iv µ · qτ e −i(c−d)φ µ Z db (III.20)
Here
v µ = ∂ ε k ∂ k µ = λ cos φ µ , 2 ε sin φ µ sin φ µ and v µ · q = λ q x cos φ µ +2q y ε sin φ µ sin φ µ .
We can further simplify the equation by expanding the fraction in the integrand to the q 2 term and
define Φ cd = dΩ µ 1 − iv µ · qτ − v µ · qτ 2 e −i(c−d)φ µ (III.21)
Then the Bethe-Salpeter equation reduces to a simple matrix equation Z = z + zΦ Φ ΦZ, and its solution is
Z = (1 − zΦ Φ Φ) −1 z (III.22)
We will only keep the most divergent term which is given as
Z 00 ≈ 1 1 z 00 − 1 + g x D x q 2 x τ + g y D y q 2 y τ g x = 1 + 2 z 11 z 00 + 10 7 z 11 z 00 , g y = 1 + 2 z 11 z 00 − 10 9 z 11 z 00 (III.23)
Here D x,y are bare diffusion constants in Eq.(III.4).
Make use the above result, we find that the summation of the Cooperon vertex function is
∑ q Γ(q) = 1 8π 2 ρ(ε F )τ 2 g x g y D x D y ln −2 0 + −2 e −2 0 + −2 φ (III.24)
Here we define a combined length parameter −2 0 = 1 z 00 − 1 (g x D x τ ) −1 , and the momentum summation is bounded by the mean free path e and the phase coherence length φ .
Substitute the Cooperon vertex function into Eq.(III.14), we find the conductivity with the quantum interference correction as
σ qi µ µ = − e 2 h D µ η 2 µ 1 + 2ξ µ 4π 2 g x g y D x D y ln −2 0 + −2 e −2 0 + −2 φ (III.25)
which displays the weak localization effects. In the above expression, we have restored the unit of universal conductance e 2 /h. When the magnetic field is applied, the quantum interference effect will be suppressed, and the conductivity is expected to increase. We will turn to the effects of magnetic field in the next section.
IV. Magneto-conductivity
Applying a magnetic field along the z direction, we expect the motion on the xy plane will become quantized as the ordinary free fermions. The vector potential in the Landau gauge is A = (0, Bx, 0). By making the Peierls substitution, the Hamiltonian Eq. (II.1) becomes
H(k + eA) = λ k x σ x + (k y + eBx) 2 σ y . (IV.1)
The eigenvalue equation for the two-components wavefunction ψ is Hψ = Eψ, which is also equivalent to a second-order differential equation as [4,5] −
λ 2 ∂ 2 x + iλ e 2 B 2 −i∂ x , x 2 σ z + e 4 B 4 x 4 ψ = E 2 ψ. (IV.2)
Clearly, the x 4 term will dominate for large x, thus we can drop the commutator to simplify the above equation. Then we can find the quantized eigenvalues by the WKB quantization condition
pdx = n + 1 2 π. (IV.3)
After finish the integral, we find that
E n = γ λ 2 B 2/3 n + 1 2 2/3 , γ = 3π √ 2π Γ (1/4) 2 2/3 ≈ 1.478, (IV.4)
Here the magnetic length is B = h/eB. Because of this Landau level quantization, in the Cooperon vertex function, the momentum q is then quantized into q n = 2γ λ 4
B −1/3 n + 1 2 2/3 .
To obtain the conductivity correction formula in a finite magnetic field B, we insert a Dirac δ function in the Cooperon vertex function Γ(q) to restrict the momentum to quantized values:
∑ q Γ(q) ⇒ ∑ q Γ(q) × ∑ n δ n + 1 2 − (2γ) −3/2 √ λ 2 B q 3/2 (IV.5)
Since the momentum summation is bounded as −1 e < q < −1 φ , correspondingly, the lower and upper bound of the Landau Levels, n L and n U , cut-off by the mean free path e and the phase coherence length φ .
n L = χ −3/2 φ − 1 2 , n U = χ −3/2 e − 1 2 (IV.6)
Here we define the abbreviation χ = (2γ) −3/2 √ λ 2 B . To carry out the Landau level summation, we first consider a simpler case with −2 0 = 0. In this case, Eq. (IV.5) simply reduce to a harmonic series and we find
σ qi µ µ (B, −2 0 = 0) = − D µ η 2 µ 1 + 2ξ µ 3π 2 g x g y D x D y ψ χ −3/2 e + 1 2 − ψ χ −3/2 φ + 1 2 (IV.7)
If the magnet field B is very small, then the magnetic length B → +∞. By making use of the asymptotic behavior of digamma function ψ x + 1 2
≈ ln x, one finds
ψ χ −3/2 e + 1 2 − ψ χ −3/2 φ + 1 2 = 3 4 ln −2 e −2 φ . (IV.8)
which reproduce Eq.(III.25) with −2 0 = 0. For non-zero −2 0 , we simply make the replacement −2 e/φ → −2 e/φ + −2 0 in Eq. (IV.7) and obtain
σ qi µ µ (B) = − D µ η 2 µ 1 + 2ξ µ 3π 2 g x g y D x D y ψ χ −2 e + −2 0 3/4 + 1 2 − ψ χ −2 φ + −2 0 3/4 + 1 2 (IV.9)
The magneto-conductivity is the change of conductivity induced by the applied magnetic field, which is defined as ∆σ (B) = σ where the U x,y,z (r) impurity potentials are absent, which means τ = τ 0 and 1/τ x,y,z = 0. According z 00 = τ /τ 0 −τ /τ z , we find z 00 ≈ 1 in this case. Since −2 0 ∝ (1/z 00 −1), we find that 0 become very large such as 0 = 1000nm. In this case, the Cooperon contribution is quite large as can be seen from the figure. Since the magnetic field breaks the weak localization, the magneto-conductivity rapidly increases with the increasing magnetic field. One can see magneto-conductivity quickly Fig. 2, we can also see that the magneto-conductivity almost stays at zero until B ≈ 0.5 T.
All the above results are obtained for T = 0. Now we briefly discuss some possible temperature dependence. In the quantum diffusive regime with small −2 0 , the weak localization correction of conductivities is proportional to ln φ / e , where the phase coherence length φ has a temperature dependence of T −p/2 with parameter p depending on the decoherence mechanisms.
Electron-electron (p = 1) and electron-phonon (p = 3) interaction are two dominant decoherence mechanisms in two-dimensional disordered metals [45,46]. At sufficiently low temperature, the phonon excitations are suppressed and the electron-electron interaction dominates the decoherence effect. Therefore, the temperature dependence of conductivities is mainly generated by φ ∝ T −1/2 , which can be verified in magneto-conductivity measurements.
V. Anomalous Hall Conductivity
After the investigation of the longitudinal conductivity, we now turn to the study of the transverse component, which is usually considered under the name of anomalous Hall effect. The anomalous Hall effect originates from the interplay between the spin-orbit coupling and timereversal symmetry breaking [33], and can be classified into intrinsic and extrinsic contributions.
The former comes from the Berry curvature of occupied bands below the Fermi surface, and the latter is the consequence of impurity scattering near the Fermi surface. The weak localization behavior reveals that the Berry phase of the bulk states are zero, and thus the intrinsic anomalous Hall conductivity is always zero. This has also be verified in our numerical simulation, where a lattice version of Hamiltonian Eq.(II.1) can be obtained by naive replacements k x → sin k x and k 2 y → 2 (1 − cos k y ). However, the doped impurities may result in non-zero anomalous Hall conductivity.
The calculation of extrinsic anomalous Hall conductivity is attributed to five representative
Feynman diagrams [32][33][34][35] Therefore, we only need to calculate the contributions of these 5 representative diagrams and take the real part of them.
For later convenience, we slightly extend our previous notations to include both occupied and un-occupied bands. The energy eigenvalues are ε s (k) = s λ 2 k 2
x + k 4 y , with s = + for conduction band and s = − for valence band, respectively. And the corresponding Bloch wavefunctions are
|ks = 1 √ 2 s ξ e iφ . (V.1)
The Born scattering amplitude is
U ss kk = d 2 r S ks|U(r) k s e −i(k−k )r , (V.2)
Then our previous defined amplitude is just one component as U kk = U ++ kk . The bare velocity operator is also extended to be v µss k = ks| ∂ H ∂ k µ |ks , and likewise, the previous velocity is also one
component v µ k = v µ++ k
. Finally, the bare Green's function in Eq. (III.1) is also extended to include two components as
G s (k, iω m ) = 1 iω m − ε sk + i/(2τ ) sgn ω m (V.3) 2 I. Anomalous Hall Conductivity V x++ k v y−+ k +k +k U ++ kk +k −k U +− k k Λ V x++ k v y+− k +k +k U ++ kk +k U −+ k k −k Λ V x++ k V y++ k +k +k +k U ++ kk −k U −+ k k +k U ++ k k +k U +− k k Λ Λ V x++ k V y++ k +k +k −k U +− kk U ++ k k +k U ++ k k +k +k U ++ k k Λ Λ V x++ k V y++ k +k +k U ++ kk +k U ++ k k +k +k U ++ k k Λ Λ
A. Side-jump Mechanism
The side-jump contribution consists of diagrams in Fig. 3
σ sj1 xy = 1 β ν n ∑ m d 2 k (2π) 2 d 2 k (2π) 2 V x++ k G + (k, iω m ) U ++ kk U +− k k imp × G + (k , iω m )G − (k, iω m )v y−+ k G + (k, iω m + iν n ) σ sj2 xy = 1 β ν n ∑ m d 2 k (2π) 2 d 2 k (2π) 2 V x++ k G + (k, iω m ) U ++ kk U −+ k k imp × G + (k , iω m )v y+− k G − (k , iω m + iν n )G + (k, iω m + iν n ) σ sj3 xy = 1 β ν n ∑ m d 2 k (2π) 2 d 2 k (2π) 2 d 2 k (2π) 2 V x++ k G + (k, iω m ) U ++ kk U −+ k k imp G + (k , iω m ) U ++ k k U +− k k imp × G + (k , iω m )V y++ k G + (k , iω m + iν n )G − (k , iω m + iν n )G + (k, iω m + iν n ) σ sj4 xy = 1 β ν n ∑ m d 2 k (2π) 2 d 2 k (2π) 2 d 2 k (2π) 2 V x++ k G + (k, iω m ) U +− kk U ++ k k imp G − (k , iω m ) U −+ k k U ++ k k imp × G + (k , iω m )G + (k , iω m )V y++ k G + (k , iω m + iν n )G + (k, iω m + iν n ) (V.4)
And after some integration we come to the fact that the contribution of side-jump mechanism is zero. The reason for vanishing side-jump contribution is because the energy band is gapless.
B. Skew-scattering Mechanism
In the skew-scattering mechanism, the Hall conductivity comes from two diagrams of third order correction. One diagram is shown in the bottom panel of Fig. 3 with inverse "Y"-type disorder correlation lines, and the other diagram is complex conjugate of previous one. Combining these two diagrams, the Hall conductivity from skew-scattering is
σ sk xy = 2 Re 1 β ν n ∑ m d 2 k (2π) 2 d 2 k (2π) 2 d 2 k (2π) 2 V x++ k G + (k, iω m ) U ++ kk U ++ k k U ++ k k imp × G + (k , iω m )V y++ k G + (k , iω m + iν n )G + (k , iω m + iν n )G + (k, iω m + iν n ) (V.5)
After the momentum summation, we find a non-zero result for the Hall conductivity as
σ sk xy = − π 3 6 η x η y 5 2π D x D y n z u 3 z ρ(ε F ) 3 τ (V.6)
Since the intrinsic and side-jump part are zero, the above result is the only contribution to the transverse conductivity.
VI. Universal Conductance Fluctuation
Universal conductance fluctuation (UCF) is another famous quantum interference phenomenon due to disorder that goes along with weak localization/anti-localization in mesoscopic physics [45,[47][48][49]. The fluctuation of conductance, or here in 2D, the conductivity shows certain statistical distribution from sample to sample, yet its root mean square is unrelated to impurity configuration and only slightly rely on the shape and dimension of the sample. The weak localization/antilocalization correction is of magnitude e 2 /h and so is the UCF. Such an effect is sensitive to temperature. The temperature will affects not only the Fermi level but also the phase coherence length φ , and excites the thermal diffusive motion of electrons, characterized by T = hβ D µ .
Hence we present the investigation of the UCF of semi-Dirac semimetal at zero temperature. The conductivity correlation function is
F µ (ε − ε ) = δ σ µ µ (ε)δ σ µ µ (ε ) imp . (VI.1)
Here δ σ = σ − σ . If one generalize the classical Einstein's relation to a random variable relation as σ = Dρ, then the conductivity correlation can be separated into two parts
δ σ (ε)δ σ (ε ) = σ 2 δ ρ(ε)δ ρ(ε ) ρ 2 + δ D(ε)δ D(ε ) D 2 . (VI.2)
We can interpret the fluctuation as the joint contribution from the fluctuations of DOS and diffusion constants, assuming the two fluctuations are uncorrelated. The explicit correspondence is shown in Fig. 4.
The diagrams in Fig. 4 (a-d) can be calculated as the product of the building blocks such as the Hikami boxes and the diffuson/Cooperon vertex function. Let us first collect the expressions for all these building blocks as follows. The bare diffuson is
Λ 0 kk = U kk U k k imp = 1 2πρτ S 1 + τ τ 0 − τ τ z cos φ − φ (VI.3)
and the Bethe-Salpeter equation for the diffuson is
Λ k α k β = Λ 0 k α k β + ∑ k µ Λ 0 k α k µ G k µ , iω m G k µ + q, iω m + iν n Λ k µ k β (VI.4)
Through a similar method we have used in solving the full Cooperon vertex function, we obtain the full diffuson vertex function as Similarly, the full Cooperon vertex function is
Λ(q, iν n ) = 1 2πρτ 2 S 1 ν n + τ τ 0 − τ τ z D x q 2 x + D y q 2 y (VI.5) 3 III. Universal Conductance Fluctuation (a) V a k V a k V a k V a k Λ(q) Λ(q) (b) V a k V a k V a k V a k Γ(q) Γ(q) (c) V a k V a k V a k V a k Λ(q) Λ(q) (d) V a k V a k V a k V a k Γ(q) Γ(q) H1 = H 0 1 + H R 1 + H A 1 H2 = H 0 2 + H R 2 + H A 2 (f) (e)Γ(q, iν n ) = 1 2πρτ 2 S 1 ν n + 1 z 00 − 1 τ −1 + g x D x q 2 x + g y D y q 2 y (VI.6)
For later convenience, we drop the prefactor of the above diffuson/Cooperon vertex and define the diffuson/Cooperon kernel as
P D (q, ν) = 1 −iν + η D x q 2 x + D y q 2 y P C (q, ν) = 1 −iν + Ω 0 + g x D x q 2 x + g y D y q 2 y (VI.7)
Here we introduced the Cooperon gap Ω 0 = z −1 00 − 1 τ −1 and also η = z 00 . We also have made the analytic continuation as iν n → ν in above expressions.
Then we consider the two types of Hikami boxes in Fig. 4 (e-f) cancel out because the additional disorder correlation lines make the two incoming momenta uncorrelated. Thus they vanish after the angular average of the incoming momenta. This is also true for H A 2 . Therefore, we find the summation of the first type of Hikami as
H 1 = H 0 1 = 1 β ν n ∑ m d 2 k (2π) 2 V µ k 2 G (k, iω m ) 2 G (k, iω m + iν n ) 2 = 4πρη 2 µ D µ τ −1 (ν n + τ −1 ) 3 (VI.8)
In the zero-temperature limit we arrived at H 1 = 4πρη 2 µ D µ τ 2 . Similarly, for H 2 we have H 0 2 = H 0 1 , H A 2 = 0 and the disorder dressed one is H R 2 = ξ µ H 0 2 . Combining all the three terms, we find H 2 = 4πρη 2 µ 1 + ξ µ D µ τ 2 . Collect all the above results, the fluctuations of diffusion constants corresponding to the sum of diagrams of Fig. 4 (a-b) is
δ σ µ µ (ε)δ σ µ µ (ε ) (1) = (H 1 ) 2 ∑ q Λ(q, ε − ε ) 2 + Γ(q, ε − ε ) 2 =4η 2 µ D 2 µ /S 2 ∑ q P D (q, ε − ε ) 2 + P C (q, ε − ε ) 2 (VI.9)
Similarly, the fluctuations of density of states corresponding to the sum of diagrams of Fig. 4 (c-d) together with their complex conjugate gives δ σ µ µ (ε)δ σ µ µ (ε )
(2) =2 (H 2 ) 2 ∑ q Re Λ(q, ε − ε ) 2 + Γ(q, ε − ε ) 2 =8η 2 µ 1 + ξ µ 2 D 2 µ /S 2 ∑ q Re P D (q, ε − ε ) 2 + P C (q, ε − ε ) 2 (VI.10)
Plug the above two results into Eq.(VI.2), we will arrive at the final results of the conductance fluctuation. Here we will apply this general formalism to two different regimes as follows
A. Universal Conductance Fluctuation in mesoscopic regime
In the mesoscopic regime, φ > L x,y where L x,y is the sample size along x, y direction. The momentum is quantized as q µ = πn µ /L µ . Applying this momentum quantization to Eq.(VI.2), we find the UCF as Other than the parameter shown in the figure, we also used η x = 3 and ξ x = −1/3 which corresponds to the case with only U 0 (r) impurity potential. The grey dashed line shows the UCF of 2D electron gas.
δ σ 2 µ µ = 4η 2 µ D 2 µ 1 + 2 1 + ξ µ 2 π 4 D x D y ∑ n x ,n y 1 η 2 n 2 x R 1 R −1/2 2 + n 2 y R −1 1 R 1/2 2 2 + 1 Ω 0 S π 2 √ D x D y + g x n 2 x R 1 R −1/2 2 + g y n 2 y R −1 1 R 1/2 2 2 , (VI.11) ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇◇ ◇◇◇ ◇◇◇ ◇◇◇◇ ◇◇◇◇ ◇◇◇◇◇ ◇◇◇◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇◇ ◇◇◇ ◇◇◇ ◇◇◇ ◇◇◇◇ ◇◇◇◇ ◇◇◇◇◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇ ◇
Here we introduced the geometrical ratio R 1 = L y /L x and the anisotropic diffusion ratio R 2 = D y /D x ∼ λ 2 /ε F . In the above formula, we only consider the case of ε = ε = ε F for simplicity.
The general case with non-zero Fermi energy difference can be achieved by applying a gate voltage in experiment. The prime above ∑ n x ,n y means that the n x = 0 term is excluded, otherwise the denominator will vanish if one also have n y = 0 at the same time. The reason for this exclusion is because the conductance is always measured by attaching leads to the sample. Thus, we can, for example, assume that the leads are attached in the x direction, corresponding to absorbing walls or the Dirichlet boundary condition where the n x = 0 mode is removed. On the other hand, in the y direction, we have hard walls or the Neumann boundary conditions where the n y = 0 mode is included. Note that the first term inside the bracket of Eq. (VI.11) comes from diffuson contribution and the other term is from Cooperon. which does not depend on R 1,2 . However, the UCF of Eq. (VI.11) shows a dependence on the sample shape, impurity scattering type, and intrinsic properties such as λ 2 /ε F . In Fig. 5, we plot the conductance fluctuation of Eq. (VI.11) as a function of R 1 for several different R 2 . One can see the fluctuation approaches to the value of 2D electron gas as R 1 increasing. We would like to mention that the contribution from the Cooperon will be suppressed if a strong magnetic field is applied. In this case, only the contribution of the diffuson remains.
B. Conductance Fluctuation in quantum diffusive regime
In the opposite quantum diffusive regime, we have φ L. In this case, one can treat the momentum as continuous variables and carry out the momentum integration to find the following result of conductance fluctuation
δ σ 2 µ µ = η 2 µ D 2 µ 1 + 2 1 + ξ µ 2 πg x g y D x D y S g x g y 2 φ − 2 e η 2 + 1 −2 φ + −2 0 − 1 −2 e + −2 0 (VI.13)
Since now the conductance fluctuation depends on the φ and e , etc, it is not universal any more.
This result can be compared with the conductivity of Eq.(III.14) which is also valid in the quantum diffusive regime.
Now we can consider the influence of magnetic field on the conductance fluctuation. Following the similar steps in the computation of magneto-conductivity, we can replace the momentum integration in the above formula by a summation of Landau level. Note that since the diffuson vertex Λ kk depend on the difference of the two momentum, the vector potential of magnetic field will cancel out if one make a Peierls substitution. Because of this, only the Cooperon vertex will depend on the vector potential and requires to take account the Landau level summation. With all above considerations, we can find the conductance fluctuation under the magnetic field as To be specific, we apply the above general formalism to a square shaped sample with the critical size L = φ . In Fig. 6, the difference ∆ δ σ 2 (B) without the prefactor is plotted as a function of B for two limiting cases with 0 = 1000 nm and 0 = 8 nm, which are the same as in Fig. 2 To understand the behavior of this difference, we first note that the diffuson does not depend on the magnetic field, thus its contribution cancels out. We expect that ∆ δ σ 2 (B) only shows the difference of Cooperon contribution with and without magnetic field. In panel (a) with large 0 , the Cooperon effect is large in the absence of magnetic field, which generates a large positive conductance fluctuation. This fluctuation is suppressed by the increasing magnetic field. Therefore, in panel (a), we see that ∆ δ σ 2 (B) is large and negative. On the other hand, in panel (b) with small 0 , the Cooperon effect is small even without magnetic field. Because of this, one can see that the overall scale of panel (b) is two order of magnitudes smaller than panel (a).
δ σ 2 µ µ (B) = η 2 µ D 2 µ 1 + 2 1 + ξ µ 2 πg x g y D x D y S g x g y 2 φ − 2 e η 2 + 4 3 χ 4 3 ζ 7 3 χ −2 φ + −2 0 3 4 + 1 2 − ζ7 3 χ −2 φ + −2
VII. Conclusion and Remarks
In this work we studied the transport properties of the so-called semi-Dirac semimetals doped with various types of impurities. We use the diagrammatic perturbation method to obtain the zerotemperature electrical conductivities in both x and y direction. More specifically, we systematically explored the weak localization phenomenon and calculated the magneto-conductivity under a finite external magnetic field. Furthermore, the anomalous Hall conductivity generated by the sidejump mechanism and skew-scattering mechanism is investigated. It is found that the side-jump contribution is zero due to the vanishing energy gap. On the other hand, the third order perturbation of impurity potential U z (r) give rise to a non-zero skew-scattering contribution. At last, in the mesoscopic regime, we found that the UCF of semi-Dirac semimetal depend on the sample shape, impurity scattering type and some other intrinsic properties. In the quantum diffusive regime, the conductance fluctuation is suppressed by applying magnetic field.
We briefly discuss some possible extensions to our theory by taking other effects into consideration. We completely ignores the electron interaction in our discussion. It is known that the electron-electron interaction self energy Σ ee (k, iω m ) can modify the DOS at the Fermi surface and induce additional conductivity correction. This is known as the Altshuler-Aronov effect [50].
In the first order approximation of Dyson equation for full Green's function, the DOS shift is given by: δ ρ(ε F ) = − 1 π Im ∑ k G(k, iω m ) 2 Σ ee (k, iω m ) iω m →ε F +i0 + , (VII.1) and the resultant conductivity correction from the electron-electron interaction is found to be σ ee = σ sc δ ρ(ε F ) ρ(ε F ) , where σ sc is the semi-classical Boltzmann conductivity aforementioned.
At last, we would like to emphasize that in the Sec. IV, the linear part of effective potential in Eq. (IV.2) coming from the commutator is left out, which means that the original asymmetric effective potential is approximated by a symmetric quartic potential. Because of this, the difference between the two components of eigen-wavefunction is eliminated. The validity of such an approximation is verified by numerical calculations and is proved to be very successful in [4].
However, the eigen-function of the lowest Landau level is fully polarized and only the lower component is non-zero. Therefore, in the ultra-quantum limit where only the lowest Landau level is partially filled (which can be achieved when the magnetic field is very strong), such an approximation may be invalid and needs further consideration. In this case, the magnetoconductivity may have a different dependence on the external magnetic field. All the possibilities discussed above are remained to be investigated in the future.
For the convenience of readers, we summarize the notations used throughout this paper and compare some quantities in 2DEG, 2D S-DSM, and 2D massless Dirac fermion in the Appendix Tabs. I and II.
Length parameter Λ(q, iν n ) diffuson vertex function Λ(q, iν n ) = 1 2πρτ 2 S 1 ν n + η D x q 2
x + D y q 2 y Γ(q, iν n ) Cooperon vertex function Γ(q, iν n ) = 1 2πρτ 2 S 1 ν n + Ω 0 + g x D x q 2
x + g y D y q 2 y P D (q, ν) diffuson kernel P D (q, ν) = 1 −iν + η D x q 2 x + D y q 2 y P C (q, ν) Cooperon kernel P C (q, ν) = 1 −iν + Ω 0 + g x D x q 2 x + g y D y q 2 y Tab. I: Summary of notations.
2DEG
S-DSM 2D massless Dirac fermion
ε(k) k 2 2m λ 2 k 2 x + k 4 y λ k ρ(ε) m 2π K(1/2) √ 2λ π 2 ε 1/2 ε 2πλ 2 D x,y 1 2 v 2 F τ D x = 2 3 λ 2 τ 1 2 λ 2 τ D y = 6π 5K(1/2) 2 |ε F | τ η x,y 1 η x = 3 2 η y = 1 −
(1/2) 2 |sin φ | .(II.6)
Electronic address: [email protected]
Fig. 1 :
1Feynman diagrams for the conductivity calculation. (a) Born approximation for the disorderaveraged self energy. The cross is the impurity and the dashed lines are the impurity potential. (b) Vertex correction to the velocity at the Boltzmann level. (c) Quantum conductivity correction from the bare and two dressed Hikami boxes. (d) The Bethe-Salpeter equation for the Cooperon diagram.
To find a qualitative picture of above results, we drop the prefactor before the large brackets of Eq. (IV.9), and choose some parameters of interest to show the magnetic field dependence of magneto-conductivity inFig. 2. The effective velocity in the x-direction is proportional to the inverse of lattice constant a, and we take a = 0.246 nm as in the Graphene. The mean free path is set as 10 nm. The phase coherence length is of order 100 nm typically, and can be tuned by the temperature in the experiment.
Fig. 2 :
2Magneto-conductivity versus magnetic field strength. In panel (a), the adjustable 0 = 1000nm and the phase coherence length φ =10, 20, 50, 100, 200, 1000nm from bottom to top. In panel (b), the adjustable 0 = 8nm and the phase coherence length φ =10, 12, 20, 50, 200nm from bottom to top. For both panles, the mean free path e = 10nm. The effective velocity λ is the inverse of the lattice constant a −1 with a = 0.246 nm as in Graphene. The magnetic length taken is to be B =25.66 nm/ √ B, with B in Tesla. In Fig. 2, the magneto-conductivity is displayed in the two limiting cases where the adjustable length parameter 0 = 1000nm in panel (a) and 0 = 8nm in panel (b). In both panels, we take a series of values for φ and plot ∆σ as a function of B. The panel (a) corresponds to the case
become saturated to some constant values at about B = 0.5T. The panel (b) corresponds to the other limiting case where τ z is close to τ 0 . It is easy to see that z 00 ≈ 0 in this case. Thus −2 0 becomes divergent or 0 is very small such as 0 = 8nm. Form Eq.(III.25), one can see that the conductivity with quantum interference is proportional become very small for divergent −2 0 . Form this, it is clear that the Cooperon correction is suppressed in this case. It can be see from the figure since the overall scale of panel (b) is much smaller than panel (a). In panel (b) of
in Fig. 3. One can exchange the position of impurity scattering correlation lines and the velocity operators to obtain the full 16 possible diagrams. However, those diagrams obtained by exchanging the lines are usually complex conjugate of the original diagrams.
Fig. 3 :
3Feynman diagrams for the extrinsic Hall conductivity calculation. Here we list five representative diagrams. Top four: Side-jump contribution. Bottom: Skew-scattering contribution.
Fig. 4 :
4Feynman diagrams (a-d) for the conductivity correlation function. The blue and red stripes denote the diffusons and Cooperons, respectively. The blocks rendered with lines are Hikami boxes in (e) and (f).(a-b) are for fluctuations of diffusion constants and (c-d) are for fluctuations of density of states. (e-f) are bared and dressed Hikami boxes, and the gray parts are vertex correction mentioned before.
◇ ℛ 2 =1. 2 ◇ ℛ 2 =0. 8 ◇ ℛ 2 Fig. 5 :
222825◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇ ◇◇◇ ◇◇◇ ◇◇◇ ◇◇◇ ◇◇◇◇ ◇ =0The universal conductance fluctuation in the x direction as a function of R 1 , for R 2 = 0.4, 0.6, 0.8.
(
VI.12)
Fig. 6 :
6) is the Hurwitz zeta function.In order to visualize the above results, we introduce the difference of the conductance fluctuations with and without magnetic field as∆ δ σ 2 µ µ (B) = δ σ 2 µ µ (B) − δ σ 2 µ µ (0) (VI.15) The difference of conductance fluctuation ∆ δ σ 2 (B) as a function of the magnetic field. In panel (a), the adjustable 0 = 1000nm and the phase coherence length φ =10, 12, 20, 50, 200 nm from top to bottom. In panel (b), the adjustable 0 = 8nm and the phase coherence length φ =20, 50, 100, 10 nm from top to bottom. Other parameters are e = 10 nm, η = g x = g y = 1.
values for φ are assumed in both panel (a) and (b).
Tab. II: Comparison between the 2DEG, S-DSM, and 2D massless Dirac fermion. The parameters are computed in the existence of U 0 (r) only.
. For H 1 , only the bare Hikamibox H 0
1 is non-zero. The other two Hikami boxes H
R/A
1
B9780444869166500077.
Appendix A: Summary of notationsNotationsMeaning Value τ 0,x,y,z Relaxation time πρ(ε F )n 0,x,y,z u 2 0,x,y,z D x,y Bare diffusion constantsVertex correction coefficientsand bare Hikami boxes ξ y = − 5π 96
. V Pardo, W E Pickett, https:/link.aps.org/doi/10.1103/PhysRevLett.102.166803Phys. Rev. Lett. 102166803V. Pardo and W. E. Pickett, Phys. Rev. Lett. 102, 166803 (2009), URL https://link.aps.org/ doi/10.1103/PhysRevLett.102.166803.
. S Banerjee, R R P Singh, V Pardo, W E Pickett, https:/link.aps.org/doi/10.1103/PhysRevLett.103.016402Phys. Rev. Lett. 10316402S. Banerjee, R. R. P. Singh, V. Pardo, and W. E. Pickett, Phys. Rev. Lett. 103, 016402 (2009), URL https://link.aps.org/doi/10.1103/PhysRevLett.103.016402.
. V Pardo, W E Pickett, https:/link.aps.org/doi/10.1103/PhysRevB.81.035111Phys. Rev. B. 8135111V. Pardo and W. E. Pickett, Phys. Rev. B 81, 035111 (2010), URL https://link.aps.org/doi/ 10.1103/PhysRevB.81.035111.
. P Dietl, F Piéchon, G Montambaux, https:/link.aps.org/doi/10.1103/PhysRevLett.100.236405Phys. Rev. Lett. 100236405P. Dietl, F. Piéchon, and G. Montambaux, Phys. Rev. Lett. 100, 236405 (2008), URL https://link. aps.org/doi/10.1103/PhysRevLett.100.236405.
. G Montambaux, F Piéchon, J.-N Fuchs, M Goerbig, 10.1140/epjb/e2009-00383-0The European Physical Journal B. 72509G. Montambaux, F. Piéchon, J.-N. Fuchs, and M. Goerbig, The European Physical Journal B 72, 509 (2009), URL https://doi.org/10.1140/epjb/e2009-00383-0.
. Z J Xiang, G J Ye, C Shang, B Lei, N Z Wang, K S Yang, D Y Liu, F B Meng, X G Luo, L J Zou, https:/link.aps.org/doi/10.1103/PhysRevLett.115.186403Phys. Rev. Lett. 115186403Z. J. Xiang, G. J. Ye, C. Shang, B. Lei, N. Z. Wang, K. S. Yang, D. Y. Liu, F. B. Meng, X. G. Luo, L. J. Zou, et al., Phys. Rev. Lett. 115, 186403 (2015), URL https://link.aps.org/doi/10.1103/ PhysRevLett.115.186403.
. R Fei, V Tran, L Yang, https:/link.aps.org/doi/10.1103/PhysRevB.91.195319Phys. Rev. B. 91195319R. Fei, V. Tran, and L. Yang, Phys. Rev. B 91, 195319 (2015), URL https://link.aps.org/doi/ 10.1103/PhysRevB.91.195319.
. P.-L Gong, D.-Y Liu, K.-S Yang, Z.-J Xiang, X.-H Chen, Z Zeng, S.-Q Shen, L.-J Zou, https:/link.aps.org/doi/10.1103/PhysRevB.93.195434Phys. Rev. B. 93195434P.-L. Gong, D.-Y. Liu, K.-S. Yang, Z.-J. Xiang, X.-H. Chen, Z. Zeng, S.-Q. Shen, and L.-J. Zou, Phys. Rev. B 93, 195434 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.93.195434.
. C Zhong, Y Chen, Y Xie, Y.-Y. Sun, S Zhang, 10.1039/C6CP08439GPhys. Chem. Chem. Phys. 193820C. Zhong, Y. Chen, Y. Xie, Y.-Y. Sun, and S. Zhang, Phys. Chem. Chem. Phys. 19, 3820 (2017), URL http://dx.doi.org/10.1039/C6CP08439G.
. H Duan, S Li, S.-H Zheng, Z Sun, M Yang, R.-Q Wang, 10.1088/1367-2630/aa833aNew Journal of Physics. 19103010H. Duan, S. Li, S.-H. Zheng, Z. Sun, M. Yang, and R.-Q. Wang, New Journal of Physics 19, 103010 (2017), URL https://doi.org/10.1088/1367-2630/aa833a.
. H.-J Duan, Y.-Y Yang, S.-H Zheng, C.-Y Zhu, M.-X Deng, R Wang, M Yang, 2009.00775H.-J. Duan, Y.-Y. Yang, S.-H. Zheng, C.-Y. Zhu, M.-X. Deng, R. Wang, and M. Yang (2020), 2009.00775, URL https://arxiv.org/abs/2009.00775.
. P Delplace, G Montambaux, https:/link.aps.org/doi/10.1103/PhysRevB.82.035438Phys. Rev. B. 8235438P. Delplace and G. Montambaux, Phys. Rev. B 82, 035438 (2010), URL https://link.aps.org/ doi/10.1103/PhysRevB.82.035438.
. L.-K Lim, J.-N Fuchs, G Montambaux, https:/link.aps.org/doi/10.1103/PhysRevLett.108.175303Phys. Rev. Lett. 108175303L.-K. Lim, J.-N. Fuchs, and G. Montambaux, Phys. Rev. Lett. 108, 175303 (2012), URL https: //link.aps.org/doi/10.1103/PhysRevLett.108.175303.
. S Banerjee, W E Pickett, https:/link.aps.org/doi/10.1103/PhysRevB.86.075124Phys. Rev. B. 8675124S. Banerjee and W. E. Pickett, Phys. Rev. B 86, 075124 (2012), URL https://link.aps.org/doi/ 10.1103/PhysRevB.86.075124.
. K Saha, R Nandkishore, S A Parameswaran, https:/link.aps.org/doi/10.1103/PhysRevB.96.045424Phys. Rev. B. 9645424K. Saha, R. Nandkishore, and S. A. Parameswaran, Phys. Rev. B 96, 045424 (2017), URL https: //link.aps.org/doi/10.1103/PhysRevB.96.045424.
. P Adroguer, D Carpentier, G Montambaux, E Orignac, https:/link.aps.org/doi/10.1103/PhysRevB.93.125113Phys. Rev. B. 93125113P. Adroguer, D. Carpentier, G. Montambaux, and E. Orignac, Phys. Rev. B 93, 125113 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB.93.125113.
. S Park, S Woo, H Min, https:/iopscience.iop.org/article/10.1088/2053-1583/ab01c32D Materials. 625016S. Park, S. Woo, and H. Min, 2D Materials 6, 025016 (2019), URL https://iopscience.iop.org/ article/10.1088/2053-1583/ab01c3.
. Z Ning, B Fu, Q Shi, X Wang, https:/link.aps.org/doi/10.1103/PhysRevB.102.134207Phys. Rev. B. 102134207Z. Ning, B. Fu, Q. Shi, and X. Wang, Phys. Rev. B 102, 134207 (2020), URL https://link.aps. org/doi/10.1103/PhysRevB.102.134207.
. Q Niu, W C Yu, E I P Aulestia, Y J Hu, K T Lai, H Kotegawa, E Matsuoka, H Sugawara, H Tou, D Sun, https:/link.aps.org/doi/10.1103/PhysRevB.99.125126Phys. Rev. B. 99125126Q. Niu, W. C. Yu, E. I. P. Aulestia, Y. J. Hu, K. T. Lai, H. Kotegawa, E. Matsuoka, H. Sugawara, H. Tou, D. Sun, et al., Phys. Rev. B 99, 125126 (2019), URL https://link.aps.org/doi/10. 1103/PhysRevB.99.125126.
. S Kim, H Min, https:/link.aps.org/doi/10.1103/PhysRevResearch.2.022045Phys. Rev. Research. 222045S. Kim and H. Min, Phys. Rev. Research 2, 022045 (2020), URL https://link.aps.org/doi/10. 1103/PhysRevResearch.2.022045.
. A Narayan, https:/link.aps.org/doi/10.1103/PhysRevB.91.205445Phys. Rev. B. 91205445A. Narayan, Phys. Rev. B 91, 205445 (2015), URL https://link.aps.org/doi/10.1103/ PhysRevB.91.205445.
. H Isobe, B.-J Yang, A Chubukov, J Schmalian, N Nagaosa, https:/link.aps.org/doi/10.1103/PhysRevLett.116.076803Phys. Rev. Lett. 11676803H. Isobe, B.-J. Yang, A. Chubukov, J. Schmalian, and N. Nagaosa, Phys. Rev. Lett. 116, 076803 (2016), URL https://link.aps.org/doi/10.1103/PhysRevLett.116.076803.
. G Y Cho, E.-G Moon, Sci. Rep. 61G. Y. Cho and E.-G. Moon, Sci. Rep. 6, 1 (2016), URL https://www.nature.com/articles/ srep19198.
. V N Kotov, B Uchoa, O P Sushkov, https:/link.aps.org/doi/10.1103/PhysRevB.103.045403Phys. Rev. B. 10345403V. N. Kotov, B. Uchoa, and O. P. Sushkov, Phys. Rev. B 103, 045403 (2021), URL https://link. aps.org/doi/10.1103/PhysRevB.103.045403.
. P.-L Zhao, J.-R Wang, A.-M Wang, G.-Z Liu, https:/link.aps.org/doi/10.1103/PhysRevB.94.195114Phys. Rev. B. 94195114P.-L. Zhao, J.-R. Wang, A.-M. Wang, and G.-Z. Liu, Phys. Rev. B 94, 195114 (2016), URL https: //link.aps.org/doi/10.1103/PhysRevB.94.195114.
. K Saha, https:/link.aps.org/doi/10.1103/PhysRevB.94.081103Phys. Rev. B. 9481103K. Saha, Phys. Rev. B 94, 081103 (2016), URL https://link.aps.org/doi/10.1103/PhysRevB. 94.081103.
. G Bergmann, 0370-1573Physics Reports. 107G. Bergmann, Physics Reports 107, 1 (1984), ISSN 0370-1573, URL https://www. sciencedirect.com/science/article/pii/0370157384901030.
. P A Lee, T V Ramakrishnan, https:/link.aps.org/doi/10.1103/RevModPhys.57.287Rev. Mod. Phys. 57P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985), URL https://link.aps.org/ doi/10.1103/RevModPhys.57.287.
. S Chakravarty, A Schmid, 0370-1573Physics Reports. 140S. Chakravarty and A. Schmid, Physics Reports 140, 193 (1986), ISSN 0370-1573, URL https: //www.sciencedirect.com/science/article/pii/037015738690027X.
. S Hikami, A I Larkin, Y Nagaoka, 10.1143/PTP.63.7070033- 068XProgress of Theoretical Physics. 63S. Hikami, A. I. Larkin, and Y. Nagaoka, Progress of Theoretical Physics 63, 707 (1980), ISSN 0033- 068X, URL https://doi.org/10.1143/PTP.63.707.
H.-Z Lu, S.-Q Shen, ; H.-J Drouhin, J.-E Wegrowe, M Razeghi, 10.1117/12.2063426International Society for Optics and Photonics (SPIE, 2014). 9167Spintronics VIIH.-Z. Lu and S.-Q. Shen, in Spintronics VII, edited by H.-J. Drouhin, J.-E. Wegrowe, and M. Razeghi, International Society for Optics and Photonics (SPIE, 2014), vol. 9167, pp. 263 -273, URL https: //doi.org/10.1117/12.2063426.
. N A Sinitsyn, A H Macdonald, T Jungwirth, V K Dugaev, J Sinova, https:/link.aps.org/doi/10.1103/PhysRevB.75.045315Phys. Rev. B. 7545315N. A. Sinitsyn, A. H. MacDonald, T. Jungwirth, V. K. Dugaev, and J. Sinova, Phys. Rev. B 75, 045315 (2007), URL https://link.aps.org/doi/10.1103/PhysRevB.75.045315.
. N Nagaosa, J Sinova, S Onoda, A H Macdonald, N P Ong, https:/link.aps.org/doi/10.1103/RevModPhys.82.1539Rev. Mod. Phys. 821539N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010), URL https://link.aps.org/doi/10.1103/RevModPhys.82.1539.
. S A Yang, H Pan, Y Yao, Q Niu, https:/link.aps.org/doi/10.1103/PhysRevB.83.125122Phys. Rev. B. 83125122S. A. Yang, H. Pan, Y. Yao, and Q. Niu, Phys. Rev. B 83, 125122 (2011), URL https://link.aps. org/doi/10.1103/PhysRevB.83.125122.
. H.-Z Lu, S.-Q Shen, https:/link.aps.org/doi/10.1103/PhysRevB.88.081304Phys. Rev. B. 8881304H.-Z. Lu and S.-Q. Shen, Phys. Rev. B 88, 081304 (2013), URL https://link.aps.org/doi/10. 1103/PhysRevB.88.081304.
. H.-Z Lu, J Shi, S.-Q Shen, https:/link.aps.org/doi/10.1103/PhysRevLett.107.076801Phys. Rev. Lett. 10776801H.-Z. Lu, J. Shi, and S.-Q. Shen, Phys. Rev. Lett. 107, 076801 (2011), URL https://link.aps. org/doi/10.1103/PhysRevLett.107.076801.
. H.-Z Lu, S.-Q Shen, https:/link.aps.org/doi/10.1103/PhysRevLett.112.146601Phys. Rev. Lett. 112146601H.-Z. Lu and S.-Q. Shen, Phys. Rev. Lett. 112, 146601 (2014), URL https://link.aps.org/doi/ 10.1103/PhysRevLett.112.146601.
. H.-J Kim, K.-S Kim, J.-F Wang, M Sasaki, N Satoh, A Ohnishi, M Kitaura, M Yang, L Li, https:/link.aps.org/doi/10.1103/PhysRevLett.111.2466031307.6990Phys. Rev. Lett. 111246603H.-J. Kim, K.-S. Kim, J.-F. Wang, M. Sasaki, N. Satoh, A. Ohnishi, M. Kitaura, M. Yang, and L. Li, Phys. Rev. Lett. 111, 246603 (2013), 1307.6990, URL https://link.aps.org/doi/10. 1103/PhysRevLett.111.246603.
. J Xiong, S K Kushwaha, T Liang, J W Krizan, M Hirschberger, W Wang, R J Cava, N P Ong, Science. 350413J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan, M. Hirschberger, W. Wang, R. J. Cava, and N. P. Ong, Science 350, 413 (2015).
. B Zhao, P Cheng, H Pan, S Zhang, B Wang, G Wang, F Xiu, F Song, 1601.05536Scientific Reports. 622377B. Zhao, P. Cheng, H. Pan, S. Zhang, B. Wang, G. Wang, F. Xiu, and F. Song, Scientific Reports 6, 22377 (2016), 1601.05536.
. H Li, H He, H.-Z Lu, H Zhang, H Liu, R Ma, Z Fan, S.-Q Shen, J Wang, 1507.06470Nature communications. 71H. Li, H. He, H.-Z. Lu, H. Zhang, H. Liu, R. Ma, Z. Fan, S.-Q. Shen, and J. Wang, Nature communications 7, 1 (2016), 1507.06470.
. A A Burkov, https:/link.aps.org/doi/10.1103/PhysRevLett.113.2472031409.0013Phys. Rev. Lett. 113247203A. A. Burkov, Phys. Rev. Lett. 113, 247203 (2014), 1409.0013, URL https://link.aps.org/doi/ 10.1103/PhysRevLett.113.247203.
. A A Burkov, https:/link.aps.org/doi/10.1103/PhysRevB.91.2451571505.01849Phys. Rev. B. 91245157A. A. Burkov, Phys. Rev. B 91, 245157 (2015), 1505.01849, URL https://link.aps.org/doi/ 10.1103/PhysRevB.91.245157.
. C.-K Chiu, J C Y Teo, A P Schnyder, S Ryu, https:/link.aps.org/doi/10.1103/RevModPhys.88.035005Rev. Mod. Phys. 8835005C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016), URL https://link.aps.org/doi/10.1103/RevModPhys.88.035005.
E Akkermans, G Montambaux, Mesoscopic Physics of Electrons and Photons. Cambridge University PressE. Akkermans and G. Montambaux, Mesoscopic Physics of Electrons and Photons (Cambridge University Press, 2007).
Electronic Transport in Mesoscopic Systems. S Datta, Cambridge Studies in Semiconductor Physics and Microelectronic Engineering. Cambridge University PressS. Datta, Electronic Transport in Mesoscopic Systems, Cambridge Studies in Semiconductor Physics and Microelectronic Engineering (Cambridge University Press, 1995).
. P A Lee, A D Stone, https:/link.aps.org/doi/10.1103/PhysRevLett.55.1622Phys. Rev. Lett. 551622P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985), URL https://link.aps.org/doi/ 10.1103/PhysRevLett.55.1622.
. P A Lee, A D Stone, H Fukuyama, https:/link.aps.org/doi/10.1103/PhysRevB.35.1039Phys. Rev. B. 351039P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Rev. B 35, 1039 (1987), URL https://link.aps. org/doi/10.1103/PhysRevB.35.1039.
. L Zhao-Guo, Z Shuai, S Feng-Qi, 1000-3290, 1710.07938Acta Physica Sinica. 6497202L. Zhao-Guo, Z. Shuai, and S. Feng-Qi, Acta Physica Sinica 64, 097202 (2015), ISSN 1000-3290, 1710.07938, URL http://wulixb.iphy.ac.cn/cn/article/id/64502.
of Modern Problems in Condensed Matter Sciences. B L Altshuler, A G Aronov, Electron Electron Interactions in Disordered Systems. A. Efros and M. PollakElsevier10B. L. Altshuler and A. G. Aronov, in Electron Electron Interactions in Disordered Systems, edited by A. Efros and M. Pollak (Elsevier, 1985), vol. 10 of Modern Problems in Condensed Matter Sciences, pp. 1-153, URL https://www.sciencedirect.com/science/article/pii/
| [] |
[
"Physical and Geometrical Parameters of CVBS XI: COU1511 (HIP12552)",
"Physical and Geometrical Parameters of CVBS XI: COU1511 (HIP12552)"
] | [
"Mashhoor A Al-Wardat \nPhysics Department\nAl al-Bayt University\nPO Box: 13004025113MafraqJordan\n",
"M H El-Mahameed \nDepartment of Physics\nAl-Hussein Bin Talal University\n71111MaanJordan\n",
"Nihad A Yusuf \nDepartment of Physics\nYarmouk University\n21163IrbidJordan\n\nPrincess Sumaya University for Technology\nAmmanJordan\n",
"Awni M Khasawneh \nRoyal Jordanian Geographic Center\n11941AmmanJordan\n",
"Suhail G Masda \nAstrophysikalisches Institut und Universitäts-Sternwarte Jena, FSU Jena\n07745JenaGermany\n"
] | [
"Physics Department\nAl al-Bayt University\nPO Box: 13004025113MafraqJordan",
"Department of Physics\nAl-Hussein Bin Talal University\n71111MaanJordan",
"Department of Physics\nYarmouk University\n21163IrbidJordan",
"Princess Sumaya University for Technology\nAmmanJordan",
"Royal Jordanian Geographic Center\n11941AmmanJordan",
"Astrophysikalisches Institut und Universitäts-Sternwarte Jena, FSU Jena\n07745JenaGermany"
] | [
"Research in Astron. Astrophys"
] | Model atmospheres of the close visual binary star COU1511 (HIP12552) are constructed using grids of Kuruz's blanketed models to build the individual synthetic SEDs for both components. These synthetic SED's are combined together for the entire system and compared with the observational one following Al-Wardat's complex method for analyzing close visual binary stars. The entire observational spectral energy distribution (SED) of the system is used as a reference for comparison between synthetic SED and the observed one. The parameters of both components are derived as: T a eff = 6180 ± 50 K, T b eff = 5865 ± 70 K, log g a = 4.35 ± 0.12, log g b = 4.45 ± 0.14, R a = 1.262 ± 0.08R ⊙ , R b = 1.006 ± 0.07R ⊙ , L a = 2.09 ± 0.10L ⊙ , L b = 1.08 ± 0.12L ⊙ , with spectral types F8 and G1 for both components (a,b) respectively, and age of 3.0 ± 0.9 Gy. A modified orbit of the system is built and the masses of the two components are calculated as M a = 1.17 ± 0.11M ⊙ , M b = 1.06 ± 0.10M ⊙ . | 10.1088/1674-4527/16/11/166 | [
"https://arxiv.org/pdf/1606.05509v1.pdf"
] | 119,263,106 | 1606.05509 | d57091b21da19209285fd2184dc5a8eafcee1d60 |
Physical and Geometrical Parameters of CVBS XI: COU1511 (HIP12552)
2016
Mashhoor A Al-Wardat
Physics Department
Al al-Bayt University
PO Box: 13004025113MafraqJordan
M H El-Mahameed
Department of Physics
Al-Hussein Bin Talal University
71111MaanJordan
Nihad A Yusuf
Department of Physics
Yarmouk University
21163IrbidJordan
Princess Sumaya University for Technology
AmmanJordan
Awni M Khasawneh
Royal Jordanian Geographic Center
11941AmmanJordan
Suhail G Masda
Astrophysikalisches Institut und Universitäts-Sternwarte Jena, FSU Jena
07745JenaGermany
Physical and Geometrical Parameters of CVBS XI: COU1511 (HIP12552)
Research in Astron. Astrophys
XNo.XX2016Received 2016 May 4; accepted 2016 June 8Research in Astronomy and Astrophysicsbinariesvisual starsfundamental parameters starsCOU1511 (Hip12552)
Model atmospheres of the close visual binary star COU1511 (HIP12552) are constructed using grids of Kuruz's blanketed models to build the individual synthetic SEDs for both components. These synthetic SED's are combined together for the entire system and compared with the observational one following Al-Wardat's complex method for analyzing close visual binary stars. The entire observational spectral energy distribution (SED) of the system is used as a reference for comparison between synthetic SED and the observed one. The parameters of both components are derived as: T a eff = 6180 ± 50 K, T b eff = 5865 ± 70 K, log g a = 4.35 ± 0.12, log g b = 4.45 ± 0.14, R a = 1.262 ± 0.08R ⊙ , R b = 1.006 ± 0.07R ⊙ , L a = 2.09 ± 0.10L ⊙ , L b = 1.08 ± 0.12L ⊙ , with spectral types F8 and G1 for both components (a,b) respectively, and age of 3.0 ± 0.9 Gy. A modified orbit of the system is built and the masses of the two components are calculated as M a = 1.17 ± 0.11M ⊙ , M b = 1.06 ± 0.10M ⊙ .
INTRODUCTION
Recent surveys of the sky showed that more than 50% of the galactic stellar systems are binaries, which raises their importance in understanding the formation and evolution of the galaxy. This role in determining precisely different stellar parameters gives the study of binary stars a special importance. The case is a bit complicated in the case of close visual binary stars (CVBS), which are not resolved as binaries by inspecting limited images, but can be resolved in space based observations, or by using modern techniques of ground-based observations, like speckle interferometry and adaptive optics.
In addition to that, Docobo et al. (2001) pointed out that the study of orbital motion of visual and interferometric pairs remains an important astronomical discipline. The visual binaries are the main key source of information about stellar masses and distances, and they define practically our understanding of stellar physical properties especially for the lower part of the main sequence stars .
For now, hundreds of CVBS with periods on the order of 10 years or less are routinely observed by different groups of high resolution techniques around the world. This has helped in determining the orbital parameters and magnitude differences for some of these CVBS. However, this is not sufficient to determine the individual physical parameters of the components of the system.
Al-Wardat's method for analyzing CVBS (Al-Wardat 2012) offers a complementary solution for this problem by implementing differential photometry, spectrophotometry, atmospheric modeling, and orbital solution in accurate determination of different physical and geometrical parameters of this category of stars. The method was successfully applied to several solar type and subgaint binary systems such as Hip70973, Hip72479 (Al-Wardat 2012), Hip689 (Al-Wardat et al. 2014), and Hip11253 (Al-Wardat & Widyan 2009).
As a consequence of the previous work, this paper (the XI in its series) presents the analysis of the nearby solar type CVBS COU1511 (Hip12552), with a modification to its parallax. Table 1 shows the basic data of the system taken from SIMBAD and NACA/IPAC catalogues, and Table 2 shows data from Hipparcos and Tycho catalogues (ESA 1997), while Table 3 shows the magnitude difference of the system along with filters used to observe them. Docobo et al. (2006), 3 Horch et al. (2011).
ATMOSPHERIC MODELING
The observational SED of the system Hip12552 obtained by Al-Wardat (2002) is used as reference for the comparison with synthetic SED. Using m V = 8. m 51 (see Table 2), ∆m = 0. m 76±0.03 (as the average of all ∆m using the different filters for V-band only (545-562 nm), see Table 3), and Hipparcos trigonometric parallax (π = 11.83 ± 1.07 mas), the individual and absolute magnitudes of both components (a, b) of the system are calculated using the following relations:
F a F b = 2.512 −△m (1) M v = m v + 5 − 5 log(d) − A v (2) to get m a v = 8. m 95 ± 0.02, m b v = 9. m 71 ± 0.05, and M a v = 4. m 07 ± 0.18, M b v = 4. m 83 ± 0.19,
where the extinction value A v was taken from Table 1.
To calculate the preliminary input parameters used to build the atmospheric modelling, we use the bolometric magnitudes, the luminosities from Lang (1992), and Gray (2005) with the following relations:
log(R/R ⊙ ) = 0.5 log(L/L ⊙ ) − 2 log(T ef f /T ⊙ ),(3)log g = log(M/M ⊙ ) − 2 log(R/R ⊙ ) + 4.43,(4)
to estimate the effective temperatures and gravity acceleration. These values for the effective temperature and gravity acceleration allow us to construct the model atmosphere for each component using grids of Kuruz's line blanketed models (ATLAS9) (Kurucz 1994). Here we used T ⊙ = 5777K and M ⊙ bol = 4. m 75 in all calculations.
The total energy flux from a binary star is due to the net luminosity of the components a, and b located at a distance d from the Earth. The total energy flux may be written as:
F λ · d 2 = H a λ · R 2 a + H b λ · R 2 b ,(5)
Rearranging equ. 5 gives
F λ = (R a /d) 2 (H a λ + H b λ · (R b /R a ) 2 ),(6)
where R a and R b are the radii of the primary and secondary components of the system in solar units, H a λ and H b λ are the fluxes at the surface of the star and F λ is the flux for the entire SED of the system. Many attempts were made to achieve the best fit ( Fig. 1) between the observed flux of Al-Wardat (2002) and the total computed one using the iteration method of different sets of parameters. The best fit is found using the following set of parameters:
T a eff = 6180 ± 50K, T b eff = 5865 ± 70K,
log g a = 4.35 ± 0.12, log g b = 4.45 ± 0.14,
R a = 1.262 ± 0.08R ⊙ , R b = 1.006 ± 0.07R ⊙
Using equ. 3 the luminosities are calculated yielding the following values: L a = 2.09 ± 0.10L ⊙ and L b = 1.08 ± 0.12L ⊙ . Using tables of Gray (2005) or the Sp − T ef f. empirical relation from Lang (1992) , the spectral types of the components (a, b) of the system are F8 and G2 respectively. Fig. 1 Best fit between the entire observed spectrum (dotted line) (Al-Wardat 2002) and the synthetic entire SED (solid line) for the system Hip12552 using the following parameters T A eff = 6180 ± 50 K, log g A = 4.35 ± 0.10, R A = 1.262 ± 0.09R ⊙ , T B eff = 5865 ± 70 K, log g B = 4.45 ± 0.15, R B = 1.006 ± 0.10R ⊙ with d = 84.53 ± 0.009 pc (π = 11.83 mas).
ORBITAL ANALYSIS
The orbit of the system is built using the positional measurements listed in Table 4, following Tokovinin's method (Tokovinin 1992). The modified orbital elements of the system along with those taken from the sixth interferometric catalogue are listed in Table 5. The table shows a good agreement between our estimated orbital period, P; eccentricity, e; semi-major axis, a; inclination, i; argument of periastron, ω; position angle of nodes, Ω; and time of primary minimum, T 0 with those previously reported results.
MASSES
Using the estimated orbital element, the masses of the system and the corresponding errors are calculated using the following relations: The preliminary result obtained using the new Hipparcos trigonometric parallax (π = 11.07 ± 1.07 mas) (van Leeuwen 2007) is M a + M b = 2.72 ± 0.75M ⊙ , while it is 3.26 ± 1.18M ⊙ when using Hipparcos trigonometric parallax (π = 10.42 ± 0.2 mas, See Table 5). But depending on our analysis (Sec. 2), we achieved the best fit between the synthetic and observational entire SED using π = 11.83 mas, this new parallax value gives a mass sum of M a + M b = 2.23 ± 0.57M ⊙ , which fits better the positions of the two components on the evolutionary tracks as shown in Fig. 3. 2.72 ± 0.75 * 2.23 ± 0.57 * * 3.26 ± 1.18 * * * * Based on new Hipparcos trigonometric parallax (π = 11.07 ± 1.07 mas) * * Based on the parallax estimated in this work (π = 11.83 mas) * * * Based on Hipparcos trigonometric parallax (π = 10.42 ± 0.2 mas)
M A + M B = ( a 3 π 3 P 2 ) M ⊙ (7) σ M M = (3 σ π π ) 2 + (3 σ a a ) 2 + (2 σ p p ) 2(8)
SYNTHETIC PHOTOMETRY
As a double-check for the best fit and to present a new synthetic photometrical data of the unseen individual components of the system, we apply the following relation (Maíz Apellániz 2006, 2007:
m p [F λ,s (λ)] = −2.5 log P p (λ)F λ,s (λ)λdλ P p (λ)F λ,r (λ)λdλ + ZP p ,(9)
to calculate total and individual synthetic magnitudes of the systems, where m p is the synthetic magnitude of the passband p, P p (λ) is the dimensionless sensitivity function of the passband p, F λ,s (λ) is the synthetic SED of the object and F λ,r (λ) is the SED of the reference star (Vega). Here the zero points (ZP p ) of Maíz Apellániz (2007) are adopted. Calculated synthetic magnitudes and color indices of the entire system and individual components of different photometrical systems are shown in Table 6.
RESULTS AND DISCUSSION
The synthetic SEDs of the individual component and the system HIP12552 are built using atmospheric modeling and the visual magnitude difference between the two components along with the total observed SED. Least square fitting with weights inversely proportional to the squares of the positional measurement errors is used to modify the orbit of the system. So, the physical and geometrical parameters of HIP12552 are estimated. Fig. 1 shows the best fit of the total synthetic SED to the observed one. Table 7 Comparison between the observational and synthetic magnitudes, colors and magnitude differences of the system Hip 12552. Table 2 b See Table 6 c As the average of all ∆m using the differnt filters under V-band (see Table 3). Table 7 shows a comparison between the observational and synthetic magnitudes, colors and magnitudes differences for the system HIP12552. This gives a good indication of the reliability of the estimated parameters of the individual components of the system which are listed in Table 8
CONCLUSION
The CVBS COU1511 (HIP12552) is analyzed using Al-Wardat's complex method for analyzing close visual binary stars, which is based on combining magnitude difference measurements from speckle interferometry, entire spectral energy distribution (SED) from spectrophotometry, atmospheres modeling and orbital analysis to estimate the individual physical and geometrical parameters of the system.
The entire and individual Johnson-Cousin UBVR, Strömgren uvby, and Tycho BV synthetic magnitudes and color indices of the system are calculated. A modified orbit and geometrical elements of the system are introduced and compared with earlier results.
The positions of the two components on the evolutionary tracks and isochrones are shown, their spectral types are estimated as F8 and G1 respectively with the age of 3.0 ± 0.9 Gy.
1
1McAlister & Hendry (1982), 2 McAlister et al. (1987), 3 Hartkopf et al. (1994), 4 Balega et al. (1994), 5 ten Brummelaar et al. (2000), 6 Balega et al. (1999), 7 Hartkopf et al. (1997), 8 Balega et al. (2006), 9 Balega et al. (2013), 10 Hartkopf et al. (2008), 11 Balega et al. (2007), 12 Docobo et al. (2006), 13 Mason et al. (2011), 14 Gili & Prieur (2012), 15 Horch et al. (2011).
Fig. 2
2Relative visual orbit of the system HIP12552 showing the epoch of the positional measurements.
Synthetic b (This work) VJ 8. m 51 8. m 51 ± 0.03 BT 9. m 24 ± 0.02 9. m 25 ± 0.03 VT 8. m 59 ± 0.01 8. m 58 ± 0.03 (B − V )J 0. m 60 ± 0.02 0. m 60 ± 0.03 △m 0. m 76 c ± 0.03 0. m 76 ± 0.04 a See
Fig. 3
3The system components on the evolutionary tracks ofGirardi et al. (2000b).The positions of the system's components on the evolutionary tracks of Girardi et al. (2000a) (Fig. 3) show that both components with masses M A = 1.20 and M B = 1.09M ⊙ belong to the main-sequence stars. And their positions on Girardi et al. (2000a) isochrones for low-and intermediate-mass stars of different metallicities and that of the solar composition [Z = 0.019, Y = 0.273] are shown in Figs 4 & 5, which give an age of the system around 3.0 ± 0.9 Gy.
Fig. 4 Fig. 5
45The systems' components on the isochrones of low-and intermediate-mass, solar composition [Z=0.019, Y =0.273] stars of Girardi et al. (2000a). The systems' components on the isochrones for low-and intermediate-mass stars of different metallicities of Girardi et al. (2000a).
Table 1
1Basic data of the system http://simbad.u-strasbg.fr/simbad/ * http://irsa.ipac.caltech.edu,Hip12552
Reference
α2000
02 h 41 m 28. s 88
SIMBAD †
δ2000
+40 • 52 ′ 50. ′′ 84
-
Tyc.
2849-1282-1
-
HD
16656
-
Sp. Typ.
G0
-
E(B − V )
0.076 ± 0.002
NASA/IPAC *
Av
0. m 24
NASA/IPAC
†
Table 2
2Data of Hipparcos and Tycho CataloguesHip12552
Source of data
VJ (Hip)
8. m 51
Hipparcos
(B − V )J (Hip) 0. m 60 ± 0.018
-
πHip (mas)
9.69 ± 1.29
-
BT
9. m 24 ± 0.016
Tycho
VT
8. m 59 ± 0.014
-
πT yc (mas)
14.7 ± 9.20
-
πHip * (mas)
11.07 ± 1.07
New Hipparcos
* (van Leeuwen 2007)
Table 3
3Magnitude difference between the components of the system Hip12552, along with the filters used to obtain the observations.△m
Filter (λ/∆λ) References
0. m 65 ± 0.06
545nm/30
1
0. m 75 ± 0.04
545nm/30
2
0. m 88 ± −
562nm/40
3
0. m 86 ± −
692nm/40
3
1 Balega et al. (2007), 2
Table 4
4Positional measurements of the system from the Fourth Interferometric Catalogue. These points were modified by 180 • to achieve consistency with nearby points.Epoch
θ ,deg ρ ,arcsec References
1979.7732
91.6
0.156
1
1982.7605
65.9
0.153
2
1982.7659
66.3
0.142
2
1983.7131
57.9
0.136
2
1984.7046
49.4
0.119
2
1985.8540
30.4
0.106
2
1991.8973 184.9*
0.105
3
1993.7652 161.0*
0.122
4
1994.7087 151.3*
0.136
5
1994.8989 143.0*
0.146
6
1995.7710 139.2*
0.135
7
1996.6912 132.7*
0.150
5
2000.8730
98.8
0.152
8
2003.9468
73.8
0.143
9
2003.9598
77.1
0.146
10
2004.8374
65.0
0.135
11
2004.9905
64.1
0.135
12
2007.6075
30.0
0.109
13
2008.861
334.9*
0.077
14
2010.0074
328.1
0.0659
15
*
Table 5
5Orbital elements of the system.Parameters
This work
Hartkopf & Mason (2001) Couteau (1996)
P , yr
21.90188 ± 0.07339
22.18
19.7
T0, yr
2010.7566 ± 0.0714
1988.64
1988.49
e
0.4599 ± 0.0087
0.474
0.43
a, arcsec
0.1209 ± 0.0017
0.124
0.121
i, deg
152.10 ± 2.71
147.0
139.7
ω, deg
274.90 ± 6.06
287.0
114.2
Ω, deg
202.50 ± 6.32
217.2
44.50
Ma + M b , M⊙
Table 6
6Synthetic magnitudes and color indices of the system.Sys.
Filter
Entire
Comp. Comp.
σ = ±0.03
a
b
Joh-
U
9.22
9.59
10.56
Cou.
B
9.11
9.51
10.38
V
8.51
8.95
9.71
R
8.18
8.64
9.36
U − B
0.11
0.08
0.18
B − V
0.60
0.57
0.66
V − R
0.33
0.31
0.36
Ström.
u
10.38
10.76
11.70
v
9.44
9.83
10.73
b
8.85
9.27
10.08
y
8.48
8.92
9.68
u − v
0.94
0.93
0.97
v − b
0.59
0.56
0.65
b − y
0.37
0.35
0.40
Tycho
BT
9.25
9.65
10.55
VT
8.58
9.01
9.79
BT − VT
0.68
0.64
0.76
Table 8
8Parameters of the components of the system HIP12552.Parameters
Comp. a
Comp. b
T eff (K)
6180 ± 50
5865 ± 70
Radius (R⊙)
1.262 ± 0.08 1.006 ± 0.07
log g
4.35 ± 0.12
4.45 ± 0.14
L(L⊙)
2.09 ± 0.10
1.08 ± 0.12
M bol
3. m 95 ± 0.18 4. m 67 ± 0.19
MV
4. m 07 ± 0.18 4. m 83 ± 0.19
Mass (M⊙) *
1.17 ± 0.11
1.06 ± 0.10
Sp. Type * *
F8
G2
Parallax (mas)
11.83 ± 1.07
( Ma+M b
M ⊙
) * * *
2.23 ± 0.57
.3.85
3.80
3.75
3.70
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.3M
1.1M
HIP 12552
Component A
Component B
1M
1.4M
1.2M
Log (L/L )
Log T
eff
Main Sequence
Parameters of CVBS XI: COU1511 (HIP12552)
van Leeuwen, F. 2007, A&A, 474, 653 2, 5
Acknowledgements This work made use of SAO/ NASA, SIMBAD database, Fourth Catalog of Interferometric Measurements of Binary Stars, IPAC data systems and CHORIZOS code of photometric and spectrophotometric data analysis. The authors thank Mrs. Kawther Al-Waqfi for her help in some orbital calculations.
. M A Al-Wardat, Bull. Special Astrophys. Obs. 534Al-Wardat, M. A. 2002, Bull. Special Astrophys. Obs., 53, 58 3, 4
. M A Al-Wardat, PASA. 29Al-Wardat, M. A. 2012, PASA, 29, 523 2
. M A Al-Wardat, Y Y Balega, V V Leushin, Astrophysical Bulletin. 69Astrophysical BulletinAl-Wardat, M. A., Balega, Y. Y., Leushin, V. V., et al. 2014, Astrophysical Bulletin, 69, 58 2 Al-Wardat, M. A., & Widyan, H. 2009, Astrophysical Bulletin, 64, 365 2
. I I Balega, A F Balega, E V Maksimov, Bull. Special Astrophys. Obs. 595A&ASBalega, I. I., Balega, A. F., Maksimov, E. V., et al. 2006, Bull. Special Astrophys. Obs., 59, 20 5 Balega, I. I., Balega, Y. Y., Belkin, I. N., et al. 1994, A&AS, 105, 503 5
. I I Balega, Y Y Balega, L T Gasanova, Astrophysical Bulletin. 685Balega, I. I., Balega, Y. Y., Gasanova, L. T., et al. 2013, Astrophysical Bulletin, 68, 53 5
. I I Balega, Y Y Balega, A F Maksimov, Astrophysical Bulletin. 625Balega, I. I., Balega, Y. Y., Maksimov, A. F., et al. 2007, Astrophysical Bulletin, 62, 339 2, 5
. I I Balega, Y Y Balega, A F Maksimov, A&AS. 2875Balega, I. I., Balega, Y. Y., Maksimov, A. F., et al. 1999, A&AS, 140, 287 5
. P Couteau, 1285IAU Commission on Double StarsCouteau, P. 1996, IAU Commission on Double Stars, 128, 1 5
. J A Docobo, V S Tamazian, Y Y Balega, A&A. 8681Docobo, J. A., Tamazian, V. S., Balega, Y. Y., et al. 2001, A&A, 366, 868 1
. J A Docobo, V S Tamazian, Y Y Balega, N D Melikian, AJ. 1325Docobo, J. A., Tamazian, V. S., Balega, Y. Y., & Melikian, N. D. 2006, AJ, 132, 994 2, 5
Esa, The Hipparcos and Tycho Catalogues (ESA). ESA. 1997, The Hipparcos and Tycho Catalogues (ESA) 2
. R Gili, J.-L Prieur, Astronomische Nachrichten. 3335Gili, R., & Prieur, J.-L. 2012, Astronomische Nachrichten, 333, 727 5
. L Girardi, A Bressan, G Bertelli, C Chiosi, A&AS. 14110Girardi, L., Bressan, A., Bertelli, G., & Chiosi, C. 2000a, A&AS, 141, 371 8, 9, 10
VizieR Online Data Catalog. L Girardi, A Bressan, G Bertelli, C Chiosi, 414Girardi, L., Bressan, A., Bertelli, G., & Chiosi, C. 2000b, VizieR Online Data Catalog, 414, 10371 8
The Observation and Analysis of Stellar Photospheres. D F Gray, 5053Gray, D. F. 2005, The Observation and Analysis of Stellar Photospheres, 505 3
. W I Hartkopf, B D Mason, IAU Commission on Double Stars. 1455Hartkopf, W. I., & Mason, B. D. 2001, IAU Commission on Double Stars, 145, 1 5
. W I Hartkopf, B D Mason, T J Rafferty, AJ. 5Hartkopf, W. I., Mason, B. D., & Rafferty, T. J. 2008, AJ, 135, 1334 5
. W I Hartkopf, H A Mcalister, B D Mason, AJ. 108Hartkopf, W. I., McAlister, H. A., Mason, B. D., et al. 1994, AJ, 108, 2299 5
. W I Hartkopf, H A Mcalister, B D Mason, AJ. 114Hartkopf, W. I., McAlister, H. A., Mason, B. D., et al. 1997, AJ, 114, 1639 5
. E P Horch, S C Gomez, W H Sherry, AJ. 1415Horch, E. P., Gomez, S. C., Sherry, W. H., et al. 2011, AJ, 141, 45 2, 5
Solar abundance model atmospheres for 0,1,2,4,8 km/s. Kurucz CD-ROM No. 19. R Kurucz, Smithsonian Astrophysical Observatory. 193Kurucz, R. 1994, Solar abundance model atmospheres for 0,1,2,4,8 km/s. Kurucz CD-ROM No. 19. Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1994., 19 3
. K R Lang, Astrophysical Data I. Planets and Stars. 1337Lang, K. R. 1992, Astrophysical Data I. Planets and Stars., 133 3, 7
. J Maíz Apellániz, AJ. Maíz Apellániz, J. 2006, AJ, 131, 1184 6
The Future of Photometric, Spectrophotometric and Polarimetric Standardization. J Maíz Apellániz, Astronomical Society of the Pacific Conference Series. C. SterkenSan Francisco3646Maíz Apellániz, J. 2007, in Astronomical Society of the Pacific Conference Series, Vol. 364, The Future of Photometric, Spectrophotometric and Polarimetric Standardization, ed. C. Sterken (San Francisco: Astronomical Society of the Pacific), 227 6
. B D Mason, W I Hartkopf, D Raghavan, AJ. 1425Mason, B. D., Hartkopf, W. I., Raghavan, D., et al. 2011, AJ, 142, 176 5
. H A Mcalister, W I Hartkopf, D J Hutter, O G Franz, AJ. 6885McAlister, H. A., Hartkopf, W. I., Hutter, D. J., & Franz, O. G. 1987, AJ, 93, 688 5
. H A Mcalister, E M Hendry, T Brummelaar, B D Mason, H A Mcalister, ApJS. 495AJMcAlister, H. A., & Hendry, E. M. 1982, ApJS, 49, 267 5 ten Brummelaar, T., Mason, B. D., McAlister, H. A., et al. 2000, AJ, 119, 2403 5
A Tokovinin, Complementary Approaches to Double and Multiple Star Research. H. A. McAlister & W. I. Hartkopf324Astronomical Society of the Pacific Conference SeriesTokovinin, A. 1992, in Astronomical Society of the Pacific Conference Series, Vol. 32, IAU Colloq. 135: Complementary Approaches to Double and Multiple Star Research, ed. H. A. McAlister & W. I. Hartkopf, 573 4
| [] |
[
"Lepton masses and mixing in a three-Higgs doublet model",
"Lepton masses and mixing in a three-Higgs doublet model"
] | [
"Joris Vergeest ",
"Bartosz Dziewit ",
"Piotr Chaber ",
"Marek Zra ",
"\nInstitute of Physics\nUniversity of Silesia\n75 Pu lku Piechoty 141-500ChorzówPoland\n",
"\nHumanitas University in Sosnowiec\nul. Kilińskiego 4341-200SosnowiecPoland\n"
] | [
"Institute of Physics\nUniversity of Silesia\n75 Pu lku Piechoty 141-500ChorzówPoland",
"Humanitas University in Sosnowiec\nul. Kilińskiego 4341-200SosnowiecPoland"
] | [] | In the three-Higgs doublet model (3HDM) frame, we search for discrete flavour symmetries that give relations among the lepton masses and their mixing angles. We explore discreet non-Abelian groups of order less than 1035, treating neutrinos as Majorana or Dirac particles. Unlike the Standard Model, the 3HDM does not require the flavor symmetry to be broken. Despite the four free dynamic parameters available in the model, none of the groups fully predicts the lepton data. However, some of the scanned groups provide either the correct neutrino masses and mixing angles or the correct masses of the charged leptons. ∆(96) is the smallest group compatible with the experimental data of neutrino masses and PMNS mixing. S4 is an approximate symmetry of Dirac neutrino mixing, with parameters staying about 3σ apart from the measured θ12, θ23, θ13 and δCP . | null | [
"https://arxiv.org/pdf/2203.03514v1.pdf"
] | 247,292,470 | 2203.03514 | 1125c95fc5534049b332c5ae72221248a87ef9f2 |
Lepton masses and mixing in a three-Higgs doublet model
7 Mar 2022 (Dated: March 8, 2022)
Joris Vergeest
Bartosz Dziewit
Piotr Chaber
Marek Zra
Institute of Physics
University of Silesia
75 Pu lku Piechoty 141-500ChorzówPoland
Humanitas University in Sosnowiec
ul. Kilińskiego 4341-200SosnowiecPoland
Lepton masses and mixing in a three-Higgs doublet model
7 Mar 2022 (Dated: March 8, 2022)
In the three-Higgs doublet model (3HDM) frame, we search for discrete flavour symmetries that give relations among the lepton masses and their mixing angles. We explore discreet non-Abelian groups of order less than 1035, treating neutrinos as Majorana or Dirac particles. Unlike the Standard Model, the 3HDM does not require the flavor symmetry to be broken. Despite the four free dynamic parameters available in the model, none of the groups fully predicts the lepton data. However, some of the scanned groups provide either the correct neutrino masses and mixing angles or the correct masses of the charged leptons. ∆(96) is the smallest group compatible with the experimental data of neutrino masses and PMNS mixing. S4 is an approximate symmetry of Dirac neutrino mixing, with parameters staying about 3σ apart from the measured θ12, θ23, θ13 and δCP .
I. INTRODUCTION
Within the Standard Model, despite its success, still there are basic open questions. For example, the question of the origin of the three fermion generations, with such diverse mass hierarchies and radically differing mixing patterns of leptons, compared to quarks is unsolved [1,2]. In the lepton sector, which is the focus of this work, various attempts have been made to devise a theory to predict the neutrino masses and the lepton mixing angles (e.g. see [3][4][5]). As the experimental data on these quantities have gained precision in the recent years (e.g. see [6][7][8]), it is getting increasingly hard to explain them satisfactorily. A common approach to this problem is to look for lepton flavor symmetries of the interaction Lagrangian, which may pin down mass values and the neutrino mixing angles, and if so, to determine whether these are consistent (even partially) with the experimental data. It is well known that an exact nontrivial flavor symmetry in the lepton sector does not exist, which follows from the fact that the lepton masses are distinct [9]. Due to Schur's first lemma, the acting of two inequivalent representations in flavor space (one on the lepton doublets, one on the lepton singlets) implies that any mass matrix is either proportional to the unit matrix or vanishes. One strategy to avoid the above problem is to break the flavor symmetry explicitly. Then the charged lepton mass matrix and the neutrino mass matrix are separately invariant under two different subgroups of a larger symmetry group G [10]. The subgroups are usually kept small, whereas in [11] groups G of order up to 1000 have been investigated. It was shown that the possible lepton mixing patterns then depend on how the two subgroups are embedded within G. Another approach is based on a Lagrangian with mass terms constructed with * [email protected] † [email protected] lepton, Higgs and scalar flavon fields. Group-invariance of the terms is looked for by systematic probing all plausible representation assignments [12]; dynamic parameters (VEVs) affect the predicted mixing angles [13] [14]. Explicit flavor symmetry breaking can be avoided in the presence of Higgs fields that transform under G. Two-Higgs doublet models [15] are obvious candidates to investigate that principle. In [16] and [17] non-Abelian groups have been identified providing lepton masses and a mixing matrix, but these were not in agreement with experiment. Although the implied lepton masses of the model were found nondegenerate, the implied mixing matrices appeared to be always monomial. However, the results are different when one more Higgs doublet is added, as will be outlined below. For an overview of the work on lepton flavor symmetry, we refer to [9], [18].
We propose to study a model in which the SM is extended with two SU (2) Higgs doublets [19,20]. The left-handed lepton doublets, the right-handed charged leptons, the right-handed neutrinos as well as the three Higgs doublets themselves are modelled as flavor triplets, each associated with a unitary three-dimensional irreducible representation of a discrete group isomorphic to a subgroup of U (3). This three-Higgs doublet model (3HDM) contains no flavons or other additional fields. The model allows the treatment of neutrinos as Dirac particles or as Majorana particles on equal footing. Since the Higgs doublets form a flavor vector transforming under G, the mass-squared matrices are not affected by Schur's first lemma: the masses can be nondegenerate and the neutrino mixing nontrivial, even when the assigned 3D irreps are inequivalent and G is non-Abelian. None of the flavor groups with order | G |< 1035 can satisfactorily reproduce the experimental data of the lepton sector in its entirety; only partial symmetries occur. A few groups turn out to be a symmetry of nontrivial Majorana neutrino mixing; the smallest is ∆(96). This symmetry favours inverse ordering of the neutrinos with masses between 0,013 and 0,05 eV.
In the following section, the 3HDM and its group-invariance is defined. Section III outlines the (almost fully automated) process of detecting flavor symmetry, and its implications for the lepton masses and mixing angles. In section IV, the results of a scan of groups with | G |< 1035 are presented. Details on the predicted relations among the neutrino mixing angles and lepton masses are provided in Section V. In section VI we summarize the results and discuss directions to modify and generalize the 3HDM.
II. THE THREE-HIGGS DOUBLET MODEL (3HDM)
In the models which we consider, the unitary symmetry transformations for the fields will not affect most of the terms in the Lagrangian. Only the Yukawa part and Higgs potential [21] are not automatically invariant. Yet we assume that also the scalar Higgs Lagrangian is invariant, following [16]. Thus, let us solely consider the Yukawa interaction for charged leptons L l , for Dirac neutrinos L ν and for Majorana neutrinos L M , using an effective dimension-five operator for the latter [22]:
L l =−(h l i ) αβ L αLΦi l βR + H.c.,(1)L ν =−(h ν i ) αβ L αL Φ i ν βR + H.c.,(2)L M =− g M (h M ij ) αβ (L αL Φ i )(Φ T j L c βR ) + H.c.,(3)
where summation over the lepton flavors α, β = e, µ, τ and over the Higgs flavors i, j = 1, 2, 3 is understood. h l i , h ν i and h M ij are three-dimensional Yukawa matrices. Φ i is an SU (2) Higgs doublet;Φ i = iσ 2 Φ * i . The L αL = (ν αL , l αL ) T are lepton doublets, andL and L c denote the adjoint and charge-conjugated lepton doublets, respectively. After spontaneous symmetry breaking, the three mass matrices M l , M ν and M M are defined by the mass Lagrangian terms:
L l mass =−l L M l l R + H.c. (4) L ν mass =−ν L M ν ν R + H.c.(5)L M mass =− 1 2 ν L M M ν c L + H.c.(6)
where l L , l R , ν L , ν R are flavor vectors for the left/righthanded charged leptons and neutrinos, respectively. Each mass matrix is linearly composed of three Yukawa matrices using the vacuum expectation values (VEVs) v i obtained from Φ i after spontaneous gauge symmetry breaking:
M l = − 1 √ 2 v * i h l i (7) M ν = 1 √ 2 v i h ν i (8) M M = g M v i v j h M ij .(9)
l L , l R , ν L , ν R and Φ = (Φ 1 , Φ 2 , Φ 3 ) T will each be assigned a three-dimensional irreducible representation of some finite group G. All representation matrices should be unitary in order to conserve the total lepton number and to ensure that Σ|v i | 2 = ( √ 2G F ) −1/2 =246 GeV, where G F is the Fermi coupling constant.
The first step in this study is to identify distinct groups G, isomorphic to a U (3) subgroup, that have one or more three-dimensional irreducible representations, and assign them to flavor vectors so that the Yukawa terms in Eq.
(1) and either (2) or (3) remain G-invariant. To this end it is determined which of the three terms
L L A † L (A * ΦΦ ) i h l i A lR l R ,(10)L L A † L (A Φ Φ) i h ν i A νR ν R ,(11)L L A † L (A Φ Φ) i (A Φ Φ ′T ) j )h M ij A * L L c R ,(12)
(if any) remain unaffected by the simultaneous matrix operators A L (g), A lR (g), A νR (g) and A Φ (g) for all g in G.
The matrix operators are defined by the representations acting on the flavor vectors.
Φ ′T = (Φ T 1 , Φ T 2 , Φ T 3 ) T .
Since the transformations are unitary, the kinetic part of the Lagrangian will be automatically invariant. It can be assumed that also the Higgs potential is unaffected [16], which justifies our analysis of the Yukawa sector in isolation.
We aim to find symmetry groups leaving both expressions (10) and (11) (and hence L l + L ν ) invariant, as we will then be able to derive the implications of the 3HDM on the lepton masses and neutrino mixing angles, in case the neutrinos are Dirac particles. Likewise we look for groups leaving both (10) and (12) (hence L l + L M ) invariant, when neutrinos have Majorana nature.
III. SOLVING THE INVARIANCE EQUATIONS
At first we treat the three Yukawa terms separately. G-invariance of L l , see Eqs.
(1) and (10), is achieved if and only if
((A Φ (g)) † ⊗ (A L (g)) † ⊗ (A lR (g)) T ) h l = h l ,(13)
as demonstrated in [23]. The Kronecker product gives a 27×27 matrix, and h l is the 27-dimensional vector built from the Yukawa matrices h l 1 , h l 2 and h l 3 , row-wise. If h l is an invariant eigenvector satisfying Eq.(13) for all g ∈ G then L l is G-invariant. It can be proven that it is sufficient to test the generators of G instead of all g ∈ G.
The invariance equations for the terms L ν and L M are (14) or (15) are computationally demanding for groups of a high order, as they may have a large number of representations and thus give rise to many combinations of the representations assigned to the flavor vectors. However, we will describe here that based on the character table of G, we can filter away the irrelevant representation assignments, thus reducing the computational task enormously.
((A Φ ) T ⊗ (A L ) † ⊗ (A νR ) T )h ν = h ν (14) ((A Φ ) T ⊗ (A Φ ) T ⊗ (A L ) † ⊗ (A L ) † )h M = h M ,(15)
Eqs. (13), (14) and (15) turn out to be equivalent to the (Clebsch-Gordan) tensor product decomposition equations [23]
A L ⊗ A * lR = A * Φ ⊕ ... (16) A L ⊗ A * νR = A Φ ⊕ ... (17) A * Φ ⊗ A L ⊗ A L = A Φ ⊕ ...(18)
Eqs.(16) (or (17)) are fulfilled if and only if the 9dimensional tensor product operator allows a decomposition containing at least one three-dimensional matrix operator. Eq. (18) requires that the 27-dimensional tensor product operator contains at least one three-dimensional matrix operator. All three cases can be verified by reading off the group's character table, as detailed in the next section. From the solutions h l , h ν or h M , the mass matrices are constructed enabling us to pin down the lepton mass ratios and/or the neutrino mixing angles to certain values or intervals.
IV. FINDING THE G-INVARIANT LAGRANGIANS FOR | G |< 1035
A candidate group G rendering a mass term Ginvariant must have at least one faithful threedimensional representation (otherwise G would not be isomorphic to a subgroup of U (3)). Unfaithful representations are also included in this analysis, despite the abundant repetition of representation assignments that can be expected in groups containing G. Our precise selection criterion is that at least one of the irreducible representations assigned to a mass term is faithful. Out of the U (3) subgroups with | G |< 1035, 749 groups provide solutions to Eq. (13) and, in these cases, also to Eq. (14). 216 groups provide solutions to Eq. (15).
The selection and processing of groups is fully automated using the computer-algebra system GAP [24]. To determine which representation assignments would solve a particular invariance equation it is sufficient to observe the group's character table, which is readily provided by GAP. The character of g ∈ G in representation A, denoted χ A (g), is defined as the trace of matrix A(g). The mapping χ A is called the character of A. Let A and B be representations of G, then
χ A , χ B := 1 |G| g∈G χ A (g) ⋆ χ B (g),(19)
defines the inner product of χ A and χ B . It can be proven that χ A⊗B (g) = χ A (g)χ B (g) for all g ∈ G. Let also C be an irreducible representation of G. Then χ A⊗B , χ C is the multiplicity of C occurring in the tensor product decomposition of A ⊗ B. For the decomposition in Eq.
(16), χ AL⊗A * lR , χ A * Φ
can take the values 0 to 3. This is the number of linearly independent solutions h l to Eq. (13). The inner products can be directly deduced from the character table of G, and thus prior to the actual generation of the representation matrices themselves and without explicitly solving Eq. (13). For brevity let us denote the representations appearing in Eq. (16) as A, B and C, respectively. Then, if A, B and C are irreducible, Eq. (13) has a nontrivial solution if and only if n C := χ A⊗B ⋆ , χ C ⋆ > 0.
In the selection procedure the representation triplet (A, B, C) is accepted only if n C = 1; as a trade-off regarding computational load, we disregard multidimensional solutions (n C > 1). We find over 6 million accepted triplets. In the following step, the explicit threedimensional matrix representations (denoted 3 A , 3 B and 3 C ) of A, B and C are obtained using the Repsn package of GAP [25]. The Kronecker product Eq.(13) is set up for each generator of G, and solved for h l , using the BaseFixedSpace function of GAP. The total number of inequivalent vectors h l from the group scan is 2130 (a set of inequivalent vectors contains no colinear pairs; colinear solutions would imply mass matrices differing by a constant only). For the Dirac neutrino term, with n C = χ A⊗B ⋆ , χ C we find the same number of solutions. For the Majorana term, only two characters are involved; let us denote them χ A and χ C . The character inner product χ C ⋆ ⊗A⊗A , χ C can take the values 0 to 9, equal to the dimension of the solution space. Again, only solutions with inner product equal to one are accepted. We find 70 inequivalent solutions h M . Using Eqs. (7), (8), (9) we obtain the mass matrices as functions of v i . For this calculation and the subsequent (mostly numerical) computations, we use the Mathematica package from Wolfram [26].
If the triplet of representations (3 A , 3 B , 3 C ) yields a G-invariant charged-lepton term, then this triplet can also render the Dirac neutrino term G-invariant. Can any charged-lepton triplet (3 A , 3 B , 3 C ) be combined with any Dirac neutrino triplet (3 D , 3 E , 3 F ) to obtain a Ginvariant charged-current lepton interaction term? The answer is no; we must require 3 D = 3 A because the lefthanded charged lepton and the left-handed Dirac neutrino are contained in the same SU (2) L doublet and hence transform equally. Furthermore, we require 3 F = 3 C since the two states in a Higgs doublet respect SU (2) L symmetry and thus differ by complex conjugation. So we only consider representation assignments of the form ((3 A , 3 B , 3 C ), (3 A , 3 D , 3 C )). The simultaneous solution of Eqs. (13) and (14) gives mass matrices M l and M ν that define the PMNS matrix. In the case where neutrinos are Majorana particles, G-invariance of the chargedcurrent interaction term requires assignments of the form ((3 A , 3 B , 3 C ), (3 A , 3 C )).
For Dirac neutrinos the PMNS matrix is calculated with the two unitary matrices U l and U ν that diagonalize the mass-squared matrices for charged leptons and for neutrinos, respectively:
U † l (M l M l † )U l = (M l d ) 2 , U † ν (M ν M ν † )U ν = (M ν d ) 2 ,(20)
where the subscript d denotes the diagonal matrix. In the case of Majorana neutrinos their mass matrix is symmetric, and is diagonalized using one unitary matrix U ν :
U T ν M M U ν = M M d .(21)
Independently of the neutrinos' nature the PMNS matrix is given by U † l U ν . For a given group we can identify group-invariant mass terms and derive masses and mixing angles as functions of the VEVs for each particular representation assignment. From group theory alone, we can neither determine the absolute scale of the Higgs couplings nor the flavor assignments, so we have v 2 /v 1 and v 3 /v 1 as four free real parameters, and we can at most determine mass ratios m i /m j . It will be only possible to determine the PMNS matrix up to permutation of rows or columns and (for the Majorana case) up to a phase for two rows.
The results from selected groups are presented in Table 1. In its first column the GAP-ID is the identifier supplied by the SmallGroups library of GAP [27]. The first index equals the group's order, the second distinguishes between the non-isomorphic groups of that order. The second column of the table shows the group structure (when informative). The number of three-dimensional irreducible representations and of faithful ones, are listed in column 3. The following cases are distinguished in the "Dirac Mass" and "Maj. Mass" columns. A minus sign "-" means that the tensor product could not be decomposed for any of the representation assignments otained for the group. "a" means that mass ratios consistent with the experimental values can be obtained. "b" indicates one, two or three vanishing or degenerate masses. "c" indicates an upper bound for m 3 /m 2 while m 2 /m 1 and m 3 /m 1 can obtain any positive value, as functions of the v i . Multiple tokens mean different results for different representation assignments for the group. The columns labeled "Dirac mixing" and "Maj. Mixing" indicate how close a predicted PMNS matrix gets to the experimental data, expressed as χ 2 /4, the average of the deviations between calculated and experimental values of the four quantities: sin 2 Θ 12 , sin 2 Θ 23 , sin 2 Θ 13 and δ CP . "d" signifies that only monomial PMNS matrices are found. χ 2 is derived from a simplified extraction of the experimental neutrino oscillation parameters, see Table II. χ 2 is the smallest value found by equidistant numerical sampling in a region of four-dimensional (v 2 /v 1 , v 3 /v 1 )-space. There is no proof that we find the global minimum.
We find no group representations implying the lepton masses and mixing angles to be simultaneously consistent with the experimental data. Group T 7 is the smallest group with invariant L l + L ν compatible either with the experimental charged-lepton masses or with the experimental neutrino mass data, with different VEVs for ei-ther case. Group ∆(48) is the smallest group with invariant L M compatible with the experimental neutrino mass data. For Majorana neutrinos, the smallest group compatible with the PMNS data is ∆(96), (that is L l + L M invariant). For Dirac neutrinos we find no group exactly compatible with the PMNS data. S 4 (and groups containing it) comes closest, with χ 2 = 5.3. In the next section, further details of the ∆(96) and S 4 are presented in subsections A and B, respectively. Other groups generating specific solutions are described in subsection C. 7) and (9) are of the form
M l = − c l √ 2 0 0 v ⋆ 3 v ⋆ 1 0 0 0 v ⋆ 2 0 , M M = − cM g M 0 v 2 2 v 2 1 v 2 2 0 v 2 3 v 2 1 v 2 3 0 ,(22)
where c l and c M are arbitrary constants inherent to the h matrices. From these two matrices follow the two mass ratios of the charged leptons, the two mass ratios of the neutrinos (or two mass-squared differences up to a common factor), and the four neutrino mixing angles. All 8 quantities are functions of the VEVs. We search for values v i yielding the mass ratios and/or mixing parameters in agreement with experimental data. The four calculated mixing angles are consistent with experimental data with high accuracy for determined regions in the four-dimensional search space. The search is implemented numerically by an equidistant sampling of (|v 2 /v 1 |, arg(v 2 /v 1 ), |v 3 /v 1 |, arg(v 3 /v 1 ). The search box is, at present, limited to [0,4000] for the |v i /v 1 | ratios. We find multiple choices of the VEVs fitting the mixing data with χ 2 < 10 −3 . In general it will be possible to detect only a subset of solutions yielding consistent mixing angles. For the charged leptons the mass ratios are equal to |v i /v 1 |, as obtained from M l in Eq. (22). The charged lepton mass ratios found until present remain far below the experimental values by approximately a factor 2 for the muon to 35 for the tau. We do obtain fits simultaneously to the Majorana neutrino masssquared data and to the mixing angles. The best fit (χ 2 = 0.3) is achieved with modulus of eigenvalues of M M equal to (0.217, 0.867, 0.879). (χ 2 here is averaged over six quantities: two neutrino mass-squared differences and four mixing angles). Scaled to the experimental data it corresponds to (inverted ordered) neutrino ) that also renders the PMNS matrix invariant, which however is monomial, ruling out any flavor mixing. It turns out that there are no further representation assignments inequivalent to the two just described; two representation assignments are equivalent if their sets of solution vectors (h l , h M ) are linearly dependent. We note that the ∆(96)-symmetry involves the unfaithful representation 3 1 . Out of the 6 3 = 216 permutations, 40 representation assignments for the charged lepton mass term can realize the tensor product decomposition (Eq.(16)) and 3 6 )). So we conclude that this representation assignment, as a provider for ∆(96) mixing symmetry, is not unique and, consequently the 8 pairs of invariant eigenvectors (h l , h M ) are mutually equivalent. Similar pairs also appear in higher-order groups, such as ∆(150) and C2 × ∆(96).
B. Group S4
S 4 is the smallest group allowing nontrivial flavor mixing when neutrinos are Dirac particles. The two representation assignments providing this symmetry are ((3 1 , 3 1 , 3 1 ), (3 1 , 3 1 , 3 1 )) and ((3 1 , 3 1 , 3 2 ), (3 1 , 3 1 , 3 2 )), where 3 1 and 3 2 are the two inequivalent threedimensional representations of S 4 . The first representation assignment leads to anti-symmetric mass matrices and trivial mixing. The second implies the mass matrices (using Eqs. (7) and (8)):
M l = − c l √ 2 0 v ⋆ 3 v ⋆ 2 v ⋆ 3 0 v ⋆ 1 v ⋆ 2 v ⋆ 1 0 , M ν = − cν c l M l * ,(23)
where c ν is an arbitrary constant. We obtain fits to the neutrino mixing data with χ 2 = 5.3. The deviation is mainly due to sin 2 Θ 23 and sin 2 Θ 13 , which end up approximately 3σ larger than the observed values. The same results are found in higher-order groups, such as C i × S 4 and C 4 × A 4 . The predicted mass ratios m ν /m e and m τ /m e are too small and do not come close to the actual charged lepton mass ratios. However, the Dirac neutrino mass ratios are consistent with the experimental data. A fit to the mass term alone -not combined with neutrino mixing -has χ 2 < 0.01 and gives (inverted ordered) neutrino masses: (m 1 , m 2 , m 3 ) = (0.733, 49.17, 49.91)×10 −3 eV; the combined fit of Dirac neutrino masses and mixing has χ 2 = 8.
C. Other groups
Group Σ(36 × 3) is the lowest-order group generating a Dirac mass matrix with 9 non-zero entries. 96 representation assignments generate a Dirac mass matrix (the set of 96 representations form 5 mutually inequivalent subsets), none being compatible with the observed masses. None of the representation assignments gives an invariant interaction term.
Further new types of mass matrices show up in group Z ′′ (3,3). For this group we find 1134 representation assignments (forming 109 inequivalent subsets), each giving an invariant Dirac mass matrix, and 2592 assignments (forming 74 inequivalent subsets) generate an invariant PMNS matrix for Dirac neutrinos. It turns out that when an unfaithful representation occurs in the assignment, a χ 2 ≈ 150 fit to the oscillation data can be obtained, whereas for all other representation assignments the PMNS matrix is monomial. The group Z ′′ (3, 4) generates similar results, with further new types of Dirac matrices.
Π(1, 2) generates a new type of Majorana mass matrix, giving neutrino masses with normal ordering: (m 1 , m 2 , m 3 ) = (17.3, 19.3, 52.81)×10 −3 eV. The charged lepton mass matrix is anti-symmetric, disabling nontrivial neutrino mixing. Another new type of Majorana mass matrix is generated by Π(1, 3), providing results similar to those of Π(1, 2).
VI. CONCLUSIONS
In the 3HDM model, we have searched for a discrete flavour symmetry that would predict the hierarchy of charged leptons and their mixing. The investigation comprised all discrete groups having three-dimensional irreducible representations up to the order of 1035. With the applied representation selection criteria, none of the studied groups is a symmetry of the entire lepton sector. There exist symmetries that separately predict the masses of charged leptons, the masses of neutrinos, and/or the elements of the PMNS mixing matrix, as can be expected for a model with four free parameters.
The most noticeable results are obtained with ∆(96) and S 4 . The smallest group compatible with the neutrino mixing data is ∆(96), when assuming that neutrinos have Majorana nature. For the same parameters v i the predicted neutrino masses are consistent with the experimental data as well. The predicted charged lepton masses are then far off the experimental values.
S 4 is the smallest group approximately compatible with the experimental neutrino mass and mixing data, in case the neutrinos are Dirac particles. The v i producing that fit imply masses of the charged leptons which are in disagreement with experiment.
It should be noted that we limited our search for proper discrete symmetry to groups with irreducible threedimensional faithful representations, and we searched for non-degenerate eigenvectors only, constructed from Yukawa matrices for leptons and neutrinos. In this context, we can say that the problem of mass for charged leptons and neutrinos and their mixing in charged currents still awaits a solution.
There are several possible improvements and extensions to the described methods. The present results are based on numerical sampling in four-dimensional VEVspace, where the choice of search interval and the sampling density is limited for practical reasons. The analysis would be highly enhanced when analytic expressions for the eigenvalues of mass matrices are used to find bounds on physical quantities. The restrictions we made on the representation assignments are mostly for practical reasons, to limit the computational work. It is therefore worthwhile to explore the inclusion of unfaithful and reducible representations. Also, the requirement that the tensor product decomposition is unique could be relaxed, allowing multi-dimensional solutions.
is a symmetry group of lepton mixing if we assume the neutrinos to be Majorana particles, and apply representation assignment ((3 1 , 3 3 , 3 6 ), (3 1 , 3 6 )). Here the representations 3 i are from those provided by Repsn (six 3D irreps in total).3 1 and 3 2 are the two unfaithful representations and 3 3 . . . 3 6 are faithful, with 3 4 = 3 ⋆ 3 and 3 6 = 3 ⋆ 5 . The mass matrices obtained from h l and h M using Eqs.(
masses m 1
1= 12.70×10 −3 eV, m 2 = 50.78×10 −3 eV, m 3 = 51.49 × 10 −3 eV. This fit is obtained with: |v 2 /v 1 | = 0.31, arg(v 2 /v 1 ) = 0.81,(|v 3 /v 1 | = 0.93, arg(v 3 /v 1 ) = 2.15. Let us point out some remarks on the uniqueness of this result. Numerical explorations suggest finite, possibly disconnected, regions in VEV space with χ 2 < 0.5, so there are multiple choices of the v i yielding neutrino mixing and masses consistent with experimental data. Besides ((3 1 , 3 3 , 3 6 ), (3 1 , 3 6 )) there is another valid representation assignment: ((3 3 , 3 4 , 3 2 ), (3 3 , 3 2 )
TABLE I .
IComparison of the 3HDM predictions to the experimental oscillation data, for selected groups.GAP-ID
Structure
3D Irreps (Faithful)
Dirac Mass
Maj. Mass
Dirac Mixing χ 2 /4
Maj. Mixing χ 2 /4
[12, 3]
A4
1 (1)
-
-
-
-
[21, 1]
T7
2 (2)
a
-
d
-
[24, 12]
S4
2 (2)
b,c
-
5.3
-
[39, 1]
T13
4 (4)
a
-
d
-
[48, 3]
∆(48)
5 (4)
a
a
d
d
[48, 30]
A4 : C4
4 (2)
b
-
d
-
[48, 48]
C2 × S4
4 (2)
b,c
-
5.3
-
[60, 5]
A5
2 (2)
b
-
d
-
[72, 42]
C3 × S4
6 (4)
b,c
-
5.3
-
[84, 11]
9 (6)
a
a
d
d
[96, 64]
∆(96)
6 (4)
a,b,c
a,c
d
0.0
[96, 68]
10(4)
a
a
d
d
[96, 186]
C4 × A4
8 (4)
b.c
-
5.3
-
[108, 15]
Σ(36 × 3)
8 (8)
b,c
-
-
-
[120, 37]
C5 × S4
10 (8)
b,c
-
5.3
-
[150, 5]
∆(150)
8 (8)
a,b,c
a,b,c
d
0.0
[192, 182]
∆(96, 2)
12 (4)
a,b,c
a,b
d
0.0
[192, 944]
C2 × ∆(96)
12 (4)
a,b,c
a,c
d
0.0
[243, 19]
Z ′′ (3, 3)
24 (18)
a,c
-
150
-
[432, 239]
Π(1, 2)
16 (8)
b
a
d
d
[729, 63]
Z ′′ (3, 4)
72 (54)
a,c
-
150
-
[864, 675]
Π(1, 3)
32 (16)
b
a
d
d
TABLE II .
IIExperimental oscillation parameters. Extracted
from [28] Table 3.
Parameter
Value
σ
∆m 2
21 (10 −5 eV 2 )
7.5
0.18
∆m 2
32 (10 −5 eV 2 )
249
3.5
sin 2 Θ12
0.310
0.015
sin 2 Θ23
0.566
0.025
sin 2 Θ13
0.0224
0.0007
δCP (°)
250
35
hence define solution vectors h l . If we remove vectors
such that no two vectors differ from each other by a con-
stant, only 5 inequivalent vectors are left. The same num-
bers come for the Dirac neutrino mass term (Eq.(17)).
For the Majorana mass term (Eq.(18)) there are 24 valid
representation assignments of which 3 are inequivalent.
Finally, for the neutrino mixing term we find 16 valid
representation assignments, 8 of which are equivalent to
((3 1 , 3 3 , 3 6 ), (3 1 ,
ACKNOWLEDGMENTSThis research was funded by the Polish National Science Center (NCN) under grant 2020/37/B/ST2/0237. We thank Jacek Holeczek for his help. We are very grateful to the GAP Support Group for their advising.
Why three generations?. M Ibe, A Kusenko, T T Yanagida, Physics Letters B. 758365M. Ibe, A. Kusenko, and T. T. Yanagida, Why three generations?, Physics Letters B 758, 365 (2016).
Neutrino Mass and Mixing: from Theory to Experiment. S F King, A Merle, S Morisi, Y Shimizu, M Tanimoto, arXiv:1402.4271New J. Phys. 1645018hep-phS. F. King, A. Merle, S. Morisi, Y. Shimizu, and M. Tani- moto, Neutrino Mass and Mixing: from Theory to Exper- iment, New J. Phys. 16, 045018 (2014), arXiv:1402.4271 [hep-ph].
X Qian, P Vogel, arXiv:1505.01891Neutrino Mass Hierarchy. 831hepexX. Qian and P. Vogel, Neutrino Mass Hierarchy, Prog. Part. Nucl. Phys. 83, 1 (2015), arXiv:1505.01891 [hep- ex].
Simple nonAbelian finite flavor groups and fermion masses. P H Frampton, T W Kephart, arXiv:hep-ph/9409330Int. J. Mod. Phys. 104689hep-phP. H. Frampton and T. W. Kephart, Simple nonAbelian finite flavor groups and fermion masses, Int. J. Mod. Phys. A10, 4689 (1995), arXiv:hep-ph/9409330 [hep-ph].
Direct neutrino-mass measurement with sub-electronvolt sensitivity. M Aker, Nature Phys. 18160M. Aker et al., Direct neutrino-mass measurement with sub-electronvolt sensitivity, Nature Phys. 18, 160 (2022).
Constraint on the matter-antimatter symmetry-violating phase in neutrino oscillations. K Abe, T2KarXiv:1910.03887Nature. 580Erratum: Nature 583, E16 (2020). hep-exK. Abe et al. (T2K), Constraint on the mat- ter-antimatter symmetry-violating phase in neutrino os- cillations, Nature 580, 339 (2020), [Erratum: Nature 583, E16 (2020)], arXiv:1910.03887 [hep-ex].
Measurement of Reactor Antineutrino Oscillation Amplitude and Frequency at RENO. G Bak, RENO)arXiv:1806.00248Phys. Rev. Lett. 121201801hep-exG. Bak et al. (RENO), Measurement of Reactor Antineu- trino Oscillation Amplitude and Frequency at RENO, Phys. Rev. Lett. 121, 201801 (2018), arXiv:1806.00248 [hep-ex].
Lepton flavor symmetries. F Feruglio, A Romanino, arXiv:1912.06028Rev. Mod. Phys. 9315007hep-phF. Feruglio and A. Romanino, Lepton flavor symmetries, Rev. Mod. Phys. 93, 015007 (2021), arXiv:1912.06028 [hep-ph].
Finite Modular Groups and Lepton Mixing. R De Adelhart Toorop, F Feruglio, C Hagedorn, arXiv:1112.1340Nucl. Phys. B. 858437hep-phR. de Adelhart Toorop, F. Feruglio, and C. Hagedorn, Finite Modular Groups and Lepton Mixing, Nucl. Phys. B 858, 437 (2012), arXiv:1112.1340 [hep-ph].
Lepton Mixing Patterns from a Scan of Finite Discrete Groups. M Holthausen, K S Lim, M Lindner, arXiv:1212.2411Phys. Lett. B. 72161hep-phM. Holthausen, K. S. Lim, and M. Lindner, Lepton Mix- ing Patterns from a Scan of Finite Discrete Groups, Phys. Lett. B 721, 61 (2013), arXiv:1212.2411 [hep-ph].
Tribimaximal Mixing From Small Groups. K M Parattu, A Wingerter, arXiv:1012.2842Phys. Rev. 8413011hep-phK. M. Parattu and A. Wingerter, Tribimaximal Mixing From Small Groups, Phys. Rev. D84, 013011 (2011), arXiv:1012.2842 [hep-ph].
Group Theory and Dynamics of Neutrino Mixing. C S Lam, arXiv:1104.0055Phys. Rev. D. 83113002hep-phC. S. Lam, Group Theory and Dynamics of Neutrino Mixing, Phys. Rev. D 83, 113002 (2011), arXiv:1104.0055 [hep-ph].
Finite Symmetry of Leptonic Mass Matrices. C S Lam, arXiv:1208.5527Phys. Rev. D. 8713001hepphC. S. Lam, Finite Symmetry of Leptonic Mass Matrices, Phys. Rev. D 87, 013001 (2013), arXiv:1208.5527 [hep- ph].
Theory and phenomenology of two-higgsdoublet models. G Branco, P Ferreira, L Lavoura, M Rebelo, M Sher, J P Silva, theory and phenomenology of two-Higgs-doublet models. 516G. Branco, P. Ferreira, L. Lavoura, M. Rebelo, M. Sher, and J. P. Silva, Theory and phenomenology of two-higgs- doublet models, Physics Reports 516, 1 (2012), theory and phenomenology of two-Higgs-doublet models.
Lepton masses and mixing in a two-Higgs-doublet model. P Chaber, B Dziewit, J Holeczek, M Richter, M Zralek, S Zajac, arXiv:1808.08384Phys. Rev. D. 9855007hep-phP. Chaber, B. Dziewit, J. Holeczek, M. Richter, M. Zralek, and S. Zajac, Lepton masses and mixing in a two-Higgs-doublet model, Phys. Rev. D 98, 055007 (2018), arXiv:1808.08384 [hep-ph].
Family symmetries and multi higgs doublet models. B Dziewit, J Holeczek, S Zajac, M Zralek, 10.3390/sym12010156Symmetry. 12B. Dziewit, J. Holeczek, S. Zajac, and M. Zralek, Family symmetries and multi higgs doublet models, Symmetry 12, 10.3390/sym12010156 (2020).
S F King, C Luhn, arXiv:1301.1340Neutrino Mass and Mixing with Discrete Symmetry. 7656201hep-phS. F. King and C. Luhn, Neutrino Mass and Mixing with Discrete Symmetry, Rept. Prog. Phys. 76, 056201 (2013), arXiv:1301.1340 [hep-ph].
Phenomenology of models with more than two Higgs doublets. Y Grossman, arXiv:hep-ph/9401311Nucl. Phys. B. 426355Y. Grossman, Phenomenology of models with more than two Higgs doublets, Nucl. Phys. B 426, 355 (1994), arXiv:hep-ph/9401311.
Three-Higgs-doublet models: symmetries, potentials and Higgs boson masses. V Keus, S F King, S Moretti, arXiv:1310.8253JHEP. 0152hep-phV. Keus, S. F. King, and S. Moretti, Three-Higgs-doublet models: symmetries, potentials and Higgs boson masses, JHEP 01, 052, arXiv:1310.8253 [hep-ph].
Classification of finite reparametrization symmetry groups in the three-higgsdoublet model. I P Ivanov, E Vdovin, 10.1140/epjc/s10052-013-2309-xThe European Physical Journal C. 73I. P. Ivanov and E. Vdovin, Classification of finite reparametrization symmetry groups in the three-higgs- doublet model, The European Physical Journal C 73, 10.1140/epjc/s10052-013-2309-x (2013).
Baryon and Lepton Nonconserving Processes. S Weinberg, Phys. Rev. Lett. 431566S. Weinberg, Baryon and Lepton Nonconserving Pro- cesses, Phys. Rev. Lett. 43, 1566 (1979).
On the finite subgroups of U(3) of order smaller than 512. P O Ludl, arXiv:1006.1479Erratum: J. Phys. 43139501J. Phys.. math-phP. O. Ludl, On the finite subgroups of U(3) of order smaller than 512, J. Phys. A43, 395204 (2010), [Erratum: J. Phys.A44,139501(2011)], arXiv:1006.1479 [math-ph].
Gap, Gap -Groups, Algorithms, and Programming. Version 4.11.1 , The GAP GroupGAP, GAP -Groups, Algorithms, and Programming, Version 4.11.1 , The GAP Group (2021).
Repsn, a GAP4 Package for constructing representations of finite groups. V Dabbaghian, Version 3.0.2. refereed GAP packageV. Dabbaghian, Repsn, a GAP4 Package for con- structing representations of finite groups, Version 3.0.2, http://www.sfu.ca/~vdabbagh/gap/repsn.html (2011), refereed GAP package.
. W R Inc, Mathematica, Version 12.3.1, champaign, IL, 2021W. R. Inc., Mathematica, Version 12.3.1, champaign, IL, 2021.
. H U Besche, B Eick, E O'brien, small, the Small Groups library, Version 2.1. GAP 4.7.6 componentH. U. Besche, B. Eick, and E. O'Brien, small, the Small Groups library, Version 2.1, http://www.icm.tu-bs.de/ag algebra/software/ small/ (2014), GAP 4.7.6 component.
. P F Salas, D V Forero, S Gariazzo, P Martínez-Miravé, O Mena, C A Ternes, M Tórtola, J W , P. F. de Salas, D. V. Forero, S. Gariazzo, P. Martínez- Miravé, O. Mena, C. A. Ternes, M. Tórtola, and J. W. F.
global reassessment of the neutrino oscillation picture. Valle, arXiv:2006.11237JHEP. 0271hep-phValle, 2020 global reassessment of the neutrino oscillation picture, JHEP 02, 071, arXiv:2006.11237 [hep-ph].
| [] |
[
"Compartmental epidemic model to assess undocumented infections: applications to SARS-CoV-2 epidemics in Brazil",
"Compartmental epidemic model to assess undocumented infections: applications to SARS-CoV-2 epidemics in Brazil"
] | [
"Guilherme S Costa \nDepartamento de Física\nUniversidade Federal de Viçosa\n36570-900ViçosaMinas GeraisBrazil\n",
"Wesley Cota \nDepartamento de Física\nUniversidade Federal de Viçosa\n36570-900ViçosaMinas GeraisBrazil\n",
"Silvio C Ferreira \nDepartamento de Física\nUniversidade Federal de Viçosa\n36570-900ViçosaMinas GeraisBrazil\n\nNational Institute of Science and Technology for Complex Systems\n22290-180Rio de JaneiroBrazil\n"
] | [
"Departamento de Física\nUniversidade Federal de Viçosa\n36570-900ViçosaMinas GeraisBrazil",
"Departamento de Física\nUniversidade Federal de Viçosa\n36570-900ViçosaMinas GeraisBrazil",
"Departamento de Física\nUniversidade Federal de Viçosa\n36570-900ViçosaMinas GeraisBrazil",
"National Institute of Science and Technology for Complex Systems\n22290-180Rio de JaneiroBrazil"
] | [] | Nowcasting and forecasting of epidemic spreading, fundamental support for policy makers' decisions, rely on incidence series of reported cases to derive the fundamental epidemiological parameters. Two relevant drawbacks for predictions are the unknown fraction of undocumented cases and levels of nonpharmacological interventions that span highly heterogeneously across different places. We describe a simple approach using a compartmental model including asymptomatic and pre-asymptomatic contagions that allows to estimate both the level of undocumented infections and the value of Rt from reported case series in terms of epidemiological parameters. The method was applied to epidemic series for of COVID-19 across different municipalities in Brazil allowing to quantify the heterogeneity level of under-reporting across different places. The reproductive number derived within the current framework is little sensitive to both diagnosis and infection rates during the asymptomatic states, while being very sensitive to variations in case count series. The methods described here are general and we expect that they can be extended to other epidemiological approaches and surveillance data. | null | [
"https://arxiv.org/pdf/2201.03476v1.pdf"
] | 245,837,257 | 2201.03476 | 295f4839de1c7324cc936bd03019c13a103040f3 |
Compartmental epidemic model to assess undocumented infections: applications to SARS-CoV-2 epidemics in Brazil
Guilherme S Costa
Departamento de Física
Universidade Federal de Viçosa
36570-900ViçosaMinas GeraisBrazil
Wesley Cota
Departamento de Física
Universidade Federal de Viçosa
36570-900ViçosaMinas GeraisBrazil
Silvio C Ferreira
Departamento de Física
Universidade Federal de Viçosa
36570-900ViçosaMinas GeraisBrazil
National Institute of Science and Technology for Complex Systems
22290-180Rio de JaneiroBrazil
Compartmental epidemic model to assess undocumented infections: applications to SARS-CoV-2 epidemics in Brazil
Nowcasting and forecasting of epidemic spreading, fundamental support for policy makers' decisions, rely on incidence series of reported cases to derive the fundamental epidemiological parameters. Two relevant drawbacks for predictions are the unknown fraction of undocumented cases and levels of nonpharmacological interventions that span highly heterogeneously across different places. We describe a simple approach using a compartmental model including asymptomatic and pre-asymptomatic contagions that allows to estimate both the level of undocumented infections and the value of Rt from reported case series in terms of epidemiological parameters. The method was applied to epidemic series for of COVID-19 across different municipalities in Brazil allowing to quantify the heterogeneity level of under-reporting across different places. The reproductive number derived within the current framework is little sensitive to both diagnosis and infection rates during the asymptomatic states, while being very sensitive to variations in case count series. The methods described here are general and we expect that they can be extended to other epidemiological approaches and surveillance data.
Nowcasting and forecasting of epidemic spreading, fundamental support for policy makers' decisions, rely on incidence series of reported cases to derive the fundamental epidemiological parameters. Two relevant drawbacks for predictions are the unknown fraction of undocumented cases and levels of nonpharmacological interventions that span highly heterogeneously across different places. We describe a simple approach using a compartmental model including asymptomatic and pre-asymptomatic contagions that allows to estimate both the level of undocumented infections and the value of Rt from reported case series in terms of epidemiological parameters. The method was applied to epidemic series for of COVID-19 across different municipalities in Brazil allowing to quantify the heterogeneity level of under-reporting across different places. The reproductive number derived within the current framework is little sensitive to both diagnosis and infection rates during the asymptomatic states, while being very sensitive to variations in case count series. The methods described here are general and we expect that they can be extended to other epidemiological approaches and surveillance data.
I. INTRODUCTION
Our contemporary society is facing an unprecedented threat imposed by the COVID-19, caused by the pathogen SARS-CoV-2, evidencing the importance, limitations, and subtleties of using compartmental epidemic models for the forecasting or even the nowcasting of pandemic scenarios [1][2][3][4][5]. After two years of intensive investigation, much has been learned with respect to the virology of SARS-CoV-2 in humans [6][7][8][9], several key aspects of the transmission were unveiled [5,[10][11][12], and efficient vaccines have been developed [13], among other achievements. Variants of SARS-CoV-2 [14,15] give rise to new and more aggressive outbreaks due to reinfection and raised contagion rates implying that natural herd immunity is definitely not an option [16]. The newest Omicron (B.1.1.529) variant [17], containing several mutations on genes of the spike proteins, a leading target of antibodies produced to combat a COVID-19 infection, emerges as a new concern. Whilst the biology of the virus and interaction with human hosts is better understood, other crucial aspects of the epidemiology, specially the behavioural ones, remains unpredictable even at a short-term, varying across time and location. In particular the non-pharmaceutical interventions (NPI), such as face masks, testing policies, and social distancing, have played a central role on spreading of COVID-19 [18][19][20][21]. The aforementioned NPIs contribute for reduction or increase of the effective contagion rate that, therefore, must be inferred from count case series via likelihood or other calibration methods [22,23].
The most stringent epidemic characteristic of the SARS-CoV-2 contagion in humans is probably its high transmission before the onset of the symptoms [5,11,24], the presymptomatic individuals, and even for those that never manifest relevant symptoms [25,26], the true asymptomatic individuals. The latter could be accessed by mass testing and contact tracing, for example, practices defended by experts but rarely implemented. Seroprevalence studies for different phases and regions [14,27] reveals population incidences of antibodies for SARS-CoV-2 in levels much higher than counts of COVID-19 cases reported in surveillance systems. So, the case fatality ratio (CFR), defined as the ratio between number of deaths and diagnosed cases, can differ substantially from the infection fatality ratio (IFR), defined as the fraction of all infections (documented or not) that evolves to death [27].
The level of under-reporting, in which the CFR is greater than the IFR, varies widely in different seroprevalence inquiries [27] due to several uncontrolled factors such as the testing policies (only symptomatic cases, contact tracing, etc.), availability (low or high income places), sensitivity (antigen or PCR), and seek for medical care, among others. The relation between seropositivity and immunity is not fully established and new emerging variants always opens path for reinfections and new outbreaks [28]. Therefore, to estimate the level of undocumented infections across different places and times remains a challenge. Epidemic models of statistical inference were developed to access the amount of undocumented infections of SARS-CoV-2. For example, Pullano et al. [29] estimated that 9 out 10 cases of symptomatic infections were not ascertained by the surveillance system in France from 11 May to 28 June 2020, suggesting that large numbers of symptomatic cases of COVID-19 did not seek for medical advice. Lu et al. [30] considered four complementary approaches to estimate the cumulative incidence of symptomatic cases of COVID-19 in the US and concluded that on April 4, 2020, the estimated case count was 5 to 50 times higher than the official positive test counts across the different states. Subramaniana et al. [31] used a model including testing information to fit the case and serology data from New York City to estimate a low proportion of symptomatic cases (13 to 18% of the total infections), and that the reproductive number could be larger than often assumed. Similarly, Irons and Raftery [32] used a similar approach to estimate that approximated 60% of the infection were not diagnosed by tests in USA as of March 7, 2021. Due to the importance of asymptomatic or presymptomatic transmission, the corresponding compartments were soon included in mathematical models for COVID-19 [11,[33][34][35][36]. However, it is concomitantly an additional source of uncertainty in the initial conditions. Predictive scenarios of the first SARS-CoV-2 outbreak were either semi-quantitative [33,37,38] or based on Bayesian inference using reported cases' series [34,39,40]. Brazil is an example, certainly not an exception, of highly heterogeneous responses to COVID-19 pandemics due to the lack of coordinated policies across different administrative layers [41], in addition to the intrinsic variability of social-economic indexes across the country impacting directly the epidemiological outcomes. Therefore, a mechanistic approach for simulation of epidemic spreading with asymptomatic transmission calls for a systematic way to determine the initial conditions.
The contribution of asymptomatic infections and testing policies to the effective reproductive number R t [42] through surveillance counts is an important issue [32]. The basic reproductive number is defined as the average number of secondary infections generated by a single infected individual introduced in a completely susceptible population, commonly represented by R 0 . The effective reproductive number is given by R t = S(t)R 0 , where S is the fraction of susceptible population (who can be infected) at time t. This definition, under the hypothesis of homogeneous mixing, is the simplest one and can be generalized to stratified compartments [42]. The reproductive number can also be estimated directly from case counts using statistical inference models [22], as commonly reported for COVID-19 pandemics across the world [19,34,41,43].
In this present work, we describe an approach to estimate the number of undocumented cases (asymptomatic or not) using the epidemic surveillance data for cases and deaths. The method is grounded on a compartmental epidemic model including both documented and undocumented compartments (asymptomatic, exposed, symptomatic, and so on), the latter not counting for surveillance reports. The present approach allows to determine the effective reproductive number, the level of under-reporting, and initial condition using the day of diagnosis. The approach can be promptly modified or generalized for other types of data and epidemic compartments. The method shares similarities with the recent approaches to estimate undocumented cases [25,29,30,32], such as the use of reported infections and deaths. The central difference is that our approach is more mechanistic and less Bayesian.
We applied the method across different geographical scales of two Brazilian states, namely Paraná (PR) and Espírito Santo (ES) using time series with dates of di-agnosis of COVID-19 counts available by the epidemic surveillance of the respective states. The time windows of investigation was from 1 January to 31 May of 2021 corresponding to the second epidemic wave driven mainly by the Gamma variant [41,44]. We observed variable levels of under-reporting across different places and times. Particularly, the analysis indicates that the fraction of undocumented cases is correlated positively with the epidemic incidence: the higher the epidemic incidence in case counts the larger the fraction of undocumented infections. We were able to estimate initial conditions for the hidden compartments and effective infection rates which gave an efficient short-time forecast for the series of confirmed cases. Despite the basic reproductive number being explicitly dependent on the asymptomatic transmission, the analysis indicates that undocumented infections seem to not alter significantly the reproductive number for the analyzed data.
The remaining of this paper is organized as follows. The methodology is detailed in section II. The epidemic compartmental model and some analytical results are presented in subsection II A. The data-driven approach to estimate the under-reporting level from surveillance counts is described in subsection II B while the eigenvalue approach to determine the initial conditions is presented in subsection II C. The application of method to epidemiological data is presented in section III and the main conclusions of the work are discussed in section IV.
II. SIMPLE APPROACH TO ESTIMATE UNDOCUMENTED CASES
A. Compartmental model
Presymptomatic, asymptomatic and undocumented transmissions are investigated using a compartmental model [42] under the homogeneous mixing hypothesis grouping individuals according to their epidemic states in the following compartments: Susceptible (S) who can be infected; exposed (E) who were infected but is not contagious yet; asymptomatic (A) who are infectious but do not present symptoms; symptomatic (I) ones who may seek for medical care and testing due to the presence of symptoms; undocumented recovered (R) who has been infected, healed but not diagnosed; deceased (D) who died due to COVID-19; two compartments of diagnosed cases for SARS-CoV-2 including individuals who were asymptomatic (C A ) or symptomatic (C I ) at moment of testing; and the corresponding recovered compartments for confirmed cases R A and R I . The epidemiological model and rates are schematically depicted in Fig. 1.
Susceptible persons in contact with infectious (asymptomatic or symptomatic) individuals become exposed with rates λ A and λ I , respectively. For sake of simplicity, confirmed cases are assumed to be isolated and do not contribute for new infections. The remaining transitions are spontaneous and expressed in Fig viduals become asymptomatic with rate µ A . The latter can evolve to a symptomatic state with rate β I , recover with rate β R , or be diagnosed by tests with rate β C moving to the confirmed compartment C A . Similarly, the undocumented symptomatic ones can recover with rate α R or be diagnosed and become C I with rate α C . The clinical state of confirmed cases evolves as does the undocumented ones. An infected confirmed case (C I ) can die (D) with rate η while undocumented deaths are neglected, again, for sake of simplicity. The true asymptomatic and the presymptomatic cases are implicitly considered with transitions A→R (C A →R A ) and A→I (A→C A →C I ), respectively. See Methods for the complete set of equations.
Consider a more intuitive parameterization in terms of the probabilities p A and p I that infected individuals are diagnosed during the asymptomatic or symptomatic phases, respectively, which can be computed from the compartmental model and are given by
p A = β C β I + β C + β R and p I = α C α C + α R .(1)
One can also show that an exposed individual ends diagnosed with probability
P C = p A + (1 − p A )p I φ,(2)
where φ = β I /(β I + β R ). The first and second terms of (2) are due to diagnosis during asymptomatic and symptomatic phases, respectively, while recovering without diagnosis happens with probability P R = 1 − P C . Therefore, we can determine a simple relation between the final number of documented (N C ) and undocumented (N R ) infections defining the under-reporting coefficient σ ur as
σ ur = N R N C = P R P C = (1 − p A )(1 − φp I ) p A + (1 − p A )p I φ ,(3)
where
N C = N CA + N CI + N RA + N RI + N D .
We can also analytically determine the model's IFR IFR , considering the probabilities that exposed individuals evolve to death passing through C A compartment or not, that are p A
φη η+αR or (1 − p A )φp I η η+αR , respectively. The IFR becomes IFR = [p A + (1 − p A )p I ] φη η + α R .(4)
B. Estimating under-reporting from epidemic surveillance counts
The rates µ A , β I , β R , α R , and η are biological and can, in principle, be found in epidemiological surveys [6-9, 12, 45]. The parameters λ A and λ I depend on behavioral aspects such as the number of potential infectious contacts per unit of time [20,34,38]; prophylactic attitudes by means of NPI such as mask wearing and adoption of social distancing [46][47][48]; infectiousness and prevalence of new variants [14,41,49]; to cite only some of the most prominent issues. Similarly, the confirmation rates β C and α C depend on several behavioral and socioeconomic factors being highly influenced by testing policies [38,50,51]. All these aspects are very heterogeneously distributed across time and different places.
We describe how testing probabilities can be estimated from surveillance count series with the aid of the compartmental model of Figure 1. Let C(t) and D(t) represent the cumulative series of confirmed cases and deaths; and let x A be the fraction of them which were confirmed during the asymptomatic stage. We equate the CFR computed for reported cases within a given time window [t cal , t cal + ∆τ ]:
CFR ≡ ∆D(t cal ) ∆C(t cal − t delay ) ,(5)
and the CFR extracted from the model such that
CFR = η η + α R x A φ + η η + α R (1 − x A ),(6)
in which t delay is a delay between reported death and positive test report. The first and second terms in the righthand side of (6) represent the probabilities of death for who were diagnosed for SARS-CoV-2 during the asymptomatic and symptomatic stages, respectively. Now, taking the ratio between Eqs. (4) and (5), one obtains
IFR CFR = [p A + (1 − p A )p I ]φ 1 − x A (1 − φ) .(7)
Despite its simplicity, Eq. (7) is very handy since it relates the testing rates (or probabilities) with epidemiological parameters ( IFR and φ) or factors (x A and CFR ) which could, in principle, be obtained directly from data. Therefore, if the ratio r = p A /p I is given, the testing rates can be estimated. We assume r ≤ 1, that is, the chance of detecting asymptomatic is lower than of symptomatic cases. Equation (7) is therefore consistent (
p A ≤ p I ≤ 1) only if ζ = [1 − x A (1 − φ)] IFR CFR ≤ φ.(8)
If inequality (8) is not satisfied, a value p I 1 (α C α R ) is fixed since p I = 1 means instantaneous transition to the C I compartment (α C = ∞) that implies numerical difficulties. The parameter x A is not commonly available in epidemic surveillance data, which usually report only the total number of confirmed cases. However, we may assume x A 1 approximating the denominator of Eq. (7) by 1, for which we expect a small error in general. Also, a sensibility analysis of the ratio r can be used to verify whether the results are little sensitive to this choice (it was the case all data analyzed in this work); otherwise the ratio must be determined using some calibration or likelihood method.
C. Assessing hidden compartments from epidemic surveillance data
Epidemic surveillance provides the number of confirmed cases, deaths, date of first symptoms, or diagnosis; nothing with respect to the other compartments is commonly available. Actually, in the real scenario, the situation is much more complicated due to delays and other complex issues on surveillance counts [53,54]. Using the case series C(t), we estimate infection rates λ A and λ I concomitantly with the initial conditions (S * , E * , A * , I * , R * ) using the calibration procedure described in Methods Section AA 2. We applied the method to two types of count series available for Brazil, hereafter named Type-I and Type-II. The former consists of count series using release dates provided by epidemic surveil- (8). Evolution of σur for the capital cities of (b) Manaus and (c) São Paulo estimated using moving time windows of 3 weeks for type-I count series (see main text) as notified by state surveillance departments [52]. Two values of the ratio pA/pI = 0.1 and 0.5 were compared. The interval of confidence of is 95% shown in the shaded region.
lance departments of Brazilian federation units 1 which are aggregated and publicly available for all 5570 Brazillian municipalities [52]. These data do not yield the date of diagnosis and may present uncontrolled bias caused by reporting delays and should be used with care. The type-II data sets contain dates of diagnosis and first symptoms onset. In this work, we use the publicly available Type-II data for Paraná (PR) [55] and Espirito Santo (ES) [56] states. The data are publicly available in the cited resources and the data aggregated for different municipalities, used in the present work, is available on [57]. A full description of these datasets is available in the Supplementary Material (SM) [58].
We fixed the average values of parameters µ −1 A = 3.2 d and β −1 I = 3.2 d so that the mean incubation time is of 6.4 d [6,34]. The mean recovery time for symptomatic individuals was taken as α −1 R = 3.2 d [59]. Following [33,34], asymptomatic cases were assumed to have the same recovering time such that β −1 R = β −1 I + α −1 R . Sensibility analysis was done drawing µ A , β I , and α R from Gamma distributions with standard deviation of 1.3 d. The IFR IFR was drawn from a uniform distribution in the range 0.5% to 1% in accordance with seroprevalence studies; 0 1 -J a n 1 5 -J a n 2 9 -J a n See Ref. [27] for collection of reports up to September 2020. We analyzed case report series for PR state and found out a delay of t delay ≈ 10 d between death and positive test report. For ES state this delay was t delay ≈ 20 d. The delay is obtained by shifting the time series so that the peaks of deaths and cases coincide, as shown in Fig. S1 of the Supplementary Material (SM) [58]. We consider t tr as January 1st 2021.
III. RESULTS
A. Under-reporting coefficient
The under-reporting coefficient σ ur is little sensitive to the choice of ratio r = p A /p I as shown in Fig 2(a). The evolution of σ ur using r = 0.1 and 0.5 for Type-I count series of two capital cities of Brazil, which were severely impacted by COVID-19 second infection wave, namely Manaus and São Paulo [41], are presented in Figs. 2(b) and (c), for which the estimated delays between case and death confirmations were t delay = 7 and 9 days, respectively; see Figs. S1 (a) and (b) in the SM [58]. The value of σ ur is practically the same or both ratios with differences lying within a confidence interval of 95%. The ratio r = 0.1 is adopted in all results presented hereafter. The values of σ ur were higher when both cities were facing high epidemic incidence.
We analyzed Type-II count series for PR and ES states aggregating data of municipality level into immediate regions defined by the Instituto Brasileiro de Geografia e Estatística (IBGE) [60] as a group of nearby municipalities of a same state with intense interchange for immediate needs (purchasing, work, healthcare, education, and so on). The evolution of σ ur since January 2021 computed with counts aggregated by states and two selected immediate regions are shown in Figures 3(a) and (b) for PR and ES, respectively. Curves for the 28 and 8 immediate regions of PR and ES, respectively, with the confidence intervals are available in Figs. S2 and S3 of the SM [58]. A strong correlation between σ ur and the CFR is observed, Fig. 3(c), since Eqs. (7) and (3) imply σ ur ∝ CFR for a fixed value of p I . The second relevant outcome is the substantial variation of undocumented infection along the time and across different places. For example, in Nova Venécia-ES, σ ur ranges from below 1 to higher than 6. The under-reporting coefficient for all immediate regions of both PR ans ES states are presented in Figs. 3(d) and (e); the chosen dates correspond to low and high CFR in the respective state counts. The differences between immediate regions can be up to 3-fold in a same time interval. The space-time variability reflects the high diversities of outbreak across different places, due to unsynchronized and unequal responses to pandemics besides demographic, economic, and developmental heterogeneity of states as predicted [33] and later observed [41] for the first epidemic wave in Brazil.
B. Determination of the initial conditions
To exemplify the calibration method, we performed the analysis for case counts of the PR state shown in Fig. 4; see Fig. S4 on SM [58] for the ES state. We further simplified the analysis assuming the same infection rate for both asymptomatic and symptomatic individuals prior diagnosis confirmation, λ A = λ I , implying in a single parameter to fit the data. Typical calibration curves are presented in Fig. 4(a)-(i) for different times using a 14-day moving window of calibration. A forecast of 1 week is also presented to verify the calibration robustness, reproducing very well the short-term progression of the cumulative case count time series. The method also performs very well for smaller geographical scales such as immediate regions; Fig. S5 of the SM [58]. The evolution of the undocumented epidemic compartments (exposed, asymptomatic, and symptomatic) yielded by the calibration method for PR state from January to May 2021 is presented in Fig. 5. Remark that the ratio between the total amount of infected individuals and the number of confirmed cases at a given day is much higher than the under-reporting coefficient shown in Fig. 3 since the latter refers to the final epidemic chain where an infection ends documented or not, whereas the former refers to the amount of infected individuals in a given day which has not been documented yet. The peaks of prevalence of infectious cases happen slightly before peaks of incidence of confirmed cases. The effective reproduction number for PR state is presented in Fig. S6 on SM [58]. The calibration is sensitive to the variations and inflections in case count series, where the mean value of R t oscillated between approximately 0.9 and 1.2. We performed a sensibility analysis of R t and verified that its value is almost independent of the testing rates of asymptomatic compartments. More precisely, the curves of R t collapses within the confidence interval when the ratios between testing probabilities p A /p I and infection rates λ A /λ I are varied by one order of magnitude. 0 1 -J a n 0 8 -J a n 1 5 -J a n 2 2 -J a n 2 9 -J a n 0
IV. DISCUSSION
The pandemic caused by the SARS-CoV-2 led to unprecedented efforts gathering scientific community, epidemic surveillance, public authorities, and communication systems to provide almost real-time updated and publicly available counts for diagnosed infections, deaths, and other important statistics for COVID-19 spread across the globe. Available epidemic series, however, are still not ideal since under-reporting, delays, and many other issues intrinsic of our limited capacity in documenting are unavoidable. Moreover, these limitations vary enormously across different places and at different moments reflecting the unequal contemporary world, from socioeconomic to educational perspectives. However, this opens new avenues for construction and improvement of tools to extract information which are not explicit in data. A common strategy, possibly the gold standard in applied epidemiology, is the use of Bayesian inference models [29,30]. A particularly promising strategy is the data-driven [33,34,37] approach where mathematical and mechanistic models are supplied by data, allowing to make predictions which are not explicitly avail-able. In the case of COVID-19 we have the important class of asymptomatic or pre-symptomatic infections, in which individuals transmit the SARS-CoV-2 pathogen even without symptoms, being very difficulty to be detected in epidemic surveillance systems.
In the present work we follow a data-driven approach using a compartmental model to estimate the amount of undocumented cases in the epidemic compartments which are not directly accessible in surveillance systems. The method allows to estimate the fraction of undocumented infections using case fatality ratio (CFR) and biological parameters estimated in controlled studies, in particular the infection fatality ratio (IFR). We applied the method to epidemic series of diagnosed cases and deaths of two Brazilian states where days of the symptoms onset were available. We selected the first semester of 2021 where Brazil was struck by a second epidemic wave of COVID-19, mainly driven the Gamma variant (lineage P.1). We calculated a under-reporting coefficient σ ur , giving the ratio between infections which ends diagnosed by tests or not. Our analysis reports a large variation of σ ur , up to one order of magnitude, along the time and also across different locations at the same period. The method allows to estimate the initial condition for the undocumented compartments, in particular the asymptomatic and exposed ones. The undocumented symptomatic cases are estimated as being approximately 1/5 of all infected individuals in a given moment. While, on the one hand, these numbers should be not interpreted as accurate estimates of actual epidemic prevalence, on the other hand, they clearly demonstrate that the infected individuals that can potentially seek for medical assistance are a minor part of all cases. Interestingly, the effective reproduction number is almost insensitive to the testing rate of asymptomatic cases, confirming that undocumented infections do not affect this important epidemic indicator.
The method can be generalized for stratified data including age contact matrices [61] or metapopulation approaches [33,34]. However, the main lesson is that initial conditions for undocumented compartments can be inferred using a simple mechanistic approach, based on compartmental models fueled by basic epidemic series of diagnosed death and cases. Nonetheless, the accuracy of methods depends on good estimates of biological parameters, mainly the IFR that changes as the epidemic scenario is altered. For example, vaccination is expected to reduce IFR while the emergence of more aggressive variants can increase it.
Appendix A: Methods
Here we describe the methods used to evolve the compartmental model and its calibration with real data.
Equations of the compartmental model
Assuming a constant population N = X N X , where N X is the number of individuals in the compartment X, the above transitions can be summarized in the following set of differential equations
dS dt = −(λ A A + λ I I)S (A1a) dE dt = (λ A A + λ I I)S − µ A E (A1b) dA dt = µ A E − (β I + β R + β C )A (A1c) dI dt = β I A − (α R + α C )I (A1d) dR dt = α R I + β R A (A1e) dC dt = β C A + α C I (A1f) dC A dt = β C A − (β R + β I )C A (A1g) dC I dt = α C I + β I C A − (α R + η)C I (A1h) dR A dt = β R C A (A1i) dR I dt = α R C I (A1j) dD dt = ηC I ,(A1k)
where X = N X /N , X ∈ {S, E,. . . , D}, is the corresponding population fraction in the compartment X. The basic reproductive number R 0 is straightforwardly computed and given by
R 0 = 1 β I + β C + β R λ A + λ I β I α C + α R . (A2)
Calibration procedure
In this section we describe the calibration procedure for the estimation of under-reporting from epidemic surveillance counts. The steps are the following:
i. Select the time interval [t cal , t cal + ∆τ ] for which the reported case series C(t) will be analyzed. This time window should be short enough to assume that infection rates λ A and λ I are approximately constant, but sufficiently large to have significant amount of data;
ii. Using the time series of case and death counts, determine the probability p I using Eq. (7) for a given ratio r = p A /p I , assumed to be a parameter of the method. Determine the under-reporting coefficient given by Eq. (3).
iii. Consider an adiabatic approximation assuming that susceptible population varies much more slowly than the other compartments such that one can neglect its variation and take S(t) ≈ S * as being constant over the investigated period.
iv. Start with guessed initial values for the products γ A = λ A S * and γ I = λ I S * (to be fitted with data).
v. Determine the number of undocumented cases N * R at t = t cal using the under-reporting coefficient calculated in step (ii) and the number of confirmed cases N * C = C(t cal ) − C(t tr ) from case counting, where t tr is a transient time.
vi. Under these conditions the compartmental model provides a closed linear systemẊ = JX for the infectious compartments X = (E, A, I) where the Jacobian is given by
J = −µ A γ A γ I µ A −(β I + β R + β C ) 0 0 β I −(α R + α C ) .(A3)
We assume that the solution is ruled by the leading term X ∼ v 1 exp[Λ 1 (t−t cal )] where v 1 = (v E , v A , v I ) is the principal eigenvector corresponding to the largest eigenvalue Λ 1 of J, providing two relation among initial conditions (E * , A * , I * )
E * A * ≈ v E v A ,(A4)I * A * ≈ v I v A .(A5)
Using again X ∼ v 1 exp[Λ 1 (t − t cal )], a closed system of initial conditions for (E * , A * , I * ) is obtained with the integration of Eq. (A1f)
∆C ≈ (β C A * + α C I * ) e Λ1∆τ − 1 Λ 1 ,(A6)
where ∆C is the increment of confirmed cases during the interval ∆τ which is given by the reported case counts. If Λ 1 ∆τ 1 we obtain
β C A * + α C I * ≈ ∆C ∆τ .(A7)
Finally, the susceptible population is determined as
N * S = N − X =S N X ,(A8)
S * = N S /N , and the infection rates self-consistently estimated as λ A = γ A /S * and λ I = γ I /S * .
vii. Equations (A1b) to (A1f) are integrated in the interval [t cal , t cal + ∆τ ] and the dispersion with respect to the case counts is computed as Ω(γ I , γ A ) = C − C .
viii. The parameters γ A and γ I are incremented interactively and steps (iv) to (vii) are implemented using a multi-parametric bisection method to minimize Ω(γ I , γ A ).
We remark that new variants and waning immunity lead to reinfections such that individuals recovered from an infection can still be susceptible, justifying to discard of cases before the transient time t tr . Another source of uncertainty is the vaccination which also confers unknown levels of immunity to infections affecting directly the susceptible population. Vaccination will impact both the IFR and CFR, such that the update values of the IFR should be considered.
Code and data: Fortran and Python codes used for calibration and processing the epidemic series were made publicly in [57]. A description of the datasets and codes can be found in the SM [58].
Author contributions: GSC wrote the codes and performed the numerical studies. WC collected epidemic data. GSC, WC, and SCF conceived the method. SCF wrote the first version of the manuscript. All authors edited the manuscript and analyzed the results.
. 1 .FIG. 1 .
11Exposed Schematic representation of the epidemic model including the following compartments: susceptible (S), exposed (E), asymptomatic (A), symptomatic (I), recovered (R, RA, and RI), deceased (D), and confirmed cases (CA and CI). The transition and respective rates are indicated by arrows. The infectious compartments are depicted with the symbol . The infection processes, represented by the dashed line, involve the interaction between susceptible and one of the infectious compartments, happening with rates λX, X=A, I, CA, and CI, which can depend on the compartment.
FIG. 2 .
2(a) Under-reporting coefficient σur as a function of pA/pI for different values of ζ = [1−xA(1−φ)] IFR CFR ; see equation
FIG. 3 .
3Evolution of under-reporting coefficient for (a) PR and (b) ES states using pA/pI = 0.1 and time windows of 3 weeks. Two immediate regions of each state are presented in the corresponding panels. (c) Evolution of the CFR computed using delays t delay = 10 d and 20 d for PR and ES states, respectively. Under-reporting coefficients for all immediate regions of (d) PR and (e) ES and the for states (indicated by arrows) computed when the CFR is low (Jan 2021) and high (April 2021).
FIG. 4 .
4Calibration curves for PR state in different time windows of 14 days indicated by the vertical lines. Initial day is in the top of each panel. One week of forecasting is also shown. Symbols are the cumulative cases' counts while lines with shaded regions represent the calibrated curves and the corresponding confidence interval of 95%.
FIG. 5 .
5Evolution of the undocumented compartments (exposed, asymptomatic and symptomatic) for the PR state since 1 January 2021.
Competing interests: Authors declare no competing interest. ACKNOWLEDGMENTS This work was partially supported by the Brazilian agencies Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -CAPES (Grant no. 88887.507046/2020-00), Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq (Grants no. 430768/2018-4 and 311183/2019-0) and Fundação de Amparoà Pesquisa do Estado de Minas Gerais -FAPEMIG (Grant no. APQ-02393-18). This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) -Brasil -Finance Code 001.
Brazil is divided into 26 states and 1 federal district. States aggregate municipalities with independent administrative structure. Cases are reported by municipalities to state's healthcare departments which release the information publicly.
. J T Wu, K Leung, G M Leung, 10.1016/S0140-6736(20)30260-9Lancet. 395689J. T. Wu, K. Leung, and G. M. Leung, Lancet 395, 689 (2020).
. J Zhang, M Litvinova, Y Liang, Y Wang, W Wang, S Zhao, Q Wu, S Merler, C Viboud, A Vespignani, M Ajelli, H Yu, 10.1126/science.abb8001Science. 3681481J. Zhang, M. Litvinova, Y. Liang, Y. Wang, W. Wang, S. Zhao, Q. Wu, S. Merler, C. Viboud, A. Vespignani, M. Ajelli, and H. Yu, Science 368, 1481 (2020).
. E Estrada, 10.1016/j.physrep.2020.07.005Phys. Rep. 8691E. Estrada, Phys. Rep. 869, 1 (2020).
. M Gilbert, G Pullano, F Pinotti, E Valdano, C Poletto, P.-Y Boëlle, E Ortenzio, Y Yazdanpanah, S P Eholie, M Altmann, B Gutierrez, M U G Kraemer, V Colizza, 10.1016/S0140-6736(20)30411-6Lancet. 395871M. Gilbert, G. Pullano, F. Pinotti, E. Valdano, C. Po- letto, P.-Y. Boëlle, E. D'Ortenzio, Y. Yazdanpanah, S. P. Eholie, M. Altmann, B. Gutierrez, M. U. G. Kraemer, and V. Colizza, Lancet 395, 871 (2020).
. R Li, S Pei, B Chen, Y Song, T Zhang, W Yang, J Shaman, 10.1126/science.abb3221Science. 368489R. Li, S. Pei, B. Chen, Y. Song, T. Zhang, W. Yang, and J. Shaman, Science 368, 489 (2020).
. Q Li, X Guan, P Wu, X Wang, L Zhou, Y Tong, R Ren, K S Leung, E H Lau, J Y Wong, X Xing, N Xiang, Y Wu, C Li, Q Chen, D Li, T Liu, J Zhao, M Liu, W Tu, C Chen, L Jin, R Yang, Q Wang, S Zhou, R Wang, H Liu, Y Luo, Y Liu, G Shao, H Li, Z Tao, Y Yang, Z Deng, B Liu, Z Ma, Y Zhang, G Shi, T T Lam, J T Wu, G F Gao, B J Cowling, B Yang, G M Leung, Z Feng, 10.1056/NEJMoa2001316N. Engl. J. Med. 3821199Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, R. Ren, K. S. Leung, E. H. Lau, J. Y. Wong, X. Xing, N. Xiang, Y. Wu, C. Li, Q. Chen, D. Li, T. Liu, J. Zhao, M. Liu, W. Tu, C. Chen, L. Jin, R. Yang, Q. Wang, S. Zhou, R. Wang, H. Liu, Y. Luo, Y. Liu, G. Shao, H. Li, Z. Tao, Y. Yang, Z. Deng, B. Liu, Z. Ma, Y. Zhang, G. Shi, T. T. Lam, J. T. Wu, G. F. Gao, B. J. Cowling, B. Yang, G. M. Leung, and Z. Feng, N. Engl. J. Med. 382, 1199 (2020).
. H Nishiura, N M Linton, A R Akhmetzhanov, 10.1016/j.ijid.2020.02.060Int. J. Infect. Dis. 93284H. Nishiura, N. M. Linton, and A. R. Akhmetzhanov, Int. J. Infect. Dis. 93, 284 (2020).
. D Baud, X Qi, K Nielsen-Saines, D Musso, L Pomar, G Favre, 10.1016/s1473-3099(20)30195-xLancet Infect. Dis. 20773D. Baud, X. Qi, K. Nielsen-Saines, D. Musso, L. Pomar, and G. Favre, Lancet Infect. Dis. 20, 773 (2020).
. X He, E H Y Lau, P Wu, X Deng, J Wang, X Hao, Y C Lau, J Y Wong, Y Guan, X Tan, X Mo, Y Chen, B Liao, W Chen, F Hu, Q Zhang, M Zhong, Y Wu, L Zhao, F Zhang, B J Cowling, F Li, G M Leung, 10.1038/s41591-020-0869-5Nat. Med. 26672X. He, E. H. Y. Lau, P. Wu, X. Deng, J. Wang, X. Hao, Y. C. Lau, J. Y. Wong, Y. Guan, X. Tan, X. Mo, Y. Chen, B. Liao, W. Chen, F. Hu, Q. Zhang, M. Zhong, Y. Wu, L. Zhao, F. Zhang, B. J. Cowling, F. Li, and G. M. Leung, Nat. Med. 26, 672 (2020).
. L C Tindale, J E Stockdale, M Coombe, E S Garlock, W Y V Lau, M Saraswat, L Zhang, D Chen, J Wallinga, C Colijn, 10.7554/eLife.571492020.03.03.200299839L. C. Tindale, J. E. Stockdale, M. Coombe, E. S. Gar- lock, W. Y. V. Lau, M. Saraswat, L. Zhang, D. Chen, J. Wallinga, and C. Colijn, eLife 9, 2020.03.03.20029983 (2020).
. J C Emery, T W Russell, Y Liu, J Hellewell, C A Pearson, K E Atkins, P Klepac, A Endo, C I Jarvis, N G Davies, E M Rees, S R Meakin, A Rosello, K Van Zandvoort, J D Munday, W J Edmunds, T Jombart, M Auzenbergs, E S Nightingale, M Jit, S Abbott, D Simons, N I Bosse, Q J Leclerc, S R Procter, C J Villabona-Arenas, D C Tully, A K Deol, F Y Sun, S Hué, A M Foss, K Prem, G Medley, A Gimma, R Lowe, S Clifford, M Quaife, C Diamond, H P Gibbs, B J Quilty, K Oreilly, G M Knight, R M Eggo, A J Kucharski, S Funk, S Flasche, R M Houben, 10.7554/eLife.5869991J. C. Emery, T. W. Russell, Y. Liu, J. Hellewell, C. A. Pearson, K. E. Atkins, P. Klepac, A. Endo, C. I. Jarvis, N. G. Davies, E. M. Rees, S. R. Meakin, A. Rosello, K. van Zandvoort, J. D. Munday, W. J. Edmunds, T. Jombart, M. Auzenbergs, E. S. Nightingale, M. Jit, S. Abbott, D. Simons, N. I. Bosse, Q. J. Leclerc, S. R. Procter, C. J. Villabona-Arenas, D. C. Tully, A. K. Deol, F. Y. Sun, S. Hué, A. M. Foss, K. Prem, G. Medley, A. Gimma, R. Lowe, S. Clifford, M. Quaife, C. Diamond, H. P. Gibbs, B. J. Quilty, K. OReilly, G. M. Knight, R. M. Eggo, A. J. Kucharski, S. Funk, S. Flasche, and R. M. Houben, eLife 9, 1 (2020).
. N G Davies, P Klepac, Y Liu, K Prem, M Jit, R M Eggo, 10.1038/s41591-020-0962-9Nat. Med. 261205N. G. Davies, P. Klepac, Y. Liu, K. Prem, M. Jit, and R. M. Eggo, Nat. Med. 26, 1205 (2020).
. M Lipsitch, N E Dean, 10.1126/science.abe5938Science. 370763M. Lipsitch and N. E. Dean, Science 370, 763 (2020).
. N G Davies, S Abbott, R C Barnard, C I Jarvis, A J Kucharski, J D Munday, C A B Pearson, T W , N. G. Davies, S. Abbott, R. C. Barnard, C. I. Jarvis, A. J. Kucharski, J. D. Munday, C. A. B. Pearson, T. W.
. D C Russell, A D Tully, T Washburne, A Wenseleers, W Gimma, K L M Waites, K Wong, J D Van Zandvoort, K Silverman, R Diaz-Ordaz, R M Keogh, S Eggo, M Funk, K E Jit, W J Atkins, Edmunds, 10.1126/science.abg3055Science. 3723055Russell, D. C. Tully, A. D. Washburne, T. Wenseleers, A. Gimma, W. Waites, K. L. M. Wong, K. van Zand- voort, J. D. Silverman, K. Diaz-Ordaz, R. Keogh, R. M. Eggo, S. Funk, M. Jit, K. E. Atkins, and W. J. Edmunds, Science 372, eabg3055 (2021).
. L F Buss, C A Prete, C M M Abrahim, A Mendrone, T Salomon, C Almeida-Neto, R F O França, M C Belotti, M P S S Carvalho, A G Costa, M A E Crispim, S C Ferreira, N A Fraiji, S Gurzenda, C Whittaker, L T Kamaura, P L Takecian, P Da, L. F. Buss, C. A. Prete, C. M. M. Abrahim, A. Mendrone, T. Salomon, C. de Almeida-Neto, R. F. O. França, M. C. Belotti, M. P. S. S. Carvalho, A. G. Costa, M. A. E. Crispim, S. C. Ferreira, N. A. Fraiji, S. Gurzenda, C. Whittaker, L. T. Kamaura, P. L. Takecian, P. da
. Silva Peixoto, M K Oikawa, A S Nishiya, V Rocha, N A Salles, A A De Souza Santos, M A Silva, B Custer, K V Parag, M Barral-Netto, M U G Kraemer, R H M Pereira, O G Pybus, M P Busch, M C Castro, C Dye, V H Nascimento, N R Faria, E C Sabino, 10.1126/science.abe9728Science. 371288Silva Peixoto, M. K. Oikawa, A. S. Nishiya, V. Rocha, N. A. Salles, A. A. de Souza Santos, M. A. da Silva, B. Custer, K. V. Parag, M. Barral-Netto, M. U. G. Krae- mer, R. H. M. Pereira, O. G. Pybus, M. P. Busch, M. C. Castro, C. Dye, V. H. Nascimento, N. R. Faria, and E. C. Sabino, Science 371, 288 (2020).
. D Sridhar, D Gurdasani, 10.1126/science.abf7921Science. 371230D. Sridhar and D. Gurdasani, Science 371, 230 (2021).
WHO), (2021), classification of Omicron (B.1.1.529): SARS-CoV-2 Variant of Concern. W H O , W. H. O. (WHO), (2021), classification of Omicron (B.1.1.529): SARS-CoV-2 Variant of Concern.
. A Pan, L Liu, C Wang, H Guo, X Hao, Q Wang, J Huang, N He, H Yu, X Lin, S Wei, T Wu, 10.1001/jama.2020.6130JAMA. 3231915A. Pan, L. Liu, C. Wang, H. Guo, X. Hao, Q. Wang, J. Huang, N. He, H. Yu, X. Lin, S. Wei, and T. Wu, JAMA 323, 1915 (2020).
. S T Ali, L Wang, E H Y Lau, X.-K Xu, Z Du, Y Wu, G M Leung, B J Cowling, 10.1126/science.abc9004Science. 3691106S. T. Ali, L. Wang, E. H. Y. Lau, X.-K. Xu, Z. Du, Y. Wu, G. M. Leung, and B. J. Cowling, Science 369, 1106 (2020).
. A Goyal, D B Reeves, E F Cardozo-Ojeda, J T Schiffer, B T Mayer, 10.7554/elife.6353710A. Goyal, D. B. Reeves, E. F. Cardozo-Ojeda, J. T. Schiffer, and B. T. Mayer, eLife 10 (2021), 10.7554/elife.63537.
. S Flaxman, S Mishra, A Gandy, H J T Unwin, T A Mellan, H Coupland, C Whittaker, H Zhu, T Berah, J W Eaton, M Monod, P N Perez-Guzman, N Schmit, L Cilloni, K E C Ainslie, M Baguelin, A Boonyasiri, O Boyd, L Cattarino, L V Cooper, Z Cucunubá, G Cuomo-Dannenburg, A Dighe, B Djaafara, I Dorigatti, S L Van Elsland, R G Fitzjohn, K A M Gaythorpe, L Geidelberg, N C Grassly, W D Green, T Hallett, A Hamlet, W Hinsley, B Jeffrey, E Knock, D J Laydon, G Nedjati-Gilani, P Nouvellet, K V Parag, I Siveroni, H A Thompson, R Verity, E Volz, C E Walters, H Wang, Y Wang, O J Watson, P Winskill, X Xi, P G T Walker, A C Ghani, C A Donnelly, S Riley, M A C Vollmer, N M Ferguson, L C Okell, S Bhatt, 10.1038/s41586-020-2405-7Nature. 584257S. Flaxman, S. Mishra, A. Gandy, H. J. T. Unwin, T. A. Mellan, H. Coupland, C. Whittaker, H. Zhu, T. Berah, J. W. Eaton, M. Monod, P. N. Perez-Guzman, N. Schmit, L. Cilloni, K. E. C. Ainslie, M. Baguelin, A. Boonyasiri, O. Boyd, L. Cattarino, L. V. Cooper, Z. Cucunubá, G. Cuomo-Dannenburg, A. Dighe, B. Djaafara, I. Dori- gatti, S. L. van Elsland, R. G. FitzJohn, K. A. M. Gaythorpe, L. Geidelberg, N. C. Grassly, W. D. Green, T. Hallett, A. Hamlet, W. Hinsley, B. Jeffrey, E. Knock, D. J. Laydon, G. Nedjati-Gilani, P. Nouvellet, K. V. Parag, I. Siveroni, H. A. Thompson, R. Verity, E. Volz, C. E. Walters, H. Wang, Y. Wang, O. J. Watson, P. Win- skill, X. Xi, P. G. T. Walker, A. C. Ghani, C. A. Donnelly, S. Riley, M. A. C. Vollmer, N. M. Ferguson, L. C. Okell, and S. Bhatt, Nature 584, 257 (2020).
. A Cori, N M Ferguson, C Fraser, S Cauchemez, 10.1093/aje/kwt133Am. J. Epidemiol. 1781505A. Cori, N. M. Ferguson, C. Fraser, and S. Cauchemez, Am. J. Epidemiol. 178, 1505 (2013).
. K M Gostic, L Mcgough, E B Baskerville, S Abbott, K Joshi, C Tedijanto, R Kahn, R Niehus, J A Hay, P M Salazar, J Hellewell, S Meakin, J D Munday, N I Bosse, K Sherrat, R N Thompson, L F , K. M. Gostic, L. McGough, E. B. Baskerville, S. Ab- bott, K. Joshi, C. Tedijanto, R. Kahn, R. Niehus, J. A. Hay, P. M. De Salazar, J. Hellewell, S. Meakin, J. D. Munday, N. I. Bosse, K. Sherrat, R. N. Thompson, L. F.
. J S White, J Huisman, S Scire, T Bonhoeffer, J Stadler, S Wallinga, M Funk, S Lipsitch, Cobey, 10.1371/journal.pcbi.1008409PLOS Comput. Biol. 161008409White, J. S. Huisman, J. Scire, S. Bonhoeffer, T. Stadler, J. Wallinga, S. Funk, M. Lipsitch, and S. Cobey, PLOS Comput. Biol. 16, e1008409 (2020).
. K Mizumoto, K Kagaya, A Zarebski, G Chowell, 10.2807/1560-7917.ES.2020.25.10.2000180Eurosurveillance. 25K. Mizumoto, K. Kagaya, A. Zarebski, and G. Chowell, Eurosurveillance 25 (2020), 10.2807/1560- 7917.ES.2020.25.10.2000180.
. O Byambasuren, M Cardona, K Bell, J Clark, M.-L Mclaws, P Glasziou, 10.3138/jammi-2020-0030Off. J. Assoc. Med. Microbiol. Infect. Dis. Canada. 5223O. Byambasuren, M. Cardona, K. Bell, J. Clark, M.-L. McLaws, and P. Glasziou, Off. J. Assoc. Med. Microbiol. Infect. Dis. Canada 5, 223 (2020).
. D Buitrago-Garcia, D Egli-Gany, M J Counotte, S Hossmann, H Imeri, A M Ipekci, G Salanti, N Low, 10.1371/journal.pmed.1003346PLoS Med. 171D. Buitrago-Garcia, D. Egli-Gany, M. J. Counotte, S. Hossmann, H. Imeri, A. M. Ipekci, G. Salanti, and N. Low, PLoS Med. 17, 1 (2020).
. J P A Ioannidis, 10.2471/BLT.20.265892Bull. World Health Organ. 9919J. P. A. Ioannidis, Bull. World Health Organ. 99, 19 (2021).
. C M Romano, A C Felix, A V De Paula, J G De Jesus, P S Andrade, D Cândido, F M De Oliveira, A C Ribeiro, F C Silva, M Inemami, A A Costa, C O D Leal, W M Figueiredo, C S Pannuti, W M Souza, N R Faria, E C Sabino, 10.1590/s1678-9946202163036Rev. Inst. Med. Trop. Sao Paulo. 630C. M. Romano, A. C. Felix, A. V. de Paula, J. G. de Jesus, P. S. Andrade, D. Cândido, F. M. de Oliveira, A. C. Ribeiro, F. C. da Silva, M. Inemami, A. A. Costa, C. O. D. Leal, W. M. Figueiredo, C. S. Pannuti, W. M. de Souza, N. R. Faria, and E. C. Sabino, Rev. Inst. Med. Trop. Sao Paulo 63, 0 (2021).
. G Pullano, L Di Domenico, C E Sabbatini, E Valdano, C Turbelin, M Debin, C Guerrisi, C Kengne, - , G. Pullano, L. Di Domenico, C. E. Sabbatini, E. Val- dano, C. Turbelin, M. Debin, C. Guerrisi, C. Kengne-
. C Kuetche, T Souty, T Hanslik, P.-Y Blanchon, J Boëlle, S Figoni, C Vaux, S Campèse, V Bernard-Stoecklin, Colizza, 10.1038/s41586-020-03095-6Nature. 590134Kuetche, C. Souty, T. Hanslik, T. Blanchon, P.-Y. Boëlle, J. Figoni, S. Vaux, C. Campèse, S. Bernard-Stoecklin, and V. Colizza, Nature 590, 134 (2021).
. F S Lu, A T Nguyen, N B Link, M Molina, J T Davis, M Chinazzi, X Xiong, A Vespignani, M Lipsitch, M Santillana, 10.1371/journal.pcbi.1008994PLOS Comput. Biol. 171008994F. S. Lu, A. T. Nguyen, N. B. Link, M. Molina, J. T. Davis, M. Chinazzi, X. Xiong, A. Vespignani, M. Lip- sitch, and M. Santillana, PLOS Comput. Biol. 17, e1008994 (2021).
R Subramanian, Q He, M Pascual, 10.1073/pnas.2019716118Proc. Natl. Acad. Sci. Natl. Acad. Sci1182019716118R. Subramanian, Q. He, and M. Pascual, Proc. Natl. Acad. Sci. 118, e2019716118 (2021).
. N J Irons, A E Raftery, 10.1073/pnas.2103272118Proc. Natl. Acad. Sci. U. S. A. 118N. J. Irons and A. E. Raftery, Proc. Natl. Acad. Sci. U. S. A. 118 (2021), 10.1073/pnas.2103272118.
. G S Costa, W Cota, S C Ferreira, 10.1103/PhysRevResearch.2.0433062011.03380Phys. Rev. Res. 243306G. S. Costa, W. Cota, and S. C. Ferreira, Phys. Rev. Res. 2, 043306 (2020), 2011.03380.
. A Arenas, W Cota, J Gómez-Gardeñes, S Gómez, C Granell, J T Matamalas, D Soriano-Paños, B Steinegger, 10.1103/PhysRevX.10.041055Phys. Rev. X. 1041055A. Arenas, W. Cota, J. Gómez-Gardeñes, S. Gómez, C. Granell, J. T. Matamalas, D. Soriano-Paños, and B. Steinegger, Phys. Rev. X 10, 041055 (2020).
. P Ashcroft, S Lehtinen, D C Angst, N Low, S Bonhoeffer, 10.7554/eLife.63704101P. Ashcroft, S. Lehtinen, D. C. Angst, N. Low, and S. Bonhoeffer, eLife 10, 1 (2021).
. L Ferretti, C Wymant, M Kendall, L Zhao, A Nurtay, L Abeler-Dörner, M Parker, D Bonsall, C Fraser, 10.1126/science.abb6936Science. 3686936L. Ferretti, C. Wymant, M. Kendall, L. Zhao, A. Nurtay, L. Abeler-Dörner, M. Parker, D. Bonsall, and C. Fraser, Science 368, eabb6936 (2020).
. A , Y Moreno, Med, 10.1186/s12916-020-01619-518157A. Aleta and Y. Moreno, BMC Med. 18, 157 (2020).
. A Aleta, D Martín-Corral, A Pastore Y Piontti, M Ajelli, M Litvinova, M Chinazzi, N E Dean, M E Halloran, I M LonginiJr, S Merler, A Pentland, A Vespignani, E Moro, Y Moreno, 10.1038/s41562-020-0931-9Nat. Hum. Behav. 4964A. Aleta, D. Martín-Corral, A. Pastore y Piontti, M. Ajelli, M. Litvinova, M. Chinazzi, N. E. Dean, M. E. Halloran, I. M. Longini Jr, S. Merler, A. Pentland, A. Vespignani, E. Moro, and Y. Moreno, Nat. Hum. Behav. 4, 964 (2020).
. J Dehning, J Zierenberg, F P Spitzner, M Wibral, J P Neto, M Wilczek, V Priesemann, 10.1126/science.abb9789Science. 3699789J. Dehning, J. Zierenberg, F. P. Spitzner, M. Wibral, J. P. Neto, M. Wilczek, and V. Priesemann, Science 369, eabb9789 (2020).
. B F Maier, D Brockmann, 10.1126/science.abb4557Science. 368742B. F. Maier and D. Brockmann, Science 368, 742 (2020).
. M C Castro, S Kim, L Barberia, A F Ribeiro, S Gurzenda, K B Ribeiro, E Abbott, J Blossom, B Rache, B H Singer, 10.1126/science.abh1558Science. 372821M. C. Castro, S. Kim, L. Barberia, A. F. Ribeiro, S. Gurzenda, K. B. Ribeiro, E. Abbott, J. Blossom, B. Rache, and B. H. Singer, Science 372, 821 (2021).
M Keeling, P Rohani, 10.1515/9781400841035Modeling Infectious Diseases in Humans and Animals. Princeton University PressM. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals (Princeton University Press, 2008).
Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand. N M Ferguson, D Laydon, G Nedjati-Gilani, N Imai, K Ainslie, M Baguelin, S Bhatia, A Boonyasiri, Z Cucunubá, G Cuomo-Dannenburg, A Dighe, I Dorigatti, H Fu, K Gaythorpe, W Green, A Hamlet, W Hinsley, L C Okell, S Van Elsland, H Thompson, R Verity, E Volz, H Wang, Y Wang, P Walker, C Walters, P Winskill, C Whittaker, C A Donnelly, S Riley, A C Ghani, 10.25561/77482Report. 9Tech. RepN. M. Ferguson, D. Laydon, G. Nedjati-Gilani, N. Imai, K. Ainslie, M. Baguelin, S. Bhatia, A. Boonyasiri, Z. Cu- cunubá, G. Cuomo-Dannenburg, A. Dighe, I. Dorigatti, H. Fu, K. Gaythorpe, W. Green, A. Hamlet, W. Hinsley, L. C. Okell, S. Van Elsland, H. Thompson, R. Verity, E. Volz, H. Wang, Y. Wang, P. Gt Walker, C. Walters, P. Winskill, C. Whittaker, C. A. Donnelly, S. Riley, and A. C. Ghani, Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID19 mortality and healthcare demand , Tech. Rep. March (2020).
. E C Sabino, L F Buss, M P S Carvalho, C A Prete, M A E Crispim, N A Fraiji, R H M Pereira, K V Parag, P Da Silva Peixoto, M U G Kraemer, M K Oikawa, T Salomon, Z M Cucunuba, M C Castro, A A De Souza Santos, V H Nascimento, H S Pereira, N M Ferguson, O G Pybus, A Kucharski, M P Busch, C Dye, N R Faria, 10.1016/s0140-6736(21)00183-5The Lancet. 397452E. C. Sabino, L. F. Buss, M. P. S. Carvalho, C. A. Prete, M. A. E. Crispim, N. A. Fraiji, R. H. M. Pereira, K. V. Parag, P. da Silva Peixoto, M. U. G. Kraemer, M. K. Oikawa, T. Salomon, Z. M. Cucunuba, M. C. Castro, A. A. de Souza Santos, V. H. Nascimento, H. S. Pereira, N. M. Ferguson, O. G. Pybus, A. Kucharski, M. P. Busch, C. Dye, and N. R. Faria, The Lancet 397, 452 (2021).
. S A Lauer, K H Grantz, Q Bi, F K Jones, Q Zheng, H R Meredith, A S Azman, N G Reich, J Lessler, 10.7326/M20-0504Ann. Intern. Med. 172577S. A. Lauer, K. H. Grantz, Q. Bi, F. K. Jones, Q. Zheng, H. R. Meredith, A. S. Azman, N. G. Reich, and J. Lessler, Ann. Intern. Med. 172, 577 (2020).
. D S Candido, I M Claro, J G De Jesus, W M Souza, F R R Moreira, S Dellicour, T A Mellan, L Plessis, R H M Pereira, F C S Sales, E R Manuli, J Thézé, L Almeida, M T Menezes, C M , D. S. Candido, I. M. Claro, J. G. de Jesus, W. M. Souza, F. R. R. Moreira, S. Dellicour, T. A. Mellan, L. du Plessis, R. H. M. Pereira, F. C. S. Sales, E. R. Manuli, J. Thézé, L. Almeida, M. T. Menezes, C. M.
. M J Voloch, T M Fumagalli, C A M Coletti, M S Silva, M R Ramundo, H H Amorim, S Hoeltgebaum, M S Mishra, L M Gill, L F Carvalho, C A Buss, J Prete, H I Ashworth, P S Nakaya, O J Peixoto, S M Brady, A Nicholls, Á D Tanuri, C K V Rossi, A L Braga, Gerber, A. P. de C. Guimarães, N. Gaburo, C. S. Alencar, A. C. S. Ferreira, C. X. Lima, J. E. Levi, C. Granato, G. M. Ferreira, R. S. Francisco, F. Granja, M. T. Garcia, M. L. Moretti, M. W. Perroud, T. M. P. P. Castiñeiras, C. S. Lazari, S. C. Hill, A. A. de Souza Santos, C. L. Simeoni, J. Forato, A. C. Sposito, A. ZVoloch, M. J. Fumagalli, T. M. Coletti, C. A. M. da Silva, M. S. Ramundo, M. R. Amorim, H. H. Hoeltgebaum, S. Mishra, M. S. Gill, L. M. Carvalho, L. F. Buss, C. A. Prete, J. Ashworth, H. I. Nakaya, P. S. Peixoto, O. J. Brady, S. M. Nicholls, A. Tanuri,Á. D. Rossi, C. K. V. Braga, A. L. Gerber, A. P. de C. Guimarães, N. Gaburo, C. S. Alencar, A. C. S. Ferreira, C. X. Lima, J. E. Levi, C. Granato, G. M. Ferreira, R. S. Francisco, F. Granja, M. T. Garcia, M. L. Moretti, M. W. Perroud, T. M. P. P. Castiñeiras, C. S. Lazari, S. C. Hill, A. A. de Souza San- tos, C. L. Simeoni, J. Forato, A. C. Sposito, A. Z.
. M N N Schreiber, C Z Santos, R P De Sá, L C Souza, M M Resende-Moreira, J Teixeira, P A F Hubner, R G Leme, M L Moreira, N M Nogueira, S F Ferguson, Costa, 10.1126/science.abd2161Science. J. L. Proenca-Modena, A. T. R. Vasconcelos, S. Bhatt, P. Lemey, C.-H. Wu, A. Rambaut, N. J. Loman, R. S. Aguiar, O. G. Pybus, E. C. Sabino, and N. R. F. and3691255Schreiber, M. N. N. Santos, C. Z. de Sá, R. P. Souza, L. C. Resende-Moreira, M. M. Teixeira, J. Hubner, P. A. F. Leme, R. G. Moreira, M. L. Nogueira, N. M. Ferguson, S. F. Costa, J. L. Proenca-Modena, A. T. R. Vasconce- los, S. Bhatt, P. Lemey, C.-H. Wu, A. Rambaut, N. J. Loman, R. S. Aguiar, O. G. Pybus, E. C. Sabino, and N. R. F. and, Science 369, 1255 (2020).
. S Moore, E M Hill, M J Tildesley, L Dyson, M J Keeling, 10.1016/S1473-3099(21)00143-2Lancet Infect. Dis. 21793S. Moore, E. M. Hill, M. J. Tildesley, L. Dyson, and M. J. Keeling, Lancet Infect. Dis. 21, 793 (2021).
. L Di Domenico, G Pullano, C E Sabbatini, P.-Y Boëlle, V Colizza, Med, 10.1186/s12916-020-01698-418240L. Di Domenico, G. Pullano, C. E. Sabbatini, P.-Y. Boëlle, and V. Colizza, BMC Med. 18, 240 (2020).
. K Kupferschmidt, M Wadman, 10.1126/science.372.6549.1375Science. 3721375K. Kupferschmidt and M. Wadman, Science 372, 1375 (2021).
. F Ahmed, N Ahmed, C Pissarides, J Stiglitz, 10.1016/S2468-2667(20)30085-2Lancet Public Heal. 5240F. Ahmed, N. Ahmed, C. Pissarides, and J. Stiglitz, Lancet Public Heal. 5, e240 (2020).
. K Sun, W Wang, L Gao, Y Wang, K Luo, L Ren, Z Zhan, X Chen, S Zhao, Y Huang, Q Sun, Z Liu, M Litvinova, A Vespignani, M Ajelli, C Viboud, H Yu, 10.1126/science.abe2424Science. 3712424K. Sun, W. Wang, L. Gao, Y. Wang, K. Luo, L. Ren, Z. Zhan, X. Chen, S. Zhao, Y. Huang, Q. Sun, Z. Liu, M. Litvinova, A. Vespignani, M. Ajelli, C. Viboud, and H. Yu, Science 371, eabe2424 (2021).
. W Cota, 10.1590/SciELOPreprints.362SciELOPreprints. 362W. Cota, SciELOPreprints.362 (2020), 10.1590/SciELO- Preprints.362.
. L S Bastos, T Economou, M F C Gomes, D A M Villela, F C Coelho, O G Cruz, O Stoner, T Bailey, C T Codeço, 10.1002/sim.8303Stat. Med. 384363L. S. Bastos, T. Economou, M. F. C. Gomes, D. A. M. Villela, F. C. Coelho, O. G. Cruz, O. Stoner, T. Bailey, and C. T. Codeço, Stat. Med. 38, 4363 (2019).
. L S Bastos, R P Niquini, R M Lana, D A M Villela, O G Cruz, F C Coelho, C T Codeço, M F C Gomes, 10.1590/0102-311x00070120Cad. Saude Publica. 36L. S. Bastos, R. P. Niquini, R. M. Lana, D. A. M. Villela, O. G. Cruz, F. C. Coelho, C. T. Codeço, and M. F. C. Gomes, Cad. Saude Publica 36 (2020), 10.1590/0102- 311x00070120.
G Do Estado Do Paraná, COVID-19Boletim -Informe Epidemiológico Coronavírus. Online; accessed 08-Sep-2021G. do Estado do Paraná, (2021), Boletim -Informe Epidemiológico Coronavírus (COVID-19), https://www. saude.pr.gov.br/Pagina/Coronavirus-COVID-19 [On- line; accessed 08-Sep-2021].
COVID-19 -Painel COVID-19 -Estado do Espírito Santo. G Do Estado Do, Espírito Santo, OnlineG. do Estado do Espírito Santo, (2021), COVID-19 - Painel COVID-19 -Estado do Espírito Santo, https:// coronavirus.es.gov.br/painel-covid-19-es [Online;
Codes and datasets are freely available at https:// github.com/ghscosta/covid19-cal. "Codes and datasets are freely available at https:// github.com/ghscosta/covid19-cal.".
Supplementary material. "Supplementary material,".
. J M Read, J R E Bridgen, D A T Cummings, A Ho, C P Jewell, 10.1098/rstb.2020.0265Philos. Trans. R. Soc. B Biol. Sci. 37620200265J. M. Read, J. R. E. Bridgen, D. A. T. Cummings, A. Ho, and C. P. Jewell, Philos. Trans. R. Soc. B Biol. Sci. 376, 20200265 (2021).
Instituto Brasileiro De Geografia E Estatística, Divisão Regional do Brasil em Regiões Geográficas Imediatas e Regiões Geográficas Intermediárias. Instituto Brasileiro de Geografia e Estatística Rio de JaneiroInstituto Brasileiro de Geografia e Estatística (IBGE), Divisão Regional do Brasil em Regiões Geográficas Ime- diatas e Regiões Geográficas Intermediárias (Instituto Brasileiro de Geografia e Estatística Rio de Janeiro, 2017).
. K Prem, A R Cook, M Jit, 10.1371/journal.pcbi.1005697PLOS Comput. Biol. 131005697K. Prem, A. R. Cook, and M. Jit, PLOS Comput. Biol. 13, e1005697 (2017).
| [] |
[
"Bose-Einstein correlations in perturbative QCD: v n dependence on multiplicity",
"Bose-Einstein correlations in perturbative QCD: v n dependence on multiplicity"
] | [
"E Gotsman \nDepartment of Particle Physics\nSchool of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Science\nTel Aviv University\n69978Tel AvivIsrael\n",
"E Levin \nDepartment of Particle Physics\nSchool of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Science\nTel Aviv University\n69978Tel AvivIsrael\n\nDepartemento de Física\nUniversidad Técnica Federico Santa María, and Centro Científico-Tecnológico de Valparaíso\nAvda. Espana 1680, Casilla 110-VValparaísoChile\n"
] | [
"Department of Particle Physics\nSchool of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Science\nTel Aviv University\n69978Tel AvivIsrael",
"Department of Particle Physics\nSchool of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Science\nTel Aviv University\n69978Tel AvivIsrael",
"Departemento de Física\nUniversidad Técnica Federico Santa María, and Centro Científico-Tecnológico de Valparaíso\nAvda. Espana 1680, Casilla 110-VValparaísoChile"
] | [] | In this paper we study the dependence of Bose-Einstein correlations on the multiplicity of an event. We found that events with large multiplicity, stem from the production of several parton showers, while the additional production of small multiplicity in the central rapidity region (central diffraction), gives a negligible contribution due to emission of soft gluons, that leads to the Sudakov suppression of the exclusive production of two gluon jets. Hence, the Bose-Einstein correlation is the main source of the azimuthal angle correlations which generates vn with odd and even n. We found, that without this suppression, the measurement of an event with given multipilicity, yields vn,n < 0 for odd n. It appears that in hadron-nucleus and nucleus-nucleus collisions, the Bose-Einstein correlations do not depend on multiplicity, while for hadron-hadron scattering, such dependence can be considerable. We proposed a simple Kharzeev-Levin-Nardi (KLN) type model, to describe the dependence of azimuthal angle correlations on the centrality of the event, in ion-ion collisions. | 10.1103/physrevd.96.074011 | [
"https://arxiv.org/pdf/1705.07406v2.pdf"
] | 119,025,743 | 1705.07406 | 339ca03818bb9d93fd5b4b98f7a6a43dd872b14a |
Bose-Einstein correlations in perturbative QCD: v n dependence on multiplicity
26 May 2017
E Gotsman
Department of Particle Physics
School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Science
Tel Aviv University
69978Tel AvivIsrael
E Levin
Department of Particle Physics
School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Science
Tel Aviv University
69978Tel AvivIsrael
Departemento de Física
Universidad Técnica Federico Santa María, and Centro Científico-Tecnológico de Valparaíso
Avda. Espana 1680, Casilla 110-VValparaísoChile
Bose-Einstein correlations in perturbative QCD: v n dependence on multiplicity
26 May 2017(Dated: September 28, 2018)numbers: 1238-t2485+p2575-q
In this paper we study the dependence of Bose-Einstein correlations on the multiplicity of an event. We found that events with large multiplicity, stem from the production of several parton showers, while the additional production of small multiplicity in the central rapidity region (central diffraction), gives a negligible contribution due to emission of soft gluons, that leads to the Sudakov suppression of the exclusive production of two gluon jets. Hence, the Bose-Einstein correlation is the main source of the azimuthal angle correlations which generates vn with odd and even n. We found, that without this suppression, the measurement of an event with given multipilicity, yields vn,n < 0 for odd n. It appears that in hadron-nucleus and nucleus-nucleus collisions, the Bose-Einstein correlations do not depend on multiplicity, while for hadron-hadron scattering, such dependence can be considerable. We proposed a simple Kharzeev-Levin-Nardi (KLN) type model, to describe the dependence of azimuthal angle correlations on the centrality of the event, in ion-ion collisions.
I. INTRODUCTION
In this paper we continue to discuss the Bose-Einstein correlations of gluons as being the main source of the strong azimuthal angle (ϕ) correlations, that have been observed experimentally, in nucleus-nucleus, hadron-nucleus and hadron hadron collisions [1][2][3][4][5][6][7][8][9][10][11]. It has been known for some time in the framework of Gribov Pomeron Calculus, that the Bose-Einstein correlations which stem from the exchange of two Pomerons lead to azimuthal angle correlations [12] (see also Ref. [13]), which do not depend on the rapidity difference between measured hadrons ( large range rapidity (LRR) correlations). In the framework of QCD, these azimuthal correlations originate from the production of two patron showers, and have been re-discovered in Refs. [14][15][16][17][18] (see also Ref. [19,20]). In Ref. [21] it was demonstrated that Bose-Einstein correlations generate v n with even and odd n, with values which are close to the experimental observed ones.
The goal of this paper is to answer three questions: (i) Is the symmetry ϕ → π − ϕ an inherent property of QCD, or of the colour glass condensate (CGC) approach, which is the effective theory of QCD at high energies, or it is based on the model assumptions ? (ii) What is the multiplicity dependence of the azimuthal angle correlations which stem from the Bose-Einstein ones? (iii) Is it possible to build a simple KLN-type [22][23][24][25][26][27][28] approach to describe azimuthal correlations in nucleus-nucleus collisions ?
The following are our answers to these questions : The symmetry ϕ → π − ϕ, is not a general feature of the QCD (or CGC) approach. It does not stem from the Bose-Einstein correlations of identical gluons, and can only appear in measurements that mix events with different multiplicities. In the case of hadron-hadron collisions, for example, such symmetry exists in the Born approximation of perturbative QCD, and could only be measured, if experimentally the central diffraction production and the event with double multiplicity ( n = 2n, wheren is the average multiplicity in inclusive production) are measured and summed. However, the emission of soft gluons for the central exclusive production in the Double Log Approximation of perturbative QCD, leads to a Sudakov form factor which suppress this contribution. Therefore, the Bose-Einstein correlations prevail, leading to v n = 0 for odd n, even in totally inclusive measurements, without selection of an event with given multiplicities.
We expect a very mild dependence of v n on the multiplicity of the observed events. We suggest a model for the Bose-Einstein correlations in heavy ion collisions in the spirit of the KLN approach, which is based on the concept of constructing the simplest model that takes into account the discussed phenomena: in our case, the saturation of the gluon density and the Bose-Einstein correlations.
The double inclusive cross section of two identical gluons has the following general form:
d 2 σ dy 1 dy 2 d 2 p T 1 d 2 p T 2 (identical gluons) = d 2 σ dy 1 dy 2 d 2 p T 1 d 2 p T 2 (different gluons) 1 + C (L c |p T 2 − p T 1 |)(1)
where C (L c |p T 2 − p T 1 |) denotes the correlation function and L c the correlation length. Eq. (1) is in accord with Hanbury Brown and Twiss formula (see Refs. [29,30])
d 2 σ dy 1 dy 2 d 2 p T 1 d 2 p T 2 (identical gluons) ∝ 1 + e irµQµ(2)
where averaging . . . includes the integration over r µ = r 1,µ − r 2,µ . There is only one difference: Q µ = p 1,µ − p 2,µ degenerates to Q ≡ p T,12 = p T 2 − p T 1 , as the production of two gluons from the two parton showers, does not depend on rapidities. Eq. (2) allows us to measure the typical r µ of the interaction, or in other words, L c in Eq. (1) is determined by the typical volume of the interaction. Therefore, we expect several typical L c : the size of the nucleus R A ; the nucleon size R N and the typical size, related to the saturation scale (r sat = 1/Q s , where Q s denotes the saturation scale [31]). Indication of all these sizes have been seen in Bose -Einstein correlations (see Ref. [17,21]). Using Eq. (1) we can find v n , since d 2 σ dy 1 dy 2 d 2 p T 1 d 2 p T 2 ∝ 1 + 2 n v n,n (p T 1 , p T 2 ) cos (n ϕ)
where ϕ is the angle between p T 1 and p T 2 . v n is determined from v n,n (p T 1 , p T 2 )
1. v n (p T ) = v n,n (p T , p T ) ; 2. v n (p T ) = v n,n p T , p Ref
Eq. (4)-1 and Eq. (4)-2 depict two methods of how the values of v n have been extracted from the experimentally measured v n,n (p T 1 , p T 2 ). p Ref T denotes the momentum of the reference trigger. These two definitions are equivalent if v n,n (p T 1 , p T 2 ) can be factorized as v n,n (p T 1 , p T 2 ) = v n (p T 1 ) v n (p T 2 ).
II. SYMMETRY ϕ → π − ϕ ( vn = 0 FOR ODD n) FOR DIFFERENT MULTIPLICITIES OF PRODUCED HADRONS
A. The Bose-Einstein correlation function for deuteron-deuteron scattering with the correlation length Lc ∝ RD First, we consider the simplest diagram in the Born approximation of perturbative QCD, which we have discussed in Ref. [17] (see Fig. 1-a) ). This diagram describes the interference between two identical gluons in the process of multiparticle production, or in other words, in the processes of the production of two parton showers. In this diagram Q T ∝ 1/R D and |Q T − p 12,T | ∝ 1/R D , where R D denotes the deuteron radius, which is much larger than the size of the proton, R N . Momenta k T , l T , p 1,T and p 2,T in this diagram are of the order of 1/R N ≫ 1/R D and, therefore, we can neglect Q T as well as p 12,T , in the diagram. Bearing this in mind, we see that the correlation function C L c |p 12,T | is equal to
C L c |p 12,T | = 1 N 2 c − 1 d 2 Q T G D (Q T ) G D Q T − p 12,T d 2 Q T G D (Q T ) G D (Q T ) with G D (Q T ) = d 2 r e ir·Q T |Ψ D (r) | 2(5)
where r denotes the distance between the proton and the neutron in the deuteron. Eq. (5) displays no symmetry with respect to ϕ → π − ϕ. However, we can add a different diagram of Fig. 1-b, which describes the central diffraction production of two different gluons in a colourless state 1 . This diagram depends on p 1,T + p 2,T and generates the correlation function
C L c |p 1,T + p 2,T | ∝ 1 N 2 c − 1 d 2 Q T G D (Q T ) G D Q T − p 1,T − p 2,T d 2 Q T G D (Q T ) G D (Q T )(6)
since in this diagram Q T and Q T − p 1,T − p 2,T are of the order of 1/R D , while k T , l T , p 1,T and p 2,T in this diagram are of the order of 1/R N ≫ 1/R D , therefore, we can neglect Q T as well as p 1,T + p 2,T in the diagram or, in other words, we can put p 1,T = −p 2,T . After this substitution, both diagrams have the same expressions. Therefore, if diagrams of Fig. 1-a and Fig. 1-b have the same weight, the sum will have the symmetry with respect to p 2,T → −p 2,T , restoring the symmetry with respect to ϕ → π − ϕ. At first sight this is the case, since all integrations over k T and l T look the same. However, in these two diagrams this is certainly not the case due to different integration with respect to k − and l − (or k + and l + ). These integrations generates 1/4 suppression of the diagram of Fig. 1-b with respect to the diagram of Fig. 1-a. It is a well known fact, which for the first time, has been discussed in the AGK paper of Ref. [32], as well as in the most reviews and books that are devoted to the high energy scattering ( in particular those, where one of us is an author [31,33,34]). For the completeness of presentation we add appendix A in which we discuss this integration.
However, we found it instructive to discuss the contribution of these two diagrams in the framework of the AGK cutting rules, which is the technique that we will use in considering the dependence of the correlation function on multiplicity of produced particles. First, accounting for emission of the gluons with rapidities larger than y 1 and smaller than y 2 , and consideringᾱ S |y 1 − y 2 | ≪ 1, we can describe the two partonic showers contribution in deuterondeuteron scattering by the diagrams of Fig. 2-a and Fig. 2
-b.
The AGK cutting rules describe the relative contributions of different processes that stem from two BFKL Pomeron [36,37] exchange. Fig. 3-a describes the elastic scattering, Fig. 3-b the one parton shower production, that is screened by the BFKL Pomeron exchange. Fig. 3-c is the production of two parton showers. The AGK cutting rules state that the cross sections of these three processes are related as 1 : −4 : 2. The sum of these processes is equal to -1, leading to the negative contribution to the total cross section of two Pomeron exchange. These rules have a rather general
G (Q ) D T G (Q ) D T k l k− p 1 Y − k + Q T 0 p 1 − l − Q T p 2 − k− p + Q 2 T −l+ p − Q 1 −l+ p 2 G (Q − p − p ) D T 1 2 k l k− p 1 Y − k + Q T 0 p 1 − l − Q T p 2 l − p 2 T −l+ p − Q 1 −k+ p + Q 2 T G (Q − p ) D T 12 a) b) T FIG. 1:
Deuteron-deuteron scattering in the Born approximation of perturbative QCD: Fig. 1-a describes the interference diagrams in the production of two identical gluons, in the process of multiparticle generation that gives rise to the correlation function C L c |p 12,T = p 1,T − p 2,T | ; Fig. 1-b corresponds to the central diffraction of two gluons with different colour charges in the colourless state.
origin based on the unitarity constraints and physical properties of the Pomerons. Indeed, the unitarity constraint has the following form
2 ImA el (s, b; i) = |A el (s, b; i) | 2 elastic cross section + G (s, b, i) contribution of inelastic processes(7)
where W = √ s denotes the energy of the collision, b is impact parameter, and i the set of other quantum numbers that diagonalize the interaction matrix.
For the BFKL Pomeron, the elastic cross section is much smaller than the exchange of a single Pomeron, and Eq. (7) takes the form [35] for two parton shower production of gluons: Fig. 2-a describes the interference diagrams in the production of two identical gluons in the process of multiparticle production that generates the correlation function C L c |p 12,T ; Fig. 2-b corresponds to central diffraction of two gluons with different colour charges in the colourless state; Fig. 2-c describes the central diffractive production with a different final state, where one deuteron remains intact. The wavy line stand for the BFKL Pomeron [36]. Helical lines correspond to gluons.
2 ImP BFKL (s, b, i) = G BFKL (s, b, i)
The vertical dashed lines show the cuts.
Using Eq. (8) one can see that Fig. 3(a-c)) and the contributions of the central (Fig. 3-d) and two parton showers production ( Fig. 3-e) of two gluons.
σ el ∝ |P BFKL (s, b, i) | 2 ; σ one parton shower ∝ − 2 P BFKL G BFKL (s, b, i) ; σ two parton showers ∝ − 1 2 G BFKL (s, b, i) G BFKL (s, b, i) ;(9)
where 1 2 in the last term stem from the fact that the two cut Pomerons are identical. Using Eq. (8) one reproduces the AGK cutting rules of Fig. 3-a -Fig. 3 -c.
The central diffraction production of two gluons is shown in the diagram of Fig. 3-a (elastic scattering), while the interference diagram, that generates the Bose-Einstein correlations, originates from Fig. 3-c with the extra factor 2, which reflects the fact that the gluon with rapidity, say, y 1 can be produced from two different parton cascades (see Fig. 3-e). The processes of central diffractive production are suppressed by a factor of 4 compared to the Bose-Einstein correlations.
To complete the discussion of the possible restoration of ϕ → π − ϕ symmetry, due the processes of the central diffraction, we note that in these processes there can be a final state in which one or two deuterons remain intact ( see for example Fig. 2-c) which leads to different correlation functions. For example for Fig. 2-c the correlation function has the form
C Fig. 2−c L c |p 1,T + p 2,T | ∝ 1 N 2 c − 1 d 2 Q T G D (Q T ) G 2 D Q T − p 1,T − p 2,T d 2 Q T G D (Q T ) G D (Q T )(10)
which differs from Eq. (6). A comment regarding the status of the AGK cutting rules in QCD. For deuteron-deuteron scattering, the cutting rules shown in Fig. 3-a - Fig. 3-c , have been proved on general grounds [38], using unitarity and the wave nature of the colliding particles. In the framework of perturbative QCD these cutting rules were proven in Refs. [33,39]. For the inclusive cross sections, the AGK cutting rules were discussed and proven in Refs. [40][41][42][43][44][45][46]. However, in Ref. [47] it is shown that the AGK cutting rules are violated for double inclusive production. This violation is intimately related to the enhanced diagrams [46,47], and reflects the fact that different cuts of the triple BFKL Pomeron vertex lead to different contributions. Recall, that we do not consider such diagrams.
Therefore, the contribution of the central diffraction process is suppressed by a factor of four, due to the longitudinal momenta integration. However, we need to compare the values of the vertices for gluon inclusive production (see Fig. 3d )and the vertex for two gluon production from the BFKL Pomeron. From Fig. 4 we can see that this vertex, is two times larger than the vertex for gluon inclusive production. Indeed, the contribution of Fig. 4-a is the same as for inclusive production, but we have to add Fig. 4-b. In appendix B we show that these two diagrams ( Fig. 4-a and Fig. 4-b) are the same. Adding these diagrams we note that for deuteron-deuteron scattering we expect, the symmetry ϕ → π − ϕ in the measurements with no selection on multiplicity. This observation supports the claim of Refs. [15,16].
In this paper as well as in Refs. [15][16][17] we discuss the caseᾱ S |y 1 − y 2 | ≤ 1. Let us consider this restriction more carefully. We start with writing the expression for the two diagrams of Fig. 3-d. The inclusive cross section for production of the gluon with rapidity y 1 and transverse momentum p 1,T due to the exchange of one BFKL Pomeron, has the following form
dσ dy 1 d 2 p T 1 ∝ᾱ S p 2 1,T d 2 k T φ BFKL (Y − y 1 , k T ) Γ µ (k T , p 1,T ) Γ µ (k T , p 1,T ) k 2 T k T − p 1,T 2 φ BFKL (y 1 , k T )(11)
The interference diagram in which the parton shower with a gluon with y 1 and p 1,T in the amplitude, is squared with the parton shower in which a gluon with y 2 and p 2,T is produced, takes the form
dσ dy 1 d 2 p T 1 ∝ᾱ S p 2 1,T d 2 k T φ BFKL (Y − y 1 , k T ) Γ µ (k T , p 1,T ) Γ ν (k T , p 2,T ) k 2 T k T − p 2,T 2 φ BFKL (y 2 , k T )(12)
In Eq. (11) and Eq. (12) we neglected p 12,T ∝ 1/R D as we have explained above. In Eq. (11) and Eq. (12) φ is the solution of the BFKL equation
∂φ BFKL (y, k T ) ∂y =ᾱ S d 2 k ′ T π 1 (k T − k ′ T ) 2 φ BFKL (y, k ′ T ) − 2ω G (k T ) G (y, k T ) ;(13)
where
ω G (k T ) = 1 2ᾱ S k 2 T d 2 k ′ T 2π 1 k ′2 T (k T − k ′ T ) 2 =ᾱ S k 2 T d 2 k ′ T 2π 1 k ′2 T + (k T − k ′ T ) 2 (k T − k ′ T ) 2(14)
Comparing Eq. (11) and Eq. (12) one can see that to neglect the difference between y 2 and y 1 in φ BFKL (y 2 , k T ) we need to assume that 2.8ᾱ S |y 1 − y 2 | ≪ 1 ( 2.8ᾱ S is the intercept of the BFKL Pomeron). However, the actual restriction turns out to be even more severe. Indeed, in all interference diagrams as well as in double gluon production between rapidities y 1 and y 2 , we have the exchange in the t-channel of two gluons in the octet state. This means that we have the additional emission of gluons with rapidities between y 1 and y 2 (see Fig. 4-c). This emission leads to the extra Sudakov form factor [49] in Eq. (12) which takes the form:
dσ dy 1 d 2 p T 1 ∝ᾱ S p 2 1,T d 2 k T e − S(δy,kT ,p1,T ) φ BFKL (Y − y 1 , k T ) Γ µ (k T , p 1,T ) Γ ν (k T , p 2,T ) k 2 T k T − p 2,T 2 φ BFKL (y 2 , k T )(15)
where δY = |y 1 − y 2 |. We recall the structure of the one parton shower that is described by the BFKL Pomeron in Fig. 4-e [36], the one parton shower is given by
n i=1 Γ µ (k i,T , p i,T ) e ωG(ki,T ) (yi−yi−1) k 2 i,T(16)
which being squared, leads to the parton density φ (y, k 1,T ). In simple words the BFKL cascade is the ladder diagram with specific vertices of gluon production, and with the exchange of the reggeized gluons with trajectories which are given by Eq. (14). Absorbing the terms in φ(y, k T ) for Eq. (15) we see that
S (δy, k T , p 1,T ) = ω k T − p 1,T + ω (k T ) δy =ᾱ S 2 ln k T − p 1,T 2 /µ 2 + ln k 2 T /µ 2 δy(17)
and it has a typical Sudakov form factor structure. µ is the typical dimensional parameter which in the DGLAP evolution, is of the order of the soft scale in the hadron, and in CGC it is a saturation scale Q s (y 1 ≈ y 2 ). For the diagrams of Fig. 4-a and Fig. 4-b we need to introduce the same suppressions. These Sudakov suppressions result from the fact that in the approximation forᾱ S δy ≪ 1 we take into account only simple diagrams with two gluons, and without extra gluon emissions; and they stipulate the size we need to take for δy. However, the two gluon production has an additional suppression of the Sudakov type , which applies even at y 1 = y 2 , where S of Eq. (17) is equal to zero: the emission of gluons that are shown in Fig. 4-d, has been discussed in detail in Ref. [48,49].
This emission leads to the value of S in the double log approximation of perturbative QCD that has the form:
S (p 1,T , k T ) =ᾱ S π M/2 kT d 2 q T q 2 T M/2 qT dq 0 q 0 =ᾱ S 4 ln 2 M 2 4 k 2 T(18)
where M denotes the mass of the produced dijet, which is given by M 2 = 2p 2 T (1 + cosh (y 1 − y 2 )) considering p 1,T = −p 2,T = p T . The limits in integration over q 0 can easily be understood in the rest frame of the two gluon jets. In this frame the minimal q 0 = q T . The lower limit in q T integration stems from the fact, that at distances less than 1/q T , the emission with two t-channel gluons have a distructive interference canceling the emission, since the total colour charge is zero. For q T ≥ q T the emission of gluons comes from the t-channel gluon, which carries color, and leads to the color coefficient in Eq. (18). Fig. 4-c shows the emission of soft gluons whose suppression leads to the Sudakov form factor. Fig. 4 shows the emission of the gluon in the DLA approximation of perturbative QCD, which leads to the Sudakov form factor in the vertex of two gluons emission .
Finally, the contribution of the diagram of Fig. 4-b has the following for y 1 = y 2 :
dσ dy 1 d 2 p T 1 (F ig. 4 − b) ∝ (19) α S p 2 1,T d 2 k T e − S(M,kT ) φ BFKL (Y − y 1 , k T ) Γ µ k T , p 1,T Γ ν k T − p 1,T , p 2,T k 2 T k T − p 1,T 2 φ BFKL (y 1 , k T )
The integration over k T of the parton densities is concentrated in the vicinity of the saturation scale, since in coordinate space φ ∝ ∇ 2 N (r, y) [50], deep in the saturation region it tends to zero. Of course, we consider not only one BFKL Pomeron, but a more complicated structure of the single parton cascade (see Fig. 5). Therefore, substituting Q s instead of k T in the Sudakov form factor, we find that Eq. (19) takes the form:
dσ dy 1 d 2 p T 1 (F ig. 4 − b) ∝ᾱ S p 2 1,T exp −ᾱ S 4 ln 2 p 2 T (1 + cosh (y 1 − y 2 )) 2 Q 2 s (y 1 ≈ y 2 )(20)d 2 k T φ BFKL (Y − y 1 , k T ) Γ µ k T , p 1,T Γ ν k T − p 1,T , p 2,T k 2 T k T − p 1,T 2 φ BFKL (y 1 , k T )
However, for discussing the current experimental data, especially for hadron-hadron interactions, for the parton densities, we can use the experimental data for DIS structure function which is well described [52], by the DGLAP evolution equations [53]. In this case, we need to put the value of Q 0 = Q s (y 1 = Y 0 ≈ 3) from the Colour Glass Condensate (CGC) motivated fit of HERA data [54,55]. This value turns out to be in the range Q 0 = 0.2 − 0.5 GeV [54,55].
Finally, we obtain the resulting correlation function is the sum of Eq. (6) and Eq. (10):
C (ϕ) = C Eq. (6); L c 2 p T sin(ϕ) + e − α S 2 ln 2 p 2 T (1+cosh(y 1 −y 2 )) 2 Q 2 s C Eq. (10); L c 2 p T cos(ϕ)(21)
where we assume that |p 1,T | = |p 2,T | = p T . The general expectation from Eq. (21) indicates that v n with odd n will peak at p T ≈ 4Q 0 , where the second term will be approximately three times smaller that the first one. The experimental data for v n in proton-proton collisions [56] show that v n reaches a maximum at p T ≈ 3 GeV , and this value is independent of the energy. Such a behaviour qualitatively supports Eq. (21) with Q 0 ≈ 0.6 GeV .
Concluding this section we would like to summarize our results: (i) we showed that at small transverse momenta the processes of exclusive(diffractive) in the central rapidity region (CED) of two gluons, are equal to the interference contributions of two parton showers, confirming the results of Refs. [15,16], this fact leads to v n = 0 for odd n, in the total inclusive measurements, without any selection on multiplicity of produced hadrons; (ii) we found the mechanism of suppression of CED of two gluon jets for large transverse momenta due to Sudakov form factor, which leads to the correlation function of Eq. (6), and to v n = 0 for odd n, in the experiments without selection on multiplicities; and (iii) only the correlation function of Eq. (5) can be measured in the processes of multiparticle generation with the multiplicities N ≥n, wheren is the average multiplicity in the collisions. The process of the central diffraction which generates the correlation function of Eq. (6) corresponds to the event with low multiplicity N <n). The last item is the best motivation for study of the identical particle correlations v n with even n, and with different multiplicities, which we will consider below.
b) a) c)
FIG. 5: The double inclusive production for dense-dense parton system scattering: the central diffraction production (see Fig. 5-a) and the Bose-Einstein correlation of the identical gluons ( Fig. 5-b). The wavy lines denote the BFKL Pomerons. Fig. 5-c shows the diagrams that do not contribute for the inclusive production of two gluons. The green blobs show the Mueller vertices for two gluon production, while the circles stand for the triple Pomeron vertices. The produced gluons are denoted by red helical lines.
B. Bose-Einstein correlation function for heavy ions scattering with the correlation length Lc ∝ RA
Inclusive measurements
Concluding the previous subsection, we claim that for deuteron-deuteron scattering, we see how the processes of the central diffraction, in the measurements that sum processes with all possible multiplicities of produced particles, can lead to the symmetry ϕ → π − ϕ for p T ≤ Q s . In this section we would like to examine, if such symmetry could be possible for ion-ion interactions, which can be described by the Glauber [57] formula (see Fig. 6-a):
A AA (s, b) = i (1 − exp (−Ω (s, b))) with Ω (s, b) = g 2 A P BFKL (s, b) T AA (b)(22)
where T AA (b) is the optical width and given by
T AA (b) = d 2 b ′ S A b − b ′ S A b ′ with S A (b) = +∞ −∞ dzρ (z, b) d 2 b S A (b) = A(23)
where ρ (z, b) denotes the nucleon density in the nucleus, and z the longitudinal coordinate of the nucleon. In Eq. (22) g N denotes the impact factor that describes the interaction of the BFKL Pomeron (whose Green function is P BFKL ), with the nucleon. . 6-a), and the first corrections to this approach due to triple BFKL Pomeron interactions ( Fig. 6-b). The wavy lines denote the BFKL Pomerons. The blobs show the triple Pomeron vertices.
We wish to stress that Eq. (22) in the framework of perturbative QCD (pQCD) has a region of applicability. Indeed, the contribution of one BFKL Pomeron in pQCD, in Eq. (22), is proportional to
g 2 N P BFKL T AA (b) ∝ α 2 S A 4/3 exp (∆ BFKL Y ) where ∆ BFKL ∝ᾱ S ,
where ∆ BFKL denotes the BFKL Pomeron intercept. The first 'fan' diagrams lead to corrections to the Glauber formula, these are shown in Fig. 6-b , and are of the order
g 2 N P BFKL (Y ) T AA (b) Y 0 dy ′ G 3I P g N P BFKL (y ′ ) S A (b) ∝ᾱ 4 S A P BFKL (Y, b) 2(24)
Comparing Eq. (24) with the exchange of two BFKL Pomerons, we see that the contribution of the fan diagrams will be smaller that 1 for ∆ BFKL Y ≪ 1 2 ln 1/(ᾱ 4 S A) , while the contribution of the BFKL Pomeron in Glauber formula will be larger than 1. In other words, for Y ≤ (1/(2 ∆ BFKL )) ln 1/(ᾱ 4 S A) we can describe the ion-ion collisions using the Glauber formula of Eq. (22).
In this formula the contributions of n-BFKL Pomeron exchanges to the total cross section is equal to
σ (n) tot = 2 (−1) n−1 n! Ω n (s, b) .(25)
Accordingly to the AGK cutting rules, the relative weight of the process with m cut Pomerons, (n − m + 1 of them are not cut) is equal To find the contribution of all possible processes of different multiplicities related to the production of m-parton showers, we need to calculate the following sum (see Fig. 7)
σ (m) n σ (n) tot = (−1) n−m n! m! (n − m)! 2 n−1 for m ≥ 1; σ (0) n σ (n) tot = (−1) n 2 n−1 − 1 ;(26)d 2 σ dy 1 dy 2 d 2 p T 1 d 2 p T 2 = (27) C A (L c |p 12,T |) dσ BFKL dy 1 d 2 p T 1 dσ BFKL dy 2 d 2 p T 2 ∞ m=2 n m=2 m(m − 1) σ (m) n Ω 2 (s, b) = 4C (L c |p 12,T |) dσ BFKL dy 1 d 2 p T 1 dσ BFKL dy 2 d 2 p T 2
In Eq. (27) Neglecting all correlations inside the nucleus, its wave function can be written as
Ψ A ({r i }) = A i=1 Ψ i (r i )
where Ψ (r i ) denotes the wave function of i-th nucleon. In this approach
C A (L c |p 12,T |) = 1 N 2 c − 1 d 2 Q T G 2 A (Q T ) G 2 A Q T − p 12,T d 2 Q T G 4 A (Q T ) with G A (Q T ) = d 2 b e ib·Q T S A (b)(28)
where S A (b) denotes the number of the nucleons at fixed impact parameter b.
Eq. (28) can be re-written in the impact parameter representation using Eq. (23): viz.
C A (L c |p 12,T |) = 1 N 2 c − 1 d 2b e ib·p 12,T T 2 A b d 2b T 2 A b where T A (b) = d 2 b ′ , S A (b ′ ) S A b − b ′(29)
The production of gluons by the BFKL Pomerons given by the Mueller diagrams in Fig. 7, generally has a more complicated form than we used in Eq. (27) ( see Eq.(38) of Ref. [17]), and cannot be reduced to the production of single inclusive cross sections. However, in the case of deuteron scattering, we can consider p 1,T = p 2,T , since the difference p 12,T ∼ 1/R D ≪ 1/R N or ≪ Q s , where 1/R N and Q s are typical momenta in the BFKL Pomeron. Bearing this in mind, we can replace the contribution of the Mueller diagram by the single inclusive production of the gluon, by the BFKL Pomeron.
The contribution to the central diffraction productions is shown in Fig. 8, and takes the following form 29), as well as p 1,T = −p 1,2 for deuteron-deuteron scattering. Actually , these estimates are correct only in the region of large Ω. The general expression for the correlation function has the following form
d 2 σ dy 1 dy 2 d 2 p T 1 d 2 p T 2 = (30) C A L c |p 1,T + p 2,T | dσ BFKL dy 1 d 2 p T 1 dσ BFKL dy 2 d 2 p T 2 n−2 2 n! 2!(n − 2)! σ (0) n Ω 2 (s, b) + ∞ n=1 n−2 m=1 n! 2!(n − m − 2)! σ (m) n Ω 2 (s, b) = C A L c |p 1,T + p 2,T | dσ BFKL dy 1 d 2 p T 1 dσ BFKL dy 2 d 2 p T 2 Ω≫1 − −− → 2 C A L c |p 1,T + p 2,T | dσ BFKL dy 1 d 2 p T 1 dσ BFKL dy 2 d 2 p T 2C A L c |p 1,T + p 2,T | = (31) 1 N 2 c − 1 d 2b e ib·(p 1,T +p 2,T ) d 2 B d 2 b S A B + 1 2b S A B − 1 2b S A b + 1 2b S A b − 1 2b 2 − exp (−Ω (b + B)) d 2b d 2 B d 2 b S A B + 1 2b S A B − 1 2b S A b + 1 2b S A b − 1 2b 2 − exp (−Ω (b + B))
We also make use of the fact that the Mueller vertex for production of two gluons by the BFKL Pomeron (see Fig. 8), is equal to the Mueller vertex for inclusive production of a single gluon (see Fig. 7). Comparing Eq. (27) and Eq. (30) we see that the contribution of the central diffraction production, is twice as large (at small p T ) than the contribution of the Bose-Einstein correlations. Therefore, the dominant contribution comes from Eq. (30) leading to the negative values of v n,n for odd n. This prediction contradicts experimental observations. Such a situation could result for two reasons:(1) the measured p T are larger than typical momentum Q 0 , and this contribution is suppressed, as has been discussed in Eq. (21); and (2) the measurements were not made in an inclusive type of the experiment, in which all events were summed without selection on multiplicities of the secondary hadron, but only events with large multiplicity were measured.
Measurements with fixed multiplicity N = mn,
First, we would like to examine what happens to the symmetry ϕ → π − ϕ in an event with given multiplicity. We need to compare the production of m parton showers which generate the event with multiplicity N = mn, with the event with the same multiplicity, but in which we produce in addition the low multiplicity events, by central diffraction production. From the point of view of the AGK cutting rules, the first process, is the process with m-cut Pomerons, while the second, is the process with the same m-cut Pomerons, plus two Pomerons which are not cut. At first sight, the second case could have a larger cross section, since it has an additional factor (σ in T A (b)) 2 , which can be large for nucleus-nucleus scattering. We need to estimate this contribution since it is suppressed by factor exp (−2Ω) in Eq. (39). In Fig. 9 we plot the b-dependence of σ (m) (b) of Eq. (39), together with the coefficient from the AGK cutting rules. From this figure we see that the processes of central diffraction in the inelastic environment is dominant, except for the process with N = 2n which needs additional consideration. This fact is a bit surprising since Inelastic production:
N = mn σ (m) in ∝ m(m − 1) m! (2Ω (b)) m−2 exp (−2Ω (b)) ;
Inelastic production + CED:
N = mn σ (m) CED ∝ 2 m! (2Ω (b)) m exp (−2Ω (b)) ;(32)
The survival probability exp (−2Ω (b)), is very small at all b less than 2 R A , and determines the value for 2Ω (b) ≤ 1. Therefore, the extra factor (2Ω) 2 , is not an enhancement, but a suppression (see Fig. 9-c). Nevertheless, it turns out that together with numerical coefficients this kind of suppression does not work.
However, we need to consider the contribution to the correlation function, which includes the additional integrations over impact parameters, Fig. 9-a for m =2 and Fig. 9-b for m = 3. Fig. 9-c shows the same contribution as Fig. 9-a after all integrations. .
C p 1,T ± p 2,T = d 2b e (p 1,T ± p 2,T )·bc b ;c b = d 2 B c b , B c b , B = d 2 b S A B + 1 2b S A B − 1 2b S A b + 1 2b S A b − 1 2b σ (m) in,CED (B)(33)
Integration over all impact parameters shows that in the event with N = 2n, the process with dijet production is also larger than the Bose-Einstein correlations (see Fig. 9-c).
One can see that the multiparticle production accompanied by exclusive production of two gluon jet prevails, leading to negative v n,n , for odd n. For nucleus-nucleus collisions, it is well known, that this statement contradicts the experimental data [6,7,11].
Measurements with multiplicity N ≥ mn,
Hence, for nucleus-nucleus scattering, the inclusive experiments, as well as the measurements with fixed multiplicity in the Leading Log(1/x) Approximation of perturbative QCD, generate negative v n,n for odd n, which contradicts the experimental data. In this subsection we examine the situation when the events with multiplicities larger that m 0n (N ≥ m 0n ) is measured, as it has been done in the most experiments. Summing Eq. (32) over all m ≥ m 0 we obtain
σ m0 in (Y ; B) = 1 − Γ (m 0 − 2, 2Ω (B; Y )) Γ (m 0 − 2) Ω≫1 − −− → 1 − (2Ω (B; Y )) m0−3 (m 0 − 3)! e −2Ω(B;Y ) ;(34)
One can see that at large Ω, that the inelastic event with additional dijet production, is larger that the inelastic event that generates the Bose-Einstein correlations. In Fig. 10-a we plot the functionc b of Eq. (33), which also shows that the inelastic contribution with dijet production prevails. Fig. 10-b shows the correlation functions of Eq. (6) and Eq. (10). Note that the Bose-Einstein correlations are smaller than the correlations due to the diffractive production of dijets. Fig. 10-a shows the contribution of inelastic event and inelastic even plus central diffraction, for m 0 = 4. In Fig. 10-b we plot the correlation functions C ± |p 1,T ± p 2,T | (see Eq. (6) and Eq. (10)). p ± T ≡ |p 1,T ± p 2,T |.
Hence, the experimental results are in direct contradiction with the theoretical predictions based on the Leading Log(1/x) Approximation of perturbative QCD. The only explanation that we can suggest is, that the Sudakov form factor suppresses the dijets production.
We believe that the p measured p T turns out to be much larger than Q 0 , and double log suppression results in a small contribution of the process of central diffraction. Indeed, for the exchange of the BFKL Pomeron our value for Q 0 ≈ Q s (y 1 ) appears to be overestimated. Our conclusions that typical k T ≈ Q s is based on the diagrams of Fig. 5-a and Fig. 5-b, in which the same diagrams contribute to central diffraction and the inclusive cross section. However, for the exclusive central production there no AGK cutting rules, and the diagrams of Fig. 5-c should be taken into account. If we remove the integral in Eq. (19) for the Sudakov form factor, the remaining expression takes the form of Eq. (20). For the BFKL Pomeron, it is just the contribution to the total cross section. The typical transverse momenta in the BFKL Pomeron, both increase and decrease as function of rapidity (see Ref [58]) and at large y 2 or Y − y 1 , the typical k T is as small as the non-pertutbative soft momentum, which could be of the order of Λ QCD . If we replace the emission of gluons by Eq. (20), the diagrams of Fig.5-c reduce to the contribution to the total cross section, supporting the idea that Q 0 is of the order of typical soft momentum. Therefore, we expect that Q 0 ≈ µ soft ≈ Λ QCD . Bearing this in mind we concentrate our efforts below on the calculating Bose-Einstein correlations, and their dependence on multiplicity of the events.
III. DEPENDENCE OF BOSE-EINSTEIN CORRELATIONS ON THE MULTIPLICITY OF THE EVENT
In this section, we consider the dependence of Bose-Einstein correlations on the multiplicity of the event, using the Glauber formula for the total cross section. In accord with the AGK cutting rules, the multiplicity of the event (N ) is intimately related to the number of parton showers (m) that are produced, where N = mn.
In the framework of this approach, the Bose-Einstein correlations in the event with multiplicity N = mn is determined by the following expression (see also Eq. (33)):
d 2 σ dy 1 dy 2 d 2 p T 1 d 2 p T 2 ∝ C A L c |p 12,T | dσ BFKL dy 1 d 2 p T 1 dσ BFKL dy 2 d 2 p T 2 ; (36) C A L c |p 12,T | = 1 N 2 c − 1 I L c |p 12,T | I (0) , I L c |p 12,T | = d 2b e ib·p 12,T I b (37) I b = d 2 B c b , B(38)σ (m) (B + b) = ∞ n=m,m≥2 m (m − 1) σ (m) n Ω 2 (s, B) = (2Ω (s, B)) m−2 (m − 2)! e −2Ω(s,B)(39)
If we assume S A (b) to have a Gaussian form i.e. S A (b) = A/ πR 2
A exp −b 2 /R 2 A ,I b = A π R 2 A 4 e −b 2 R 2 A d 2 B d 2 b e −2 (B 2 +b 2 ) R 2 A σ (m) n (B + b) Ω 2 (s, B + b)(40)
and the correlation function does not depend on m or, in other words, it does not depend on the multiplicity of the event. However, this result is the specific property of the Gaussian approximation, which cannot be correct even for hadron-hadron collisions, since it does not lead to the correct exponential behaviour of the scattering amplitude at large impact parameters b. Considering the Glauber model for the description of the proton-proton scattering at high energies, we replace S A and T A in Eq. (28) and Eq. (29) by
S N (b) = m 2 2 π K 0 (mb) ; T N = d 2 b ′ S N (b ′ ) S N b − b ′ ; Ω = σ 0 e ∆ Y T N (b)(41)
where σ 0 = 4 1/GeV 2 , m = 1 GeV and ∆ = 0.1, were chosen to describe the value and energy behaviour of the total cross section for the proton-proton interaction at high energy. In Fig. 11-a the behaviour of I (b) is shown for the events with different multiplicities. We see that the correlation length L c decreases as function of the multiplicity. In other words, the typical momentum in the correlation function C (L c p 12,T ) increases with N , as can be seen from Fig. 11-b, where the value of the correlation function C (L c p 12,T ) is plotted. The correlation length of the correlation function in nucleus-nucleus collisions, shows only mild dependence on the multiplicity of the events, (see Fig. 12 -b, while the value of I crucially depends on N (see Fig. 12-a). Fig. 12-c shows that the correlation function C A (L c p 12,T ) does not depend on the multiplicity of the event. 11: Fig. 11-a shows I (b) for proton-proton scattering with the parameters, that are given in Eq. (41), as a function of b, for the events with different multiplicities normalized to 1 at b = 0. In Fig. 11-b the correlation function C (p 12,T ) is plotted versus p 12,T .n is the average multiplicity in the single inclusive production.
For completeness of presentation we calculated both I (b) and C pA (p 12,T ) for proton-gold scattering.
The results of these calculations are plotted in Fig. 13. The first observation is that the correlation length does not depend on the size of the nucleus, and is determined by the typical impact parameter in proton-proton scattering. The dependence on multiplicity of the event is rather mild. 28), as a function of b, for the events with different multiplicities. In Fig. 12-a I (b) are normalized to their values at b = 0.n is the average multiplicity in the single inclusive production. The correlation function C (p 212,T ) is plotted in Fig. 12-b.
Concluding this section, we would like to emphasis that the dependence on multiplicity due to the production of several parton showers, turns out to be mild, except for the case of hadron-hadron collisions. For this collision the larger multiplicity of the event, the shorter is the correlation length L c , or, in other words, the typical momentum increases in the events with large multiplicities. On the other hand, such an increase is not very pronounced, and even for hadron-hadron collisions, we can expect that the main source of the multiplicity dependence is from the structure of one parton shower. In the next section, we discuss the saturation of the parton density in the one parton shower for nucleus-nucleus collisions, and we develop a simple model in the spirit of the KLN approach. Fig. 13-a shows I (b) for proton-gold scattering with the parameters that are given in Eq. (28), and with the typical b = 1 1/GeV in proton-proton scattering, as a function of b, for the events with different multiplicities, normalized to 1 at b = 0. In Fig. 13-b the correlation function C pA (p 12,T ) is plotted versus p 12,T .n denotes the average multiplicity in single inclusive production.
IV. A SIMPLE KLN -TYPE MODEL FOR THE STRUCTURE OF ONE PARTON CASCADE IN CGC
A. Momentum dependence of the BFKL Pomeron in a nucleus.
As we have seen, the diagrams in which the structure of the one parton shower is described by the BFKL Pomeron, lead to the correlation length of azimuthal angle correlations L c ∝ 1/R A or, in other words, to the typical transverse momentum which is very small (see Fig. 12). Therefore, we need to discuss a more complicated structure of the single parton shower, which is related, for example, to 'fan' diagrams shown in Fig. 5-b. We expect that the interaction of the BFKL Pomeron will lead to the value of L c ∼ 1/Q s,A , where Q s,A denotes the nucleus saturation momentum. In particular, we consider the diagrams of Fig. 14-a and Fig. 14-b. The diagram of Fig. 14-a is the first diagram that leads to the correlation function which depends on the saturation momentum of the nucleon, as shown in Ref. [17,21]. We will show that the interaction of the BFKL Pomerons with the nucleus, examples of which are shown in Fig. 14
-b, will lead to L c ∝ 1/Q s,A . (y ,p ) 1 2T (y ,p ) 1 1T (y ,p ) 1 1T a) Y Y' 0 b) c)
FIG. 14: The double inclusive production for ion-ion collisions which lead to the azimuthal correlations with the correlation length L c ∝ 1/Q s : the first diagram is displayed in Fig. 14-a, while Fig. 14 of the non-linear Balitsky-Kovchegov equation of Fig. 15-c , respectively, the equations take the forms
T A (Y, Q T ; Y ′ Q ′ T ) = T (Y − Y ′ , Q T ) − ∆ Y 0 dY ′′ d 2 Q ′′ T T (Y − Y ′′ , Q T ) G A Y ′′ , Q T − Q ′′ T T A (Y ′′ , Q ′′ T ; Y ′ Q ′ T ) ;(42) ∂T A (Y, Q T ; Y ′ Q ′ T ) ∂Y = ∆ T A (Y, Q T ; Y ′ Q ′ T ) − d 2 Q ′′ T G A Y, Q T − Q ′′ T T A (Y, Q ′′ T ; Y ′ Q ′ T ) ;(43)T (Y − Y ′ , Q T ) = g (Q T ) exp (∆ (Y − Y ′ )) ; T A (Y = Y ′ , Q T ; Y ′ Q ′ T ) = g (Q T ) ;(44)G A (Y, Q T ) = G (Y − Y ′ , Q T ) − ∆ Y 0 dY ′′ d 2 Q ′′ T G 0 (Y − Y ′′ , Q T ) G A Y ′′ , Q T − Q ′′ T G A (Y ′′ , Q ′′ T ; Y ′ Q ′ T ) ; (45) ∂G A (Y, Q T ) ∂Y = ∆ G A (Y, Q T ) − d 2 Q ′′ T G A Y, Q T − Q ′′ T G A (Y, Q ′′ T ) ; (46) G 0 (Y − Y ′ , Q T ) = exp (∆ (Y − Y ′ )) ; G A (Y = 0, Q T ) = S A (Q T ) with S A (Q T ) = d 2 be iQ T ·b S A (b) ;(47)
The main idea of solution, is the observation that in
G A (Y, Q T ) the typical Q T ∼ 1/R A ≪ 1/R N or Q s , where R N is the nucleon size. Therefore, in Eq. (42)-Eq. (47) we can replace G A (Y, Q T ) by d 2 Q T G A (Y, Q T ) δ 2 (Q T ). At Y = 0, d 2 Q T G A (Y, Q T ) = S A (b = 0) ∝ 2ρR A ,
where ρ denotes the density of the nucleons in a nucleus. Plugging this expression in the above equations, they reduce to the following form
dT A (Y, Q T ; Y ′ Q ′ T ) dY = ∆ T A (Y, Q T ; Y ′ Q ′ T ) −G A (Y ) T A (Y, Q T ; Y ′ Q ′ T ) ; (48) dG A (Y ) dY = ∆ G A (Y ) −G 2 A (Y ) whereG A (Y ) = d 2 Q T G A (Y, Q T )(49)
Solving Eq. (49) and Eq. (48) we obtaiñ
G A (Y ) = S A (b = 0) e ∆Y 1 + S A (b = 0) (e ∆Y − 1) ; T A (Y, Q T ; Y ′ Q T ) = g (Q T ) e ∆(Y −Y ′ ) 1 + S A (b = 0) e ∆Y ′ − 1 1 + S A (b = 0) (e ∆Y − 1) ;(50)
In the general case, the equations have a more complicated structure, and include the dependence on the transverse momenta, which are the Fourier images of the dipole sizes. However, in the vicinity of the saturation scale, the scattering amplitude displays a geometric scaling behaviour [60], and depends only on one variable Q 2 s /p 2 T . In the vicinity of the saturation scale the equations take the form:
dT A (z; z ′ ) dY = (1 − γ cr ) T A (z, z ′ ) −G A (z) T A (z, z ′ ) ; (51) dG A (z) dz = (1 − γ cr ) G A (z) −G 2 A (z)(52)
Solutions of these equations have the following form:
G A (z) = φ 0 e (1−γcr) z 1 + φ 0 e (1−γcr) z − 1 ; T A (z, z ′ ) = g (Q T ) e (1−γcr )(z−z ′ ) 1 + φ 0 e (1−γcr) z ′ − 1 1 + φ 0 e (1−γcr) z − 1 ;(53)
where φ 0 denotes the value of the scattering amplitude at z = 0 and
z = ln Q 2 s,A (Y ) p 2 T with Q 2 s,A (Y ) = S A (b = 0) Q 2 s (Y )(54)
where Q S (Y ) denotes the proton saturation momentum.
The principle feature of all these solutions is that, the interaction with nucleus, which is shown in Fig. 14-b and in Fig. 15, does not affect the dependence on Q T , which determines the angular correlations. The only diagrams that could depend on the nuclear saturation momentum, are shown in Fig. 14-c. Generally speaking the BFKL Pomeron from the rapidity 0 to rapidity Y ′ , should be replaced by the dressed BFKL Pomeron (see Fig. 16). B. The model.
The general formulae.
The diagram for the interference of two parton showers is shown in Fig. 16, and can be written in the form:
d 2 σ interference diagram dy 1 dy 2 d 2 p 1,T d 2 p 2,T ∝ (55) α 2 S V 2 (p 1,T , p 2,T , y 1 − y 2 ) p 2 1,T p 2 2,T z1≈z2 0 dz ′ G A (z Y − z 1 ) G A (z Y − z 2 ) T A (z 1 − z ′ ) T A (z 2 − z ′ ) Γ 3I P (Q T ; Q s,A (Y ′ )) G A (z ′ ) Assumingᾱ S (y 1 − y 2 ) ≪ 1, V (p 1,T , p 2,T , y 1 − y 2 ) takes the simple form V (p 1,T , p 2,T , y 1 − y 2 ) = Γ µ (p 1,T , k T ) Γ µ (p 2,T , k T )(56)
with integration over k T . Since this function does not depend on Q T , we are not interested in its exact structure. The only function which determines the Q T , is the triple Pomeron vertex (see Ref. [17]). However, we recall that in inclusive production, the contributions of the BFKL Pomerons with rapidities Y − y 1 (y 2 ) and y 1 (y 2 ) − Y ′ vanish in deep saturation region, as they are proportional to ∇ 2 N (r, . . .), ( where r denotes the dipoles size [31,50]), and N → 1 in the saturation region. This means that the contributions of these Pomerons have maximum at z → 0. Therefore, we can use the solutions of Eq. (53) to estimate the value of the cross section.
To specify the Q T dependence, we need to find which values of z ′ ( or Y ′ ) contribute to the integral. Plugging in T A from Eq. (54), we can take the integral over z ′ resulting in the following expression
d 2 σ interference diagram dy 1 dy 2 d 2 p 1,T d 2 p 2,T ∝ e 2(1−γcr) z 1 1 + φ 0 e 2(1−γcr ) z − 1 2 (1 − φ 0 ) 1 − γ cr + φ 0 z 1(57)
The two terms in Eq. (57) stem from different region of integration over z ′ . The first one originates from z ′ → 0 or Y ′ ∝ 1/ᾱ S . The second term comes from the region of integration in the entire kinematic region. The typical saturation momentum for such an integration is equal toQ 2 s,A = Q 2 s,A (Y 0 ) Q 2 s,A (y 1 ≈ y 2 ). The dependence on Q T only comes from the triple Pomeron vertex. Since G A ∝ S A (b), the typical Q T along two upper BFKL Pomerons are equal to zero Q T ∼ 1/R A ≪ 1/Q s , and the dependence on azimuthal angle ϕ stems from p 2 12,T = 4p 2 T sin (ϕ/2). Finally, the general formula for the angular correlations has the form The triple Pomeron vertex has been calculated in Ref. [17], and at large Q T it has the form (see Eq.(45) and Eq.(A12))
d 2 σ dy 1 dy 2 d 2 p 1,T d 2 p 2,T ∝ (1 − φ 0 ) 1 − γ cr Γ 3I P (Q T = 0, Q s,A (Y 0 )) + φ 0 z 1 Γ 3I P Q T = 0,Q s,A (y 1 ≈ y 2 )(58)+ 1 N 2 c − 1 (1 − φ 0 ) 1 − γ cr Γ 3I P (Q T = p 12,T , Q s,A (Y 0 )) + φ 0 z 1 Γ 3I P Q T = p 12| > 1.0} η ∆ {2, | n v 0 0.1 0.2 5.02 TeV |>1} η ∆ {2, | 2 v |>1} η ∆ {2, | 3 v |>1} η ∆ {2, | 4 v 2.76 TeV |>1} η ∆ {2, | 2 v |>1} η ∆ {2, | 3 v |>1} η ∆ {2, |Γ 3I P (Q T , Q s,A ) QT ≫ Qs,A − −−−−−− → 1 k T − 1 2 Q T 2γcr (Q 2 T ) 1−2γcr 2 QT ≫ kT ≈ Qs,A −−−−−−−−−− → 1 (Q 2 T ) 2(1−γcr )(59)
where k T denotes the momentum inside of the triple Pomeron vertex, which is of the order of the typical saturation momentum of the lower BFKL Pomeron in Fig. 16. To specify dependence of the triple Pomeron vertex, we recall FIG. 20: Comparison of the estimates from our model for v 3 , with the experimental data of ALICE collaboration [7].
that at large impact parameters, the scattering amplitude should decreases exponentially [61] . Bearing this in mind we suggest that
Γ 3I P (Q T , Q s,A ) = Q 2 s Q 2 T + Q 2 s 2(1−γcr )(60)
which reproduces Eq. (59) at large Q T , and has the exponential decrease at large b.
Plugging Eq. (60) into Eq. (58) we obtain the correlation function in the form
C A (p 12,T ) = 1 N 2 c − 1 (1−φ0) 1−γcr Γ 3I P (Q T = p 12,T , Q s,A (Y 0 )) + φ 0 z 1 Γ 3I P Q T = p 12,T ,Q s,A (y 1 ≈ y 2 ) (1−φ0) 1−γcr Γ 3I P (Q T = 0, Q s,A (Y 0 )) + φ 0 z 1 Γ 3I P Q T = 0,Q s,A (y 1 ≈ y 2 )(61)
The multiplicity dependence stems from Eq. (61), where we replace Q S,A by the value of the saturation momentum, which corresponds to the given number of participants, this in the spirit of the KLN approach [27,28]. In Fig. 17 the correlation functions are shown for W = 5.02 TeV, and for the choice Y 0 = ln (W 0 /m) with W = 130 GeV and m = 1 GeV . This function has an essential dependence on N part , or on centrality.
v n can be calculated for |p 1,T | = |p 2,T | as v n = dϕ cos (nϕ) C Npart (2p T sin (ϕ/2)) 2π + dϕ C Npart (2p T sin (ϕ/2)) 1 2
(62)
2. The Choice of parameters.
The formulae of Eq. (61) and Eq. (62) depend only on the value of the saturation momentum, and consequently, it depends on rapidity, and N part . We follow the KLN-approach [23,26,27] in finding these dependences. We assume that
Q 2 s (Y ; N part ) = ρ part 2 Q 2 0 e λ(Y −Y0) ;(63)
The value of Q 0 we fix from the gold-gold scattering at W = 130 GeV and for centrality 0 − 5% Q 2 s (Y = Y 0 ) = 2.02 GeV 2 . Y = ln (W/W 0 ) and Y − Y 0 = ln (W/130). ρ part have been calculated in Ref. [23] for the LHC energies, and in Ref. [27] for W 0 = 130 GeV . The choice Y 0 = ln (W 0 /m) in Eq. (61) is not theoretically determined, note that the value of typical ∆Y ′ in the integral over Y ′ , is about ∆Y ∼ 1/ᾱ S , and forᾱ S = 0.2, this results in a value which is close to the chosen Y 0 . Finally, we take λ = 0.25 as it is done in Refs. [23][24][25][26][27]. 3. Comparison with the experimental data.
Using the parameters, discussed above, we evaluate the correlation function (see Fig. 17, and the values of v n which are plotted in Fig. 18, Fig. 19 and Fig. 20). First, we note that the correlation function depends strongly on the centrality, leading to a correlation length which increases for large centralities.
However, v n show only mild dependence on centralities (compare Fig. 18 and Fig. 19 ). Such a behaviour at first sight is in disaccord with the experimental data. v 2 turns out to be smaller that the experimental values for both centralities. On the other hand, the value for v 2 , as well as for other even n, is not very decisive, since in QCD there are many other sources of v n with even n, beside Bose-Einstein correlations. However, we have not found the other sources for v n with even n. Fig. 20 presents our estimate for v 3 together with the experimental data. We see that our predictions for v 3 describe the experimental data fairly well. Not extremely well, but the model that we develop here, is very simple. These estimates encourage us to develop a more complete description of v n for even n, with different multiplicities, based on the Bose-Einstein correlations.
V. CONCLUSIONS
We summarize the main results of this paper. The main goal of this paper is to investigate the dependence of Bose-Einnstein correlations on the multiplicity of the events. We view these correlations as the major source of the azimuthal angle correlations, and the only known origin of v n with odd n in the framework of the Color Glass Condensate. Indeed, the correlation of identical gluons produces the correlation function that depends on |p 1,T − p 2,T | which gives v n with odd n. However, in Refs. [15,16] it was noted, that the diffractive central production of two different gluons in the colorless state leads to dependence on |p 1,T + p 2,T |. If these two sources have the same strength, the totally inclusive experiments without any selection on multiplicities, will give v n = 0 for odd n. In this paper, we showed in the Leading Log(1/x) Approximation of perturbative QCD, the amplitude of two gluon exclusive production turns out to be equal to the interference diagram, that is the source of the Bose-Einstein correlation, in accord with Refs. [15,16]. However, the emission of soft gluons for the central exclusive production in the Double Log Approximation of perturbative QCD, leads to the Sudakov form factor which suppress this contribution. Therefore, the Bose-Einstein correlations prevail leading to v n = 0 for odd n. It should be stressed, that without this suppression, the measurement of an event with given multiplicity, yields v n,n < 0 for odd n.
We demonstrated that the Bose-Einstein correlation function does not depend on the number of produced parton showers for hadron-nucleus and nucleus-nucleus collisions, but for hadron-hadron collisions such dependence turns out to be considerable.
Finally, we developed a simple KLN-type model to describe the Bose-Einstein correlation in one parton cascade, as a function of centralities. The predicted dependence reflects the main features of the observed data, reproduces the value of v n with odd n, but, much work is still needed to develop a more complete approach. This paper encourages us to search for such an approach.
We view this paper as an argument that the description of v n is possible due to interactions in the initial state, and that these interactions should not be neglected. takes the following form (see Fig. 21-c) G (s, t = 0) = g 4 C 4s 2 dk + dk − d 2 k T (2 pi) 4 i 1 (k + k − − k 2 T − iǫ) 2 2πδ (P 1 −?k) 2 2π δ (P 2 + k) 2
= 16 C α 2 S s 2 dk + dk − d 2 k T 1 (k + k − − k 2 T − iǫ) 2 δ −P 1 + k − − k 2 T δ P 2,− k + − k 2 T = 16 C α 2 S s d 2 k T k 4 T (A1)
In Eq. (A1) C is the colour coefficient which is the same for all diagrams, factor 4s 2 (s = (P 1 + P 2 ) 2 = 2P 1,µ P µ 2 at high energy) stems from the summation over polarization of the t-channel gluon of the gluon current of quarks 2P 1,µ (2P 2,µ ). α S = g 2 /4π. Integrating the δ-functions, one can see that k + k − ≪ k 2 T . The scattering amplitude is equal to
A (s, t = 0) = g 4 C 4 s 2 dk + dk − d 2 k T (2 pi) 4 i 1 (k + k − − k 2 T − iǫ) 2 1 −P 1 + k − − k 2 T − i ǫ × 1 P 2,− k + − k 2 T − i ǫ F ig. 21−a + 1 −P 2,− k + − k 2 T − i ǫ F ig. 21−b(A2)
For k + > 0 we can take the integral over the pole: k 0 − = −k 2 T −iǫ P + 1 closing around this pole, the contour of integration in lower semi-plane in complex k − plane, since the integral over large circle decreases at large k − . The other pole
k 1 − = k 2 T +iǫ k +
is located in the upper semi-plane. For k + < 0 all singularities are situated in lower semi-plane leading to vanishing of the integral. Bearing this in mind we reduce Eq. (A2) to the following expression:
A (s, t = 0) = 8 α 2 S π C s 2 ∞ 0 d k + d 2 k T 1 k 4 T 1 −P + 1 1 P 2,− k + − k 2 T − i ǫ + 1 −P 2,− k + − k 2 T − i ǫ = 8 α 2 S π C s 2 ∞ −∞ d k + d 2 k T 1 k 4 T 1 −P + 1 1 P 2,− k + − k 2 T − i ǫ (A3)
Taking the integral over k + using contour C in Fig. 21-d, and taking into account that the integral over a large circle is equal to i π we obtain A (s, t = 0) = i 8 α 2 S C s
d 2 k T k 4 T (A4)
The diagram Fig. 21-c gives the same contribution as the imaginary part of diagram of Fig. 21-a, multiplied by factor 2, since in this diagram we have 2πδ P 2,− k + − k 2 T . Therefore, we obtain that 2 Im A (s, t = 0) = G (F ig. 21 − c) which proves Eq. (8) in Born approximation of pQCD.
For the amplitude of the two gluon production (see Fig. 22-a and Fig. 22-b) as well as for the cross section of the one gluon production which is shown in Fig. 22-c, we have the following hierarchy of the longitudinal momenta:
P + 1 ≫ p + 1 ∼ p + 2 ≫ k + ; P 2,− ≫ p 1,− ∼ p 2,− ≫ k − ;(A5)
assuming that both gluons are produced with almost equal rapidities (y 1 ≈ y 2 ) in the central rapidity region (y 1 ≈ y 2 ≪ 1) in c.m.f. Using Eq. (A5) we can reduce the amplitude to the following expresion: Fig. 21-a and Fig. 21-b are the diagrams for the scattering amplitude at high energy in theᾱ 2 S order of pQCD. Fig. 21-c is the cross section for two quarks production (cut Pomeron). Fig. 21-d shows the contour of integration over k + . Helical lines denote gluons, the solid lines indicate quarks.
A (F ig. 22 − a + F ig. 22 − b) = (A6) 32 π α 3 S C s 2 ∞ 0 d k + d 2 k T 1 k 4 T Γ µ (p 2,T , k T ) p 2,T + k T 2 Γ ν (p 1,T , k T ) p 1,T − k T 2 1 −P + 1 1 P 2,− k + − k 2 T − i ǫ
In Eq. (A6) we use the same contour of integration over k + (see Fig. 21-d) as calculating the elastic amplitude (see Eq. (A3)). The Lipatov vertices Γ µ for the gluon emission depend only on transverse momenta, and do not influence the integration over longitudinal momenta. The cross section of Fig. 22-c differs from the amplitude by factor 2, which has the same origin as has been discussed above (see Eq. (A1). Fig. 22-a and Fig. 22-b are the diagrams for the amplitude for the production of two gluons with momenta p 1 and p 2 , inᾱ 3 S order of pQCD. Fig. 22-c is the cross section for two quarks and two gluons production (cut Pomeron). Helical lines denote gluons, the solid lines indicate quarks. The blobs denote the Lipatov vertices for gluon production (Γ ν ).
T v n,n p Ref T , p Ref T ;
FIG. 3 :
3AGK cutting rules for the exchange of two BFKL Pomerons (
FIG. 4 :
4Vertex for emission of two gluons by the BFKL Pomeron.
FIG. 6 :
6Nucleus-nucleus scattering in the Glauber[57] approach (Fig
FIG. 7 :
7The contribution of different processes of production of the number of parton showers (more than 2), to the Bose-Einstein correlation. The wavy lines denote the BFKL Pomerons. The blobs show the Mueller vertices for two gluon production . The produced identical gluons are denoted in red helical lines.
we use Eq. (26), Eq. (25), the function C A is determined by an equation which is similar to Eq. (5).
FIG. 8 :
8The contribution of different processes of production of the number of parton showers to the central diffraction production. The wavy lines denote the BFKL Pomerons. The blobs show the Mueller vertices for two gluons production . The produced gluons are denoted by red helical lines. In Eq. (30) we use Eq. (26), Eq. (25) and the function C A L c |p 1,T + p 2,T | from Eq. (
FIG. 9 :
9Comparison of the inelastic events with the multiplicity N = mn: for the production of two identical gluons from the m-parton showers, and central diffraction production in the event:
σ m0 CED (Y ; B) = 2 1 − Γ (m 0 , 2Ω (B; Y )) Γ (m 0 ) Ω≫1 − −− → 2 1 − (2Ω (B; Y )) m0−1 (m 0 − 1)! e −2Ω(B;Y ) ;
FIG. 10 :
10Comparison of the inelastic events with the multiplicity N ≥ m 0n in gold-gold collision at W = 7 T eV : for the production of two identical gluons for larger than m 0 -parton showers, and central diffraction production in the event with the multiplicity not smaller than m 0n .
Fig. 11-a Fig. 11-b FIG. 11: Fig. 11-a shows I (b) for proton-proton scattering with the parameters, that are given in Eq. (41), as a function of b, for the events with different multiplicities normalized to 1 at b = 0. In Fig. 11-b the correlation function C (p 12,T ) is plotted versus p 12,T .n is the average multiplicity in the single inclusive production.
FIG. 12 :
12I (b) for nucleus-nucleus (gold-gold) scattering with S A (b) given in Eq. (
Fig. 13-a Fig. 13-b FIG. 13: Fig. 13-a shows I (b) for proton-gold scattering with the parameters that are given in Eq. (28), and with the typical b = 1 1/GeV in proton-proton scattering, as a function of b, for the events with different multiplicities, normalized to 1 at b = 0. In Fig. 13-b the correlation function C pA (p 12,T ) is plotted versus p 12,T .n denotes the average multiplicity in single inclusive production.
FIG. 15 :
15-b shows the interaction of the BFKL Pomerons which results in L c ∝ 1/Q s,A , where Q s,A denotes the saturation momentum of the nucleus. The wavy lines denote the BFKL Pomerons. The red blobs show the Mueller vertices for two gluons production, while the gray circles stand for the triple Pomeron vertices. The white circles show the vertex of the interaction of the BFKL Pomeron with the nucleon in the nucleus.The produced gluons are denoted by red helical lines. For simplicity we draw the diagrams at y 1 = y 2 . The general equation for the propagator of the BFKL Pomeron in a nucleus is shown in Fig. 15. The simplest form these equation have in the framework of Gribov Pomeron Calculus [59] with α ′ I P = 0 and the Pomeron intercept ∆. Denoting by T A (Y, Q T ; Y ′ Q ′ T ) and G A (Y, Q T ) the dressed (resulting) propagator of the Pomeron, and the solution Equations for BFKL Pomeron propagator in the nucleus. Fig. 15-a shows the first simple diagrams. Fig. 15-b presents the equation for the propagator. Fig. 15-c describes the Balitsky-Kovchegov equation. Wavy lines describes the BFKL Pomerons. The double wavy lines denote the resulting propagator. The bold wavy lines stand for the solution of Balitsky-Kovchegov equation in the nucleus. The blobs denote the triple Pomeron vertices.
FIG. 16 :
16Double inclusive cross section. The double wavy lines denote the propagator of the dressed BFKL Pomeron. The bold wavy lines stand for the solution of Balitsky-Kovchegov equation in the nucleus. Helical line denote gluons.
,T ,Q s,A (y 1 ≈ y 2 ) FIG. 17: The correlation function C Npart (p 12,T ) at different centralities: 0-5% and 30-40% , versus p
FIG. 18 :
18Experimental data for v n versus p T [7] at two different centralities: 0-5% in the upper figure and 30-40% in the lower one. FIG. 19: Our model for v n versus p T for different centralities: 0-5% in the upper and 30-40% in the lower figures.
FIG. 21 :
21+ d 2 k T 1 k 4 T Γ µ (p 2,T , k T ) p 2,T + k T 2 Γ ν (p 1,T , k T ) p 1,T − k 2,− k + − k 2 T − i ǫ = 32 π i α 3 S C s Γ µ (p 2,T , k T ) Γ ν (p 1,T , k T ) d 2 k T k 4T p 2,T + k T 2 p 1,T − k T Born Approximation of pQCD: longitudinal momenta integration.
FIG. 22 :
22Born Approximation of pQCD: longitudinal momenta integration.
We are grateful to Alex Kovner and Michael Lublinsky who drew our attention to this diagram, and explained that in their approach[16] this diagram restores the symmetry φ → π − φ.
Acknowledgements We thank our colleagues at Tel Aviv University and UTFSM for encouraging discussions. Our special thanks go to Carlos Cantreras, Alex Kovner and Michael Lublinsky for elucidating discussions on the subject of this paper. This research was supported by the BSF grant 2012124, by Proyecto Basal FB 0821(Chile) , Fondecyt (Chile) grant 1140842, and by CONICYT grant PIA ACT1406.Appendix A: Integration over longitudinal momentaIn this appendix we recall the calculation that results in Eq.(8). For simplicity we restrict ourselves to calculate both the scattering amplitude at high energies ( Pomeron, seewhile forFig. 4-b it can be written asWe need to calculate these vertices for p 1,T = − p 2,T , since |p 1,T + p 2,T | ∝ 1/R D ≪ 1/R N . The vertices Γ µ has the following expressions:We need to convolute these vertices with Γ µ l T , p 1,T and Γ ν l T − p 1,T , p 2,T for the different Pomerons, where the integration is over l T . In such convolution the terms that are proportional to p 1,T · k T or to p 1,T · l T vanish due to angular integrations. Only the term which is proportional to (k T · l T ) 2 survives and yelds 1 2 l 2 T k 2 T . It is easy to see that this term is the same in both vertices of Eq. (B1) and Eq. (B2). Now we need to comparewhere we denote p 1,T = p T = −p 2,T . The direct calculations gives the same expression for both terms:Therefore, both diagrams give the same contribution.
. V Khachatryan, CMS CollaborationarXiv:1510.03068nucl-exV. Khachatryan et al. [CMS Collaboration], arXiv:1510.03068 [nucl-ex];
. arXiv:1009.4122arXiv:1510.03068JHEP. 100991hep-ex. nucl-exJHEP 1009 (2010) 091 [arXiv:1009.4122 [hep-ex]]. arXiv:1510.03068 [nucl-ex].
. J Adams, STAR Collaborationnucl-ex/0501016Phys. Rev. Lett. 95152301J. Adams et al. [STAR Collaboration], Phys. Rev. Lett. 95 (2005) 152301 [nucl-ex/0501016].
. B Alver, PHOBOS CollaborationarXiv:0903.2811Phys. Rev. Lett. 10462301nucl-exB. Alver et al. [PHOBOS Collaboration], Phys. Rev. Lett. 104 (2010) 062301 [arXiv:0903.2811 [nucl-ex]].
. H Agakishiev, STAR CollaborationarXiv:1010.0690nucl-exH. Agakishiev et al. [STAR Collaboration], arXiv:1010.0690 [nucl-ex].
. S Chatrchyan, CMS CollaborationarXiv:1210.5482Phys. Lett. B. 718795nucl-exS. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 718 (2013) 795 [arXiv:1210.5482 [nucl-ex]];
. V Khachatryan, CMS CollaborationarXiv:1009.4122JHEP. 100991hep-exV. Khachatryan et al. [CMS Collaboration], JHEP 1009 (2010) 091, [arXiv:1009.4122 [hep-ex]].
. S Chatrchyan, CMS CollaborationarXiv:1312.1845JHEP. 140288nucl-exS. Chatrchyan et al. [CMS Collaboration], JHEP 1402 (2014) 088, [arXiv:1312.1845 [nucl-ex]];
Centrality dependence of dihadron correlations and azimuthal anisotropy harmonics in PbPb collisions at √ sNN = 2.76 TeV. 10.1007/JHEP02(2014)088arXiv:1310.8651arXiv:1312.1845Eur. Phys. J. C. 89488Phys. Rev. C. nucl-exPhys. Rev. C 89 (2014) no.4, 044906; [arXiv:1310.8651 [nucl-ex]]; "Centrality dependence of dihadron correlations and azimuthal anisotropy harmonics in PbPb collisions at √ sNN = 2.76 TeV," Eur. Phys. J. C 72 (2012) 2012 [arXiv:1201.3158 [nucl-ex]]; JHEP 1402 (2014) 088 doi:10.1007/JHEP02(2014)088 [arXiv:1312.1845 [nucl-ex]].
. J Adam, ALICE CollaborationarXiv:1604.07663nucl-exJ. Adam et al. [ALICE Collaboration], arXiv:1604.07663 [nucl-ex];
. arXiv:1602.01119Phys. Rev. Lett. 11613nucl-exPhys. Rev. Lett. 116 (2016) no.13, 132302, [arXiv:1602.01119 [nucl-ex]];
. L Milano, ALICE CollaborationarXiv:1407.5808Nucl. Phys. A. 931hepexL. Milano [ALICE Collaboration], Nucl. Phys. A 931 (2014) 1017, [arXiv:1407.5808 [hep- ex]];
. Y Zhou, ALICE CollaborationarXiv:1309.3237J. Phys. Conf. Ser. 50912029nucl-exY. Zhou [ALICE Collaboration], J. Phys. Conf. Ser. 509 (2014) 012029, [arXiv:1309.3237 [nucl-ex]].
. B B Abelev, ALICE CollaborationarXiv:1406.2474Phys. Rev. C. 905nucl-exB. B. Abelev et al. [ALICE Collaboration], Phys. Rev. C 90 (2014) no.5, 054901, [arXiv:1406.2474 [nucl-ex]];
. B B Abelev, ALICE Collaboration10.1016/j.physletb.2013.08.024arXiv:1307.3237Phys. Lett. B. 726164nuclexB. B. Abelev et al. [ALICE Collaboration], Phys. Lett. B 726 (2013) 164 doi:10.1016/j.physletb.2013.08.024 [arXiv:1307.3237 [nucl- ex]];
. B Abelev, ALICE CollaborationarXiv:1212.2001Phys. Lett. B. 719nucl-exB. Abelev et al. [ALICE Collaboration], Phys. Lett. B 719 (2013) 29, [arXiv:1212.2001 [nucl-ex]].
. G Aad, ATLAS CollaborationarXiv:1509.04776Phys. Rev. Lett. 116hep-exG. Aad et al. [ATLAS Collaboration], Phys. Rev. Lett. 116 (2016) 172301, [arXiv:1509.04776 [hep-ex]].
. G Aad, ATLAS CollaborationarXiv:1409.1792Phys. Rev. C. 904hep-exG. Aad et al. [ATLAS Collaboration], Phys. Rev. C 90 (2014) no.4, 044906; [arXiv:1409.1792 [hep-ex]];
. B Wosiek, ATLAS CollaborationAnnals Phys. 352117B. Wosiek [ATLAS Collaboration], Annals Phys. 352 (2015) 117;
. G Aad, ATLAS CollaborationarXiv:1303.2084Phys. Lett. B. 725hep-exG. Aad et al. [ATLAS Collaboration], Phys. Lett. B 725 (2013) 60, [arXiv:1303.2084 [hep-ex]].
. B Wosiek, ATLAS CollaborationarXiv:1203.3087Phys. Rev. C. 8614907hep-exB. Wosiek [ATLAS Collaboration], Phys. Rev. C 86 (2012) 014907, [arXiv:1203.3087 [hep-ex]].
. E M Levin, M G Ryskin, S I Troian, Sov. J. Nucl. Phys. 23222Yad. Fiz.E. M. Levin, M. G. Ryskin and S. I. Troian, Sov. J. Nucl. Phys. 23 (1976) 222 [Yad. Fiz. 23 (1976) 423];
. A Capella, A Krzywicki, E M Levin, Phys. Rev. D. 44704A. Capella, A. Krzywicki and E. M. Levin, Phys. Rev. D 44 (1991) 704.
. E Gotsman, E Levin, U Maor, 10.1103/PhysRevD.95.034005arXiv:1604.04461Phys. Rev. D. 95334005hep-phE. Gotsman, E. Levin and U. Maor, Phys. Rev. D 95, no. 3, 034005 (2017) doi:10.1103/PhysRevD.95.034005 [arXiv:1604.04461 [hep-ph]].
. A Kovner, M Lublinsky, arXiv:1012.3398Phys. Rev. D. 8334017hep-phA. Kovner and M. Lublinsky, Phys. Rev. D 83, 034017 (2011), [arXiv:1012.3398 [hep-ph]].
. Y V Kovchegov, D E Wertepny, arXiv:1212.1195Nucl. Phys. A. 906hep-phY. V. Kovchegov and D. E. Wertepny, Nucl. Phys. A 906 (2013) 50, [arXiv:1212.1195 [hep-ph]];
. T Altinoluk, N Armesto, G Beuf, A Kovner, M Lublinsky, arXiv:1509.03223Phys. Lett. B. 752hepphT. Altinoluk, N. Armesto, G. Beuf, A. Kovner and M. Lublinsky, Phys. Lett. B 752 (2016) 113, [arXiv:1509.03223 [hep- ph]];
. T Altinoluk, N Armesto, G Beuf, A Kovner, M Lublinsky, arXiv:1503.07126Phys. Lett. B. 751hep-phT. Altinoluk, N. Armesto, G. Beuf, A. Kovner and M. Lublinsky, Phys. Lett. B 751 (2015) 448, [arXiv:1503.07126 [hep-ph]].
. E Gotsman, E Levin, arXiv:1611.01653Phys. Rev. D. 95114034hep-phE. Gotsman and E. Levin, Phys. Rev. D 95 (2017) no.1, 014034 [arXiv:1611.01653 [hep-ph]].
Exploring correlations in the CGC wave function: odd azimuthal anisotropy. A Kovner, M Lublinsky, V Skokov, arXiv:1612.07790hep-phA. Kovner, M. Lublinsky and V. Skokov, "Exploring correlations in the CGC wave function: odd azimuthal anisotropy," arXiv:1612.07790 [hep-ph].
. K Dusling, R Venugopalan, arXiv:1302.7018Phys. Rev. D. 879hep-ph. and reference thereinK. Dusling and R. Venugopalan, Phys. Rev. D 87 (2013) no.9, 094034, [arXiv:1302.7018 [hep-ph]] and reference therein.
. A Kovner, M Lublinsky, arXiv:1211.1928Int. J. Mod. Phys. E. 221330001hep-ph. and references thereinA. Kovner and M. Lublinsky, Int. J. Mod. Phys. E 22, 1330001 (2013), [arXiv:1211.1928 [hep-ph]] and references therein.
. E Gotsman, E Levin, U Maor, arXiv:1607.00594Eur. Phys. J. C. 7611607hep-phE. Gotsman, E. Levin and U. Maor, Eur. Phys. J. C 76 (2016) no.11, 607, arXiv:1607.00594 [hep-ph].
. A Dumitru, D E Kharzeev, E M Levin, Y Nara, arXiv:1111.3031Phys. Rev. C. 85hep-phA. Dumitru, D. E. Kharzeev, E. M. Levin and Y. Nara, Phys. Rev. C 85 (2012) 044920, [arXiv:1111.3031 [hep-ph]].
. D Kharzeev, E Levin, M Nardi, hep-ph/0408050Nucl. Phys. A. 747D. Kharzeev, E. Levin and M. Nardi, Nucl. Phys. A 747, 609 (2005), [hep-ph/0408050].
. D Kharzeev, E Levin, M Nardi, hep- ph/0212316Nucl. Phys. A. 730329Nucl. Phys. AD. Kharzeev, E. Levin and M. Nardi, Nucl. Phys. A 730, 448 (2004) Erratum: [Nucl. Phys. A 743, 329 (2004)], [hep- ph/0212316].
. D Kharzeev, E Levin, M Nardi, hep-ph/0111315Phys. Rev. C. 7154903D. Kharzeev, E. Levin and M. Nardi, Phys. Rev. C 71, 054903 (2005), [hep-ph/0111315].
. D Kharzeev, E Levin, L Mclerran, hep-ph/0210332Phys. Lett. B. 56193D. Kharzeev, E. Levin and L. McLerran, Phys. Lett. B 561, 93 (2003), [hep-ph/0210332].
. D Kharzeev, E Levin, nucl-th/0108006Phys. Lett. B. 52379D. Kharzeev and E. Levin, Phys. Lett. B 523, 79 (2001), [nucl-th/0108006].
. D Kharzeev, M Nardi, 10.1016/S0370-2693(01)00457-9nucl-th/0012025Phys. Lett. B. 507121D. Kharzeev and M. Nardi, Phys. Lett. B 507, 121 (2001) doi:10.1016/S0370-2693(01)00457-9 [nucl-th/0012025].
. R , Hanbury Brown, R Q Twiss, Nature. 1781046R. Hanbury Brown and R. Q. Twiss, Nature 178 (1956) 1046.
. G Goldhaber, W B Fowler, S Goldhaber, T F Hoang, Phys. Rev. Lett. 3181G. Goldhaber, W. B. Fowler, S. Goldhaber and T. F. Hoang, Phys. Rev. Lett. 3, 181 (1959);
Podgoretsky. G I Kopylov, M , Sov. J. Nucl. Phys. 15Yad. Fiz.G. I. Kopylov and M. I. Pod- goretsky, Sov. J. Nucl. Phys. 15, 219 (1972) [Yad. Fiz. 15, 392 (1972)];
. G Alexander, hep-ph/0302130Rept. Prog. Phys. 66G. Alexander, Rept. Prog. Phys. 66 (2003) 481, [hep-ph/0302130].
V Yuri, Eugene Kovchegov, Levin, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Cambridge University PressQuantum Choromodynamics at High EnergiesYuri V Kovchegov and Eugene Levin, " Quantum Choromodynamics at High Energies", Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology, Cambridge University Press, 2012 .
. V A Abramovsky, V N Gribov, O V Kancheli, Sov. J. Nucl. Phys. 18595Yad. Fiz.V. A. Abramovsky, V. N. Gribov and O. V. Kancheli, Yad. Fiz. 18, 595 (1973) [Sov. J. Nucl. Phys. 18, 308 (1974)].
. L V Gribov, E M Levin, M G Ryskin, Phys. Rep. 1001L. V. Gribov, E. M. Levin and M. G. Ryskin, Phys. Rep. 100 (1983) 1.
. E M Levin, M G Ryskin, Phys. Rept. 189267E. M. Levin and M. G. Ryskin, Phys. Rept. 189, 267 (1990).
. A H Mueller, Phys. Rev. 22963A. H. Mueller, Phys. Rev. D2 (1970) 2963.
. E A Kuraev, L N Lipatov, F S Fadin, Sov. Phys. JETP. 45199E. A. Kuraev, L. N. Lipatov, and F. S. Fadin, Sov. Phys. JETP 45, 199 (1977);
. Ya, Ya, L N Balitsky, Lipatov, Sov. J. Nucl. Phys. 2822Ya. Ya. Balitsky and L. N. Lipatov, Sov. J. Nucl. Phys. 28, 22 (1978).
. L N Lipatov, Phys. Rep. 286131L. N. Lipatov, Phys. Rep. 286 (1997) 131;
and references therein. 63904Sov. Phys. JETP 63 (1986) 904 and references therein.
. E M Levin, M G Ryskin, hep-ph/9710546Yad. Fiz. 25849E. M. Levin and M. G. Ryskin, Yad. Fiz. 25, 849 (1977); hep-ph/9710546.
. J Bartels, M G Ryskin, hep-ph/9612226Z. Phys. C. 76241J. Bartels and M. G. Ryskin, Z. Phys. C 76, 241 (1997), [hep-ph/9612226].
. Y V Kovchegov, arXiv:hep-ph/0107256Phys. Rev. D. 64114016Erratum-ibid. D 68, 039901 (2003)Y. V. Kovchegov, Phys. Rev. D 64, 114016 (2001) [Erratum-ibid. D 68, 039901 (2003)] [arXiv:hep-ph/0107256].
. Y V Kovchegov, K Tuchin, arXiv:hep-ph/0111362Phys. Rev. D. 6574026Y. V. Kovchegov and K. Tuchin, Phys. Rev. D 65, 074026 (2002) [arXiv:hep-ph/0111362].
. J Jalilian-Marian, Y V Kovchegov, arXiv:hep-ph/0405266Phys. Rev. D. 70114017Erratum-ibid. D 71, 079901 (2005)J. Jalilian-Marian and Y. V. Kovchegov, Phys. Rev. D 70, 114017 (2004) [Erratum-ibid. D 71, 079901 (2005)] [arXiv:hep- ph/0405266].
. M A Braun, arXiv:hep-ph/0603060Eur. Phys. J. C. 48501M. A. Braun, Eur. Phys. J. C 48, 501 (2006) [arXiv:hep-ph/0603060].
. C Marquet, arXiv:hep-ph/0409023Nucl. Phys. B. 705319C. Marquet, Nucl. Phys. B 705, 319 (2005) [arXiv:hep-ph/0409023].
. A Kovner, M Lublinsky, arXiv:hep-ph/0609227JHEP. 061183A. Kovner and M. Lublinsky, JHEP 0611, 083 (2006) [arXiv:hep-ph/0609227].
. E Levin, A Prygarin, arXiv:0804.4747Phys. Rev. C. 7865202hep-phE. Levin and A. Prygarin, Phys. Rev. C 78, 065202 (2008), [arXiv:0804.4747 [hep-ph]].
. J Jalilian-Marian, Y V Kovchegov, arXiv:hep-ph/0405266Phys. Rev. D. 7079901Erratum-ibid. DJ. Jalilian-Marian and Y. V. Kovchegov, Phys. Rev. D 70, 114017 (2004) [Erratum-ibid. D 71, 079901 (2005), [arXiv:hep- ph/0405266].
. Y L Dokshitzer, D Diakonov, S I Troian, Phys. Rept. 58269Y. L. Dokshitzer, D. Diakonov and S. I. Troian, Phys. Rept. 58 (1980) 269.
. V A Khoze, A D Martin, M G Ryskin, 10.1140/epjc/s10052-006-0025-5[hep-ph/0605113hep-ph/9705258Eur. Phys. J. C. 485867Phys. Rev. DV. A. Khoze, A. D. Martin and M. G. Ryskin, Eur. Phys. J. C 48 (2006) 467 doi:10.1140/epjc/s10052-006-0025-5 [hep- ph/0605113], Eur. Phys. J. C 14 (2000) 525 doi:10.1007/s100520000359 [hep-ph/0002072]. Phys. Rev. D 56 (1997) 5867 [hep-ph/9705258];
. hep-ph/9701419Phys. Lett. B. 401330Phys. Lett. B 401, 330 (1997) [hep-ph/9701419].
. Y V Kovchegov, K Tuchin, 10.1103/PhysRevD.65.074026hep-ph/0111362Phys. Rev. D. 6574026Y. V. Kovchegov and K. Tuchin, Phys. Rev. D 65 (2002) 074026 doi:10.1103/PhysRevD.65.074026 [hep-ph/0111362].
. A H Mueller, D N Triantafyllopoulos, hep-ph/0205167Nucl. Phys. B. 640331A. H. Mueller and D. N. Triantafyllopoulos, Nucl. Phys. B 640, 331 (2002); [hep-ph/0205167].
. D N Triantafyllopoulos, hep-ph/0209121Nucl. Phys. B. 648293D. N. Triantafyllopoulos, Nucl. Phys. B 648, 293 (2003) [hep-ph/0209121].
. H Abramowicz, ZEUS CollaborationarXiv:1603.09628Phys. Rev. D. 93992002hep-exH. Abramowicz et al. [ZEUS Collaboration], Phys. Rev. D 93 (2016) no.9, 092002, [arXiv:1603.09628 [hep-ex]];
. H Abramowicz, H1 and ZEUS CollaborationsarXiv:1506.06042Eur. Phys. J. C. 7512hep-exH. Abramowicz et al. [H1 and ZEUS Collaborations], Eur. Phys. J. C 75 (2015) no.12, 580, [arXiv:1506.06042 [hep-ex]].
. V N Gribov, L N Lipatov, Sov. J. Nucl. Phys. 15438V.N. Gribov and L.N. Lipatov: Sov. J. Nucl. Phys. 15 (1972) 438;
. L N Lipatov, Yad. Fiz. 20181L.N. Lipatov: Yad. Fiz. 20 (1974) 181;
. G Altarelli, G Parisi, Nucl. Phys. 126298G. Altarelli and G. Parisi: Nucl. Phys. B126 (1977) 298;
. Yu L Dokshitser, Sov. Phys. JETP. 46641Yu.L. Dokshitser:Sov. Phys. JETP 46 (1977) 641.
. J L Albacete, N Armesto, J G Milhano, P Quiroga-Arias, C A Salgado, arXiv:1012.4408Eur. Phys. J. C. 711705hep-phJ. L. Albacete, N. Armesto, J. G. Milhano, P. Quiroga-Arias and C. A. Salgado, Eur. Phys. J. C 71 (2011) 1705, [arXiv:1012.4408 [hep-ph]].
. E Iancu, J D Madrigal, A H Mueller, G Soyez, D N Triantafyllopoulos, arXiv:1507.03651Phys. Lett. B. 750hep-phE. Iancu, J. D. Madrigal, A. H. Mueller, G. Soyez and D. N. Triantafyllopoulos, Phys. Lett. B 750 (2015) 643, [arXiv:1507.03651 [hep-ph]].
. G Aad, ATLAS CollaborationarXiv:1509.04776Phys. Rev. Lett. 11617172301hep-exG. Aad et al. [ATLAS Collaboration], Phys. Rev. Lett. 116, no. 17, 172301 (2016), [arXiv:1509.04776 [hep-ex]].
. R J Glauber, Lectures in Theor. Phys. v. 1, ed. W.E. Brittin and L.G. Duham. NY: IntersciencesR.J. Glauber, In: Lectures in Theor. Phys., v. 1, ed. W.E. Brittin and L.G. Duham. NY: Intersciences, 1959
. J Bartels, J.Phys. 191611J. Bartels, J.Phys. G19 (1993) 1611.
. V N Gribov, Zh. Eksp. Teor. Fiz. 26414Sov. Phys. JETPV. N. Gribov, Sov. Phys. JETP 26 (1967) 414 [Zh. Eksp. Teor. Fiz. 53 (1967) 654].
. J Bartels, E Levin, ; E Iancu, K Itakura, L Mclerran, arXiv:hep-ph/0203137Nucl. Phys. 387327Nucl. Phys.J. Bartels and E. Levin, Nucl. Phys. B387 (1992) 617, E. Iancu, K. Itakura and L. McLerran, Nucl. Phys. A708 (2002) 327 [arXiv:hep-ph/0203137].
. M Froissart, Phys. Rev. 1231053M. Froissart, Phys. Rev. 123 (1961) 1053;
Scattering Theory: Unitarity, Analitysity and Crossing. A Martin, Lecture Notes in Physics. Springer-VerlagA. Martin, "Scattering Theory: Unitarity, Analitysity and Crossing." Lecture Notes in Physics, Springer-Verlag, Berlin- Heidelberg-New-York, 1969.
| [] |
[
"SHARP -III: FIRST USE OF ADAPTIVE OPTICS IMAGING TO CONSTRAIN COSMOLOGY WITH GRAVITATIONAL LENS TIME DELAYS",
"SHARP -III: FIRST USE OF ADAPTIVE OPTICS IMAGING TO CONSTRAIN COSMOLOGY WITH GRAVITATIONAL LENS TIME DELAYS"
] | [
"Geoff C ",
"-F Chen \nInstitute of Astronomy and Astrophysics\nAcademia Sinica\nP.O. Box 23-14110617TaipeiTaiwan\n\nDepartment of Physics\nNational Taiwan University\n10617TaipeiTaiwan\n\nDepartment of Physics\nUniversity of California\n95616DavisCAUSA\n",
"Sherry H Suyu \nInstitute of Astronomy and Astrophysics\nAcademia Sinica\nP.O. Box 23-14110617TaipeiTaiwan\n",
"Kenneth C Wong \nInstitute of Astronomy and Astrophysics\nAcademia Sinica\nP.O. Box 23-14110617TaipeiTaiwan\n\nNational Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan\n",
"Christopher D Fassnacht \nDepartment of Physics\nUniversity of California\n95616DavisCAUSA\n",
"Tzihong Chiueh \nDepartment of Physics\nNational Taiwan University\n10617TaipeiTaiwan\n\nInstitute of Astrophysics\nNational Taiwan University\n10617TaipeiTaiwan\n\nCenter for Theoretical Sciences\nNational Taiwan University\n10617TaipeiTaiwan\n",
"Aleksi Halkola ",
"Shing Hu \nDepartment of Mathematics\nNational Taiwan University\n10617TaipeiTaiwan\n",
"Matthew W Auger \nInstitute of Astronomy\nUniversity of Cambridge\nMadingley RdCB3 0HACambridgeUK\n",
"Leon V E Koopmans \nKapteyn Astronomical Institute\nUniversity of Groningen\nP.O.Box 8009700 AVGroningenThe Netherlands\n",
"David J Lagattuta \nCRAL\nObservatoire de Lyon\n\nUniversit Lyon 1\n9 Avenue Ch. AndrF-69561Saint Genis Laval CedexFrance\n",
"John P Mckean \nKapteyn Astronomical Institute\nUniversity of Groningen\nP.O.Box 8009700 AVGroningenThe Netherlands\n\nNetherlands Institute for Radio Astronomy (ASTRON)\nP.O. Box 27990 AADwingelooThe Netherlands\n",
"Simona Vegetti \nMax Planck Institute for Astrophysics\nKarl-Schwarzschild-Strasse 1D-85740GarchingGermany\n"
] | [
"Institute of Astronomy and Astrophysics\nAcademia Sinica\nP.O. Box 23-14110617TaipeiTaiwan",
"Department of Physics\nNational Taiwan University\n10617TaipeiTaiwan",
"Department of Physics\nUniversity of California\n95616DavisCAUSA",
"Institute of Astronomy and Astrophysics\nAcademia Sinica\nP.O. Box 23-14110617TaipeiTaiwan",
"Institute of Astronomy and Astrophysics\nAcademia Sinica\nP.O. Box 23-14110617TaipeiTaiwan",
"National Astronomical Observatory of Japan\n2-21-1 Osawa181-8588MitakaTokyoJapan",
"Department of Physics\nUniversity of California\n95616DavisCAUSA",
"Department of Physics\nNational Taiwan University\n10617TaipeiTaiwan",
"Institute of Astrophysics\nNational Taiwan University\n10617TaipeiTaiwan",
"Center for Theoretical Sciences\nNational Taiwan University\n10617TaipeiTaiwan",
"Department of Mathematics\nNational Taiwan University\n10617TaipeiTaiwan",
"Institute of Astronomy\nUniversity of Cambridge\nMadingley RdCB3 0HACambridgeUK",
"Kapteyn Astronomical Institute\nUniversity of Groningen\nP.O.Box 8009700 AVGroningenThe Netherlands",
"CRAL\nObservatoire de Lyon",
"Universit Lyon 1\n9 Avenue Ch. AndrF-69561Saint Genis Laval CedexFrance",
"Kapteyn Astronomical Institute\nUniversity of Groningen\nP.O.Box 8009700 AVGroningenThe Netherlands",
"Netherlands Institute for Radio Astronomy (ASTRON)\nP.O. Box 27990 AADwingelooThe Netherlands",
"Max Planck Institute for Astrophysics\nKarl-Schwarzschild-Strasse 1D-85740GarchingGermany"
] | [
"MNRAS"
] | Accurate and precise measurements of the Hubble constant are critical for testing our current standard cosmological model and revealing possibly new physics. With Hubble Space Telescope (HST ) imaging, each strong gravitational lens system with measured time delays can allow one to determine the Hubble constant with an uncertainty of ∼7%. Since HST will not last forever, we explore adaptive-optics (AO) imaging as an alternative that can provide higher angular resolution than HST imaging but has a less stable point spread function (PSF) due to atmospheric distortion. To make AO imaging useful for time-delay-lens cosmography, we develop a method to extract the unknown PSF directly from the imaging of strongly lensed quasars. In a blind test with two mock data sets created with different PSFs, we are able to recover the important cosmological parameters (time-delay distance, external shear, lens mass profile slope, and total Einstein radius). Our analysis of the Keck AO image of the strong lens system RXJ 1131−1231 shows that the important parameters for cosmography agree with those based on HST imaging and modeling within 1-σ uncertainties. Most importantly, the constraint on the model time-delay distance by using AO imaging with 0.045 ′′ resolution is tighter by ∼50% than the constraint of time-delay distance by using HST imaging with 0.09 ′′ when a power-law mass distribution for the lens system is adopted. Our PSF reconstruction technique is generic and applicable to data sets that have multiple nearby point sources, enabling scientific studies that require high-precision models of the PSF. | 10.1093/mnras/stw991 | [
"https://arxiv.org/pdf/1601.01321v2.pdf"
] | 119,259,632 | 1601.01321 | 91cf4dab2129369786ad8d991517d6e452378546 |
SHARP -III: FIRST USE OF ADAPTIVE OPTICS IMAGING TO CONSTRAIN COSMOLOGY WITH GRAVITATIONAL LENS TIME DELAYS
Aug 2016. 2015
Geoff C
-F Chen
Institute of Astronomy and Astrophysics
Academia Sinica
P.O. Box 23-14110617TaipeiTaiwan
Department of Physics
National Taiwan University
10617TaipeiTaiwan
Department of Physics
University of California
95616DavisCAUSA
Sherry H Suyu
Institute of Astronomy and Astrophysics
Academia Sinica
P.O. Box 23-14110617TaipeiTaiwan
Kenneth C Wong
Institute of Astronomy and Astrophysics
Academia Sinica
P.O. Box 23-14110617TaipeiTaiwan
National Astronomical Observatory of Japan
2-21-1 Osawa181-8588MitakaTokyoJapan
Christopher D Fassnacht
Department of Physics
University of California
95616DavisCAUSA
Tzihong Chiueh
Department of Physics
National Taiwan University
10617TaipeiTaiwan
Institute of Astrophysics
National Taiwan University
10617TaipeiTaiwan
Center for Theoretical Sciences
National Taiwan University
10617TaipeiTaiwan
Aleksi Halkola
Shing Hu
Department of Mathematics
National Taiwan University
10617TaipeiTaiwan
Matthew W Auger
Institute of Astronomy
University of Cambridge
Madingley RdCB3 0HACambridgeUK
Leon V E Koopmans
Kapteyn Astronomical Institute
University of Groningen
P.O.Box 8009700 AVGroningenThe Netherlands
David J Lagattuta
CRAL
Observatoire de Lyon
Universit Lyon 1
9 Avenue Ch. AndrF-69561Saint Genis Laval CedexFrance
John P Mckean
Kapteyn Astronomical Institute
University of Groningen
P.O.Box 8009700 AVGroningenThe Netherlands
Netherlands Institute for Radio Astronomy (ASTRON)
P.O. Box 27990 AADwingelooThe Netherlands
Simona Vegetti
Max Planck Institute for Astrophysics
Karl-Schwarzschild-Strasse 1D-85740GarchingGermany
SHARP -III: FIRST USE OF ADAPTIVE OPTICS IMAGING TO CONSTRAIN COSMOLOGY WITH GRAVITATIONAL LENS TIME DELAYS
MNRAS
000Aug 2016. 2015Accepted -. Received -; in original form 16 August 2016Preprint 16 August 2016 Compiled using MNRAS L A T E X style file v3.0gravitational lensing:strong -cosmolgy:distance scale -methods:data analysis -adaptive optics ⋆
Accurate and precise measurements of the Hubble constant are critical for testing our current standard cosmological model and revealing possibly new physics. With Hubble Space Telescope (HST ) imaging, each strong gravitational lens system with measured time delays can allow one to determine the Hubble constant with an uncertainty of ∼7%. Since HST will not last forever, we explore adaptive-optics (AO) imaging as an alternative that can provide higher angular resolution than HST imaging but has a less stable point spread function (PSF) due to atmospheric distortion. To make AO imaging useful for time-delay-lens cosmography, we develop a method to extract the unknown PSF directly from the imaging of strongly lensed quasars. In a blind test with two mock data sets created with different PSFs, we are able to recover the important cosmological parameters (time-delay distance, external shear, lens mass profile slope, and total Einstein radius). Our analysis of the Keck AO image of the strong lens system RXJ 1131−1231 shows that the important parameters for cosmography agree with those based on HST imaging and modeling within 1-σ uncertainties. Most importantly, the constraint on the model time-delay distance by using AO imaging with 0.045 ′′ resolution is tighter by ∼50% than the constraint of time-delay distance by using HST imaging with 0.09 ′′ when a power-law mass distribution for the lens system is adopted. Our PSF reconstruction technique is generic and applicable to data sets that have multiple nearby point sources, enabling scientific studies that require high-precision models of the PSF.
vations of the Cosmic Microwave Background (CMB; e.g., Hinshaw et al. 2012;Planck Collaboration et al. 2015) have established a standard cosmological paradigm where our Universe is spatially flat and is dominated by cold dark matter (CDM) and dark energy: the so-called flat ΛCDM model, where Λ represents a constant dark energy density. While the CMB provides strong constraints on the parameters of this model, a relaxation of the assumptions in this model, such as spatial flatness or constant dark energy density, leads to a strong degeneracy between the cosmological parameters, particularly those with the Hubble constant H0. Therefore, independent and accurate measurements of H0 provide one of the most useful complements to the observations of the CMB in constraining the spatial curvature of the Universe, dark energy equation of state, and the number of neutrino species (e.g., Hu 2005;Riess et al. 2009Riess et al. , 2011Freedman et al. 2012;?). The recent inferred value of Hubble constant H0 = 67.8 ± 0.9 km s −1 Mpc −1 , based on the Planck satellite data of the CMB and the assumption of the flat ΛCDM model, is low in comparison to several direct measurements including those from the Cepheids distance ladder with H0 = 74.3 ± 1.5(stat.) ± 2.1(sys.) km s −1 Mpc −1 (Freedman et al. 2012) and H0 = 73.8 ± 2.4 km s −1 Mpc −1 (Riess et al. 2011). If this indication of tension is not ruled out by systematic effects, then this could indicate new physics beyond the standard flat ΛCDM model. Therefore, pinning down the Hubble constant with independent methods is a key approach to better understand our Universe.
Strong gravitational lensing with time delays provides a one-step measurement of a cosmological distance in the Universe. The background source is composed of a centrally varying source, such as an active galactic nucleus (AGN), and its host galaxy. The time delays between the multiple images of the source, induced by the foreground lens, are given by ∆t = 1 c D∆t∆τ . Here, ∆τ is dependent on the geometry and the gravitational potential of the lens system; ∆τ can be tightly constrained by the spatially extended images (we usually call them "arcs") of the lensed background galaxy (e.g., Kochanek et al. 2001;Suyu et al. 2009), together with stellar kinematics of the foreground lens galaxy (e.g., Treu & Koopmans 2002;Koopmans et al. 2003;Suyu et al. , 2014 and studies of the lens environment combined with ray-tracing through numerical simulations (e.g., Hilbert et al. 2007Hilbert et al. , 2009Fassnacht et al. 2011;Greene et al. 2013;Collett et al. 2013). The stellar kinematics and lens environment studies are important for overcoming the mass-sheet degeneracy and source-position transformations in lensing (Falco et al. 1985;Schneider & Sluse 2013Xu et al. 2015). Therefore, by measuring the time delays between the multiple images and modeling the lens and line-of-sight mass distributions, we can constrain D∆t, which is the so-called timedelay distance that encompasses cosmological dependences and is particularly sensitive to the Hubble constant (e.g., . The time delays in combination with the stellar velocity dispersion measurements of the lens galaxy further allow us to infer the angular diameter distance to the lens galaxy (Paraficz & Hjorth 2009;Jee et al. 2014).
? have shown that for each lens system we can measure H0 to ∼7% precision. Hubble Space Telescope (HST ) imaging is imperative for this analysis because it not only provides high angular resolution but also a stable point spread function (PSF) for the lens mass modeling. However, HST 's lifetime is finite 1 , and the angular resolution is also limited by its aperture size. Given the dozens of time-delay lenses from COSMOGRAIL 2 (e.g. Vuissoz et al. 2007Vuissoz et al. , 2008Courbin et al. 2011;Tewes et al. 2013b,a;Rathna Kumar et al. 2013;Eulaers et al. 2013), and hundreds of new lenses to be discovered in the near future (e.g., Oguri & Marshall 2010;Agnello et al. 2015;Chan et al. 2015;Marshall et al. 2015;More et al. 2015), finding an alternative long-term solution for this promising method is timely.
One alternative approach is imaging from the ground via adaptive optics (AO), which is a technology used to improve the performance of optical systems by reducing the effect of wavefront distortions (e.g., Rousset et al. 1990;Beckers 1993;Watson 1997;Brase 1998). In other words, it aims at correcting the deformations of an incoming wavefront by deforming a mirror and thus compensating for the distortion. The advantages of using AO imaging are (1) the angular resolution obtained with telescopes that are larger than HST can be higher than that of HST since a perfect AO system would lead to a diffraction limited PSF, (2) ground-based telescopes are more accessible. The disadvantage is that we do not have a stable PSF model a priori, since the atmospheric distortion varies both temporally and spatially across the image. Lens targets typically do not have a nearby bright star within ∼ 10 arcseconds, and stars at further angular distance from the target may be insufficient in providing an accurate PSF model given the spatial variation of the PSF across the field.
In HST imaging, we can use the lensing arcs to constrain the lens mass model by using the stable PSF of HST to separate the arc from the bright AGN, but we cannot do so in AO imaging. The contamination of the AGN light on the lensing arcs in AO imaging makes it difficult to constrain the lens model, and consequently H0. One therefore needs to obtain a good PSF model for the AO data, and there are recent studies that aim to do so directly from the AO imaging. Lagattuta et al. (2010) use three Gaussian components as the PSF model to subtract the AGN light which is sufficient to study the lensing galaxy and its substructures. However, the analytical model is not sufficient to describe the complexity of the PSF (see Figure 1 of Lagattuta et al. 2010) which could potentially impact the cosmographic measurements. Rusu et al. (2015) use either an analytic or a hybrid PSF to study the host galaxies of the lensed AGNs (see also Rusu et al. 2014). The hybrid PSF is built from elliptical Moffat profiles (Moffat 1969) with central parts iteratively tuned to match a single AGN image. While this hybrid PSF is useful for extracting properties of the AGN host galaxy, the central parts of the PSF model could manifest the noise pattern in the image (see Figure B.7 of Rusu et al. 2015) which also could potentially impact cosmographic measurements. Agnello et al. (2015) use an iterative method to reconstruct the PSF directly from lens imaging by averaging the doubly lensed AGN. This method is valid only when the lensed AGN are separated far enough from each other. For 1 And no equivalent optical space-based telescope might be forthcoming soon. 2 COSmological MOnitoring of GRAvItational Lenses typical quad (four-image) lens systems, the lensed AGNs are often close in separation (within 2 ′′ ), leading to overlaps in the wings of the AGN images that are smeared by the PSF. Our goal is to provide a general method to overcome the unknown PSF model problem by extracting the PSF directly from strong lensing imaging and simultaneously modeling the lens mass distribution. We test our method on simulated AO images, and apply the method to the known gravitational lens RXJ 1131−1231 with Keck AO imaging, a part of data from SHARP 3 , which is a project that focuses on studying known quadruple-image and Einstein ring lenses using high-resolution AO imaging, in order to probe their mass distributions in unprecedented detail (e.g., Lagattuta et al. 2010Lagattuta et al. , 2012Vegetti et al. 2012, Hsueh et al. submitted). The gravitational lens system RXJ 1131−1231 was discovered by Sluse et al. (2003) who also measured the lens and source redshifts to be 0.295 and 0.658, respectively. The HST observations of the system RXJ 1131−1231 have been modeled by ? Suyu et al. (2014) for cosmography and more recently by Birrer et al. (2015).
The outline of the paper is as follows. In Section 2, we describe the observation of RXJ 1131−1231 with the adaptive optics imaging system at the Keck Observatory. We briefly recap in Section 3 the basics of cosmography with time-delay lenses. In Section 4, we describe our new procedure to analyze AO images without information on the PSF in advance. In Section 5, we use simulated data to test and verify the method. In Section 6, we demonstrate the results from real data and provide a comparison between the results from HST imaging and AO imaging. Finally, we summarize in Section 7.
OBSERVATION
The RXJ 1131−1231 system was observed on the nights of UT 2012 May 16 and May 18 with the Near Infrared Camera 2 (NIRC2) on the Keck-2 Telescope (e.g., Wizinowich et al. 2003). This image was a part of SHARP data. The adaptive optics corrections were achieved through the use of a R = 15.8 tip-tilt star located 54.5 arcseconds from the lens system and a laser guide star. The system was observed in the "Wide Camera" mode, which provides a roughly 40 ′′ × 40 ′′ field of view and a pixel scale of 0.0397 arcseconds. This pixel scale slightly undersamples the point spread function (PSF), but the angular extent of the lens system and the distance from the tip-tilt star made the use of the Wide Camera the preferable approach.
The observations consisted of 61 exposures, each consisting of 6 coadded 10 s exposures, for a total on-source integration time of 3660 s. The data were reduced by a python-based pipeline that has steps that do the flat-field correction, subtract the sky, correct for the optical distortions in the raw images, and combine the calibrated data frames (for details, see Auger et al. 2011). The final image has a pixel-scale of 0.04 arcseconds and is shown in Figure 1. In this section we briefly explain the relation between gravitational time delays and cosmology. When a light ray passes near a massive object, it experiences a deflection in its trajectory and acquires a time delay by the gravitational field with respect to the travel time without the massive object. Therefore, the time delay has two contributions: (1) the geometric delay, ∆tgeom, which is caused by the bent trajectory being longer than the unbent one, and (2) the gravitational delay, ∆tgrav, which is due to the fact that the space and time are affected around the gravitational field, so after integrating the gravitational potential along the path, a far away observer receives the light later by a Shapiro delay (Shapiro 1964;Refsdal 1964).
The combination of the two delays is
∆t = D∆t c 1 2 (θ − β) 2 − ψ(θ) ,(1)
where θ, β, and ψ(θ) are the image coordinates, the source coordinates, and the lens potential respectively. The timedelay distance is defined as
D∆t ≡ (1 + z d ) D d Ds D ds ∝ H −1 0 ,(2)
where D d , Ds and D ds are the angular diameter distances to the lens, to the source, and between the lens and the source, respectively. Thus, we can measure D∆t via gravitational lensing with time delays. Notice that the gradient of the term in the square brackets in Equation (1) vanishes at the positions of the lensed images and yields the lens equation
β = θ − ∇ψ(θ),(3)
which governs the deflection of light rays. We refer the reader to, e.g., Schneider et al. (2006), Bartelmann (2010), Treu (2010), , Treu & Ellis (2014) for more details.
Probability Theory
A meaningful measurement should have an uncertainty as a reference and it is also the key to confirm or rule out possible models. Thus, we need to analyze our data based on a probability theory that can present this idea. Bayes' theorem provides the conditional probability distribution, so we can obtain the posterior probability distribution of the model parameters given the data from Bayes' rule. For example, if we are interested in the posterior of the parameters π of the hypothesis model H given the data d, it can be expressed as
posterior P (π|d, H) = likelihood P (d|π, H) prior P (π|H) P (d|H) evidence (marginalized likelihood) ,(4)
where the Bayesian evidence can be used to rank the model and our prior based on the data (e.g., MacKay 1992; Hobson et al. 2002;Marshall et al. 2002) In addition, if we are interested in the posterior of a specific parameter, πN , the posterior distribution can be obtained by marginalizing over other parameters P (πN |d, H) = P (π|d, H)
N−1 i=1 dπi.(5)
Markov chain Monte Carlo
Obtaining the probability distribution function of the parameters in a model can be non-trivial, especially when the number of parameters is high. It is computationally unfeasible to explore a high-dimensional parameter space on a regular grid since the number of the grid points for the task exponentially increases with the number of dimensions. Due to the fact that the parameter space is typically large in strong lensing analyses, one can bypass the use of grids by obtaining samples in the multi-dimensional parameter space that represent the probability distribution (i.e., the number density of the samples is proportional to the probability density). A Markov Chain Monte Carlo (MCMC) provides an efficient way to draw samples from the posterior probability density function (PDF) of the lens parameters, because of the approximately linear relation between the computational time and the dimension of the parameter space. We use MCMC sampling that is implemented in Glee, a strong lens modeling software developed by S. H. Suyu and A. Halkola (Suyu & Halkola 2010;Suyu et al. 2012b). It is based on Bayes' theorem and follows Dunkley et al. (2005) to achieve efficient sampling and to test convergence. The pragmatic procedure for convergence is described in Suyu & Halkola (2010). We use Bayesian language in the following sections.
METHOD: PSF RECONSTRUCTION AND LENS MODELING
In this section, we describe a novel procedure to analyze the AO imaging without a PSF model a priori. Readers who are not planning to use this method may wish to proceed directly to Section 5 on the scientific results enabled by the method.
The assumption of this strategy is that the PSF does not change significantly within several arcseconds, which is typically valid in AO imaging (van Dam et al. 2006;Wizinowich et al. 2006). We show an overall flow chart in Figure 2 to illustrate how to obtain iteratively the PSF, background source intensity, the lens mass and light model.
In Section 4.1, we decompose the observed light from the lens system into three components (lens galaxy, lensed arcs of the background source galaxy, and the lensed background AGN) and introduce the notation that we will use in the subsequent discussion. In Section 4.2, we obtain the preliminary global structure of AGN light model, while separating the lens light and arc light. In Section 4.3, we obtain the fine structure of the AGN light and incorporate it into the preliminary AGN light model. This is accomplished by correcting the PSF model. In Section 4.4, we update the PSF and use it to model the lens mass and source intensity distributions. Since the lens galaxy light is quite smooth and less sensitive to the PSF model, we use the PSF built from the AGN light for the lens galaxy light model. The PSF updating and lens mass modeling are repeated until the corrections to the PSF become insignificant. (See the criteria in Section 4.3.3 and Section 4.4.3.)
Light components of the lens system
As shown in Figure 3, our model for the observed light in the lens system on the image plane has three contributions: the lens galaxy light, the arc light (the lensed background source, i.e., the host galaxy of the AGN), and the light of the multiple AGNs on the image plane. We define
d = d P + n,(6)
where d is the vector of observed data (image pixel values),
d P = lens light Kg + arc light KLs + AGN light Mw ,(7)
and n is the noise in the data characterized by the covariance matrix CD (we use subscript D to indicate "data"). The blurring matrix K accounts for the PSF convolution, the vector g is the image pixel values of the lens galaxy light, the matrix L maps source intensity to the image plane, the vector s describes the source surface brightness on a grid of pixels, the matrix M is composed using the positions and the intensities of the AGNs, and w is the vector of pixel values of the PSF grid. We refer to Treu & Koopmans (2004) for constructing K and L, and illustrate the effect of M in Figure 3. The flow-chart describes the overall procedures in Section 4. We use the procedures to reconstruct the PSF directly from lens image and do the lens modeling. In step 1, we use a nearby star (or one of the lensed AGN itself) as the initial PSF; in step 2, we sequentially obtain the lens light, arc light, AGNs light, and the positions and relative amplitudes of AGNs; steps 3 to 5 form an inner loop to add the correction (fine structures) into the PSF and accumulate the correction uncertainties; in step 6, we enter the outer loop which updates the image covariance matrix, PSF of all light model, and then repeat the full procedure until the image χ 2 no longer decreases.
At first, since we do not know the AO PSF a priori, K and w are just the initial blurring matrix and PSF grid values, respectively. As we iteratively model the light components and correct the PSF, we update w (and subsequently K).
Determining the light components
The goal in this section is to obtain the preliminary model of each of the three light components. In step 1 of Figure 2, we input the observed image into the lens modeling software Glee with a nearby star as our initial PSF model. If there is no nearby star, any star in the field can be used as the initial PSF or we can use one of the AGN images. A different initial PSF does not affect the final results, although we note that a good initial PSF would be helpful as they would require fewer iterations of PSF corrections. In step 2, we decompose the predicted total light sequentially into lens light, arc light, and AGN light. We detail this process in Section 4.2.1 to Section 4.2.3 below.
Lens Light Model (Step 2)
For modeling the light distribution of the lens galaxy, we use parametrized profiles, such as the elliptical Sérsic profile, where Is is the amplitude, k is a constant such that R eff is the effective radius, qL is the minor-to-major axis ratio, and n sérsic is the Sérsic index (Sérsic 1968).
IS(θ1, θ2) = Is exp −k θ 2 1 + θ 2 2 /q 2 L R eff 1/n sérsic − 1 ,(8)
In order to get a preliminary model of the lens light, we mask out the arc light and AGN light region; that is, we boost the uncertainty of the region where the arc light and the AGN light are apparently dominant. Thus, in the fitting region, equation (7) becomes effectively
d P = Kg.(9)
By Bayes' rule, we have
P (η|d) ∝ P (d|η)P (η),(10)
where η represents the parameters of lens light (such as Is, qL, n sérsic , R eff ). We assume uniform prior on the lens light parameters, so we want to obtain
P (d|η) = exp[−ED,mArcAGN(d|η)] ZD,mArcAGN ,(11)
where, ED,mArcAGN(d|η)
= 1 2 (d − Kg) T C −1 D,mArcAGN (d − Kg) = 1 2 χ 2 mArcAGN ,(12)
and
ZD,mArcAGN = (2π) N d /2 (det CD,mArcAGN) 1/2(13)
is the normalization for the probability. The covariance matrix, CD,mArcAGN, is the original covariance matrix with entries corresponding to the arc and AGN mask region boosted (see Appendix A), and N d is the number of image pixels. We denoteη as the maximum likelihood parameters (which maximizes equation 11).
Since the initial PSF is a prototype, usually there are some significant residuals in the lens light center when maximizing the posterior of lens light parameters. However, this does not affect the subsequent lens light prediction in the arc region, because the residuals are far from the arc regions. To recap, we can obtainη by masking out the arc light and AGN light regions.
Arc Light Model (Step 2)
For modeling the arc light, we describe the source intensity on a grid of pixels on the source plane and map the source intensity values onto the image plane using a lens mass model (via the operation KLs in equation (7)). We use elliptically symmetric power-law distributions to model the dimensionless surface mass density of lens galaxies,
κ pl (θ1, θ2) = 3 − γ ′ 1 + q θE θ 2 1 + θ 2 2 /q 2 γ ′ −1 ,(14)
where γ ′ is the radial power-law slope (γ ′ = 2 corresponding to isothermal), θE is the Einstein radius, and q is the axis ratio of the elliptical isodensity contour. In addition to the lens galaxies, we include a constant external shear with the following lens potential in polar coordinates θ and ϕ:
ψext(θ, ϕ) = 1 2 γextθ 2 cos 2(ϕ − φext),(15)
where γext is shear strength and φext is the shear angle. The shear position angle of φext = 0 • corresponds to a shearing along θ1 whereas φext = 90 • corresponds to shearing along θ2. 4 We model the arc light with the lens light fixed. Since the AGN light dominates near the AGN image positions, we mask out the region where the arc is hard to be seen; that is, we want to minimize the contribution to the source intensity reconstruction from the AGN light. Since the regions of the AGN are masked out, we temporarily 5 drop the AGN component, Mw, in Equation (7), which givenη becomes
d P = Kĝ + KLs,(16)
whereĝ = g(η). The posterior based on the arc light is
P (ζ|d, ∆t,η) ∝ P (d, ∆t|η, ζ)P (ζ),(17)
where ζ are the parameters of the lens mass distributions (such as γ ′ , θE, γext). The likelihood of the data can be expressed as
P (d, ∆t|η, ζ) = ds P(d, ∆t|η, ζ, s)P(s),(18)
where
P (d, ∆t|η, ζ, s) = exp[−ED,mAGN(d|η, ζ, s)] ZD,mAGN · N AGN i=1 1 √ 2πσAGN,i exp − |θAGN,i − θ p AGN,i | 2 2σ 2 AGN,i · i=1 1 √ 2πσ∆t,i exp − (∆ti − ∆t p i ) 2 2σ 2 ∆t,i ,(19)
ED,mAGN(d|η, ζ, s) 4 Our (right-handed) coordinate system (θ 1 , θ 2 ) has θ 1 along the East-West direction and θ 2 along the North-South direction. 5 We will put the AGN component back in next section, 4.2.3
= 1 2 (d − Kĝ − KLs) T C −1 D,mAGN (d − Kĝ − KLs),(20)
and
ZD,mAGN = (2π) N d /2 (det CD,mAGN) 1/2(21)
is the normalization for the probability. We discuss the "mAGN"regions in Appendix A. In the second term of Equation (19), θAGN,i is the measured AGN image position and σAGN,i is the estimated positional uncertainty of AGN image i; in the third term, ∆ti is the measured time delay with uncertainty σ∆t,i for image pair i = AB, CB, or DB. After we maximize the likelihood of the data, we obtainζ, and also the predicted arc light of the reconstructed background source intensity,ŝ, of the AGN host galaxy. Note that if there is no time-delay information, one can remove the last term in Equation (19).
AGN Light Model (Step 2)
In Equation (7), we use Mw to represent the AGN light.
In the next section, we further decompose the PSF, w, into the global structure and the fine structure that are shown in Figure 4. In particular, we define
w = w [0] + T [0] δw [0] ,(22)
where w [0] is the vector of global structure, δw [0] is the vector of fine structure and the subscript, [0], represents the zero-th iteration. Since, in this section, we focus on the global structure of the PSF, we postpone the discussion of T to Equation (29) and let
w = w [0] .(23)
By usingη,ζ, andŝ from the previous two sections and keeping them fixed, we model the global structure of the PSF with Gaussian profiles, each of the form
IG(θ1, θ2) = Ig exp − θ 2 1 + (θ 2 2 /q 2 g ) 2σ 2 g ,(24)
where Ig is the amplitude, qg is the axis ratio, and σg is the width. We find that a few Gaussians (∼ 2 − 4) with a common centroid are sufficient in describing the global structure of the PSF. 6 Substituting Equation (23) into Equation (7), givenη,ζ andŝ, we obtain
d P = Kĝ + KLŝ + Mw [0] ,(25)
whereL = L(ζ), which is kept fixed at this step. Note that the K matrix here is based on the initial PSF model, before the multi-Gaussian fitting. The posterior of the PSF and AGN parameters is given by
P (ν, ξ|d,η,ζ) = P (d|η,ζ, ν, ξ)P (ν, ξ) P (d|η,ζ) ,(26)
where ν represents the parameters of the Gaussian profiles in Equation (24) (26) is
P (d|η,ζ, ν, ξ) = exp[−ED(d|η,ζ, ν, ξ)] ZD ,(27)
where ED(d|η,ζ, ν, ξ)
= 1 2 (d − Kĝ − KLŝ − Mw [0] ) T · C −1 D (d − Kĝ − KLŝ − Mw [0] ),(28)
6 The different Gaussian components can vary their amplitudes, position angles and axis ratios.
and ZD = (2π) N d /2 (det CD) 1/2 . We denoteν andξ as the maximum likelihood parameters (that maximizes Equation (26)) from which we can obtain the optimal AGN light on the image, given the optimized source and lens mass models from the previous sections.
Pixelated Fine Structure of AGN light
In this section, we introduce the inner loop which aims at extracting the fine structure, δw [0] , in Equation (22)
PSF Correction for Each Iteration (Inner Loop:
Step 3)
In general, givenη [n] ,ζ [n] ,ŝ [n] ,ν [n,m] 7 , andξ [n,m] , where m is the iteration number of the inner loop and n is the 7ν [n,m] is only present when n = m = 0, which corresponds to parameters of the Gaussian profiles in Equation (24) iteration number of the outer loop, we can write out the equation as
d P = K [n]ĝ[n] + K [n]L[n]ŝ[n] +M [n,m] (ŵ [n,m] + T [n,m] δw [n,m] ) ≡ d P correction ,(29)
where P (δw [n,m]
= 1 2 (d − d P correction ) T C −1 D,mAc (d − d P correction ),(32)
and ZD,mAc = (2π) N d /2 (det CD,mAc) 1/2 is the normalization for the probability. We discuss the mAc (maskAGNcenter) regions in Appendix A The prior/regularization we impose in Equation (30) on the correction grid (fine structure of PSF) is to prevent the correction grid from absorbing the noise in the observed image. We express the prior in the following form P (δw [n,m] [n,m] ) is the normalization of the prior probability distribution (note that the optimal λ δw,[n,m] is not determined yet), and N δw, [n,m] is the number of pixels of the correction grid. We use the curvature form for the function E δw, [n,m] , which is discussed in Suyu et al. (2006).
Again, it is easy to understand that we want to maximize Equation (30). We obtain the most probable solution δw [n,m]
= (F + λ δw,[n,m] H) −1 (M [n,m] T [n,m] ) T C −1 D,mAc u,(34)H = ∇∇E δw,[n,m] ,(35)u = d − K [n]ĝ[n] − K [n]L[n] s [n] −M [n,m]ŵ[n,m] ,(36)
and ∇ ≡ ∂ ∂δw [n,m] .
Now, we go back to find the optimal regularization constant; that is, we want to maximize P (λ δw, [n,m]
where Tr denotes the trace andλ δw, [n,m] is the optimal regularization constant. If we set m = 0 (zeroth iteration of the fine structure), we obtain δw [n,0] . Due to the sharp intensity of the AGN center, the residuals there are much stronger than the peripheral area. If we directly extract the full correction grid, the regularization intends to under-regularize on the peripheral area and over-regularize on the center. To avoid this problem, at first, we extract the correction only around the AGN center; that is, we start from small N δw,[n,m] (half light radius or smaller) and increase it gradually (around 1.2 times previous size each time) as we obtain δw [n,m] . We show the idea in Figure 5 (note that the indices on δw in the figure are labeling the pixels, rather than the iteration numbers).
Since every iteration of δw [n,m] has their own fine structure (correction) uncertainty, according to Suyu et al. (2006), we also take as estimates of the 1σ uncertainty on each pixel value the square root of the corresponding diagonal element of the covariance matrix given by
C δw,[n,m] = (F +λ δw,[n,m] H) −1 .(41)
Add Fine Structure into Global Structure (Inner Loop: Step 4)
We start with the zeroth inner loop iteration, by setting m = 0, of the global structure, w [n,0] , and fine structure, δw [n,0] (which we can obtain by following the previous two sections). We then add the fine structure into the global structure by defining
w [n,1] = w [n,0] + T [n,0] δw [n,0] ,(42)
where w [n,1] is the first iteration in inner loop. More generally, we define the m + 1 th iteration of the PSF as
w [n,m+1] = w [n,m] + T [n,m] δw [n,m] .(43)
We recalculate the AGN parameters every time after getting a new w [n,m+1] , so given the sameη [n] andζ [n] in Equation (29), the posterior of the AGN parameters is given by
P (ξ [n,m+1] |d,η [n] ,ζ [n] ) = P (d|η [n] ,ζ [n] , ξ [n,m+1] )P (ξ [n,m+1] ) P (d|η [n] ,ζ [n] ) .(44)
(recall that ξ [n,m+1] represents the relative amplitudes and the positions of the AGNs in the n th outer loop iteration, and m+1 th inner loop iteration). The likelihood of Equation (44) is
P (d|η [n] ,ζ [n] , ξ [n,m+1] ) = exp[−E D,[n,m+1] (d|η [n] ,ζ [n] , ξ [n,m+1] )] ZD ,(45)
where
E D,[n,m+1] (d|η [n] ,ζ [n] , ξ [n,m+1] ) = 1 2 (d − Ω) T C −1 D (d − Ω)(46)
with
Ω = K [n]ĝ[n] + K [n]L[n]ŝ[n] + M [n,m+1] w [n,m+1] ,(47)
and ZD, as usual, is (2π) N d /2 (det CD) 1/2 . After maximizing Equation (44), we obtainξ [n,m+1] . We then replace thê ξ [n,m] from the previous iteration with theξ [n,m+1] we just obtained, and conduct the next inner loop iteration.
The Criteria to Stop the Inner Loop.
During every inner loop, we gradually increase the size, N δw, [n,m] , of the correction grid. Then, if (1) there is no residuals outside the correction grid, (2) Equation (34)
Lens Modeling with updated PSF
The goal of the outer loop in Figure 2 is to remodel all the light components with the updated PSF; that is, we want to obtain a better lens light model and arc light model with the new blurring matrix, and the underlying fine structure can then be revealed.
Update the Blurring Matrix and the Image Covariance Matrix (Outer Loop: Step 6)
Blurring matrix : After obtaining the last version of the PSF from Section 4.3.3, we update the blurring matrix, K, in Equation (7). In order to accelerate modeling speed, which highly depends on the size of the PSF for convolution of the extended images, we choose the central l
where T [n,m],ki is the element at k row and i column of T [n,m] , C δw,[n,m],ij is the element at i row and j column of C δw, [n,m] , and δij is the Kronecker delta. The element of the n + 1 th noise vector is defined as
n [n+1],µ = n 2 µ + kM [n+1],µk n 2 δw,[n+1],k ,(51)
which is characterized by the covariance matrix C D,[n+1] 9 . Note that nµ is the element of the original data noise vector.
Lens Modeling with All Light Components (Outer loop: Step 2)
In general, when executing the next iteration of outer loop, we can express Equation (7) as
d P = K [n+1] g [n+1] + K [n+1] L [n+1] s [n+1] + M [n+1] w [n+1] ≡ d P total .(52)
The posterior can be written as
P (η [n+1] , ζ [n+1] , ξ [n+1] |d, ∆t) ∝ P (d, ∆t|η [n+1] , ζ [n+1] , ξ [n+1] )P (η [n+1] , ζ [n+1] , ξ [n+1] ).(53)
8 For example, we choose l [n] = nl [1] 9 The purpose of updating the the image covariance matrix is to speed up the modeling to the final answer since the correction uncertainty that we add into the image covariance matrix is around AGN; that is, we weight the arc region more. However, in the end, if there is no "correction", Equation (50) is close to zero.
The likelihood of the data can be expressed as
P (d, ∆t|η [n+1] , ζ [n+1] , ξ [n+1] ) = ds [n+1] P(d, ∆t|η [n+1] , ζ [n+1] , ξ [n+1] , s [n+1] )P(s [n+1] ),(54)where P (d, ∆t|η [n+1] , ζ [n+1] , ξ [n+1] , s [n+1] ) = exp[−E D,[n+1] (d|η [n+1] , ζ [n+1] , ξ [n+1] , sn+1)] Z D,[n+1] · N AGN i=1 1 √ 2πσAGN,i exp − |θ AGN,i,[n+1] − θ P AGN,i,[n+1] | 2 2σ 2 AGN,i · i=1 1 √ 2πσ∆t,i exp − (∆ti − ∆t P i,[n+1] ) 2 2σ 2 ∆t,i ,(55)E D,[n+1] (d|η [n+1] , ζ [n+1] , ξ [n+1] , s [n+1] ) = 1 2 (d − d P total ) T C −1 D,[n+1] (d − d P total ),(56)
where
Z D,[n+1] = (2π) N d /2 (det C D,[n+1] ) 1/2(57)
The Criteria to Stop the Outer Loop.
We iterate the outer loop until Equation (56) does not decrease. 10 We also ensure that the size of the PSF (l [n] × l [n] ) for convolution of the lens light and arc light is big enough. Since the AO PSF can have substantial wings that contribute significantly, the size of the PSF in AO image is usually substantially larger than those of HST images. We set the size of the PSF (l [n] × l [n] ) such that the modeling results remain stable beyond this PSF size.
DEMONSTRATION AND BLIND TEST
In this section, we demonstrate the method using two mock data sets that are created with different PSFs, and show that we can recover the input parameters in both mocks by using
10 E D,[n] −E D,[n+1] E D,[n] < 0.2%
the strategy in Section 4 together with Glee. S.H.S. simulates AO images that mimic the strong lensing system, RXJ 1131−1231, with two foreground lens galaxies and a background source comprised of an AGN and its host galaxy. S.H.S. uses an elliptically symmetric power-law profile to describe the main lens mass distribution and a pseudoisothermal elliptic mass profile to describe the mass distribution of the satellite galaxy. The background host galaxy of the AGN is described by a Sérsic profile with additional starforming regions superposed, and the lens light distribution is based on a composite of two Sérsic light profiles. The simulated lensed images and background sources are shown in the third and second column, respectively, of the first (mock #1) and third (mock #2) rows of Figure 6. The difference between the two mocks is their PSFs. In mock #1, the PSF is taken to be a star observed with Keck's laser guide star adaptive optics system (LGSAO) that is relatively sharp and with a lot of structures (FWHM is ∼ 0.03 ′′ ). In mock #2, the PSF is relatively diffuse and without structures, which is similar to the PSF in the real data (FWHM is ∼ 0.045 ′′ ). We show them in the first column of the first and third rows of Figure 6. G.C.F.C. does a blind test of the PSF reconstruction method on mock #1; that is, G.C.F.C. does not know the input value at the beginning, and S.H.S. only reveals the input value when G.C.F.C. has completed the analysis of mock #1. Since the input value is the same in mock #2, G.C.F.C. models mock #2 by using the same strategy although the mock #2 test is performed after mock #1 and therefore is not blinded.
Mock #1: a sharp and rich structured PSF
The mock #1 image has 200 × 200 surface brightness pixels as constraints. The pixel size is 0.04 ′′ . The simulated time delays in days relative to image B are: ∆tAB = 1.5 ± 1.5, ∆tCB = −0.5 ± 1.5, ∆tDB = 90.5 ± 1.5. We follow the procedure described in Section 4 and Figure 2. The reconstructions are shown in the second row of Figure 6. To demonstrate the iterative processs visually, we show the process in Figure 7. The first column shows each PSF correction grid in different iteration, the second column shows the cumulative PSF correction from iteration 1 to iteration 18, the third column is the PSF model at each iteration, and the right-most column shows the best fitting residuals with current PSF model. It is obvious that we get better and better normalized residuals as the iterative PSF corrections proceed. We follow Section 4.3.3 and increase gradually the size of the PSF; the size of the final PSF is 85×85 (which corresponds to 3.4 ′′ ×3.4 ′′ ). However, since the PSF is very sharp in mock #1, the PSF size with 19 × 19 (which corresponds to 0.76 ′′ × 0.76 ′′ ) for the blurring matrix is enough. While 19 × 19 is sufficient for the extended source/lens light, it is not for the AGNs; 85 × 85 is needed for describing the AGNs.
We try a series of source resolutions from coarse to fine, and the parameter constraints stabilize starting at ∼ 52×52 source pixels, corresponding to source pixel size of ∼ 0.045 ′′ . In order to quantify the systematic uncertainty, we consider the following set of source resolutions: 52×52, 54×54, 56×56, 58 × 58, 60 × 60, and 62 × 62. We weight each choice of the Figure 6. The simulation (input), reconstruction (output), and normalized residuals of mock #1 and mock #2. The left column shows the input/output PSF, the middle left column shows the input/output sources (host galaxy of the AGN), the middle right column shows the input/output images, and the right column shows the normalized image residules (in units of the estimated pixel uncertainties). Our PSF reconstruction method is able to reproduce both the global and fine structures in the PSF, yielding successful reconstructions of the background source intensity and the lensed images. Both reduced χ 2 are ∼ 1.
source resolution equally 11 , and combine the Markov chains together. The time delays are also reproduced by the model: for the various source resolutions, the total χ 2 is ∼ 3 for the three delays. We demonstrate the important parameters for cosmography in the upper panel of Figure 8 (time-delay distance, external shear, radial slope of the main lens galaxy, Einstein radii of the main galaxy and its satellite, total Einstein radius). The white dots represent the input values. The results show that we can recover the important parameters for cosmography. There is a strong degeneracy between the Einstein radii of the main galaxy and the satellite galaxy, as expected since these two galaxies are both located within the arcs. However, the effect on time-delay distance due to the presence of the satellite is less than 1% (?). Despite the 11 We weight the chains by the same weight because the source evidence are similar, and the lens parameterizations are the same. degeneracy, we can recover the total Einstein radius within 1σ, where the total Einstein radius, θE,tot, is defined by
θ E,tot 0 2π 0 κtot(θ, ϕ) dϕ dθ πθ 2 E,tot = 1,(58)
κtot is the total projected mass density including the main galaxy and its satellite, and ϕ is the polar angle on the image plane. The total Einstein radius in here is only a circular approximation for the elliptical galaxy plus its satellite.
Mock #2: a diffuse and smooth PSF
The mock #2 image has 300 × 300 surface brightness pixels as constraints (the larger dimensions of the image are necessary for modeling the diffuse PSF). The pixel size and time delays are the same as in mock #1. The size of the final PSF is 127 × 127 (which corresponds to 5.08 ′′ × 5.08 ′′ ). Since the PSF is very diffuse in mock #2, the PSF size with Figure 7. We demonstrate the iterative reconstruction process. From the left to the right, we show the PSF correction, cumulative PSF correction, current PSF model, and normalized residuals after using the current PSF model at iteration 1, 9, and 18. Since we sequentially increase the PSF correction grid as we iterate, the size of the PSF correction grid at iteration 1 is smaller than that of other iterations.
59 × 59 (which corresponds to 2.36 ′′ × 2.36 ′′ ) for the blurring matrix is needed to convolve the spatially extended images. We show the reconstruction in the fourth row of Figure 6.
We also try a series of source resolutions from coarse to fine, and the parameter constraints stabilize starting at ∼ 59 × 59. To quantify systematic uncertainties due to source resolution, we consider the following set of source resolutions: 59 × 59, 60 × 60, 61 × 61, 62 × 62, and 63 × 63. We also weight each source resolution equally, and combine the Markov chains together. We show the constraints on the same important parameters as mock #1 for cosmography in the lower panel of Figure 8. The white dots represent the input values. The results show that we can recover the important parameters for cosmography. Again, although we cannot recover the individual Einstein radius due to the strong degeneracy between these two Einstein radii, we can still recover the total Einstein radius.
We use the source-intensity-weighted regularization in the source reconstruction to prevent the source from fitting to the noise. The noise-overfitting problem is due to the fact that the outer region of the source plane is underregularized. We do two tests which show its negligible impact on cosmographic inference: (1) We test it by changing the image covariance, CD, such that the uncertainties corresponding to low surface brightness areas are boosted (which is a similar effect as allowing the source to be more regularized at low surface brightness regions). The results show that the relative posteriors of lens/cosmological parameters are insensitive to such changes of CD (hence the source regularization); (2) We impose the source-intensity weighted regularization on the source plane, which can regularize more on the low surface brightness area on the source plane (see e.g., Tagore & Keeton 2014, for another type of regularization based on analytic profile). Specifically, we obtain the first version of the source intensity distribution s f on a grid of pixels following the method of Suyu et al. (2006) with a constant regularization for all source pixels. We then repeat the source reconstruction but with the regularization constant λ scaled inversely proportional to s 4 f , allowing high/low source intensity regions to be less/more regularized. The relative posteriors between the different MCMC samples in the chains are the same between the uniform and sourceintensity-weighted source regularizations. Furthermore, even with different source reconstrcution methods, the Einstein radius, which also plays an important role in cosmographic inference, is still robust.
REAL DATA MODELING
We apply our newly-developed PSF reconstruction method to the real AO imaging shown in Section 2, and use the time delays from Tewes et al. (2013b). For the lens light, we use two Sérsic profiles with common centroids and position angles for the main lens galaxy, and use one circular Sérsic profile for the small satellite (whereas in the mock data in We can recover the key lens parameters for cosmography such as the modeled time-delay distance, total Einstein radius, and external shear, despite the strong degeneracy between the Einstein radii of the main and satellite galaxies (which consequently we do not recover in mock #2). Section 5 we describe the light of the satellite as a point source with PSF, w). We find that, in this AO image, 4 concentric Gaussian profiles provide a good description of the initial global structure of the PSF 12 , which is the procedure we discussed in Section 4.2.3. For modeling the main lens mass, we use an elliptical symmetric distribution with 12 Due to unknown PSF, we do not have prior information on PSF. Thus, we test multiple concentric Gaussian profiles to fit the AGN. However, we find that the initial PSF model does not affect the final results which is shown in Section 5, because the iterative method will correct it in the end. power-law profile and an external shear which are described in Section 4.2.2; for modeling the mass distribution of the satellite, we use a pseudo-isothermal mass distribution.
After we increase the PSF grid during the iterative reconstruction scheme, the final PSF size is 127 × 127 (which corresponds to 5.08 ′′ × 5.08 ′′ ). We try a series of source resolutions from coarse to fine and a series of PSF sizes for the blurring matrix from small to large. The parameter constraints stabilize starting at ∼ 71 × 71 for the source resolution and at ∼ 59 × 59 for the PSF size for the blurring matrix, corresponding to a source pixel size of ∼ 0.05 ′′ and a PSF size of 2.36 ′′ × 2.36 ′′ . Note, again, that while a PSF cutout of 59 × 59 is sufficient for the extended source, the AGNs require a larger PSF grid of 127 × 127. We show the reconstructions of AO imaging in Figure 9 13 and the reconstructed PSF in Figure 10. To quantify the systematic uncertainty, we show in Figure 11 the parameter constraints of different sizes of the source grid, 71 × 71, 73 × 73, 75 × 75, 77 × 77, and 79 × 79, with the PSF size, 59 × 59, for the blurring matrix. After combining all the chains with different source resolutions, we overlap the contours from the 59 × 59 PSF with the contours from the 69 × 69 PSF (for the blurring matrix) in Figure 12; the results agree with each other within 1 − σ uncertainty.
Since the PSF in RXJ 1131−1231 AO imaging is similar to the PSF of mock #2, the results from Figure 8 provide a valuable reference. Thus, note that the Einstein radii of the main galaxy and the satellite galaxy inferred from the Keck AO image are also degenerate with each other, as we saw in the case of mock #2.
By using the same time delay meausurements from Tewes et al. (2013b) as in ?, we compare the results of modeling the AO image with the results of modeling the HST image from ?. 14 We show the comparison in Figure 13 and list all the lens model parameters in Table 1. Except for the highly degenerate Einstein radius of the main galaxy, other important parameters are overlapping within 1-σ uncertainty. Furthermore, the constraint of time-delay distance by using AO imaging with 0.045 ′′ resolution is tighter than the constraint of time-delay distance by using HST imaging with 0.09 ′′ by around 50%.
For cosmographic measurement from time-delay lenses, 13 We use the source-intensity-weighted regularization in the source reconstruction 14 The mass model parameterization is the same as ? except for a slight difference in the definition of θ E due to ellipticity. In this paper, we compare the θ E as defined in equation (14). Thus the θ E shown in this paper is slightly different from that of ?.
we need to break the mass-sheet degeneracy in gravitational lensing (e.g., Falco et al. 1985;Schneider & Sluse 2013Xu et al. 2015) that can change the modeled time-delay distance. This would involve considerations of mass profiles, lens stellar kinematics and external convergence (e.g., Treu & Koopmans 2002; Barnabè et al. 2011;?;Suyu et al. 2014) that are beyond the scope of this paper. The focus of this paper is to investigate the feasibility of AO imaging for follow up. As illustrated in Figure 13, AO imaging together with our new PSF reconstruction technique (especially of quad lens systems) is a competitive alternative to HST imaging for following up time-delay lenses for accurate lens modeling.
SUMMARY
In this paper we develop a new method, iterative PSF correction scheme, which can overcome the unknown PSF problem, to constrain cosmology by modeling the strong lensing AO imaging with time delays. We elaborate the procedures in Section 4 and draw an overall flow chart in Figure 2. We test the method on two mock systems, mock #1 (blindly) and mock #2, which are created by using a sharp PSF and diffuse PSF, respectively, and apply this method to the high-resolution AO RXJ 1131−1231 image taken with the Keck telescope as part of the SHARP AO observation. We draw the following conclusions.
• We perform a blind test on mock #1, which mimics the appearance of RXJ 1131−1231 but with a sharp and richly structured PSF (based on a star observed with Keck's LGSAO). Afterward, we model the mock #2, which is created by a diffuse PSF that is similar with the PSF in AO RXJ 1131−1231 image, using the same strategy. The results show that the more diffuse PSF the AO imaging has, the larger the PSF is needed for representing the AGN; similarly, the larger the PSF for representing the AGNs, the larger the The reference of the position is in Figure 9. All the position angles are measured conterclockwise from positive θ 2 (north). The amplitude is in equation (8).
Note: There are total 39 parameters that are optimized or sampled. The optimal parameters have little effect on the key parameters for cosmology (such as D model ∆t ). For the lens light, two Sérsic profiles with common centroid and position angle are used to describe the main lens galaxy G. They are denoted as G1 and G2 above. The source pixel parameters (s) are marginalized and are thus not listed.
PSF is needed for convolution of the spatially extended lens and arcs. By performing MCMC sampling, we can recover the important parameters for cosmography (time-delay distance, external shear, slope, and total Einstein radius of the main galaxy plus its satellite). Although we cannot recover the individual Einstein radius, the effect on time-delay distance due to the presence of the satellite is less than 1% (?).
• We model the AO RXJ 1131−1231 image by the iterative PSF correction scheme. We compare the results of important parameters with the results from modeling the HST imaging in ?. Except for the highly degenerate Einstein radius of the main galaxy, other important parameters for cosmography agree with each other within 1-σ ( Figure 13). Furthermore, the constraint of time-delay distance by using AO imaging with 0.045 ′′ resolution is tighter than the constraint of time-delay distance by using HST imaging with 0.09 ′′ by around 50%.
The iterative PSF reconstruction method that we have developed is general and widely applicable to studies that require high-precision PSF reconstruction from multiple nearby point sources in the field (e.g., the search of faint companions of stars in star clusters). For the case of gravitational lens time delays, this method lifts the restriction of using HST strong lensing imaging, and opens a new series of AO imaging data set to study cosmology. From the upcoming surveys, hundreds of new lenses are predicted to be discovered; this method not only can motivate more telescopes to be equipped with AO technology, but also facilitate the goal to reveal possible new physics by beating down the uncertainty on H0 to 1% via strong lensing (?). Figure A1. The three different mask regions which are circled in red, and the white arrows indicate the special area which need to be masked out (that is, we boost the uncertainty in that region) while we extract the PSF correction. The left panel shows the maskArcAGN region for fitting the lens light, and the middle panel shows the maskAGN region for fitting the arc light. When obtaining the PSF corrections, the white circles in the right panel need to be masked out when the PSF grid is small. As we increase the PSF grid around each AGN image such that the grid contains other AGN images (shown in the right panel of Figure 5), we mask out the red circles associated with these other AGN images and also the white circles. Figure A2. The matrix T [n,m] for making δw [n,m] the same length as w [n,m] . The indices of δw i and 0 k are for the pixels (rather than the PSF correction iterations). T [n,m] is a matrix at the n th outer loop and the m th inner loop. Suyu S. H., et al., 2013, ApJ, 766, 70 Suyu S. H., et al., 2014 This paper has been typeset from a T E X/L A T E X file prepared by the author.
Figure 1 .
1Keck AO image (K ′ band) of the gravitational lens RXJ1131-1231. The lensed AGN image of the spiral source galaxy are marked by A, B, C and D, and the star-forming regions in the background spiral galaxy form plentiful lensed features. The foreground main lens and the satellite are indicated by G and S
Figure 2 .
2Figure 2. The flow-chart describes the overall procedures in Section 4. We use the procedures to reconstruct the PSF directly from lens image and do the lens modeling. In step 1, we use a nearby star (or one of the lensed AGN itself) as the initial PSF; in step 2, we sequentially obtain the lens light, arc light, AGNs light, and the positions and relative amplitudes of AGNs; steps 3 to 5 form an inner loop to add the correction (fine structures) into the PSF and accumulate the correction uncertainties; in step 6, we enter the outer loop which updates the image covariance matrix, PSF of all light model, and then repeat the full procedure until the image χ 2 no longer decreases.
Figure 3 .
3Top panel: we decompose the image into lens light, arc light, and AGN light sequentially. Bottom panel: we model the AGN light by placing the PSF grid (described by vector w) at each of the AGN positions and scaling each PSF by its respective AGN amplitude. This procedure can be characterized by a matrix M, such that the AGN light model on the image plane can be expressed as Mw.
Figure 4 .
4The left panel is the global structure of the PSF. The middle panel is the cumulative fine structure of the PSF. We add the fine structure to global structure to get the PSF model in the right panel.
Figure 5 .
5The PSF correction grids of the iterative PSF correction scheme. In the inner loop of the PSF correction scheme (same n but different m), we start with a small correction grid δw and increase it sequentially. This accommodates for the larger corrections needed in the central parts of the AGN. Left panel: a small PSF correction grid is placed at each of the four AGN images A, B, C and D inFigure 1via the matrix M, and the values of the PSF correction grid is determined via a linear inversion to reduce the overall image residuals. Since the AGN centroids are typically non-integral pixel values, we linearly interpolate the correction grid onto the image plane. Right panel: the enlarged corrections grids after several iterations of PSF correction, showing overlap between the grids. When the peripheral area of a correction grid overlaps with the central parts of another AGN image (e.g., AGN image C in the lower-right parts of the correction grid of image A), we mask out the center of the AGN region in order to prevent the correction grids from absorbing the residuals which come from the mismatch of the sharp intensity of AGN center (see Appendix A for more details).
that yield w [ 0 ]
0; ξ are the amplitudes and the positions of the AGN, which are coded in M. The likelihood of Equation
by using a correction grid. We show it visually inFigure 5. The goal of the inner loop is to incorporate most of the fine structures into the PSF model; then in the outer loop, we can use the updated PSF model obtained from the inner loop to remodel all the light components (which require a given PSF model). Since this section is the starting point of the inner loop and outer loop, theη,ζ,ŝ,ν, andξ we get by optimizing Equations (10), (17), and (26) are actually the zero-th outer loop iteration and the zero-th inner loop iteration, which we denote byη[0] ,ζ[0] ,ŝ[0] ,ν[0,0] , andξ[0,0] .
|λ δw,[n,m] , R) = exp(−λ δw,[m] E δw,[n,m] (δw [n,m] |R)) Z δw,[n,m] (λ δw,[n,m] ) ,
where F = ∇∇E D,mAc,[n,m] = T T [n,m] (M T [n,m] C −1 D,mAc M [n,m] )T [n,m] ,
Figure 9 .Figure 10 .
910RXJ 1131−1231 AO image reconstruction of the most probable model with a source grid of 79 × 79 pixels and 69 × 69 PSF for convolution of spatially extended images. Top left: RXJ 1131−1231 AO image. Top middle: predicted lensed image of the background AGN host galaxy. Top right: predicted light of the lensed AGNs, the bright compact region: lensed images of a bright compact region in the AGN host galaxy, and the lens galaxies. Bottom left: predicted image from all components, which is a sum of the top-middle and top-right panels. Bottom middle: image residual, normalized by the estimated 1-σ uncertainty of each pixel. Bottom right: the reconstructed host galaxy of the AGN in the source plane The left panel is the reconstructed AO PSF. The right panel is the radial average intensity of the PSF, which shows the core plus its wings.
Figure 13 .
13Left panel: comparison of posterior of the key lens model parameters between AO imaging (dashed) and HST imaging (shades). The AO constraints are from the combination of both the 59 × 59 and 69 × 69 chains containing the series of source resolutions (e.g.,Figure 12for 59 × 59). The contours/shades mark the 68.3%, 95.4%, and 99.7% credible regions. The AO constraints are consistent with the HST constraints, and are in fact ∼ 50% tighter on the modeled time-delay distance. Right panel: PDFs for D ∆t , showing the constraints from HST image and AO image.
, L35 Tagore A. S., Keeton C. R., 2014, MNRAS, 445, 694 Tewes M., Courbin F., Meylan G., 2013a, A&A, 553, A120 Tewes M., et al., 2013b, A&A, 556, A22 Treu T., 2010, ARA&A, 48, 87 Treu T., Ellis R. S., 2014, preprint, (arXiv:1412.6916) Treu T., Koopmans L. V. E., 2002, MNRAS, 337, L6 Treu T., Koopmans L. V. E., 2004, ApJ, 611, 739 Vegetti S., Lagattuta D. J., McKean J. P., Auger M. W., Fassnacht C. D., Koopmans L. V. E., 2012, Nature, 481, 341 Vuissoz C., et al., 2007, A&A, 464, 845 Vuissoz C., et al., 2008, A&A, control. http://www.osti.gov/scitech/servlets/purl/647009 Wizinowich P. L., Le Mignant D., Stomski Jr P. J., Acton D. S., Contos A. R., Neyman C. R., 2003, in Astronomical Telescopes and Instrumentation. pp 9-20 Wizinowich P. L., et al., 2006, Publications of the Astronomical Society of the Pacific, 118, 297 Xu D., Sluse D., Schneider P., Springel V., Vogelsberger M., Nel-son D., Hernquist L., 2015, ArXiv:1507.07937, van Dam M. A., et al., 2006, Publications of the Astronomical Society of the Pacific, 118, 310
|λ δw,[n,m] , R) is the prior on δw[n,m] given {λ δw,[n,m] , R} with R denoting a particular form of "regular-E D,mAc,[n,m] (d|δw[n,m] ,η[n] ,ζ[n] ,ν[n,m] ,ξ[n,m] )ization" on δw [n,m] and λ δw,[n,m] characterising the strength
of the regularization. We can write the likelihood in Equa-
tion (30) as
P (d|δw [n,m] ,η [n] ,ζ [n] ,ν [n,m] ,ξ [n,m] )
=
exp[−E D,mAc,[n,m] (d|δw [n,m ,η [n] ,ζ [n] ,ν [n,m] ,ξ [n,m] )]
ZD,mAc
,
(31)
where "mAc" stands for maskAGNcenter,
[n] ×l[n] pixels of the updated PSF grid (that has N δw,[n,Ninner] pixels) as the new PSF to construct K [n+1] for the spatially extended images.8 Image covariance matrix : We accumulate the uncertainty of the PSF pixel grid from every inner loop. The accumulated uncertainty isn 2
δw,[n+1],k =
N inner
m=0
i
T [n,m],ki C δw,[n,m],ij δij ,
ζ[n+1] , andξ[n+1] . Then, we replace theη[n] ,ζ[n] , andξ[n] in Section 4.3 withη[n+1] ,ζ[n+1] , andξ[n+1] and then execute the next set of inner loop iterations. If we have a total of N outer loop iterations, we obtain the final K[N] and w[N] .is the normalization for the probability, andθ AGN,i,[n+1] =
θAGN,i(ξ [n+1] ).
After maximizing Equation (53), we obtainη [n+1] ,
Figure 11. Posterior of the key lens model parameters for RXJ 1131−1231 and the time delays. We use the PSF size, 59 × 59, for convolution of the spatially extended lens and arcs. We show the constraints from Markov chains of different source resolutions: 71 × 71, 73 × 73, 75 × 75, 77 × 77, and 79 × 79. The contours mark the 68.3%, 95.4%, and 99.7% credible regions for each source resolution. The spread in the constraints from different chains allow us to quantify the systematic uncertainty due to the pixelated source resolution.1.54 1.56 1.58 1.60
1.96 1.98 2.00
ϑ
E,G
γ'
0.26 0.28 0.30 0.32
ϑ
E,S
0.075 0.080 0.085
γ
ext
1.54 1.56 1.58 1.60
1800 1900 2000 2100
ϑ
E,G
D
Δt
[Mpc]
1.96 1.98 2.00
γ'
1.96 1.98 2.00
γ'
0.26 0.28 0.30 0.32
ϑ
E,S
0.26 0.28 0.30 0.32
ϑ
E,S
0.075
0.075
0.080
0.080
0.085
0.085
γ
ext
γ
ext
79x79
77x77
75x75
73x73
71x71
1.54 1.56 1.58 1.60
1.96 1.98 2.00 2.02
ϑ
E,G
γ'
0.240.260.280.300.320.34
ϑ
E,S
0.075 0.080 0.085 0.090
γ
ext
1.54 1.56 1.58 1.60
1800 1900 2000 2100
ϑ
E,G
D
Δt
model
[Mpc]
1.96 1.98 2.00 2.02
γ'
1.96 1.98 2.00 2.02
γ'
0.240.260.280.300.320.34
ϑ
E,S
0.240.260.280.300.320.34
ϑ
E,S
0.075
0.075
0.080
0.080
0.085
0.085
0.090
0.090
γ
ext
γ
ext
59x59
69x69
Figure 12. Posterior of the key lens model parameters for RXJ1131 and the time delays. We compare the PSF size, 59 × 59 and 69 × 69,
for convolution of the spatially extended lens and arcs. The constraints correspond to the combination of Markov chains of different
source resolutions (71 × 71, 73 × 73, 75 × 75, 77 × 77, and 79 × 79) in both PSF sizes. The contours mark the 68.3%, 95.4%, and 99.7%
credible regions. The constraints of the two PSF sizes are in good agreement, indicating that PSF sizes larger than ∼ 59 × 59 are sufficient
to capture the PSF features for convolving the spatially extended images.
Table 1 .
1Lens Model ParameterDescription
Parameter
Marginalized
or Optimized
Constraints
Time-delay distance (Mpc)
D model
∆t
1970 +40
−43
Lens mass distribution
Centroid of G in θ 1 (arcsec)
θ 1,G
6.306 +0.004
−0.008
Centroid of G in θ 2 (arcsec)
θ 2,G
5.955 +0.005
−0.005
Axis ratio of G
q G
0.753 +0.008
−0.007
Position angle of G ( • )
φ G
113.4 +0.4
−0.5
Einstein radius of G (arcsec)
θ E,G
1.57 +0.01
−0.01
Radial slope of G
γ ′
1.98 +0.07
−0.02
Centroid of S in θ 1 (arcsec)
θ 1,S
6.27 +0.02
−0.03
Centroid of S in θ 2 (arcsec)
θ 2,S
6.56 +0.01
−0.01
Einstein radius of S (arcsec)
θ E,S
0.282 +0.003
−0.003
External shear strength
γext
0.083 +0.003
−0.003
External shear angle ( • )
φext
93 +1
−1
Lens light as Sérsic profiles
Centroid of S in θ 1 (arcsec)
θ 1,GL
6.3052 +0.0002
−0.0002
Centroid of S in θ 2 (arcsec)
θ 2,GL
6.0660 +0.0002
−0.0002
Position angle of G ( • )
φ GL
116.9 +0.4
−0.4
Axis ratio of G1
q G
0.912 +0.004
−0.004
Amplitude of G1
I s,GL1
1.47 +0.02
−0.02
Effective radius of G1 (arcsec)
R eff,GL1
2.37 +0.01
−0.01
Index of G1
n sérsic,GL1
0.63 +0.01
−0.01
Axis ratio of G2
q GL2
0.867 +0.002
−0.002
Amplitude of G2
I s,GL2
18.1 +0.3
−0.3
Effective radius of G2 (arcsec)
R eff,GL2
0.404 +0.005
−0.005
Index of G2
n sérsic,GL2
1.97 +0.02
−0.02
Centroid of S in θ 1 (arcsec)
θ 1,SL
6.210 +0.001
−0.001
Centroid of S in θ 2 (arcsec)
θ 2,SL
6.605 +0.001
−0.001
Axis ratio of S
q SL
≡ 1
Amplitude of S
I s,SL
69 +6
−6
Effective radius of S (arcsec)
R eff,SL
0.027 +0.001
−0.001
Index of S
n sérsic,SL
0.42 +0.04
−0.02
Lensed AGN light
Position of image A in θ 1 (arcsec)
θ 1,A
4.256
Position of image A in θ 2 (arcsec)
θ 2,A
6.652
Amplitude of image A
a A
21880
Position of image B in θ 1 (arcsec)
θ 1,B
4.288
Position of image B in θ 2 (arcsec)
θ 2,B
4.348
Amplitude of image B
a B
38555
Position of image C in θ 1 (arcsec)
θ 1,C
4.871
Position of image C in θ 2 (arcsec)
θ 2,C
4.348
Amplitude of image C
a C
11565
Position of image D in θ 1 (arcsec)
θ 1,D
7.378
Position of image D in θ 2 (arcsec)
θ 2,D
6.340
Amplitude of image D
a D
3215
Strong-lensing High Angular Resolution Program(Fassnacht et al. in prep.)
MNRAS 000, 1-20(2015)
ACKNOWLEDGMENTSWe thank Giuseppe Bono, James Chan, Thomas Lai, Anja von der Linden, Eric Linder, Phil Marshall, and David Spergel for the useful discussions. G.C.F.C. and S.H.S. are grateful to Bau-Ching Hsieh for computing support on the SuMIRe computing cluster. G.C.F.C. and S.H.S. acknowledge support from the Ministry of Science and Technology in Taiwan via grant MOST-103-2112-M-001-003-MY3. LVEK is supported in part through an NWO-VICI career grant (project number 639.043.308). CDF and DJL acknowledge support from NSF-AST-0909119. The data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.APPENDIX A: ARC AND AGN MASK REGIONSWe show the three different mask regions, maskArcAGN (mArcAGN), maskArc (mArc), and maskAGNcenter (mAc) inFigure A1. For modeling the lens light in Section 4.2.1, we mask out the region which contains significant arc light and AGN light in the left panel. For modeling the arc light in Section 4.2.2, we mask out the region with significant AGN light shown in the middle panel. For extracting the PSF correction, we show the residuals in the right panel (which is the image with the lens light, arc light, and AGN light subtracted). When the size of the correction grid is small such that the correction grids do not overlap other AGN center, we only need to mask out the area where it comes obviously from the host galaxy of AGN. For instance, if the background AGN has compact bright blobs in its host galaxy, due to the limit of the resolution on the source plane, the predicted arc cannot reconstruct the compact blobs, so there are residuals around these compact blobs on the image plane (shown in the right panel with red arrows). In order to prevent the correction grid from absorbing the light due to the resolution problem and adding non-PSF features into the PSF, we mask them out. When the correction grid is enlarged and covers other AGN centers, we need to mask out both regions (AGN centers and lensed compact blob).APPENDIX B: T [n,m] MATRIXSince δw[n,m]has different length in each iteration of inner/outer loop [n, m], we use a matrix T [n,m] to make δw[n,m]the same length as w[n,m]by padding the twodimensional boundaries of the PSF correction grid with zeros, as illustrated inFigure A2.
. A Agnello, B C Kelly, T Treu, P J Marshall, 10.1093/mnras/stv037MNRAS. 4481446Agnello A., Kelly B. C., Treu T., Marshall P. J., 2015, MNRAS, 448, 1446
. M W Auger, T Treu, B J Brewer, P J Marshall, 10.1111/j.1745-3933.2010.00980.xMNRAS. 4116Auger M. W., Treu T., Brewer B. J., Marshall P. J., 2011, MNRAS, 411, L6
. M Barnabè, O Czoske, L V E Koopmans, T Treu, A S Bolton, 10.1111/j.1365-2966.2011.18842.xMNRAS. 4152215Barnabè M., Czoske O., Koopmans L. V. E., Treu T., Bolton A. S., 2011, MNRAS, 415, 2215
. M Bartelmann, 10.1088/0264-9381/27/23/233001Classical and Quantum Gravity. 27233001Bartelmann M., 2010, Classical and Quantum Gravity, 27, 233001
. J M Beckers, 10.1146/annurev.aa.31.090193.000305ARA&A. 3113Beckers J. M., 1993, ARA&A, 31, 13
ArXiv e-prints (1511.03662). S Birrer, A Amara, A Refregier, Wavefront control system for the Keck telescope. Birrer S., Amara A., Refregier A., 2015, ArXiv e-prints (1511.03662), Brase J. M., 1998, Wavefront con- trol system for the Keck telescope. http://www.osti.gov/scitech/servlets/purl/302840
. J H H Chan, S H Suyu, T Chiueh, A More, P J Marshall, J Coupon, M Oguri, P Price, 10.1088/0004-637X/807/2/138ApJ. 807138Chan J. H. H., Suyu S. H., Chiueh T., More A., Marshall P. J., Coupon J., Oguri M., Price P., 2015, ApJ, 807, 138
. V Chantry, P Magain, 10.1051/0004-6361:20066839A&A. 470467Chantry V., Magain P., 2007, A&A, 470, 467
. D Coe, L A Moustakas, 10.1088/0004-637X/706/1/45ApJ. 70645Coe D., Moustakas L. A., 2009, ApJ, 706, 45
. T E Collett, 10.1093/mnras/stt504MNRAS. 432679Collett T. E., et al., 2013, MNRAS, 432, 679
. F Courbin, 10.1051/0004-6361/201015709A&A. 53653Courbin F., et al., 2011, A&A, 536, A53
. J Dunkley, M Bucher, P G Ferreira, K Moodley, C Skordis, 10.1111/j.1365-2966.2004.08464.xMNRAS. 356925Dunkley J., Bucher M., Ferreira P. G., Moodley K., Skordis C., 2005, MNRAS, 356, 925
. E Eulaers, 10.1051/0004-6361/201321140A&A. 553121Eulaers E., et al., 2013, A&A, 553, A121
. E E Falco, M V Gorenstein, I I Shapiro, 10.1086/184422ApJ. 2891Falco E. E., Gorenstein M. V., Shapiro I. I., 1985, ApJ, 289, L1
. C D Fassnacht, L V E Koopmans, K C Wong, 10.1111/j.1365-2966.2010.17591.xMNRAS. 4102167Fassnacht C. D., Koopmans L. V. E., Wong K. C., 2011, MNRAS, 410, 2167
. W L Freedman, B F Madore, V Scowcroft, C Burns, A Monson, S E Persson, M Seibert, J Rigby, 10.1088/0004-637X/758/1/24ApJ. 75824Freedman W. L., Madore B. F., Scowcroft V., Burns C., Monson A., Persson S. E., Seibert M., Rigby J., 2012, ApJ, 758, 24
. Z S Greene, 10.1088/0004-637X/768/1/39ApJ. 76839Greene Z. S., et al., 2013, ApJ, 768, 39
. S Hilbert, S D M White, J Hartlap, P Schneider, 10.1111/j.1365-2966.2007.12391.xMNRAS. 382121Hilbert S., White S. D. M., Hartlap J., Schneider P., 2007, MNRAS, 382, 121
. S Hilbert, J Hartlap, S D M White, P Schneider, 10.1051/0004-6361/200811054A&A. 49931Hilbert S., Hartlap J., White S. D. M., Schneider P., 2009, A&A, 499, 31
. G Hinshaw, ArXiv e-prints (1212.5226Hinshaw G., et al., 2012, ArXiv e-prints (1212.5226),
. M P Hobson, S L Bridle, O Lahav, 10.1046/j.1365-8711.2002.05614.xMNRAS. 335377Hobson M. P., Bridle S. L., Lahav O., 2002, MNRAS, 335, 377
. J.-W Hsueh, C D Fassnacht, S Vegetti, J P Mckean, C Spingola, M W Auger, L V E Koopmans, D J Lagattuta, arXiv:1601.01671preprintHsueh J.-W., Fassnacht C. D., Vegetti S., McKean J. P., Spingola C., Auger M. W., Koopmans L. V. E., Lagattuta D. J., 2016, preprint, (arXiv:1601.01671)
W Hu, arXiv:astro-ph/0407158Astronomical Society of the Pacific Conference Series. Wolff S. C., Lauer T. R.339215Hu W., 2005, in Wolff S. C., Lauer T. R., eds, Astronomical Soci- ety of the Pacific Conference Series Vol. 339, Observing Dark Energy. p. 215 (arXiv:astro-ph/0407158)
. I Jee, E Komatsu, S H Suyu, ArXiv e-prints (1410.7770Jee I., Komatsu E., Suyu S. H., 2014, ArXiv e-prints (1410.7770),
. I Jee, E Komatsu, S H Suyu, D Huterer, arXiv:1509.03310preprintJee I., Komatsu E., Suyu S. H., Huterer D., 2015, preprint, (arXiv:1509.03310)
. C S Kochanek, C R Keeton, B A Mcleod, 10.1086/318350ApJ. 54750Kochanek C. S., Keeton C. R., McLeod B. A., 2001, ApJ, 547, 50
. L V E Koopmans, T Treu, C D Fassnacht, R D Blandford, G Surpi, 10.1086/379226ApJ. 59970Koopmans L. V. E., Treu T., Fassnacht C. D., Blandford R. D., Surpi G., 2003, ApJ, 599, 70
. D J Lagattuta, M W Auger, C D Fassnacht, 10.1088/2041-8205/716/2/L185ApJ. 716185Lagattuta D. J., Auger M. W., Fassnacht C. D., 2010, ApJ, 716, L185
. D J Lagattuta, S Vegetti, C D Fassnacht, M W Auger, L V E Koopmans, J P Mckean, 10.1111/j.1365-2966.2012.21406.xMNRAS. 4242800Lagattuta D. J., Vegetti S., Fassnacht C. D., Auger M. W., Koop- mans L. V. E., McKean J. P., 2012, MNRAS, 424, 2800
. E V Linder, 10.1103/PhysRevD.70.043534Phys.Rev.D. 7043534Linder E. V., 2004, Phys.Rev.D, 70, 043534
. D J C Mackay, Neural Computation. 4415MacKay D. J. C., 1992, Neural Computation, 4, 415
. P Magain, F Courbin, S Sohy, 10.1086/305187ApJ. 494472Magain P., Courbin F., Sohy S., 1998, ApJ, 494, 472
. P J Marshall, M P Hobson, S F Gull, S L Bridle, 10.1046/j.1365-8711.2002.05685.xMNRAS. 3351037Marshall P. J., Hobson M. P., Gull S. F., Bridle S. L., 2002, MNRAS, 335, 1037
. P J Marshall, arXiv:1504.06148A&A. 3455preprintMarshall P. J., et al., 2015, preprint, (arXiv:1504.06148) Moffat A. F. J., 1969, A&A, 3, 455
. A More, arXiv:1504.05587preprintMore A., et al., 2015, preprint, (arXiv:1504.05587)
. M Oguri, P J Marshall, 10.1111/j.1365-2966.2010.16639.xMNRAS. 4052579Oguri M., Marshall P. J., 2010, MNRAS, 405, 2579
. D Paraficz, J Hjorth, 10.1051/0004-6361/200913307A&A. 50749Paraficz D., Hjorth J., 2009, A&A, 507, L49
. S Perlmutter, 10.1086/307221ApJ. 517565Perlmutter S., et al., 1999, ApJ, 517, 565
. 10.1051/0004-6361/201322116arXiv:1502.01589A&A. 55744preprintPlanck Collaboration et al., 2015, preprint, (arXiv:1502.01589) Rathna Kumar S., et al., 2013, A&A, 557, A44
. S Refsdal, MNRAS. 128307Refsdal S., 1964, MNRAS, 128, 307
. A G Riess, 10.1086/300499AJ. 1161009Riess A. G., et al., 1998, AJ, 116, 1009
. A G Riess, 10.1088/0004-637X/699/1/539ApJ. 699539Riess A. G., et al., 2009, ApJ, 699, 539
. A G Riess, 10.1088/0004-637X/730/2/119ApJ. 730119Riess A. G., et al., 2011, ApJ, 730, 119
. G Rousset, J C Fontanella, P Kern, P Gigan, F Rigaut, A&A. 23029Rousset G., Fontanella J. C., Kern P., Gigan P., Rigaut F., 1990, A&A, 230, L29
. C E Rusu, M Oguri, Y Minowa, M Iye, A More, N Inada, S Oya, 10.1093/mnras/stu1621MNRAS. 4442561Rusu C. E., Oguri M., Minowa Y., Iye M., More A., Inada N., Oya S., 2014, MNRAS, 444, 2561
. C E Rusu, 10.1051/0004-6361/2013218821506.05147A&A. 55937Rusu C. E., et al., 2015, ArXiv e-prints (1506.05147), Schneider P., Sluse D., 2013, A&A, 559, A37
. P Schneider, D Sluse, 10.1051/0004-6361/201322106A&A. 564103Schneider P., Sluse D., 2014, A&A, 564, A103
P Schneider, C S Kochanek, J Wambsganss, 10.1007/978-3-540-30310-7.Gravitational Lensing: Strong, Weak and Micro. SpringerSchneider P., Kochanek C. S., Wambsganss J., 2006, Grav- itational Lensing: Strong, Weak and Micro (Springer), doi:10.1007/978-3-540-30310-7.
Atlas de galaxias australes. J L Sérsic, Observatorio AstronomicoCordoba, ArgentinaSérsic J. L., 1968, Atlas de galaxias australes. Cordoba, Ar- gentina: Observatorio Astronomico, 1968
. I I Shapiro, 10.1103/PhysRevLett.13.789Physical Review Letters. 13789Shapiro I. I., 1964, Physical Review Letters, 13, 789
. D Sluse, 10.1051/0004-6361:20030904A&A. 40643Sluse D., et al., 2003, A&A, 406, L43
. S H Suyu, A Halkola, 10.1051/0004-6361/201015481A&A. 52494Suyu S. H., Halkola A., 2010, A&A, 524, A94
. S H Suyu, P J Marshall, M P Hobson, R D Blandford, 10.1111/j.1365-2966.2006.10733.xMNRAS. 371983Suyu S. H., Marshall P. J., Hobson M. P., Blandford R. D., 2006, MNRAS, 371, 983
. S H Suyu, P J Marshall, R D Blandford, C D Fassnacht, L V E Koopmans, J P Mckean, T Treu, 10.1088/0004-637X/691/1/277ApJ. 691277Suyu S. H., Marshall P. J., Blandford R. D., Fassnacht C. D., Koopmans L. V. E., McKean J. P., Treu T., 2009, ApJ, 691, 277
. S H Suyu, P J Marshall, M W Auger, S Hilbert, R D Blandford, L V E Koopmans, C D Fassnacht, T Treu, 10.1088/0004-637X/711/1/201ApJ. 711201Suyu S. H., Marshall P. J., Auger M. W., Hilbert S., Blandford R. D., Koopmans L. V. E., Fassnacht C. D., Treu T., 2010, ApJ, 711, 201
. S H Suyu, 10.1088/0004-637X/750/1/10ApJ. 75010ArXiv e-prints (1202.4459Suyu S. H., et al., 2012a, ArXiv e-prints (1202.4459), Suyu S. H., et al., 2012b, ApJ, 750, 10
| [] |
[
"A Semi-supervised Learning Approach with Two Teachers to Improve Breakdown Identification in Dialogues",
"A Semi-supervised Learning Approach with Two Teachers to Improve Breakdown Identification in Dialogues"
] | [
"Qian Lin [email protected] \nDepartment of Computer Science\nNational University of Singapore\n\n",
"Tou Hwee \nDepartment of Computer Science\nNational University of Singapore\n\n",
"Ng \nDepartment of Computer Science\nNational University of Singapore\n\n"
] | [
"Department of Computer Science\nNational University of Singapore\n",
"Department of Computer Science\nNational University of Singapore\n",
"Department of Computer Science\nNational University of Singapore\n"
] | [] | Identifying breakdowns in ongoing dialogues helps to improve communication effectiveness. Most prior work on this topic relies on human annotated data and data augmentation to learn a classification model. While quality labeled dialogue data requires human annotation and is usually expensive to obtain, unlabeled data is easier to collect from various sources. In this paper, we propose a novel semi-supervised teacher-student learning framework to tackle this task. We introduce two teachers which are trained on labeled data and perturbed labeled data respectively. We leverage unlabeled data to improve classification in student training where we employ two teachers to refine the labeling of unlabeled data through teacher-student learning in a bootstrapping manner. Through our proposed training approach, the student can achieve improvements over singleteacher performance. Experimental results on the Dialogue Breakdown Detection Challenge dataset DBDC5 and Learning to Identify Follow-Up Questions dataset LIF show that our approach outperforms all previous published approaches as well as other supervised and semi-supervised baseline methods. | 10.1609/aaai.v36i10.21349 | [
"https://arxiv.org/pdf/2202.10948v2.pdf"
] | 247,025,719 | 2202.10948 | 3cd02d5e8421612aff2525c1fb294ed229cfaa68 |
A Semi-supervised Learning Approach with Two Teachers to Improve Breakdown Identification in Dialogues
Qian Lin [email protected]
Department of Computer Science
National University of Singapore
Tou Hwee
Department of Computer Science
National University of Singapore
Ng
Department of Computer Science
National University of Singapore
A Semi-supervised Learning Approach with Two Teachers to Improve Breakdown Identification in Dialogues
Identifying breakdowns in ongoing dialogues helps to improve communication effectiveness. Most prior work on this topic relies on human annotated data and data augmentation to learn a classification model. While quality labeled dialogue data requires human annotation and is usually expensive to obtain, unlabeled data is easier to collect from various sources. In this paper, we propose a novel semi-supervised teacher-student learning framework to tackle this task. We introduce two teachers which are trained on labeled data and perturbed labeled data respectively. We leverage unlabeled data to improve classification in student training where we employ two teachers to refine the labeling of unlabeled data through teacher-student learning in a bootstrapping manner. Through our proposed training approach, the student can achieve improvements over singleteacher performance. Experimental results on the Dialogue Breakdown Detection Challenge dataset DBDC5 and Learning to Identify Follow-Up Questions dataset LIF show that our approach outperforms all previous published approaches as well as other supervised and semi-supervised baseline methods.
Introduction
In recent years, interactive virtual conversational agents have been developed rapidly and used widely in daily lives. The information exchange between a user and an agent is done via a conversational dialogue. To achieve effective communication, the agent is expected to generate a proper and rational response based on not only the last turn but also all previous utterances in the dialogue history to continue the dialogue. The user's trust in the agent is damaged when the agent fails to identify the user's intent and generates an inappropriate response, which confuses the user and causes a breakdown in the dialogue. Therefore, identifying breakdowns in dialogues is essential for improving the effectiveness of conversational agents, so that the agent is able to avoid generating responses which cause the breakdowns.
Much prior work on breakdown identification in dialogues has focused on supervised learning on human annotated data. One line of work relies on feature-engineered machine learning methods including decision trees and random forests (Wang, Kato, and Sakai 2019). Another line of work utilizes non-Transformer based neural networks Copyright © 2022, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. such as LSTM (Hendriksen, Leeuwenberg, and Moens 2019;Wang, Kato, and Sakai 2019;Shin, Dirafzoon, and Anshu 2019). Transformer-based methods involve pre-trained language models which are pre-trained on large corpora (Devlin et al. 2019;Conneau et al. 2020). Sugiyama (2019) and utilize BERT with input consisting of the text and textual features from the dialogue. Lin, Kundu, and Ng (2020) introduce multilingual transfer learning through a cross-lingual pre-trained language model and co-attention modules to reason between the dialogue history and the last utterance.
A recent work proposes to perform pretraining on BERT with conversational data and apply selfsupervised data augmentation on labeled data. Although good performance has been reported, we observe that the gain of either continued pre-training or data augmentation on labeled data is marginal over the conventional BERT classification scheme. Moreover, pre-training of a pre-trained language model on large corpora is resource-intensive.
We believe that training with dialogue data from other sources introduces diversity and enables the trained model to generalize better. Since annotated dialogue data is expensive to obtain, we propose using unlabeled data through semisupervised learning and self-training, such that the training data is enriched and more diverse.
In this work 1 , we propose a novel semi-supervised teacherstudent learning framework to improve the performance of pre-trained language models with unlabeled data. We leverage unlabeled data from other sources to enrich the training set through self-training, which is a general case of domain adaptation where source data and target data are sampled from different data sources. Self-training uses a trained classifier to assign label score vectors on unlabeled data instances. However, such labeling process tends to generate labels under the assumption that a similar distribution is shared by labeled data and unlabeled data. Since the distribution of unlabeled data is difficult to estimate, we introduce two teachers to improve the labeling of unlabeled data. The student model is encouraged to integrate the knowledge from two teachers in a bootstrapping manner.
We leverage a data augmentation method (Yavuz et al. 2020) Language Model (MLM) pre-training objective of pre-trained language models (PLM). It is a natural fit to incorporate such augmented data with PLM like RoBERTa and XLM-R (Conneau et al. 2020), since the PLMs have adapted to the masking patterns during the pre-training process on large corpora. The GOLD teacher learns knowledge from only labeled data and it tends to generate labels following the distribution of labeled data. The MASKED teacher is trained with only perturbed labeled data which is augmented by randomly replacing tokens from the labeled data with the [MASK] tokens based on a predefined probability.
We construct the training data for the student model as the combination of two segments: labeled data and [MASK]perturbed unlabeled data. We explicitly impose a difference between masking probabilities applied to labeled data (for training the MASKED Teacher) and unlabeled data (for training the student). Two teachers can provide proper distribution estimation on these two data segments separately. Therefore the student is optimized to distill the knowledge from the two teachers to improve self-training on the combined training set.
We evaluate our proposed approach on two multi-turn dialogue breakdown detection datasets and a large-scale followup question identification dataset. Experimental results show that our semi-supervised teacher-student learning framework outperforms all previous published approaches and competitive supervised and semi-supervised baselines. We also conduct further analysis to verify the effectiveness of the training strategies proposed in our framework.
Task Overview
Given a dialogue history H consisting of a sequence of alternating user and system utterances and the succeeding target system utterance T , the task is to determine whether or not the target utterance causes a certain dialogue breakdown type. Each instance (H, T ) is associated with a soft label vector y ∈ R |C| which corresponds to the probability distribution over the set C covering all possible breakdown types.
Proposed Approach
We give a detailed description of the proposed approach in this section.
Pre-trained Language Model
Assume that the dialogue history H = [H 1 , ..., H h ] consists of h tokens after tokenization and the target system utterance T = [T 1 , ..., T t ] has t tokens. We first obtain a sequence of tokens by concatenating the two sequences with [CLS] and [SEP] tokens. The input to the pre-trained language model is:
x = [CLS]H 1 ...H h [SEP]T 1 ...T t(1)
The combined sequence x includes n tokens. x is first converted to embedding x by the PLM embedding layer. The output of the pre-trained language model is a sequence of hidden states from the last layer of the model:
f (x; θ m ) = PLM(x)(2)
where θ m denotes the parameters of PLM, d is the hidden size of the pre-trained language model and the output shape is n × d.
Data Augmentation with [MASK] Tokens
We leverage the pre-trained language models with Masked Language Model (MLM) training objective (Devlin et al. 2019;Liu et al. 2019;Conneau et al. 2020) for data augmentation with [MASK] tokens. We perform data augmentation on the available data by randomly replacing the tokens in the instances with [MASK] tokens (Yavuz et al. 2020).
For each labeled or unlabeled data instance, we generate a certain number of new data instances with [MASK] tokens based on a predefined replacement probability ρ. For instance, we replace 10% of tokens with [MASK] tokens with ρ = 0.1.
For the unlabeled data
X U = {(x i , ·)} |X U | i=1 , we eventu- ally obtain the augmented data {(x j , ·)} k|X U | j=1 , where k is the number of augmented instances per original instance. Sim- ilarly, for the labeled data X L = {(x i , y i )} |X L | i=1 , we obtain the augmented dataset {(x j , y j )} k|X L | j=1
where the label of the augmented instance remains the same as the original instance. We use subscript L to indicate that the dataset is labeled and use subscript U for an unlabeled dataset.
Teacher Models
We introduce two teacher models, namely Gold Teacher (GT) and MASKED Teacher (MT). Both teacher models share the same neural network architecture. Specifically, GOLD Teacher is trained with labeled dataset (gold data) while MASKED Teacher is trained with [MASK] augmented data.
A teacher model is formed by a pre-trained language model and a classification layer. We denote parameters of the teacher model as
θ (T) = {θ (T) m , θ (T) c } where θ (T)h = f (x; θ (T) m )[0](3)
The classification layer consists of two linear functions connected by tanh activation.
g = W 2 (tanh(W 1 h + b 1 )) + b 2 (4) where W 1 ∈ R d×d , h, b 1 ∈ R d , W 2 ∈ R |C|×d and g, b 2 ∈ R |C| . The prediction is calculated by: y = f (x; θ (T) ) = softmax(g)(5)
The loss function is a weighted sum of three objectives: cross-entropy loss L CE , supervised contrastive learning loss L SCL (Gunel et al. 2021), and mean squared error loss L MSE .
L = β 1 L CE + β 2 L SCL + β 3 L MSE(6)
The supervised contrastive learning loss is defined as:
L SCL = N b i=1 − 1 N b,yi − 1 N b j=1 1 i =j 1 yi=yj log exp (Φ(x i ) · Φ(x j )/τ ) N b k=1 1 i =k exp (Φ(x i ) · Φ(x k )/τ )(7)
where y i = argmax(y i ), N b is the batch size, and N b,yi is the number of instances with the same label as the i-th instance within the batch. 1 denotes the indicator function. τ is a temperature parameter. Φ(·) corresponds to the encoder function described in Eqn. 3. We fine-tune all parameters in θ (T) . We use θ (GT) and θ (MT) for Gold Teacher and MASKED Teacher respectively.
GOLD Teacher
The GOLD Teacher is fine-tuned on labeled dataset
X L = {(x i , y i )} |X L | i=1
. It learns knowledge purely from quality annotated data.
Given the unlabeled dataset
X U = {(x i , ·)} |X U | i=1 , we aug- ment X U to A U = {(x i , ·)} k|X U | i=1
with k augmented instances per original unlabeled instance and the [MASK] token replacement probability ρ U .
We use the fine-tuned GOLD Teacher to assign soft labels to the augmented unlabeled dataset A U .
MASKED Teacher Given the labeled dataset
X L = {(x i , y i )} |X L | i=1
which is the training data for GOLD Teacher, we prepare the training data for MASKED Teacher by generating an augmented dataset on X L .
The [MASK] augmentation on X L is determined by k and the [MASK] token replacement probability ρ L which results in the augmented dataset
B L = {(x i , y i )} k|X L | i=1 . The MASKED Teacher is fine-tuned on B L .
The MASKED Teacher learns from training data with [MASK] tokens which adapts to the situation of predicting labels on instances with [MASK] tokens.
We use the fine-tuned MASKED Teacher to assign soft labels to unlabeled instances from A U which is the same augmented unlabeled dataset described for Gold Teacher.
We set [MASK] token replacement probability ρ L to be larger than ρ U (ρ L > ρ U ) such that the MASKED Teacher is robust to produce more confident label scores on A U .
Student Model
The student model follows the same architecture as the teacher model, consisting of a pre-trained language model and a classification layer.
The parameters of the student model are denoted as
θ (S) = {θ (S) m , θ (S)
c } which correspond to the pre-trained language model and the classification layer. The pre-trained language model inherits the weights from the GOLD Teacher fine-tuned on X L , that is, θ
(S) m := θ (GT) m . We do not perform fine-tuning of θ (S)
m during the training of the student model. The training objective of the student model is the same as the teacher model, which is defined in Eqn. 6.
Training Process
As mentioned in the last section, the GOLD Teacher is finetuned on the labeled dataset X L and the MASKED Teacher is fine-tuned on B L where B L is augmented based on X L . We also have the unlabeled dataset A U which is augmented based on the unlabeled dataset X U . We present the overall training process in Figure 1.
Joint Scoring on Unlabeled Data
For each x ∈ A U , we assign label score vectors by both the GOLD Teacher and the MASKED Teacher respectively. We define a joint scoring function to compute the label score vector with weights determined by hyperparameter γ.
y GT = f (x ; θ (GT) )(8)y MT = f (x ; θ (MT) )(9)y = γŷ GT + (1 − γ)ŷ MT(10)
We then obtain the labeled set
G L = {(x i , y i )} |A U | i=1
where x ∈ A U and y is calculated by Eqn. 10.
Bootstrapping Strategy
We consider a bootstrapping strategy to refine the labeling of unlabeled data to improve classification. Continuing with G L obtained by the process mentioned in the last subsection, we describe the bootstrapping strategy for student model fine-tuning as follows.
We use the combined dataset X (S) L = X L ∪ G L as the initial training set for the student model. After a complete training iteration consisting of k e epochs, the fine-tuned student model predicts label score vectors for each unlabeled data x ∈ A U .ŷ
S = f (x ; θ (S) )(11)
The refined label score vector after each training iteration is calculated by:
λ = i/N (i = 1, 2, ..., N − 1) y = α[(1 + λ)ŷ S + (1 − λ)ŷ MT ](12)
where i is the iteration index and N the total number of iterations. Therefore, λ ranges between 0 and 1 (0 < λ < 1). We set α = 0.5 in our experiments, such that we ramp up the weight ofŷ S from 0.5 to 1.0 while progressively decreasing the contribution from the MASKED Teacher to produce better predictions for the unlabeled data. As a result, the label score vectors in G L and the combined training set X (S) L are updated after each training iteration. For training of the student model in the succeeding iteration, we retain the parameters from the best epoch in the last iteration based on development set performance. We use the trained student model to make predictions on test sets.
Experiments Datasets
We evaluate our proposed approach on two multi-turn dialogue datasets DBDC5 English Track (Higashinaka et al. 2020) and DBDC5 Japanese Track (Higashinaka et al. 2020) (Higashinaka et al. 2020), and one much larger Learning to Identify Follow-Up Questions dataset LIF . DBDC5 English Track This is a multi-turn dialogue dataset which requires identification of the predefined dialogue breakdown type of the last system utterance given the dialogue history. Based on the annotation quality (Higashinaka et al. 2020), we use the re-annotated DBDC4 data as the labeled dataset. For the unlabeled data, we use the English data released in Higashinaka et al. (2017). DBDC5 Japanese Track This is a Japanese dataset with the same format as DBDC5 English Track. For the Japanese track, we use datasets released in previous DBDC tasks as training set, including DBDC1, DBDC2, DBDC3, and DBDC4 development sets, as well as DBDC5 development set. We use DBDC4 evaluation set for validation. These data were annotated by 15-30 annotators per instance. We use Chat dialogue corpus as the source of unlabeled data, which were annotated by only 2-3 annotators. (Higashinaka et al. 2019) LIF LIF is a conversational question answering dataset for the task of follow-up question identification, which requires the model to identify whether or not the last question follows up on the context passage and previous conversation history. Since LIF is derived from QuAC (Choi et al. 2018), we select the training set of CoQA (Reddy, Chen, and Manning 2019) which is a similar conversational QA dataset, as the source of unlabeled data. 2 We present the statistics of both DBDC5 datasets in Table 1 and the statistics of the LIF dataset in Table 2. The numbers reported do not include augmented data.
Evaluation Metrics
DBDC5 English Track and DBDC5 Japanese Track require classification-based metrics including accuracy and F1 scores, and distribution-based metrics including Jensen-Shannon divergence (JSD) and Mean Squared Error (MSE). 3 LIF dataset requires classification-based metrics including precision, recall, and F1 of class Valid, and macro F1. 4
Experimental Setup
We experiment with RoBERTa ) as the pre-trained language model in DBDC5 English Track and LIF. We use multilingual pre-trained language model XLM-R (Conneau et al. 2020) for DBDC5 Japanese Track. We use the large version with hidden size d = 1024. The maximum input length is set to 256.
For experiments on LIF, we concatenate the context passage and the conversation history into H, and T corresponds to the candidate question.
The weights β 1 , β 2 , β 3 are set to 1e-2, 1e-3, and 1.0 in the loss function for both DBDC5 English Track and Japanese Table 4: Experimental results on the LIF dataset. V-P, V-R, and V-F1 correspond to precision, recall, and F1 score on class Valid. denotes the percentage of the LIF training dataset used.
Track. Since LIF does not require distribution-based metrics, the weights β 1 , β 2 , β 3 are set to 1.0, 0.1, and 0 in experiments on LIF. We set temperature τ to 1.0 in L SCL and γ to 0.5 in Eqn. 10. We optimize the loss using AdamW (Loshchilov and Hutter 2019) with 0.01 weight decay. For data augmentation, we generate 6 instances for each labeled or unlabeled instance. We set [MASK] token replacement probability ρ U = 0.15 aligning to (Devlin et al. 2019;Liu et al. 2019) and ρ L = 0.25.
To train two teacher models, we use a batch size of 16, 8, and 12 for experiments on DBDC5 English Track, DBDC5 Japanese Track, and LIF respectively. The learning rate during training is set to 1e-5, 1e-5, and 2e-6 respectively. To train the student model, we use a batch size of 128 and learning rate 2e-6 for experiments on all three datasets. We set the maximum number of iterations N to 5 and the number of epochs k e to 5 per iteration. Models are trained on a single Tesla V100 GPU.
Compared Models
BERT+SSMBA The model consists of pre-trained language model BERT-base and a classification layer. The BERT parameters are further pretrained on largescale Reddit dataset. The labeled training data is augmented based on SSMBA (Ng, Cho, and Ghassemi 2020) and original labels are assigned to augmented instances. This is the best-performing model published to date on the DBDC5 En-glish track. We implement a baseline RoBERTa+SSMBA using RoBERTa classification model with SSMBA augmentation. For fairer comparison with our proposed approach on DBDC5 English dataset, we adopt RoBERTa-large model and generate data with SSMBA which follows BERT+SSMBA. XLMR+CM The model uses cross-lingual language model XLM-R with context matching (CM) modules. This is the best-performing model published to date on the DBDC5 Japanese track (Higashinaka et al. 2020). Three-way AP The model applies an attentive pooling network to capture interactions among the context passage, conversation history, and the candidate follow-up question. This is the best performing-model published to date on the LIF dataset. PLM Baseline We build a simple but effective baseline model consisting of a pre-trained language model and a classification layer. We select RoBERTa for experiments on English tasks (DBDC5 English Track and LIF) and XLM-R for experiments on DBDC5 Japanese track. We use the output from [CLS] as the representation for classification. We adopt the large version of the pre-trained language model (RoBERTa-large or XLM-R-large) unless stated otherwise. PLM+CoAtt We build another baseline by applying a coattention network on the output from a pre-trained language model. The selection of pretrained language models follows PLM Baseline. Since the co-attention network applies to two sequences of represen-tations corresponding to conversation history and the last utterance, we prepend the context passage to the conversation history and the candidate question is treated as the last utterance for experiments on LIF.
We also experiment with recently proposed semisupervised methods UDA (Xie et al. 2020) and Mix-Text (Chen, Yang, and Yang 2020). We adopt RoBERTa-large for the English datasets and XLM-R-large for the Japanese dataset. Labeled and unlabeled data (before augmentation) used are the same as our proposed method. For English datasets DBDC5 English and LIF, we use back-translation with German and Russian as intermediate languages for augmentation on unlabeled data following Chen, Yang, and Yang (2020). For DBDC5 Japanese dataset, we apply [MASK] augmentation used in our proposed method due to the nonavailability of Japanese round-trip back-translation model. UDA and MixText are trained on both labeled and unlabeled data, while the other compared models are trained on labeled data in a supervised manner.
Results
Main Results
We present the experimental results of DBDC5 (both English Track and Japanese Track) and LIF in Table 3 and Table 4, respectively. Results of BERT+SSMBA and XLMR+CM are retrieved from Higashinaka et al. (2020) and results of Threeway AP are retrieved from Kundu, Lin, and Ng (2020). For results of both DBDC5 datasets, we report Accuracy, F1(B), JS Divergence (JSD), and Mean Squared Error (MSE). For metrics MSE and JSD, the reported percentage of improvement is calculated as 100 − (ours/other model) × 100.
In the DBDC5 English Track, our proposed approach outperforms the prior best-performing model (BERT+SSMBA) by 4.0%, 4.2%, 17.1%, and 22.2% on metrics Accuracy, F1(B), JSD, and MSE. It also performs better than all supervised and semi-supervised baselines by at least 2.2%, 1.9%, 1.7%, and 6.7% on the reported four metrics. The results of RoBERTa+SSMBA show that based on the large pre-trained language model setting, adding SSMBA augmented data does not contribute improvement on this task.
In the DBDC5 Japanses Track, our proposed approach outperforms the prior best-performing model (XLMR+CM) by 2.2%, 6.0%, 19.5%, and 22.5% on metrics Accuracy, F1(B), JSD, and MSE. The improvements are at least 2.4%, 3.9%, 10.1%, and 11.4% when compared to all other baseline models. We notice that models with large version of PLM perform generally better on these datasets.
In the much larger LIF dataset, we sample different sizes of labeled training data from the full training dataset to verify the robustness of our approach. The smallest sampled training set consists of only 6,000 (4.74% of 126,632) labeled training instances, similar to the sizes of Test-I and Test-II. In this case, we sample 18,000 instances from CoQA as unlabeled data and increase the sample size accordingly for the larger training sets. With only 6,000 labeled training instances, our method achieves competitive performance which outperforms the previous best-performing model (Three-way AP) on all three LIF test sets except V-P of Test-I and Test-II. Table 5: Performance on the test set after removing (-) different components. We report Accuracy on DBDC5 English (D-EN) and DBDC5 Japanese (D-JP) and Macro F1 on LIF Test-I. GT: GOLD Teacher. MT: MASKED Teacher.
We also experiment with 25%, 50%, and 100% of full training data and observe further performance improvement. With 100% labeled training data, our approach outperforms Threeway AP on all metrics by a wide margin and also performs better than other supervised and semi-supervised baselines on all metrics except for V-P.
We conduct experiments comparing the use of X U and A U when training the student model. We replace A U as X U in our original approach and denote this variation as Ours (X U , no A U ). The results on the test sets indicate that utilizing A U (augmentations on unlabeled data) is more effective.
We perform statistical significance tests with regards to Accuracy (DBDC5 English and Japanese datasets) and Macro F1 (LIF) on test sets. Our proposed method is significantly better (p < 0.05) than all baseline methods.
Analysis
Based on our implementation of teacher and student models where teachers and student use the same architecture, our proposed approach is in line with the idea of selfdistillation (Mobahi, Farajtabar, and Bartlett 2020;Furlanello et al. 2018). It has been observed that self-distillation helps to improve test performance (Liu, Shen, and Lapata 2021;Furlanello et al. 2018;Zhang et al. 2019). Allen-Zhu and Li (2020) show that self-distillation performs implicit ensemble with knowledge distillation. In traditional self-distillation, the student is distilled from a single trained teacher. In our approach, we distill the knowledge from two different trained teachers with additional unlabeled data. We evaluate the effectiveness of two teachers by removing components in the proposed approach and show the results in Table 5. Performance drops when we remove either teacher but keep the self-training on combined training set. We observe that performance drops further if we continue to remove the selftraining process. This shows that both teachers contribute to the performance improvement on these tasks.
We conduct analysis of different settings on the development set, in order to select hyperparameters as well as to better understand the effectiveness of our proposed approach. For development set performance, we report accuracy score in the DBDC5 English Track and DBDC5 Japanese Track, and Macro F1 score in LIF6000 in which the model is provided with 6,000 labeled training instances sampled from LIF.
We select the number of augmented samples k from {4,6,8} and observe that the sample size 6 performs con- sistently better than the other two on all three datasets.
We investigate the impact of [MASK] token replacement probability of labeled data (ρ L ) on the model performance. Given a constant ρ U = 0.15, we vary the value of ρ L from 0.15 to 0.30 with step size of 0.05. The best development set performance is achieved at ρ L = 0.25.
We also explore different training strategies and compare them with our proposed approach. GOLD denotes that we only use GOLD Teacher which is trained on labeled data X L only to make predictions. MASKED denotes that we only use MASKED Teacher which is trained on [MASK] augmentation of labeled data B L to make predictions. Combined means the model is trained from scratch on the combination of labeled data and [MASK] augmentation of labeled data (X L ∪ B L ) without bootstrapping. EqualW indicates the training method in which we use trained GOLD Teacher and trained MASKED Teacher to make predictions on unlabeled data and score equally for the final label scores. That is,ŷ = 0.5ŷ GT + 0.5ŷ MT in Eqn. 10 for generating labels for unlabeled data and obtaining G L . We then train the student model on X L ∪ G L without bootstrapping. RefGold denotes a variant of our bootstrapping approach where the score refinement in Eqn. 12 is altered toŷ = α[(1 + λ)ŷ S + (1 − λ)ŷ GT ] in which we refer to GOLD Teacher. We present performance comparison in Figure 2.
Our proposed approach outperforms all other mentioned training strategies on the development set. GOLD achieves the best performance among non-bootstrapping settings, indicating that preserving knowledge from GOLD Teacher is important, which validates the initialization of our proposed student model. RefGold produces slightly lower scores than our proposed approach, probably because the number of masked training instances is more than instances without [MASK] tokens during bootstrapping training, so using predictions (Eqn. 12) from MASKED Teacher is better. But it still performs better than other non-bootstrapping settings. This finding suggests that the proposed bootstrapping is essential for further performance improvement. Our proposed method (Ours) is also significantly better (p < 0.05) than GOLD and RefGold.
Related Work
Data Augmentation Recent unsupervised data augmentation methods have shown the effectiveness on classification tasks with short text instances. Wei and Zou (2019) introduce random word-level operations including replacement, insertion, deletion, and swapping. Xie et al. (2020) add noise to the unlabeled data and generate new training data by backtranslation. These augmentation methods tend to generate unnatural text samples as the text sequence becomes longer such as multi-turn dialogues and conversations. A recent data augmentation method based on self-supervised learning is proposed to tackle the out-of-domain issue . Another line of recent work proposes to augment data by randomly replacing word tokens with [MASK] tokens while working with pre-trained language models pre-trained with Masked Language Model objective (Yavuz et al. 2020). In our work, we leverage this idea and further investigate how different probabilities of [MASK] token replacement affect the model performance. Domain Adaptation Domain adaptation is a general method which transfers the knowledge from a source domain to a target domain. Domain adaptation is usually applied to text classification tasks when labeled source domain data is more abundant than target domain data. It enables a classifier trained on a source domain to be generalized to another target domain (Jiang and Zhai 2007;Chen, Weinberger, and Blitzer 2011;Chen et al. 2012). Recent works incorporate output features from pre-trained language models to improve domain adaptation (Nishida et al. 2020;Ye et al. 2020). In our work, we sample unlabeled data from sources other than the available labeled training set to enrich the training data. We leverage the idea of domain adaptation with source data and target data sampled from different data sources. Semi-supervised Learning In our work, we utilize unlabeled data from other sources for model training via semisupervised learning. Semi-supervised learning involves both labeled and unlabeled data during training. The general idea is to train a model with labeled data in a supervised learning manner and then enrich the labeled set with the most confident predictions on unlabeled data (Kehler et al. 2004;McClosky, Charniak, and Johnson 2006;Oliver et al. 2018;Li et al. 2019). Regularization techniques are applied to obtain better decision boundaries of unlabeled data with unknown distribution, including adversarial training (Miyato, Dai, and Goodfellow 2017), adding dropout, adding noise, and bootstrapping (Laine and Aila 2017). We consider the bootstrapping strategy to refine the labeling of unlabeled data. Prior work shows that bootstrapping improves the labeling of unlabeled data (Reed et al. 2015;Laine and Aila 2017;He et al. 2018).
Conclusion
In this work, we propose a novel semi-supervised teacherstudent learning framework with two teachers. We leverage both labeled and unlabeled data during training in a bootstrapping manner. We show that bootstrapping with the proposed re-labeling method is essential to improve performance. Evaluation results on two multi-turn dialogue breakdown detection datasets and a large-scale follow-up question identification dataset show that our proposed method achieves substantial improvements over prior published methods and competitive baselines.
Figure 1 :
1Overview of the proposed training process.
m denotes parameters of the pre-trained language model and θ (T) c the parameters of the classification layer. We use the output at the first position ([CLS]) as the representation of the input x:
Figure 2 :
2Performance comparison on different training strategies on the development set.
incorporating [MASK] tokens derived from a Masked 1 The source code and trained models of this paper are available at https://github.com/nusnlp/S2T2. [CLS]...I don't cook but i love producing music, that's what i do for a living... LABEL 2 [CLS]...Hello. Everything is nice. I was playing with my dog the whole day... LABEL 0 [CLS]...I don't cook [MASK] i love [MASK] music, that's [MASK] i do for [MASK] living... LABEL 2 [CLS]...I don't [MASK] but [MASK] love producing music, [MASK] what [MASK] do for a living... LABEL 2 [CLS]...I [MASK] cook but i love producing [MASK], that's what [MASK] do [MASK] a living... LABEL 2 [CLS]...[MASK]. Everything is [MASK]. I was [MASK] with my dog the whole [MASK]... LABEL 0 [CLS]...Hello. Everything [MASK] nice. I [MASK] playing [MASK] my dog [MASK] whole day... LABEL 0 [CLS]...Hello. [MASK] is nice. [MASK] was playing [MASK] [MASK] dog the whole day... LABEL 0 [CLS]...At the end of the day , the administration is handling it roughly right... [CLS]...No matter what you think, you did come up with something unique... [CLS]...At the [MASK] of the day , the administration is handling [MASK] roughly right... [CLS]...At the end of [MASK] day , the administration is [MASK] it roughly right... [CLS]...At [MASK] end of the day , the administration is handling it [MASK] right... [CLS]...No [MASK] what you think, you did [MASK] up with something unique... [CLS]...No matter what you [MASK], you did come up [MASK] something unique... [CLS]...No matter [MASK] you think, you did come up with [MASK] unique...Labeled Data
[MASK] Data Augmentation with
high replacement probability
Unlabeled Data
GOLD
Teacher
Augmented Data
Augmented Data
MASKED
Teacher
Assign soft labels
to
Assign soft labels
to
Calculate joint soft labels
on
to produce
.
Combine
and
for
Student Model training
Student
Model
Repeat
iterations
[MASK] Data Augmentation with
low replacement probability
Table 1 :
1Statistics of DBDC5 English and Japanese datasets.LIF
Train/Dev/Test-I/Test-II/Test-III
#instances 126,632/5,861/5,992/5,247/2,685
#unlabeled 101,448
Table 2 :
2Statistics of LIF dataset.
Table 3 :
3Experimental results on the DBDC5 English and Japanese track. ↓ the lower the better. # subscript denotes the base version of PLM.Test-I
Test-II
Test-III
Models
V-P/-R/-F1/Macro F1 V-P/-R/-F1/Macro F1 V-P/-R/-F1/Macro F1
Three-way AP
74.4/75.7/75.0/81.4
89.0/75.7/81.8/86.2
81.9/75.7/78.7/65.0
PLM Baseline
75.6/85.4/80.2/84.9
88.2/85.4/86.8/89.6
84.2/85.4/84.8/72.0
PLM+CoAtt
76.6/80.6/78.5/83.9
87.7/80.6/84.0/87.6
85.8/80.6/83.1/71.8
UDA
79.2/83.9/81.5/86.1
91.4/83.9/87.5/90.3
85.6/83.9/84.7/73.2
MixText
74.8/86.6/80.3/84.8
87.8/86.6/87.2/89.9
83.4/86.6/85.0/71.5
Ours (XU , no AU )
78.0/83.9/80.9/85.6
89.5/83.9/86.6/89.6
85.9/83.9/84.9/73.5
Ours (4.74% )
73.9/83.3/78.3/83.5
87.5/83.3/85.4/88.6
82.6/83.3/83.0/69.0
Ours (25% )
74.6/86.9/80.3/84.8
88.7/86.9/87.8/90.4
82.4/86.9/84.6/70.1
Ours (50% )
77.6/87.3/82.1/86.4
89.3/87.3/88.3/90.8
85.5/87.3/86.4/74.8
Ours (100% )
78.1/86.6/82.2/86.5
90.6/86.6/88.6/91.0
85.0/86.6/85.8/73.8
As we use CoQA samples without modification, the samples do not include the cases where the candidate question is from other conversations, we suggest these samples still contribute to the generalization.3 Refer toHigashinaka et al. (2020) for details. 4 Refer toKundu, Lin, and Ng (2020) for details.
AcknowledgementsThis research is supported by the National Research Foundation, Singapore under its AI Singapore Programme (AISG Award No: AISG-RP-2018-007). The computational work for this article was partially performed on resources of the National Supercomputing Centre, Singapore (https://www.nscc.sg).
Z Allen-Zhu, Y Li, arXiv:2012.09816Towards Understanding Ensemble, Knowledge Distillation and Self-Distillation in Deep Learning. CoRR. Allen-Zhu, Z.; and Li, Y. 2020. Towards Understanding Ensemble, Knowledge Distillation and Self-Distillation in Deep Learning. CoRR, arXiv:2012.09816.
Mixtext: Linguistically-Informed Interpolation of Hidden Space for Semi-Supervised Text Classification. J Chen, Z Yang, Yang , D , ACL. Chen, J.; Yang, Z.; and Yang, D. 2020. Mixtext: Linguistically-Informed Interpolation of Hidden Space for Semi-Supervised Text Classification. In ACL, 2147-2157.
Co-Training for Domain Adaptation. M Chen, K Q Weinberger, J Blitzer, NIPS. Chen, M.; Weinberger, K. Q.; and Blitzer, J. 2011. Co- Training for Domain Adaptation. In NIPS, 2456-2464.
Marginalized Denoising Autoencoders for Domain Adaptation. M Chen, Z Xu, K Q Weinberger, F Sha, ICML. Chen, M.; Xu, Z.; Weinberger, K. Q.; and Sha, F. 2012. Marginalized Denoising Autoencoders for Domain Adap- tation. In ICML, 1627-1634.
QuAC: Question Answering in Context. E Choi, H He, M Iyyer, M Yatskar, W Yih, Y Choi, P Liang, L Zettlemoyer, EMNLP. Choi, E.; He, H.; Iyyer, M.; Yatskar, M.; Yih, W.-t.; Choi, Y.; Liang, P.; and Zettlemoyer, L. 2018. QuAC: Question Answering in Context. In EMNLP, 2174-2184.
Unsupervised Cross-lingual Representation Learning at Scale. A Conneau, K Khandelwal, N Goyal, V Chaudhary, G Wenzek, F Guzmán, E Grave, M Ott, L Zettlemoyer, V Stoyanov, ACL. Conneau, A.; Khandelwal, K.; Goyal, N.; Chaudhary, V.; Wenzek, G.; Guzmán, F.; Grave, E.; Ott, M.; Zettlemoyer, L.; and Stoyanov, V. 2020. Unsupervised Cross-lingual Repre- sentation Learning at Scale. In ACL, 8440-8451.
BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. J Devlin, M.-W Chang, K Lee, K Toutanova, NAACL. Devlin, J.; Chang, M.-W.; Lee, K.; and Toutanova, K. 2019. BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. In NAACL, 4171-4186.
Born Again Neural Networks. T Furlanello, Z Lipton, M Tschannen, L Itti, A Anandkumar, ICML. Furlanello, T.; Lipton, Z.; Tschannen, M.; Itti, L.; and Anand- kumar, A. 2018. Born Again Neural Networks. In ICML, 1607-1616.
Supervised Contrastive Learning for Pre-trained Language Model Fine-tuning. B Gunel, J Du, A Conneau, V Stoyanov, ICLR. Gunel, B.; Du, J.; Conneau, A.; and Stoyanov, V. 2021. Super- vised Contrastive Learning for Pre-trained Language Model Fine-tuning. In ICLR.
Adaptive Semi-supervised Learning for Cross-domain Sentiment Classification. R He, W S Lee, H T Ng, D Dahlmeier, EMNLP. He, R.; Lee, W. S.; Ng, H. T.; and Dahlmeier, D. 2018. Adap- tive Semi-supervised Learning for Cross-domain Sentiment Classification. In EMNLP, 3467-3476.
LSTM for Dialogue Breakdown Detection: Exploration of Different Model Types and Word Embeddings. M Hendriksen, A Leeuwenberg, M.-F Moens, IWSDS Workshop. Hendriksen, M.; Leeuwenberg, A.; and Moens, M.-F. 2019. LSTM for Dialogue Breakdown Detection: Exploration of Different Model Types and Word Embeddings. In IWSDS Workshop.
Overview of the Dialogue Breakdown Detection Challenge 4. R Higashinaka, L F Haro, B A Shawar, R E Banchs, K Funakoshi, M Inaba, Y Tsunomori, T Takahashi, J Sedoc, IWSDS Workshop. Higashinaka, R.; D'Haro, L. F.; Shawar, B. A.; Banchs, R. E.; Funakoshi, K.; Inaba, M.; Tsunomori, Y.; Takahashi, T.; and Sedoc, J. 2019. Overview of the Dialogue Breakdown Detec- tion Challenge 4. In IWSDS Workshop.
Overview of Dialogue Breakdown Detection Challenge 3. R Higashinaka, K Funakoshi, M Inaba, Y Tsunomori, T Takahashi, N Kaji, DSTC6 Workshop. Higashinaka, R.; Funakoshi, K.; Inaba, M.; Tsunomori, Y.; Takahashi, T.; and Kaji, N. 2017. Overview of Dialogue Breakdown Detection Challenge 3. In DSTC6 Workshop.
Overview of Dialogue Breakdown Detection Challenge 5. R Higashinaka, Y Tsunomori, T Takahashi, H Tsukahara, M Araki, J Sedoc, R E Banchs, L F Haro, IWSDS Workshop. Higashinaka, R.; Tsunomori, Y.; Takahashi, T.; Tsukahara, H.; Araki, M.; Sedoc, J.; Banchs, R. E.; and D'Haro, L. F. 2020. Overview of Dialogue Breakdown Detection Challenge 5. In IWSDS Workshop.
Instance Weighting for Domain Adaptation in NLP. J Jiang, C Zhai, ACL. Jiang, J.; and Zhai, C. 2007. Instance Weighting for Domain Adaptation in NLP. In ACL, 264-271.
Competitive Self-Trained Pronoun Interpretation. A Kehler, D Appelt, L Taylor, A Simma, NAACL. Kehler, A.; Appelt, D.; Taylor, L.; and Simma, A. 2004. Com- petitive Self-Trained Pronoun Interpretation. In NAACL, 33- 36.
Learning to Identify Follow-Up Questions in Conversational Question Answering. S Kundu, Q Lin, H T Ng, ACL. Kundu, S.; Lin, Q.; and Ng, H. T. 2020. Learning to Identify Follow-Up Questions in Conversational Question Answering. In ACL, 959-968.
Temporal Ensembling for Semi-Supervised Learning. S Laine, Aila , T , ICLR. Laine, S.; and Aila, T. 2017. Temporal Ensembling for Semi- Supervised Learning. In ICLR.
Learning to Self-Train for Semi-Supervised Few-Shot Classification. X Li, Q Sun, Y Liu, Q Zhou, S Zheng, T.-S Chua, B Schiele, NeurIPS. Li, X.; Sun, Q.; Liu, Y.; Zhou, Q.; Zheng, S.; Chua, T.-S.; and Schiele, B. 2019. Learning to Self-Train for Semi-Supervised Few-Shot Classification. In NeurIPS, 10276-10286.
A Co-Attentive Cross-Lingual Neural Model for Dialogue Breakdown Detection. Q Lin, S Kundu, H T Ng, COLING. Lin, Q.; Kundu, S.; and Ng, H. T. 2020. A Co-Attentive Cross- Lingual Neural Model for Dialogue Breakdown Detection. In COLING, 4201-4210.
Y Liu, M Ott, N Goyal, J Du, M Joshi, D Chen, O Levy, M Lewis, L Zettlemoyer, V Stoyanov, arXiv:1907.11692RoBERTa: A Robustly Optimized BERT Pretraining Approach. CoRR. Liu, Y.; Ott, M.; Goyal, N.; Du, J.; Joshi, M.; Chen, D.; Levy, O.; Lewis, M.; Zettlemoyer, L.; and Stoyanov, V. 2019. RoBERTa: A Robustly Optimized BERT Pretraining Ap- proach. CoRR, arXiv:1907.11692.
Noisy Self-Knowledge Distillation for Text Summarization. Y Liu, S Shen, M Lapata, NAACL. Liu, Y.; Shen, S.; and Lapata, M. 2021. Noisy Self- Knowledge Distillation for Text Summarization. In NAACL, 692-703.
Decoupled Weight Decay Regularization. I Loshchilov, F Hutter, ICLR. Loshchilov, I.; and Hutter, F. 2019. Decoupled Weight Decay Regularization. In ICLR.
Effective Self-Training for Parsing. D Mcclosky, E Charniak, Johnson , M , NAACL. McClosky, D.; Charniak, E.; and Johnson, M. 2006. Effective Self-Training for Parsing. In NAACL, 152-159.
Adversarial Training Methods for Semi-Supervised Text Classification. T Miyato, A M Dai, I Goodfellow, ICLR. Miyato, T.; Dai, A. M.; and Goodfellow, I. 2017. Adversarial Training Methods for Semi-Supervised Text Classification. In ICLR.
Self-Distillation Amplifies Regularization in Hilbert Space. H Mobahi, M Farajtabar, P Bartlett, NeurIPS. Mobahi, H.; Farajtabar, M.; and Bartlett, P. 2020. Self- Distillation Amplifies Regularization in Hilbert Space. In NeurIPS, 3351-3361.
SSMBA: Self-Supervised Manifold Based Data Augmentation for Improving Out-of-Domain Robustness. N Ng, K Cho, M Ghassemi, EMNLP. Ng, N.; Cho, K.; and Ghassemi, M. 2020. SSMBA: Self- Supervised Manifold Based Data Augmentation for Improv- ing Out-of-Domain Robustness. In EMNLP, 1268-1283.
Improving Dialogue Breakdown Detection with Semi-Supervised Learning. N Ng, M Ghassemi, N Thangarajan, J Pan, Q Guo, NeurIPS Workshop on Human in the Loop Dialogue Systems. Ng, N.; Ghassemi, M.; Thangarajan, N.; Pan, J.; and Guo, Q. 2020. Improving Dialogue Breakdown Detection with Semi-Supervised Learning. In NeurIPS Workshop on Human in the Loop Dialogue Systems.
Unsupervised Domain Adaptation of Language Models for Reading Comprehension. K Nishida, K Nishida, I Saito, H Asano, J Tomita, LREC. Nishida, K.; Nishida, K.; Saito, I.; Asano, H.; and Tomita, J. 2020. Unsupervised Domain Adaptation of Language Models for Reading Comprehension. In LREC, 5392-5399.
Realistic Evaluation of Deep Semi-Supervised Learning Algorithms. A Oliver, A Odena, C A Raffel, E D Cubuk, I Goodfellow, NeurIPS. Oliver, A.; Odena, A.; Raffel, C. A.; Cubuk, E. D.; and Goodfellow, I. 2018. Realistic Evaluation of Deep Semi- Supervised Learning Algorithms. In NeurIPS, 3239-3250.
CoQA: A Conversational Question Answering Challenge. S Reddy, D Chen, C D Manning, Reddy, S.; Chen, D.; and Manning, C. D. 2019. CoQA: A Conversational Question Answering Challenge. TACL, 249- 266.
Contextenriched Attentive Memory Network with Global and Local Encoding for Dialogue Breakdown Detection. S E Reed, H Lee, D Anguelov, C Szegedy, D Erhan, A Rabinovich, IWSDS Workshop. Dirafzoon, A.; and Anshu, A. 2019Reed, S. E.; Lee, H.; Anguelov, D.; Szegedy, C.; Erhan, D.; and Rabinovich, A. 2015. Training Deep Neural Networks on Noisy Labels with Bootstrapping. In ICLR Workshop. Shin, J.; Dirafzoon, A.; and Anshu, A. 2019. Context- enriched Attentive Memory Network with Global and Local Encoding for Dialogue Breakdown Detection. In IWSDS Workshop.
Dialogue breakdown detection using BERT with traditional dialogue features. H Sugiyama, IWSDS Workshop. Sugiyama, H. 2019. Dialogue breakdown detection using BERT with traditional dialogue features. In IWSDS Work- shop.
RSL19BD at DBDC4: Ensemble of Decision Tree-based and LSTM-based Models. C Wang, S Kato, T Sakai, IWSDS Workshop. Wang, C.-h.; Kato, S.; and Sakai, T. 2019. RSL19BD at DBDC4: Ensemble of Decision Tree-based and LSTM-based Models. In IWSDS Workshop.
EDA: Easy Data Augmentation Techniques for Boosting Performance on Text Classification Tasks. J Wei, K Zou, EMNLP. Wei, J.; and Zou, K. 2019. EDA: Easy Data Augmentation Techniques for Boosting Performance on Text Classification Tasks. In EMNLP, 6382-6388.
Unsupervised Data Augmentation for Consistency Training. Q Xie, Z Dai, E Hovy, T Luong, Q Le, NeurIPS. Xie, Q.; Dai, Z.; Hovy, E.; Luong, T.; and Le, Q. 2020. Un- supervised Data Augmentation for Consistency Training. In NeurIPS, 6256-6268.
Simple Data Augmentation with the Mask Token Improves Domain Adaptation for Dialog Act Tagging. S Yavuz, K Hashimoto, W Liu, N S Keskar, R Socher, C Xiong, EMNLP. Yavuz, S.; Hashimoto, K.; Liu, W.; Keskar, N. S.; Socher, R.; and Xiong, C. 2020. Simple Data Augmentation with the Mask Token Improves Domain Adaptation for Dialog Act Tagging. In EMNLP, 5083-5089.
. H Ye, Q Tan, R He, J Li, H T Ng, L Bing, Ye, H.; Tan, Q.; He, R.; Li, J.; Ng, H. T.; and Bing, L.
Feature Adaptation of Pre-Trained Language Models across Languages and Domains with Robust Self-Training. EMNLP. Feature Adaptation of Pre-Trained Language Models across Languages and Domains with Robust Self-Training. In EMNLP, 7386-7399.
Be Your Own Teacher: Improve the Performance of Convolutional Neural Networks via Self Distillation. L Zhang, J Song, A Gao, J Chen, C Bao, K Ma, ICCV. Zhang, L.; Song, J.; Gao, A.; Chen, J.; Bao, C.; and Ma, K. 2019. Be Your Own Teacher: Improve the Performance of Convolutional Neural Networks via Self Distillation. In ICCV, 3713-3722.
| [
"https://github.com/nusnlp/S2T2."
] |
[
"Interacting two-particle states in the symmetric phase of the chiral Nambu-Jona-Lasinio model",
"Interacting two-particle states in the symmetric phase of the chiral Nambu-Jona-Lasinio model"
] | [
"A Jakovác \nInstitute of Physics\nEötvös University\nH-1117BudapestHungary\n",
"A Patkós \nInstitute of Physics\nEötvös University\nH-1117BudapestHungary\n"
] | [
"Institute of Physics\nEötvös University\nH-1117BudapestHungary",
"Institute of Physics\nEötvös University\nH-1117BudapestHungary"
] | [] | The renormalisation group flow of the chiral Nambu-Jona-Lasinio (NJL) model with one fermion flavor is mapped out in the symmetric phase with the help of the Functional Renormalisation Group (FRG) method using a physically motivated non-local trial effective action. The well-known infrared unstable strongly coupled fixed point characterized by a set of pointlike four-fermion couplings is reproduced. The Gaussian infrared end-point of the flow of the four-fermion couplings is now accompanied by non-zero limiting composite couplings characteristic for interacting two-particle states with finite energy and physical size. The negative interaction energy of the constituents is extracted as a function of the physical size of the composite object. This function reaches a minimum in the accessible range of physical sizes, mildly depending on the set of initial values of the couplings. The propagation of a two-particle state minimizing the interaction energy has a natural bound state interpretation.Here the function f stands for the bound state-constituent vertex (the Bethe-Salpeter wave function), while D k is the propagator of the bound state composed with the two constituent fields. One expects the solution of f to display essential momentum dependence (e.g. non-local features) and D k to produce the resonance pole when the scale pushed to the infrared limit (k → 0). A next step was the proposition for introducing a field σ representing the bound state into the effective arXiv:1902.06492v1 [hep-th] | 10.1142/s0217732320501308 | [
"https://arxiv.org/pdf/1902.06492v1.pdf"
] | 119,062,722 | 1902.06492 | 30ea9c8496b9ab95f0324418e9befcb3a91244bf |
Interacting two-particle states in the symmetric phase of the chiral Nambu-Jona-Lasinio model
(Dated: February 19, 2019)
A Jakovác
Institute of Physics
Eötvös University
H-1117BudapestHungary
A Patkós
Institute of Physics
Eötvös University
H-1117BudapestHungary
Interacting two-particle states in the symmetric phase of the chiral Nambu-Jona-Lasinio model
(Dated: February 19, 2019)
The renormalisation group flow of the chiral Nambu-Jona-Lasinio (NJL) model with one fermion flavor is mapped out in the symmetric phase with the help of the Functional Renormalisation Group (FRG) method using a physically motivated non-local trial effective action. The well-known infrared unstable strongly coupled fixed point characterized by a set of pointlike four-fermion couplings is reproduced. The Gaussian infrared end-point of the flow of the four-fermion couplings is now accompanied by non-zero limiting composite couplings characteristic for interacting two-particle states with finite energy and physical size. The negative interaction energy of the constituents is extracted as a function of the physical size of the composite object. This function reaches a minimum in the accessible range of physical sizes, mildly depending on the set of initial values of the couplings. The propagation of a two-particle state minimizing the interaction energy has a natural bound state interpretation.Here the function f stands for the bound state-constituent vertex (the Bethe-Salpeter wave function), while D k is the propagator of the bound state composed with the two constituent fields. One expects the solution of f to display essential momentum dependence (e.g. non-local features) and D k to produce the resonance pole when the scale pushed to the infrared limit (k → 0). A next step was the proposition for introducing a field σ representing the bound state into the effective arXiv:1902.06492v1 [hep-th]
I. INTRODUCTION
Determination of bound state spectra is of central importance in theories of fundamental interactions. The possible bound state nature of the Higgs-field is one of the actual hot questions of the physics beyond the standard model [1]. Also, low energy observables of the physics of strong interactions refer exclusively to bound states, where lattice field theory represents a most successful approach to the first principle determination of the QCD spectra [2,3]. Considerable progress has been achieved also with this technique in the non-perturbative investigation of composite Higgs models [4].
The Bethe-Salpeter integral equation (BSE) [5,6] is the classic "tool" of quantum field theory in solving the relativistic bound state problem. Its success depends critically on the quality of the kernel-function. Ladder-type resummations of some perturbative kernel, involving the exchange of a single force field quantum, possibly completed by additional exchange of two quanta in the crossed (t-or u-) channels, are the most frequently applied approaches. Finding the appropriate interaction vertex of the constituent field with the force field represents a further challenge. Despite of these difficulties, starting from the 1990-ies a rather systematic exploration of the mesonic and barionic bound states has been realized within the BSE-framework combined with Dyson-Schwinger equations for the quark propagators [7][8][9][10].
At about the same time was also initiated the application of the non-perturbative FRG-approach [11,12] to the determination of bound state excitations of quantum field theories. In Ref. [13] the renormalisation group equation (RGE) of the scale-dependent four-point function Γ (4) k has been considered. In the framework of the Wetterich-equation [11] one writes for its evolution with the scale k ∂ k Γ (4) k (p 1 , p 2 , p 3 , p 4 ) = − q Γ (4) k (p 1 , p 2 , q, −q − p 1 − p 2 ) ×∂ k G k ((p 1 + p 2 + q) 2 )G k (q 2 ) Γ (4) k (−q, q + p 1 + p 2 , p 3 , p 4 ).
(1)
Here on the right hand side only the partial k-derivative of the constituent propagators G k (k) is taken through the k-dependence of its regulator. For the bound state solution of this equation a factorized ansatz has been proposed, whose formal equivalence to the BSE has been demonstrated: Γ (4) k (p 1 , p 2 , p 3 , p 4 ) = f (p 2 1 + p 2 2 , p 2 1 − p 2 2 )D k ((p 1 + p 2 ) 2 )f (p 2 3 + p 2 4 , p 2 3 − p 2 4 ). (2) action by adding a quadratic form to Γ k [14]:
Γ k [ϕ, σ] = Γ k [ϕ] + 1 2 O † G O − σ † O + 1 2 σ † G −1 σ.(3)
The field σ couples via Yukawa-type convolution (symbolically denoted by ) with the nonlocal field O(q), formed by the constituent fields ϕ:
O(q) = p1 p2 g(p 1 , p 2 )ϕ † (−p 1 )ϕ(p 2 )(2π) 4 δ(q − p 1 − p 2 ).(4)
The strategic goal is to find the non-local Yukawa-coupling g(p 1 , p 2 ) and the propagator of the composite fieldG from the requirement that the quadratic form 1 2 O † G O maximally cancels the resonant piece developing in Γ k when reaching the scale characteristic for the bound state.
A rather practical "recipe" for the realisation of this strategy has been put forward in 2002 [15]. A composite field was introduced into the original action by replacing the pointlike four-fermion interaction by a pointlike Yukawa interaction with the bound state field using a local Hubbard-Stratonovich (HS-) transformation [16,17]. On the example of a gauged chiral NJL model with one fermion species the authors have demonstrated the existence of two (partial) fixed points connected along a one dimensional "renormalized trajectory" following the variation of the effective four-fermion coupling. The neighbourhood of the infrared unstable fixed point represents a strongly coupled interactive theory, which describes via the HS-transformation the dynamics of an elementary scalar field. The other, infrared stable fixed point corresponds to the electrodynamics of massless fermions. It was conjectured that during its evolution towards the infrared the scalar field transmutates into the representative of the positronium-like fermion-antifermion bound state. Although this last conjecture has not been confirmed explicitly, since then the extension of any effective theory with fields representing composite (bound) objects with appropriate quantum numbers has become a common usage. The generalisation of the local HS-transformation to larger sets of composite fields has been worked out [18,19] and important steps were made towards the application of this FRG-technique to n-point functions of QCD [20].
The basic steps of the procedure called "dynamic hadronisation" [21][22][23] can be summarized as follows: The composite local field is constructed from the constituent fields with an arbitrary three-variable weight function. Its scale dependence is deduced from the "no-run" requirement enforced on the 4-fermion couplings. Therefore the three-point function has no direct contact to the Bethe-Salpeter wave function of the two-particle composite. The usual initial condition for the RG-equations consists of setting 4fermion couplings equal to zero at the cut-off. Is this physical? Can one start at a scale where already no individual fermions, but only correlated fermion pairs do interact? Although the mapping of the 4fermion theory on the Yukawa-type theory with such a supplementary requirement is in principle exact, but its adequacy is questionable when only a subset of the operators is included into the renormalisation flow.
Another sort of intriguing question is related to the set of composite fields. Can one rely on a unique field representing the complete spectra in a given channel? A careful detailed FRG-analysis of the full s-channel spectra provides evidence for a negative answer to this question, using the example of QED [24]. What concerns strong interactions, the lowest meson masses might be satisfactorily fitted within a quark-meson effective model with pointlike Yukawa terms [25][26][27]. Despite of this fact, at present we do not know if the s-channel singularity structure of the 4-fermion function consistently reproduces them.
We propose to return to the investigation of non-local extensions of the effective action [14]. We follow a slightly different strategy and shall concentrate on the renormalisation flow in an extended coupling space towards the infrared. The set of couplings contains those which one suspects intuitively to signal bound state formation. In particular, the scale evolution of the composite-2fermion vertex function (Yukawacoupling) will be determined dynamically, not by a constraint equation. Since the resonant state cannot exhaust the spectral function of a certain channel, we trace also the evolution of the pointlike 4-fermion vertices towards zero, as expected from the simplified RG-analysis. They fade away in the neighbourhood of the IR fixed point.
The numerical solution starts in the ultraviolet regime (much above the characteristic momentum scale of the composite state) from the neighbourhood of the ultraviolet fixed point [28] characterized by a nonvanishing set of 4-fermion couplings. It will be argued that along some unstable directions the system enters the regime with non-zero Yukawa coupling describing the association of two fermions into a bosonic composite and also a non-zero pointlike 4-point function of the composites.
One reaches very quickly a scaling regime for the dimensionful couplings. One finds that the Gaussian fixed point is the only IR stable fixed point in the subspace of the pointlike fermionic couplings as it is the case of the 4-fermion NJL theory [28]. In particular, the strength of the constituent-composite 3-point function tends logarithmically to zero as well as the four-boson coupling. The truly new feature is that along the trajectories passing in the subspace of the dimensionful composite couplings one observes the existence of a sequence of limiting finite masses and composite object sizes, corresponding to interacting two-particle states of different physical size and well defined interaction energy. The structure of this paper is the following. In section 2 we extend the effective action of the model with the contributions reflecting the presence of two-particle states composed from two fermions. Via a non-local Hubbard-Stratonovich transformation we arrive at the form of the effective action for which the RG-equations are formulated in section 3. In section 4 the features of the RG flow are analyzed numerically and interpreted in favor of the existence of a bound state in the symmetric phase.
II. THE MODEL AND ITS PHYSICALLY MOTIVATED EFFECTIVE ACTION ANSATZ
The Fierz complete definining action of the one-flavor chiral NJL-model is the following:
Γ N JL = d 4 x iψγ m ∂ m ψ + 2λ σ ψ L ψ R ψ R ψ L − 1 2 λ V (ψγ m ψ) 2 .(5)
We shall investigate the emergence of a bound state in the scalar-pseudoscalar channel by explicitly introducing into the scale dependent effective action a four-point term which describes the propagation of a non-local composite two-particle operator in the s-channel. One still keeps non-resonating pointlike four-fermion interactions with strength δλ σ , λ V :
Γ N L−N JL = d 4 x ψ iγ m ∂ m ψ − 1 2 λ V (ψγ m ψ) 2 + 2δλ σ (ψ R ψ L )(ψ L ψ R ) + 2 d 4 x d 4 y d 4 x 1 d 4 x 2 d 4 y 1 d 4 y 2 ×ψ R (x 1 )ψ L (x 2 )∆ C (x − x 1 , x − x 2 )G C (x − y)∆ C (y − y 1 , y − y 2 )ψ L (y 1 )ψ R (y 2 ).(6)
The binding function ∆ C (x − x 1 , x − x 2 ) propagates the U A (1) × U V (1) (globally) invariant fermion and anti-fermion elementary fields from the location (x 1 , x 2 ) to the location x of the composite. The propagation of the composite field is accounted by the propagator G C (x − y). Both functions have to be determined dynamically.
One can easily construct a non-local Hubbard-Stratonovich transform introducing the auxiliary scalar Φ S (x) and pseudoscalar Φ 5 (x) fields with the following correspondence to nonlocal composites:
Φ S (y) ↔ i dx dx 1 dx 2 G C (y − x)∆ C (x − x 1 , x − x 2 )ψ(x 1 )ψ(x 2 ), Φ 5 (y) ↔ dx dx 1 dx 2 G C (y − x)∆ C (x − x 1 , x − x 2 )ψ(x 1 )γ 5 ψ(x 2 ).(7)
If one also adds to the effective action a local potential term which is fourth power in the composite fields, then one has
Γ N L−N JL = d 4 x ψ iγ m ∂ m ψ − 1 2 λ V (ψγ m ψ) 2 + 2δλ σ (ψ R ψ L )(ψ L ψ) + λ 24 Φ 2 S (x) + Φ 2 5 (x) 2 + 1 2 dx dy Φ S (x)Γ (2) C (x − y)Φ S (y) + Φ 5 (x)Γ (2) C (x − y)Φ 5 (y) − i dx dx 1 dx 2 ∆ C (x − x 1 , x − x 2 ) Φ S (x)ψ(x 1 )ψ(x 2 ) − iΦ 5 (x)ψ(x 1 )γ 5 ψ(x 2 ) .(8)
The two-point function of Φ S and Φ 5 is the same inverse function of the chiral propagator:
dzG C (x − y)Γ (2) C (y − z) = δ(x − z).(9)
In the symmetric phase the scalar and the pseudoscalar spectra are degenerate. We shall use a simplified one-particle form:
Γ (2) S (q) = Z C q 2 + M 2 C = Γ(2)5 (q).(10)
By the above construction this analytic structure shows up also in the four-fermion function. If the RGequations display a set of solutions characterized by a continuous variation of the limiting IR-value of M 2 C then this ansatz corresponds rather to a certain (imperfect) description of the two-particle continuum, than to some real bound state. A bound state might be signaled by the existence of well distinguishable minima in the effective squared "mass". The concept will be elaborated more accurately below in the main text. The other quantity which needs explicit modeling is the 3-point bosonic composite-fermion-fermion function. The simplest parametrisation used in a first investigation is to assume a Gaussian shape both in the center-of-mass momentum of the composite and in the relative momentum of the two fermion constituents:
∆ C (q 1 , q 2 ) = g C e −β(q1+q2−Q) 2 e −α(q1−q2) 2 ,(11)
with scale dependent parameters g C , β, α. For the sake of reducing the dimension of the coupling space the widths of the two Gaussians are assumed to be equal: β = α. The parameter α can be interpreted as the square of the physical size of the two-particle state, α ∼ R 2 phys . Its scale dependence will be investigated with RG-equations. It will be shown that for a decreasing series of its initial values defined at the maximum scale Λ, for each of them a well-defined limiting behavior is obtained in the infrared. The IR-value of the squared propagator mass M 2 C depends on the infrared size of the system and on one additional renormalisation condition to be specified in section IV. Since it varies also continuously, one is tempted to conclude that our trial effective action actually represents a continuous set of two-particle states. The question is if some specific characterisation could still suggest bound state formation. There an affirmative answer will be claimed to this question.
In the next section, the Wetterich-equtions will be deduced for the couplings δλ σ , λ V , g C , α, M C , λ with a general regulator function.
III. THE RENORMALISATION GROUP EQUATIONS
It is convenient to split the right hand side of the Wetterich equation [11,12] into three pieces:
∂ t Γ = −Tr log Γ (2) F + 1 2
Tr log Γ
(2) B + 1 2 Tr log(I − G B Γ (2) BF G F Γ (2) F B ).(12)
where Γ (2) refers to the second functional derivative matrix of the effective action, Γ
B/F represents the purely bosonic/fermionic derivative matrix, while the mixed matrices are denoted by Γ (2) BF or Γ (2) F B . The variable t = ln(k/Λ) relates the actual scale k to the initial (cut-off) scale Λ.
The first term on the right hand side can be rewritten with the help of the massless fermion propagator G
(0) ψ : −Tr log Γ (2) F = −Tr log Γ (2) F (m ψ = 0) − Tr log(I + G (0) ψ ∆Γψ ψ ).(13)
The quantity ∆Γψ ψ is independent of the fermi-fields, and depends linearly on the composite fields. Therefore its second derivative contributes a fermion-bubble to the RGE of the composite two-point function and its fourth derivative to the running of the pointlike quartic composite coupling (the fermion quadrangle). The purely bosonic contribution on the right hand side of (12) has the following explicit form on constant composite field background:
1 2
Tr log Γ
(2) B = 1 2 q log Γ S + λ 2 Φ 2 S + λ 6 Φ 2 5 Γ 5 + λ 2 Φ 2 5 + λ 6 Φ 2 S − λ 2 9 Φ 2 S .(14)
The second derivative with respect to Φ S gives a tadpole contribution to the RGE of M 2 C . Since the tadpole is momentum independent, it does not contribute to the field renormalisation. Therefore there is no need to consider the pure bosonic term in non-constant background. The fourth derivative gives the composite one-loop contribution to the running of itself.
The RG-equation of the composite scalar two-point function is as follows:
∂ t Γ (2) S (P ) = q −∆ C (q, −q − P )∆ C (−P + q, q)∂ t Tr D G (0) ψ (q)G (0) ψ (q − P ) + λ 2∂ t G S (q) + 1 3 G 5 (q) .
(15) The RG-equation of M 2 C is arrived by setting P = 0 on both sides, while η C = −∂ t ln Z C is found by taking first the second derivative ∂ 2 Γ
(2) S (P )/∂P 2 m and setting P = 0 in the derivative. The RGE of λ is the sum of the pure composite and the pure fermion loop:
∂ t λ = 24 q∂ t 1 Z 4 ψ (1 + r F (q)) 4 q 4 ∆ 2 C (q, −q)∆ 2 C (−q, q) − 3λ 2 2 q∂ t G 2 S (q) + 1 9 G 2 5 (q) .(16)
Since one needs the anomalous scaling of the fermion propagator, also the expression of the fermionic self-energy contribution is found from the third term of (12), when the logarithm is expanded to linear power:
∂ t Γ (2) ψ (P ) = 1 2 q ∆ C (P, q − P )∆ C (−q + P, −P )∂ t (G S (q) + G 5 (q))G (0) ψ (q − P ) .(17)
There are two contributions to the composite boson -fermion -fermion three-point function. In the first the fermion legs interact via t-channel exchange of the composite field itself, in the second a fermion loop is generated by taking into account the four-fermion interactions:
∂ t Γ (3) ψψS (P 1 , P 2 , P 3 ) = 1 8 q ∆ C (P 1 , q − P 1 )∆ C (P 1 − q, q + P 2 )∆ C (−q − P 2 , P 2 )∂ t tr D G (0) ψ (q − P 1 )G (0) ψ (q + P 2 ) (G 5 (q) − G S (q)) + (δλ σ + λ V ) q ∆ C (−P 3 − q, q)∂ t tr D G (0) ψ (q)G (0) ψ (q + P 3 ) .(18)
The RG-equation of the strength g C is determined by setting all external momenta zero. The running of the width α of the composite "wave function" is determined first setting P 1 = P 2 = −P 3 /2 ≡ −P/2, next taking the second derivative with respect to P m on both sides and eventually setting P = 0. Finally, we need the RG-equations of the two-types of pointlike four-fermion couplings. One has three contributions all coming from the third term on the right hand side of (12). The exchange of two composite mesons gives
− 1 16∂ t q ∆ 2 C (q, 0)∆ 2 C (−q, 0) Z 2 ψ (1 + r F (q)) 2 q 2 [(G 2 S (q) + G 2 5 (q))(ψγ m ψ) 2 + 2G s (q)G 5 (q)(ψγ m γ 5 ψ) 2 ] = − 1 16∂ t q ∆ 2 C (q, 0)∆ 2 C (−q, 0) Z 2 ψ (1 + r F (q)) 2 q 2 [(G s (q) + G 5 (q)) 2 (ψγ m ψ) 2 + 4G S (q)G 5 (q)((ψψ) 2 − (ψγ 5 ψ) 2 )].(19)
The second equality is arrived at after exploiting Fierz-identity (ψγ m γ 5 ψ) 2 = 2((ψψ) 2 − (ψγ 5 ψ) 2 ) + (ψγ m ψ) 2 . The second contribution comes from combining the exchange of one composite boson and the pointlike (non-resonant) four-fermion interaction:
1 4∂ t q ∆ C (0, q)∆ C (0, −q) Z 2 ψ (1 + r F (q)) 2 q 2 × ((ψψ) 2 + (ψγ 5 ψ) 2 )(G S (q) − G 5 (q))(−2δλ σ + λ V ) + ((ψψ) 2 − (ψγ 5 ψ) 2 )(G S (q) + G 5 (q))(−δλ σ + λ V ) − (ψγ m ψ) 2 (G S (q) + G 5 (q)) (δλ σ + λ V ) .(20)
It is clear that the explicit symmetry breaking in the boson propagators would induce symmetry breaking in the four-fermion couplings. In the following we shall substitute here and in all RG-equations G s (q) = G 5 (q).
The third contribution comes from the fermion bubble of the nonresonant fermion-fermion scattering which formally coincides with FRG contributions of the original NJL model:
−∂ t q 2 Z 2 ψ (1 + r F (q)) 2 q 2 (ψψ) 2 − (ψγ 5 ψ) 2 )(δλ 2 σ + 4δλ σ λ V + 3λ 2 V ) + 1 2 (ψγ m ψ) 2 (λ 2 V + 2δλ σ λ V + δλ 2 σ ) .
(21) This expression leads to the RGE of the NJL theory with Fierz-complete pointlike couplings [28].
The explicit equations of the RG-flow with specific regulator function choice appear in the Appendix.
IV. RG-FLOW NEAR THE FIXED POINT SOLUTIONS
The extended set of RG-equations in the Appendix admits the same two fixed points like the 4-fermion model with pointlike couplings. Below we shall characterize the behavior of complete set of couplings in their respective neighbourhood (for the definitions of the dimensionless couplings, see (40) in the Appendix).
The fixed points of the Fierz-complete chiral symmetric NJL-model (with all bosonic couplings set zero) are well-known [28,30]. The flow-equations for the rescaled dimensionless 4-fermion couplings (see Appendix) read as
∂ t δλ σr = 2δλ σr − 1 8π 2 δλ 2 σr + 4δλ σr λ V r + 3λ 2 V r , ∂ t λ V r = 2λ V r − 1 16π 2 (δλ σr + λ V r ) 2(22)
have three fixed points: the IR-attractive trivial
(δλ * σr , λ * V r ) = (0, 0)(23)
the partially UV-repulsive, strongly interacting and stable (δλ * σr , λ * V r ) = (6π 2 , 2π 2 ), (24) and the partially UV-repulsive, strongly interacting and unstable (δλ * σr , λ * V r ) = (−64π 2 , 32π 2 ).
The Gaussian fixed point is absolute infrared attractive, the other two have one attractive and one repulsive direction.
A. The strongly coupled IR-unstable fixed point
In the framework of the extended effective action we investigate the behaviour of the composite bosonic couplings around the fixed point (24). Since the anomalous dimensions η C , η ψ are both proportional tõ g 2 Cr they can be neglected in the equation ofg 2 Cr , which arises from linearizing (43) around the fermionic fixed point:
∂ tg 2 Cr ≈ −(δλ * σr + λ * V r ) 2F 1 (0) πg 2 Cr = −4πg 2 Cr ,(26)
with the solutiong
2 Cr k Λ 4π = 1(27)
(in the stability analysis we set α r = 0). This means that the Yukawa-coupling is IR-relevant around this fixed point. One notes that as a (small) non-zero initial value of the Yukawa coupling starts to grow around this fixed point, it induces an increase of µ 2 C . The linear part of (45) can be written in this regime as
∂ t µ 2 C = −2µ 2 C + 1 4π 2g 2 Cr(28)
which in addition to the solution of the homogeneous equation suggests the following proportionality
µ (inh)2 C (t) = Ag 2 Cr (t).(29)
With the coefficient
A = 1 8π 2 (1 − 2π)(30)
one finds for the solution
µ 2 C (t) = µ 2 C (t = 0) 1 + µ −2 C (t = 0) 8π 2 (1 − 2π) −1 k Λ −2 + µ −2 C (t = 0) 8π 2 (1 − 2π) k Λ −4π .(31)
The four-boson coupling stays zero at linear order (see (46)!). The width of the bosonic wave-function follows the leading order equation (the second equation of (43))
∂ t (g Cr α r ) = 2g Cr α r −g Cr 2 ,(32)
which implies that g Cr α r ∼ Bg Cr , if it starts with zero at t = 0. After substitution one finds
α r = 1 4(1 + π) ,(33)
which means in physical units a quadratically increasing value with k. The scaled value is rather small, confirming the selfconsistency of evaluating F n (#α r ), H n (#α r ) at α r = 0. One can conclude that the fixed point (24) is fully IR-unstable also along the directions of the composite couplings.
B. Analytic features of the mass-distorted Gaussian fixed point
Assuming first η C , η ψ ≈ 0, one instantly finds that the irrelevant nature of the dimensionless pointlike 4-fermion couplings δλ σr and λ V r is unchanged, and similarly α r → 0 for k → 0. The rescaled µ 2 C increases as ∼ k −2 and its physical value M 2 C remains close to its initial value, only slightly modified due to the interactions.
In the neigbourhood of the Gaussian fixed point, when the system is driven below the mass scales the RG-flow ofg 2 Cr is understood with the help of the approximate expression of η C (η ψ becomes very quickly negligible and α r ≈ 0 is also a very good approximation):
∂ tg 2 Cr ≈ η Cg 2 Cr , η C ≈g 2 Cr 4π 2 ,(34)
which leads to the following IR-behavior when starting the solution at k = k 0 :
g 2 Cr (k) =g 2 Cr (k 0 ) 1 + g 2 Cr (k0) 4π 2 ln k0 k .(35)
It is logarithmically approaching zero, independently of the initial coupling. We could check that the variation ofg 2 Cr found numerically (see next subsection) indeed follows this behavior and alsog 2 Cr (t) is proportional to η C (t). In this way, the anomalous fermionic and bosonic dimensions are both negligible in the immediate neighborhood of the fixed point.
The asymptotic flow of λ r is governed by the following approximate equation, after the composite bubble diagram is stopped to contribute by the increase of µ 2
C ∂ t λ r ≈g 2 Cr 2π 2 λ r − 3g 4 Cr π 2 .(36)
The second term is negligible for |t| → ∞ and substituting the extreme asymptotic from of (35) one arrives at
λ r (t) ≈ |t| −2 .(37)
C. Detailed numerical study of the flow near the Gaussian fixed point
The RG flow was studied around the Gaussian fixed point numerically. Our intention was to set the initial values of the couplings at "macroscopic" distance from the Gaussian fixed point, though much closer than to the strongly coupled fixed point. A major question arises concerning the number of freely allowed initial choices among the dimensionful parameters.
The starting value of α(t = 0) ∼ R 2 phys (t = 0) controls the initial size of the system. Very large values (e.g. α r = 100 − 1000) correspond essentially to uniform spatial distribution of the constituents. In this limiting case the effect of the quantum fluctuations on M 2 C should be interpreted as the sum of the self-energies of the separate constituents. One expects that the initial contributions (from the far ultraviolet) should follow the same t-dependence when α r (0) is lowered. This is expressed by the following renormalisation condition:
dM 2 C dt (α (1) r (0), µ 2(1) C (0)) = dM 2 C dt (α (2) r (0), µ 2(2) C (0)).(38)
Whenever one fixes the first pair containing a large value of α (1) r (0) then for the next (lower) α (2) r (0) value the corresponding initial value of µ 2(2) C should be found by solving the above equation. This procedure seems to be somewhat arbitrary. We argue that through this renormalisation condition we are able to introduce some physical features of the two-particle state which cannot be built in explicitly into the assumed oversimplified form of the composite two-point function.
A generic "reference" run has been started with the initial conditions α r (0) = 100, µ 2 C (0) = 0.5, λ r (0) = 10, g 2 Cr (0) = 10, Q 2 r (0) = 1, δλ σr (0) = 1, λ V r (0) = 1. Keeping all other initial values fixed, a lowered choice α r (0) = 1.5 requires by (38) µ 2 C (0) = 0.37, meaning that the initial value of µ 2 C varies rather slowly under the variation of the initial size. For each set we start at t = 0 and follow the evolution to t = −10 (restricted by the numerical stability of the applied Mathematica procedure). This is more than sufficient to recognize the asymptotic tendencies in all couplings for k → 0.
The features of the RG-flow for the representative set of the initial parameters can be summarized as follows.
• First peek in the dimensionless quantities. If we plot the rescaled Yukawa coupling Z Cg 2 Cr after a transient increase a clear constant asymptotics is seen (see the left figure of FIG.1). This is consistent with and indirectly checks numerically the validity of both asymptotic equations in (34). Concerning the scalar self-coupling λ r , we can observe only a very slow variation (see the right figure of FIG.1), making hard to draw any quantitative conclusion on its compatibility with the limiting behavior (37).
• The anomalous dimension of the composite scalar field converges to zero as displayed in FIG.2 and in quantitative agreement with the expectation based on the estimated asymptotics in (34). The anomalous dimension of the fermi-field stays very small from the very start of the RG-evolution.
• The 4-fermion couplings are, by their physical dimension, irrelevant in the IR around the Gaussian fixed point. By rescaling with the canonical dimension we find an almost constant behavior (see FIG. 3). We remark that starting with too large values for these couplings would drive the solution of the RG-equations to instability.
• Other dimensionful quantities are M 2 C and α. We plot in FIG. 4 the physical value of M 2 C and the dimensionless combination αM 2 C . As it can be seen from the figures, both quantities converge very quickly to their asymptotic nonzero values. If one fixes the value of M 2 C (k = 0) to some value in GeV 2 then the physical size R 2 phys (k = 0) is determined by the asymptotics as a well-defined value in GeV −2 .
• Next, we explore the behavior of the effect of quantum fluctuations on the squared mass parameter by displaying δM 2 C (α(k = Λ)) ≡ M 2 C (k = 0, α(k = Λ)) − M 2 C (k = Λ, α(k = Λ)) as a function of t for gradually decreasing values of α(k = Λ) (accompanied by the corresponding choice of µ 2 C (0)). In FIG. 5 one notices the equal slope starting rise (required by the renormalisation condition) is followed by a drop which gets apparently larger as α r (k = Λ) is lowered. The decrease in δM 2 C (α(k = Λ) relative to the case of the completely unbound particles (α r (k = Λ) ≈ ∞) lends itself to a physical identification with an attractive interaction energy.
• One can display the interaction energies ∆M 2 C ≡ δM 2 C (α(k = Λ)) − δM 2 C (α(k = Λ) = ∞) and the squared physical size α(t = −∞) as functions of the initial squared physical size α(k = Λ). By the previous figure a sequence of monotonically increasing (less and less negative) interaction energy function is expected with increasing α(t = 0). But, in the range 2.5 > α(t = 0) > 1.0 an opposite tendency takes over (see FIG. 6), leading to a minimum in the dependence of the interaction energy on the initial size of the two-particle composite. Similar behaviour is observed in α(t = −∞) as a function of the initial size. In the figure on the right one notices that quantum fluctuations diminish the final size for large initial values, but it gets larger than the initial value if one "squeezes" it beyond a certain size. The minimum of the final size qualitatively coincides with the maximum of the negative interaction energy. When states are compared along a different initial slope dM 2 C /dt the locations of the minima are shifted in a correlated way and also the minimal values change mildly. The most natural conclusion is that the composite which corresponds to the minimum It is worthwhile to observe the following mapping among the "potentials" derived with different starting dM 2 C /dt. In the left figure of FIG.7 a set of the potentials is shown for which this derivative has been fixed by choosing at α(t = 0) = 1000 the values µ 2 C (t = 0) = 0.08, 0.1, 0.2, ..., 1.2, respectively. Using (38) for each of them one finds the corresponding α(t = 0), µ 2 C (t = 0) values and in the same steps as for FIG.6 one constructs the interaction energy vs. initial squared size curve. The smooth deformation due to the change of the initial data allows to map the potentials onto each other (figure on the right). The multiplicative scaling of ∆M 2 C compensates is fixed by the requirement ∆M 2 Cmin /C = −1. Then the coefficients of the linear mapping of the α-scale α → A 1 α + A 2 are determined completely by fixing the location of the maximally negative interaction energy to A 1 α min + A 2 = 1, which leads to
A i = q 1i C + q 2i + q 3i √ C.(39)
In this way the size of the two-particle system with maximally negative interaction energy is determined by the maximal interaction energy (binding energy) itself. The whole curve is now parametrized with the help of |∆M 2 C | max .
V. CONCLUSIONS
On the basis of the stability investigation around the UV and the IR fixed point one might attempt to sketch the global RG-flow. If one starts from the neighbourhood of the strongly coupled stable fixed point (24) one enters unavoidably into the subspace of composite couplings and at the same time the projection of the RG-flow into the subspace of local 4-fermion couplings evolves towards the Gaussian point. After passing the mass scale of the composite spectra the composite couplings enter a scaling regime. The direct four-fermion interaction is washed out from the system, and a weakly interacting bosonic field theory represents the symmetric phase of the NJL model at the longest wavelengths.
The main new result of the present investigation is to provide a numerical method to extract the interaction energy of the two particles as function of the initial physical size of the composite they constitute. For this we solved the renormalisation group equations derived from an ansatz for the effective quantum action with physically motivated parametrisation.
It is important to emphasize that by varying the initial width-parameter α(0) in the RG-equations we could explore a continuously infinite set of the approximate two-particle eigenmodes of the 4-fermion function of the original model. The resulting squared mass for any single solution in itself was not sufficient to decide whether one deals with a bound-state or only a component of the continuum. Only, by considering the complete set, we were able to single out the state with maximally negative interaction energy, and suggest a bound state interpretation for it. This way of processing is a fully consistent (though approximate) realisation of the strategy which emerged from our general discussion in Ref. [24].
APPENDIX
The renormalisation group equations are written for the dimensionless couplings
µ 2 C = M 2 C Z C k 2 ,g 2 Cr = g 2 C e −2αQ 2 Z 2 ψ Z C , α r = αk 2 , λ r = λ Z 2 C , δλ σr = k 2 δλ σ Z 2 ψ , λ V r = k 2 λ V Z 2 ψ .(40)
The dimensionful quantities are reconstructed by the formulae:
M 2 phys (t) = µ 2 C (t)e 2t e −η C (t) Λ 2 , α phys = α r (t)e −2t 1 Λ 2 ,(41)
where Λ is the scale where the solution of the RG-equations is starting. The explicit equations were derived using an optimized regulator [29].
The RG-equations of scaled four fermion couplings λ V r and δλ σr contain terms which coincide with those appearing in the RGE's of the chiral NJL-model [28]. In addition two more contributions come when evaluating (19) and (20):
∂ t δλ σr = 2(1 + η ψ )δλ σr − 1 8π 2 (δλ 2 σr + 4δλ σr λ V r + 3λ 2 V r ) +g 4 Cr 16π 2 [(2 − η C )F 1 (8α r ) + η c F 2 (8α r )] 1 (1 + µ 2 C ) 3 +g 4 Cr 16π 2 [(1 − η ψ )F 1 (8α r ) + η ψ H 2 (8α r )] 1 (1 + µ 2 C ) 2 +g 2 Cr 16π 2 (δλ S − λ V ) [(2 − η C )F 1 (4α r ) + η C F 2 (4α r )] 1 (1 + µ 2 C ) 2 , ∂ t λ V r = 2(1 + η ψ )λ V r − 1 16π 2 δλ 2 σr + 2δλ σr λ V r + λ 2 V r +g 4 Cr16π 2 (δλ S + λ V ) [(2 − η C )F 1 (4α r ) + η C F 2 (4α r )] 1 (1 + µ 2 C ) 2 .(42)
The amplitude of the bound state "wave function" figures in the equations always in the combinatioñ g Cr = g Cr exp(−2α r Q 2 r ). The corresponding RG-equations read:
∂ tg 2 Cr = (2η ψ + η C )g 2 Cr − (δλ σr + λ V r )g 2 Cr 2 π 2 [F 1 (4α r )(1 − η ψ ) + H 2 (4α r )η ψ ] , ∂ t g Cr α r (2 − α r Q 2 r ) = η ψ + 1 2 η C + 2 g Cr α r (2 − α r Q 2 r ) −g Cr (δλ σr + λ V r ) 4π 2
1 2 e −4αr − 16α 2 r [(1 − η ψ )F 2 (4α r ) + η ψ H 3 (4α r )] + 4α r (4 − α r Q 2 r ) [(1 − η ψ )F 1 (4α r ) + η ψ H 2 (4α r )] + 3 [(1 − η ψ )F 0 (4α r ) + η ψ H 1 (4α r )] .(43)
Here the functions F n (κ) and H n (κ) are explicit weighted phase space integrals:
Finally, the couplings describing the effective potential of the composite field obey the following equations:
∂ t µ 2 C = −(2 − η C )µ 2 C +g 2 Cr π 2 [(1 − η ψ )F 1 (8α r ) + η ψ H 2 (8α r )] − λ r 24π 2 1 (1 + µ 2 C ) 2 1 − η C 6 ,(45)
∂ t λ r = 2η C λ r + 5λ 2 r 24π 2
1 (1 + µ 2 C ) 3 1 − η C 6 − 12g 4
Cr π 2 [(1 − η ψ )F 1 (16α r ) + η ψ H 2 (16α r )] .
The expressions of the anomalous field dimensions are given with the expressions:
η ψ =g 2 Cr 8π 2 1 (1 + µ 2 C ) 2 (2 − η C ) 3 4 H 1 (4α r ) − 2α r H 2 (4α r ) + η C 3 4 H 2 (4α r ) − 2α r H 3 (4α r ) , η C =g 2 Cr 8π 2 1 2 e −8αr − (1 − η ψ )[64α 2 r F 2 (8α r ) − 32α r F 1 (8α r ) − 3F 0 (8α r )] − η ψ [64α 2 r H 3 (8α r ) − 32α r H 2 (8α r ) − 3H 1 (8α r )] .(47)
FIG. 1 .FIG. 2 .
12RG-evolution of the rescaled composite-to-constituents Yukawa coupling and the four-RG-variation of the anomalous dimension ηC demonstrates the evolution of the composite field towards the canonical kinetic term.
λ Vr /k 2 FIG. 3 .
23RG evolution of the rescaled 4-fermion couplings.
FIG. 4 .
4RG-variation of the squared boson mass and of the width of the composite "wave function".
FIG. 5 .
5RG-variation of the quantum contribution to the squared composite mass δM 2C (α(k = Λ)) = M 2 C (t = −∞, α(k = Λ)) − M 2 C (t = 0, α(k = Λ)). From the top to the bottom curves with diminishing αr(k = Λ) are presented.
FIG. 6 .
6Dependence of the interaction energy and of the squared physical size of the composite on the cut-off value of the size parameter (negative maximum) of the interaction energy is a stable configuration, in other words it is natural to interpret it as a bound state.
FIG. 7 .
7Left: Dependence of the interaction energy on the input value of the size parameter, for different dM 2 C /dt at t = 0, Right: the set of potentials scaled together with the mapping M 2 C → M 2 C /C, α(k = Λ) → A1α(k = Λ) + A2.
16π 2 [( 2
22− η C )F 1 (8α r ) + η c F 2 (8α r 2 [(1 − η ψ )F 1 (8α r ) + η ψ H 2 (8α r
The composite Nambu-Goldstone Higgs. G Panico, A Wulzer, 978-3-319-22616-3Lecture Notes in Physics. 913Springer Int. PublG. Panico and A. Wulzer, The composite Nambu-Goldstone Higgs, Lecture Notes in Physics, vol. 913, Springer Int. Publ., 2016, ISBN 978-3-319-22616-3
. Z Fodor, C Hoelbling, Rev. Mod. Phys. 84449Z. Fodor and C. Hoelbling, Rev. Mod. Phys. 84 (2012) 449
. C B Lang, AIP Conf. Proc. 173520002C.B. Lang, AIP Conf. Proc. 1735 (2016) 020002
. J Kuti, PoS. 5J. Kuti, PoS(Lattice2013)005;
. O Witzel, arXiv:1901.08216PoS(. O. Witzel PoS(Lattice 2018) arXiv:1901.08216
. H Bethe, E E Salpeter, Phys. Rev. 82309H. Bethe and E.E. Salpeter, Phys. Rev. 82 (1951) 309
. E E Salpeter, H Bethe, Phys. Rev. 841232E.E. Salpeter and H. Bethe, Phys. Rev. 84 (1951) 1232
. P Maris, C D Roberts, P Tandy, Phys.Lett. 420267P.Maris, C.D. Roberts and P. Tandy, Phys.Lett. B420 (1998) 267
. G Eichmann, A Krassnigg, M Schwinzerl, R Alkofer, Annals Phys. 3232505G.Eichmann, A.Krassnigg, M. Schwinzerl, and R. Alkofer, Annals Phys.323, 2505 (2008)
. A Krassnigg, Phys. Rev. 80114010A. Krassnigg, Phys. Rev. D80 (2009) 114010
. G Eichmann, H Sanchis-Alepuz, R Williams, R Alkofer, C S Fischer, Prog. Part. Nucl. Phys. 911G.Eichmann, H.Sanchis-Alepuz, R.Williams,R. Alkofer, C.S. Fischer, Prog. Part. Nucl. Phys.91, 1 (2016)
. C Wetterich, Phys. Lett. 30190C. Wetterich, Phys. Lett. B301 (1993) 90
. T R Morris, Int. J. Mod. Phys. 92411T.R. Morris, Int. J. Mod. Phys. A9 (1994) 2411
. U Ellwanger, Z. f. Physik. 62503U. Ellwanger, Z. f. Physik C62 (1994)503
. U Ellwanger, C Wetterich, Nucl.Phys. 423137U.Ellwanger and C. Wetterich, Nucl.Phys. B423 (1994) 137
. H Gies, C Wetterich, Phys. Rev. 6565001H. Gies and C. Wetterich, Phys. Rev. D65 (2002) 065001
. R L Stratonovich, Soviet Physics Doklady. 2416Stratonovich, R.L. Soviet Physics Doklady. 2(1958) 416,
. J Hubbard, J. Phys. Rev. Lett. 377J. Hubbard, J. Phys. Rev. Lett. 3 (1959) 77.
. S Floerchinger, C Wetterich, Phys. Lett. 680371S. Floerchinger and C. Wetterich, Phys. Lett. B680 (2009) 371
. J M Pawlowski, Ann. Phys. (N.Y.). 3222831J.M. Pawlowski, Ann. Phys. (N.Y.) 322 (2007) 2831
. A K Cyrol, M Mitter, J M Pawlowski, N Strodthoff, Phys. Rev. 9754006A.K. Cyrol, M. Mitter, J.M. Pawlowski and N. Strodthoff, Phys. Rev. D97 (2018) 054006
. M Mitter, J M Pawlowski, N Strodthoff, Phys. Rev. 9154035M. Mitter, J.M. Pawlowski and N. Strodthoff, Phys. Rev. D91 (2015) 054035
. J Braun, L Fister, J M Pawlowski, F Rennecke, Phys Rev. D. 9334016J. Braun, L. Fister, J.M. Pawlowski and F. Rennecke, Phys Rev. D 93 (2016) 034016
. R Alkofer, A Maas, W A Mian, M Mitter, J Paris-Lopez, J M Pawlowski, N Wink, arXiv:1810.07955R. Alkofer, A. Maas, W.A. Mian, M. Mitter, J. Paris-Lopez, J.M. Pawlowski and N. Wink, arXiv:1810.07955
A Jakovác, A Patkós, arXiv:1811.09418Bound states in Functional Renormalisation Group. A. Jakovác and A. Patkós, Bound states in Functional Renormalisation Group arXiv:1811.09418
. A Jakovác, A Patkós, Zs Szép, P Szépfalusy, Phys. Lett. 582179A. Jakovác, A. Patkós, Zs. Szép and P. Szépfalusy, Phys. Lett. B582 (2004) 179
. P Kovács, Zs, Szép, Phys. Rev. 7525015P. Kovács and Zs. Szép, Phys. Rev. D75 (2007) 025015
. P Kovács, Zs, Szép, Phys. Rev. 7765016P. Kovács and Zs. Szép, Phys. Rev. D77 (2008) 065016
. J Braun, J. Phys. 3933001J. Braun, J. Phys. G39 (2012) 033001
. D Litim, Phys. Rev. 64105004D. Litim, Phys. Rev. D64, 105004 (2001)
. J Braun, M Leonhardt, M Pospiech, Phys. Rev. D. 9676003J. Braun, M. Leonhardt, and M. Pospiech, Phys. Rev. D 96 (2017) 076003
| [] |
[
"First measurement of the forward-backward charge asymmetry in top quark pair production",
"First measurement of the forward-backward charge asymmetry in top quark pair production",
"First measurement of the forward-backward charge asymmetry in top quark pair production",
"First measurement of the forward-backward charge asymmetry in top quark pair production"
] | [
"V M Abazov ",
"B Abbott ",
"M Abolins ",
"B S Acharya ",
"M Adams ",
"T Adams ",
"E Aguilo ",
"S H Ahn ",
"M Ahsan ",
"G D Alexeev ",
"G Alkhazov ",
"A Alton ",
"G Alverson ",
"G A Alves ",
"M Anastasoaie ",
"L S Ancu ",
"T Andeen ",
"S Anderson ",
"B Andrieu ",
"M.SAnzelc ",
"Y Arnoud ",
"M Arov ",
"M Arthaud ",
"A Askew ",
"B Åsman ",
"A C S Assis Jesus ",
"O Atramentov ",
"C Autermann ",
"C Avila ",
"C Ay ",
"F Badaud ",
"A Baden ",
"L Bagby ",
"B Baldin ",
"D V Bandurin ",
"S Banerjee ",
"P Banerjee ",
"E Barberis ",
"A.-F Barfuss ",
"P Bargassa ",
"P Baringer ",
"J Barreto ",
"J F Bartlett ",
"U Bassler ",
"D Bauer ",
"S Beale ",
"A Bean ",
"M Begalli ",
"M Begel ",
"C Belanger-Champagne ",
"L Bellantoni ",
"A Bellavance ",
"J A Benitez ",
"S B Beri ",
"G Bernardi ",
"R Bernhard ",
"I Bertram ",
"M Besançon ",
"R Beuselinck ",
"V A Bezzubov ",
"P C Bhat ",
"V Bhatnagar ",
"C Biscarat ",
"G Blazey ",
"F Blekman ",
"S Blessing ",
"D Bloch ",
"K Bloom ",
"A Boehnlein ",
"D Boline ",
"T A Bolton ",
"G Borissov ",
"T Bose ",
"A Brandt ",
"R Brock ",
"G Brooijmans ",
"A Bross ",
"D Brown ",
"N J Buchanan ",
"D Buchholz ",
"M Buehler ",
"V Buescher ",
"V Bunichev ",
"S Burdin ",
"S Burke ",
"T H Burnett ",
"C P Buszello ",
"J M Butler ",
"P Calfayan ",
"S Calvet ",
"J Cammin ",
"W Carvalho ",
"B C K Casey ",
"N M Cason ",
"H Castilla-Valdez ",
"S Chakrabarti ",
"D Chakraborty ",
"K M Chan ",
"K Chan ",
"A Chandra ",
"F Charles ",
"E Cheu ",
"F Chevallier ",
"D K Cho ",
"S Choi ",
"B Choudhary ",
"L Christofek ",
"T Christoudias ",
"S Cihangir ",
"D Claes ",
"Y Coadou ",
"M Cooke ",
"W E Cooper ",
"M Corcoran ",
"F Couderc ",
"M.-C Cousinou ",
"S Crépé-Renaudin ",
"D Cutts ",
"M Ćwiok ",
"H Da Motta ",
"A Das ",
"G Davies ",
"K De ",
"S J De Jong ",
"E De ",
"La Cruz-Burelo ",
"C De Oliveira Martins ",
"J D Degenhardt ",
"F Déliot ",
"M Demarteau ",
"R Demina ",
"D Denisov ",
"S P Denisov ",
"S Desai ",
"H T Diehl ",
"M Diesburg ",
"A Dominguez ",
"H Dong ",
"L V Dudko ",
"L Duflot ",
"S R Dugad ",
"D Duggan ",
"A Duperrin ",
"J Dyer ",
"A Dyshkant ",
"M Eads ",
"D Edmunds ",
"J Ellison ",
"V D Elvira ",
"Y Enari ",
"S Eno ",
"P Ermolov ",
"H Evans ",
"A Evdokimov ",
"V N Evdokimov ",
"A V Ferapontov ",
"T Ferbel ",
"F Fiedler ",
"F Filthaut ",
"W Fisher ",
"H E Fisk ",
"M Ford ",
"M Fortner ",
"H Fox ",
"S Fu ",
"S Fuess ",
"T Gadfort ",
"C F Galea ",
"E Gallas ",
"E Galyaev ",
"C Garcia ",
"A Garcia-Bellido ",
"V Gavrilov ",
"P Gay ",
"W Geist ",
"D Gelé ",
"C E Gerber ",
"Y Gershtein ",
"D Gillberg ",
"G Ginther ",
"N Gollub ",
"B Gómez ",
"A Goussiou ",
"P D Grannis ",
"H Greenlee ",
"Z D Greenwood ",
"E M Gregores ",
"G Grenier ",
"Ph Gris ",
"J.-F Grivaz ",
"A Grohsjean ",
"S Grünendahl ",
"M W Grünewald ",
"J Guo ",
"F Guo ",
"P Gutierrez ",
"G Gutierrez ",
"A Haas ",
"N J Hadley ",
"P Haefner ",
"S Hagopian ",
"J Haley ",
"I Hall ",
"R E Hall ",
"L Han ",
"K Hanagaki ",
"P Hansson ",
"K Harder ",
"A Harel ",
"R Harrington ",
"J M Hauptman ",
"R Hauser ",
"J Hays ",
"T Hebbeker ",
"D Hedin ",
"J G Hegeman ",
"J M Heinmiller ",
"A P Heinson ",
"U Heintz ",
"C Hensel ",
"K Herner ",
"G Hesketh ",
"M D Hildreth ",
"R Hirosky ",
"J D Hobbs ",
"B Hoeneisen ",
"H Hoeth ",
"M Hohlfeld ",
"S J Hong ",
"S Hossain ",
"P Houben ",
"Y Hu ",
"Z Hubacek ",
"V Hynek ",
"I Iashvili ",
"R Illingworth ",
"A S Ito ",
"S Jabeen ",
"M Jaffré ",
"S Jain ",
"K Jakobs ",
"C Jarvis ",
"R Jesik ",
"K Johns ",
"C Johnson ",
"M Johnson ",
"A Jonckheere ",
"P Jonsson ",
"A Juste ",
"D Käfer ",
"E Kajfasz ",
"A M Kalinin ",
"J R Kalk ",
"J M Kalk ",
"S Kappler ",
"D Karmanov ",
"P Kasper ",
"I Katsanos ",
"D Kau ",
"R Kaur ",
"V Kaushik ",
"R Kehoe ",
"S Kermiche ",
"N Khalatyan ",
"A Khanov ",
"A Kharchilava ",
"Y M Kharzheev ",
"D Khatidze ",
"H Kim ",
"T J Kim ",
"M H Kirby ",
"M Kirsch ",
"B Klima ",
"J M Kohli ",
"J.-P Konrath ",
"M Kopal ",
"V M Korablev ",
"A V Kozelov ",
"D Krop ",
"T Kuhl ",
"A Kumar ",
"S Kunori ",
"A Kupco ",
"T Kurča ",
"J Kvita ",
"F Lacroix ",
"D Lam ",
"S Lammers ",
"G Landsberg ",
"P Lebrun ",
"W M Lee ",
"A Leflat ",
"F Lehner ",
"J Lellouch ",
"J Leveque ",
"P Lewis ",
"J Li ",
"Q Z Li ",
"L Li ",
"S M Lietti ",
"J G R Lima ",
"D Lincoln ",
"J Linnemann ",
"V V Lipaev ",
"R Lipton ",
"Y Liu ",
"Z Liu ",
"L Lobo ",
"A Lobodenko ",
"M Lokajicek ",
"P Love ",
"H J Lubatti ",
"A L Lyon ",
"A K A Maciel ",
"D Mackin ",
"R J Madaras ",
"P Mättig ",
"C Magass ",
"A Magerkurth ",
"P K Mal ",
"H B Malbouisson ",
"S Malik ",
"V L Malyshev ",
"H S Mao ",
"Y Maravin ",
"B Martin ",
"R Mccarthy ",
"A Melnitchouk ",
"A Mendes ",
"L Mendoza ",
"P G Mercadante ",
"M Merkin ",
"K W Merritt ",
"J Meyer ",
"A Meyer ",
"T Millet ",
"J Mitrevski ",
"J Molina ",
"R K Mommsen ",
"N K Mondal ",
"R W Moore ",
"T Moulik ",
"G S Muanza ",
"M Mulders ",
"M Mulhearn ",
"O Mundal ",
"L Mundim ",
"E Nagy ",
"M Naimuddin ",
"M Narain ",
"N A Naumann ",
"H A Neal ",
"J P Negret ",
"P Neustroev ",
"H Nilsen ",
"H Nogima ",
"A Nomerotski ",
"S F Novaes ",
"T Nunnemann ",
"V O'dell ",
"D C O'neil ",
"G Obrant ",
"C Ochando ",
"D Onoprienko ",
"V M Abazov ",
"B Abbott ",
"M Abolins ",
"B S Acharya ",
"M Adams ",
"T Adams ",
"E Aguilo ",
"S H Ahn ",
"M Ahsan ",
"G D Alexeev ",
"G Alkhazov ",
"A Alton ",
"G Alverson ",
"G A Alves ",
"M Anastasoaie ",
"L S Ancu ",
"T Andeen ",
"S Anderson ",
"B Andrieu ",
"M.SAnzelc ",
"Y Arnoud ",
"M Arov ",
"M Arthaud ",
"A Askew ",
"B Åsman ",
"A C S Assis Jesus ",
"O Atramentov ",
"C Autermann ",
"C Avila ",
"C Ay ",
"F Badaud ",
"A Baden ",
"L Bagby ",
"B Baldin ",
"D V Bandurin ",
"S Banerjee ",
"P Banerjee ",
"E Barberis ",
"A.-F Barfuss ",
"P Bargassa ",
"P Baringer ",
"J Barreto ",
"J F Bartlett ",
"U Bassler ",
"D Bauer ",
"S Beale ",
"A Bean ",
"M Begalli ",
"M Begel ",
"C Belanger-Champagne ",
"L Bellantoni ",
"A Bellavance ",
"J A Benitez ",
"S B Beri ",
"G Bernardi ",
"R Bernhard ",
"I Bertram ",
"M Besançon ",
"R Beuselinck ",
"V A Bezzubov ",
"P C Bhat ",
"V Bhatnagar ",
"C Biscarat ",
"G Blazey ",
"F Blekman ",
"S Blessing ",
"D Bloch ",
"K Bloom ",
"A Boehnlein ",
"D Boline ",
"T A Bolton ",
"G Borissov ",
"T Bose ",
"A Brandt ",
"R Brock ",
"G Brooijmans ",
"A Bross ",
"D Brown ",
"N J Buchanan ",
"D Buchholz ",
"M Buehler ",
"V Buescher ",
"V Bunichev ",
"S Burdin ",
"S Burke ",
"T H Burnett ",
"C P Buszello ",
"J M Butler ",
"P Calfayan ",
"S Calvet ",
"J Cammin ",
"W Carvalho ",
"B C K Casey ",
"N M Cason ",
"H Castilla-Valdez ",
"S Chakrabarti ",
"D Chakraborty ",
"K M Chan ",
"K Chan ",
"A Chandra ",
"F Charles ",
"E Cheu ",
"F Chevallier ",
"D K Cho ",
"S Choi ",
"B Choudhary ",
"L Christofek ",
"T Christoudias ",
"S Cihangir ",
"D Claes ",
"Y Coadou ",
"M Cooke ",
"W E Cooper ",
"M Corcoran ",
"F Couderc ",
"M.-C Cousinou ",
"S Crépé-Renaudin ",
"D Cutts ",
"M Ćwiok ",
"H Da Motta ",
"A Das ",
"G Davies ",
"K De ",
"S J De Jong ",
"E De ",
"La Cruz-Burelo ",
"C De Oliveira Martins ",
"J D Degenhardt ",
"F Déliot ",
"M Demarteau ",
"R Demina ",
"D Denisov ",
"S P Denisov ",
"S Desai ",
"H T Diehl ",
"M Diesburg ",
"A Dominguez ",
"H Dong ",
"L V Dudko ",
"L Duflot ",
"S R Dugad ",
"D Duggan ",
"A Duperrin ",
"J Dyer ",
"A Dyshkant ",
"M Eads ",
"D Edmunds ",
"J Ellison ",
"V D Elvira ",
"Y Enari ",
"S Eno ",
"P Ermolov ",
"H Evans ",
"A Evdokimov ",
"V N Evdokimov ",
"A V Ferapontov ",
"T Ferbel ",
"F Fiedler ",
"F Filthaut ",
"W Fisher ",
"H E Fisk ",
"M Ford ",
"M Fortner ",
"H Fox ",
"S Fu ",
"S Fuess ",
"T Gadfort ",
"C F Galea ",
"E Gallas ",
"E Galyaev ",
"C Garcia ",
"A Garcia-Bellido ",
"V Gavrilov ",
"P Gay ",
"W Geist ",
"D Gelé ",
"C E Gerber ",
"Y Gershtein ",
"D Gillberg ",
"G Ginther ",
"N Gollub ",
"B Gómez ",
"A Goussiou ",
"P D Grannis ",
"H Greenlee ",
"Z D Greenwood ",
"E M Gregores ",
"G Grenier ",
"Ph Gris ",
"J.-F Grivaz ",
"A Grohsjean ",
"S Grünendahl ",
"M W Grünewald ",
"J Guo ",
"F Guo ",
"P Gutierrez ",
"G Gutierrez ",
"A Haas ",
"N J Hadley ",
"P Haefner ",
"S Hagopian ",
"J Haley ",
"I Hall ",
"R E Hall ",
"L Han ",
"K Hanagaki ",
"P Hansson ",
"K Harder ",
"A Harel ",
"R Harrington ",
"J M Hauptman ",
"R Hauser ",
"J Hays ",
"T Hebbeker ",
"D Hedin ",
"J G Hegeman ",
"J M Heinmiller ",
"A P Heinson ",
"U Heintz ",
"C Hensel ",
"K Herner ",
"G Hesketh ",
"M D Hildreth ",
"R Hirosky ",
"J D Hobbs ",
"B Hoeneisen ",
"H Hoeth ",
"M Hohlfeld ",
"S J Hong ",
"S Hossain ",
"P Houben ",
"Y Hu ",
"Z Hubacek ",
"V Hynek ",
"I Iashvili ",
"R Illingworth ",
"A S Ito ",
"S Jabeen ",
"M Jaffré ",
"S Jain ",
"K Jakobs ",
"C Jarvis ",
"R Jesik ",
"K Johns ",
"C Johnson ",
"M Johnson ",
"A Jonckheere ",
"P Jonsson ",
"A Juste ",
"D Käfer ",
"E Kajfasz ",
"A M Kalinin ",
"J R Kalk ",
"J M Kalk ",
"S Kappler ",
"D Karmanov ",
"P Kasper ",
"I Katsanos ",
"D Kau ",
"R Kaur ",
"V Kaushik ",
"R Kehoe ",
"S Kermiche ",
"N Khalatyan ",
"A Khanov ",
"A Kharchilava ",
"Y M Kharzheev ",
"D Khatidze ",
"H Kim ",
"T J Kim ",
"M H Kirby ",
"M Kirsch ",
"B Klima ",
"J M Kohli ",
"J.-P Konrath ",
"M Kopal ",
"V M Korablev ",
"A V Kozelov ",
"D Krop ",
"T Kuhl ",
"A Kumar ",
"S Kunori ",
"A Kupco ",
"T Kurča ",
"J Kvita ",
"F Lacroix ",
"D Lam ",
"S Lammers ",
"G Landsberg ",
"P Lebrun ",
"W M Lee ",
"A Leflat ",
"F Lehner ",
"J Lellouch ",
"J Leveque ",
"P Lewis ",
"J Li ",
"Q Z Li ",
"L Li ",
"S M Lietti ",
"J G R Lima ",
"D Lincoln ",
"J Linnemann ",
"V V Lipaev ",
"R Lipton ",
"Y Liu ",
"Z Liu ",
"L Lobo ",
"A Lobodenko ",
"M Lokajicek ",
"P Love ",
"H J Lubatti ",
"A L Lyon ",
"A K A Maciel ",
"D Mackin ",
"R J Madaras ",
"P Mättig ",
"C Magass ",
"A Magerkurth ",
"P K Mal ",
"H B Malbouisson ",
"S Malik ",
"V L Malyshev ",
"H S Mao ",
"Y Maravin ",
"B Martin ",
"R Mccarthy ",
"A Melnitchouk ",
"A Mendes ",
"L Mendoza ",
"P G Mercadante ",
"M Merkin ",
"K W Merritt ",
"J Meyer ",
"A Meyer ",
"T Millet ",
"J Mitrevski ",
"J Molina ",
"R K Mommsen ",
"N K Mondal ",
"R W Moore ",
"T Moulik ",
"G S Muanza ",
"M Mulders ",
"M Mulhearn ",
"O Mundal ",
"L Mundim ",
"E Nagy ",
"M Naimuddin ",
"M Narain ",
"N A Naumann ",
"H A Neal ",
"J P Negret ",
"P Neustroev ",
"H Nilsen ",
"H Nogima ",
"A Nomerotski ",
"S F Novaes ",
"T Nunnemann ",
"V O'dell ",
"D C O'neil ",
"G Obrant ",
"C Ochando ",
"D Onoprienko "
] | [] | [] | We present the first measurement of the integrated forward-backward charge asymmetry in topantitop quark pair (tt) production in proton-antiproton (pp) collisions in the lepton+jets final state. | 10.1063/1.3052037 | [
"https://arxiv.org/pdf/0712.0851v1.pdf"
] | 6,440,951 | 0712.0851 | 8c0396f1e3fa0bbac57b705a834ffc1c4dd53314 |
First measurement of the forward-backward charge asymmetry in top quark pair production
5 Dec 2007
V M Abazov
B Abbott
M Abolins
B S Acharya
M Adams
T Adams
E Aguilo
S H Ahn
M Ahsan
G D Alexeev
G Alkhazov
A Alton
G Alverson
G A Alves
M Anastasoaie
L S Ancu
T Andeen
S Anderson
B Andrieu
M.SAnzelc
Y Arnoud
M Arov
M Arthaud
A Askew
B Åsman
A C S Assis Jesus
O Atramentov
C Autermann
C Avila
C Ay
F Badaud
A Baden
L Bagby
B Baldin
D V Bandurin
S Banerjee
P Banerjee
E Barberis
A.-F Barfuss
P Bargassa
P Baringer
J Barreto
J F Bartlett
U Bassler
D Bauer
S Beale
A Bean
M Begalli
M Begel
C Belanger-Champagne
L Bellantoni
A Bellavance
J A Benitez
S B Beri
G Bernardi
R Bernhard
I Bertram
M Besançon
R Beuselinck
V A Bezzubov
P C Bhat
V Bhatnagar
C Biscarat
G Blazey
F Blekman
S Blessing
D Bloch
K Bloom
A Boehnlein
D Boline
T A Bolton
G Borissov
T Bose
A Brandt
R Brock
G Brooijmans
A Bross
D Brown
N J Buchanan
D Buchholz
M Buehler
V Buescher
V Bunichev
S Burdin
S Burke
T H Burnett
C P Buszello
J M Butler
P Calfayan
S Calvet
J Cammin
W Carvalho
B C K Casey
N M Cason
H Castilla-Valdez
S Chakrabarti
D Chakraborty
K M Chan
K Chan
A Chandra
F Charles
E Cheu
F Chevallier
D K Cho
S Choi
B Choudhary
L Christofek
T Christoudias
S Cihangir
D Claes
Y Coadou
M Cooke
W E Cooper
M Corcoran
F Couderc
M.-C Cousinou
S Crépé-Renaudin
D Cutts
M Ćwiok
H Da Motta
A Das
G Davies
K De
S J De Jong
E De
La Cruz-Burelo
C De Oliveira Martins
J D Degenhardt
F Déliot
M Demarteau
R Demina
D Denisov
S P Denisov
S Desai
H T Diehl
M Diesburg
A Dominguez
H Dong
L V Dudko
L Duflot
S R Dugad
D Duggan
A Duperrin
J Dyer
A Dyshkant
M Eads
D Edmunds
J Ellison
V D Elvira
Y Enari
S Eno
P Ermolov
H Evans
A Evdokimov
V N Evdokimov
A V Ferapontov
T Ferbel
F Fiedler
F Filthaut
W Fisher
H E Fisk
M Ford
M Fortner
H Fox
S Fu
S Fuess
T Gadfort
C F Galea
E Gallas
E Galyaev
C Garcia
A Garcia-Bellido
V Gavrilov
P Gay
W Geist
D Gelé
C E Gerber
Y Gershtein
D Gillberg
G Ginther
N Gollub
B Gómez
A Goussiou
P D Grannis
H Greenlee
Z D Greenwood
E M Gregores
G Grenier
Ph Gris
J.-F Grivaz
A Grohsjean
S Grünendahl
M W Grünewald
J Guo
F Guo
P Gutierrez
G Gutierrez
A Haas
N J Hadley
P Haefner
S Hagopian
J Haley
I Hall
R E Hall
L Han
K Hanagaki
P Hansson
K Harder
A Harel
R Harrington
J M Hauptman
R Hauser
J Hays
T Hebbeker
D Hedin
J G Hegeman
J M Heinmiller
A P Heinson
U Heintz
C Hensel
K Herner
G Hesketh
M D Hildreth
R Hirosky
J D Hobbs
B Hoeneisen
H Hoeth
M Hohlfeld
S J Hong
S Hossain
P Houben
Y Hu
Z Hubacek
V Hynek
I Iashvili
R Illingworth
A S Ito
S Jabeen
M Jaffré
S Jain
K Jakobs
C Jarvis
R Jesik
K Johns
C Johnson
M Johnson
A Jonckheere
P Jonsson
A Juste
D Käfer
E Kajfasz
A M Kalinin
J R Kalk
J M Kalk
S Kappler
D Karmanov
P Kasper
I Katsanos
D Kau
R Kaur
V Kaushik
R Kehoe
S Kermiche
N Khalatyan
A Khanov
A Kharchilava
Y M Kharzheev
D Khatidze
H Kim
T J Kim
M H Kirby
M Kirsch
B Klima
J M Kohli
J.-P Konrath
M Kopal
V M Korablev
A V Kozelov
D Krop
T Kuhl
A Kumar
S Kunori
A Kupco
T Kurča
J Kvita
F Lacroix
D Lam
S Lammers
G Landsberg
P Lebrun
W M Lee
A Leflat
F Lehner
J Lellouch
J Leveque
P Lewis
J Li
Q Z Li
L Li
S M Lietti
J G R Lima
D Lincoln
J Linnemann
V V Lipaev
R Lipton
Y Liu
Z Liu
L Lobo
A Lobodenko
M Lokajicek
P Love
H J Lubatti
A L Lyon
A K A Maciel
D Mackin
R J Madaras
P Mättig
C Magass
A Magerkurth
P K Mal
H B Malbouisson
S Malik
V L Malyshev
H S Mao
Y Maravin
B Martin
R Mccarthy
A Melnitchouk
A Mendes
L Mendoza
P G Mercadante
M Merkin
K W Merritt
J Meyer
A Meyer
T Millet
J Mitrevski
J Molina
R K Mommsen
N K Mondal
R W Moore
T Moulik
G S Muanza
M Mulders
M Mulhearn
O Mundal
L Mundim
E Nagy
M Naimuddin
M Narain
N A Naumann
H A Neal
J P Negret
P Neustroev
H Nilsen
H Nogima
A Nomerotski
S F Novaes
T Nunnemann
V O'dell
D C O'neil
G Obrant
C Ochando
D Onoprienko
First measurement of the forward-backward charge asymmetry in top quark pair production
5 Dec 2007
We present the first measurement of the integrated forward-backward charge asymmetry in topantitop quark pair (tt) production in proton-antiproton (pp) collisions in the lepton+jets final state.
Using a b-jet tagging algorithm and kinematic reconstruction assuming tt+X production and decay, a sample of 0.9 fb −1 of data, collected by the D0 experiment at the Fermilab Tevatron Collider, is used to measure the asymmetry for different jet multiplicities. The result is also used to set upper limits on tt + X production via a Z ′ resonance.
PACS numbers: 12.38.Qk,13.87.Ce At lowest order in quantum chromodynamics (QCD), the standard model (SM) predicts that the kinematic distributions in pp → tt + X production are charge symmetric. But this symmetry is accidental, as the initial pp state is not an eigenstate of charge conjugation. Next-to-leading order (NLO) calculations predict forward-backward asymmetries of (5-10)% [1,2], but recent next-to-next-to-leading order (NNLO) calculations predict significant corrections for tt production in association with a jet [3]. The asymmetry arises mainly from interference between contributions symmetric and antisymmetric under the exchange t ↔t [1], and depends on the region of phase space being probed and, in particular, on the production of an additional jet [2]. The small asymmetries expected in the SM make this a sensitive probe for new physics [4].
A charge asymmetry in pp → tt + X can be observed as a forward-backward production asymmetry. The signed difference between the rapidities [5] of the t andt, ∆y ≡ y t − yt, reflects the asymmetry in tt production. We define the integrated charge asymmetry as
A fb = (N f − N b ) / (N f + N b ) , where N f (N b )
is the number of events with a positive (negative) ∆y.
This Letter describes the first measurement of A fb in pp → tt + X production. The 0.9 fb −1 data sample used was collected at √ s = 1.96 TeV with the D0 detector [6], using triggers that required a jet and an electron or muon. In the lepton+jets final state of the tt system, one of the two W bosons from the tt pair decays into hadronic jets and the other into leptons, yielding a signature of two bjets, two light-flavor jets, an isolated lepton, and missing transverse energy (/ E T ). This decay mode is well suited for this measurement, as it combines a large branching fraction (∼ 34%) with high signal purity, the latter a consequence of requiring an isolated electron or muon with large transverse momentum (p T ). The main background is from W +jets and multijet production. This channel allows accurate reconstruction of the t andt directions in the collision rest frame, and the charge of the electron or muon distinguishes between the t andt quarks.
The dependence of A fb on the region of phase space, as calculated by the mc@nlo event generator [7], is demonstrated in Fig. 1. The large dependence on the fourthhighest jet p T is not available in the calculations of Refs. [1,2,3], as these do not consider decays of the top quarks, and include only acceptance for jets from additional radiation.
We conclude that acceptance can strongly affect the asymmetry. To facilitate comparison with theory, the analysis is therefore designed to have an acceptance which can be described simply. Event selection is limited to either: (i) selections on directions and momenta that can be described at the particle level (which refers to produced particles before they start interacting with material in the detector) or (ii) criteria with high signal efficiency, so that their impact on the region of acceptance is negligible. In addition, the observable quantity and the fitting procedure are chosen to ensure that all events have the same weight in determining the asymmetry. The measurement is not corrected for acceptance and reconstruction effects, but a prescription provides the acceptance at the particle level. Reconstruction effects are also accommodated at the particle level by defining the asymmetry as a function of the generated |∆y|:
A fb (|∆y|) = g(|∆y|) − g(−|∆y|) g(|∆y|) + g(−|∆y|) ,(1)
where g is the probability density for ∆y within the acceptance. This asymmetry can be folded with the "geometric dilution," D, which is described later:
A pred fb = ∞ 0 A fb (∆y) D (∆y) [g (∆y) + g (−∆y)] d∆y.
(2) This procedure yields the predictions in Table I. The values are smaller than those of Ref. [1,2], because of the inclusion of jet acceptance and dilution.
We select events with at least four jets reconstructed using a cone algorithm [8] with an angular radius R = 0.5 (in rapidity and azimuthal angle). All jets must have p T > 20 GeV and pseudorapidity (relative to the reconstructed primary vertex) |η| < 2.5. The leading jet
Njet A pred fb (in %) 4 0.8 ± 0.2(stat.) ±1.0(accept.) ± 0.0(dilution) 4 2.3 ± 0.2(stat.) ±1.0(accept.) ± 0.1(dilution) 5 −4.9 ± 0.4(stat.) ±1.0(accept.) ± 0.2(dilution) must have p T > 35 GeV.
Events are required to have / E T > 15 GeV and exactly one isolated electron with p T > 15 GeV and |η| < 1.1 or one isolated muon with p T > 18 GeV and |η| < 2.0. More details on lepton identification and trigger requirements are given in Ref. [9]. Events in which the lepton momentum is mismeasured are suppressed by requiring that the direction of the / E T not be along or opposite the azimuth of the lepton. To enhance the signal, at least one of the jets is required to be identified as originating from a long-lived b hadron by a neural network b-jet tagging algorithm [10]. The variables used to identify such jets rely on the presence and characteristics of a secondary vertex and tracks with high impact parameter inside the jet.
The top quark pair is reconstructed using a kinematic fitter [11], which varies the four-momenta of the detected objects within their resolutions and minimizes a χ 2 statistic, constraining both W boson masses to exactly 80.4 GeV and top quark masses to exactly 170 GeV. The b-tagged jet of highest p T and the three remaining jets with highest p T are used in the fit. The b-tagging information is used to reduce the number of jet-parton assignments considered in the fit. Only events in which the kinematic fit converges are used, and for each event only the reconstruction with the lowest χ 2 is retained.
The jet-p T selection criteria strongly affect the observed asymmetry (see Fig. 1), and this must be considered when comparing a model to data. Fortunately, these effects can be approximated by simple cuts on particle-level momenta without changing the asymmetry by more than 2% (absolute). This is verified using several simulated samples with generated asymmetries and particle jets clustered using the pxcone algorithm [12] ("E" scheme and R = 0.5). The particle jet cuts are p T > 21 GeV and |η| < 2.5, with the additional requirement on the leading particle jet p T > 35 GeV and the lepton requirements detailed above. Systematic uncertainties on jet energy calibration introduce possible shifts of the particle jet thresholds. The shifts are +1.3 −1.5 GeV for the leading jet and +1.2 −1.3 GeV for the other jets, for ±1 standard deviation (sd) changes in the jet energy calibration. The resulting changes in the asymmetry predicted using mc@nlo are of the order of 0.5%. The effect of all other selections on the asymmetry is negligible. The predictions in Table I use a more complete description of the acceptance based on efficiencies factorized in p T and η, accurate to < 1% (absolute).
Misreconstructing the sign of ∆y dilutes the asymme- try. Such dilution can arise from misidentifying lepton charge or from misreconstructing event geometry. The rate for misidentification of lepton charge is taken from the signal simulation and verified using data. False production asymmetries arising from asymmetries in the rate for misidentification of lepton charge are negligible owing to the frequent reversal of the D0 solenoid and toroid polarities. The dilution, D, depends mainly on |∆y|. It is defined as D = 2P − 1, where P is the probability of reconstructing the correct sign of ∆y. It is obtained from tt + X events generated with pythia [13] and passed through a geant-based simulation [14] of the D0 detector, and is parametrized as:
D (|∆y|) = c 0 ln 1 + c 1 |∆y| + c 2 |∆y| 2 ,(3)
with the parameters given in Table II (see Fig. 2). As this measurement is integrated in |∆y|, the dependence of the dilution on |∆y| introduces a model dependence into any correction from observed asymmetry (A obs fb ) to a particle-level asymmetry. Such a correction factor would depend not only on the model's |∆y| distribution, but also on its prediction of A fb (|∆y|). Furthermore, such a correction would be sensitive to small new physics components of the selected sample. We therefore present a measurement uncorrected for reconstruction effects and provide the reader with a parametrization of D that describes these effects, to be applied to any model.
The dilution depends weakly on other variables correlated with A fb , such as the number of jets. This possible bias is included in the systematic uncertainties. Nonstandard production mechanisms can affect reconstruction quality, primarily due to changes in the momenta of the top quarks. By studying extreme cases, we find that when comparing non-standard tt + X production to data an additional 15% relative uncertainty on A fb is needed.
The main background is from W +jets production. To estimate it, we define a likelihood discriminant L using variables that are well-described in our simulation, provide separation between signal and W +jets background, and do not bias |∆y| for the selected signal. The following variables are used: the p T of the leading b-tagged jet, the χ 2 statistic from the kinematic fit, the invariant mass of the jets assigned to the hadronic W boson decay, and k min T = p min T R min , where R min is the smallest angular distance between any two jets used in the kinematic fit, and p min T is the smaller of the corresponding jets' transverse momenta.
The next largest background after W +jets is from multijet production, where a jet mimics an isolated electron or muon. Following the procedure described in Ref. [9], the distributions in likelihood discriminant and reconstructed asymmetry for this background are derived from samples of data that fail lepton identification. The normalization of this background is estimated from the size of those samples and the large difference in efficiencies of lepton identification for true and false leptons. The effects of additional background sources not considered explicitly in extracting A fb ; namely Z+jets, single top quark, and diboson production; are evaluated using ensembles of simulated datasets and found negligible.
The sample composition and A fb are extracted from a simultaneous maximum-likelihood fit to data of a sum of contributions to L and to the sign of the reconstructed ∆y (∆y reco ) from forward signal, backward signal, W +jets, and multijet production. Both signal contributions are generated with pythia, have the same distribution in L, and differ only in their being reconstructed as either forward or backward. The W +jets contribution is generated with alpgen [15] interfaced to pythia and has its own reconstructed asymmetry. Although W boson production is inherently asymmetric, the kinematic reconstruction to the tt+ X hypothesis reduces its reconstructed asymmetry to [4.4 ± 1.6 (stat.)] %. The multijet contribution is derived from data, as described above. The fitted parameters are shown in Table III. Correlations between the asymmetry and the other parameters are < 10%. The fitted asymmetries in data are consistent with the SM predictions given in Table I. In Fig. 3 we compare the fitted distributions to data for events with 4 jets.
The dominant sources of systematic uncertainty for the measured asymmetry are the relative jet energy calibration between data and simulation (±0.5%), the asymme- try reconstructed in W +jets events (±0.4%), and the modeling of additional interactions during a single pp bunch crossing (±0.4%). The total systematic uncertainty for the asymmetry is ±1%, which is negligible compared to the statistical uncertainty.
We check the simulation of the production asymmetry, and of the asymmetry reconstructed under the tt + X hypothesis in the W +jets background, by repeating the analysis in a sample enriched in W +jets events. The selection criteria for this sample are identical to the main analysis, except that we veto on any b-tags. Both the fully reconstructed asymmetry and the forward-backward lepton asymmetry are consistent with expectations. We also find that the fitted sample composition (Table III) is consistent with the cross section for tt + X production obtained in a dedicated analysis on this dataset. We check the validity of the fitting procedure, its calibration, and its statistical uncertainties using ensembles of simulated datasets.
To demonstrate the measurement's sensitivity to new physics, we examine tt production via neutral gauge bosons (Z ′ ) that are heavy enough to decay to on-shell top and antitop quarks. Direct searches have placed limits on tt production via a heavy narrow resonance [17], while the asymmetry in tt production may be sensitive to production via both narrow and wide resonances. The Z ′ → tt channel is of interest in models with a "leptophobic" Z ′ that decays dominantly to quarks. We study the scenario where the coupling between the Z ′ boson and quarks is proportional to that between the Z boson and quarks, and interference effects with SM tt production are negligible. Using pythia we simulate tt production via Z ′ resonances with decay rates chosen to yield narrow resonances as in Ref. [17], and find large positive asymmetries [(13-35)%], which are a consequence of the predominantly left-handed decays. We predict the distribution of A fb as a function of the fraction (f ) of tt events produced via a Z ′ resonance of a particular mass from ensembles of simulated datasets. We use the procedure of Ref. [18] to arrive at the limits shown in Fig. 4. These limits can be applied to wide Z ′ resonances by averaging over the distribution of Z ′ mass.
In summary, we present the first measurement of the integrated forward-backward charge asymmetry in tt+X production. We find that acceptance affects the asymmetry and must be specified as above, and that correc- tions for reconstruction effects are too model-dependent to be of use. We observe an uncorrected asymmetry of A obs fb = [12 ± 8 (stat.) ± 1 (syst.)] % for tt + X events with 4 jets that are within our acceptance, and we provide a dilution function (Eq. 3) that can be applied to any model (through Eq. 2). For events with only four jets and for those with 5 jets, we find A obs fb = [19 ± 9 (stat.) ± 2 (syst.)] % and A obs fb = −16 +15 −17 (stat.) ± 3 (syst.) %, respectively, where most of the systematic uncertainty is from migrations of events between the two subsamples. The measured asymmetries are consistent with the mc@nlo predictions for standard model production.
FIG. 1 :
1Forward-backward tt charge asymmetry predicted by mc@nlo as a function of the fourth-highest particle jet pT .
FIG. 2 :
2The geometric dilution and its uncertainty band as a function of generated |∆y| for standard model tt + X production and 4 jets.
FIG. 3 :FIG. 4 :
34Comparison of data for 4 jets with the fitted model as a function of L for events reconstructed (a) as forward (∆yreco > 0) and (b) as backward (∆yreco < 0). The number of events from each source is listed with its statistical uncertainty. 95% C.L. limits on the fraction of tt produced via a Z ′ resonance as a function of the Z ′ mass, under assumptions detailed in the text. Limits expected in the absence of a Z ′ resonance are shown by the dashed curve, with the shaded bands showing limits one and two standard deviations away. The observed limits are shown by the solid curve, and the excluded region is hatched.
TABLE I :
IPredictions based on mc@nlo.
TABLE II :
IIParameters of the dilution. The ±1 sd values include both statistical and systematic uncertainties.Variation
c0
c1
c2
Njet 4
0.262
14.6
−1.5
+1 sd variation
0.229
20.3
1.2
−1 sd variation
0.289
11.4
−2.2
Njet = 4
0.251
17.6
−1.4
+1 sd variation
0.201
30.3
7.7
−1 sd variation
0.293
11.6
−2.3
Njet 5
0.254
9.6
0
+1 sd variation
0.206
17.4
2.4
−1 sd variation
0.358
5.0
−0.9
y|
∆
|
TABLE III :
IIINumber of selected events and fit results in data.4 Jets
4 Jets
5 Jets
No. Events
376
308
68
tt + X
266 +23
−22
214±20
54 +10
−12
W +jets
70±21
61 +19
−18
7 +11
−5
Multijets
40±4
32.7 +3.5
−3.3
7.1 +1.6
−1.5
A fb
(12±8)%
(19±9)%
(−16 +15
−17 )%
We thank the staffs at Fermilab and collaborating institutions, and acknowledge support from the DOE and NSF (USA); CEA and CNRS/IN2P3 (France); FASI, Rosatom and RFBR (Russia); CAPES, CNPq, FAPERJ, FAPESP and FUNDUNESP (Brazil); DAE and DST
Rijssenbeek 73 , I. Ripp-Baudot 19 , F. Rizatdinova 77 , S. Robinson 44 , R.F. Rodrigues 3 , M. Rominsky 76 , C. Royon 18 , P. Rubinov 51 , R. Ruchti 56 , G. Safronov 37 , G. Sajot 14 , A. Sánchez-Hernández 33. S.J. Wimpenny. R. Unalan 66 , S. Uvarov 40 , L. Uvarov 40 , S. Uzunyan 53 , B. Vachon 6 , P.J. van den Berg 34 , R. Van Kooten 55 , W.M. van Leeuwen 34 , N. Varelas 52 , E.W. Varnes 46 , I.A. Vasilyev 39 , M. Vaupel 26 , P. Verdier 20 , L.S. Vertogradov 36 , M. Verzocchi 51 , F. Villeneuve-Seguier 44 , P. Vint 44 , P. Vokac 10 , E. Von Toerne 60 , M. Voutilainen 68,e , R. Wagner 69 , H.D. Wahl 50 , L. Wang 62 , M.H.L.S Wang 5117T.R. Wyatt. Y.A. Yatsunenko 36 , K. Yip 74 , H.D. Yoo 78 , S.W. Youn 54 , J. Yu 79 , A. Zatserklyaniy 53 , C. Zeitnitz 26 , T. Zhao 83 , B. Zhou 65 , J. Zhu 73 , M. Zielinski 72 , D. Zieminska 55 , A. Zieminski 55, ‡ , L. Zivkovic 71 , V. Zutshi 53 , and E.G. Zverev. IndiaN. Oshima 51 , J. Osta 56 , R. Otec 10 , G.J. Otero y Garzón 51 , M. Owen 45 , P. Padley 81 , M. Pangilinan 78 , N. Parashar 57 , S.-J. Park 72 , S.K. Park 31 , J. Parsons 71 , R. Partridge 78 , N. Parua 55 , A. Patwa 74 , G. Pawloski 81 , B. Penning 23 , M. Perfilov 38 , K. Peters 45 , Y. Peters 26 , P. Pétroff 16 , M. Petteni 44 , R. Piegaia 1 , J. Piper 66 , M.-A. Pleier 22 , P.L.M. Podesta-Lerma 33,c , V.M. Podstavkov 51 , Y. Pogorelov 56 , M.-E. Pol 2 , P. Polozov 37 , B.G. Pope 66 , A.V. Popov 39 , C. Potter 6 , W.L. Prado da Silva 3 , H.B. Prosper 50 , S. Protopopescu 74 , J. Qian 65 , A. Quadt 22,d , B. Quinn 67 , A. Rakitine 43 , M.S. Rangel 2 , K. Ranjan 28 , P.N. Ratoff 43 , P. Renkel 80 , S. Reucroft 64 , P. Rich 45 , M. Rijssenbeek 73 , I. Ripp-Baudot 19 , F. Rizatdinova 77 , S. Robinson 44 , R.F. Rodrigues 3 , M. Rominsky 76 , C. Royon 18 , P. Rubinov 51 , R. Ruchti 56 , G. Safronov 37 , G. Sajot 14 , A. Sánchez-Hernández 33 , M.P. Sanders 17 , A. Santoro 3 , G. Savage 51 , L. Sawyer 61 , T. Scanlon 44 , D. Schaile 25 , R.D. Schamberger 73 , Y. Scheglov 40 , H. Schellman 54 , P. Schieferdecker 25 , T. Schliephake 26 , C. Schwanenberger 45 , A. Schwartzman 69 , R. Schwienhorst 66 , J. Sekaric 50 , H. Severini 76 , E. Shabalina 52 , M. Shamim 60 , V. Shary 18 , A.A. Shchukin 39 , R.K. Shivpuri 28 , V. Siccardi 19 , V. Simak 10 , V. Sirotenko 51 , P. Skubic 76 , P. Slattery 72 , D. Smirnov 56 , J. Snow 75 , G.R. Snow 68 , S. Snyder 74 , S. Söldner-Rembold 45 , L. Sonnenschein 17 , A. Sopczak 43 , M. Sosebee 79 , K. Soustruznik 9 , M. Souza 2 , B. Spurlock 79 , J. Stark 14 , J. Steele 61 , V. Stolin 37 , D.A. Stoyanova 39 , J. Strandberg 65 , S. Strandberg 41 , M.A. Strang 70 , M. Strauss 76 , E. Strauss 73 , R. Ströhmer 25 , D. Strom 54 , L. Stutte 51 , S. Sumowidagdo 50 , P. Svoisky 56 , A. Sznajder 3 , M. Talby 15 , P. Tamburello 46 , A. Tanasijczuk 1 , W. Taylor 6 , J. Temple 46 , B. Tiller 25 , F. Tissandier 13 , M. Titov 18 , V.V. Tokmenin 36 , T. Toole 62 , I. Torchiani 23 , T. Trefzger 24 , D. Tsybychev 73 , B. Tuchming 18 , C. Tully 69 , P.M. Tuts 71 , R. Unalan 66 , S. Uvarov 40 , L. Uvarov 40 , S. Uzunyan 53 , B. Vachon 6 , P.J. van den Berg 34 , R. Van Kooten 55 , W.M. van Leeuwen 34 , N. Varelas 52 , E.W. Varnes 46 , I.A. Vasilyev 39 , M. Vaupel 26 , P. Verdier 20 , L.S. Vertogradov 36 , M. Verzocchi 51 , F. Villeneuve-Seguier 44 , P. Vint 44 , P. Vokac 10 , E. Von Toerne 60 , M. Voutilainen 68,e , R. Wagner 69 , H.D. Wahl 50 , L. Wang 62 , M.H.L.S Wang 51 , J. Warchol 56 , G. Watts 83 , M. Wayne 56 , M. Weber 51 , G. Weber 24 , A. Wenger 23,f , N. Wermes 22 , M. Wetstein 62 , A. White 79 , D. Wicke 26 , G.W. Wilson 59 , S.J. Wimpenny 49 , M. Wobisch 61 , D.R. Wood 64 , T.R. Wyatt 45 , Y. Xie 78 , S. Yacoob 54 , R. Yamada 51 , M. Yan 62 , T. Yasuda 51 , Y.A. Yatsunenko 36 , K. Yip 74 , H.D. Yoo 78 , S.W. Youn 54 , J. Yu 79 , A. Zatserklyaniy 53 , C. Zeitnitz 26 , T. Zhao 83 , B. Zhou 65 , J. Zhu 73 , M. Zielinski 72 , D. Zieminska 55 , A. Zieminski 55, ‡ , L. Zivkovic 71 , V. Zutshi 53 , and E.G. Zverev 38 (India);
Colciencias (Colombia). Colciencias (Colombia);
. CONACyT (Mexico). CONACyT (Mexico);
. Kosef (korea); Krf, Ubacyt Conicet, ArgentinaKRF and KOSEF (Korea); CONICET and UBACyT (Argentina);
The Netherlands); Science and Technology Facilities Council. MSMT and GACR. United Kingdom; Czech RepublicFOM (The Netherlands); Science and Tech- nology Facilities Council (United Kingdom); MSMT and GACR (Czech Republic);
Crc Program, Westgrid Cfi, Project, BMBF and DFG. GermanyCRC Program, CFI, NSERC and WestGrid Project (Canada); BMBF and DFG (Ger- many);
. Sfi (ireland, SFI (Ireland);
The Swedish Research Council (Sweden); CAS and CNSF (China). The Swedish Research Council (Sweden); CAS and CNSF (China);
Alexander von Humboldt Foundation; and the Marie Curie Program. Alexander von Hum- boldt Foundation; and the Marie Curie Program.
Visitor from Augustana College. Sioux Falls, SD, USAVisitor from Augustana College, Sioux Falls, SD, USA.
Visitor from ICN-UNAM. Mexico City, MexicoVisitor from ICN-UNAM, Mexico City, Mexico.
Visitor from II. Physikalisches Institut. Göttingen, GermanyGeorg-August-UniversityVisitor from II. Physikalisches Institut, Georg-August- University, Göttingen, Germany.
. Fermilab International Fellow. Fermilab International Fellow.
. Deceased, Deceased.
. J H Kühn, G Rodrigo, Phys. Rev. D. 5954017J. H. Kühn and G. Rodrigo, Phys. Rev. D 59, 054017 (1999).
. M T Bowen, Phys. Rev. D. 7314008M. T. Bowen et al., Phys. Rev. D 73, 14008 (2006).
. S Dittmaier, Phys. Rev. Lett. 98262002S. Dittmaier et al., Phys. Rev. Lett. 98, 262002 (2007).
. O For Example, J H Antuñano, G Kühn, Rodrigo, arXiv:0709.1652For example, O. Antuñano, J. H. Kühn and G. Rodrigo, arXiv:0709.1652.
. V M Abazov, D0 CollaborationNucl. Instrum. Methods A. 565463V. M. Abazov et al. (D0 Collaboration), Nucl. Instrum. Methods A 565, 463 (2006).
. S Frixione, B R Webber, JHEP. 020629S. Frixione and B.R. Webber, JHEP 0206, 29 (2002);
. S Frixione, JHEP. 03087S. Frixione et al., JHEP 0308, 7 (2003).
G C Blazey, Fermilab-Pub- 00/297Proceedings of the Workshop: QCD and Weak Boson Physics in Run II. U. Baur, R.K. Ellis, and D. Zeppenfeldthe Workshop: QCD and Weak Boson Physics in Run IIG. C. Blazey et al., in Proceedings of the Workshop: QCD and Weak Boson Physics in Run II, edited by U. Baur, R.K. Ellis, and D. Zeppenfeld, Fermilab-Pub- 00/297 (2000).
. V M Abazov, D0 CollaborationPhys. Rev. D. 7692007V. M. Abazov et al. (D0 Collaboration), Phys. Rev. D 76, 092007 (2007).
. T Scanlon, University of LondonFermilab-ThesisPh.D. thesisT. Scanlon, Ph.D. thesis, University of London (2006), Fermilab-Thesis-2006-43.
. S Snyder, State University of New York at Stony BrookPh.D. ThesisS. Snyder, Ph.D. Thesis, State University of New York at Stony Brook (1995).
. C Adloff, H1 CollaborationNucl. Phys. 5453C. Adloff et al. (H1 Collaboration), Nucl. Phys. B545, 3 (1999).
. T Sjöstrand, Comput. Phys. Commun. 135238T. Sjöstrand et al., Comput. Phys. Commun. 135, 238 (2001).
R Brun, F Carminati, CERN Program Library Long Writeup W5013. unpublishedR. Brun and F. Carminati, CERN Program Library Long Writeup W5013, 1993 (unpublished).
. M L Mangano, JHEP. 03071M. L. Mangano et al., JHEP 0307, 1 (2003);
. S Höche, arXiv:hep-ph/0602031S. Höche et al., arXiv:hep-ph/0602031.
. W.-M Yao, J. Physics G. 331W.-M. Yao et al., J. Physics G 33, 1 (2006).
. V M Abazov, D0 CollaborationPhys. Rev. Lett. 92221801V. M. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 92, 221801 (2004);
T Aaltonen, CDF CollaboraarXiv:0709.0705tion). T. Aaltonen et al. (CDF Collabora- tion), arXiv:0709.0705.
. G Feldman, R Cousins, Phys. Rev. D. 573873G. Feldman and R. Cousins, Phys. Rev. D 57, 3873 (1998).
| [] |
[
"Towards a Heavy-ion Transport Capability in the MARS15 Code * TOWARDS A HEAVY-ION TRANSPORT CAPABILITY IN THE MARS15 CODE *",
"Towards a Heavy-ion Transport Capability in the MARS15 Code * TOWARDS A HEAVY-ION TRANSPORT CAPABILITY IN THE MARS15 CODE *"
] | [
"N V Mokhov \nFermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA\n\nFermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA\n",
"K K Gudima \nInstitute of Applied Physics\nAcademy of Sciences of Moldova\nMD-2028Kishinev, Moldova\n\nInstitute of Applied Physics, Academy of Sciences of Moldova\n2028Kishinev, MoldovaMD\n",
"S G Mashnik \nLos-Alamos National Laboratory\nB283, 87545Los-AlamosMS, New-MexicoUSA\n\nLos-Alamos National Laboratory\nB283, 87545Los-AlamosMS, New-MexicoUSA\n",
"I L Rakhno \nUniversity of Illinois at Urbana-Champaign\n1110 W. Green St61801-3080UrbanaIllinoisUSA\n\nUniversity of Illinois at Urbana-Champaign\n1110 W. Green Street61801-3080UrbanaIllinoisUSA\n",
"S I Striganov \nFermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA\n\nFermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA\n",
"N V Mokhov \nFermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA\n\nFermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA\n",
"K K Gudima \nInstitute of Applied Physics\nAcademy of Sciences of Moldova\nMD-2028Kishinev, Moldova\n\nInstitute of Applied Physics, Academy of Sciences of Moldova\n2028Kishinev, MoldovaMD\n",
"S G Mashnik \nLos-Alamos National Laboratory\nB283, 87545Los-AlamosMS, New-MexicoUSA\n\nLos-Alamos National Laboratory\nB283, 87545Los-AlamosMS, New-MexicoUSA\n",
"I L Rakhno \nUniversity of Illinois at Urbana-Champaign\n1110 W. Green St61801-3080UrbanaIllinoisUSA\n\nUniversity of Illinois at Urbana-Champaign\n1110 W. Green Street61801-3080UrbanaIllinoisUSA\n",
"S I Striganov \nFermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA\n\nFermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA\n"
] | [
"Fermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA",
"Fermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA",
"Institute of Applied Physics\nAcademy of Sciences of Moldova\nMD-2028Kishinev, Moldova",
"Institute of Applied Physics, Academy of Sciences of Moldova\n2028Kishinev, MoldovaMD",
"Los-Alamos National Laboratory\nB283, 87545Los-AlamosMS, New-MexicoUSA",
"Los-Alamos National Laboratory\nB283, 87545Los-AlamosMS, New-MexicoUSA",
"University of Illinois at Urbana-Champaign\n1110 W. Green St61801-3080UrbanaIllinoisUSA",
"University of Illinois at Urbana-Champaign\n1110 W. Green Street61801-3080UrbanaIllinoisUSA",
"Fermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA",
"Fermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA",
"Fermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA",
"Fermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA",
"Institute of Applied Physics\nAcademy of Sciences of Moldova\nMD-2028Kishinev, Moldova",
"Institute of Applied Physics, Academy of Sciences of Moldova\n2028Kishinev, MoldovaMD",
"Los-Alamos National Laboratory\nB283, 87545Los-AlamosMS, New-MexicoUSA",
"Los-Alamos National Laboratory\nB283, 87545Los-AlamosMS, New-MexicoUSA",
"University of Illinois at Urbana-Champaign\n1110 W. Green St61801-3080UrbanaIllinoisUSA",
"University of Illinois at Urbana-Champaign\n1110 W. Green Street61801-3080UrbanaIllinoisUSA",
"Fermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA",
"Fermi National Accelerator Laboratory\n220, 60510-0500BataviaMS, IllinoisUSA"
] | [
"10 th International Conference on Radiation Shielding"
] | In order to meet the challenges of new accelerator and space projects and further improve modelling of radiation effects in microscopic objects, heavy-ion interaction and transport physics have been recently incorporated into the MARS15 Monte Carlo code. A brief description of new modules is given in comparison with experimental data.Abstract. In order to meet the challenges of new accelerator and space projects and further improve modelling of radiation effects in microscopic objects, heavy-ion interaction and transport physics have been recently incorporated into the MARS15 Monte Carlo code. A brief description of new modules is given in comparison with experimental data. | 10.1093/rpd/nci147 | [
"https://export.arxiv.org/pdf/nucl-th/0404085v1.pdf"
] | 28,609,673 | nucl-th/0404085 | dc2bdcb2f4bdce1569c718fa79e86e1fe3f23e78 |
Towards a Heavy-ion Transport Capability in the MARS15 Code * TOWARDS A HEAVY-ION TRANSPORT CAPABILITY IN THE MARS15 CODE *
May 9-14, 2004
N V Mokhov
Fermi National Accelerator Laboratory
220, 60510-0500BataviaMS, IllinoisUSA
Fermi National Accelerator Laboratory
220, 60510-0500BataviaMS, IllinoisUSA
K K Gudima
Institute of Applied Physics
Academy of Sciences of Moldova
MD-2028Kishinev, Moldova
Institute of Applied Physics, Academy of Sciences of Moldova
2028Kishinev, MoldovaMD
S G Mashnik
Los-Alamos National Laboratory
B283, 87545Los-AlamosMS, New-MexicoUSA
Los-Alamos National Laboratory
B283, 87545Los-AlamosMS, New-MexicoUSA
I L Rakhno
University of Illinois at Urbana-Champaign
1110 W. Green St61801-3080UrbanaIllinoisUSA
University of Illinois at Urbana-Champaign
1110 W. Green Street61801-3080UrbanaIllinoisUSA
S I Striganov
Fermi National Accelerator Laboratory
220, 60510-0500BataviaMS, IllinoisUSA
Fermi National Accelerator Laboratory
220, 60510-0500BataviaMS, IllinoisUSA
N V Mokhov
Fermi National Accelerator Laboratory
220, 60510-0500BataviaMS, IllinoisUSA
Fermi National Accelerator Laboratory
220, 60510-0500BataviaMS, IllinoisUSA
K K Gudima
Institute of Applied Physics
Academy of Sciences of Moldova
MD-2028Kishinev, Moldova
Institute of Applied Physics, Academy of Sciences of Moldova
2028Kishinev, MoldovaMD
S G Mashnik
Los-Alamos National Laboratory
B283, 87545Los-AlamosMS, New-MexicoUSA
Los-Alamos National Laboratory
B283, 87545Los-AlamosMS, New-MexicoUSA
I L Rakhno
University of Illinois at Urbana-Champaign
1110 W. Green St61801-3080UrbanaIllinoisUSA
University of Illinois at Urbana-Champaign
1110 W. Green Street61801-3080UrbanaIllinoisUSA
S I Striganov
Fermi National Accelerator Laboratory
220, 60510-0500BataviaMS, IllinoisUSA
Fermi National Accelerator Laboratory
220, 60510-0500BataviaMS, IllinoisUSA
Towards a Heavy-ion Transport Capability in the MARS15 Code * TOWARDS A HEAVY-ION TRANSPORT CAPABILITY IN THE MARS15 CODE *
10 th International Conference on Radiation Shielding
Funchal (Madeira), PortugalMay 9-14, 2004FERMILAB-Conf-04/052-AD April 2004 __________________________ * Presented paper at the 2
In order to meet the challenges of new accelerator and space projects and further improve modelling of radiation effects in microscopic objects, heavy-ion interaction and transport physics have been recently incorporated into the MARS15 Monte Carlo code. A brief description of new modules is given in comparison with experimental data.Abstract. In order to meet the challenges of new accelerator and space projects and further improve modelling of radiation effects in microscopic objects, heavy-ion interaction and transport physics have been recently incorporated into the MARS15 Monte Carlo code. A brief description of new modules is given in comparison with experimental data.
INTRODUCTION
The MARS Monte Carlo code (1) is widely used in numerous accelerator, detector, shielding and cosmic ray applications. The needs of the Relativistic Hevy-Ion Collider, Large Hadron Collider, Rare Isotope Accelerator and NASA projects have recently induced adding heavy-ion interaction and transport physics to the MARS15 code. The key modules of the new implementation are described below along with their comparisons to experimental data.
HEAVY-ION NUCLEAR INTERACTIONS
The 2003 version (2) of the LAQGSM code (3) was implemented into MARS15 after substantial revisions and merging with CEM03 and native MARS14 modules. It can now be used in full transport simulations in complex macro-systems for modelling all heavy-ion and hadron nuclear interactions from 10 MeV/A to 800 GeV/A as well as photo-nuclear interactions (currently in a limited energy range). LAQGSM03 includes an improved version of the Dubna intra-nuclear cascade model (4) that makes use of experimental elementary cross sections (or those calculated with the Quark-Gluon String Model for energies above 4.5 GeV/A), the pre-equilibrium model from the improved Cascade-Exciton Model (5) as realized in the CEM03 code (6,7) , refined versions of the Fermi break-up and coalescence models (4) , and an improved version of the Furihata's Generalized Evaporationfission Model (8) (GEM2). This implementation provides a power of full theoretically consistent modelling of exclusive and inclusive distributions of secondary particl-________________________________________
NUCLEUS-NUCLEUS CROSS SECTIONS
Inelastic and elastic cross sections for heavy ion nuclear interactions have been extensively studied both theoretically and experimentally for the past decades.
Nevertheless, quality of experimental data is still far from perfect and measurements by different groups in many cases contradict each other (13) . Several empirical prescriptions have been developed for inelastic cross sections. Most recent and comprehensive studies were performed at NASA (14) and JINR (15) . Both models show reasonable agreement with experimental data for energy range from a few A MeV to 200 A GeV and for ions, both projectiles and targets, ranging from deuteron to lead. A typical comparison between data and these models is presented in Fig. 5 for 20 Ne ions. Situation is similar for other projectiles. The JINR model was chosen for MARS15 because it describes also elastic cross sections needed for full particle transport. ions vs JINR (15) (solid) and NASA (14) (dashed ) models.
ELECTROMAGNETIC INTERACTIONS
A latest version of the algorithm used in MARS15 for modelling of correlated ionization energy loss and multiple Coulomb scattering is described in Ref. (16). It is based on a separate treatment of "soft" and "hard" interactions. A large number of "soft" collisions are described using a "continuous scattering approximation"; a small number of "hard" collisions are simulated directly. A transition angle between "soft" and "hard" collisions is determined as a function of a step-size providing a possibility for fast and precise simulation. "Hard" collisions are sampled taking into account projectile and target form-factors and exact kinematics of projectileelectron interactions. Corresponding energy losses are calculated from a simulated scattering angle. Energy loss in "soft" projectile-electron collisions is described by a modified Vavilov function, approximated by a lognormal distribution. Calculated correlations between energy loss and scattering are quite substantial for low-Z targets. The mean stopping power used for a normalization of ionization energy loss on a step is described in the next section. This algorithm is directly applicable for heavy ions of a charge z . Measurements on various targets (17) confirm this, although there is an evidence that the width of energy loss distributions is narrower for z close to 100. Radiative processes for heavy ionsbremsstrahlung and direct pair production -are modelled directly (1) .
MEAN STOPPING POWER
The mean ionization energy loss, , / dx dE is calculated using the Bethe-Bloch formalism (18) in combination with various corrections (1,18) . Two additional corrections have been implemented into the code to describe better the Barkas effect and take into account electron capture by low-energy ions. An improved Barkas term is described according to Ref. (19). Electron capture reveals itself as a significant decrease in an ion charge at low kinetic energies. The effective ion charge, eff z , is determined according to semi-empirical formulae (20) and used instead of a bare projectile charge. Thus, the regular ionization logarithm (18)
) ( 1 Z V F z eff + where ) (V F
is a tabulated function, V is a dimensionless parameter equal to , / Z α βγ α is the fine-structure constant, and all the other terms have their usual meaning (18)(19)(20) . A comparison with data is shown in Figs. 6-7. For light ions like 12 C our model is well justified -in aluminium the difference in the calculated dx dE / values between our approach and a more comprehensive (from formal standpoint) one (19) is less than 1% in the energy region from 10 -3 up to 10 5 GeV/A. For Pb ions the difference increases up to 6% above 0.1 GeV/A. The advantage of our approach is in its speed (a factor of 10 3 when compared to Ref. (19)).
Bremsstrahlung and direct pair production by heavy ions are modelled directly, as mentioned in previous section, or -alternatively -can be treated in a continuous slowdown approximation with corresponding MARS modules for dx dE / (see Ref. (1,18)).
MARS15 TEST FOR A THICK LEAD TARGET
Results of a heavy-ion implementation test in MARS15 are presented in Fig.8 for a lead cylinder of 20-cm diameter and 60-cm thick irradiated by 0.5 to 3.65 GeV/A light ion beams. Our results are in a good agreement with data (23) and predictions of the latest version of the SHIELD code (24) presented in Ref. (25). Figure 8. Calculated and measured (23) total neutron yield (E<14.5 MeV) from a lead cylinder vs ion beam energy.
*
This work was supported by the Universities Research Association, Inc., under contract DE-AC02-76CH03000 with the U.S. Department of Energy, and in part by the Moldovan-U.S. Bilateral Grants Program, CRDF Projects MP2-3025 and MP2-3045, the NASA ATP01 Grant NRA-01-01-ATP-066, and the Higher Education Cooperative Act Grant of the Illinois Board of Higher Education. ** [email protected] es, spallation, fission, and fragmentation products. Although benchmarking results, shown in Figs. 1 and 2 for nuclide production and in Figs. 3 and 4 for inclusive spectra, are quite impressive, further development of this package is underway(6,7) .
Figure 1 .
1Mass yield in d+ 208 Pb interaction at 1 GeV/A as calculated with original LAQGSM03 and that implemented into MARS15, and measured in Ref.(9).
Figure 2 .
2Mass yield in 86 Kr +9 Be reaction at 1 GeV/A as calculated with LAQGSM and measured in Ref.(10).
Figure 3 .
3Comparison of measured(11) differential cross sections of neutrons with LAQGSM and calculations with QMD and HIC models for 560 MeV/A Ar+Pb reaction.
Figure 4 .
4Invariant proton yield per central Au+Au collision at 8 GeV/A as calculated with LAQGSM03 (histograms) and measured in Ref. (12) (symbols). Solid lines and open circles is forward production, dashed lines and open triangles is backward production. Midrapidity (upper set) is shown unscaled, while the 0.1 unit rapidity slices are scaled down by successive factors of 10.
Figure 5 .
5Data(13) (symbols) on inelastic cross sections of 20 Ne
Figure 6 .
6Measured(21) (symbols) and calculated (lines) ionization energy losses by various ions in aluminium.
Figure 7 .
7Measured(22) and calculated (full and open symbols, respectively) ionization losses by 780-MeV/A Xe ions.
The MARS code system user's guide. Fermilab-FN-628. N V Mokhov, Fermi National Accelerator LaboratoryMokhov, N. V. The MARS code system user's guide. Fermilab-FN-628, Fermi National Accelerator Laboratory (1995);
N V Mokhov, O E Krivosheev, Fermilab-Conf-00/181MARS code status. Proc. Monte Carlo. Lisbon943Mokhov, N. V., Kri- vosheev, O. E. MARS code status. Proc. Monte Carlo 2000 Conf., p. 943, Lisbon, October 23-26, 2000, Fermilab-Conf-00/181 (2000);
N V Mokhov, Status of MARS code. Fermilab-Conf-03/053. Fermi National Accelerator LaboratoryMokhov, N. V. Status of MARS code. Fermilab-Conf-03/053, Fermi National Accelerator Laboratory (2003);
Recent developments in LAQGSM. S G Mashnik, K K Gudima, R E Prael, A J Sierk, In these proceedingsMashnik, S.G., Gudima, K.K., Prael, R.E., Sierk, A.J. Recent developments in LAQGSM. In these proceedings.
User manual for the code LAQGSM. K K Gudima, S G Mashnik, A J Sierk, LA-UR-01-6804Los Alamos National LaboratoryGudima, K. K., Mashnik, S. G., and Sierk, A. J. User manual for the code LAQGSM, LA-UR-01-6804, Los Alamos National Laboratory (2001).
Particle emission in light and heavy-ion reactions. V D Toneev, K K Gudima, Nucl. Phys. 400Toneev, V. D., Gudima, K. K. Particle emission in light and heavy-ion reactions. Nucl. Phys. A400, 173c-190c (1983).
CEM2k-recent developments in CEM. S G Mashnik, A J Sierk, E-print: nucl-th/0011064Proc. Conf. on Accelerator Applications. Conf. on Accelerator ApplicationsWashington DC, USAMashnik, S. G., and Sierk, A. J. CEM2k-recent developments in CEM. Proc. Conf. on Accelerator Applications, Washington DC, USA, pp. 328-341 (2000), E-print: nucl-th/0011064.
Analysis of the GSI A+p and A+A spallation, fission, and fragmentation measurements with the LANL CEM2k and LAQGSM codes. S G Mashnik, K K Gudima, R E Prael, A J Sierk, E-print : nucl-th/0404018Proc. Workshop on Nuclear Data for the Transmutation of Nuclear Waste, GSI, Darmstad, September 1-5. Workshop on Nuclear Data for the Transmutation of Nuclear Waste, GSI, Darmstad, September 1-5Los Alamos LAMashnik, S.G., Gudima, K.K., Prael, R.E., Sierk, A.J. Analysis of the GSI A+p and A+A spallation, fission, and fragmentation measurements with the LANL CEM2k and LAQGSM codes. Proc. Workshop on Nuclear Data for the Transmutation of Nuclear Waste, GSI, Darmstad, Septem- ber 1-5, 2003, Los Alamos LA-UR-04-1873 (2004), E-print : nucl-th/0404018.
Improved intranuclear cascade models for the codes CEM2k and LAQGSM. S G Mashnik, K K Gudima, A J Sierk, R E Prael, UR-04-0039Los Alamos LAMashnik, S.G., Gudima, K.K., Sierk, A.J, Prael, R.E. Improved intranuclear cascade models for the codes CEM2k and LAQGSM, Los Alamos LA-UR-04-0039 (2004).
Statistical analysis of light fragment production from medium energy proton-induced reactions. S Furihata, Nucl. Instrum. Methods. 1713Furihata, S. Statistical analysis of light fragment production from medium energy proton-induced reactions. Nucl. Instrum. Methods. B171(3), 251-258 (2000);
The GEM code version 2. User's manual. S Furihata, Mitsubishi Research Institute, IncTokyo, JapanFurihata, S. The GEM code version 2. User's manual, Mitsubishi Research Institute, Inc., Tokyo, Japan (2001).
Primary-residue production cross sections and kinetic energy in 1 A GeV 208 Pb + deuteron reactions. Nucl. Phys. A703. T Enqvist, 17Enqvist, T., et al. (17 authors). Primary-residue production cross sections and kinetic energy in 1 A GeV 208 Pb + deuteron reactions. Nucl. Phys. A703, pp. 435-465 (2002).
. B Voss, D Ph, Thesis, Darmstadt, Voss, B. Ph.D. thesis, KTH Darmstadt (1995). http://www-wnt.gsi.de/kschmidt/theses.htm.
Double-differential cross sections for the neutron production from heavy-ion reactions at energies. Y Iwata, 11 authorsIwata, Y., et al. (11 authors). Double-differential cross sections for the neutron production from heavy-ion reactions at energies 290-600
. / A Mev, Phys. Rev. 6454609MeV/A. Phys. Rev. C64, 054609 (2001).
Longitudinal flow of protons from (2-8) A GeV central Au+Au collisions. J L Klay, Phys. Rev. Lett. 8810230153 authorsKlay, J.L., et al. (53 authors). Longitudinal flow of protons from (2-8) A GeV central Au+Au collisions. Phys. Rev. Lett. 88, 102301 (2002).
Cross-sections of particle and nuclei interactions with nuclei. (Dubna: Joint Institute for Nuclear Research. V S Barashenkov, In RussianBarashenkov, V. S. Cross-sections of particle and nuclei interactions with nuclei. (Dubna: Joint Institute for Nuclear Research) (1993) (In Russian).
Accurate universal parametrization of absorption cross sections III -light systems. R K Tripathi, F A Cucinotta, J W Wilson, Nucl. Instrum. Methods. 1554Tripathi, R. K., Cucinotta, F. A., Wilson, J. W. Accurate universal parametrization of absorption cross sections III -light systems. Nucl. Instrum. Methods. B155(4), 349-356 (1999).
Integral nucleus-nucleus cross-sections. V S Barashenkov, H Kumawat, Joint Inst. for Nuclear Research. JINR E2-2003-128Barashenkov, V. S., Kumawat, H. Integral nucleus-nucleus cross-sections. JINR E2-2003-128, Joint Inst. for Nuclear Research (2003).
On the theory and simulation of multiple Coulomb scattering of heavy particles. S I Striganov, In these proceedingsStriganov, S.I. On the theory and simulation of multiple Coulomb scattering of heavy particles. In these proceedings.
Effect of nuclear size on the stopping power of ultrarelativistic ions. S Datz, H F Krause, C R Vane, H Knudse, P Grafstrom, R H Schuch, Phys. Rev. Lett. 77Datz, S., Krause, H.F., Vane, C.R., Knudse, H., Grafstrom, P., Schuch, R.H. Effect of nuclear size on the stopping power of ultrarelativistic ions. Phys. Rev. Lett., 77, 2925-2928 (1996).
Muon stopping power and range tables 10 MeV-100 TeV. Atomic Data and Nucl. Data Tables. D E Groom, N V Mokhov, S I Striganov, 78Groom, D. E., Mokhov, N. V., and Striganov, S. I. Muon stopping power and range tables 10 MeV-100 TeV. Atomic Data and Nucl. Data Tables. 78(2), 183-356 (2001).
Energy loss of relativistic heavy ions in matter. B A Weaver, A J Westphal, Nucl. Instrum. Meth. 187Weaver, B. A., Westphal, A. J. Energy loss of relativistic heavy ions in matter. Nucl. Instrum. Meth. B187, 285-301 (2002).
Semi-empirical formulae for heavy ion stopping powers in solids in the intermediate energy range. F Hubert, R Bimbot, H Gauvin, Nucl. Instrum, Methods. 364Hubert, F., Bimbot, R., and Gauvin, H. Semi-empirical formulae for heavy ion stopping powers in solids in the intermediate energy range. Nucl. Instrum, Methods. B36(4), 357-363 (1989).
ISBN 0 08 021604 8The stopping and range of ions in matter. Pergamon Press2The stopping and range of ions in matter (Ed. by J.F. Ziegler). Vol. 2, Pergamon Press) (1977) ISBN 0 08 021604 8.
Investigation of the passage of 10-1000-MeV/nucleon superheavy ions through homogeneous media. S D Bogdanov, S S Bogdanov, E E Zhurkin, V F Kosmach, JETP. 882Bogdanov, S. D., Bogdanov, S. S., Zhurkin, E. E., and Kosmach, V. F. Investigation of the passage of 10-1000-MeV/nucleon superheavy ions through homogeneous media. JETP. 88(2), 220-226 (1999).
Neutron emission from an extended lead target under the action of light ions in GeV region. R G Vassil'kov, V I Yurevich, Proc. ICANS-XI Conf, KEK. ICANS-XI Conf, KEKTsukuba, Japan1340Vassil'kov, R.G., Yurevich, V.I. Neutron emission from an extended lead target under the action of light ions in GeV region. Proc. ICANS- XI Conf, KEK, Tsukuba, Japan, October 22-26, 1990, vol. 1, p. 340.
SHIELD -universal Monte Carlo hadron transport code: scope and applications. Radiation Measurements. A V Dementyev, N M Sobolevsky, 30Dementyev, A.V., Sobolevsky, N.M. SHIELD -universal Monte Carlo hadron transport code: scope and applications. Radiation Measure- ments 30, 553-557 (1999).
Monte Carlo calculation of neutron yield from extended iron and lead targets irradiated by 1 and 3.65 GeV/u ion beams. N M Sobolevsky, E Mustafin, GSI-Acc-Note-2003- 09-001Proc. Workshop on Nuclear Data for the Transmutation of Nuclear Waste, GSI, Darmstad. Workshop on Nuclear Data for the Transmutation of Nuclear Waste, GSI, DarmstadDarmstadtSobolevsky, N.M., Mustafin, E. Monte Carlo calculation of neutron yield from extended iron and lead targets irradiated by 1 and 3.65 GeV/u ion beams. Proc. Workshop on Nuclear Data for the Transmutation of Nuclear Waste, GSI, Darmstad, September 1-5, 2003, GSI-Acc-Note-2003- 09-001, Darmstadt (2004).
| [] |
[
"On the role of the subsidiary fields in the momentum and angular momentum in covariantly quantized QED and QCD",
"On the role of the subsidiary fields in the momentum and angular momentum in covariantly quantized QED and QCD"
] | [
"Elliot Leader \nBlackett laboratory Imperial College London Prince Consort Road\nSW7 2AZLondonUK\n"
] | [
"Blackett laboratory Imperial College London Prince Consort Road\nSW7 2AZLondonUK"
] | [] | The covariant quantization of QED and QCD requires the introduction of subsidiary gauge-fixing and ghost fields and it is crucial to understand the role of these in the energy-momentum tensor, and in the momentum and angular momentum operators. These issues were studied in [1], and key results from this study were utilized in the major review of angular momentum in[2]. Damski [3] has rightly criticized as incorrect the derivation of certain equations in [1]. We show, however, that the key results of [1] which are utilized in [2] are unaffected by Damski's criticism. | 10.1103/physrevd.105.036005 | [
"https://arxiv.org/pdf/2108.07730v1.pdf"
] | 237,142,312 | 2108.07730 | 4344be80bf6e733c437a28060d225f2a4027c4ce |
On the role of the subsidiary fields in the momentum and angular momentum in covariantly quantized QED and QCD
17 Aug 2021
Elliot Leader
Blackett laboratory Imperial College London Prince Consort Road
SW7 2AZLondonUK
On the role of the subsidiary fields in the momentum and angular momentum in covariantly quantized QED and QCD
17 Aug 2021(Dated: August 18, 2021)numbers: 1115-q1220-m1238Aw1238Bx1238-t1420Dh * eleader@imperialacuk
The covariant quantization of QED and QCD requires the introduction of subsidiary gauge-fixing and ghost fields and it is crucial to understand the role of these in the energy-momentum tensor, and in the momentum and angular momentum operators. These issues were studied in [1], and key results from this study were utilized in the major review of angular momentum in[2]. Damski [3] has rightly criticized as incorrect the derivation of certain equations in [1]. We show, however, that the key results of [1] which are utilized in [2] are unaffected by Damski's criticism.
INTRODUCTION
It is well known that the covariant quantization of QED and QCD i.e. in which the photon vector potential A µ (x) and the gluon vector potential A µ a transform as genuine Lorentz 4-vectors, is a non-trivial task [4][5][6][7] involving the introduction of a scalar gauge-fixing field (Gf) in QED and both a gauge-fixing field and Faddeev-Popov ghosts fields (Gf+Gh) in QCD. In both QED and QCD the expressions for the linear and angular momentum operators include terms involving all these fields.
As explained in detail in [1] it is necessary to work in an indefinite-metric space i.e. one in which the "length" or norm of a vector can be either positive or negative and the definition of the "physical states" with positive norm has to specified with care. It is also necessary to specify the condition for an operator to represent a physically measurable quantity i.e. to be an "observable".
The situation is further complicated by the fact that it is possible to deal with different versions of the energy-momentum tensor t µν , of which the two most important are the Canonical version t µν can which follows from Noether's theorem and the Bellinfante version t µν bel which is symmetric under (µ ↔ ν) and which differs from t µν can by a divergence term, as will be spelled out in detail below. Both these versions of the energy-momentum tensor are conserved quantities. Based on these one can define, in the standard way, the momentum operators P µ can and P µ bel and the angular momentum operators J i can and J i bel . Because the energy-momentum tensors are related by a divergence one can show that for the matrix elements between any normalizable physical states, Φ| P µ bel |Ψ = Φ| P µ can |Ψ and Φ| J i bel |Ψ = Φ| J i can |Ψ . Of key physical interest in QCD is the question of the fraction of the momentum and angular momentum carried by quarks and gluons in a hadron, with analogous questions about electrons and photons in QED. Clearly to answer these questions one has to know the contributions of the subsidiary fields to the physical matrix elements of the above operators. In all papers dealing with the QCD case it is assumed, without comment, that the contribution from the subsidiary fields is zero. A proof of this was given in [1], which was based on the demonstration that for physical matrix elements in QED Φ ′ | t µν bel (Gf ) |Φ = Φ ′ | t µν can (Gf ) |Φ = 0 and for QCD Φ ′ | t µν bel (Gf + Gh) |Φ = Φ ′ | t µν can (Gf + Gh) |Φ = 0. Damski [3] has pointed out that the proof for both the Canonical and Bellinfante cases in QED is wrong, because it assumes that the physical states alone form a complete set and thus uses 1 = |Φ Φ|, which is an incorrect "resolution of the identity", because it leaves out the states of negative norm. Although he does not comment on QCD, Damski's argument shows that the proof that Φ ′ | t µν can (Gf + Gh) |Φ = 0 in [1] is also incorrect. This criticism seems to imply that the conclusion reached in [1] that the subsidiary fields do not contribute to the momentum and angular momentum is false, but as will be shown, this implication is wrong.
We shall show the following: a) while, indeed, for the Canonical case
Φ ′ | t µν can (Gf ) |Φ = 0 and Φ ′ | t µν can (Gf + Gh) |Φ = 0(1)
on the contrary, for the Bellinfante case, as claimed in [1]
Φ ′ | t µν bel (Gf ) |Φ = 0 and Φ ′ | t µν bel (Gf + Gh) |Φ = 0. (2) b) despite Eq. (1) one has ∂ µ Φ ′ | t µν can (Gf ) |Φ = ∂ µ Φ ′ | t µν can (Gf + Gh) |Φ = 0(3)
and obviously an analogous result for the Bellinfante case as a consequence of (2), which are key results utilized in the major review [2] of the "Angular Momentum Controversy". c) despite Eq, (1) it turns out that the subsidiary fields do not contribute to the physical matrix elements of either the Canonical or Bellinfante versions of the momentum or angular momentum. This crucial result, as mentioned above, is normally assumed without comment in papers on QCD.
II. PHYSICAL MATRIX ELEMENTS OF THE BELLINFANTE ENERGY-MOMENTUM
TENSOR
We shall show, as claimed in [1], that the physical matrix elements of the gauge-fixing and ghost contributions t µν bel (Gf + Gh) in QCD and the gauge-fixing contribution t µν bel (Gf ) in QED, vanish. The proof given for the QED case in [1], as pointed out by Damski [3], is incorrect, but the result is actually true . Surprisingly it turns out that the QCD case is simpler to deal with than the QED case , which can be derived as a special case.
A. Quantum Chromodynamics
The pure quark-gluon Lagrangian L qG is
L qG = − 1 4 G a µν G µν a + 1 2ψ l [δ lm i ( − → ∂ − ← − ∂ ) − 2 gt a lm A a ]ψ m .(4)
In order to quantize the theory covariantly one has to introduce both a gauge-fixing field B(x) and Fadeev-Popov anti-commuting fermionic ghost fields c(x),c(x). The Kugo-Ojima
Lagrangian [8] for the covariantly quantized theory is then
L = L qG + L Gf +Gh (5) where L Gf +Gh = −i(∂ µca )D ab µ c b − (∂ µ B a )A a µ + a 2 B a B a .(6)
The physical states |Ψ are defined by the subsidiary conditions
Q B |Ψ = 0 (7) Q c |Ψ = 0 (8)
where the conserved, hermitian charge Q B is given by
Q B = d 3 x[B a ← → ∂ 0 c a − gB a f abc A b 0 c c − i(g/2)(∂ 0c a )f abc c b c c ].(9)
and the conserved charge Q c
Q c = d 3 x[c a ← → ∂ 0 c a − gc a f abc A b 0 c c ](10)
"measures" the ghost number
i[Q c , φ] = Nφ(11)
where N = 1 for φ = c, −1 for φ =c and 0 for all other fields.
The Bellinfante energy-momentum tensor is (13) and the gauge-fixing and ghost terms are given by
t µν bel = t µν bel (qG) + t µν bel (Gf + Gh) (12) where t µν bel (qG) = i 4 [ψ l γ µ ← → D ν ψ l + (µ ↔ ν)] − G µβ a G ν aβ − g µν L qGt µν bel (Gf + Gh) = −(A µ a ∂ ν B a + A ν a ∂ µ B a ) − i[(∂ µc a )D ν ab c b + (∂ νc a )D µ ab c b ] − g µν L Gf +Gh . (14)
This can be rewritten [7] as an anti-commutator with Q B
t µν bel (Gf + Gh) = −{Q B , (∂ µc a )A ν a + (∂ νc a )A µ a + g µν [ a 2c a B a − (∂ ρc a )A a ρ ] }.(15)
It follows from Eqs. (7) and (15) that t µν bel (Gf +Gh) does not contribute to physical matrix elements i.e.
Φ ′ | t µν bel (Gf + Gh) |Φ = 0 (16) so that Φ ′ | t µν bel,QCD |Φ = Φ ′ | t µν bel (qG) |Φ(17)
and hence that the subsidiary fields do not contribute to the QCD expressions for either P µ bel or J bel i.e.
Φ| P µ bel |Φ ≡ d 3 x Φ| t 0µ bel |Φ = d 3 x Φ| t 0µ bel (qG) |Φ = Φ| P µ bel (qG) |Φ(18)
and
Φ| J i bel |Φ ≡ 1 2 ǫ ijk d 3 x Φ| x j t 0k bel − x k t 0j bel |Φ = 1 2 ǫ ijk d 3 x Φ| x j t 0k bel (qG) − x k t 0j bel (qG) |Φ = Φ| J i bel (qG) |Φ .(19)
B. Quantum Electrodynamics
The most general covariantly quantized version of QED is given by the Lautrup-Nakanishi
Lagrangian density [4,5], which is a combination of the Classical Lagrangian (Clas) and a Gauge Fixing part (Gf )
L = L Clas + L Gf(20)
where
L Clas = − 1 4 F µν F µν + 1 2 [ψ(i ∂ − m + e A)ψ + h.c.](21)
and
L Gf = −∂ µ B(x).A µ (x) + a 2 B 2 (x)(22)
where B(x) is the gauge-fixing field and the parameter a determines the structure of the photon propagator and is irrelevant for the present discussion 1
The physical states |Φ of the theory are defined to satisfy
B (+) (x)|Φ = 0 (23) where B(x) = B (+) (x) + B (−) (x)(24)
with B (±) (x) the positive/negative frequency parts of B(x).
For the conserved Bellinfante density one finds,
t µν bel = θ µν bel + t µν bel (Gf )(25)
where θ µν bel , which is referred to as the classical energy momentum tensor density, is
θ µν bel = i 4ψ (γ µ ← → D ν + γ ν ← → D µ ) ψ − F µβ F ν β − g µν L Clas (26) where ← → D ν = ← → ∂ ν − 2ieA ν , and t µν bel (Gf ) = −A ν ∂ µ B − A µ ∂ ν B − g µν L Gf(27)
As explained in Kugo-Ojima [8] the QED expressions can be obtained from the QCD case by putting the structure constants to zero and suppressing the colour group labels, in whch case the gauge-fixing field B and the ghost fields c andc become free fields. The expression Eq. (9) for the charge Q B then becomes, in terms of creation and annihilation operators
Q B = i d 3 k (2π) 3 2E k [c † k B k − B † k c k ].(28)
Because the Fadeev-Popov ghosts are here free, the state vector space V can be decomposed into a direct product V = V phys ⊗ V F P where V phys is the usual QED physical state vector space. Moreover the ghosts are redundant, so that one can work in the sector containing neither c norc ghosts i.e. V phys ⊗ |0 F P . Hence the physical states in QED can be taken to be |Φ ⊗ |0 F P . Using this and (28), Eq. (7) can then be shown to imply (23). Hence
Φ ′ | t µν bel,QED (Gf ) |Φ = F P 0| ⊗ Φ ′ | t µν bel,F ree (Gf + Gh) |Φ ⊗ |0 F P = 0(29)
where t µν bel,F ree (Gf + Gh) is t µν bel,QCD (Gf + Gh) in which the structure constants are put to zero and the gauge-fixing and ghost fields are free. Thus
Φ ′ | t µν bel,QED |Φ = Φ ′ | θ µν bel |Φ(30)
and hence, as in QCD, the subsidiary fields do not contribute to the QED expressions for the physical matrix elements of either P µ bel or J bel . show that the concrete expression for the gauge fixing contribution to the Bellinfante angular momentum, given in Damski's paper [3], which deals only with the free electromagnetic case, actually vanishes i.e. we shall show directly that
Φ| J bel (Gf) |Φ = 0(31)
where, in Damski's notation
J i bel (Gf) = J i div + J i ξ .(32)
Consider Φ|J i bel (Gf)|Φ = Φ|J i div + J i ξ |Φ = d 3 zǫ imn Φ|z m I n (z) − z n I m (z)|Φ
with, following Damski,
I n (z) = A n ∂ j F 0j + (∂ · A)∂ n A 0 .(34)
Defining
B(z) ≡ −∂ · A(35)
and using the fact that in Damski the fields are free, one obtains
I n (z) = A n ∂ 0 B − B∂ n A 0 .(36)
We split the fields into their positive and negative frequency parts
B(z) = B (+) + B (−) B (−) = [B (+) ] † and A µ (z) = A (+) µ + A (−) µ A (−) µ = [A (+) µ ] † (37) with A (+) µ (z) = [dk ′ ] σ=3 σ=0 ǫ µ (k ′ , σ)c k ′ σ e −ik ′ ·z(38)
where the ǫ µ are polarization vectors and the c k ′ σ are annihilation operators and we use the
shorthand [dk ′ ] ≡ d 3 k ′ (2π) 3/2 √ 2ω k ′ .(39)
In Damski's notation one has
B (+) (z) = −i [dk]ω k L k ωe −ik·z (40) where L k = c k3 − c k0 .(41)
The physical states, as usual, are defined to satisfy
B (+) (z)|Φ = 0 Φ|B (−) (z) = 0.(42)
Using these and the fact that the commutators [B (+) , A The relevant commutators and polarization vectors are
[L k , c † k ′ 0 ] = −[c k0 , c † k ′ 0 ] = δ 3 (k ′ − k) [L k , c † k ′ 3 ] = [c k3 , c † k ′ 3 ] = δ 3 (k ′ − k)(44)
and
ǫ µ (k, 0) = (1, 0) ǫ µ (k, 3) = (0, k/ω k ).(45)
Substituting Eqs. (38, 40, 44, and 45) into Eq. (43) , we find, after some labour, that
[A (+) n , ∂ 0 B (−) ] − [B (+) , ∂ n A (−) 0 ] = 0(46)
and thus that in QED
Φ|J i bel,QED (Gf)|Φ = 0 (47)
in agreement with the more general derivation based on the QCD case.
D. Bellinfante summary
To summarize, despite the incorrect proof for the QED case given in [1] , the physical matrix elements of the gauge-fixing and ghost contributions to the Bellinfante version of the energy-momentum tensor, in both QED and QCD, actually do vanish. Hence the only contributions to the momentum P µ bel and angular momentum J bel are from photons and electrons in the QED case and from quarks and gluons in QCD. This latter property was thus correctly utilized in the review paper [2]. Also, in writing down the most general structure for the matrix elements of Φ ′ | t µν bel (qG) |Φ in [2], use was made of the claim that
∂ µ Φ ′ | t µν bel (qG) |Φ = 0.(48)
This follows because the total t µν bel is a conserved operator and, via (17),
∂ µ Φ ′ | t µν bel (qG) |Φ = ∂ µ Φ ′ | t µν bel |Φ = 0.(49)
Note also that this justifies the results in several papers in the literature, e.g. Ji [10][11][12], Jaffe and Manohar [13], Bakker, Leader and Trueman (BLT) [14] and Wakamatsu [15,16],
where the general structure of the physical matrix elements of t µν bel (qG) (or its QED analogue) is derived under the unstated assumption that Eq. (49) holds.
III. PHYSICAL MATRIX ELEMENTS OF THE CANONICAL ENERGY-MOMENTUM TENSOR
In [1] it was claimed that the physical matrix elements of the gauge-fixing and ghost contributions t µν can (Gf + Gh) in QCD and the gauge-fixing contribution t µν can (Gf ) in QED, vanish. The result given for the QED case in [1], as pointed out by Damski [3], is wrong because of an incorrect use of the "resolution of the identity" for a space with an indefinite metric, and, although not discussed by Damski, also the QCD result is wrong for the same reason.
Thus, in contrast to the Bellinfante case and contrary to the claims made in [1], in QCD
Φ ′ | t µν can (Gf + Gh) |Φ = 0 (50) and in QED Φ ′ | t µν can (Gf ) |Φ = 0.(51)
We shall analyze the consequences of these for the QCD case. Completely analogous arguments hold for the case of QED.
There are two questions which have to be answered. Given (50):
1) Is the analysis of the general structure of the physical matrix elements of the quark-gluon t µν can (qG) given in [2] correct? 2) Do the subsidiary fields contribute to the physical matrix elements of the Canonical momentum P µ can and angular momentum J can ?
A. Structure of the physical matrix elements of the quark-gluon t µν can (qG)
The Bellinfante and Canonical energy-momentum tensors differ from each other by a divergence term of the following form:
t µν can = t µν bel − ∂ λ G λµν ,(52)
where the so-called superpotential reads
G λµν = 1 2 M λµν spin + M µνλ spin + M νµλ spin ,(53)
and, crucially, is antisymmetric w.r.t. its first two indices
G λµν = −G µλν .(54)
The M spin involves a sum over all fields and is given in terms of the Lagrangian by
M µνρ spin (x) = −i allf ields ∂L ∂(∂ µ φ r ) (Σ νρ ) s r φ s (x).(55)
where (Σ µν ) s r = −(Σ νµ ) s r is an operator related to the spin of the field. For example, for particles with the most common spins, one has
spin-0 particle φ(x) (Σ µν ) s r = 0,(56)
spin-1/2 Dirac particle
ψ r (x) (Σ µν ) s r = 1 2 (σ µν ) s r ,(57)spin-1 particle A α (x) (Σ µν ) β α = i δ µ α g νβ − δ ν α g µβ .(58)
Examination of the structure of the Lagrangians in Eqs. (4,6) shows that since L qG does not contain any gauge-fixing or ghost fields and L Gf +Gh does not contain any derivatives of
A µ a we may write M µνρ spin (x) = M µνρ spin (x)| qG + M µνρ spin (x)| Gf +Gh(59)
and thus in Eq. (52)
G λµν = G λµν | qG + G λµν | Gf +Gh(60)
so that separately
t µν can (qG) = t µν bel (qG) − ∂ λ G λµν | qG(61)
and
t µν can (Gf + Gh) = t µν bel (Gf + Gh) − ∂ λ G λµν | Gf +Gh .(62)
Hence,
∂ µ t µν can (Gf + Gh) = ∂ µ t µν bel (Gf + Gh) − ∂ µ ∂ λ G λµν | Gf +Gh = ∂ µ t µν bel (Gf + Gh) via (54)(63)
and thus for the physical matrix elements, using (16),
∂ µ Φ ′ | t µν can (Gf + Gh) |Φ = ∂ µ Φ ′ | t µν bel (Gf + Gh) |Φ = 0.(64)
Finally, then, since the total t µν can is a conserved operator, we obtain the key result
∂ µ Φ ′ | t µν can (qG) |Φ = ∂ µ Φ ′ | t µν can |Φ = 0(65)
and the analysis of the general structure of the physical matrix elements of the quark-gluon t µν can (qG) given in [2], which relied on this property, is correct.
Φ| P µ bel |Φ = d 3 x Φ| t 0µ bel (qG) |Φ = d 3 x Φ| t 0µ can (qG) |Φ + d 3 x Φ| ∂ λ G λ0ν | qG |Φ .(66)
By the antisymmetry property (54) the last term is actually a 3-dimensional divergence ∂ i G i0ν | qG , yielding a surface term at infinity, which, as always, is assumed to vanish. Thus and the subsidiary fields also do not contribute to the physical matrix elements of J i can .
Φ| P µ bel |Φ = d 3 x Φ| t 0µ can (qG) |Φ = Φ| P µ can (qG) |Φ .(67)
C. Canonical summary
To summarize, despite the incorrect proof given in [1] concerning the physical matrix elements of the gauge-fixing contributions in QED and the gauge-fixing and ghost contributions in QCD to the Canonical version of the energy-momentum tensor, the essential property used in writing down the most general structure for the QCD matrix elements Φ ′ | t µν can (qG) |Φ and the QED matrix elements Φ ′ | Θ µν can |Φ in [2], namely, that ∂ µ Φ ′ | t µν can (qG) |Φ = 0 and ∂ µ Φ ′ | Θ µν can |Φ = 0
is correct.
Moreover, despite the incorrect derivation in [1], the only contributions to the physical matrix elements of the momentum P µ can and angular momentum J can are from photons and electrons in the QED case and from quarks and gluons in QCD. This latter property was thus correctly utilized in the review paper [2].
IV. CONCLUSIONS
Damski's criticism [3] of the proof given in [1] for certain properties of the physical matrix elements of the energy-momentum tensor t µν in QED is valid, and it is also applicable to QCD, so that, contrary to the assertions in [1], for the Canonical case Φ ′ | t µν can (Gf ) |Φ = 0 and Φ ′ | t µν can (Gf + Gh) |Φ = 0.
Nonetheless the following crucial features for QED and QCD, utilized in the angular momentum review [2], are in fact correct: a) for the Bellinfante case, as claimed in [1] Φ ′ | t µν bel (Gf ) |Φ = 0 and Φ ′ | t µν bel (Gf + Gh) |Φ = 0.
b) despite Eq. (72) one has for the Canonical case, ∂ µ Φ ′ | t µν can (Gf ) |Φ = 0 and ∂ µ Φ ′ | t µν can (Gf + Gh) |Φ = 0.
and obviously an analogous result for the Bellinfante case as a consequence of (73). c) despite Eq. (72) the subsidiary fields do not contribute to the physical matrix elements of either the Canonical or Bellinfante versions of the momentum or angular momentum, a result which is normally assumed without comment in papers on QCD.
C. QED: direct study of subsidiary fieldsThe argument showing that the gauge fixing field in QED does not contribute to the physical expectation value of the Bellinfante version of the energy-momentum tensor and hence does not contribute to the expectation values of the Bellinfante versions of the momentum and angular momentum, based on the QCD case, is rather abstract, so we here
n (z)|Φ = Φ|A n ∂ 0 B (−) − B (+) ∂ n A 0 |Φ = Φ|[A (+) n , ∂ 0 B (−) ] − [B (+) , ∂ n A (−) 0 ]|Φ = Φ|Φ {[A (+) n , ∂ 0 B (−) ] − [B (+), ∂ n A step following since the commutators are c-numbers.
B. Contribution of the subsidiary fields to the physical matrix elements of the Canonical momentum P µ can and angular momentum J canFrom Eqs. (18) and (61)
and the subsidiary fields do not contribute to the physical matrix elements of P µ can . A similar argument shows thatBut
Φ| P µ
bel |Φ = Φ| P µ
can |Φ
(68)
so that, indeed,
Φ| P µ
can |Φ = Φ| P µ
can (qG) |Φ
(69)
Φ| J i
can |Φ = Φ| J i
can (qG) |Φ
(70)
The case a = 1 corresponds to the Gupta-Bleuler approach (see e.g.[9]) based on the Fermi Lagrangian.Note also that in order to conform to the conventions used in the QCD case, the expression for L Gf differs from Nakanishi-Lautrup by a 4-divergence.
. E Leader, arXiv:1101.5956Phys.Rev. 8396012hep-phE. Leader, Phys.Rev., D83, 096012 (2011), arXiv:1101.5956 [hep-ph].
. E Leader, C Lorce, arXiv:1309.4235Phys.Rep. 541163hep-phE. Leader and C. Lorce, Phys.Rep., 541, 163 (2013), arXiv:1309.4235 [hep-ph].
. B Damski, arXiv:2105.01072hep-phB. Damski, (2021), arXiv:2105.01072 [hep-ph].
. B Lautrup, Kgl. Danske Videnskap. Selskab, Mat.-fys. Medd. 351B. Lautrup, Kgl. Danske Videnskap. Selskab, Mat.-fys. Medd., 35, 1 (1967).
. N Nakanishi, Prog. Theor. Phys. 351111N. Nakanishi, Prog. Theor. Phys., 35, 1111 (1966).
. N Nakanishi, Suppl. Prog. Theor. Phys. 511N. Nakanishi, Suppl. Prog. Theor. Phys., 51, 1 (1972).
. T Kugo, I Ojima, Prog. Theor. Phys. Suppl. 661T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl., 66, 1 (1979).
. T Kugo, I Ojima, Prog. Theor. Phys. 601869T. Kugo and I. Ojima, Prog. Theor. Phys., 60, 1869 (1978).
J M Jauch, F Rohrlich, The Theory of Photons and Electrons. Cambridge, USAAddison-WesleyJ. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley, Cam- bridge, USA, 1955).
. X.-D Ji, arXiv:hep-ph/9603249Phys. Rev. Lett. 78610X.-D. Ji, Phys. Rev. Lett., 78, 610 (1997), arXiv:hep-ph/9603249.
. X.-D Ji, arXiv:hep-ph/9609381Phys. Rev. 55X.-D. Ji, Phys. Rev., D55, 7114 (1997), arXiv:hep-ph/9609381.
. X.-D Ji, arXiv:hep-ph/9710290Phys. Rev. 5856003X.-D. Ji, Phys. Rev., D58, 056003 (1998), arXiv:hep-ph/9710290.
. R L Jaffe, A Manohar, Nucl. Phys. 337509R. L. Jaffe and A. Manohar, Nucl. Phys., B337, 509 (1990).
. B L G Bakker, E Leader, T L Trueman, arXiv:hep-ph/0406139Phys. Rev. 70114001B. L. G. Bakker, E. Leader, and T. L. Trueman, Phys. Rev., D70, 114001 (2004), arXiv:hep- ph/0406139.
. M Wakamatsu, arXiv:1004.0268Phys. Rev. 81114010hep-phM. Wakamatsu, Phys. Rev., D81, 114010 (2010), arXiv:1004.0268 [hep-ph].
. M Wakamatsu, arXiv:1007.5355Phys.Rev. 8314012hep-phM. Wakamatsu, Phys.Rev., D83, 014012 (2011), arXiv:1007.5355 [hep-ph].
| [] |
[
"Some Complexity Results for Stateful Network Verification",
"Some Complexity Results for Stateful Network Verification"
] | [
"Kalev Alpernas [email protected] ",
"Panda ·· Aurojit ",
"Alexander Rabinovich ",
"Mooly Sagiv ",
"Scott Shenker ",
"Sharon Shoham ",
"Yaron Velner ",
"K Alpernas ",
"A Panda ",
"Nyu A Rabinovich ",
"M Sagiv ",
"S Shenker ",
"S Shoham ",
"Y Velner ",
"\nTel Aviv University\nTel Aviv University\nTel Aviv University\nBerkeley\n",
"\nTel Aviv University\nHebrew University of Jerusalem\n\n"
] | [
"Tel Aviv University\nTel Aviv University\nTel Aviv University\nBerkeley",
"Tel Aviv University\nHebrew University of Jerusalem\n"
] | [] | In modern networks, forwarding of packets often depends on the history of previously transmitted traffic. Such networks contain stateful middleboxes, whose forwarding behaviour depends on a mutable internal state. Firewalls and load balancers are typical examples of stateful middleboxes.This work addresses the complexity of verifying safety properties, such as isolation, in networks with finite-state middleboxes. Unfortunately, we show that even in the absence of forwarding loops, reasoning about such networks is undecidable due to interactions between middleboxes connected by unbounded ordered channels. We therefore abstract away channel ordering. This abstraction is sound for safety, and makes the problem decidable. Specifically, safety checking becomes EXPSPACE-complete in the number of hosts and middleboxes in the network. To tackle the high complexity, we identify two useful subclasses of finite-state middleboxes which admit better complexities. The simplest class includes, e.g., firewalls and permits polynomial-time verification. The second class includes, e.g., cache servers and learning switches, and makes the safety problem coNP-complete.Finally, we implement a tool for verifying the correctness of stateful networks.A preliminary version of this work appeared in [41].2 Kalev Alpernas et al. | 10.1007/s10703-018-00330-9 | [
"https://arxiv.org/pdf/2106.01030v1.pdf"
] | 67,890,157 | 2106.01030 | 432a9714d5d64230368bf4f27095b1e0eda80b0b |
Some Complexity Results for Stateful Network Verification
2 Jun 2021
Kalev Alpernas [email protected]
Panda ·· Aurojit
Alexander Rabinovich
Mooly Sagiv
Scott Shenker
Sharon Shoham
Yaron Velner
K Alpernas
A Panda
Nyu A Rabinovich
M Sagiv
S Shenker
S Shoham
Y Velner
Tel Aviv University
Tel Aviv University
Tel Aviv University
Berkeley
Tel Aviv University
Hebrew University of Jerusalem
Some Complexity Results for Stateful Network Verification
2 Jun 2021Safety Verification · Stateful Networks · Middleboxes · Channel Systems · Petri Nets · Complexity Bounds
In modern networks, forwarding of packets often depends on the history of previously transmitted traffic. Such networks contain stateful middleboxes, whose forwarding behaviour depends on a mutable internal state. Firewalls and load balancers are typical examples of stateful middleboxes.This work addresses the complexity of verifying safety properties, such as isolation, in networks with finite-state middleboxes. Unfortunately, we show that even in the absence of forwarding loops, reasoning about such networks is undecidable due to interactions between middleboxes connected by unbounded ordered channels. We therefore abstract away channel ordering. This abstraction is sound for safety, and makes the problem decidable. Specifically, safety checking becomes EXPSPACE-complete in the number of hosts and middleboxes in the network. To tackle the high complexity, we identify two useful subclasses of finite-state middleboxes which admit better complexities. The simplest class includes, e.g., firewalls and permits polynomial-time verification. The second class includes, e.g., cache servers and learning switches, and makes the safety problem coNP-complete.Finally, we implement a tool for verifying the correctness of stateful networks.A preliminary version of this work appeared in [41].2 Kalev Alpernas et al.
Introduction
Modern computer networks are extremely complex, leading to many bugs and vulnerabilities which affect our daily life. Therefore, network verification is an increasingly important topic addressed by the programming languages and networking communities (e.g., see [21,9,18,19,17,37,26,14]). Previous network verification tools leverage a simple network forwarding model which renders the datapath immutable; i.e., normal packets going through the network do not change its forwarding behaviour, and the control plane explicitly alters the forwarding state at relatively slow time scales. Thus, invariants can be verified before each control-plane initiated change and these invariants will be enforced until the next such change. While the notion of an immutable datapath supported by an assemblage of routers makes verification tractable, it does not reflect reality. Modern enterprise networks are comprised of roughly 2/3 routers 1 and 1/3 middleboxes [38]. A simple example of a middlebox is a stateful hole-punching firewall which permits traffic from untrusted hosts only after they have received a message from a trusted host. Middleboxes -such as firewalls, WAN optimizers, transcoders, proxies, load-balancers, intrusion detection systems (IDS) and the like -are the most common way to insert new functionality in the network datapath, and are commonly used to improve network performance and security. While useful, middleboxes are a common source of errors in the network [31], with middleboxes being responsible for over 40% of all major incidents in networks.
This work addresses the problem of verifying safety of networks with middleboxes, referred to as stateful networks. We model such a network as a finite undirected graph with two types of nodes: (i) hosts which can send packets, (ii) middleboxes which react to packet arrivals and forward modified packets. Each node in the network has a fixed number of ports, connected by network edges (links).
From a verification perspective, it is possible to view a middlebox as a procedure with local mutable state which is atomically changed every time a packet is transmitted. The local state determines the forwarding behaviour. 2 Thus, the problem of network verification amounts to verifying the correctness of a specialized distributed system where each of the middleboxes operates atomically and the order of packet processing by different middleboxes is arbitrary.
Real middleboxes are generally complex software programs implemented in several hundreds of thousands of lines of code. We follow [29,28] in assuming that we are provided with middlebox models in the form of finite-state transducers. In our experience one can naturally model the behaviour of most middleboxes this way. For every incoming packet, the transducer uses the packet header and the local state to compute the forwarding behaviour (output) and to update its state for future packets. The transducer can be non-deterministic to allow modelling of middleboxes like load-balancers whose behaviour depends not just on the state, but also on a random number source. We symbolically represent the local state of each middlebox by a fixed set of relations on finite elements, each with a fixed arity.
The Verification Problem We define network safety by means of avoiding "bad" middlebox states (e.g., states from which a middlebox forwards a packet in a way that violates a network policy). Given a set of bad middlebox states, we are interested in showing that for all packet scenarios the bad states cannot be reached. This problem is hard since the number of packets is unbounded and the states of one middlebox can affect another via transmitted packets.
What is Decidable About Middlebox Verification
In Section 2.4, we prove that for general stateful networks the verification problem is undecidable. This result relies on the observation that packet histories can be used to count, similarly to results in model checking of infinite ordered communication channels [8]. Simulating counting is immediate when the network configuration admits forwarding loops. However, such configurations are usually avoided in real networks. In order to address realistic networks, we show that the verification problem is undecidable even for networks without forwarding loops.
In order to obtain decidability, we introduce an abstract semantics of networks where the order of packet processing on each channel (connecting two middleboxes or a middlebox and a host) is arbitrary, rather than first-in, first-out (FIFO). Thus, middlebox inputs are multisets of packets which can be processed in any order. This abstraction is conservative, i.e., whenever we verify that the network does not reach a bad state, it is indeed the case. However, the verification may fail even in correct networks, resulting in false alarms. Since packets are atomically processed, we note that network designers can impose ordering even in this abstract model by sending acknowledgments for received packets, and dropping outof-order packets.
In fact, the abstraction of the packet order over channels closely corresponds to assumptions made by network engineers: since packets in modern networks can traverse multiple paths, be buffered, or be chosen for more complex analysis, network software cannot assume that packets sent from a source to a server are received by a server in order. Network protocols therefore commonly build on TCP, a protocol which uses acknowledgments and other mechanisms to ensure that servers receive packets in order. Since packet ordering is enforced by causality (by sending acknowledgments) and by software on the receiving end, rather than by the network semantics, correctness of such networks typically does not rely on the order of packet processing. Therefore we can successfully verify a majority of network applications despite our abstraction.
Complexity of Stateful Verification
In Section 5, we show that the problem of network verification when assuming a nondeterministic order of packet processing is complete for exponential space, i.e., it is decidable, and in the worst case, the decision procedure can take exponential space in terms of hosts and middleboxes. This is proved by showing that the network safety problem is equivalent to the coverability problem of Petri nets, which is known to be EXPSPACE-complete [32,22]. This result is not surprising, and resembles previous work on message passing systems with unordered communication channels [22,36].
Since the problem is complete, it is impossible to improve this upper-bound without further assumptions. Therefore, we consider limited cases of middleboxes permitting more Figure 1. We identify four classes of middleboxes with increasing expressive power and verification complexity: (i) stateless middleboxes whose forwarding behaviour is constant over time, (ii) increasing middleboxes whose forwarding behaviour increases over time, i.e., as the history of packets is extended, the set of forwarded packets may never decrease, (iii) progressing middleboxes whose forwarding behaviour cannot regress to a previous state, i.e., the transition relation of the transducer does not include cycles besides self-cycles (or cycles of "equivalent states"), and hence the forwarding behavior stabilizes after some time, and (iv) arbitrary middleboxes without any restriction. For example, NATs, Switches and simple ACL-based firewalls are stateless; hole-punching stateful firewalls are increasing -as time proceeds more hosts become "trusted" and hence more packets are being forwarded rather than dropped; and learningswitches and cache-proxies are progressing and not increasing -information that is learnt is never unlearned, but the forwarding behaviour may decrease as a result of learning (e.g., a learning switch would forward a packet to the right port rather than broadcasting it).
For stateless and increasing middleboxes, we prove that any packet which arrives once can arrive any number of times, leading to a polynomial-time verification algorithm, using a fixed-point computation. We note that efficient near linear-time algorithms for stateless verification are known (e.g., see [19]). Our result generalizes these results to increasing networks and is in line with the recent work in [13,23].
For progressing middleboxes, we show that verification is coNP-complete. The main insight is that if a bad state is reachable then there exists a small (polynomial) input scenario leading to a bad state. This means that tools like SAT solvers which are frequently used for verification can be used to verify large networks in many cases but it also means that we cannot hope for a general efficient solution unless P=NP.
Finally, we note that unlike the known results in stateless networks, the absence of forwarding loops does not improve the upper bound, i.e., we show that our lower bounds also hold for networks without forwarding loops.
Packet Space Assumption Previous works in stateless verification [18,14] assume that packet headers have n-bits, simulating realistic packet headers which can be large in practice. This makes the complexity of checking safety of stateless networks PSPACE-hard. Our model avoids packet space explosion by only supporting three fields: source, destination, and packet tags. We make this simplification since our work primarily focuses on middlebox policies (rather than routing). As demonstrated in Section 4.2, middlebox policies are commonly specified in terms of the source and destination hosts of a packet and the network port (service) being accessed. For example, at the application level, firewalls may decide how to handle a packet according to a small set of application types (e.g., skype, ssh, etc.). Source, destination and packet tag are thus sufficient for reasoning about safety with respect to these policies. This simplification is also supported by recent works (e.g. [19]) which suggest that in practice the forwarding behaviour depends only on a small set of bits.
Lossless Channels Previous works on infinite ordered communication channels have introduced lossy channel systems [2] as an abstraction of ordered communication that recovers decidability. Lossy channel systems allow messages to be lost in transit, making the reachability problem decidable, but with a non-elementary lower bound on time complexity. In our model, packets cannot be lost. On the other hand, the order of packets arrival becomes nondeterministic. With this abstraction, we manage to obtain elementary time complexity for verification.
Initial Experience We implemented a tool which accepts symbolic representations of middleboxes and a network configuration and verifies safety. For increasing (and stateless) networks, the tool generates a Datalog program and a query which holds iff a bad state is reachable. Then, the query is evaluated using existing Datalog engines [27].
For arbitrary networks (and for progressing networks), the tool generates a petri-net and a coverability property which holds iff the network reaches a bad state. To verify the coverability property we use LOLA [34,1] -a Petri-Net model checker.
Main Results and Outline
This work addresses the complexity of verifying the safety of stateful networks. It makes the following main contributions:
-We introduce a formal model for stateful networks with finite-state middleboxes, inspired by communicating finite state machines [8] (Section 2). We further propose a symbolic representation of middleboxes, resulting in some cases in an exponentially more succinct description compared to an explicit representation as a finite state machine (Section 2.1). We use the formal model to show that verifying safety properties in stateful networks is undecidable, even when the network configuration does not admit forwarding loops (Section 2.4). -We adopt an unordered abstraction inspired by [22,36] to define a conservative abstraction of networks in which packets can be processed out of order (Section 3). Under this abstraction, the safety problem of stateful networks becomes decidable, but EXPSPACE-complete. Interestingly, we show that for a certain class of networks (namely, increasing networks) this abstraction is in fact precise for safety (Section 4). -We identify four classes to which we classify networks, characterized by the forwarding behaviours of their middleboxes: stateless, increasing, progressing and arbitrary (Section 4). We characterize these classes both semantically and syntactically via restrictions on the symbolic representation of the middleboxes (Section 4.1), and demonstrate that these classes capture real-world middleboxes (Section 4.2). -We show that different network classes admit better complexity results than the EX-PSPACE complexity of arbitrary networks: PTIME for stateless and increasing networks (Section 6.1), and coNP for progressing networks (Section 6.2). The upper bounds are made more realistic by stating them in terms of a symbolic representation of middleboxes, i.e., the middlebox code, rather than the explicit state space. We match the upper bounds with lower bounds (Section 5), which are obtained with standard middleboxes, and thus reflect the complexity of realistic networks. -We present initial empirical results using Petri nets and Datalog engines to verify safety of networks (Section 7). Finally, we discuss related work and conclude in Section 8.
A Formal Model for Stateful Networks
In this section, we present a formal model of networks with stateful middleboxes. We define a concrete network semantics, and present the safety verification problem, as well as the special case of isolation. Finally, we show that the safety verification problem is undecidable under the concrete semantics.
A network N is a finite undirected graph of hosts and middleboxes, equipped with a packet domain. Formally, N = (H ∪ M, E, P), where H is a finite set of hosts, M is a finite set of middleboxes, E ⊆ {{u, v} | u, v ∈ H ∪ M} is the set of (undirected) edges and P is a set of packets.
Packets In real networks, a packet consists of a packet header and a payload. The packet header contains a source and a destination host ids and additional arbitrary stream of control bits. The payload is the content of the packet and may consist of any arbitrary sequence of bits. The cardinality of the set of packets is determined by the possible range of control bits and the possible space of payloads, and need not be finite.
In this work, P is a set of abstract packets. An abstract packet p ∈ P consists of a header only, in the form of a triple (s, d,t), where s, d ∈ H are the source and destination hosts (respectively) and t is a packet tag that ranges over a finite domain T . Intuitively, T stands for an abstract set of services or security policies. Therefore, P = H × H × T is a finite set.
Middlebox behaviour in our model is defined with respect to abstract packets and is oblivious of the underlying concrete packets.
Each host h ∈ H is associated with a set of packets that it can send, denoted P h ⊆ P.
Stateful Middleboxes
A middlebox m ∈ M in a network N has a set of ports Pr and a forwarding transducer F. The set of ports Pr consists of all the adjacent edges of m in the network N, The forwarding transducer of a middlebox is a tuple F = (Σ ,Γ , Q m , q 0 m , δ m ) where: -Σ = P × Pr is the input alphabet in which each letter consists of a packet and an input port, -Γ = 2 P×Pr is the output alphabet in which each letter describes (possibly empty) sets of packets sent over the different ports, -Q m is a possibly infinite set of states, q 0 m ∈ Q m is the initial state, and δ m ⊆ Q m × Σ × Γ × Q m is the transition relation, which describes both the output and the change of state in response to an input. Note that the alphabet Σ is finite (since abstract packets are considered). We often refer to the transition relation δ m as a function δ m :
Q m × Σ → 2 Γ ×Q m , where δ m (q, (p, pr)) = {(o, q ) | (q, (p, pr), o, q ) ∈ δ m }.
If δ m (q, (p, pr)) = / 0, we say that δ m is undefined for the packet p arriving on port pr in state q.
We extend δ m to sequences h ∈ (P × Pr) * in the natural way:
δ m (q, ε) = {(ε, q)} and δ m (q, h · (p, pr)) = {(γ i · o , q ) | ∃q i ∈ Q m . (γ i , q i ) ∈ δ m (q, h) ∧ (o , q ) ∈ δ m (q i , (p, pr))}. The language of a state q ∈ Q m is L(q) = {(h, γ) ∈ (P × Pr) * × (2 P×Pr ) * | ∃q ∈ Q m . (γ, q ) ∈ δ m (q, h)}.
The language of F, denoted L(F), is the language of q 0 m . We also define the set of histories leading to q ∈ Q m as h(q) = {h ∈ (P × Pr) * | ∃γ ∈ (2 P×Pr ) * . (γ, q) ∈ δ m (q 0 m , h)}. F is deterministic if for every q ∈ Q m and every (p, pr) ∈ Σ , |δ m (q, (p, pr))| ≤ 1. If F is deterministic, then every history leads to at most one state and output, in which case F defines a (possibly partial) forwarding function f : (P × Pr) * × (P × Pr) → 2 P×Pr where f(h, (p, pr)) = o for the (unique) output o ∈ 2 P×Pr such that (h · (p, pr), γ · o) ∈ L(F) for some γ ∈ (2 P×Pr ) * . If no such output o exists, then f is undefined. The forwarding function f defines the (possibly empty) set of output packets (paired with output ports) that m will send to its neighbors following a history h of packets that m received in the past and input packet p arriving on input port pr. We note that f(h, (p, pr)) = / 0 should not be confused with the case where f(h, (p, pr)) is undefined.
If F is nondeterministic, a forwarding relation f r ⊆ (P × Pr) * × (P × Pr) × 2 P×Pr is defined in a similar way.
Note that every forwarding function f can be defined by an infinite-state deterministic transducer: Q m will include a state for every possible history, with ε as the initial state. The transition relation δ m will map a state and an input packet to the set of output packets as defined by f, and will change the state by appending the packet to the history.
Finite-State Middleboxes
Arbitrary middlebox functionality, defined via infinite-state transducers, makes middleboxes Turing-complete, and hence impossible to analyze. To make the analysis tractable, we focus on abstract middleboxes, whose forwarding behaviour is defined by finite-state transducers. Nondeterminsm can then be used to overapproximate the behaviour of a concrete, possibly infinite-state, middlebox via a finite-state abstract middlebox, allowing a sound abstraction w.r.t. safety.
In the sequel, unless explicitly stated otherwise, we consider finite-state middleboxes. We identify a middlebox with its forwarding relation and the transducer that implements it, and use m to denote each of them.
Symbolic Representation of Middleboxes
To allow a more succinct representation, we use a symbolic representation of finite-state middleboxes, where a state of a middlebox m is described by the valuation of a finite set of relations R 1 , . . . , R k defined over finite domains (e.g., hosts). The transition relation δ m is also described symbolically using (nondeterministic) update operations of the relations and output. The syntax for the symbolic representation is described in Figure 2.
Technically, we describe δ m on an input packet (src, dst, tag) arriving from port prt by a sequence of loop-free guarded commands, which we call a guarded command block. Each guarded command in the block consists of a command and a guard, which determines whether the command should be executed. Guards are Boolean expressions over relation membership predicates of the form e in R (where e = (e 1 , . . . , e n ) for an n-ary relation R) and element equalities e 1 = e 2 . Each e i is either a constant or a variable that refers to packet fields: src, dst, tag, prt. Commands are of the form: (i) output set of tuples, (ii) abort, (iii) insert tuple e to relation R, (iv) remove tuple e from relation R, (v) sequential composition, and (vi) guarded command block. The semantics of insert, remove and sequential composition is straightforward. An output command produces output. In case more than one output is executed, e.g., in the case of a sequential composition of output commands, the output of the execution is the union of all output commands. Blocks of guarded commands are executed non deterministically. That is, all the guards in the block are evaluated, and one command whose guard is evaluated to true is executed. If no guard evaluates to true then the empty set is produced as output, and no relation changes are made. The abort command is used to signify that δ m is not defined on the given input.
A symbolic middlebox program represents a finite-state middlebox where each state represents an interpretation (state) of all the relations, and the transition relation is defined by the main guarded command block in the natural way. Note that since all the relations in the program are over finite domains, the set of states is indeed finite.
Lemma 1 Every finite-state middlebox has a symbolic representation.
Proof Let Q = {q 0 , . . . , q n } be the finite set of states of m, and q 0 be the initial state. We construct a symbolic middlebox program A over the constants q 0 , . . . , q n with a single unary relation R. Initially, R = {q 0 }. Each transition (q , o) ∈ δ m (q, (p, pr)) of m is represented by a guarded command in the main guarded command block. The guard checks whether q ∈ R and whether the packet is (p, pr). The command is a sequential composition of three commands: The first command removes the (only) current state q from R. The second inserts the new state q and the third outputs the tuples in o according to δ m . If δ m (q, (p, pr)) = / 0, the abort command is used.
Remark 1 We note that the construction of a symbolic representation in Lemma 1 results in a linear blowup of the representation, whereas the construction of the explict-state middlebox represented by a symbolic representation potentially results in an exponential blowup, suggesting that the symbolic representation is at least as succinct and is potentially exponentially more succinct than the explicit state representation.
Example 1 Figure 3a contains a symbolic representation of a hole-punching Firewall which uses a unary relation trusted. It assumes that port 1 connects hosts inside a private organization to the firewall and that port 2 connects public hosts. By default, messages from public hosts are considered untrusted and are dropped. trusted is a unary relation which stores public hosts that become trusted once they receive a packet from private hosts. Figure 3b contains a simplified, nondeterministic, version of a Proxy server (or cache server). A proxy stores copies of documents (packet payloads) that passed through it. Subsequent requests for those documents are provided by the proxy, rather than being forwarded. Technically, the middlebox has two ports, namely, a request port from which requests are received and a response port from which responses arrive. Our modelling abstracts away the packet payloads and keeps only their types. Consequently we use nondeterminism to also account for different requests with the same type. The internal relation cache stores responses for packet types.
Concrete (FIFO) Network Semantics
The concrete semantics of a network N = (H ∪ M, E, P) is given by a transition system defined over a set of configurations. In order to define the semantics we first need to define the notion of channels which capture the transmission of packets in the network. Formally, each (undirected) edge {u, v} ∈ E in the network induces two directed channels: (u, v) and (v, u). The channel (v, u) is an ingress channel of u, as well as an egress channel of v. It consists of the sequence of packets that were sent from v to u and were not yet received by u (and similarly for the channel (u, v)). The capacity of channels is unbounded, that is, the sequence of packets may be arbitrarily long. In the concrete semantics, the channels admit a first-in-first-out (FIFO) behavior: Whenever a middlebox forwards a packet p from a certain port it removes it from the head of the corresponding ingress channel and adds the generated packets to the tails of the corresponding egress channels (note that the mapping between channels and middlebox ports is unique).
Configurations and Runs
A configuration of a network consists of the content of each channel and the state of every middlebox. Channels have an unbounded capacity, resulting in an infinite number of configurations even for finite state middleboxes. The initial configuration of a network consists of empty channels and initial states for all middleboxes. A configuration c 2 is a successor of configuration c 1 if it can be obtained by either: (i) some host h sending a packet p ∈ P h to a neighbor, thus appending the packet p to the corresponding channel; or (ii) some middlebox m processing a packet p from the head of one of its ingress channels, changing its state to q and appending output o to its egress channels where q , o are defined in accordance with δ m , i.e., if q is the current state of m and pr is the port associated with the ingress channel then (o, q ) ∈ δ m (q, (p, pr)). This model corresponds to asynchronous networks with non-deterministic event order.
A run of a network from configuration c 0 is a sequence of configurations c 0 , c 1 , c 2 , . . . such that c i+1 is a successor configuration of c i . A run is a run from the initial configuration. The set of reachable configurations from a configuration c i is the set of all configurations that reside on a run from c i . The set of reachable configurations of a network is the set of reachable configurations from the initial configuration.
Safety Verification of Stateful Networks
In this section we define the safety verification problem in stateful networks, as well as the special case of isolation.
To describe safety properties, we augment middleboxes with a special abort state that is reached whenever δ m (q, (p, pr)) = / 0, i.e., the forwarding behaviour is undefined (not to be confused with the case where ( / 0, q ) ∈ δ m (q, (p, pr)) for some q ∈ Q m ). This lets middleboxes function as "monitors" for safety properties. If δ m (q, (p, pr)) = / 0, and h ∈ h(q), i.e., h is a history leading to q, we say that m aborts on h · (p, pr) (and every extension thereof). In the symbolic representation, this is captured by the abort command.
We define abort configurations as network configurations where at least one middlebox is in an abort state.
Safety The input to the safety problem consists of a network N. The output is True if no abort configuration is reachable in N, and False otherwise.
Isolation and Reachability An important example of a safety property is isolation. Informally, isolation is the requirement that certain packets (e.g., packets from a certain host) never reach some host. In the isolation problem, the input is a network N, a set of hosts H i ⊆ H and a forbidden set of packets P i ⊆ P. The output is True if there is no run of N in which a host from H i receives a packet from P i , and False otherwise.
The isolation problem can be formulated as a safety problem by introducing an isolation middlebox m h i for every host h i ∈ H i . The role of m h i is to monitor all traffic to h i , and abort if a forbidden packet p ∈ P i arrives. All other packets are forwarded to h i . (Figure 4 shows a symbolic representation of such a middlebox connected to h i on port 0 and to the rest of the network on port 1.) Clearly, isolation holds if and only if the resulting network is safe.
The Reachability problem is the dual of the isolation problem (i.e., the output is flipped).
Example 2 Figure 5 shows several examples of interesting middlebox topologies for verification. In all of the topologies shown we want to verify a variant of the isolation property. In Figure 5a we want to verify that A, a host, cannot send more than a fixed number of packets to B. Here r 1 and r 2 are rate limiters, i.e., they count the number of packets they have seen going from one host to the other, and lb is a load balancer that evenly spreads packets from input(src, dst,tag, prt) : prt = 0 ⇒ output {(src, dst,tag, 1)} prt = 1 ∧ (src, dst,tag) in forbidden ⇒ abort prt = 1 ∧ ¬((src, dst,tag) in forbidden) ⇒ output {(src, dst,tag, 0)} Fig. 4: Isolation checking middlebox. A along both paths (to minimize the load on any one path). In Figure 5b we want to ensure that host A cannot access data that originates in S 1 , but should be allowed to access data from S 2 , where f is a firewall and c is a proxy (cache) server. Finally in Figure 5c we show a multi-tenant datacenter (e.g., Amazon EC2), where many independent tenants insert rules into firewalls ( f 1 and f 2 ) and we want to ensure that the overall behaviour of these rules is correct. For example, we would like to ensure that pri 1 cannot communicate with pri 2 , and pub 2 communicates with pri 1 only if pri 1 initiates the connection.
Undecidability of Safety w.r.t. the FIFO Semantics
In this section, we prove undecidability of the safety problem by showing that (the specific example of) checking isolation w.r.t. the FIFO semantics is undecidable, even when the network does not have forwarding loops. Forwarding loops occur in networks when a packet reaches the same middlebox (or router) multiple times during the packet's traversal of the network. In the case of deterministic networks (e.g., stateless networks that consist solely of routers), forwarding loops result in the packet traversing the network in an infinite path, never reaching the packet's destination. In general, the existence of forwarding loops is considered a fault in the network design [15].
It is well known that an automaton with an ordered channel of messages (also known as a communicating FSM) can simulate a Turing machine [8]. This can be used to show that the isolation problem over ordered channels is undecidable in the presence of forwarding loops: a forwarding loop allows a packet to traverse the network and reach the same middlebox any number of times. Therefore, it allows one middlebox in the network to simulate a communicating FSM by having all packets rerouted to it. However, it turns out that forwarding loops are not the root cause for undecidability. In this work, we prove that the isolation problem is still undecidable even in the absence of forwarding loops.
To formally define forwarding loops, we augment every packet sent by a host with a unique packet id (e.g., the host id combined with a time stamp). Middlebox forwarding is oblivious to this augmentation: forwarding relations do not depend on the packet id, nor do they change it. We say that a network has a forwarding loop if there is a run in which a packet with the same packet id is received by the same middlebox twice (i.e., a run in which a packet that originates from a middlebox is received by the same middlebox again, possibly after modifications).
We now prove the undecidability result. For completeness of the presentation, our proof shows a reduction from the halting problem of 2-counter machines rather than from reachability of communicating FSMs. However, the same idea of avoiding forwarding loops could be applied to the reduction from commmunicating FSMs sketched above.
Theorem 1 The isolation problem under the FIFO network semantics is undecidable even for networks with finite-state middleboxes and without forwarding loops.
Proof We prove undecidability by a reduction from the (undecidable) halting problem of a two-counter machine to the reachability problem, which is the complement of the isolation problem. A two-counter machine M consists of a finite set of control states Q, an initial state q 0 ∈ Q, a final state q f ∈ Q, and a set of instructions per state (state transitions). An instruction determines the next state and manipulates the value of the counters c 1 , c 2 (initially the value of the two counters is 0). An instruction is in one of the two following forms [25]:
-q 1 : c i = c i + 1 ; GOTO q 2 .
The instruction increments c i and changes the state from q 1 to q 2 .
-q 1 : If c i = 0 GOTO q 2 Else c i := c i − 1 ; GOTO q 3 .
The instruction changes the state to q 2 if the counter value is zero; otherwise it decrements the counter and goes to state q 3 . We first describe a reduction that constructs a network with forwarding loops and allows discarding of packets. We then describe how to get rid of the forwarding loops and the discard operation. Our reduction constructs a network with three middleboxes: a controller middlebox that simulates the state in Q, a c 1 middlebox that helps simulate the value of the first counter, and a c 2 middlebox that helps simulate the value of the second counter, as illustrared in Figure 6. The network has two hosts: initiator and target. Intuitively, the initiator host initiates the simulation of the counter machine, and the target host receives a packet if and only if the counter machine reaches the final state q f . Isolation holds if and only if the target host receives no packet. Both hosts are connected to the controller, which is also connected to c 1 and c 2 . The set of packet tags is T = {#, 1}. Recall that this determines the set of (abstract) packets. The simulation is done by making sure that the total number of 1 packets on the ingress and egress channels of each c i corresponds the value of the simulated counter.
In our construction, the middleboxes decide on forwarding based on the packet tag only. Middlebox c i forwards all of its received packets back to the controller host. We now describe the forwarding behaviour of the controller. Initially, the initiator sends two # packets to the controller. From that point on, the initiator sends only 1 packets. As our network model does not allow to restrict the order in which hosts send packets, this scheme is enforced by the controller: if any other packet arrives, the controller goes to a sink state in which it discards all received packets. The controller forwards the first # to c 1 and the second # to c 2 . When the controller gets a 1 packet from the initiator it simulates a single step of the counter machine, as follows. In an increment operation of c i , the controller sends a 1 packet to c i . To simulate a zero test of c i , the controller receives two packets from c i (if packets from other hosts or middleboxes are received, then the controller goes to a sink state). If the first received packet is #, then the controller forwards it back to c i . If the second one is also #, then the value of the counter is zero. If it is 1, then it is discarded (the value of c i is decremented by 1). If both packets are 1, then the first one is discarded and the second is forwarded back to c i . The simulation of the states of the counter machine is performed by the states of the controller middlebox in a straightforward manner. Finally, if the controller simulates a transition to q f , then it forwards the packet to the target host. Hence, the counter machine halts if and only if the target host is not isolated.
Construction without discard operation. To avoid packet discarding we add a dummy host, and packets that should be discarded are forwarded to the dummy host.
Construction without forwarding loops. To avoid forwarding loops, we add a repeater host to every middlebox. In the new construction, if a middlebox receives a packet with tag t and needs to forward it to port p, then it discards it, and (i) if the next packet that it receives is not from its repeater with tag t, then it goes to a sink state. (ii) otherwise, it forwards the packet it got from its repeater to port p.
Abstract Network Semantics
In this section we define an abstract network semantics, called the unordered semantics, which recovers decidability of the safety problem. A similar setting was explored in [22,36] to recover decidability and obtain the same complexity results we show in Theorem 4 In the concrete (FIFO) network semantics channels are ordered. In an ordered channel, if a packet p 1 precedes a packet p 2 in an ingress channel of some middlebox, then the middlebox will receive packet p 1 before it receives packet p 2 . We abstract this semantics by an unordered network semantics, where the channels are unordered, i.e., there is no restriction on the order in which a middlebox receives packets from an ingress channel. In this case, the sequence of pending packets in a channel is abstracted by a multiset of packets. Namely, the only relevant information is how many occurrences each packet has in the channel. The definitions of configurations and runs w.r.t. the unordered semantics are adapted accordingly. Note that this change does not affect the capacity of the network edges. Consequently the set of network configurations remains infinite.
Remark 2 Every run with respect to the FIFO network semantics is also a run with respect to the unordered semantics. Therefore, if safety holds with respect to the unordered semantics, then it also holds for the FIFO semantics, making the unordered semantics a sound abstraction of the FIFO semantics with respect to safety.
The abstraction can introduce false alarms, where a violation exists with respect to the unordered semantics but not with respect to the concrete semantics. This is demonstrated by Example 3 which presents a network that violates isolation with respect to the unordered semantics, but satisfies isolation with respect to the FIFO semantics. Still, in many cases, Fig. 7: A network with two hosts and two authentication middleboxes. Isolation in this network is preserced under the FIFO semantics, but is violated under the unordered semantics.
the abstraction is precise enough to enable verification. In particular, in Lemma 4 we show that for an important class of networks, the two semantics coincide with respect to safety.
Lossy channel semantics is another overapproximation of the FIFO network semantics considered in the literature, where the order on the channels is maintained, but packets can be lost. We note that the unordered semantics also over-approximates the lossy semantics with respect to safety, as any violating run with respect to the lossy semantics can be simulated by a run with respect to the unordered semantics where "lost" packets are starved until the violation occurs.
Example 3 Consider a network with two hosts (h 1 and h 2 ), each connected to an authentication middlebox (m 1 and m 2 respectively), as depicted in Figure 7. The authentication middleboxes are connected to each other as well. Each authentication middlebox forwards all packets from a host only if the first packet seen from that host is an authentication key (k 1 and k 2 for m 1 and m 2 respectively), otherwise it drops all packets from that host. We would like to verify isolation between h 1 and h 2 . Namely, we would like to verify that no packet with source h 1 arrives at h 2 and vice versa.
A possible scenario violating isolation w.r.t the unordered semantics is: (i) h 1 sends k 1 and then sends k 2 ; (ii) m 1 receives k 1 and then receives k 2 (and forwards both packets in that order). (iii) m 2 receives k 2 before it receives k 1 (i.e., the order on the channel between m 1 and m 2 was not maintained). m 2 forwards k 2 to h 2 and isolation is violated.
On the other hand, if all channels are FIFO, then if h 1 first sends k 2 , it and all subsequent packets from h 1 will be dropped by m 1 . If h 1 first sends k 1 instead, m 1 will forward it to m 2 , which in turn will drop it and all subsequent packets from h 1 . Consequently, isolation between h 1 and h 2 is preserved under the FIFO semantics.
Decidability of Safety w.r.t. the Unordered Semantics In the unordered semantics, the network forms a special case of monotone transition systems: We define a partial order ≤ between network configurations such that c 1 ≤ c 2 if the middlebox states in c 1 and c 2 are the same and c 2 has at least the same packets (for every packet type) in every channel. The network is monotone in the sense that for every run from c 1 there is a corresponding run from any bigger c 2 , since more packets over a channel can only add possible scenarios. The partial order is trivially a well-quasi-order (as the number of packets cannot be negative), and the predecessor relation is clearly computable. The classical results of Abdulla et al. [3] and Finkel et al. [12] prove that in monotone transition systems a backward reachability algorithm always terminates and thus, the safety problem is decidable. Formal arguments and complexity bounds are provided by Theorem 4.
Classification of Stateful Middleboxes
Encouraged by the decidability of safety w.r.t. the unordered semantics, we are now interested in investigating its complexity. As a first step, in this section, we identify three special classes of forwarding behaviours of middleboxes within the class of arbitrary middleboxes. Namely, stateless, increasing, and progressing middleboxes. We show that these classes capture the behaviours of real world middleboxes. The classes naturally extend to classes of networks: a network is stateless (respectively, increasing, progressing or arbitrary) if all of its middleboxes are. As we show in Section 5 and Section 6, each of these classes results in a different complexity of the safety problem. Our definitions apply both for finite-state and infinite-state middleboxes.
Stateless Middlebox A middlebox m is stateless if it can be implemented as a transducer with a single state (in addition to the abort state), i.e., its forwarding behaviour does not depend on its history (with the exception of abort). Formally, a middlebox m is stateless if for every two histories h 1 , h 2 ∈ (P × Pr) * , packet p ∈ P, port pr ∈ Pr and output set
o ∈ 2 P×Pr , (h 1 , (p, pr), o) ∈ f r iff (h 2 , (p, pr), o) ∈ f r .
Increasing Middlebox A middlebox m is increasing if its forwarding relation f r is monotonically increasing w.r.t. its history, where histories are ordered by the subsequence relation 3 , denoted by . Formally, a middlebox m is increasing if for every two histories h 1 , h 2 ∈ (P × Pr) * : if h 1 h 2 , then for every packet p, port pr and output sets o 1 ,
o 2 ∈ 2 P×Pr , if (h 1 , (p, pr), o 1 ) ∈ f r and (h 2 , (p, pr), o 2 ) ∈ f r then o 1 ⊆ o 2 .
Intuitively, this means that new information can only expand the forwarding policy of an increasing middlebox, or lead to an abort.
Remark 3
The "increasing" property implies that the forwarding relation of an increasing middlebox is in fact a function. Hence, the middlebox can be implemented by a deterministic transducer. In the following we will refer to the forwarding function f of increasing middleboxes instead of the forwarding relation f r .
The following lemma ensures that the behaviour of an increasing middlebox can be precisely captured by a finite-state deterministic transducer. Its proof uses Higman's lemma [16] (based on well quasi ordering).
Lemma 2 Any infinite-state increasing middlebox has an implementation as a deterministic finite-state increasing middlebox.
Proof Consider an infinite-state increasing middlebox m, and its forwarding function f : (P × Pr) * × (P × Pr) → 2 P×Pr . Recall that f might be a partial function.
Let f(h) denote an × k output matrix for the middlebox m and history h, where |P| = and |Pr| = k. We further denote P = {p 1 , . . . , p } and Pr = {pr 1 , . . . , pr k }. Every entry in the output matrix f(h) contains the output set for the corresponding pair of packet and port,
or if it is undefined. Formally f(h) i, j = f(h, (p i , pr j )) or f(h) i, j = when f is undefined for the input.
As P and Pr are finite, we get that there is a finite number of different output matrices. We denote them by A 1 , . . . , A n . With every output matrix A i we associate the set of matching
histories h(A i ) = {h | f(h) = A i }. Note that h(A 1 ) ∪ . . . ∪ h(A n ) = (P × Pr) * and that h(A i ) ∩ h(A j ) = /
0 for every i = j (since the forwarding function is deterministic). Therefore, for every history h there exists a unique i such that h ∈ h(A i ).
In the following, we will show that for every A i , the set h(A i ) is regular and thus we can implement the forwarding function f of m by using finite-state automata to recognize the matrix that corresponds to the current history and then forwarding the current packet accordingly.
We show that for every output matrix A, h(A) is regular. We define a partial order ≤ over matrices as: A ≤ B iff A i, j ⊆ B i, j for every pair of indices i, j, (where X ⊆ for every X ∈ 2 P×Pr ). We denote by UP(A) the upwards closure of {A} with respect to the ≤ order on matrices. We extend the definition of h(A) to sets of matrices: for a (possibly infinite) set of
matrices A we define h(A ) = A∈A {h | f(h) = A}.
We note that since m is increasing, the set h(UP({A})) is upwards closed with respect to the subsequence relation over histories.
Indeed, if h 1 ∈ h(UP(A)), then f(h 1 ) ≥ A. For every h 2 h 1 , f(h 1 ) ≤ f(h 2 ) (as m is increas- ing)
, and thus f(h 2 ) ≥ A, which means that h 2 ∈ h(UP(A)) as well. Hence, by Higman's lemma and the finite basis property of wqo, we get that h(UP(A)) has a finite basis (which consists of histories). We denote the basis
{h 1 , . . . , h o }. Then h ∈ h(UP(A)) if and only if h h i for some i = 1, . . . , o.
We further observe that for a given history h i , the (infinite) set {h | h i h} is a regular language, and as regular languages are closed under finite union, we get that the (infinite) set of histories h(UP(A)) is regular. Finally, we note that
h(A) = h(UP(A)) \ {h(UP(A )) | A ≥ A ∧ A = A}.
Since there are finitely many output matrices, closure properties of regular languages imply that h(A) is regular.
To complete the proof, we describe the transducer contruction. Let D i be a finite-state automaton that recognizes h(A i ). We construct a finite-state transducer m for m, as follows. m runs D 1 , . . . , D n in parallel. They all start from their initial states, and on every new packet p that arrives from port pr, m updates the states of D 1 , . . . , D n in parallel based on (p, pr). Exactly one of them, say D i , will reach an accepting state, in which case m will process the packet as defined by A i . Correctness is ensured since for every history h, D i accepts h if and only if h ∈ h(A i ), which by definition ensures that f(h) = A i . In addition, the construction results in a finite-state transducer since the number of matrices is finite.
Precision of Abstract Semantics in Increasing Networks
Recall that in general, safety w.r.t. the FIFO semantics and the unordered semantics do not coincide. However, the following lemmas show that for increasing networks (with either finite-state or infinite-state middleboxes) they must coincide, making the abstraction precise for such networks. Intuitively, this is because in increasing networks if a packet p reaches a middlebox m once, then unless a middlebox in the network reaches an abort state, the packet p can reach m again, thus enabling the simulation of unordered channels with ordered ones. The following lemma formalizes this claim.
Lemma 3 Let N be an increasing network. For every middlebox m, packet p and port pr, if there exists a run r of N from the initial configuration in the FIFO semantics such that in the last step m receives p from pr, then from any configuration there exists a run, in the FIFO semantics, that ends in a step in which m receives p from pr (or in abort).
Proof We prove the assertion by induction on |r| (the length of the run from the initial configuration). We fix m, p, pr, r, and an arbitrary configuration c from which we wish to show a run.
If |r| = 1, then it must be the case that m received the packet from a neighbor host. Hence, c has a run in which the same neighbor host sends the same packet to m, and after all the previous packets in the ingress channel of m are processed, the packet p arrives from port pr.
If |r| > 1, then we consider two distinct cases. In the first case, the packet was sent to m by a neighbor host, and by the same arguments as before the assertion holds. In the second case, the packet was sent to m by a neighbor middlebox m . Let h = (p 1 , pr 1 ), . . . , (p n , pr n ) be the history of packets received by m before it sent the packet, and let (p , pr ) be the packet that triggered the forwarding of p from m to m. Since these packets were received by m before the last step of r it must be the case that there exist n + 1 runs r 1 , . . . , r n , r such that run r i ends when m receives packet (p i , pr i ), and run r ends when m receives (p , pr ). Each of the runs r 1 , . . . , r n , r has a length of at most |r| − 1, since they are subsequences of the prefix of r that excludes the packet (p, pr) sent from m to m.
Hence, by the induction hypothesis there is a run over N that begins in c and ends in some configuration c 1 after m received the packet (p 1 , pr 1 ). Similarly, for every i = 1, . . . , n there is a run that begins in c i and ends in some configuration c i+1 after m received the packet (p i , pr i ). Finally, there is a run from c n+1 to a configuration c that ends after m received (p , pr ). Consider the history h of m that is formed in the run c ; c 1 ; . . . c n+1 ; c . Regardless of the history of m in c (which is the prefix of h ), we get that h is a subsequence of h (as (p i+1 , pr i+1 ) is added after (p i , pr i )). Hence, after m receives (p , pr ), it must forward p to m (due to the fact that f m (h, (p , pr )) ⊆ f m (h , (p , pr ))). Hence, after m processes all the packets in its ingress channel, it will receive (p, pr) (or will get to an abort state).
In Lemma 4, we use the property shown in Lemma 3 to prove that any reachable configuration in an unordered network, is also a reachable configuration of a FIFO network. Given an unordered violating run, we use the construction described in the proof of Lemma 3 to build a FIFO run that ends in the packet that caused the violation in the unordered run, or in a FIFO violating run in case the construction from Lemma 3 resulted in an abort state.
Lemma 4 Let N be an increasing network. Then the output of the safety problem in N w.r.t. the FIFO semantics and the unordered semantics is identical.
Proof Recall that any (violating) run w.r.t. the FIFO semantics is also a viable (violating) run w.r.t. the unordered semantics. Therefore, in order to prove the assertion of the lemma, it suffices to prove that for every violating run w.r.t. the unordered semantics there is a violating run w.r.t. the FIFO semantics.
We prove that for every unordered run r and every middlebox m there exists an ordered run r s.t. r| m r | m where r| m is the history of middlebox m in run r.
The proof is by induction on the length of the unordered run r. The base case, where |r| = 0, is clear as the history is necessarily empty.
For |r| > 0, the induction hypothesis guarantees that for the prefix of r of length |r − 1|, denoted r −1 , there exists an ordered run r −1 s.t. r −1 | m r −1 | m . If m is not the recipient of the last packet, then we consider r = r −1 . The resulting history for middlebox m is r | m = r −1 | m , and because r| m = r −1 | m in this case, we have that r| m r | m .
If m is the recipient of the last packet, we consider two distinct cases. In the first case, the final packet (p, pr) in r was sent by a neighbor host. Since hosts can send packets in any configuration, we append the last event of r to r −1 , resulting in the ordered run r . The resulting history for middlebox m is r | m = r −1 | m · (p, pr), and because r| m = r −1 | m · (p, pr), we have that r| m r | m .
In the second case, the final packet (p, pr) in r was sent by a neighbor middlebox m . We consider the history of middlebox m for r −1 -the prefix of r of length |r − 1|, denoted h = r −1 | m = (p 0 , pr 0 ), · · · , (p l , pr l ) . By the induction hypothesis, there exists an ordered run r −1 s.t. r −1 | m r −1 | m , and by Lemma 3 we get that for every packet (p i , pr i ) in h from any configuration there exists an ordered run that ends in middlebox m receiving (p i , pr i ), or there exists a run that leads to a safety violation (in which case we have reached the goal of this construction and are done).
We proceed by constructing the run r . We first construct the runr = r −1 · r 1 · · · r l where r −1 is the ordered run guaranteed by the induction hypothesis s.t. r −1 | m r −1 | m , and r i is the ordered run ending in the middlebox m receiving the packet (p i , pr i ), starting from the configuration at the end of the previous run. The construction ensures that r −1 | m r| m (since r −1 | m r −1 | m ). In addition, because r −1 | m = (p 0 , pr 0 ), · · · , (p l , pr l ) r| m and m is increasing, m can send the packet (p, pr) to m afterr. We obtain r by appending tõ r the final event of r, where m sends the packet (p, pr) to m. Since r | m =r| m · (p, pr), r| m = r −1 | m · (p, pr) and r −1 | m r| m , we get that r| m r | m .
In particular, we can construct an ordered run in which m has an aborting history.
Progressing Middlebox In order to define progressing middleboxes, we define an equivalence relation between middlebox states based on their forwarding behaviour. States q 1 , q 2 are equivalent, denoted q 1 ≈ q 2 , if L(q 1 ) = L(q 2 ). A middlebox m is progressing if it can be implemented by a transducer in which whenever the state is changed into a non-equivalent state, it will never return to an equivalent state. Formally, if (o , q ) ∈ δ m (q, (p, pr)) and q ≈ q (where q, q are reachable states of m) then for any future sequence of packets h ∈ (P × Pr) * , if (γ , q ) ∈ δ m (q , h) for some γ and q , then q ≈ q. As opposed to increasing middleboxes, progressing middleboxes might require infinitely many states. In this case nondeterminism is essential as it allows to support the abstraction of infinite-state middleboxes via finite-state transducers.
Example 4 (Infinite-state progressing middlebox) Consider the packet space H ×H ×{0, 1}, and a deterministic middlebox m with a single port whose forwarding function is defined as follows. As long as all received packets have tag 0, then each packet is forwarded (as is) back to the single port. When a packet with tag 1 arrives for the first time, if the number of previous packets is prime, then all future packets are discarded. Otherwise, all future packets are forwarded back to the single port. Prime numbers are not recognizable by finitestate machines. Hence, there is no finite-state implementation of m. On the other hand, m is progressing since its state always progresses (from counting to always discarding or always forwarding).
Finite-state progressing middleboxes have the following useful property:
Lemma 5 Every finite-state progressing middlebox has an implementation as a finite-state transducer whose underlying state graph has a tree structure, except for, possibly, self-loops.
Proof We show an implementation as a directed acyclic graph (DAG), possibly with self loops. The transformation to a tree is then straightforward. Let m be a minimal transducer that implements the progressing middlebox. We consider the language L(q) of each state q in m. Minimality ensures that no two states in m have the same language (otherwise they are equivalent and can be merged). Therefore, each state q represents a unique language L(q).
Towards a contradiction we assume that there is a directed loop that is not a self-loop in m. A loop implies that there are two states q 1 ≈ q 2 in m such that q 1 transitions to q 2 by some sequence h 2 and q 2 transitions back to q 1 by some sequence h 3 . Further, by minimality of m, q 1 is reachable by some sequence h 1 .
Since m is progressing, contradiction is obtained.
The next lemma summarizes the hierarchy of the classes (as illustrated by Figure 1).
Lemma 6 -Any stateless middlebox is also increasing.
-Any increasing middlebox is also progressing.
Proof The first part of the lemma is straightforward. Consider the second part of the lemma. Let m be a deterministic transducer of an increasing middlebox and f is its forwarding function. Towards a contradiction assume that m is not progressing, i.e. there exist two states q 1 ≈ q 2 and three histories h 0 , h 1 , h 2 s.t.
(γ 0 , q 1 ) ∈ δ m (q 0 , h 0 ), (γ 1 , q 2 ) ∈ δ m (q 0 , h 0 · h 1 ) and (γ 2 , q 1 ) ∈ δ m (q 0 , h 0 · h 1 · h 2 ). Because q 1 ≈ q 2 , there exists a history h s.t. f(h 0 · h) = f(h 0 · h 1 · h), and since m is increasing it must be the case that f(h 0 · h) ⊂ f(h 0 · h 1 · h).
However, since m is deterministic and h 0 and h 0 · h 1 · h 2 lead to the same state, namely
q 1 , it must be that f(h 0 · h) = f(h 0 · h 1 · h 2 · h) and we get that f(h 0 · h 1 · h) ⊃ f(h 0 · h 1 · h 2 · h),
in contradiction to the fact that m is increasing.
Syntactic Characterization of Middlebox Classes
The classes of middleboxes defined above can be characterized via syntactic restrictions on their symbolic representation. In Section 6 we will use the syntactic characterization to obtain more realistic complexity upper bounds, stated in terms of the symbolic representation rather than the explicit state-space of middleboxes.
A middlebox representation is syntactically stateless if it does not use any insert or remove command on any relation. A middlebox representation is syntactically increasing if it does not use the remove command on any relation, does not use negated membership predicates in the guards and all guards are mutually exclusive (i.e. no two guards can be true at the same time). A middlebox representation is syntactically progressing if it does not use the remove command on any relation.
Lemma 7 Every stateless finite-state middlebox has an equivalent syntactically stateless symbolic representation and vice versa.
Proof The lemma is trivial for stateless middleboxes, as both the transducer and the symbolic representation simply describe a fixed forwarding table.
Lemma 8 Every increasing finite-state middlebox has an equivalent syntactically increasing symbolic representation and vice versa.
Proof We first show that every increasing finite-state middlebox has an equivalent syntactically increasing symbolic representation. Let m be an increasing finite-state middlebox implemented by a deterministic transducer with state set Q = {q 0 , . . . , q n }, where q 0 is the initial state. By Lemma 6 and Lemma 5 we may assume w.l.o.g that the underlying graph of m is a tree. We construct a symbolic program A with one unary relation R over the constants q −1 , q 0 , . . . , q n . Initially R = {q 0 }. To describe A we need the next three notations. To reduce the notational burden, we use packets p instead of pairs (p, pr) of a packet and an input port. For a state q i and a packet p we denote the successor state of q i according to packet p by q i → p (we note that possibly q i → p = q i ). The successor state is unique since the transducer is deterministic. We denote by q i (p) the output of m when m is in state q i and packet p is received. We denote the (single) predecessor of q i in the tree by pre(q i ) (we note that in case the state q i has a self loop, the predecessor function returns the unique predecessor of q i that is not q i . i.e., pre(q i ) = q j s.t. q i = q j ). For uniformity, we assume that the root q 0 also has a predecessor, namely, q −1 with q −1 (p) = / 0 for every packet p. We now describe how A processes a packet p: -Relation update. For every q i ∈ R: insert q i → p to R.
-Output. For every q i ∈ R: output q i (p) \ pre(q i )(p). We first observe that A can be implemented as a syntactically increasing program. Indeed, the "for every" loops can be replaced by a sequential composition of finitely many guarded commands consisting of positive relation membership queries, and only insert update operations.
We now show that the forwarding behaviours of A and m are identical and hence A is indeed a correct symbolic representation of m. Let h be an arbitrary history and let p be an arbitrary packet. By a simple induction we get that the states in the relation R are exactly the states that m visited while processing the history h. We assume w.l.o.g that the set of visited states (after history h) is {q 0 , . . . , q k } and that q i = pre(q i+1 ). We prove, by induction on k, that the outputs of m and A are identical. In the base case k = 0, and the proof follows as we defined pre(q 0 )(p) = q −1 and q −1 (p) = / 0. For k > 1, we observe that since m is increasing and a prefix is also a subsequence, then q k−1 (p) ⊆ q k (p). Hence, q k (p) = (q k (p) \ q k−1 (p)) ∪ q k−1 (p). By the induction hypothesis, we get that A first outputs q k−1 (p), and by the implementation of A, we get that it then outputs q k (p) \ q k−1 (p). Hence, overall A outputs q k (p), and the proof of the claim is complete.
To conclude, we proved that A is a syntactically increasing symbolic representation of m.
For the converse direction, we show that the forwarding behaviour of a middlebox given via a syntactically increasing symbolic representation is increasing. Let A be a syntactically increasing symbolic program. For simplicity we assume that A has only one relation R. The mutually exlusive guard requirement implies determinstic execution. Consequently, for a history h we can denote by R h the unique content of relation R after h. We claim that if h 1 h 2 , then R h 1 ⊆ R h 2 . The proof follows from the fact that all the guards in A have positive conditions and from the fact that elements are only added to the relation. As the forwarding behaviour depends only on the state of the relation, and since all conditions are positive, we get that the forwarding behaviour is increasing.
Lemma 9 Every progressing finite-state middlebox has an equivalent syntactically progressing symbolic representation and vice versa.
Proof We first show that every progressing finite-state middlebox has an equivalent syntactically progressing symbolic representation. Let m be a progressing finite-state middlebox, and by Lemma 5 we may assume w.l.o.g that the underlying state graph of m is a tree. Let Q = {q 0 , . . . , q n } be the states of m. We construct a symbolic program A similarly to the construction in the proof of Lemma 8 (with one unary relation R over the constants q 0 , . . . , q n , where initially R = {q 0 }, and where R accumulates the traversed states). When a packet p is processed, the program identifies the current state by computing a maximal (according to topological order) state q i in R (this is implemented using a guard for every path from the tree root to each state in the state tree). It then adds q i → p to R and outputs q i (p). Since m is a tree, there always exists exactly one maximal state in R, and we get that A always simulates m correctly.
For the converse direction, we show that the forwarding behaviour of a middlebox given via a syntactically progressing symbolic representation is progressing. Let A be a syntactically progressing symbolic program. For simplicity we assume that A has only one relation R. We recall that the domain of R is always finite, and thus it has only a finite number of different states (interpretations). We construct a middlebox m whose states are exactly the states of R, and the forwarding function is exactly according to those states. As A is progressing, we get that elements are only added to R, and thus the underlying graph of m is progressing.
Examples
In this section, we introduce several middleboxes, each of which resides in one of the classes of the hierarchy presented above.
ACL Switch An ACL switch has a fixed access control list (ACL) that indicates which packets it should forward and which packets it should discard. Typically the rules in the list refer to the port number or to hosts that are allowed to use a certain service. As such, the forwarding policy of an ACL switch is based only on the source host and/or ingress port of the current packet, and does not depend on previous packets. Hence, an ACL switch can be implemented by a stateless middlebox.
Hole-Punching Firewall A hole-punching firewall is described in Example 1. As the set of trusted hosts depends on the history of the middlebox, a hole punching firewall cannot be captured by a stateless middlebox. (Formally, given two different histories, the forwarding function might produce a different output for the same packet and port.) However, a hole punching firewall is an increasing middlebox. This follows since for every source host s and two histories h 1 h 2 , if s is trusted according to h 1 , then it is also trusted according to h 2 . The proof of the latter is by induction on |h 1 |. In the base case |h 1 | = 0, and therefore s is in the initial list of trusted hosts (and therefore, it is trusted also in h 2 ). If |h 1 | > 0, then h 1 = h 1 · (p, pr). We consider two distinct cases: In the first case s was trusted before the last packet p in h 1 was received. Hence, by the induction hypothesis we get that s is trusted also in h 2 . In the second case s became trusted only after the last packet p was processed. In this case, p had a trusted source host s 1 (according to h 1 ) with destination s. Since h 1 h 2 , there exist h 2 , h 2 such that h 2 = h 2 · (p, pr) · h 2 and h 1 h 2 . By the induction hypothesis, the source host s 1 of the last packet p is also trusted according to h 2 , and therefore s is trusted also in h 2 · (p, pr). As the set of trusted hosts never decreases, s remains trusted in h 2 .
Learning Switch A learning switch dynamically learns the topology of the network and constructs a routing table accordingly. Initially, the routing table of the switch is empty. For every host h the switch remembers the first port from which a packet with source h has arrived. When a packet arrives, if the port of the destination host is known, then the packet is forwarded to that port; otherwise, the packet is forwarded to all connected ports excluding the input-port.
A learning switch is a progressing middlebox. Intuitively, after the middlebox's forwarding function has changed to incorporate the destination port for a certain host h, it will never revert to a state in which it has to flood a packet destined to h. A learning switch is however, not an increasing middlebox, as packets destined to a host whose location is not known are input(src, dst,tag, prt) :
¬ ((dst, prt) in connected) ⇒ connected.insert(src, prt); // remember src's port initially flooded, but after the location of the host is learned, a single copy of all subsequent packets is sent. Figure 8 depicts a symbolic representation of a learning switch that uses a binary relation connected storing connections between hosts and ports. If the port of the destination host is known, then the packet is forwarded to that port; otherwise, the packet is forwarded to all connected ports excluding the input-port. The last command in the program is a syntactic shorthand used to avoid the explicit enumeration of incoming ports required to correctly perform the flood operation.
(dst, 1) in connected ⇒ output {(src, dst,tag, 1)} (dst, 2) in connected ⇒ output {(src, dst,tag, 2)} (dst, 3) in connected ⇒ output {(src, dst,tag, 3)} ¬ ((dst, 1) in connected) ∧ ¬ ((dst, 2) in connected) ∧ ¬ ((dst, 3) in connected) ⇒ output {(src,
Proxy Server The Proxy server as described in Example 1 is a progressing middlebox. After it has stored a response, it nondeterministically replies with the stored response, or sends the request to the server again. Once a new request is responded by a proxy the forwarding behaviour changes as it takes into account the new response, and it never returns to the previous forwarding behaviour (as it does not "forget" the response). This example demonstrates how nondeterminism is used to model middleboxes whose concrete behaviour depends on packet payloads. In a concrete network model that does not abstract away the packet payload, the proxy middlebox would always reply to a request with a stored response and never forward it to the server.
Round-Robin Load Balancer
A load balancer is a device that distributes network traffic across a number of servers. In its simplest implementation, a round-robin load balancer with n out-ports (each connected to a server) forwards the i-th packet it receives to outport i (mod n). Round-robin load balancers are not progressing middleboxes, as the same forwarding behaviour repeats after every cycle of n packets. Figure 9 depicts a symbolic representation of a round-robin load balancer with 3 ports: port 0 is an 'input' port, and ports 1 and 2 are 'output' ports on which the load balancer splits the incoming traffic. It uses a unary relation nextport to hold the port to which the next packet is to be sent.
Remark 4
In practice, middlebox behaviour can also be affected by timeouts and session termination. For example, in a firewall, a trusted host may become untrusted when a session terminates (which makes the firewall behaviour no longer increasing). Similarly, cached content of a cache server expires after a certain period of time (which violates progress). In this work, we do not model timeouts and session termination.
input(src, dst,tag, prt) :
(prt = 0 ∧ (1) in nextport) ⇒ output {(src, dst,tag, 1)} (prt = 0 ∧ (1) in nextport) ⇒ nextport.remove 1 (prt = 0 ∧ (1) in nextport) ⇒ nextport.insert 2 (prt = 0 ∧ (2) in nextport) ⇒ output {(src, dst,tag, 2)} (prt = 0 ∧ (2) in nextport) ⇒ nextport.remove 2 (prt = 0 ∧ (2) in nextport) ⇒ nextport.insert 1 (prt = 1 ∨ prt = 2) ⇒ output {(src, dst,tag, 0)} Fig. 9: A 3-port round-robin load-balancer.
Lower Bounds on Complexity of Safety w.r.t. the Unordered Semantics
When considering the unordered network semantics, the safety problem becomes decidable for networks with finite-state middleboxes. In this section, we analyze its complexity lower bounds. The complexity bounds are w.r.t the input size, namely, (i) the number of hosts;
(ii) number of middleboxes; and (iii) the encoding size of the middleboxes functionality, i.e., the size of the explicit state machine (if the encoding is explicit) or the number of characters in the symbolic representation (if the encoding is symbolic). In Section 6 we present matching upper bounds for networks represented symbolically. Since symbolic representations are at least as succinct as explicit-state descriptions of finitestate middleboxes, all the lower bounds obtained for the explicit finite-state model apply for the symbolic one as well, and all the upper bounds obtained for the symbolic model are applicable to the explicit finite-state model, resulting in tight complexity bounds, both for explicit finite-state middleboxes and for symbolic ones.
We obtain lower bounds for the safety verification problem by considering the isolation problem. Recall that the isolation problem reduces to a safety problem by the introduction of isolation middlebox. Since isolation middleboxes are stateless, they do not change the class of the input network. We can therefore deduce that the same lower bounds also hold for the more general safety problem.
Unordered Safety in Progressing Networks is coNP-hard.
Lemma 10 The isolation problem w.r.t. the unordered network semantics for a progressing network is coNP-hard.
Proof We show a reduction from the (NP-hard) Hamiltonian Path problem to the reachability problem, which is the complement of the isolation problem. Recall that the Hamiltonian Path problem is given a directed graph G(V, E), a source vertex s ∈ V and a target vertex t ∈ V , and it determines whether there is a simple path from s to t in G with length |V |.
In the reduction, we use flood-once middleboxes that upon receiving a packet with a numeric tag (from a finite domain) increment the packet tag and flood the new packet. All following packets that arrive at the middlebox are discarded. These flood-once middleboxes are finite-state progressing middleboxes.
We construct a network with a single flood-once middlebox for every vertex in the graph and connect them in accordance with the edges in the graph. In addition, we create two hosts h s and h t and connect them to the middleboxes representing the source and target in the graph. We use packet tags {0, . . . , |V |}. Host h s sends packets with tag 0. The reachability problem is to determine whether h t can receive a packet with tag |V |.
The flood once middleboxes ensure that the packet tags 'count' the length of the path. Thus, a Hamiltonian Path corresponds to a packet with the tag |V | arriving at the destination host h t , and the correctness of the reduction follows.
The following lemma shows that a similar result can be obtained using more "standard" middleboxes, namely, stateless middleboxes and learning switches.
Lemma 11
The isolation problem w.r.t. the unordered network semantics for a network where each middlebox is either stateless or a learning switch is coNP-hard.
Proof The proof is by reduction from the (NP-hard) Hamiltonian Path problem to the reachability problem. We use the same notation as used in the proof of Lemma 10. W.l.o.g we assume that the out-degree of all vertices of the directed graph G is two. For the reduction, we construct a network with three hosts, namely, h s , h t and h d , and 4|V | middleboxes, as described below. The topology of the resulting network is illustrated in Figure 11. The set of packet tags is {0, . . . , |V |}. As before, the reachability problem is to determine whether host h t can receive a packet with tag |V |.
We now describe the network in more detail. With every vertex v we associate three stateless middleboxes, namely, v A , v B and v C , and a learning switch v LS , illustrated in Figure 10. Intuitively, these middleboxes will simulate a "flood once" middlebox. The middlebox v A is connected to v B , v C and v LS . The middlebox v LS is connected to v B and v C as well as to v A , and if (v, u 1 ) ∈ E and (v, u 2 ) ∈ E, then v B has a link to (u 1 ) A and v C is connected to (u 2 ) A . Host h s is connected to (v s ) A and is allowed to send only the packet (h s , h t , 0) (source h s , destination h t , and tag 0). Host h t is connected to (v t ) B and (v t ) C . Host h d is a dummy host, disconnected from any middlebox. Its purpose is merely to allow three distinct host ids. The forwarding function of the learning switch is as described in Section 4.2. The forwarding function of the stateless middleboxes is defined as follows:
packets received by v A from some u B or u C : if the packet is (h s , h t , n), namely, source is h s , destination is h t and packet tag is n, then forward it to v LS .
-packets received by v A from v LS : if the packet is (h d , h s , n), then forward packet (h t , h d , n) to v LS . -packets received by v B , v C from v LS : if the packet is (h s , h t , n) forward packet (h d , h s , n)
to v LS . If the packet is (h t , h d , n) forward packet (h s , h t , n + 1) to the appropriate u A . Otherwise, discard. All other packets are discarded.
We first give an informal description of how a packet is processed and then turn to formally prove the correctness of the reduction. When v A receives a (h s , h t , n) packet from some u B or u C it sends it to the learning switch. When v LS first receives the packet it forwards it to all of its neighbors except for v A (from which it was received) and marks the port connected to v A as the destination port to h s . v B and v C reply with (h d , h s , n), and when the first of these packets arrives to v LS , then it marks either v B or v C as the destination of h d . In addition, as the port connected to v A is marked as the destination to h s , the learning switch sends the packets
(h d , h s , n) to v A . v A responds with (h t , h d , n).
When v LS receives the packet it marks the port connected to v A as the destination for h t and forwards the packet to v B or v C (depending on which was marked as the destination for h d ). v B or v C increments the tag and forwards the packet to a neighbor u A . All additional packets of the form (h s , h t , n ) that will arrive to v A after v B or v C has already incremented the tag will be forwarded by v LS back to v A (as it was marked as the destination port to h t ), and in v A they will be discarded. We now give a formal proof. We claim two assertions: (i) For every v ∈ V , at most one of the middleboxes v B and v C forwards a packet to an adjacent node (other than v LS ). (ii) Both v B and v C will never forward the same packet twice. The proof of item (i) is due to the fact that every packet passes through the learning switch and the learning switch will mark only one of v B or v C as the destination of h d . The proof of item (ii) is due to the fact that if a packet p is generated as a result of v B (v C ) sending a packet to an adjacent middlebox, then at this stage v A is already marked by the learning switch as the destination of h t . Therefore, when the packet p reaches v A , it will be forwarded from the learning switch back to v A and will be discarded. Hence, it can never reach v B (v C ) again. By the two assertions we get that reachability holds if and only if a packet visited |V | different middleboxes (v 1 ) X 1 , . . . , (v |V | ) X |V | for X i ∈ {B,C}, and each such middlebox was visited exactly once. Hence, reachability holds iff a Hamiltonian path exists.
Unordered Safety in arbitrary networks is EXPSPACE-hard.
The result in this section is similar to previous work on message passing systems with unordered communication channels [22,36], and is included here for completeness of presentation.
The lower bound is obtained by a reduction from the VASS control state reachability problem. We first present the problem and its known complexity results. A vector addition system with states (VASS) is a weighted directed graph (V, E, v 0 , w : E → Z k ), where V is a finite set of vertices (Control States), E ⊆ V × V is a set of directed edges, v 0 is the initial vertex, and w is a weight function that assigns a k-dimensional weight vector to every edge. A (finite) path π in the directed graph is valid if it begins in v 0 and every prefix of π has a non-negative sum of weights in every dimension.
The VASS control state reachability problem gets as input a VASS and a reachability set R ⊆ V , and checks whether there exists a valid path in the VASS to (at least) one vertex in R. Lemma 12 ([10,22,32]) The VASS control state reachability problem is EXPSPACEcomplete. Moreover, it is EXPSPACE-hard even when the coefficients of every vector in the image of the weight function are bounded by ±1, and even when every vector has at most one non-zero dimension.
To simplify our proofs we define the class of simple VASSs as all VASSs that satisfy: -Every weight vector has exactly one non-zero coefficient which is either +1 or −1.
-All the outgoing edges of every vertex v have different weight vectors. Formally, for
every v 1 , v 2 , v 3 ∈ V , if (v 1 , v 2 ), (v 1 , v 3 ) ∈ E and w(v 1 , v 2 ) = w(v 1 , v 3 ), then v 2 = v 3 .
The next claim is a simple corollary of Lemma 12.
Corollary 1 The control state reachability problem over simple VASS systems is EXPSPACE-hard.
Next, we show a reduction from control state reachability over simple VASS systems to stateful network reachability.
The reduction is straightforward: given a VASS system (V, E, v 0 , w : E → Z k ) and a reachability set R ⊆ V we construct a network with two hosts, namely h 1 and h 2 and one middlebox m (see Figure 12). The network reachability problem is to determine whether a packet with source host h 1 can reach h 2 . The set of packet tags is T = {1, . . . , k} (where k is the number of dimensions in the VASS system). We denote by p t = (h 1 , h 2 ,t), and P T = {p t | t ∈ T } the packets host h 1 sends. We associate each packet p t with a vector t ∈ N k that consists of 1 in dimension t and the rest of the dimensions are zero. The set of states of m is V (with initial state v 0 ) with the addition of one sink state. When in sink state, the middlebox discards all incoming packets and remains in sink state. We now describe the transitions of the middlebox m from state v ∈ V :
-Upon receipt of a packet p t from port 1: -If v ∈ R, then forward the packet to port 3 (reachability is obtained).
-If there exists u ∈ V such that (v, u) ∈ E (of the VASS) and w(v, u) = t, then:
• Forward p t to port 2 • Change state to u -Else (such u does not exists), discard packet and go to sink state.
-Upon receipt of a packet p t from port 2:
-If v ∈ R, then forward the packet to port 3 (reachability is obtained).
-If there exists u ∈ V such that (v, u) ∈ E (of the VASS) and w(v, u) = − t, then:
• Discard the packet • Change state to u -Upon receipt of a packet from port 3, go to sink state.
-Upon receipt of a packet p ∈ P T from any port, go to sink state.
In order to prove the correctness of the reduction we give the next definitions and notations. A VASS configuration is a tuple (v, c) ∈ V × N k which consists of a vertex and a vector. A configuration is reachable in n steps if there exists a valid path in the VASS with length exactly n and total sum of weights c. We denote by S VASS (n) the (finite) set of all configurations that are reachable in n steps.
A VASS-network configuration is a tuple (v, c) ∈ V × N k ,
where v is the state of the middlebox m and c corresponds to the multiplicity of the packets of P T in the multiset of packets in port 2. That is, if the multiplicity of packet p t in the multiset is r, then dimension t of c is r. We say that a VASS-network configuration is reachable in n steps if there exists a scenario that consists of exactly n middlebox packet processing events that forms the configuration. We denote by S Network (n) the (finite) set of all VASS-network configurations that are reachable in n steps.
Lemma 13
For every n ≥ 0:
S VASS (n) = S Network (n) − ({sink} × N k ).
Proof The proof is by induction over n. The proof for n = 0 is trivial. For n > 0, let (v, c) be an arbitrary VASS configuration in S VASS (n−1). We claim that every successor configuration of (v, c) is also in S Network (n). The proof is straightforward. If the successor is reachable by an addition of positive vector r, then a corresponding successor in the network is obtained when h 1 sends a packet of type r and m processes the packet. If the successor is reachable by an addition of negative vector r, then by the induction hypothesis there exists a pending packet in port 2 with type r, and a successor in the network is obtained when m processes one packet from port 2 with type r. Hence, we get that S VASS (n) ⊆ S Network (n) − ({sink} × N k ). The proof that S Network (n) − ({sink} × N k ) ⊆ S VASS (n) follows from similar arguments.
The next lemma follows immediately from Lemma 13 and Corollary 1.
Lemma 14
The reachability problem w.r.t. the unordered network semantics for an arbitrary network is EXPSPACE-hard.
Upper Bounds on Complexity of Safety w.r.t. the Unordered Semantics
This section provides complexity upper bounds for the safety problem of stateful networks w.r.t. the unordered semantics of networks. Our complexity analysis considers symbolic representations of middleboxes (which might be exponentially more succinct than explicitstate representations). The obtained upper bounds match the lower bounds from Section 5 (hence, the bounds are tight). (polynomial in the size of the network and the size of the middlebox representation). To enforce this limitation we assume that the arity of relations is constant. If the arity of the relation is bounded by a constant c, then the number of elements is bounded by the polynomial n c , where n is the size of the network.
In all of our examples we use relations with arity at most three, and since abstract packets have only three attributes, we believe that most applications will use relations with small arity.
The Input to the Safety Verification Problem The input to the safety verification problem is given in the form of a network topology description, and the symbolic representations of the middleboxes in the network.
The complexity results in this section are given in terms of the number of hosts in the network |H|, the size of the type domain |T |, the total number of ports in the network |Pr|, the number of middleboxes in the network |M|, and the total size of the symbolic representation |S| = ∑ |S i | where |S i | is the size of the symbolic representation of middlebox m i .
In our complexity analysis we sometime refer to the set of packets in the networks. Recall that the set of packets in the networks is P = H × H × T , and so the size of P is |P| = |H| 2 |T |. Finally, in our complexity analysis we also refer to ∑ |R i | which denotes the total size of the domains of relations of middleboxes in the network where R i is the domain of relation R i . Note that |R i | is polynomial in the in the size of |H|, |Pr| and |T |, as the arity of R i is fixed and the domains of its dimensions are taken from H, Pr and T .
Unordered Safety of Increasing Networks is in PTIME
In this section, we show that safety of syntactically increasing networks is in PTIME. Figure 13 presents a polynomial algorithm for determining safety of a syntactically increasing network. The algorithm performs a fixed-point computation of the set of all tuples present in middlebox relations in reachable middlebox states, as well as the set of all different packets transmitted in the network. For every middlebox m ∈ M, the algorithm maintains the following sets:
-StateData(m): a set of pairs of the form (R, d) where R is a relation of m, and d is a tuple in the domain of R, indicating that there is a run in which d ∈ R. -PacketData(m): a set of pairs of the form (p, pr), where p is a packet and pr is a port of m, indicating that p can reach m from port pr.
StateData(m) is initialized to reflect the initial values of all middlebox relations.
PacketData(m) is initialized to include the packets P h that can be sent from neighbor hosts h ∈ H. As long as a fixed-point is not reached, the algorithm iterates over all middleboxes and their packet data. For each middlebox m and (p, pr) ∈ PacketData(m), m is run over (p, pr) from a state q in which every relation R contains all the tuples d such that (R, d) ∈ StateData(m). The sets StateData(m) and PacketData(m ) for every neighbor m of m, are updated to reflect the discovery of more elements in the relations (more reachable states), and more packets that can be transmitted.
As the algorithm only adds relation elements and packets, the number of additions is bounded by (|P||Pr| + ∑ |R i |). At every iteration of the while loop, at least one relation element or packet is added to StateData or PacketData respectively. The number of foreach iterations in every single while iteration is bounded by |P||Pr|. The runtime of every foreach iteration is linear in the runtime of the corresponding middlebox, which is linear in the size of its symbolic representation. This is because the computation of δ m (q, (p, pr)) consists of executing the middlebox program, and since the symbolic representation does not have loops, the runtime is linear. Hence, the runtime of a single iteration of the foreach loop can be bounded by |S|.
The total running time of the algorithm is then bounded by (|P||Pr| + ∑ |R i |)|P||Pr||S|, and hence polynomial.
The correctness of the algorithm relies on the next lemma, which is a variation of Lemma 3.
Lemma 15
For every increasing network, if there is a run in the unordered semantics in which packet p arrives to port pr of middlebox m, then any run r in the unordered semantics has an extension in which packet p arrives to m from port pr. Moreover, if there is a run in which element d is in a relation R, then any run has an extension in which element d is in the relation R.
We now use Lemma 15 to prove that in every iteration the data structure of the algorithm under-approximates PacketData and StateData.
Lemma 16
For every iteration of the algorithm there is a run r, such that if (p, pr) ∈ PacketData(m), then in r there is a step in which p arrived to m from port pr, and if (R, d) ∈ StateData(m), then in r there is a step in which d was added to R.
Proof The proof is by induction on the number of iterations performed by the algorithm. The proof for the base case (zero iterations performed) is trivial -the initial state of the PacketData and StateData matches the initial state of the network.
For the n-th iteration, let (p, pr) ∈ PacketData(m). We consider two distinct cases. In the first case, after the n − 1-th iteration, (p, pr) ∈ PacketData(m). Then by the induction hypothesis, there exists a run r such that in r there is a step in which p arrived to m from port pr. In the second case, (p, pr) was added to PacketData in the n-th iteration. In this case, after iteration n − 1 there must have existed a middlebox m adjacent to m, a state q in which {(R 1 , d 1 ), · · · , (R k , d l )} ⊆ StateData(m ), and (p , pr ), such that as a result of running m over (p , pr ) from state q, (p, pr) was sent to m. By the induction hypothesis, there exist runs r 1,1 , · · · , r k,l in which (R 1 , d 1 ), · · · , (R k , d l ) (respectively) are added to StateData(m ), as well as a run r 0 in which p arrives to m from pr . Then by Lemma 15 we can constructs a run r in which m is in state q and p has arrived to m from pr . The configuration c, which is obtained by m processing p , is a successor of the last configuration of r . We denote the resulting run by r, and note that in the last step of r, p arrived to m from port pr.
The proof for (R, d) ∈ StateData(m) follows from similar arguments. Finally we use Lemma 15 to construct a witness run for the n-th iteration.
The next lemma shows that when fixed-point occurs the data structure over-approximate PacketData and StateData.
Lemma 17 When the algorithm reaches a fixed-point, if (p, pr) / ∈ PacketData(m) (respectively., (R, d) / ∈ StateData), then there is no run in which m receives p from port pr (resp., d is added to R).
Proof Let r be the witness run that the fixed-point under-approximates (r exists by Lemma 16). Towards a contradiction we assume that there is a run r in which m receives p from port pr (respectively, d was added to R), but such an event did not occur in r. By Lemma 15, we get that r has an extension in which the event does happen. But such an extension contradicts the fact that a fixed-point occurred. Hence, the data structure overapproximates all runs.
Lemma 16 and Lemma 17 imply that the algorithm determines the safety problem, and the next theorem follows.
Theorem 2
The safety problem of syntactically increasing networks w.r.t. the unordered semantics is in PTIME.
Proof Safety is violated iff there exists a run r that ends in a configuration c where some middlebox is in state q with packet p pending on its port pr such that δ m (q, (p, pr)) = / 0. By lemmas 16 and 17, the latter holds iff at some iteration of the algorithm (p, pr) ∈ PacketData(m), and the values pf m's relations in state q are included in StateData(m), in which case the algorithm identifies the safety violation.
Remark 6
Recall that for increasing networks, safety w.r.t. the unordered semantics and the FIFO semantics coincide. As such, the polynomial upper bound applies to both.
Remark 7
The complexity analysis of the algorithm used the property that |P| is polynomial in the network representation. If n-tag packet headers are allowed, i.e. P = H × H × T 1 . . . × T n , then |P| is no longer polynomial in the network representation, damaging the complexity analysis of the algorithm. In fact, in this case the safety problem w.r.t. the unordered semantics becomes PSPACE-hard even for stateless middleboxes.
Intuitively, n-tag packet headers allow a middlebox to maintain the state of n automata in the packet header, supporting a reduction from the emptiness problem of the intersection of n automata, which is PSPACE-hard [20].
Proof The PSPACE-hardness proof is by reduction from the problem of deciding the emptiness of intersection of n automata [20], which is formally defined as:
-Input: n automata A 1 , . . . , A n over alphabet {0, 1} with state set Q (w.l.o.g. all automata have the same set of states). -Question: is there a word w ∈ {0, 1} * that is accepted by all n automata? The reduction is as follows. Given n automata with state set Q we define a network with one host and one middlebox. The packets consist of n + 1-tuples of tags from the domain T = Q ∪ {0, 1}. Intuitively, the first n tags hold the states of the n automata, and the last tag is an input symbol for the automata. The middlebox has two ports. Port 0 is connected to the host and port 1 is a self loop.
The symbolic representation of the middlebox has four parts:
1. Initial state verifier. The first part handles packets from port 0. If the packet's first n tags do not correspond to the n initial states, then the middlebox discards the packet.
Otherwise it sends the packet to port 1. 2. Advance state. The second part handles packets from port 1. In a sequence of n|Q| commands, the program advances the state of each automaton (i.e., changes the corresponding packet tag) according to the symbol in tag n + 1. After the sequence, the program continues to the third part. 3. Accepting state verifier. If the packet's tag corresponds to n accepting states, then the program aborts. Otherwise the program continues to the fourth part. 4. New symbol generator. In the fourth part the program generates two packets that differ only in their n + 1 tag. In one packet the tag has value 0 and in the second it has value 1. Both packets are sent back to port 1. It is an easy observation that the intersection of the n automata is non-empty iff abort is invoked.
Unordered Safety of Progressing Networks is in coNP
We prove coNP-membership of the safety problem in syntactically progressing networks by proving that there exists a witness run for safety violation if and only if there exists a "short" witness run, where a witness run for safety violation is a run from the initial configuration in which at least one middlebox reaches an abort state.
The proof considers the network states that arise in a run. A network state captures the states of all middleboxes (not to be confused with a network configuration, which also includes the content of every channel). Formally, let N be a network whose middleboxes are defined symbolically via (in total) n relations, namely R 1 , . . . , R n . Then the network state consists of the values of (R 1 , . . . , R n ).
In order to construct a "short" witness run, we wish to bound both the number of different network states in a run and the number of steps in which a run stays in the same state. The former is bounded due to the progress of the network: once the state of some middlebox changes along a run, it will not change back to the previous state. The latter is more tricky. To provide a bound, we wish to analyze the packets that "affect" the run. We define the notion of active packets. The active packets are a superset of the packets that actually affect the run.
Active packets Let r be a finite run of a network. We say that a packet p is active in step i of r, if it resides in the ingress channel of some middlebox m and it is processed (i.e., received by m) in some future step of r. A packet is inactive, if it is pending in the ingress channel of m until the end of the run.
The next lemmas show that only a few active packets are needed to reach a certain state in the network. Intuitively, the proof of the lemma traverses the run from the last configuration to the first, and removes inactive packets (and steps that produce only inactive packets), which in turn makes other, earlier, packets inactive. For a run r and a network state s that appears in r, we denote by r[s] an interval of the run that includes all consecutive occurrences of s (for runs of progressing networks, the interval is unique).
Lemma 18
Let r be a run in which the network state changes exactly k times, and the different states are s 1 , s 2 , . . . , s k (in this order). Then for every prefix r s i of r that ends in a state s i , there is an extension e s i to r s i such that: (i) e s i visits the network states s i , . . . , s k ;
(ii) e s i has at most k − i active packets in every step; and (iii) the number of active packets in e s i may decrease only after a change in the network state.
Proof The proof is by induction over |r| − |r s i |. For the base case r = r s i and the proof is trivial. For |r| > |r s i |, we extend the prefix r s i by one step according to r. We denote this extended prefix by r . Let p be the last packet that was processed in r , and let m be the middlebox that processes p. That is, m and p are responsible for the step that extends r s i to r .
We consider two distinct cases. In the first case, the network state in the last configuration of r is still s i . Then by the induction hypothesis we get that there is an extension e s i with at most k − i active packets in interval e s i [s i ]. We consider the set of packets that were created by m after processing p. If this set has at least one active packet in e s i , then we define e s i to be e s i prepended by the last step of r , where p is marked as active and all the active packets of e s i remain active. Surely, there are no more than k − i active packets in the first step of e s i since at least one of the active packets in e s i resulted from p and hence did not yet exist in this step, so it balances out the addition of p as an active packet. In addition, the total number of active packets is not decreased in this step (thus, the claim holds). Otherwise, we define e s i to be e s i , i.e. we skip the processing of p, and turn it to inactive.
In the second case, the last state in r is s i+1 . Then by the induction hypothesis we get that there is an extension e s i+1 with at most k − i − 1 active packets. In this case we construct e s i simply by prepending to e s i the last step of r . That is, p is marked as active and all the active packets of e s i+1 remain active. There are only k − i − 1 + 1 = k − i active packets. Hence, the claim holds. This completes the proof.
Lemma 19
Let r be a run in which the network state changes exactly k times, and the different states are s 1 , s 2 , . . . , s k (in this order). Then there exists a run r such that: (i) r visits the network states s 1 , s 2 , . . . , s k ; and (ii) r stays in state s i at most (k − i) 2 |P||M| steps.
Proof For the sake of the proof we give a unique id to every active packet according to the following rules:
-If a host sends an active packet, then the packet gets some unique id (for example, maximal id assigned so far + 1). -If an active packet p 1 was processed by a middlebox, and the middlebox forwards only one active packet p 2 , then p 2 gets the id of p 1 . -If an active packet p 1 was processed by a middlebox, and the middlebox forwards more than one active packet, then each active packet gets a unique id (for example, maximal id assigned so far + 1). We now return to the proof. Let e be the shortest extension for the prefix of r that consists of the initial configuration that satisfies the assertions of Lemma 18. The extension e clearly visits s 1 , . . . , s k . We claim that it stays in state s i at most (k − i) 2 |P||M| steps. The proof of the claim follows from the fact that if there are two steps j 1 < j 2 in e [s i ] such that in both steps a middlebox m received an active packet p with id id, and no new active packet (i.e., an active packet with a new packet id) was generated between those rounds, then a run in which m does not process packet p with id id is shorter by one step, and reaches the same configuration in step j 2 − 1. Hence, if a certain middlebox processed more than |P|(k − i) packets, then it must be the case that either a new active packet was created, or it processed the same packet twice. The proof is complete by the pigeonhole principle and by the fact that there are at most k − i active packets and |M| middleboxes.
The next lemma shows that there is a short witness for reachability of a state in progressing networks.
Lemma 20 Let N be a syntactically progressing network whose middleboxes are defined symbolically via relations R 1 , . . . , R n (in total). Then there is a run ending in an abort state if and only if there is such a run whose length is at most
(∑ n i=1 |R i |) 3 |P||M|.
Proof The proof is an immediate corollary of Lemma 19. If there is a run r that leads to a certain state of R 1 , . . . , R n , then since all middleboxes are progressing we get that the number of intermediate network states k is at most (∑ n i=1 |R i |). We denote the intermediate states by s 1 , . . . , s k . By Lemma 19, there is also a run r that visits the same k states and stays in state s i at most (k − i) 2 |P||M| ≤ k 2 |P||M| steps. Therefore |r | ≤ k 3 |P||M|.
Since the size of each relation is polynomial in the size of the network, we conclude:
Theorem 3
The safety problem w.r.t. the unordered semantics for progressing networks is coNP-complete.
Proof The lower bound follows from Lemma 10. The upper bound is obtained by first observing that the complement of the safety problem is polynomially reducible to the reachability of a state in the network (by adding a special abort state). In addition, the state reachability problem is in NP: since the arity of each relation in the considered middlebox programs is fixed, its size is polynomial in the size of the network. Hence, by Lemma 20, there is a witness run for reachability whose length is polynomial. Thus, the NP procedure is to guess the short run and verify it, in time linear in the length of the run multiplied by |S| (the size of the symbolic representation of the middleboxes which also bounds the time it takes to compute their transitions).
Unordered Safety of Arbitrary Networks is in EXPSPACE
In this section we show how to solve the reachability problem of symbolic networks by a reduction to the coverability problem of Petri Nets, which is EXPSPACE-complete [22,32].
Similarly to the lower bound result (Section 5.2), the upper bound result on the complexity of safety of arbitrary networks is similar to previous work ( [22,36]), and is included here for completeness of presentation.
A Petri Net is a four-tuple C = (P, T , I , O) where P is a set of places, T is a set of transitions, I : T → N |P| is an input function and O : T → N |P| is an output function. A marking µ ∈ N |P| denotes the number of tokens assigned to each place. Given a marking, a transition t ∈ T can be fired (equivalently enabled) if I (t) ≤ µ. Firing a transition t ∈ T from marking µ produces a new marking µ = µ − I (t) + O(t) [30]. We denote a firing of a transition by µ → t µ . In the following, we will refer to non-zero dimensions in I (t) as consumed tokens, and non-zero dimensions in O(t) as produced tokens. A finite run in a Petri Net from a marking µ 0 is a series of transitions and resulting markings µ 0 → t 0 µ 1 → t 1 · · · → t k µ k s.t. t 0 can be fired from µ 0 and each following transition can be fired from the previous marking.
The coverability problem asks, given a Petri Net C , an initial marking µ 0 and a target marking µ, whether there is a finite run leading to a marking µ s.t. µ ≥ µ.
We now show how we encode a symbolic network as a Petri Net, and how we formulate the reachability problem as a Petri Net coverability problem. We first describe the role of every place and the initial marking, and then we describe the set of transitions used to simulate a run of the network.
Places The places are partitioned to sets of places in the following way:
-Channel places. To keep track of the packets over the unbounded channels, we assign a place to every pair of packet p ∈ P and channel. The number of tokens in the place corresponds to the number of instances of packet p on the channel. The initial marking for each packet place is 0. -Active and non-active relation places. For every element d in every relation R in every middlebox we have two places. The active place will have the marking 1 when the element is in the relation. When the element is not in the relation the non-active place will the marking 1. The initial marking for the active (respectively, non-active) place is 1 if initially the element is in the relation (resp., not in the relation). Otherwise, the initial marking is 0. The markings for both places will only be 0 or 1. We need two places since the Petri Net semantics does not allow to encode negative (i.e., non-membership) conditions. -Global command place. We have a single place that is used to make sure that at most one middlebox is processing a packet in every step. The initial marking for the place is 1; it is consumed whenever a packet processing starts, and produced when it ends. -Command places. We have a place for every triple of command, processed packet and input port in every middlebox in the network. The markings on the places are used to keep track of the next command to be executed. In particular, each guarded command block has a single place (for every combination of packet and input port) rather than a place for each guarded command in the block. This ensures that only one of the guarded commands in the block whose guards evaluate to true is executed. Having a separate command place for every packet processed and every input port allows us to evaluate variables that appear in the command (including the guards). The initial marking for the topmost guarded command block in each middlebox (with every combination of packet and input port) is 1. The initial marking for the rest is 0. -Auxiliary guard places. To allow conjunction and disjunction in the guard we add auxiliary guard places. The initial marking for each of these places is 0. -Abort place. To keep track of the safety state of the network, we assign a single place for all abort calls made during the network run. The initial marking for the place is 0.
Transitions For each middlebox in the network we define a "command transition" for each combination of processed command, input packet, input port, and next command, as explained below. For some commands only a single "next" command exists, however, since we allow non-determinism, some commands (specifically, guarded command blocks with overlapping guards) have multiple "next" commands, in which case a separate transition is defined for each one of them. For a guarded command block we define a set of "command transitions". This allows us to handle complex guards (i.e. guards which contain conjunction and disjunction in addition to atomic propositions). To do so, we recursively decompose each guard while producing a sequence of transitions that simulates the evaluation of the boolean formula in the guard.
To correctly simulate cases in which no guard in a guarded command block is evaluated to true, and as a result no command is processed, we add a default guarded command to each guarded command block. The guard of the default guarded command is a conjunction of the negations of the guards of the other guarded commands in the block. The command of the default guarded command is output / 0. Each of the command transitions of the first command in the middlebox (i.e. the topmost guarded command block) consumes a token from the global command place, and each terminating command that can be executed in the middlebox run produces a token in the global command place. Note that the addition of default guarded commands as described above means that the terminating commands are well defined (i.e. for every command in the middlebox, if it is terminating in some run then it is a terminating command in every run that it is executed in). Each of the command transitions of the first command in the middlebox also consumes a token from the corresponding channel place. Furthermore, every command transition consumes its command place, and produces the command place of the following command, specifically the place corresponding to the combination of the next command to be executed and the same input packet and input port as the packet and port processed in the current command (or the first command in case it is a terminating command).
In addition to the above, the command transition associated with a command, input packet, input port and next command consumes and produces tokens in the places relevant to the corresponding command, as well as the guards (in the case of a guarded command block), as described below.
Since we have a command transition for every combination of command, input packet and input port, when we translate the command to a transition we consider the values of the variables (src, dst, type and port) at that transition based on the packet and port currently processed by the middlebox, and simplify the command (and guards) accordingly. For example, for the command trusted.insert dst, packet (h 0 , h 1 ,t 0 ) and port pr 0 , the command simplifies to trusted.insert h 1 . In particular, atomic equality predicates are now essentially equalities between constants, and are trivially simplified.
The transition for each guarded command in a guarded command block consumes a token from the command place for the guarded command block, and produces a token in the command place of the first command in the guarded command, as well as consuming and producing the tokens of the guard as described below.
We begin by describing the tokens consumed and produced by the atomic propositions of the guards (after simplification). Note that since guards do not change the state of the network, all tokens consumed by the guard must also be produced by the guard.
-Relation membership (d ∈ R). Consume (and produce) tokens in the active place for element d in relation R. -Negated relation membership (d / ∈ R). Consume (and produce) tokens in the inactive place for element d in relation R. Next, we describe how disjunction and conjunction are handled: In the case of a guarded command whose guard's formula ϕ contains a disjunction or conjunction, we produce a series of transitions by recursively decomposing the formula, and producing a set of transitions for every decomposition step. Each decomposition step introduces new auxiliary guard places. We denote by c i =⇒ ϕ c j an intermediate step in the decomposition process where c i is the place that initiates the evaluation of ϕ and c j is the place of the next step in the execution. Specifically, initially, c i is the command place for the guarded command and c j is the command place of the command. The recursive decomposition of guard c i =⇒ ϕ c j is as follows:
-Conjunction (ϕ = ϕ 1 ∧ ϕ 2 ). We introduce five auxiliary places, denoted c 1 , c 2 , c 3 , c 4 and c 5 , two intermediate steps, and four new transitions. The first transition consumes one token from c i and produces two tokens in c 1 . The second and third transitions consume one token each from c 1 and produce a token in c 2 and c 3 respectively. We produce two intermediate steps: c 2 =⇒ ϕ 1 c 4 and c 3 =⇒ ϕ 2 c 5 . Finally, we produce a final transition that consumes one token from both c 4 and c 5 , and produces a token in c j . -Disjunction (ϕ = ϕ 1 ∨ ϕ 2 ). We introduce four auxiliary places, denoted c 1 , c 2 , c 3 and c 4 , two intermediate steps, and four new transitions. The first transition consumes a token from c i and produces a token in c 1 . Likewise, the second transition consumes a token from c i and produces a token in c 2 . We produce two intermediate steps: c 1 =⇒ ϕ 1 c 3 and c 2 =⇒ ϕ 2 c 4 . The third transition consumes a token from c 3 and produces a token in c j . Likewise, the fourth transition consumes a token from c 4 and produces a token in c j . The process is performed recursively on c i =⇒ ϕ 1 c j and c i =⇒ ϕ 2 c j . The process terminates for c i =⇒ ϕ c j once ϕ is an atomic proposition, in which case a single transition is produced, which consumes a token from c i , consumes and produces the tokens for the atomic proposition as described above, and produces a token in c j .
Finally, we describe the dimensions consumed and produced by the commands output, insert, remove and abort.
-output. Produce: the appropriate packets in the egress channel. We note that in the special case of output / 0 no tokens are produced. -insert. We replace every insert command with a guarded command block consisting of two guarded commands. The first guarded command represents the case where the element is already in the relation, in which case the guard will be a relation membership predicate, and the command will be output / 0. The second guarded command represents the case where the element is not in the relation. The guard of the command will be a negated relation membership predicate to the guard, and the transition produced from the command will consume and produce the following: Consume: a token from the appropriate non-active place of the new element. Produce: a token in the appropriate active place of the new element.
-remove. Analogous to insert. -abort. Produce: a token in the abort place. This concludes the description of the command transitions.
Finally, for every host h and every packet p ∈ P h we have a "host transition" that produces a token in the corresponding ingress channel place of the neighbor middlebox.
From Network Safety to Petri Net Coverability Non-safety of the network amounts to a run in the Petri Net where an abort place gets a token. The target marking for the coverability problem is therefore a vector of 0s, with 1 in the abort place.
As the reduction is polynomial, we get that the stateful network reachability problem is in EXPSPACE.
The reduction, combined with the lower bound implies:
Theorem 4 The safety problem of arbitrary stateful networks w.r.t. the unordered semantics is EXPSPACE-complete.
Implementation and Case Studies
In this section, we present several examples of networks consisting of stateful middleboxes and their safety properties. We describe a prototype implementation of a tool for verification of stateful networks, and describe our initial experience while running the tool on the networks listed in Example 2 and illustrated in Figure 5. For the experiments we used a machine equipped with a quad core Intel Core i7-4790 CPU and 32GB of memory, running Ubuntu Linux 14.04.
Network Examples
Load Balancer and IDS As an example consider the network shown in Figure 5a. Here A is a host, lb is a load balancer, which can send a packet received from A to either r 1 or r 2 . Both r 1 and r 2 are rate limiters, i.e., they count and limit the number of packets sent between host pairs. Let us consider a case where the administrator wants to ensure that exactly 8 packets sent by A can be received by B. If the load balancer in this case sends packets from A to both r 1 and r 2 , then this rate limit does not hold.
Firewall and Proxy Consider the network in Figure 5b. Here, c is a content addressable cache, which on receiving a packet checks if it has previously seen either server S 1 or S 2 respond to a packet of the same type; if so it sends back the previously observed response, otherwise it forwards the request to the packets original destination. f is a learning firewall. We want to ensure that A cannot receive data from S 1 , while B should be able to receive data from both S 1 and S 2 . This is complicated by the fact that c's response is based on the packet type: in the current configuration if B sends a request for type t to server S 1 then A can access the response by subsequently sending a request with the same type t addressed to server S 2 . In general this problem is not solvable without changing the cache to be policy aware.
Multi-Tenant Datacenter Consider a multi-tenant datacenter such as Amazon EC2 shown in Figure 5c. In such datacenters each tenant (customer who purchase VMs from the provider) gets to add rules about their VMs, to the firewall to which their VMs are connected. For example in Figure 5c, each tenant i owns VMs pub i 1 and pri i 1 , and programs the rules for firewall f i . Given a set of rules for firewall f 1 and f 2 we verify that VMs of the same tenant can communicate with each other and that pri VMs of one tenant can send packets to pub VMs of the other.
results
Increasing Middleboxes Increasing networks are verified using LogicBlox, a Datalog based database system [5]. The Multi-Tenant Datacenter example is an increasing network. Our tool produced a datalog program with 35 predicates, 153 rules and 29 facts. LogicBlox successfully reached a fixed point in 3s, and proved all required properties.
Arbitrary Middleboxes Progressing and Arbitrary networks are verified using LOLA, a Petri-Net model checker [34,1]. In the Load Balancer and Rate Limiter example our tool created a P/T net with 243 places and 663 transitions; it was successfully verified in 30ms. In the Firewall and Proxy example our tool produced a P/T net with 530 places and 4447 transitions. LOLA successfully verified the resulting petri-net in 0.2s.
Conclusion and Related Work
In this work, we investigated the complexity of reasoning about stateful networks. We developed three algorithms and several lower bounds. In the future we hope to develop practical verification methods utilizing the results in this work. Below we survey some of the most closely related work and conclude with open questions and future work.
Related Work
Topology-Independent Verification The earliest use of formal verification in networking focused on proving correctness and checking security properties for protocols [11,33]. Recent works such FlowLog [26] and VeriCon [6] also aim to verify the correctness of a given middlebox implementation w.r.t any possible network topology and configuration, e.g., flow table entries only contain forwarding rules from trusted hosts.
Immutable Topology-Dependent Verification Recent efforts in network verification [24,9,18,19,39,37,4,14] have focused on verifying network properties by analyzing forwarding tables. Some of these tools including HSA [17], Libra [42] and VeriFlow [19]. These tools perform near real-time verification of simple properties, but they cannot handle dynamic (mutable) datapaths.
Mutable Topology-Dependent Verification SymNet [40] has suggested the need to extend these mechanisms to handle mutable datapath elements. In their mechanism the mutable middlebox states are encoded in the packet header. This technique is only applicable when state is not shared across a flow (i.e., the middlebox can punch holes, but do no more), and will not work for cache servers or learning switches.
The work in [29] is the most similar to our model. Their work considers Python-like syntax enriched with uninterpreted functions that model complicated functionality. However [29] do not define formal network semantic (e.g., FIFO vs ordered channels) and do not give any formal claim on the complexity of the solution.
Channel Systems Channel systems, also called Finite State Communicating Machines, are systems of finite state automata that communicate via asynchronous unbounded FIFO channels [7,8]. They are a natural model for asynchronous communication protocolsand, indeed, they form the semantic basis of protocol specification languages such as SDL and Estelle. Unbounded FIFO channels can simulate unbounded Turing machine tape and therefore all verification problems are undecidable. Abdulla and Jonsson [2] introduced lossy channel systems where messages can be lost in transit. In their model the reachability problem is decidable but has a non-primitive lower bound [35].
In this work we use unordered (non-lossy) channels as a different relaxation for channel systems. The unordered semantics over-approximates the lossy semantics w.r.t. safety, as any violating run w.r.t. the lossy semantics can be simulated by a run w.r.t. the unordered semantics where "lost" packets are starved until the violation occurs.
The unordered semantics admits verification procedures with elementary complexity, and turns out to be sufficiently precise for many network protocols in which order is not guaranteed and hence not relied on.
Future Work
Exploration of Network Semantics In this work we have outlined two possible network semantics, namely FIFO and Unordered packet processing order. Various other network semantics could be considered, along with their effect on expressibility and complexity results, and the precision loss in safety analysis. One such network semantics is the Sticky Channel semantics, where packets can be added by the sending middlebox and read by the receiving middlebox but cannot be removed. This network semantics corresponds to networks in which middleboxes can arbitrarily retransmit messages.
Modelling Packet Payload In this work we have only considered packet headers. However, some middlebox behaviour depends on the content of the packet payload (Intrusion Detection Systems are one such example). A potential approach to bridging this gap could be to model middleboxes using register automata. This would allow us to reason about letters from an infinite alphabet, thus modelling the arbitrary nature of packet payloads, while potentially retaining the decidability of reasoning about such systems.
Liveness In this work we have limited ourselves to reasoning about safety properties. However, various liveness and performance properties are just as important when approaching the creation of networks. Reasoning about liveness properties such as guarantees on packet arrival, or performance properties such as load estimates or packet traversal times would require the development of a new model for describing the network semantics and middlebox behaviour. In particular, unordered semantics are ill suited for most sorts of reasoning on liveness properties.
Further Aspects of Network Security In addition to safety properties that can be expressed by checker middleboxes and liveness properties there are various other network security properties that can be considered when reasoning about networks. Non-interference and information leakage are two examples of security properties which cannot be modeled by our current approach.
Reasoning About Progressing Networks Under the FIFO Semantics
We've seen that in arbitrary networks reasoning is undecidable under the FIFO semantics but EXPSPACEcomplete under the unordered semantics, and that for increasing networks the two semantics coincide. This leaves the question of reasoning about progressing network under the FIFO semantics open.
Fig. 1 :
1Middlebox hierarchy with worst-case time complexity for each category. efficient verification procedures, as shown in
Fig. 3 :
3Symbolic representation of middleboxes.
Fig. 5 :
5Interesting network topologies for verification.
Fig. 6 :
6The network resulting from the reduction from the halting problem for Two Counter Machines.
dst,tag, oprt) | oprt in allPorts and oprt = prt} // flood Fig. 8: A learning switch with three ports.
Fig. 10 :
10The network 'gadget' associated with vertex v in the reduction from the Hamiltonian Path problem to network reachaility. The vertex v has an incoming edge from u i and an outgoing edge to vertex u j in the input graph G.
Fig. 11 :
11The network resulting from the reduction from the Hamiltonian Path problem to network reachability.
Fig. 12 :
12The network resulting in the reduction from the VASS control state reachability problem.
Remark 5
5The complexity upper bounds we present are under the assumption that all relations used to define middlebox states may have at most polynomial number of elementsStateData := {m → InitialRelationValues(m) | m ∈ M} PacketData := {m → NeighborHostPackets(m) | m ∈ M} while fixed-point not reached foreach m ∈ M, (p, pr) ∈ PacketData(m) let q = StateData(m) if δ m (q, (p, pr)) = / 0 then return violation // abort state reached let (q , o) ∈ δ m (q, (p, pr)) StateData := AddData(m, q ) PacketData := AddPacketsToNeighbors(m, o) return safe Fig. 13: Safety checking of increasing networks.
mbox ::= input(src, dst,tag, prt) : gcmd [ gcmd ] * gcmd ::= grd ⇒ cmd guarded command cmd ::= output { exp [, exp ] * }output a packet
| abort
terminate-abnormally
| id.insert exp
add tuple to relation id
| id.remove exp
remove tuple from id
| cmd ; cmd
sequence of commands
| gcmd [ gcmd ] *
guarded command block
exp
::= src | dst | tag | prt
variable
| constant
constant
grd
::= grd and grd
| grd or grd
| not atom
| atom
atom ::= exp = exp
equality
| exp in id
membership test
Fig. 2: A simple language for representing finite state middleboxes. exp denotes a vector
of exp separated by commas.
In this work we do not distinguish between routers and switches, since they obey similar forwarding models.2 Routers may be considered a degenerate case of middleboxes, whose state is constant and hence their forwarding behaviour does not change over time.
A subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements.
Verifying programs with unreliable channels. P Abdulla, B Jonsson, Logic in Computer Science (LICS). IEEEAbdulla, P., Jonsson, B.: Verifying programs with unreliable channels. In: Logic in Computer Science (LICS), pp. 160-170. IEEE (1993)
General decidability theorems for infinite-state systems. P A Abdulla, K Čerāns, B Jonsson, Y K Tsay, Logic in Computer Science (LICS). IEEEAbdulla, P.A.,Čerāns, K., Jonsson, B., Tsay, Y.K.: General decidability theorems for infinite-state sys- tems. In: Logic in Computer Science (LICS), pp. 313-321. IEEE (1996)
NetKAT: Semantic foundations for networks. C J Anderson, N Foster, A Guha, J B Jeannin, D Kozen, C Schlesinger, D Walker, POPLAnderson, C.J., Foster, N., Guha, A., Jeannin, J.B., Kozen, D., Schlesinger, C., Walker, D.: NetKAT: Semantic foundations for networks. In: POPL (2014)
Design and implementation of the logicblox system. M Aref, B Ten Cate, T J Green, B Kimelfeld, D Olteanu, E Pasalic, T L Veldhuizen, G Washburn, ACM SIGMOD International Conference on Management of Data. Aref, M., ten Cate, B., Green, T.J., Kimelfeld, B., Olteanu, D., Pasalic, E., Veldhuizen, T.L., Washburn, G.: Design and implementation of the logicblox system. In: ACM SIGMOD International Conference on Management of Data, pp. 1371-1382 (2015)
Vericon: towards verifying controller programs in software-defined networks. T Ball, N Bjørner, A Gember, S Itzhaky, A Karbyshev, M Sagiv, M Schapira, A Valadarsky, In: ACM SIGPLAN Conference on Programming Language Design and Implementation. 31PLDIBall, T., Bjørner, N., Gember, A., Itzhaky, S., Karbyshev, A., Sagiv, M., Schapira, M., Valadarsky, A.: Vericon: towards verifying controller programs in software-defined networks. In: ACM SIGPLAN Con- ference on Programming Language Design and Implementation, PLDI, p. 31 (2014)
Finite state description of communication protocols. G V Bochmann, Computer Networks. 24Bochmann, G.V.: Finite state description of communication protocols. Computer Networks (1976) 2(4), 361-372 (1978)
On communicating finite-state machines. D Brand, P Zafiropulo, Journal of the ACM (JACM). 302Brand, D., Zafiropulo, P.: On communicating finite-state machines. Journal of the ACM (JACM) 30(2), 323-342 (1983)
A nice way to test openflow applications. M Canini, D Venzano, P Peres, D Kostic, J Rexford, 9th USENIX Symposium on Networked Systems Design and Implementation (NSDI'12. Canini, M., Venzano, D., Peres, P., Kostic, D., Rexford, J.: A nice way to test openflow applications. In: 9th USENIX Symposium on Networked Systems Design and Implementation (NSDI'12) (2012)
Exponential space complete problems for petri nets and commutative semigroups (preliminary report). E Cardoza, R Lipton, A R Meyer, Proceedings of the eighth annual ACM symposium on Theory of computing. the eighth annual ACM symposium on Theory of computingACMCardoza, E., Lipton, R., Meyer, A.R.: Exponential space complete problems for petri nets and commuta- tive semigroups (preliminary report). In: Proceedings of the eighth annual ACM symposium on Theory of computing, pp. 50-54. ACM (1976)
Using state space exploration and a natural deduction style message derivation engine to verify security protocols. E M Clarke, S Jha, W R Marrero, Programming Concepts and Methods, IFIP TC2/WG2.2,2.3 International Conference on Programming Concepts and Methods (PROCOMET '98) 8-12. Shelter Island, New York, USAClarke, E.M., Jha, S., Marrero, W.R.: Using state space exploration and a natural deduction style mes- sage derivation engine to verify security protocols. In: Programming Concepts and Methods, IFIP TC2/WG2.2,2.3 International Conference on Programming Concepts and Methods (PROCOMET '98) 8-12 June 1998, Shelter Island, New York, USA, pp. 87-106 (1998)
Well-structured transition systems everywhere. A Finkel, P Schnoebelen, ! Theoretical Computer Science. 2561Finkel, A., Schnoebelen, P.: Well-structured transition systems everywhere! Theoretical Computer Sci- ence 256(1), 63-92 (2001)
A general approach to network configuration analysis. A Fogel, S Fung, L Pedrosa, M Walraed-Sullivan, R Govindan, R Mahajan, T D Millstein, 12th USENIX Symposium on Networked Systems Design and Implementation. Oakland, CA, USA15Fogel, A., Fung, S., Pedrosa, L., Walraed-Sullivan, M., Govindan, R., Mahajan, R., Millstein, T.D.: A general approach to network configuration analysis. In: 12th USENIX Symposium on Networked Systems Design and Implementation, NSDI 15, Oakland, CA, USA, May 4-6, 2015, pp. 469-483 (2015)
A coalgebraic decision procedure for netkat. N Foster, D Kozen, M Milano, A Silva, L Thompson, Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages. the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming LanguagesMumbai, IndiaFoster, N., Kozen, D., Milano, M., Silva, A., Thompson, L.: A coalgebraic decision procedure for netkat. In: Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Program- ming Languages, POPL 2015, Mumbai, India, January 15-17, 2015, pp. 343-355 (2015)
Detection and analysis of routing loops in packet traces. U Hengartner, S Moon, R Mortier, C Diot, Proceedings of the 2nd ACM SIGCOMM Workshop on Internet measurment. the 2nd ACM SIGCOMM Workshop on Internet measurmentACMHengartner, U., Moon, S., Mortier, R., Diot, C.: Detection and analysis of routing loops in packet traces. In: Proceedings of the 2nd ACM SIGCOMM Workshop on Internet measurment, pp. 107-112. ACM (2002)
Ordering by divisibility in abstract algebras. G Higman, Proceedings of the London Mathematical Society. Higman, G.: Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society pp. 326-336 (1952)
Real time network policy checking using header space analysis. P Kazemian, M Chang, H Zeng, G Varghese, N Mckeown, S Whyte, 10th USENIX Symposium on Networked Systems Design and Implementation (NSDI '13. Kazemian, P., Chang, M., Zeng, H., Varghese, G., McKeown, N., Whyte, S.: Real time network policy checking using header space analysis. In: 10th USENIX Symposium on Networked Systems Design and Implementation (NSDI '13) (2013)
Header space analysis: Static checking for networks. P Kazemian, G Varghese, N Mckeown, 9th USENIX Symposium on Networked Systems Design and Implementation (NSDI '12). Kazemian, P., Varghese, G., McKeown, N.: Header space analysis: Static checking for networks. In: 9th USENIX Symposium on Networked Systems Design and Implementation (NSDI '12) (2012)
Veriflow: verifying network-wide invariants in real time. A Khurshid, W Zhou, M Caesar, B Godfrey, Computer Communication Review. 424Khurshid, A., Zhou, W., Caesar, M., Godfrey, B.: Veriflow: verifying network-wide invariants in real time. Computer Communication Review 42(4), 467-472 (2012)
Lower bounds for natural proof systems. D Kozen, 18th Annual Symposium on Foundations of Computer Science. IEEEKozen, D.: Lower bounds for natural proof systems. In: 18th Annual Symposium on Foundations of Computer Science, pp. 254-266. IEEE (1977)
A soft way for openflow switch interoperability testing. M Kuzniar, P Peresini, M Canini, D Venzano, D Kostic, CoNEXT. Kuzniar, M., Peresini, P., Canini, M., Venzano, D., Kostic, D.: A soft way for openflow switch interop- erability testing. In: CoNEXT, pp. 265-276 (2012)
The reachability problem requires exponential space. R Lipton, 62Department of Computer Science. Yale UniversityLipton, R.: The reachability problem requires exponential space. Department of Computer Science. Yale University 62 (1976)
Checking beliefs in dynamic networks. N P Lopes, N Bjørner, P Godefroid, K Jayaraman, G Varghese, 12th USENIX Symposium on Networked Systems Design and Implementation. Oakland, CA, USA15Lopes, N.P., Bjørner, N., Godefroid, P., Jayaraman, K., Varghese, G.: Checking beliefs in dynamic net- works. In: 12th USENIX Symposium on Networked Systems Design and Implementation, NSDI 15, Oakland, CA, USA, May 4-6, 2015, pp. 499-512 (2015)
Debugging the Data Plane with Anteater. H Mai, A Khurshid, R Agarwal, M Caesar, B Godfrey, S T King, SIGCOMMMai, H., Khurshid, A., Agarwal, R., Caesar, M., Godfrey, B., King, S.T.: Debugging the Data Plane with Anteater. In: SIGCOMM (2011)
Recursive unsolvability of post's problem of "tag" and other topics in theory of turing machines. M L Minsky, Annals of Mathematics. Minsky, M.L.: Recursive unsolvability of post's problem of "tag" and other topics in theory of turing machines. Annals of Mathematics pp. 437-455 (1961)
Tierless programming and reasoning for software-defined networks. T Nelson, A D Ferguson, M J G Scheer, S Krishnamurthi, Proceedings of the 11th USENIX Symposium on Networked Systems Design and Implementation. the 11th USENIX Symposium on Networked Systems Design and ImplementationSeattle, WA, USA27Nelson, T., Ferguson, A.D., Scheer, M.J.G., Krishnamurthi, S.: Tierless programming and reasoning for software-defined networks. In: Proceedings of the 11th USENIX Symposium on Networked Systems Design and Implementation, NSDI 2014, Seattle, WA, USA, April 2-4, 2014, pp. 519-531 (2014) 27. OpenStack: LogicBlox. http://www.logicblox.com/ retrieved 07/07/2015
New directions for network verification. A Panda, K J Argyraki, M Sagiv, M Schapira, S Shenker, 1st Summit on Advances in Programming Languages. Asilomar, California, USAPanda, A., Argyraki, K.J., Sagiv, M., Schapira, M., Shenker, S.: New directions for network verifica- tion. In: 1st Summit on Advances in Programming Languages, SNAPL 2015, May 3-6, 2015, Asilomar, California, USA, pp. 209-220 (2015)
Verifying isolation properties in the presence of middleboxes. A Panda, O Lahav, K Argyraki, M Sagiv, S Shenker, arXiv:1409.7687arXiv preprintPanda, A., Lahav, O., Argyraki, K., Sagiv, M., Shenker, S.: Verifying isolation properties in the presence of middleboxes. arXiv preprint arXiv:1409.7687 (2014)
Petri nets. J L Peterson, ACM Computing Surveys (CSUR). 93Peterson, J.L.: Petri nets. ACM Computing Surveys (CSUR) 9(3), 223-252 (1977)
Demystifying the dark side of the middle: a field study of middlebox failures in datacenters. R Potharaju, N Jain, Proceedings of the 2013 Internet Measurement Conference, IMC 2013. the 2013 Internet Measurement Conference, IMC 2013Barcelona, SpainPotharaju, R., Jain, N.: Demystifying the dark side of the middle: a field study of middlebox failures in datacenters. In: Proceedings of the 2013 Internet Measurement Conference, IMC 2013, Barcelona, Spain, October 23-25, 2013, pp. 9-22 (2013)
The covering and boundedness problems for vector addition systems. C Rackoff, Theoretical Computer Science. 62Rackoff, C.: The covering and boundedness problems for vector addition systems. Theoretical Computer Science 6(2), 223-231 (1978)
Using model checking to analyze network vulnerabilities. R W Ritchey, P Ammann, Security and Privacy. Ritchey, R.W., Ammann, P.: Using model checking to analyze network vulnerabilities. In: Security and Privacy (2000)
Lola a low level analyser. In: Application and Theory of Petri Nets. K Schmidt, SpringerSchmidt, K.: Lola a low level analyser. In: Application and Theory of Petri Nets 2000, pp. 465-474. Springer (2000)
Verifying lossy channel systems has nonprimitive recursive complexity. P Schnoebelen, Information Processing Letters. 835Schnoebelen, P.: Verifying lossy channel systems has nonprimitive recursive complexity. Information Processing Letters 83(5), 251-261 (2002)
Model checking multithreaded programs with asynchronous atomic methods. K Sen, M Viswanathan, International Conference on Computer Aided Verification. SpringerSen, K., Viswanathan, M.: Model checking multithreaded programs with asynchronous atomic methods. In: International Conference on Computer Aided Verification, pp. 300-314. Springer (2006)
Abstractions for model checking sdn controllers. D Sethi, S Narayana, S Malik, FMCADSethi, D., Narayana, S., Malik, S.: Abstractions for model checking sdn controllers. In: FMCAD (2013)
Making middleboxes someone else's problem: Network processing as a cloud service. J Sherry, S Hasan, C Scott, A Krishnamurthy, S Ratnasamy, V Sekar, SIGCOMMSherry, J., Hasan, S., Scott, C., Krishnamurthy, A., Ratnasamy, S., Sekar, V.: Making middleboxes some- one else's problem: Network processing as a cloud service. In: SIGCOMM (2012)
A verification platform for sdn-enabled applications. R Skowyra, A Lapets, A Bestavros, A Kfoury, HiCoNSSkowyra, R., Lapets, A., Bestavros, A., Kfoury, A.: A verification platform for sdn-enabled applications. In: HiCoNS (2013)
Symnet: static checking for stateful networks. R Stoenescu, M Popovici, L Negreanu, C Raiciu, Proceedings of the 2013 workshop on Hot topics in middleboxes and network function virtualization. the 2013 workshop on Hot topics in middleboxes and network function virtualizationACMStoenescu, R., Popovici, M., Negreanu, L., Raiciu, C.: Symnet: static checking for stateful networks. In: Proceedings of the 2013 workshop on Hot topics in middleboxes and network function virtualization, pp. 31-36. ACM (2013)
Some complexity results for stateful network verification. Y Velner, K Alpernas, A Panda, A Rabinovich, M Sagiv, S Shenker, S Shoham, International Conference on Tools and Algorithms for the Construction and Analysis of Systems. SpringerVelner, Y., Alpernas, K., Panda, A., Rabinovich, A., Sagiv, M., Shenker, S., Shoham, S.: Some complex- ity results for stateful network verification. In: International Conference on Tools and Algorithms for the Construction and Analysis of Systems, pp. 811-830. Springer (2016)
Libra: Divide and conquer to verify forwarding tables in huge networks. H Zeng, S Zhang, F Ye, V Jeyakumar, M Ju, J Liu, N Mckeown, A Vahdat, NSDIZeng, H., Zhang, S., Ye, F., Jeyakumar, V., Ju, M., Liu, J., McKeown, N., Vahdat, A.: Libra: Divide and conquer to verify forwarding tables in huge networks. In: NSDI (2014)
| [] |
[
"Operads of Wiring Diagrams",
"Operads of Wiring Diagrams"
] | [
"Donald Yau [email protected] \nTHE OHIO STATE UNIVERSITY AT NEWARK\nNEWARKOHUSA\n"
] | [
"THE OHIO STATE UNIVERSITY AT NEWARK\nNEWARKOHUSA"
] | [] | Wiring diagrams and undirected wiring diagrams are graphical languages for describing interconnected processes and their compositions. These objects have enormous potentials for applications in many different disciplines, including computer science, cognitive neuroscience, dynamical systems, network theory, and circuit diagrams. It is known that the collection of wiring diagrams is an operad and likewise for undirected wiring diagrams.This monograph is a comprehensive study of the combinatorial structure of various operads of wiring diagrams and undirected wiring diagrams. Our first main objective is to prove a finite presentation theorem for each operad of wiring diagrams, describing each one in terms of just a few operadic generators and a small number of generating relations. For example, the operad of wiring diagrams has 8 generators and 28 generating relations, while the operad of undirected wiring diagrams has 6 generators and 17 generating relations.Our second main objective is to prove a corresponding finite presentation theorem for algebras over each operad of wiring diagrams. As applications we provide finite presentations for the propagator algebra, the algebra of discrete systems, the algebra of open dynamical systems, and the (typed) relational algebra. We also provide a partial verification of Spivak's conjecture regarding the quotient-freeness of the relational algebra.Our third main objective is to construct explicit operad maps among the several operads of wiring diagrams. In particular, there is a surjective operad map from the operad of all wiring diagrams, including delay nodes, to the operad of undirected wiring diagrams.This monograph is intended for graduate students, mathematicians, scientists, and engineers interested in operads and wiring diagrams. Assuming no prior knowledge of categories, operads, and wiring diagrams, this monograph is selfcontained and can be used as a supplement in a graduate course and for independent study. There are over 100 graphical illustrations and a chapter with a list of problems. | 10.1007/978-3-319-95001-3 | [
"https://arxiv.org/pdf/1512.01602v3.pdf"
] | 119,156,593 | 1512.01602 | 6b770190c29a48e4430bbaa75eeba0a7d80f371f |
Operads of Wiring Diagrams
7 Feb 2017
Donald Yau [email protected]
THE OHIO STATE UNIVERSITY AT NEWARK
NEWARKOHUSA
Operads of Wiring Diagrams
7 Feb 2017and phrases Wiring diagramsundirected wiring diagramsoperadscolored operadsoperad algebrasfinite presentationpropagator algebradiscrete systemsopen dynamical systemsrelational algebra
Wiring diagrams and undirected wiring diagrams are graphical languages for describing interconnected processes and their compositions. These objects have enormous potentials for applications in many different disciplines, including computer science, cognitive neuroscience, dynamical systems, network theory, and circuit diagrams. It is known that the collection of wiring diagrams is an operad and likewise for undirected wiring diagrams.This monograph is a comprehensive study of the combinatorial structure of various operads of wiring diagrams and undirected wiring diagrams. Our first main objective is to prove a finite presentation theorem for each operad of wiring diagrams, describing each one in terms of just a few operadic generators and a small number of generating relations. For example, the operad of wiring diagrams has 8 generators and 28 generating relations, while the operad of undirected wiring diagrams has 6 generators and 17 generating relations.Our second main objective is to prove a corresponding finite presentation theorem for algebras over each operad of wiring diagrams. As applications we provide finite presentations for the propagator algebra, the algebra of discrete systems, the algebra of open dynamical systems, and the (typed) relational algebra. We also provide a partial verification of Spivak's conjecture regarding the quotient-freeness of the relational algebra.Our third main objective is to construct explicit operad maps among the several operads of wiring diagrams. In particular, there is a surjective operad map from the operad of all wiring diagrams, including delay nodes, to the operad of undirected wiring diagrams.This monograph is intended for graduate students, mathematicians, scientists, and engineers interested in operads and wiring diagrams. Assuming no prior knowledge of categories, operads, and wiring diagrams, this monograph is selfcontained and can be used as a supplement in a graduate course and for independent study. There are over 100 graphical illustrations and a chapter with a list of problems.
(X,Y)
171 an undirected 2-cell λ (X,x±) 171 a loop σ (X, Wiring diagrams form a kind of graphical language that describes operations or processes with multiple inputs and multiple outputs and how such operations are wired together to form a larger and more complicated operation. Some visual examples of wiring diagrams are in (2.2.18.1), Chapter 3, and Example 4.2.3. The first type of wiring diagrams that we are going to study in this monograph was first introduced in [RS13], with variants studied in [Spi15,Spi15b,VSL15]. In [Spi15,Spi15b] wiring diagrams without delay nodes (Def. 5.3.1), which we call normal wiring diagrams, were used to study mode-dependent networks , discrete systems, and dynamical systems. In [VSL15] wiring diagrams without delay nodes and whose supplier assignments are bijections (Def. 5.4.1), which we call strict wiring diagrams, were used to study open dynamical systems.
Wiring diagrams are by nature directed, in the sense that every operation has a finite set of inputs and a finite set of outputs, each element of which is allowed to carry a value of some kind. There is also an undirected version of wiring diagrams [Spi13,Spi14]. Unlike a wiring diagram, in an undirected wiring diagram, each operation is a finite set, each element of which is again allowed to carry a value.
Some visual examples of undirected wiring diagrams are in (7.1.7.1), Example 7.3.8, and Chapters 8 and 9. For those familiar with operad theory, the distinction between wiring diagrams and undirected wiring diagrams is similar to that between operads and cyclic operads. Just as cyclic operads are not simpler than operads, undirected wiring diagrams are not really simpler than wiring diagrams and have their own subtlety. 1
Introduction
The main reason that wiring diagrams and undirected wiring diagrams are important is that they have enormous potential for applications in many different disciplines. Wiring diagrams and undirected wiring diagrams allow one to consider a finite collection of related operations, wired together in some way, as an operation itself. Such an operation can then be considered as a single operation within a yet larger collection of operations, and so forth. For instance, a finite collection of related operations may be a group of neurons in a certain region of the brain, a collection of codes within a large computer program, or a few related agents within a large supply-chain. In fact, the authors of [RS13] cited both computer science and cognitive neuroscience as potential applications of wiring diagrams. Furthermore, in [Spi15b,VSL15] wiring diagrams were used to study dynamical systems and to model certain kinds of differential equations. Many potential fields of applications are mentioned in the introduction of [VSL15]. In [Spi13] undirected wiring diagrams were used to study database relational queries, plug-and-play devices, recursion, and circuit diagrams.
The substitution process involving wiring diagrams and undirected wiring diagrams described in the previous paragraph can be captured precisely using the notion of colored operads. A colored operad, or just an operad, is a mathematical object that describes operations with multiple inputs and one output and their compositions. A colored operad in which there are only unary operations is exactly a category. If one restricts even further to just the 1-colored case in which the unary operations form a set, then one has exactly a monoid, such as the set of integers under addition. Therefore, a colored operad is a multiple-input generalization of a category, and in fact colored operads are also called symmetric multicategories. Multicategories without symmetric group actions were introduced by Lambek [Lam69]. One-colored operads, together with the name operad, were introduced by May [May72] in the topological setting. See [May97] for the definition of a one-colored operad in a symmetric monoidal category. The book [Yau16] is an elementary introduction to colored operads in symmetric monoidal categories. The book [YJ15] has more in-depth discussion of colored operads and related objects.
In [RS13] Rupel and Spivak observed that the collection of wiring diagrams is a colored operad WD, in which the operadic composition corresponds precisely to the substitution process described above. Each colored operad has associated algebras, on which the colored operad acts. The operadic action is similar to the action of an associative algebra on a left module. The authors of [RS13] defined a WD-algebra, called the propagator algebra, that models a certain kind of inputoutput process.
Closely related colored operads of sub-classes of wiring diagrams were introduced in [Spi15,Spi15b,VSL15]. We will denote by WD • (Def. 5.3.1) the operad of normal wiring diagrams, meaning those without delay nodes. In [Spi15b] Spivak introduced a WD • -algebra, called the algebra of discrete systems, that is closely related to a Moore machine, also known as a discrete state machine. Also we will write WD 0 (Def. 5.4.1) for the operad of strict wiring diagrams, meaning those without delay nodes and whose supplier assignments are bijections. In [VSL15] a WD 0 -algebra, called the algebra of open dynamical systems, was defined that models a certain kind of differential equations. Likewise, in [Spi13] Spivak constructed the colored operad UWD of undirected wiring diagrams. Spivak also defined a UWDalgebra called the relational algebra, which was used to model relational queries in database.
Purposes of this Monograph
This monograph is a comprehensive study of the combinatorial structure of the operads WD, WD • , WD 0 , and UWD of (normal/strict/undirected) wiring diagrams, their algebras, and the relationships between these operads. Specifically, our main results are of the following three types.
Finite Presentation for Operads:
For each of the operads WD, WD • , WD 0 , and UWD, we prove a finite presentation theorem that describes the operad in terms of just a few operadic generators and a small number of generating relations. For the operad of wiring diagrams WD, there are 8 generating wiring diagrams and 28 generating relations. For the smaller operads WD • and WD 0 of normal and strict wiring diagrams, the numbers of operadic generators and of generating relations are (7, 28) and (4, 8), respectively. For the operad of undirected wiring diagrams UWD, there are 6 operadic generators and 17 generating relations.
Finite Presentation for Algebras: For each of the operads WD, WD • , WD 0 , and UWD, we prove a corresponding finite presentation theorem for their algebras.
To be more precise, we describe WD-algebras using 8 generating structure maps and 28 generating axioms. So finite presentation refers to the WD-algebra structure maps, not the elements in the underlying set. Similar finite presentations are also obtained for the algebras over the operads WD • of normal wiring diagrams, WD 0 of strict wiring diagrams, and UWD of undirected wiring diagrams. As applications we provide finite presentations for the propagator algebra over WD, the algebra of discrete systems over WD • , the algebra of open dynamical systems over WD 0 , and the (typed) relational algebra over UWD. Along the way, we provide a partial verification of Spivak's conjecture [Spi13] regarding the quotientfreeness of the relational algebra. and the right units, and their inverses. The generating relations are the Pentagon Axiom for 4-fold iterated tensor products and two unity axioms.
(3) In the linear setting, the operads for associative algebras, commutative algebras, Lie algebras, Leibniz algebras, Poisson algebras, and many others, are finitely presented [GK94]. For example, the associative operad has one generator, which in its algebras corresponds to the usual multiplication A ⊗ A G G A of an associative algebra. The associative operad has one generating relation, which in its algebras corresponds to the usual associativity axiom, (ab)c = a(bc), of an associative algebra.
(4) In [BE14,BSZ14] a finite presentation is given for the symmetric monoidal category of signal-flow graphs. In applications signal-flow graphs form another kind of graphical language that describes processes with inputs and outputs and relations between them. One main difference between (undirected) wiring diagrams and signal-flow graphs is that the composition of signal-flow graphs is done by grafting. This means that the outputs of one signal-flow graph are connected to the inputs of another signal-flow graph. This is similar to the situation in string diagrams [JSV96,SSR15] On the other hand, the operadic composition of (undirected) wiring diagrams is done by substitution, which is pictorially depicted in (2.3.3.1) for wiring diagrams and in (7.3.3.1) for undirected wiring diagrams. In more conceptual terms, the collection of signal-flow graphs is a prop, hence an algebra over the operad for props [YJ15] (Theorem 14.1). On the other hand, the collection of (undirected) wiring diagrams is an operad.
(5) Closer to the topic of this monograph is [YJ15] (Ch.7), where finite presentations are given for various graph groupoids including those for colored operads, colored props, and colored wheeled props. In fact, the way we phrase and verify our finite presentation theorems for the operads of (undirected) wiring diagrams and for their algebras is conceptually similar to the presentation in [YJ15] (Ch.7). One way to explain this conceptual similarity is that, for both (undirected) wiring diagrams and the graphs for, say, colored wheeled props, the composition is done by substitution. However, wiring diagrams are in several ways more complicated than the graphs in [YJ15]. In fact, the graphs in [YJ15] do not have delay nodes, internal and external wasted wires, and split wires, all of which can happen in a wiring diagram. See, for example, the wiring diagram in (2.2.18.1).
mathematicians with an interest in operads and (undirected) wiring diagrams. Furthermore, due to the wide variety of potential applications, we also intend to make this monograph and this subject accessible to scientists and engineers.
With such a large audience in mind, the prerequisite for this monograph has been kept to an absolute minimum. In particular, we assume the reader is comfortable with basic concepts of sets, functions, and mathematical induction. No prior knowledge of categories, operads, and (undirected) wiring diagrams is assumed. The presentation of the material proceeds at a fairly leisurely pace and is roughly at the advanced undergraduate to beginning graduate level. To motivate various constructions and concepts, we have many examples and a lot of discussion that explains the intuition behind the scene. Furthermore, there are over 100 pictures throughout this monograph that help the reader visualize (undirected) wiring diagrams. Finally, to solidify one's understanding of the subject, the reader may work through the problems in Chapter 14. There are enough problems there to keep one busy for a few days.
Chapter Summaries
This monograph is divided into three parts.
Part 1: Wiring Diagrams (Chapters 2-6)
This part contains the finite presentation theorems for the operad WD of wiring diagrams, the operad WD • of normal wiring diagrams, the operad WD 0 of strict wiring diagrams, and their algebras.
Part 2: Undirected Wiring Diagrams (Chapters 7-11)
This part contains the finite presentation theorems for the operad UWD of undirected wiring diagrams and for their algebras.
Part 3: Maps Between Operads of Wiring Diagrams (Chapters 12-15)
This part contains the construction and description of various operad maps between the operads WD, WD • , WD 0 , and UWD. It also contains a chapter with a list of problems and a chapter with references for further reading.
Each part begins with a brief introduction and a reading guide. Below is a brief description of each chapter.
In Chapter 2, to keep this document self-contained, we first recall two equivalent definitions of a colored operad. The first definition is in terms of May's operad structure map γ, and the other one is in terms of the ○ i -compositions. After recalling the definition of a wiring diagram, we provide a proof of the fact from [RS13] that the collection of wiring diagrams WD is a colored operad. The main difference here is that we use the definition of a colored operad based on the ○ i -compositions. In this monograph, we prefer to work with the ○ i -compositions rather than May's operad structure map γ because the ○ i -compositions are more convenient in phrasing and verifying our finite presentation theorems.
In Chapter 3 we introduce 8 generating wiring diagrams and 28 elementary relations among them. On the one hand, one may regard this chapter as a long list of concrete examples of wiring diagrams and their operadic compositions. On the other hand, in later chapters we will see that these finite collections of generating wiring diagrams and elementary relations are sufficient to describe the operad WD of wiring diagrams, its variants WD • and WD 0 , and their algebras.
For the finite presentation theorems for the operad WD of wiring diagrams and its variants WD • and WD 0 , we will need to be able to decompose every wiring diagram in terms of the 8 generating wiring diagrams in a highly structured way. The purpose of Chapter 4 is to supply all the steps needed to establish such a decomposition.
The finite presentation theorems for the operad WD of wiring diagrams as well as its two variants WD • and WD 0 are given in Chapter 5; see Theorems 5.2.11, 5.3.7, and 5.4.8. Since we are not working in the linear setting (e.g., of modules) where we can take quotients, we need to be extra careful in phrasing our finite presentations for the operads WD, WD • , and WD 0 . For this purpose, a crucial concept is that of a stratified presentation, which is the highly structured decomposition mentioned in the previous paragraph. The results in Chapter 4 guarantees the existence of a stratified presentation for each wiring diagram. This implies the finite generation parts of our finite presentation theorems for WD, WD • , and WD 0 . The relation parts of the finite presentation theorems are phrased in terms of our concept of an elementary equivalence. Roughly speaking, an elementary equivalence means replacing one side of either (i) an elementary relation in Chapter 3 or (ii) an operad associativity/unity axiom for the generating wiring diagrams, by the other side.
In Chapter 6 we discuss finite presentations for algebras over the operads WD, WD • , and WD 0 . In each case, the finite presentation for algebras is a consequence of the finite presentation theorem for the corresponding operad of wiring diagrams. To illustrate the finite presentation for WD-algebras, we will describe the propagator algebra in terms of 8 generating structure maps and 28 generating axioms. To illustrate our finite presentation for WD • -algebras, we will describe the algebra of discrete systems in terms of 7 generating structure maps and 28 generating axioms. To illustrate our finite presentation for WD 0 -algebras, we will similarly describe the algebra of open dynamical systems in terms of 4 generating structure maps and 8 generating axioms. This finishes Part 1 on wiring diagrams. Part 2 begins with Chapter 7, where we first recall the notion of an undirected wiring diagram. Then we give a proof of the fact that the collection of undirected wiring diagrams forms an operad UWD. As in Chapter 2, the operad structure on UWD as well as its proof are both given in terms of the ○ i -compositions because 1. Introduction the finite presentation theorems are easier to phrase this way. One subtlety about the operad UWD is that undirected wiring diagrams may have wasted cables (Def. 7.1.2), which are cables that are not soldered to any wires. As opposed to what was stated in [Spi14] (Example 7.4.2.10), wasted cables cannot be excluded from the definition of undirected wiring diagrams. In fact, wasted cables can actually be created from operadic composition of undirected wiring diagrams without wasted cables. We will make this point precise in Example 7.3.8.
Chapter 8 is the undirected analogue of Chapter 3. In this chapter, we describe 6 generating undirected wiring diagrams and 17 elementary relations among them. On the one hand, one may regard this chapter as a long list of concrete examples of undirected wiring diagrams and their operadic compositions. On the other hand, in later chapters we will see that these finite collections of generating undirected wiring diagrams and elementary relations are sufficient to describe the operad UWD of undirected wiring diagrams.
Chapter 9 is the undirected analogue of Chapter 4. In this chapter, we show that each undirected wiring diagram can be decomposed in terms of the generating undirected wiring diagrams in a highly structured way. Such a decomposition is needed to establish the finite presentation theorem for the operad UWD.
Chapter 10 is the undirected analogue of Chapter 5. In this chapter, we establish the finite presentation theorem for the operad UWD of undirected wiring diagrams; see Theorem 10.2.7. This result is phrased in terms of the generating undirected wiring diagrams and an undirected version of an elementary equivalence.
Chapter 11 contains the finite presentation theorem for UWD-algebras. This result is a consequence of the finite presentation theorem for the operad UWD. It describes each UWD-algebra in terms of 6 generating structure maps and 17 generating axioms, almost all of which are trivial to check in practice. We will illustrate this point with the relation algebra and the typed relational algebra from [Spi13]. We will also provide a partial verification of Spivak's conjecture regarding the quotient-freeness of the relational algebra. This finishes Part 2 on undirected wiring diagrams.
Part 3 begins with Chapter 12, in which we first construct the operad inclusions WD 0 G G WD • G G WD. Recall that WD • is the operad of normal wiring diagramsi.e., those without delay nodes-and that WD 0 is the operad of strict wiring diagramsi.e., those without delay nodes and whose supplier assignments are bijections. Then we construct an operad map χ ∶ WD • G G UWD, essentially by forgetting directions, and its restriction χ 0 ∶ WD 0 G G UWD. For each of the operad maps χ and χ 0 , we compute precisely the image in UWD. In the terminology of Notation 9.1.1, the image of the operad map χ consists of precisely the undirected wiring diagrams without wasted cables and (0, ≥ 2)-cables. The image of the operad map χ 0 consists of precisely the undirected wiring diagrams with only (1, 1)-cables and (2, 0)-cables.
In Chapter 13 we extend the operad map χ ∶ WD • G G UWD to an operad map ρ ∶ WD G G UWD that is defined for all wiring diagrams. We prove that the operad map ρ is surjective, so every undirected wiring diagram is the image of some wiring diagram. The operad map ρ is slightly subtle because wiring diagrams may have delay nodes, while undirected wiring diagrams do not seem to have an exact analogue of delay nodes. In fact, for this reason Rupel and Spivak [RS13] (Section 4.1) expressed doubt about the possibility that there be an operad map WD G G UWD. We will see that delay nodes, far from being an obstruction, play a critical role in the surjectivity of the operad map ρ.
Chapter 14 contains some problems about operads and (undirected) wiring diagrams arising from the earlier chapters. Chapter 15 contains some relevant references on categories, operads, props, and their applications in the sciences. This finishes Part 3.
References for the Main Results and Examples
The following table summaries the key references for the various operads of (undirected) wiring diagrams and their finite presentation theorems. The following table summarizes the key references for the finite presentation theorems for algebras over the various operads of (undirected) wiring diagrams. The following table summarizes the key references for the operad maps between the various operads of (undirected) wiring diagrams.
Operad Map
Reference Note/Image
Wiring Diagrams
The main purpose of this part is to describe the combinatorial structure of (1) the operad WD of wiring diagrams, (2) the operad WD • of normal wiring diagrams, and
(3) the operad WD 0 of strict wiring diagrams.
A normal wiring diagram is a wiring diagram without delay nodes. A strict wiring diagram is a normal wiring diagram whose supplier assignment is a bijection. For each of these three operads, we prove a finite presentation theorem that describes the operad in terms of a few operadic generators and a small number of generating relations.
Operads and wiring diagrams are recalled in Chapter 2. Operadic generators and generating relations for the operad WD of wiring diagrams are presented in Chapter 3. Various decompositions of wiring diagrams are given in Chapter 4. Using the results in Chapters 3 and 4, the finite presentation theorems for the operads WD, WD • , and WD 0 are proved in Chapter 5. In Chapter 6 we prove the corresponding finite presentation theorems for WD-algebras, WD • -algebras, and WD 0 -algebras and discuss the main examples of the propagator algebra, the algebra of discrete systems, and the algebra of open dynamical systems. Each finite presentation theorem for algebras describes the algebras in terms of finitely many generating structure maps and relations among these maps.
Reading Guide: In this reading guide we describe what can be safely skipped in Part 1 during the first reading. The purpose is to help the reader get to the main results and examples quicker without getting bogged down by all the technical details.
In Chapter 2, the reader who already knows about colored operads and categories may skip Section 2.1 and start reading about wiring diagrams from Def. 2.2.6. In Section 2.3 about the operad structure on WD, the reader may wish to skip the proofs of the Lemmas and just study the pictures. Section 3.2 about internal wasted wires may be skipped during the first reading.
In Chapter 4 the various decompositions of wiring diagrams are outlined in the introduction. The reader may read that introduction, followed by Motivation 4.1.4 and 4.1.9 and Examples 4.2.3 and 4.3.2, which provide pictures that illustrate the decompositions.
In Section 5.2, after the initial definitions and examples, the reader may wish to skip the proofs of Lemmas 5.2.8, 5.2.9, and 5.2.10 and go straight to Theorem 5.2.11, the finite presentation theorem for wiring diagrams.
The reader who already knows about operad algebras may skip Section 6.1.
Chapter 2
Wiring Diagrams
This purpose of this chapter is to recall the definitions of colored operads and wiring diagrams. In Section 2.1 we recall two equivalent definitions of a colored operad, one in terms of the structure map γ (2.1.3.2) and the other in terms of the ○ i -compositions (2.1.10.1).
Wiring diagrams are defined in Section 2.2. The main difference between our definition of a wiring diagram and the original one in [RS13] is that we allow the wires to carry values in an arbitrary class S instead of just the class of pointed sets. This added flexibility will be important in later chapters when we discuss operad algebras. Indeed, in Section 6.3 when we discuss the propagator algebra, we will take S to be the class of pointed sets. On the other hand, in Section 6.7 when we discuss the algebra of open dynamical systems, we will take S to be a set of representatives of isomorphism classes of second-countable smooth manifolds.
In Section 2.3 we define the operad structure on wiring diagrams in terms of ○ icompositions. Although we could also have defined this operad structure in terms of γ as in [RS13], we chose to use ○ i -compositions because the finite presentation theorems in Chapter 5 can be phrased and proved more easily using the latter.
Colored Operads
For brief discussion about classes in the set-theoretic sense, the reader is referred to [Hal74,Pin04]. In this monograph, the reader can safely take the word class to just mean a collection of objects, such as sets, pointed sets, and real functions. First we introduce some notations for the colors in a colored operad. An element in Prof(S) is called an S-profile or just a profile if S is clear from the context.
(2) A typical S-profile of length n is denoted by s = (s 1 , . . . , s n ) with s denoting its length.
(3) The empty S-profile is denoted by ∅.
(4) For n ≥ 0 denote by Prof ≥n (S) ⊆ Prof(S) the sub-class of S-profiles of length at least n.
Motivation 2.1.2. Before we define an operad, let us first motivate its definition with a simple but important example. Suppose X is a set and Map(X n , X) is the set of functions from X n = X × ⋯ × X, with n ≥ 0 factors of X, to X. If f ∈ Map(X n , X) with n ≥ 1 and g i ∈ Map(X m i , X) for each 1 ≤ i ≤ n, then one can form the new function f ○ (g 1 , . . . , g n ) ∈ Map X m 1 +⋯+mn , X by first applying the g ′ i s simultaneously and then applying f . Moreover, we may even allow the inputs and the output of each function to be from different sets, i.e., functions X c 1 × ⋯ × X cn G G X d . In this case, the above composition is defined if and only if the outputs of the g i 's match with the inputs of f .
A function f ∈ Map(X c 1 × ⋯ × X cn , X d ) may be depicted as follows. , X c i , n ≥ 1, and each m i ≥ 0. Together with permutations of the inputs, the above collection of functions satisfies some associativity, unity, and equivariance conditions. An operad is an abstraction of this example that allows one to encode operations with multiple, possibly zero, inputs and one output and their compositions.
With the above motivation in mind, next we define colored operads. See, for example, [Yau16] for more in-depth discussion of colored operads. For each integer n ≥ 0, the symmetric group on n letters is denoted by Σ n . is the right permutation of c by σ.
(3) For each c ∈ S, O is equipped with a specific element 1 c ∈ O c c , called the c-colored unit.
(4) For This data is required to satisfy the following associativity, unity, and equivariance axioms.
Associativity Axiom: Suppose that:
• in (2.1.3.2)
b j = b j 1 , . . . , b j k j ∈ Prof(S)
has length k j ≥ 0 for each 1 ≤ j ≤ n such that at least one k j > 0; • a j i ∈ Prof(S) for each 1 ≤ j ≤ n and 1 ≤ i ≤ k j ;
• for each 1 ≤ j ≤ n,
a j = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ a j 1 , . . . , a j k j if k j > 0, ∅ if k j = 0;
(2.1.3.3)
• a = (a 1 , . . . , a n ) is their concatenation. Then the associativity diagram
O d c × ⎡ ⎢ ⎢ ⎢ ⎣ n ∏ j=1 O c j b j ⎤ ⎥ ⎥ ⎥ ⎦ × n ∏ j=1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ k j ∏ i=1 O b j i a j i ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (γ,Id) G G permute ≅ O d b × n ∏ j=1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ k j ∏ i=1 O b j i a j i ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ γ O d c × n ∏ j=1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ O c j b j × k j ∏ i=1 O b j i a j i ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (Id, j γ) O d c × n ∏ j=1 O c j a j γ G G O d a (2.1.3.4)
is commutative.
Unity Axioms: Suppose d ∈ S.
(1) If c = (c 1 , . . . , c n ) ∈ Prof(S) has length n ≥ 1, then the right unity diagram
O d c × { * } n (Id, 1c j ) ≅ G G O d c = O d c × n ∏ j=1 O c j c j γ G G O d c (2.1.3.5)
is commutative. Here { * } is the one-point set, and { * } n is its n-fold product.
(2) If b ∈ Prof(S), then the left unity diagram { * } × O d b (1 d ,Id) ≅ G G O d b = O d d × O d b γ G G O d b (2.1.3.6)
is commutative.
Equivariance Axioms: Suppose that in (2.1.3.2) b j = k j ≥ 0.
(1) For each permutation σ ∈ Σ n , the top equivariance diagram
O d c × n ∏ j=1 O c j b j γ (σ,σ −1 ) G G O d cσ × n ∏ j=1 O c σ(j) b σ(j) γ O d b 1 ,...,b n σ⟨k 1 ,...,kn⟩ G G O d b σ(1) ,...,b σ(n) (2.1.3.7)
is commutative. Here σ⟨k 1 , . . . , k n ⟩ ∈ Σ k 1 +⋯+kn is the block permutation induced by σ that permutes the n consecutive blocks of lengths k 1 , . . . , k n , leaving the relative order within each block unchanged.
(2) Given permutations τ j ∈ Σ k j for 1 ≤ j ≤ n, the bottom equivariance dia- gram O d c × n ∏ j=1 O c j b j γ (Id, τ j ) G G O d c × n ∏ j=1 O c j b j τ j γ O d b 1 ,...,b n τ 1 ⊕⋯⊕τn G G O d b 1 τ 1 ,...,b n τn
graph with one vertex, which is both the input and the output, and no edges. Here is a graph operation with two vertices and four edges, two of which are loops:
in out 1 2 3 4
There is an operad structure on graph operations given by edge substitution as follows. Suppose G ∈ GrOp n with n ≥ 1 and G i ∈ GrOp m i for 1 ≤ i ≤ n. Then the operadic composition G(G 1 , . . . , G n ) ∈ GrOp m 1 +⋯+mn is obtained from G by replacing the ith edge e i in G by G i and by identifying the initial (resp., terminal) vertex of e i with the input (resp., output) of G i . The edge ordering of the operadic composition is induced by those of G and of the G i 's. The input and the output are inherited from G. The symmetric group action on GrOp n is given by permutation of the edge ordering. The operadic unit is the graph operation in G G out with two vertices and one edge from the input to the output.
For example, suppose G, H, and K are the following graph operations in GrOp 2 :
in out in out in out Then the operadic composition G(H, K) is the graph operation with four edges above. The algebras of the operad GrOp are closely related to traffic spaces and non-commutative probability, as we will discuss in Example 6.1.8 below.
Due to the presence of the colored units, a colored operad can also be defined in terms of a binary product, called the ○ i -composition, which leads to axioms that are sometimes easier to check in practice and that we will use in most of the rest of this monograph. In the one-colored linear setting, this alternative formulation of an operad was first made explicit in [Mar96].
f ... g ... d c 1 c n c i b 1 b k Then their ○ i -composition f ○ i g = f ○ Id i−1 , g, Id n−i is the picture f g ... c i ... ... d c 1 cn b 1 b k
in which the output of g is used as the ith input of f . The operadic ○ i -composition is an abstraction of this f ○ i g of functions.
With the above motivation in mind, we now recall this alternative formulation of an operad. The elementary relations in Section 3.3 are almost all stated in terms of the ○ i -compositions in the following definition. (1) It has the same data as in (1) (2) For each d ∈ S, c = (c 1 , . . . , c n ) ∈ Prof(S) with length n ≥ 1, b ∈ Prof(S), and
1 ≤ i ≤ n, it is equipped with a map O d c × O c i b ○ i G G O d c○ i b (2.1.10.1) called the ○ i -composition, where c ○ i b = c 1 , . . . , c i−1 ∅ if i=1 , b, c i+1 , . . . , c n ∅ if i=n . (2.1.10.2)
This data is required to satisfy the following associativity, unity, and equivariance axioms. Suppose d ∈ S, c = (c 1 , . . . , c n ) ∈ Prof(S), b ∈ Prof(S) with length b = m, and a ∈ Prof(S) with length a = l.
Associativity Axioms: There are two associativity axioms.
(4) There is a category Fin whose objects are finite sets and whose morphisms are functions between finite sets. Given any disjoint finite sets X 1 , . . . , X n , their coproduct ∐ n i=1 X i is the finite set given by their disjoint union. If the X i 's are not disjoint, we can still define their coproduct, but we must first replace each X i (or just the ones with i ≥ 2) by an X ′ i equipped with a bijection to X i such that the resulting X ′ 1 , . . . , X ′ n are disjoint. Then the coproduct ∐ n i=1 X i is defined as the disjoint union of the X ′ i for 1 ≤ i ≤ n, and it is well-defined up to isomorphism. If n = 0 then the coproduct is defined as the empty set ∅.
In what follows, if the identity morphisms and the composition are obvious, then we will omit mentioning them.
Example 2.2.4.
A monoid (A, µ, 1) (Example 2.1.6) is a category with one object, whose only morphism set is A. Composition and identity are those of A, i.e., the multiplication µ and the unit element 1. Therefore, one can think of a category as a monoid with multiple objects.
Example 2.2.5. Suppose O is an S-colored operad (Def. 2.1.3).
Then O determines a category C whose objects are the elements in S. For c, d ∈ S the morphism object C(c, d) is O d c , and the identity morphism of c is the c-colored unit of O. Composition in C is the restriction of the operadic composition in O. Therefore, one can think of a colored operad as a generalization of a category in which the domain of each morphism is a finite sequence of objects.
For wiring diagrams, we will usually consider finite sets in which each element is allowed to carry a value of some kind. The precise notion is given in the following definition. Definition 2.2.6. Suppose S is a non-empty class, and Fin is the category of finite sets and functions between them.
(1) Denote by Fin S the category in which:
• an object is a pair
(X, v) with X ∈ Fin and v ∶ X G G S a function; • a map (X, v X ) G G (Y, v Y ) is a function X G G Y such that the diagram X G G v X 3 3 ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ Y v Y S (2.2.6.1)
is commutative.
(2) An object (X, v) ∈ Fin S is called an S-finite set.
(
3) For (X, v) ∈ Fin S , we call v the value assignment for X. For each x ∈ X, v(x) ∈ S is called the value of x. (4) If (X i , v i ) ∈ Fin S for 1 ≤ i ≤ n, then their coproduct X = ∐ n i=1 X i ∈ Fin S has value assignment ∐ n i=1 v i . (5)
The empty S-finite set is denoted by ∅. Definition 2.2.7. Suppose S is a non-empty class.
(1) An S-box is a pair X = (X in , X out ) ∈ Fin S × Fin S . If S is clear from the context, then we will drop S and call X a box.
• An element of X in is called an input of X.
• An element of X out is called an output of X.
• We write v = v X ∶ X in ∐ X out G G S for the value assignment for X.
(2) The class of S-boxes is denoted by Box S .
(3) The empty S-box, denoted by ∅, is the S-box with ∅ in = ∅ out the empty set. On the left, the inputs of X are depicted as arrows going into the box, and the outputs of X are depicted as arrows leaving the box. Alternatively, if we do not need to specify the sizes of X in and X out , then we depict them using a generic arrow ⇒, as in the picture on the right. The value of each x ∈ X in ∐ X out is either not depicted in the picture for simplicity, or it is written near the corresponding arrow.
Definition 2.2.11. Suppose X 1 , . . . , X n are S-boxes for some n ≥ 0. Define the S-box X = ∐ n i=1 X i , called the coproduct, as follows. (1) If n = 0, then X = ∅, the empty S-box.
(2) If n ≥ 1, then:
X in = n ∐ i=1 X in i , X out = n ∐ i=1 X out i , and v X = n ∐ i=1 v X i .
Motivation 2.2.12. Before we define wiring diagrams, let us first provide a motivating example. A typical wiring diagram looks like this:
ϕ x 1 x 2 x 3 x 1 x 2 d y 1 y 2 y 1 y 2 y 3
There is an output box Y (the outermost box in the picture), a finite number of input boxes X (which the above picture only has one), and a finite number of delay nodes (which the above picture again only has one). To make sense of such a picture, first of all, for each output y i of Y we need to specify where the arrow is coming from. In the example above, y 1 comes from x 1 , and both y 2 and y 3 come from the delay node d. We will say that each y i is a demand wire, and y 1 (resp., y 2 and y 3 ) is supplied by the supply wire x 1 (resp., d).
Similarly, for each input x i of X and the delay node d, we again need to specify where the arrow is coming from. So d is also a demand wire, and by tracing the arrow ending at d backward, we see that it is supplied by the supply wire y 1 . Starting at the input x 1 (resp., x 2 and x 3 ) and tracing the arrow backward, we see that it is supplied by x 1 (resp., y 1 and d). We will come back to this example precisely in Example 2.2.18.1 below.
The above exercise tells us that the outputs of Y, the inputs of X, and the delay nodes are demand wires, in the sense that each of their elements demands a supply wire. The supply wires consist of the inputs of Y, the outputs of X, and the delay nodes. Each supply wire may supply multiple demand wires or none at all. For instance, in the above example, the supply wire x 1 supplies both the demand wires x 1 and y 1 . On the other hand, the supply wires y 2 and x 2 do not supply to any demand wires at all. We will call them wasted wires.
To avoid pathological situations, one requirement of a wiring diagram is that a demand wire in the outputs of Y should not be supplied by a supply wire in the inputs of Y. In other words, we exclude pictures like this a bad wire y y ′ ⋯ in which the demand wire y ′ is directly supplied by the supply wire y. So we insist that an output of Y be supplied by either an output of an input box X or a delay node d. We will call this the non-instantaneity requirement.
Wiring diagrams will be defined as equivalence classes of prewiring diagrams, as defined below. Definition 2.2.13. Suppose S is a class. An S-prewiring diagram is a tuple
ϕ = X, Y, DN, v, s (2.2.13.1)
consisting of the following data.
(1) Y = (Y in , Y out ) ∈ Box S , called the output box of ϕ.
• An element in Y in is called a global input for ϕ.
• An element in Y out is called a global output for ϕ.
(2) X = (X 1 , . . . , X n ) is a Box S -profile for some n ≥ 0; i.e., each X i ∈ Box S . • We call X i the ith input box of ϕ. • Denote by X = ∐ n i=1 X i ∈ Box S the coproduct. (3) (DN, v) ∈ Fin S is an S-finite set. An element of DN is called a delay node. Define: • Dm = Y out ∐ X in ∐ DN ∈ Fin S . An element of Dm is called a demand wire in ϕ. • Sp = Y in ∐ X out ∐ DN ∈ Fin S .
An element of Sp is called a supply wire in ϕ. Furthermore:
• When DN is regarded as a subset of Dm, an element in X in ∐ DN is called an internal input for ϕ. • When DN is regarded as a subset of Sp, an element in X out ∐ DN is called an internal output for ϕ.
(4) With a slight abuse of notation, we write
Dm ∐ DN Sp = Y in ∐ Y out ∐ X in ∐ X out ∐ DN v G G S
for the coproduct of the value assignments for X, Y, and DN.
(5) s ∶ Dm G G Sp ∈ Fin S is a map, called the supplier assignment for ϕ, such that
y ∈ Y out ⊆ Dm implies s(y) ∈ Sp ∖ Y in = X out ∐ DN. (2.2.13.2)
• The condition (2.2.13.2) is called the non-instantaneity requirement.
• For w ∈ Dm, we call s(w) ∈ Sp the supplier or the supply wire of w. So non-instantaneity says that the supply wire of a global output cannot be a global input. • A supply wire w ∈ Y in that does not belong to the image of the supplier assignment s is called an external wasted wire. The set of external wasted wires in ϕ is denoted by ϕ w − . • A supply wire w ∈ X out ∐ DN that does not belong to the image of the supplier assignment s is called an internal wasted wire. The set of internal wasted wires in ϕ is denoted by ϕ w + . If S is clear from the context, then we will drop S and call ϕ a prewiring diagram. If we need to emphasize ϕ, then we will use subscripts such as Dm ϕ , Sp ϕ , and s ϕ . Remark 2.2.14. In the constructions that follow, the compatibility of the value assignments with the various supplier assignments are usually immediate because, in each prewiring diagram, the supplier assignment s ∶ Dm G G Sp is a map in Fin S . Therefore, we will omit checking such compatibility.
Definition 2.2.15. Suppose S is a class, ϕ = (X, Y, DN, v, s) is an S-prewiring di- agram as in (2.2.13.1), and ϕ ′ = (X, Y, DN ′ , v ′ , s ′ )
is another S-prewiring diagram with the same input boxes X and output box Y.
(1) An equivalence f ∶ ϕ G G ϕ ′ is an isomorphism f 0 ∶ (DN, v) G G (DN ′ , v ′ ) ∈ Fin S such that the diagram Y out ∐ X in ∐ DN = Dm ϕ sϕ Id ∐ Id ∐ f 0 G G Dm ϕ ′ = Y out ∐ X in ∐ DN ′ s ϕ ′ Y in ∐ X out ∐ DN = Sp ϕ Id ∐ Id ∐ f 0 G G Sp ϕ ′ = Y in ∐ X out ∐ DN ′ in Fin S is commutative.
(2) We say that ϕ and ϕ ′ are equivalent if there exists an equivalence ϕ G G ϕ ′ .
(3) An S-wiring diagram is an equivalence class of S-prewiring diagrams. If S is clear from the context, we will drop S and just say wiring diagram.
(4) The class of S-wiring diagrams with input boxes X = (X 1 , . . . , X n ) and output box Y is denoted by
WD Y X or WD Y X 1 ,...,Xn . (2.2.15.1)
The class of all S-wiring diagrams is denoted by WD. If we want to emphasize S, then we will write WD S .
Convention 2.2.16.
To simplify the presentation, we usually suppress the difference between a prewiring diagram and a wiring diagram. In any given wiring diagram, DN is a finite set in which each element d is equipped with a value v(d) ∈ S. We will suppress the difference between each delay node d and its value v(d). In other words, each wiring diagram has a unique representative in which:
• each delay node is an element in S;
• v ∶ DN G G S sends each delay node to itself.
Unless otherwise specified, in the rest of this book, we will always use this representative of a wiring diagram. For a wiring diagram ϕ ∈ WD Y X , we will sometimes draw it as Y in Y out ϕ or ϕ without drawing the input boxes X, the delay nodes, and the supplier assignment.
• X in = {x 1 , x 1 , x 3 }, X out = {x 1 , x 2 }, Y in = {y 1 , y 2 }, Y out = {y 1 , y 2 , y 3 }, and DN = {d}; • v(x 2 ) = v(x 3 ) = v(y 1 ) = v(y 2 ) = v(y 3 ) = v(d) ∈ S; • v(x 1 ) = v(x 1 ) = v(y 1 ) ∈ S, and v(x 2 )
and v(y 2 ) in S are arbitrary;
• s(x 1 ) = s(y 1 ) = x 1 , s(x 2 ) = s(d) = y 1 , and s(x 3 ) = s(y 2 ) = s(y 3 ) = d.
Then the above data defines a wiring diagram ϕ ∈ WD Y X with one input box X and output box Y, which can be depicted as follows:
ϕ x 1 x 2 x 3 x 1 x 2 d y 1 y 2 y 1 y 2 y 3 (2.2.18.1)
The output box Y is drawn as the outermost box. The single input box X is the smaller box. The delay node d is drawn as a circle, which will be our convention from now on. The supply wires y 2 ∈ Y in and x 2 ∈ X out are not in the image of the supplier assignment s ∶ Dm G G Sp, so y 2 (resp., x 2 ) is an external (resp., internal) wasted wire.
is as in (2.1.10.2).
(1) DN ϕ○ i ψ = DN ϕ ∐ DN ψ ∈ Fin S , so v ϕ○ i ψ = v ϕ ∐ v ψ .
(2) The supplier assignment for ϕ ○ i ψ,
Dm ϕ○ i ψ = Y out ∐ ∐ j =i X in j ∐ r ∐ k=1 W in k ∐ DN ϕ ∐ DN ψ s ϕ○ i ψ Sp ϕ○ i ψ = Y in ∐ ∐ j =i X out j ∐ r ∐ k=1 W out k ∐ DN ϕ ∐ DN ψ (2.3.4.1) in which the coproduct ∐ j =i is indexed by all j ∈ {1, . . . , i − 1, i + 1, . . . , n}, is defined as follows. Suppose z ∈ Dm ϕ○ i ψ . (a) If z ∈ Y out ∐ ∐ j =i X in j ∐ DN ϕ ⊆ Dm ϕ , then s ϕ○ i ψ (z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ s ϕ (z) if s ϕ (z) ∈ X out i ; s ψ s ϕ (z) if s ϕ (z) ∈ X out i . (2.3.4.2) (b) If z ∈ ∐ r k=1 W in k ∐ DN ψ ⊆ Dm ψ , then s ϕ○ i ψ (z) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ s ψ (z) if s ψ (z) ∈ X in i ; s ϕ s ψ (z) if s ψ (z) ∈ X in i and s ϕ s ψ (z) ∈ X out i ; s ψ s ϕ s ψ (z) if s ψ (z) ∈ X in i and s ϕ s ψ (z) ∈ X out i . (2.3.4.3)
This finishes the definition of ϕ ○ i ψ.
s ϕ○ i ψ (y) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s ϕ (y) ∈ ∐ j =i X out j ∐ DN ϕ if s ϕ (y) ∈ X out i ; s ψ s ϕ (y) ∈ r ∐ k=1 W out k ∐ DN ψ if s ϕ (y) ∈ X out i .
Here we have used the non-instantaneity requirement for both ϕ and ψ. So in either case we have that
s ϕ○ i ψ (y) ∈ Y in .
Many examples of the ○ i -composition in WD will be given in Chapter 3. We now prove that, equipped with the structure above, WD is a colored operad.
(ϕ ○ j ζ) ○ i ψ. Y in i Y out i ψ Y in j Y out j ζ Z in Z out ϕ ⋯ Y i Y j Y k Z in Z out (ϕ ○ j ζ) ○ i ψ ⋯ ψ ζ Y k (2.3.7.1)
Note that on the right side, ψ and ζ are depicted as gray boxes because their output boxes-namely Y i and Y j -are no longer input boxes in (ϕ ○ j ζ) ○ i ψ. Furthermore, for simplicity the delay nodes are not drawn. Proof. Suppose:
• ϕ ∈ WD Z Y with Y = n ≥ 2 and 1 ≤ i < j ≤ n; • ψ ∈ WD Y i W with W = l; • ζ ∈ WD Y j X with X = m.
We must show that
ϕ ○ j ζ ○ i ψ = (ϕ ○ i ψ) ○ j−1+l ζ ∈ WD Z (Y○ j X)○ i W .(Dm = Z out ∐ ∐ p =i,j Y in p ∐ l ∐ q=1 W in q ∐ m ∐ r=1 X in r ∐ DN ϕ ∐ DN ψ ∐ DN ζ in which the coproduct ∐ p =i,j is indexed by all 1 ≤ p ≤ n such that p = i, j.
Similarly, both sides in (2.3.8.1) have supply wires
Sp = Z in ∐ ∐ p =i,j Y out p ∐ l ∐ q=1 W out q ∐ m ∐ r=1 X out r ∐ DN ϕ ∐ DN ψ ∐ DN ζ .
Using the definitions (2.3.4.2) and (2.3.4.3), it follows from direct inspection that both sides in (2.3.8.1) have the following supplier assignment s ∶ Dm G G Sp.
(1) If v ∈ Z out ∐ ∐ p =i,j Y in p ∐ DN ϕ ⊆ Dm ϕ , then s(v) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s ϕ (v) if s ϕ (v) ∈ Y out i ∐ Y out j ; s ψ s ϕ (v) if s ϕ (v) ∈ Y out i ; s ζ s ϕ (v) if s ϕ (v) ∈ Y out j . (2) If v ∈ ∐ l q=1 W in q ∐ DN ψ ⊆ Dm ψ , then s(v) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s ψ (v) if s ψ (v) ∈ Y in i ; s ϕ s ψ (v) if s ψ (v) ∈ Y in i and s ϕ s ψ (v) ∈ Y out i ∐ Y out j ; s ψ s ϕ s ψ (v) if s ψ (v) ∈ Y in i and s ϕ s ψ (v) ∈ Y out i ; s ζ s ϕ s ψ (v) if s ψ (v) ∈ Y in i and s ϕ s ψ (v) ∈ Y out j . (3) Finally, if v ∈ ∐ m r=1 X in r ∐ DN ζ ⊆ Dm ζ , then s(v) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s ζ (v) if s ζ (v) ∈ Y in j ; s ϕ s ζ (v) if s ζ (v) ∈ Y in j and s ϕ s ζ (v) ∈ Y out i ∐ Y out j ; s ψ s ϕ s ζ (v) if s ζ (v) ∈ Y in j and s ϕ s ζ (v) ∈ Y out i ; s ζ s ϕ s ζ (v) if s ζ (v) ∈ Y in j and s ϕ s ζ (v) ∈ Y out j .
This finishes the proof of the desired equality (2.3.8.1).
Motivation 2.3.9.
For the vertical associativity axiom, one should keep the following picture of ϕ ○ i (ψ ○ j ζ) in mind.
○ j ○ i X in j X out j ζ Y in i Y out i ψ ⋯ X j X k Z in Z out ϕ ⋯ Y i Y l ϕ ○ i (ψ ○ j ζ) Z in Z out ⋯ ⋯ ⋯ ζ X k Y l (2.3.9.1)
Once again on the right side, ζ is depicted as a gray box because its output box X j is no longer an input box in ϕ ○ i (ψ ○ j ζ). Furthermore, for simplicity the delay nodes are not drawn. Proof. Suppose:
• ϕ ∈ WD Z Y with Y = n ≥ 1 and 1 ≤ i ≤ n; • ψ ∈ WD Y i X with X = m ≥ 1 and 1 ≤ j ≤ m; • ζ ∈ WD X j W with W = l. We must show that (ϕ ○ i ψ) ○ i−1+j ζ = ϕ ○ i ψ ○ j ζ ∈ WD Z (Y○ i X)○ i−Dm = Z out ∐ ∐ p =i Y in p ∐ ∐ q =j X in q ∐ l ∐ r=1 W in r ∐ DN ϕ ∐ DN ψ ∐ DN ζ
in which ∐ p =i is indexed by all 1 ≤ p ≤ n such that p = i, and ∐ q =j is indexed by all 1 ≤ q ≤ m such that q = j. Similarly, both sides in (2.3.10.1) have supply wires
Sp = Z in ∐ ∐ p =i Y out p ∐ ∐ q =j X out q ∐ l ∐ r=1 W out r ∐ DN ϕ ∐ DN ψ ∐ DN ζ .
Using the definitions (2.3.4.2) and (2.3.4.3), it follows from direct inspection that both sides in (2.3.10.1) have the following supplier assignment s ∶ Dm G G Sp.
(1) If v ∈ Z out ∐ ∐ p =i Y in p ∐ DN ϕ ⊆ Dm ϕ , then s(v) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s ϕ (v) if s ϕ (v) ∈ Y out i ; s ψ s ϕ (v) if s ϕ (v) ∈ Y out i and s ψ s ϕ (v) ∈ X out j ; s ζ s ψ s ϕ (v) if s ϕ (v) ∈ Y out i and s ψ s ϕ (v) ∈ X out j . (2) If v ∈ ∐ q =j X in q ∐ DN ψ ⊆ Dm ψ , then s(v) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s ψ (v) if s ψ (v) ∈ X out j ∐ Y in i ; s ζ s ψ (v) if s ψ (v) ∈ X out j ; s ϕ s ψ (v) if s ψ (v) ∈ Y in i and s ϕ s ψ (v) ∈ Y out i ; s ψ s ϕ s ψ (v) if s ψ (v) ∈ Y in i , s ϕ s ψ (v) ∈ Y out i , and s ψ s ϕ s ψ (v) ∈ X out j ; s ζ s ψ s ϕ s ψ (v) if s ψ (v) ∈ Y in i , s ϕ s ψ (v) ∈ Y out i , and s ψ s ϕ s ψ (v) ∈ X out j . (3) If v ∈ ∐ l r=1 W in r ∐ DN ζ ⊆ Dm ζ , then s(v) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ s ζ (v) if s ζ (v) ∈ X in j ; s ψ s ζ (v) if s ζ (v) ∈ X in j and s ψ s ζ (v) ∈ X out j ∐ Y in i ; s ζ s ψ s ζ (v) if s ζ (v) ∈ X in j and s ψ s ζ (v) ∈ X out j ; sϕs ψ s ζ (v) if s ζ (v) ∈ X in j , s ψ s ζ (v) ∈ Y in i , and sϕs ψ s ζ (v) ∈ Y out i ; s ψ sϕs ψ s ζ (v) if s ζ (v) ∈ X in j , s ψ s ζ (v) ∈ Y in i , sϕs ψ s ζ (v) ∈ Y out i , and s ψ sϕs ψ s ζ (v) ∈ X out j ; s ζ s ψ sϕs ψ s ζ (v) if s ζ (v) ∈ X in j , s ψ s ζ (v) ∈ Y in i , sϕs ψ s ζ (v) ∈ Y out i , and s ψ sϕs ψ s ζ (v) ∈ X out j .
Summary of Chapter 2
(1) An S-colored operad consists of a class O d c 1 ,...,cn for each d, c 1 , . . . , c n ∈ S and n ≥ 0 together with symmetric group actions, an operadic composition, and colored units that satisfy the associativity, unity, and equivariance axioms.
(2) Every operad is determined by the ○ i -compositions.
(3) An S-wiring diagram has a finite number of input boxes, an output box, an S-finite set of delay nodes, and a supplier assignment that satisfies the non-instantaneity requirement.
(4) For each class S, the collection of S-wiring diagrams WD is a Box S -colored operad.
Fix a class S. The purpose of this chapter is to describe a finite number of wiring diagrams that we will later show to be sufficient to describe the entire operad WD of wiring diagrams (Theorems 5.2.11) as well as its variants WD • (Theorem 5.3.7) and WD 0 (Theorem 5.4.8). One may also regard this chapter as consisting of a long list of examples of wiring diagrams.
In Section 3.1 we describe eight wiring diagrams, called the generating wiring diagrams. In Theorem 5.1.11 we will show that they generate the operad WD of wiring diagrams. This means that every wiring diagram can be obtained from finitely many generating wiring diagrams via iterated operadic compositions. For now one may think of the generating wiring diagrams as examples of wiring diagrams.
In Section 3.2 we explain why a wiring diagram with an internal wasted wire is not among the generating wiring diagrams. More concretely, we will observe in Prop. 3.2.3 that an internal wasted wire can be generated using two generating wiring diagrams.
In Section 3.3 we describe 28 elementary relations among the generating wiring diagrams. In Theorem 5.2.11 we will show that these elementary relations together with the operad associativity and unity axioms-(2.1.10.3), (2.1.10.4), (2.1.10.5), and (2.1.10.6)-for the generating wiring diagrams generate all the relations in the operad WD of wiring diagrams. In other words, suppose an arbitrary wiring diagram can be built in two ways using the generating wiring diagrams. Then there exists a finite sequence of steps connecting them in which each step is given by one of the 28 elementary relations or an operad associativity/unity axiom for the generating wiring diagrams. For now one may think of the elementary relations as examples of the operadic composition in the operad WD.
Generating Wiring Diagrams
Recall the definition of a wiring diagram (Def. 2.2.15). In this section, we introduce 8 wiring diagrams, called the generating wiring diagrams. They will be used in later chapters to give a finite presentation for the operad WD of wiring diagrams. (1) no input boxes;
(2) the empty box ∅ (Def. 2.2.7) as the output box;
(3) no delay nodes (i.e., DN = ∅);
(4) supplier assignment s ∶ Dm = ∅ G G ∅ = Sp the trivial function.
The next wiring diagram has a delay node as depicted in the following picture, where we use the convention that delay nodes are drawn as circles as in (2.2.18.1).
δ d d Definition 3.1.2. Suppose d ∈ S.
Denote also by d ∈ Box S the box with one input and one output, both also denoted by d and have values d ∈ S. Define the 1-delay node δ d ∈ WD d as the wiring diagram with:
(1) no input boxes; (3) DN = {d}, in which d has value d ∈ S;
(4) supplier assignment
Dm = Y out ∐ DN = {d} ∐ {d} s G G {d} ∐ {d} = Y in ∐ DN = Sp the identity function that takes d ∈ Y out to d ∈ DN and d ∈ DN to d ∈ Y in .
Next we define the wiring diagram:
τ X,Y X Y Definition 3.1.3. Suppose X, Y ∈ Box S together with isomorphisms f in ∶ X in ≅ Y in and f out ∶ Y out ≅ X out in Fin S . Define the wiring diagram τ f ∈ WD Y X with:(1)
one input box X and output box Y;
(2) no delay nodes;
(3) supplier assignment
Dm = Y out ∐ X in s = f out ∐ f in G G Y in ∐ X out = Sp
the coproduct of the given isomorphisms.
We will often suppress the given isomorphisms and simply write τ X,Y or even τ, which will be called a name change.
Next we define the wiring diagram:
X Y θ X,Y Definition 3.1.4. Suppose X, Y ∈ Box S . Define the wiring diagram θ X,Y ∈ WD X∐Y X,Y with:
(1) two input boxes (X, Y) and output box X ∐ Y;
(2) no delay nodes;
(3) supplier assignment
Dm = [X out ∐ Y out ] ∐ X in ∐ Y in s G G X in ∐ Y in ∐ [X out ∐ Y out ] = Sp
the identity map.
We will call θ X,Y a 2-cell.
Next we define the wiring diagram:
X x− x+ (X ∖ x) in (X ∖ x) out λ X,x
Definition 3.1.5. Suppose:
• X ∈ Box S , and (x + , x − ) ∈ X out × X in such that v(x + ) = v(x − ) ∈ S. • X ∖ x ∈ Box S is obtained from X by removing x ± , so (X ∖ x) in = X in ∖ {x − } and (X ∖ x) out = X out ∖ {x + }.
Define the wiring diagram λ X,x ∈ WD X∖x X with:
(1) one input box X and output box X ∖ x;
(2) no delay nodes;
(3) supplier assignment
Dm = (X∖x) out [X out ∖ {x + }] ∐ X in X in ∖ {x − } ∐ {x − } s Sp = X in ∖ {x − } (X∖x) in ∐ [X out ∖ {x + }] ∐ {x + } X out
given by s(x − ) = x + and the identity function everywhere else.
We will call the wiring diagram λ X,x a 1-loop.
Next we define the wiring diagram:
X x 1 x 2 Y in x 12 Y out σ X,x 1 ,x 2
Definition 3.1.6. Suppose:
• X ∈ Box S , and x 1 , x 2 ∈ X in are distinct elements such that v(x 1 ) = v(x 2 ) ∈ S. • Y = X (x 1 = x 2 ) ∈ Box S is obtained from X by identifying x 1 and x 2 , so Y in = X in (x 1 = x 2 )
and Y out = X out . The identified element of x 1 and x 2 in Y in will be denoted by x 12 .
Define the wiring diagram σ X,x 1 ,x 2 ∈ WD Y X with:
(1) one input box X and output box Y;
(2) no delay nodes;
(3) supplier assignment
Dm = Y out ∐ X in = X out ∐ X in s ∋ x 1 , x 2 ❴ Sp = Y in ∐ X out = X in (x 1 = x 2 ) ∐ X out ∋ x 12
that sends both x 1 , x 2 ∈ X in to x 12 ∈ Y in and is the identity function everywhere else.
We will call the wiring diagram σ X,x 1 ,x 2 an in-split.
Next we define the wiring diagram:
X y 12 Y in Y out y 1 y 2 σ Y,y 1 ,y 2
Definition 3.1.7. Suppose:
• Y ∈ Box S , and y 1 , y 2 ∈ Y out are distinct elements such that v(y 1 ) = v(y 2 ) ∈ S.
• X = Y (y 1 = y 2 ) ∈ Box S is obtained from Y by identifying y 1 and y 2 , so X in = Y in and X out = Y out (y 1 = y 2 ). The identified element of y 1 and y 2 in X out will be denoted by y 12 .
Define the wiring diagram σ Y,y 1 ,y 2 ∈ WD Y X with:
(1) one input box X and output box Y;
(2) no delay nodes;
(3) supplier assignment
Dm = Y out ∐ X in = Y out ∐ Y in s ∋ y 1 , y 2 ❴ Sp = Y in ∐ X out = Y in ∐ Y out (y 1 = y 2 )
∋ y 12 that sends both y 1 , y 2 ∈ Y out to y 12 ∈ X out and is the identity function everywhere else.
We will call the wiring diagram σ Y,y 1 ,y 2 an out-split.
Next we define the following wiring diagram with an external wasted wire:
X Y in y Y out ω Y,y Definition 3.1.8. Suppose: • Y ∈ Box S , and y ∈ Y in . • X ∈ Box S is obtained from Y by removing y, so X in = Y in ∖ {y} and X out = Y out .
Define the wiring diagram ω Y,y ∈ WD Y X with:
(1) one input box X and output box Y;
(2) no delay nodes;
(3) supplier assignment
Dm = Y out ∐ X in = Y out ∐ Y in ∖ {y} s Sp = Y in ∐ X out = Y in ∐ Y out the inclusion.
We will call the wiring diagram ω Y,y a 1-wasted wire. (2) A 1-wasted wire ω Y,y (Def. 3.1.8) is the only wiring diagram that has an external wasted wire, namely y ∈ Y in .
(3) None has an internal wasted wire (Def. 2.2.13). As we will see in Prop. 3.2.3 below, an internal wasted wire can be generated using a 1-loop and a 1-wasted wire, hence is not needed as a generator.
(4) The empty wiring diagram ǫ (Def. 3.1.1) and a 1-delay node δ d are 0-ary elements in WD.
(5) A name change τ (Def. 3.1.3), a 1-loop λ X,x (Def. 3.1.5), an in-split σ X,x 1 ,x 2 (Def. 3.1.6)
, an out-split σ Y,y 1 ,y 2 (Def. 3.1.7), and a 1-wasted wire ω Y,y are unary elements in WD.
(6) A 2-cell θ X,Y (Def. 3.1.4) is a binary element in WD.
Internal Wasted Wires
Recall from Def. 2.2.13 that an internal wasted wire is an internal output, hence a supply wire, that does not belong to the image of the supplier assignment. The purpose of this section is to explain why the wiring diagram
X X in = Y in x Y out = X out ∖ {x} ω X,x
that has an internal wasted wire x is not needed as a generating wiring diagram. First we define this wiring diagram.
Definition 3.2.1. Suppose:
• X ∈ Box S , and x ∈ X out . • Y = X ∖ {x} ∈ Box S is obtained from X by removing x.
Define the wiring diagram ω X,x ∈ WD Y X with: (1) one input box X and output box Y;
(2) no delay nodes;
(3) supplier assignment
Dm = Y out ∐ X in = [X out ∖ {x}] ∐ X in s Sp = Y in ∐ X out = X in ∐ X out the inclusion.
We will call the wiring diagram ω X,x a 1-internal wasted wire.
Motivation 3.2.2. The following observation says that a 1-internal wasted wire can be obtained as the substitution of a 1-wasted wire into a 1-loop, both of which are generating wiring diagrams. This is expressed in the following picture
X X in = Y in Y out = X out ∖ {x} x w
in which the intermediate gray box will be called W below.
Proposition 3.2.3. Suppose:
• ω X,x ∈ WD Y X is a 1-internal wasted wire (Def. 3.2.1). • W = X ∐ {w} ∈ Box S such that w ∈ W in satisfies v(w) = v(x) ∈ S. • λ W,{w,x} ∈ WD Y W is the 1-loop (Def. 3.1.5) in which x ∈ X out = W out is the supply wire of w ∈ W in . • ω W,w ∈ WD W X is a 1-wasted wire (Def. 3.1.8).
Then
λ W,{w,x} ○ 1 (ω W,w ) = ω X,x ∈ WD Y X . (3.2.3.1)
Proof. By definition both sides of (3.2.3.1) belong to WD Y X and have no delay nodes. It remains to check that their supplier assignments are equal. By the definitions of ○ 1 (Def. 2.3.4), 1-loop, and 1-wasted wire, the supplier assignment of the left side λ W,{w,x} ○ 1 (ω X,x ), namely
Dm = Y out ∐ X in = [X out ∖ {x}] ∐ X in s Sp = Y in ∐ X out = X in ∐ X out
is the inclusion. By Def. 3.2.1 this is also the supplier assignment of the 1-internal wasted wire ω X,x .
As a consequence of (3.2.3.1), the 1-internal wasted wire ω X,x is not needed as a generating wiring diagram.
Elementary Relations
The purpose of this section is to introduce 28 elementary relations among the generating wiring diagrams (Def. 3.1.9). Each elementary relation is proved by a simple inspection of the relevant definitions of the generating wiring diagrams and operadic compositions, similar to the proofs of Lemma 2.3.8, Lemma 2.3.10, and Prop. 3.2.3 above. Therefore, we will prove only the first one and omit the proofs for the rest, providing a picture instead in most cases. We will frequently use the ○ i -composition (2.1.10.1) in describing these elementary relations.
ϕ ○ φ = ϕ ○ 1 φ.
(3.3.1.1)
(2) Suppose ϕ 1 , . . . , ϕ k ∈ O such that each of ϕ 1 , . . . , ϕ k−1 belongs to an entry of O whose input profile has length 1. Then we write
ϕ 1 ○ ⋯ ○ ϕ k = ⋯(ϕ 1 ○ 1 ϕ 2 ) ○ 1 ⋯ ○ 1 ϕ k (3.3.1.2)
whenever the right side is defined, in which each pair of parentheses starts on the left. For example, we have
ϕ 1 ○ ϕ 2 ○ ϕ 3 = (ϕ 1 ○ 1 ϕ 2 ) ○ 1 ϕ 3 , ϕ 1 ○ ϕ 2 ○ ϕ 3 ○ ϕ 4 = (ϕ 1 ○ 1 ϕ 2 ) ○ 1 ϕ 3 ○ 1 ϕ 4 .
(3) Write T for the cardinality of T.
The first six relations are about the name change wiring diagrams (Def. 3.1.3). The first relation says that two consecutive name changes can be composed into one name change.
Proposition 3.3.2. Suppose:
• τ Y,Z ∈ WD Z Y and τ X,Y ∈ WD Y X are name changes. • τ X,Z ∈ WD Z
X is the name change given by composing the isomorphisms that define τ Y,Z and τ X,Y .
Then (τ Y,Z ) ○ (τ X,Y ) = τ X,Z ∈ WD Z X . (3.3.2.1) Proof. We are given isomorphisms f in ∶ X in ≅ Y in and f out ∶ Y out ≅ X out for τ X,Y and isomorphisms g in ∶ Y in ≅ Z in and g out ∶ Z out ≅ Y out for τ Y,Z .
The name change τ X,Z given by composing these isomorphisms is defined by the isomorphisms
g in ○ f in ∶ X in ≅ Z in and f out ○ g out ∶ Z out ≅ X out .
On the other hand, by Def. 2.3.4 the composition τ Y,Z ○ τ X,Y has no delay nodes, since neither τ X,Y nor τ Y,Z has a delay node. Its supplier assignment is the function
Dm = Z out ∐ X in s G G Z in ∐ X out = Sp
given by
s(z) = s τ X,Y s τ Y,Z (z) = f out g out (z) for z ∈ Z out ; s(x) = s τ Y,Z s τ X,Y (x) = g in f in (x) for x ∈ X in .
This is the same supplier assignment as that of the name change τ X,Z above.
The next relation says that name changes inside a 2-cell (Def. 3.1.4) can be rewritten as a name change outside of a 2-cell.
• τ X,X ′ ∈ WD X ′ X and τ Y,Y ′ ∈ WD Y ′ Y are name changes. • τ X∐Y,X ′ ∐Y ′ ∈ WD X ′ ∐Y ′ X∐Y is the name change induced by τ X,X ′ and τ Y,Y ′ . • θ X ′ ,Y ′ ∈ WD X ′ ∐Y ′ X ′ ,Y ′ and θ X,Y ∈ WD X∐Y X,Y are 2-cells.
Then
θ X ′ ,Y ′ ○ 1 τ X,X ′ ○ 2 τ Y,Y ′ = (τ X∐Y,X ′ ∐Y ′ ) ○ (θ X,Y ) ∈ WD X ′ ∐Y ′ X,Y . (3.3.3.1)
The next relation says that a name change inside a 1-loop (Def. 3.1.5) can be rewritten as a name change of a 1-loop.
• X ∈ Box S , (x + , x − ) ∈ X out × X in such that v(x + ) = v(x − ) ∈ S. • X ∖ x ∈ Box S is obtained from X by removing x ± . • λ X,x ∈ WD X∖x X is the corresponding 1-loop. • τ X,Y ∈ WD Y X is a name change such that (y + , y − ) ∈ Y out × Y in corresponds to (x + , x − ). • λ Y,y ∈ WD Y∖y Y is the corresponding 1-loop. • τ X∖x,Y∖y ∈ WD Y∖y X∖x is the name change induced by τ X,Y . Then λ Y,y ○ (τ X,Y ) = τ X∖x,Y∖y ○ (λ X,x ) ∈ WD Y∖y X . (3.3.4.1)
The next relation says that a name change inside an in-split (Def. 3.1.6) can be rewritten as a name change of an in-split. Proposition 3.3.5. Suppose:
• X ∈ Box S , and x 1 , x 2 ∈ X in are distinct elements such that v(x 1 ) = v(x 2 ) ∈ S. • X ′ = X (x 1 = x 2 ) ∈ Box S
is obtained from X by identifying x 1 and x 2 .
• σ X,x 1 ,x 2 ∈ WD X ′ X is the corresponding in-split. • τ X,Y ∈ WD Y X is a name change with y 1 , y 2 ∈ Y in corresponding to x 1 , x 2 ∈ X in . • Y ′ = Y (y 1 = y 2 ) ∈ Box S
is obtained from Y by identifying y 1 and y 2 .
• σ Y,y 1 ,y 2 ∈ WD Y ′ Y is the corresponding in-split. • τ X ′ ,Y ′ ∈ WD Y ′ X ′ is the name change induced by τ X,Y . Then σ Y,y 1 ,y 2 ○ (τ X,Y ) = (τ X ′ ,Y ′ ) ○ (σ X,x 1 ,x 2 ) ∈ WD Y ′ X . (3.3.5.1)
The next relation is the out-split (Def. 3.1.7) analogue of (3.3.5.1).
Proposition 3.3.6. Suppose:
• Y ∈ Box S , and y 1 , y 2 ∈ Y out are distinct elements such that v(y 1 ) = v(y 2 ) ∈ S.
• Y ′ = Y (y 1 = y 2 ) ∈ Box S is obtained from Y by identifying y 1 and y 2 .
• σ Y,y 1 ,y 2 ∈ WD Y Y ′ is an out-split. • τ X,Y ∈ WD Y X is a name change with x 1 , x 2 ∈ X out corresponding to y 1 , y 2 ∈ Y out . • X ′ = X (x 1 = x 2 ) ∈ Box S is obtained from X by identifying x 1 and x 2 . • σ X,x 1 ,x 2 ∈ WD X X ′ is the corresponding out-split. • τ X ′ ,Y ′ ∈ WD Y ′ X ′ is the name change induced by τ X,Y .
Then
σ Y,y 1 ,y 2 ○ (τ X ′ ,Y ′ ) = (τ X,Y ) ○ σ X,x 1 ,x 2 ∈ WD Y X ′ . (3.3.6.1)
The next relation says that a name change inside a 1-wasted wire (Def. 3.1.8) can be rewritten as a name change of a 1-wasted wire.
Proposition 3.3.7. Suppose:
• Y ∈ Box S , y ∈ Y in , and Y ′ = Y ∖ {y} ∈ Box S is obtained from Y by removing y. • ω Y,y ∈ WD Y Y ′ is the corresponding 1-wasted wire. • τ X,Y ∈ WD Y X is a name change with x ∈ X in corresponding to y ∈ Y in . • X ′ ∈ Box S is obtained from X by removing x. • ω X,x ∈ WD X X ′ is the corresponding 1-wasted wire. • τ X ′ ,Y ′ ∈ WD Y ′ X ′ is the name change induced by τ X,Y .
Then
ω Y,y ○ (τ X ′ ,Y ′ ) = (τ X,Y ) ○ (ω X,x ) ∈ WD Y X ′ .• X ∈ Box S with X-colored unit 1 X ∈ WD X X . • ǫ ∈ WD ∅ is the empty wiring diagram.
• θ X,∅ ∈ WD X X,∅ is the 2-cell with input boxes (X, ∅) and output box X.
Then
θ X,∅ ○ 2 ǫ = 1 X ∈ WD X X . (3.3.8.1)
The next relation is the associativity property of 2-cells. It says that, in the picture below, the wiring diagram in the middle can be constructed using two 2cells, either as the operadic composition on the left or the one on the right.
• θ X∐Y,Z ∈ WD X∐Y∐Z X∐Y,Z and θ X,Y ∈ WD X∐Y X,Y are 2-cells. • θ X,Y∐Z ∈ WD X∐Y∐Z X,Y∐Z and θ Y,Z ∈ WD Y∐Z Y,Z are 2-cells. Then (θ X∐Y,Z ) ○ 1 (θ X,Y ) = (θ X,Y∐Z ) ○ 2 (θ Y,Z ) ∈ WD X∐Y∐Z X,Y,Z . (3.3.9.1)
The next relation is the commutativity property of 2-cells and uses the equivariant structure in WD (2.3.1.1). Proposition 3.3.10. Suppose:
• θ X,Y ∈ WD X∐Y X,Y is a 2-cell. • (1 2) ∈ Σ 2 is the non-trivial permutation.
Then
θ X,Y (1 2) = θ Y,X ∈ WD Y∐X Y,X . (3.3.10.1)
The next relation says that substituting a 1-loop inside a 2-cell can be rewritten as substituting a 2-cell inside a 1-loop. It gives two different ways to construct the wiring diagram in the middle in the picture below using a 1-loop and a 2-cell, either as the operadic composition on the left or the one on the right.
x− x+ X X in ∖ {x−} X out ∖ {x+} Y = X Y X Y =
Proposition 3.3.11. Suppose:
• X ∈ Box S , and (x + , x − ) ∈ X out × X in such that v(x + ) = v(x − ) ∈ S. • X ∖ x ∈ Box S is obtained from X by removing x ± . • θ X∖x,Y ∈ WD (X∐Y)∖{x}
X∖x,Y and θ X,Y ∈ WD X∐Y X,Y are 2-cells.
• λ X,x ∈ WD X∖x X and λ X∐Y,x ∈ WD (X∐Y)∖{x} X∐Y are the corresponding 1-loops.
Then (θ X∖x,Y ) ○ 1 (λ X,x ) = (λ X∐Y,x ) ○ (θ X,Y ) ∈ WD (X∐Y)∖{x} X,Y .
(3.3.11.1)
All the relations in the rest of this section can be illustrated with pictures similar to the two previous pictures, each one showing how a wiring diagram can be built in two different ways using operadic compositions. So we will mostly just draw the picture of the wiring diagram being built without the accompanying pictures of the operadic compositions.
The next relation says that substituting an in-split inside a 2-cell can be rewritten as substituting a 2-cell inside an in-split. It gives two different ways to construct the following wiring diagram using an in-split and a 2-cell:
x 1 x 2 X X in (x 1 = x 2 )
Y Proposition 3.3.12. Suppose:
• X ∈ Box S , and x 1 , x 2 ∈ X in are distinct elements such that v(x 1 ) = v(x 2 ) ∈ S. • X ′ = X (x 1 = x 2 ) ∈ Box S
is obtained from X by identifying x 1 and x 2 .
• θ X ′ ,Y ∈ WD X ′ ∐Y X ′ ,Y and θ X,Y ∈ WD X∐Y X,Y are 2-cells. • σ X,x 1 ,x 2 ∈ WD X ′ X and σ X∐Y,x 1 ,x 2 ∈ WD X ′ ∐Y X∐Y are in-splits. Then (θ X ′ ,Y ) ○ 1 (σ X,x 1 ,x 2 ) = σ X∐Y,x 1 ,x 2 ○ (θ X,Y ) ∈ WD X ′ ∐Y X,Y . (3.3.12.1)
The next relation says that substituting an out-split inside a 2-cell can be rewritten as substituting a 2-cell inside an out-split. It gives two different ways to construct the following wiring diagram using an out-split and a 2-cell: • X ∈ Box S , and x 1 , x 2 ∈ X out are distinct elements such that v(
x 12 X ′ x 1 x 2 Y X outx 1 ) = v(x 2 ) ∈ S. • X ′ = X (x 1 = x 2 ) ∈ Box S
is obtained from X by identifying x 1 and x 2 .
• θ X,Y ∈ WD X∐Y X,Y and θ X ′ ,Y ∈ WD X ′ ∐Y X ′ ,Y are 2-cells. • σ X,x 1 ,x 2 ∈ WD X X ′ and σ X∐Y,x 1 ,x 2 ∈ WD X∐Y X ′ ∐Y are out-splits. Then (θ X,Y ) ○ 1 σ X,x 1 ,x 2 = σ X∐Y,x 1 ,x 2 ○ (θ X ′ ,Y ) ∈ WD X∐Y X ′ ,Y . (3.3.13.1)
The next relation says that substituting a 1-wasted wire inside a 2-cell can be rewritten as substituting a 2-cell inside a 1-wasted wire. It gives two different ways to construct the following wiring diagram using a 1-wasted wire and a 2-cell:
X ′ x 0 Y X in
Proposition 3.3.14. Suppose:
• X ∈ Box S , x 0 ∈ X in , and X ′ = X ∖ {x 0 } ∈ Box S is obtained from X by removing x 0 . • θ X,Y ∈ WD X∐Y X,Y and θ X ′ ,Y ∈ WD X ′ ∐Y X ′ ,Y are 2-cells. • ω X,x 0 ∈ WD X X ′ and ω X∐Y,x 0 ∈ WD X∐Y X ′ ∐Y are 1-wasted wires.
Then
(θ X,Y ) ○ 1 (ω X,x 0 ) = ω X∐Y,x 0 ○ (θ X ′ ,Y ) ∈ WD X∐Y X ′ ,Y .
(3.3.14.1)
The following six relations are about 1-loops. The next relation is the commutativity property of 1-loops. It gives two different ways to construct the following wiring diagram, which we will call a double-loop, using two 1-loops:
X x 1 − x 1 + x 2 − x 2 + X in ∖ {x 1 − , x 2 − } X out ∖ {x 1 + , x 2 + }
Proposition 3.3.15. Suppose: 2 , and X ∖ x ∈ Box S are obtained from X by removing x 1 ± , x 2 ± , and {x 1 ± , x 2 ± }, respectively.
• X ∈ Box S , x 1 − = x 2 − ∈ X in , and x 1 + = x 2 + ∈ X out such that v(x 1 + ) = v(x 1 − ) ∈ S and v(x 2 + ) = v(x 2 − ) ∈ S. • X ∖ x 1 , X ∖ x
• λ X∖x 1 ,x 2 ∈ WD X∖x X∖x 1 and λ X,x 1 ∈ WD X∖x 1 X are 1-loops.
• λ X∖x 2 ,x 1 ∈ WD X∖x X∖x 2 and λ X,x 2 ∈ WD X∖x 2 X are 1-loops.
Then
λ X∖x 1 ,x 2 ○ λ X,x 1 = λ X∖x 2 ,x 1 ○ λ X,x 2 ∈ WD X∖x X . (3.3.15.1)
The next relation is the commutativity property between 1-loops and in-splits. It gives two different ways to construct the following wiring diagram using one 1-loop and one in-split:
x− x+ x 1 x 2 X X in ∖{x−} (x 1 = x 2 ) X out ∖ {x + } Proposition 3.3.16. Suppose: • X ∈ Box S , x − , x 1 , x 2 ∈ X in are distinct, and x + ∈ X out such that v(x + ) = v(x − ) ∈ S and v(x 1 ) = v(x 2 ) ∈ S. • X ∖ x ∈ Box S is obtained from X by removing x ± . • X ′ = X (x 1 = x 2 ) ∈ Box S is obtained from X by identifying x 1 and x 2 . • X ′ ∖ x ∈ Box S is obtained from X ′ by removing x ± . • λ X ′ ,x ∈ WD X ′ ∖x X ′
and λ X,x ∈ WD X∖x X are 1-loops.
• σ X,x 1 ,x 2 ∈ WD X ′ X and σ X∖x,x 1 ,x 2 ∈ WD X ′ ∖x X∖x are in-splits.
Then (λ X ′ ,x ) ○ (σ X,x 1 ,x 2 ) = (σ X∖x,x 1 ,x 2 ) ○ (λ X,x ) ∈ WD X ′ ∖x X . (3.3.16.1)
The next relation is the commutativity property between 1-loops and out-splits. It gives two different ways to construct the following wiring diagram using one 1loop and one out-split:
x− x+ x 12 X ′ X in ∖ {x − } X out ∖ {x + } x 1 x 2
Proposition 3.3.17. Suppose:
• X ∈ Box S , and (x + , x − ) ∈ X out × X in such that v(x + ) = v(x − ) ∈ S. • x 1 = x 2 ∈ X out ∖ {x + } such that v(x 1 ) = v(x 2 ) ∈ S.
• X ′ = X (x 1 = x 2 ) ∈ Box S is obtained from X by identifying x 1 and x 2 .
• X ∖ x ∈ Box S is obtained from X by removing x ± . • X ′ ∖ x ∈ Box S is obtained from X ′ by removing x ± . • λ X ′ ,x ∈ WD X ′ ∖x X ′ and λ X,x ∈ WD X∖x X are 1-loops. • σ X∖x,x 1 ,x 2 ∈ WD X∖x X ′ ∖x and σ X,x 1 ,x 2 ∈ WD X X ′ are out-splits.
Then
σ X∖x,x 1 ,x 2 ○ (λ X ′ ,x ) = (λ X,x ) ○ σ X,x 1 ,x 2 ∈ WD X∖x X ′ . (3.3.17.1)
The next relation is the commutativity property between 1-loops and 1-wasted wires. It gives two different ways to construct the following wiring diagram using one 1-loop and one 1-wasted wire:
x− x+ X ′ X in ∖ {x − } X out ∖ {x + } x 0
Proposition 3.3.18. Suppose:
• X ∈ Box S , (x + , x − ) ∈ X out × X in such that v(x + ) = v(x − ) ∈ S, and x 0 ∈ X in ∖ {x − }. • X ′ = X ∖ {x 0 } ∈ Box S is obtained from X by removing x 0 . • X ′ ∖ x ∈ Box S is obtained from X ′ by removing x ± . • X ∖ x ∈ Box S is obtained from X by removing x ± . • λ X ′ ,x ∈ WD X ′ ∖x X ′
and λ X,x ∈ WD X∖x X are 1-loops.
• ω X∖x,x 0 ∈ WD X∖x X ′ ∖x and ω X,x 0 ∈ WD X X ′ are 1-wasted wires.
Then (ω X∖x,x 0 ) ○ (λ X ′ ,x ) = (λ X,x ) ○ (ω X,x 0 ) ∈ WD X∖x X ′ . (3.3.18.1)
The next relation involves 1-loops, in-splits, and out-splits. It says that the following two wiring diagrams are equal:
x 1 x 2 x 12 x 12 X = x 1 x 2 x 12 x 1 x 2 X
The wiring diagram on the left, in which the gray box is called X ′ below, is created by substituting an in-split into a 1-loop. The wiring diagram on the right is created by substituting an out-split into a 1-loop, which is then substituted into another 1-loop. The inner gray box is called Y, and the outer gray box is called Y ∖ x(1) below. In both wiring diagrams, the outermost box is called X * . Proposition 3.3.19. Suppose:
• Y ∈ Box S , and x 1 = x 2 ∈ Y out such that v(x 1 ) = v(x 2 ) ∈ S. • X = Y (x 1 = x 2 ) ∈ Box S is obtained from Y by identifying x 1 and x 2 , called x 12 ∈ X out . • x 1 = x 2 ∈ X in = Y in such that v(x 12 ) = v(x 1 ) = v(x 2 ) ∈ S. • X ′ = X (x 1 = x 2 ) ∈ Box S is obtained from X by identifying x 1 and x 2 , called x 12 in X ′ . • σ X,x 1 ,x 2 ∈ WD X ′ X is an in-split. • X * = X ∖ {x 12 , x 1 , x 2 } ∈ Box S
is obtained from X by removing x 12 , x 1 , and x 2 .
• λ X ′ ,x ∈ WD X * X ′ is the 1-loop in which x 12 is the supply wire of x 12 . • σ Y,x 1 ,x 2 ∈ WD Y X is an out-split. • λ Y,x(1) ∈ WD Y∖x(1) Y is a 1-loop, where Y ∖ x(1) ∈ Box S is obtained from Y by removing {x 1 , x 1 }. • λ Y∖x(1),x(2) ∈ WD X * Y∖x(1) is a 1-loop, in which x 2 is the supply wire of x 2 . Then (λ X ′ ,x ) ○ (σ X,x 1 ,x 2 ) = λ Y∖x(1),x(2) ○ λ Y,x(1) ○ σ Y,x 1 ,x 2 ∈ WD X * X . (3.3.19.1)
The next relation says that the colored unit of a box can be rewritten as the substitution of an out-split into a 1-wasted wire and then into a 1-loop. This is depicted in the picture
x 2 x 1 x 1 x 2 X
in which the outer gray box is called Z and the inner gray box is called Y below.
• Z ∈ Box S , and (x 1 , x 1 ) ∈ Z in × Z out such that v(x 1 ) = v(x 1 ) ∈ S. • Y = Z ∖ {x 1 } ∈ Box S is obtained from Z by removing x 1 ∈ Z in . • x 1 = x 2 ∈ Y out = Z out such that v(x 1 ) = v(x 2 ) ∈ S. • X = Z ∖ {x 1 , x 1 } ∈ Box S is obtained from Z by removing {x 1 , x 1 }. • σ Y,x 1 ,x 2 ∈ WD Y X is an out-split in which both x 1 , x 2 ∈ Y out have supply wire x 2 ∈ X out . • ω Z,x 1 ∈ WD Z Y is a 1-wasted wire. • λ Z,x ∈ WD X Z is a 1-loop in which x 1 ∈ Z out is the supply wire of x 1 ∈ Z in .
Then
(λ Z,x ) ○ (ω Z,x 1 ) ○ σ Y,x 1 ,x 2 = 1 X ∈ WD X X . (3.3.20.1)
The following five relations are about in-splits. The next one is the associativity property of in-splits. It gives two different ways to construct the following wiring diagram using two in-splits: • X ∈ Box S , and x 1 ,
X x 1 x 2 x 3 Y in Y outx 2 , x 3 ∈ X in are distinct elements such that v(x 1 ) = v(x 2 ) = v(x 3 ) ∈ S.
• X 12 = X (x 1 = x 2 ) ∈ Box S is obtained from X by identifying x 1 and x 2 , called x 12 ∈ X in 12 . • X 23 = X (x 2 = x 3 ) ∈ Box S is obtained from X by identifying x 2 and x 3 , called
x 23 ∈ X in 23 . • Y = X (x 1 = x 2 = x 3 ) ∈ Box S is obtained from X by identifying x 1 , x 2 , and x 3 .
• σ X 12 ,x 12 ,x 3 ∈ WD Y X 12 and σ X,
x 1 ,x 2 ∈ WD X 12 X are in-splits. • σ X 23 ,x 1 ,x 23 ∈ WD Y X 23 and σ X,x 2 ,x 3 ∈ WD X 23 X are in-splits. Then (σ X 12 ,x 12 ,x 3 ) ○ (σ X,x 1 ,x 2 ) = (σ X 23 ,x 1 ,x 23 ) ○ (σ X,x 2 ,x 3 ) ∈ WD Y X . (3.3.21.1)
The next relation is the commutativity property of in-splits. It gives two different ways to construct the following wiring diagram using two in-splits:
X x 1 x 2 x 3 x 4 Y in Y out
Proposition 3.3.22. Suppose:
• X ∈ Box S , and x 1 , x 2 , x 3 , x 4 ∈ X in are distinct elements such that v(x 1 ) = v(x 2 ) and v(x 3 ) = v(x 4 ) ∈ S. • X 12 = X (x 1 = x 2 ) ∈ Box S is obtained from X by identifying x 1 and x 2 , called x 12 ∈ X in 12 . • X 34 = X (x 3 = x 4 ) ∈ Box S is obtained from X by identifying x 3 and x 4 , called x 34 ∈ X in 34 . • Y = X (x 1 = x 2 ; x 3 = x 4 ) ∈ Box S is obtained from X by (i) identifying x 1 and x 2
and (ii) identifying x 3 and x 4 .
• σ X 12 ,x 3 ,x 4 ∈ WD Y X 12 and σ X,
x 1 ,x 2 ∈ WD X 12 X are in-splits. • σ X 34 ,x 1 ,x 2 ∈ WD Y X 34 and σ X,x 3 ,x 4 ∈ WD X 34 X are in-splits. Then (σ X 12 ,x 3 ,x 4 ) ○ (σ X,x 1 ,x 2 ) = σ X 34 ,x 1 ,x 2 ○ (σ X,x 3 ,x 4 ) ∈ WD Y X . (3.3.22.1)
The next relation is the commutativity property between an in-split and an outsplit. It gives two different ways to construct the following wiring diagram using one in-split and one out-split:
X z 1 z 2 z 12 Y in = W in Y out = Z out z 1 z 2
Proposition 3.3.23. Suppose:
• Z ∈ Box S , and z 1 = z 2 ∈ Z out such that v(z 1 ) = v(z 2 ) ∈ S. • X = Z (z 1 = z 2 ) ∈ Box S
is obtained from Z by identifying z 1 and z 2 .
• z 1 = z 2 ∈ Z in such that v(z 1 ) = v(z 2 ) ∈ S.
• Y = Z (z 1 = z 2 ) ∈ Box S is obtained from Z by identifying z 1 and z 2 .
• W = Z (z 1 = z 2 ; z 1 = z 2 ) ∈ Box S is obtained from Z by (i) identifying z 1 and z 2 and (ii) identifying z 1 and z 2 .
• σ Y,z 1 ,z 2 ∈ WD Y W is an out-split, and σ X,z 1 ,z 2 ∈ WD W X is an in-split. • σ Z,z 1 ,z 2 ∈ WD Y Z is an in-split, and σ Z,z 1 ,z 2 ∈ WD Z X is an out-split.
Then σ Y,z 1 ,z 2 ○ (σ X,z 1 ,z 2 ) = (σ Z,z 1 ,z 2 ) ○ σ Z,z 1 ,z 2 ∈ WD Y X . (3.3.23.1)
The next relation is the commutativity property between an in-split and a 1wasted wire. It gives two different ways to construct the following wiring diagram using one in-split and one 1-wasted wire:
X z 1 z 2 Y in z Proposition 3.3.24. Suppose: • Z ∈ Box S , and z, z 1 , z 2 ∈ Z in are distinct elements such that v(z 1 ) = v(z 2 ) ∈ S.
• Y = Z (z 1 = z 2 ) ∈ Box S is obtained from Z by identifying z 1 and z 2 .
• X = Z ∖ {z} ∈ Box S is obtained from Z by removing z ∈ Z in .
• W = X (z 1 = z 2 ) ∈ Box S is obtained from X by identifying z 1 and z 2 .
• σ X,z 1 ,z 2 ∈ WD W X and σ Z,z 1 ,z 2 ∈ WD Y Z are in-splits. • ω Y,z ∈ WD Y W and ω Z,z ∈ WD Z X are 1-wasted wires. Then (ω Y,z ) ○ (σ X,z 1 ,z 2 ) = (σ Z,z 1 ,z 2 ) ○ (ω Z,z ) ∈ WD Y X . (3.3.24.1)
The next relation says that the colored unit of a box X can be rewritten as the substitution of a 1-wasted wire into an in-split. This is depicted in the picture
• Y ∈ Box S , and x, y ∈ Y in are distinct elements such that v(x) = v(y) ∈ S. • X = Y (x = y) ∈ Box S is obtained from Y by identifying x and y. • ω Y,y ∈ WD Y X is a 1-wasted wire. • σ Y,x,y ∈ WD X Y is an in-split. Then σ Y,x,y ○ ω Y,y = 1 X ∈ WD X X .
(3.3.25.1)
The following three relations are about out-splits. The next one is the associativity property of out-splits. It gives two different ways to construct the following wiring diagram using two out-splits: • Y ∈ Box S , and y 1 , y 2 , y 3 ∈ Y out are distinct elements such that v(
X Y in y 1 y 2 y 3 Y outy 1 ) = v(y 2 ) = v(y 3 ) ∈ S.
• Y 12 = Y (y 1 = y 2 ) ∈ Box S is obtained from Y by identifying y 1 and y 2 , called y 12 in Y 12 .
• Y 23 = Y (y 2 = y 3 ) ∈ Box S is obtained from Y by identifying y 2 and y 3 , called y 23 in Y 23 .
• X = Y (y 1 = y 2 = y 3 ) ∈ Box S is obtained from Y by identifying y 1 , y 2 , and y 3 .
• σ Y,y 1 ,y 2 ∈ WD Y Y 12 and σ Y 12 ,y 12 ,y 3 ∈ WD Y 12 X are out-splits.
• σ Y,y 2 ,y 3 ∈ WD Y Y 23 and σ Y 23 ,y 1 ,y 23 ∈ WD Y 23 X are out-splits.
Then σ Y,y 1 ,y 2 ○ σ Y 12 ,y 12 ,y 3 = σ Y,y 2 ,y 3 ○ σ Y 23 ,y 1 ,y 23 ∈ WD Y X .
(3.3.26.1)
The next relation is the commutativity property of out-splits. It gives two different ways to construct the following wiring diagram using two out-splits: • Y ∈ Box S , and y 1 , y 2 , y 3 , y 4 ∈ Y out are distinct elements such that v(
X Y in y 1 y 2 y 3 y 4 Y outy 1 ) = v(y 2 )
and v(y 3 ) = v(y 4 ) ∈ S.
• Y 12 = Y (y 1 = y 2 ) ∈ Box S is obtained from Y by identifying y 1 and y 2 .
• Y 34 = Y (y 3 = y 4 ) ∈ Box S is obtained from Y by identifying y 3 and y 4 .
• X = Y (y 1 = y 2 ; y 3 = y 4 ) ∈ Box S is obtained from Y by (i) identifying y 1 and y 2 and (ii) identifying y 3 and y 4 .
• σ Y,y 1 ,y 2 ∈ WD Y Y 12 and σ Y 12 ,y 3 ,y 4 ∈ WD Y 12 X are out-splits.
• σ Y,y 3 ,y 4 ∈ WD Y Y 34 and σ Y 34 ,y 1 ,y 2 ∈ WD Y 34 X are out-splits.
Then σ Y,y 1 ,y 2 ○ σ Y 12 ,y 3 ,y 4 = σ Y,y 3 ,y 4 ○ σ Y 34 ,y 1 ,
y 2 ∈ WD Y X . (3.3.27.1)
The next relation is the commutativity property between an out-split and a 1wasted wire. It gives two different ways to construct the following wiring diagram using one out-split and one 1-wasted wire:
X Y in y y 1 y 2 Y out
Proposition 3.3.28. Suppose:
• Y ∈ Box S , y ∈ Y in , and y 1 , y 2 ∈ Y out such that v(y 1 ) = v(y 2 ) ∈ S.
• W = Y (y 1 = y 2 ) ∈ Box S is obtained from Y by identifying y 1 and y 2 .
• Z = Y ∖ {y} ∈ Box S is obtained from Y by removing y ∈ Y in .
• X = Z (y 1 = y 2 ) ∈ Box S is obtained from Z by identifying y 1 and y 2 .
• σ Z,y 1 ,y 2 ∈ WD Z X and σ Y,y 1 ,y 2 ∈ WD Y W are out-splits. • ω Y,y ∈ WD Y Z and ω W,y ∈ WD W X are 1-wasted wires.
Then ω Y,y ○ σ Z,y 1 ,y 2 = σ Y,y 1 ,y 2 ○ ω W,y ∈ WD Y X .
(3.3.28.1)
The final relation is the commutativity property of 1-wasted wires. It gives two different ways to construct the following wiring diagram using two 1-wasted wires: • Y ∈ Box S , and y 1 , y 2 ∈ Y in are distinct elements.
X Y in y 1 y 2 Y out• Y 1 = Y ∖ {y 1 } ∈ Box S is obtained from Y by removing y 1 . • Y 2 = Y ∖ {y 2 } ∈ Box S is obtained from Y by removing y 2 . • X = Y ∖ {y 1 , y 2 } ∈ Box S is
obtained from Y by removing y 1 and y 2 .
• ω Y,y 1 ∈ WD Y Y 1 and ω Y 1 ,y 2 ∈ WD Y 1 X are 1-wasted wires.
• ω Y,y 2 ∈ WD Y Y 2 and ω Y 2 ,y 1 ∈ WD Y 2 X are 1-wasted wires.
Then
ω Y,y 1 ○ ω Y 1 ,y 2 = ω Y,y 2 ○ ω Y 2 ,y 1 ∈ WD Y X .
Summary of Chapter 3
(1) There are eight generating wiring diagrams in WD.
(2) Each internal wasted wire can be generated using a 1-wasted wire and a 1-loop.
(3) There are twenty-eight elementary relations in WD.
Chapter 4
Decomposition of Wiring Diagrams
As part of the finite presentation theorem for the operad WD of wiring diagrams (Theorem 5.2.11), in Theorem 5.1.11 we will observe that each wiring diagram has a highly structured decomposition into generating wiring diagrams (Def. 3.1.9), called a stratified presentation. Stratified presentations are also needed to establish the second part of the finite presentation theorem for WD regarding relations. The purpose of this chapter is to provide all the steps needed to establish the existence of a stratified presentation for each wiring diagram. We remind the reader about Notation 3.3.1 for (iterated) operadic compositions.
In Section 4.1 we show that each wiring diagram ψ has a specific operadic decomposition (4.1.7.1)
ψ = α ○ ϕ.
An explanation of this decomposition is given just before Def. 4.1.5. The idea of this decomposition is that we are breaking the complexity of a general wiring diagram into two simpler parts. On the one hand, the inner wiring diagram ϕ contains all the input boxes and the delay nodes of ψ, but its supplier assignment is as simple as possible, namely the identity map. See Lemmas 4.1.8 and 4.1.10. On the other hand, the outer wiring diagram α has only one input box and no delay nodes, but its supplier assignment is equal to that of ψ.
In Section 4.2 we observe that the outer wiring diagram α in the previous decomposition of ψ can be decomposed as (4.2.6.1)
α = π 1 ○ π 2 .
63 Example 4.2.3 has a concrete wiring diagram that illustrates this decomposition. The idea of this decomposition is that in a wiring diagram there are usually wires that go backward (i.e., "point to the left"), as in (2.2.18.1), in a 1-loop (Def. 3.1.5), and in the pictures just before Prop. 3.3.11 and Prop. 3.3.15. This decomposition breaks the complexity of the wiring diagram α into two simpler parts. On the one hand, the outer wiring diagram π 1 contains all the backward-going wires in α but no wasted wires or split wires (Lemma 4.2.7). On the other hand, the inner wiring diagram π 2 contains no backward-going wires, but it has all the wasted wires and split wires in α.
In Section 4.3 we observe that the wiring diagram π 2 in the previous decomposition of α can be decomposed further as (4.3.5.1)
π 2 = β 1 ○ β 2 ○ β 3 .
Example 4.3.2 has a concrete wiring diagram that illustrates this decomposition. In this decomposition:
• The outermost wiring diagram β 1 is an iterated operadic composition of 1-wasted wires (Lemma 4.3.6).
• The middle wiring diagram β 2 is an iterated operadic composition of insplits (Lemma 4.3.9).
• The innermost wiring diagram β 3 is an iterated operadic composition of out-splits (Lemma 4.3.12).
By convention an empty operadic composition means a colored unit. In summary, for a wiring diagram ψ, we will decompose it as
ψ = π 1 ○ β 1 ○ β 2 ○ β 3 ○ ϕ.
Factoring Wiring Diagrams
Assumption 4.1.1. Throughout this chapter, fix a class S. Suppose
ψ = DN ψ , v ψ , s ψ ∈ WD Y X (4.1.1.1)
is a wiring diagram with:
• output box Y ∈ Box S and input boxes X = (X 1 , . . . , X N ) for some N ≥ 0;
• r delay nodes DN ψ = {d 1 , . . . , d r } for some r ≥ 0;
• value assignment v ψ ∶ Y in ∐ Y out ∐ X in ∐ X out ∐ DN ψ G G S, where X in = ∐ N i=1 X in i and X out = ∐ N i=1 X out i ; • supplier assignment s ψ ∶ Dm ψ G G Sp ψ .
Since N = 0 and r = 0 are both allowed, ψ is a general wiring diagram. Furthermore:
(1) To simplify notation, we will write v ψ (d i ) ∈ S simply as d i , so each δ d i ∈ WD d i is a 1-delay node (Def. 3.1.2). i.e., ψ ∈ WD Y has no input boxes and no delay nodes. Then one of the following two statements is true.
(2) X = ∐ N i=1 X i ∈ Box S is the coproduct of the X i 's. (3) Define X ′ ∈ Box S as X ′in = X in ∐ DN ψ and X ′out = X out ∐ DN ψ .(1) ψ = ǫ ∈ WD ∅ , the empty wiring diagram (Def. 3.1.1). (2) There exist 1-wasted wires (Def. 3.1.8) ω 1 , . . . , ω m , where m = Y in > 0, such that ψ = ω 1 ○ ⋯ ○ ω m ○ ǫ. (4.1.3.1)
Proof. Since X in = X out = DN ψ = ∅, the supplier assignment of ψ is a function
Dm ψ = Y out s G G Y in = Sp ψ .
The non-instantaneity requirement (2.2.13.2) then implies Y out = ∅. If m = Y in = 0, then Y is the empty box and ψ = ǫ, the empty wiring diagram, by definition.
If m > 0, then every global input y ∈ Y in = {y 1 , . . . , y m } is an external wasted wire, and the supplier assignment
s ∶ Y out = ∅ G G Y in is the trivial map. For each 1 ≤ j ≤ m, define the box Y j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Y if j = 1; Y ∖ {y 1 , . . . , y j−1 } if 2 ≤ j ≤ m; ∅ if j = m + 1.
Each ω Y j ,y j ∈ WD Y j Y j+1 is a 1-wasted wire. Using the notation (3.3.1.2), the iterated composition
ω Y 1 ,y 1 ○ ⋯ ○ ω Ym,ym ○ ǫ ∈ WD Y
is then a wiring diagram with output box Y 1 = Y, no input boxes and no delay nodes, and supplier assignment ∅ = Y out G G Y in the trivial map. This is the same as ψ.
Next, for wiring diagrams ψ not necessarily covered by Lemma 4.1.3, we define two relatively simple wiring diagrams that will be shown to provide a decomposition for ψ. Each of these two simpler wiring diagrams will then be analyzed further.
Motivation 4.1.4. This decomposition for ψ is depicted in the following picture:
X 1 ⋮ X N d 1 ⋮ d r ⋮ ⋮ ψ = α ○ ϕ
Here the intermediate gray box is X ′ (4.1.1.2). In this decomposition, the inside wiring diagram ϕ has all the input boxes and the delay nodes of ψ, but its supplier assignment is the identity function. The outside wiring diagram α has a single input box X ′ and no delay nodes, but it has the same supplier assignment as ψ.
Definition 4.1.5. Suppose ψ is as in Assumption 4.1.1. Define the wiring diagram
ϕ = DN ϕ , v ϕ , s ϕ ∈ WD X ′ X (4.1.5.1)
with:
• output box X ′ (4.1.1.2) and input boxes X = (X 1 , . . . , X N );
• delay nodes DN ϕ = DN ψ = {d 1 , . . . , d r };
• supplier assignment
Dm ϕ = X ′out ∐ X in ∐ DN ϕ = X out ∐ DN ψ ∐ X in ∐ DN ψ sϕ Sp ϕ = X ′in ∐ X out ∐ DN ϕ = X in ∐ DN ψ ∐ X out ∐ DN ψ the identity function that sends X ′out to (X out ∐ DN ψ ) and (X in ∐ DN ψ ) to X ′in .
Definition 4.1.6. Suppose ψ is as in Assumption 4.1.1. Define the wiring diagram
α = DN α , v α , s α ∈ WD Y X ′ (4.1.6.1)
with:
• one input box X ′ (4.1.1.2) and output box Y;
• no delay nodes;
• supplier assignment
Dm α = Y out ∐ X ′in = Y out ∐ X in ∐ DN ψ = Dm ψ sα Sp α = Y in ∐ X ′out = Y in ∐ X out ∐ DN ψ = Sp ψ equal to s ψ .ψ = α ○ ϕ ∈ WD Y X (4.1.7.1)
in which α ∈ WD Y X ′ is as in (4.1.6.1) and ϕ ∈ WD X ′ X is as in (4.1.5.1).
Proof. By the definition of ○ 1 (Def. 2.3.4), α ○ ϕ = α ○ 1 ϕ belongs to WD Y X and has DN ϕ = DN ψ as its set of delay nodes. It remains to check that its supplier assignment is equal to that of ψ. This follows from a direct inspection because s ϕ is the identity function, while s α = s ψ .
To obtain the desired stratified presentation of ψ, we now begin to analyze the wiring diagram ϕ. Proof. All three cases are checked by direct inspection.
Motivation 4.1.9. Next we observe that, for higher values of N + r, the wiring diagram ϕ is generated by 2-cells (Def. 3.1.4) and 1-delay nodes via iterated operadic compositions, as depicted in the following picture.
X 1 ⋮ X N d r−1 ⋮ d r ϕ
The operadic composition γ (2.1.3.2) is used in the following observation.
ϕ = γ θ; 1 X i N i=1 , δ d j r j=1 .
(4.1.10.1)
Here
θ = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ θ 1 if N + r = 2, θ 1 ○ 2 ⋯ ○ 2 θ N+r−2 ○ 2 θ N+r−1 if N + r > 2 (4.1.10.2)
with each θ k a 2-cell, and each δ d j ∈ WD d j is a 1-delay node as in Assumption 4.1.1.
Proof.
Recall that X = (X 1 , . . . , X N ) and DN ϕ = DN ψ = {d 1 , . . . , d r }.
For N + 1 ≤ j ≤ N + r define the box X j = d j ∈ Box S as X in j = {d j } = X out j . For 1 ≤ i ≤ N + r define the box X ≥i = N+r ∐ p=i X p ∈ Box S . Note that X ≥1 = X ′ (4.1.1.2). Next, for 1 ≤ k ≤ N + r − 1, define the 2-cell θ k = θ X k ,X ≥k+1 ∈ WD X ≥k X k , X ≥k+1 . Then we have a wiring diagram θ ∈ WD X ′ X 1 ,...,X N+r
in which θ * is defined as in (4.1.10.2). Since 1 X i ∈ WD X i X i and δ d j ∈ WD d j , the operadic composition on the right side of (4.1.10.1) is defined and belongs to WD
X ′ X .
Since the two sides of (4. 1
Unary Wiring Diagrams
In this section, we analyze wiring diagrams with exactly one input box and no delay nodes, such as α (4.1.6.1). We will show that such a wiring diagram can be generated by the generating wiring diagrams (Def. 3.1.9) of name changes (Def. 3.1.3), 1-loops (Def. 3.1.5), 1-wasted wires (Def. 3.1.8), in-splits (Def. 3.1.6), and out-splits (Def. 3.1.7), in this order. We remind the reader of Notation 3.3.1 for (iterated) ○ 1 .
We will need a few definitions and notations. (1) A loop element in π is an element x ∈ X out such that there exists x ′ ∈ X in with x as its supply wire. The set of loop elements in π is denoted by π lp .
(2) An element
x ′ ∈ X in is said to be internally supplied if s π (x ′ ) ∈ X out . The set of such elements in π is denoted by π sp + . (3) An element x ′ ∈ X in is said to be externally supplied if s π (x ′ ) ∈ Y in . The set of such elements in π is denoted by π sp − .
Recall from Definition 2.2.13 the concepts of external wasted wires π w − and of internal wasted wires π w + of a wiring diagram π.
Lemma 4.2.2. Suppose π ∈ WD Y X is a wiring diagram with one input box X and no delay nodes. Then:
(1) π sp + ∐ π sp − = X in . (2) π w + ∐ π lp ⊆ X out . (3) s π (π sp + ) = π lp . (4) s π (π sp − ) ∐ π w − = Y in . (5) s π (Y out ) ⊆ X out ∖ π w + .
Proof. All the statements are immediate from the definitions.
Example 4.2.3. Consider the wiring diagram π ∈ WD Y X as depicted in the picture
x 1 x 2 x 3 x 4 x 1 x 2 x 3 π y 1 y 2 y 1 y 2 with • X in = {x 1 , x 2 , x 3 , x 4 }, X out = {x 1 , x 2 , x 3 }, Y in = {y 1 , y 2 }, and Y out = {y 1 , y 2 }; • π lp = {x 1 }, π sp + = {x 1 , x 2 }, π sp − = {x 3 , x 4 }, π w + = {x 3 }, and π w − = {y 1 }.
Note that we may operadically decompose π as follows.
x 1 x 2 x 3 x 4 x 1 x 2 x 3 π = π 1 ○ π 2 y 1 y 2 y 1 y 2 (4.2.3.1)
The point of this decomposition is that the inner wiring diagram π 2 is generated by:
• two 1-wasted wires, one for the external wasted wire y 1 and the other for the internal wasted wire x 3 ;
• two in-splits, one for {x 1 , x 2 } and the other for {x 3 , x 4 };
• one out-split for x 1 , which is the supply wire of y 1 , x 1 , and x 2 .
At the same time, the outer wiring diagram π 1 is generated by two 1-loops, one for the loop element x 1 and the other for the internal wasted wire x 3 .
With this example as a guide, next we will factor a general wiring diagram with one input box and no delay nodes into two wiring diagrams. The outer wiring diagram will be generated by name changes and 1-loops. The inner wiring diagram will be generated by 1-wasted wires, in-splits, and out-splits. The intermediate gray box in (4.2.3.1) will be called Z below. With this in mind, we will mostly not mention name changes.
For a wiring diagram with one input box and no delay nodes, we will decompose it using the wiring diagrams in the next definition.
Definition 4.2.5. Suppose π ∈ WD Y X is a wiring diagram with one input box X and no delay nodes.
(1) Define the box Z ∈ Box S as Z in = Y in ∐ π w + ∐ π lp ; Z out = Y out ∐ π w + ∐ π lp .
(2) Define the wiring diagram π 1 ∈ WD Y Z with: • one input box Z, output box Y, and no delay nodes; • supplier assignment
Dm π 1 = Y out ∐ Z in = Y out ∐ Y in ∐ π w + ∐ π lp sπ 1 Sp π 1 = Y in ∐ Z out = Y in ∐ Y out ∐ π w + ∐ π lp (4.2.5.1)
the identity function.
(3) Define the wiring diagram π 2 ∈ WD Z X with: • one input box X, output Z, and no delay nodes; • supplier assignment
Dm π 2 = Z out ∐ X in = Y out ∐ π w + ∐ π lp ∐ [π sp + ∐ π sp − ] sπ 2 Sp π 2 = Z in ∐ X out = Y in ∐ π w + ∐ π lp ∐ X out (4.2.5.2) whose restriction to: -Y out is s π ∶ Y out G G X out ; -π w + ∐ π lp is the subset inclusion into X out ; -π sp + is s π ∶ π sp + G G π lp ; -π sp − is s π ∶ π sp − G G Y in .
This is well-defined by the non-instantaneity requirement (2.2.13.2) for π and Lemma 4.2.2.
An example of the following decomposition is the picture (4.2.3.1) above.
Lemma 4.2.6. Suppose π ∈ WD Y X is a wiring diagram with one input box X and no delay nodes. Then it admits a decomposition π = π 1 ○ π 2 (4.2.6.1)
in which π 1 ∈ WD Y Z and π 2 ∈ WD Z X are as in Def. 4.2.5.
Proof. Both sides of (4.2.6.1) belong to WD Y X and have no delay nodes. So it remains to check that the supplier assignment s of π 1 ○ π 2 is equal to s π . Note that
Dm π 1 ○π 2 = Y out ∐ X in = Y out ∐ [π sp + ∐ π sp − ]
. By the definitions of ○ = ○ 1 (Def. 2.3.4), s π 1 (4.2.5.1), and s π 2 (4.2.5.2):
• on Y out the supplier assignment s is s π 2 s π 1 = s π Id = s π .
• on π sp + ∐ π sp − the supplier assignment s is s π 2 s π 1 = Id s π = s π . So the supplier assignment of π 1 ○ π 2 is equal to s π .
To obtain the desired stratified presentation of π, next we observe that π 1 in Def. 4.2.5 is either a colored unit (2.3.2.1) or an iterated operadic composition of 1-loops (Def. 3.1.5). An example of π 1 is the outer wiring diagram in the example (4.2.3.1).
Lemma 4.2.7. Suppose:
• Y, Z ∈ Box S such that Z in = Y in ∐ T and Z out = Y out ∐ T for some T ∈ Fin S . • ζ ∈ WD Y Z
is a wiring diagram with no delay nodes and with supplier assignment
Dm ζ = Y out ∐ Z in = Y out ∐ Y in ∐ T s ζ Sp ζ = Y in ∐ Z out = Y in ∐ [Y out ∐ T]
the identity function.
Then the following statements hold.
(1) ζ = 1 Y if T = ∅.
(2) If p = T > 0, then there exist 1-loops λ 1 , . . . , λ p such that
ζ = λ 1 ○ ⋯ ○ λ p . (4.2.7.1)
Proof. If T = ∅, then ζ ∈ WD Y Y has no delay nodes and has supplier assignment s ζ = Id. So ζ is the Y-colored unit.
Next suppose T = {t 1 , . . . , t p } with p > 0. For the definitions below, it is convenient to keep in mind the following picture of ζ:
Z tp tp t 1 t 1 Y in Y out ⋮ For 0 ≤ j ≤ p define the box Y j ∈ Box S as Y j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Y if j = 0; Y ∐ {t 1 , . . . , t j } if 1 ≤ j ≤ p. Here Y ∐ {t 1 , . . . , t j } means a copy of t i for 1 ≤ i ≤ j is added to each of Y in and Y out . In particular, we have Y p = Z. For 1 ≤ j ≤ p define the 1-loop λ j = λ Y j ,t j ∈ WD Y j−1 Y j in which t j ∈ Y in j has supply wire t j ∈ Y out j . The iterated operadic composition λ 1 ○ ⋯ ○ λ p ∈ WD Y 0 Yp = WD Y Z
has no delay nodes. To see that it is equal to ζ, it remains to check that its supplier assignment is equal to s ζ = Id. This holds because each 1-loop λ j has identity supplier assignment.
Observe that Lemma 4.2.7 applies to π 1 ∈ WD Y Z in Def. 4.2.5 with T = π w + ∐ π lp . So π 1 is either the Y-colored unit or an iterated operadic composition of 1-loops.
Unary Wiring Diagrams with No Loop Elements
In order to show that the wiring diagram π 2 ∈ WD Z X in Def. 4.2.5 is generated by 1-wasted wires, in-splits, and out-splits, first we identify its external wasted wires, internal wasted wires, and loop elements. (1) The set of external wasted wires in π 2 is π w
2− = π w + ∐ π w − . (2) The set of internal wasted wires in π 2 is π w 2+ = ∅. (3) The set of loop elements in π 2 is π lp 2 = ∅. (4) s π 2 (Z out ) = X out . (5) Z in = π w + ∐ π w − ∐ s π 2 (X in ).
Proof.
(1) By definition an external wasted wire in π 2 is an element in Z in that is not in the image of s π 2 (4.2.5.2). By the definition of s π 2 , this is the subset
π w + ∐ [Y in ∖ s π (π sp − )] ⊆ Z in . It follows from the non-instantaneity requirement (2.2.13.2) for π that Y in ∖ s π (π sp − ) = π w − .
(2) By definition an internal wasted wire in π 2 is an element in X out that is not in the image of s π 2 . An element of X out is either an internal wasted wire in π, or else it is the s π -image of an element in π sp
+ ∐ Y out . Since X out = π w + ∐ s π π sp + ∐ Y out = π w + ∐ π lp ∪ s π (Y out )
, an inspection of the definition of s π 2 (4.2.5.2) reveals that all of X out is in the image of s π 2 . So π 2 has no internal wasted wires.
(3) By definition a loop element in π 2 is an element in X out that is the supply wire, under s π 2 , of some element in X in . Since X in = π sp + ∐ π sp − , the definition of s π 2 (4.2.5.2) yields
s π 2 (π sp + ∐ π sp − ) = s π (π sp + ) ∐ s π (π sp − ) ⊆ π lp ∐ Y in ⊆ Z in = Sp π 2 ∖ X out .
So π 2 has no loop elements.
(4) By (2) π 2 has no internal wasted wires, so X out = s π 2 (Z out ∐ π sp 2+ ). But by (3) π 2 has no loop elements, so π sp
2+ = ∅ and X out = s π 2 (Z out ). (5) Since π 2 has no loop elements by (3), s π 2 (X in ) ⊆ Z in . An element in Z in that is not in s π 2 (X in )
is precisely an external wasted wire in π 2 . By (1) the set of external wasted wires in π 2 is π w
+ ∐ π w − .
Continuing our analysis of wiring diagrams with one input box and no delay nodes, our next goal is to construct a decomposition for π 2 (Def. 4.2.5) involving 1-wasted wires, in-splits, and out-splits.
Example 4.3.2.
Consider the inner wiring diagram π 2 ∈ WD Z X in the example (4.2.3.1), which is depicted in the following picture.
x 1 x 2 x 3 x 4 x 1 x 2 x 3 π 2 y 1 y 2 y 1 y 2
For this wiring diagram, the desired decomposition is depicted in the picture:
x 1 x 2 x 3 x 4 x 1 x 2 x 3 y 1 y 2 π 2 = β 1 ○ β 2 ○ β 3 y 1 y 2
The inner gray box will be called V, and the outer gray box will be called W below. Note that:
• The outermost wiring diagram β 1 ∈ WD Z W is generated by two 1-wasted wires.
• The middle wiring diagram β 2 ∈ WD W V is generated by two in-splits.
• The innermost wiring diagram β 3 ∈ WD V X is an out-split.
For a general wiring diagram with one input box, no delay nodes, no loop elements, and no internal wasted wires, such a decomposition uses the following definitions.
Definition 4.3.3. Suppose X, Z ∈ Box S and β ∈ WD Z X is a wiring diagram with no delay nodes and no loop elements.
(1) Suppose the box W = Z ∖ β w − ∈ Box S is obtained from Z ∈ Box S by removing the external wasted wires of β, so W in = Z in ∖ β w − and W out = Z out . (2) Define the wiring diagram β 1 ∈ WD Z W as having: • no delay nodes; • supplier assignment Dm β 1 = Z out ∐ W in = Z out ∐ Z in ∖ β w − s β 1 Sp β 1 = Z in ∐ W out = Z in ∐ Z out (4.3.3.1)
the identity function on Z out and the subset inclusion on
Z in ∖ β w − ⊆ Z in . (3) Define the box V ∈ Box S as V in = X in and V out = Z out = W out .
(4) Define the wiring diagram β 2 ∈ WD W V as having: • no delay nodes; • supplier assignment
Dm β 2 = W out ∐ V in = W out ∐ X in s β 2 Sp β 2 = W in ∐ V out = Z in ∖ β w − ∐ W out (4.3.3.2)
the coproduct of the identity function on W out and the restriction of the supplier assignment s β ∶ X in G G Z in ∖ β w − . This is well-defined because β has no delay nodes and no loop elements.
(5) Define the wiring diagram β 3 ∈ WD V X as having: • no delay nodes;
• supplier assignment
Dm β 3 = V out ∐ X in = Z out ∐ X in s β 3 Sp β 3 = V in ∐ X out = X in ∐ X out (4.3.3.3)
the coproduct of the identity function on X in and s β ∶ Z out G G X out . This is well-defined because β has no delay nodes and because of the noninstantaneity requirement (2.2.13.2) for β.
(1) The map s β ∶ X in G G Z in ∖ β w − , which is part of s β 2 , is surjective. (2) If β has no internal wasted wires (such as π 2 in Def. 4.2.5), then the map s β ∶ Z out G G X out , which is part of s β 3 , is surjective.
Proof. The first assertion is true because β has no delay nodes and because of the non-instantaneity requirement (2.2.13.2). The second assertion is true because β has no delay nodes and no loop elements.
Lemma 4.3.5. In the context of Def. 4.3.3, there is a decomposition
β = β 1 ○ β 2 ○ β 3 ∈ WD Z X . (4.3.5.1)
Proof. By construction the iterated operadic composition β 1 ○ β 2 ○ β 3 also belongs to WD Z X and has no delay nodes. So it remains to check that its supplier assignment s is equal to s β . A direct inspection of (4.3.3.1), (4.3.3.2), and (4.3.3.3) reveals that:
• on X in ⊆ Dm β 3 the supplier assignment s is given by Id s β 2 Id X in = s β ;
• on Z out ⊆ Dm β 1 the supplier assignment s is given by
s β 3 Id Z out Id Z out = s β .
So the supplier assignment of β 1 ○ β 2 ○ β 3 is equal to s β .
Note that the decomposition in Lemma 4.3.5 applies to π 2 because π 2 has one input box, no delay nodes, and no loop elements (by Lemma 4.3.1).
Next we show that in the decomposition 4.3.5.1:
(1) β 1 is either a colored unit (Def. 2.3.2) or an iterated operadic composition of 1-wasted wires (Def. 3.1.8). See Lemma 4.3.6.
(2) β 2 is either a colored unit or an iterated operadic composition of in-splits (Def. 3.1.6). See Lemma 4.3.9.
(3) If β has no internal wasted wires, such as π 2 by Lemma 4.3.1, then β 3 is either a colored unit or an iterated operadic composition of out-splits (Def. 3.1.7). See Lemma 4.3.12.
During the first reading, the reader may wish to skip the proofs of the following Lemmas and simply look at the pictures. The following observation deals with the first statement.
Lemma 4.3.6. Consider the wiring diagram
β 1 ∈ WD Z W in Def. 4.3.3. (1) If β w − = ∅ (i.e.
, β has no external wasted wires), then
β 1 = 1 Z , the Z-colored unit. (2) If q = β w − > 0, then there exist 1-wasted wires ω 1 , . . . , ω q such that β 1 = ω 1 ○ ⋯ ○ ω q . (4.3.6.1)
Proof. Recall that β 1 has no delay nodes and has supplier assignment (4.3.3.1)
Dm β 1 = Z out ∐ W in = Z out ∐ Z in ∖ β w − s β 1 Sp β 1 = Z in ∐ W out = Z in ∐ Z out
that is the identity function on Z out and the subset inclusion on Z in ∖ β w − . If β w − = ∅, then s β 1 = Id and, therefore, β 1 is the colored unit.
Next suppose β w − = {w 1 , . . . , w q } ⊆ Z in with q > 0. Recall that W = Z ∖ β w − . For 0 ≤ j ≤ q define the box Z j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Z if j = 0; Z ∖ {w 1 , . . . , w j } if 1 ≤ j ≤ q.
So in particular Z q = W. The iterated operadic composition on the right side of (4.3.6.1) is represented in the following picture.
W Z in ∖ β w − w 1 w 2 ⋮ wq Z out ⋱
Here the outermost box is Z, the outermost gray box is Z 1 , and the innermost gray box is Z q−1 .
For 1 ≤ j ≤ q define the 1-wasted wires (Def. 3.1.8) ω j = ω Z j−1 ,w j ∈ WD Z j−1 Z j .
The iterated operadic composition
ω 1 ○ ⋯ ○ ω q ∈ WD Z 0 Zq = WD Z W
has no delay nodes. So to prove (4.3.6.1), it remains to check that its supplier assignment s is equal to s β 1 .
• On Z out ⊆ Dm ω 1 the supplier assignment s is the composition of q copies of the identity function, hence is the identity function.
• On W in ⊆ Dm ωq the supplier assignment s is the composition of the inclu-
sions Z in j G G Z in j−1 for 1 ≤ j ≤ q, which is the inclusion W in G G Z in .
Motivation 4.3.7. To show that β 2 is either a colored unit or an iterated composition of in-splits, we first need a lemma that says that the following wiring diagram is generated by in-splits.
X x 1 ⋮ x k Y in y ⋮ Y out Lemma 4.3.8. Suppose: • X, Y ∈ Box S such that X out = Y out .
• There exist y ∈ Y in and distinct elements x 1 , . . . , x k ∈ X in with k ≥ 1 such that
X in = Y in ∖ {y} ∐ {x 1 , . . . , x k } and v(y) = v(x i ) ∈ S for all i. • σ ∈ WD Y X
is a wiring diagram with no delay nodes and with supplier assignment
Dm σ = Y out ∐ X in = Y out ∐ Y in ∖ {y} ∐ {x 1 , . . . , x k } sσ Sp σ = Y in ∐ X out = Y in ∐ Y out
given by
s σ (z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y if z = x 1 , . . . , x k ; z if z ∈ Y out ∐ Y in ∖ {y} .
Then:
(1) σ is the Y-colored unit if k = 1; (2) σ is an iterated operadic composition of (k − 1) in-splits if k ≥ 2. (k − 1) in-splits by induction on k. If k = 2, then by definition σ is the in-split σ X,x 1 ,x 2 (Def. 3.1.6).
Suppose k ≥ 3. We will factor σ into two wiring diagrams as depicted in the picture
X x 1 ⋮ x k−1 x k Y in y x ′ ⋮ Y out σ = σ 1 ○ σ 2
in which the intermediate gray box will be called W below. The outer wiring diagram σ 1 will be an in-split, and the inner wiring diagram σ 2 will be an iterated operadic composition of (k − 2) in-splits. To define such a decomposition, we will need the following definitions.
(1) Suppose W ∈ Box S such that W out = Y out = X out and W in = Y in ∖ {y} ∐ {x ′ , x k } for some x k = x ′ such that v(x ′ ) = v(x k ) ∈ S.
In particular, we have
X in = W in ∖ {x ′ } ∐ {x 1 , . . . , x k−1 }. (4.3.8.1)
(2) Define the wiring diagram σ 1 ∈ WD Y W with no delay nodes and with supplier assignment
Dm σ 1 = Y out ∐ W in = Y out ∐ Y in ∖ {y} ∐ {x ′ , x k } sσ 1 Sp σ 1 = Y in ∐ W out = Y in ∖ {y} ∐ {y} ∐ Y out
given by
s σ 1 (z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y if z = x ′ , x k ; z otherwise.
(3) Define the wiring diagram σ 2 ∈ WD W X with no delay nodes and with supplier assignment
Dm σ 2 = W out ∐ X in = W out ∐ Y in ∖ {y} ∐ {x 1 , . . . , x k−1 } ∐ {x k } sσ 2 Sp σ 2 = W in ∐ X out = Y in ∖ {y} ∐ {x ′ , x k } ∐ W out given by s σ 2 (z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x ′ if z = x 1 , . . . , x k−1 ; z otherwise.
Then σ 1 ○ σ 2 ∈ WD Y X is a wiring diagram with no delay nodes. To see that it is equal to σ, it suffices to check that the supplier assignment of σ 1 ○ σ 2 is equal to s σ . This is true by a direct inspection of s σ 1 and s σ 2 .
By definition σ 1 is the in-split σ W,x ′ ,x k . By (4.3.8.1) the induction hypothesis applies to σ 2 , which says that it is an iterated operadic composition of (k − 2) insplits. Combined with the previous paragraph, it follows that σ = σ 1 ○ σ 2 is the iterated operadic composition of (k − 1) in-splits, finishing the induction.
Next we consider β 2 .
Proof. Recall that W in = Z in ∖ β w − , V in = X in , and V out = Z out = W out . The wiring diagram β 2 ∈ WD W
V has no delay nodes and has supplier assignment (4.3.3.2)
Dm β 2 = W out ∐ V in = Z out ∐ X in s β 2 Sp β 2 = W in ∐ V out = Z in ∖ β w − ∐ Z out the coproduct of the identity function on Z out and s β ∶ X in G G Z in ∖ β w − . Write W in = Z in ∖ β w − = {z 1 , . . . , z p }, so each s −1 β (z i ) is non-empty by Lemma 4.3.4.
If p = 0, then Z in ∖ β w − = ∅ = X in , and β 2 is the W-colored unit. Suppose p > 0. We will write β 2 as an iterated composition as in the following picture.
V s −1 β (z 1 ) ⋮ s −1 β (zp) W out ⋮ ⋱ β 2 = σ 1 ○ ⋯ ○ σ p z 1 ⋮ zp W in
There are p − 1 gray boxes. The outermost gray box will be called W 1 , and the innermost gray box will be called W p−1 below. To define such a decomposition, we will need the following definitions.
(1) For 0 ≤ j ≤ p define the box W j ∈ Box S with W out j = W out and W in j = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ j ∐ i=1 s −1 β (z i ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ∅ if j = 0 ∐ {z j+1 , . . . , z p } ∅ if j = p . Note that W in 0 = W in by definition, while W in p = X in = V in by Lemma 4.3.4. So W 0 = W and W p = V.
(2) For 1 ≤ j ≤ p define the wiring diagram σ j ∈ WD W j−1 W j with no delay nodes and with supplier assignment
Dm σ j = W out j−1 ∐ W in j = W out ∐ j ∐ i=1 s −1 β (z i ) ∐ {z j+1 , . . . , z p } sσ j Sp σ j = W in j−1 ∐ W out j = j−1 ∐ i=1 s −1 β (z i ) ∐ {z j , . . . , z p } ∐ W out
given by
s σ j (x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ z j if x ∈ s −1 β (z j ); x otherwise. Since W in j = W in j−1 ∖ {z j } ∐ s −1 β (z j ), by Lemma 4.3.8 σ j is: • a colored unit if s −1 β (z j ) = 1;
• an iterated operadic composition of
s −1 β (z j ) − 1 in-splits if s −1 β (z j ) ≥ 2.
Therefore, to show that β 2 is either a colored unit or an iterated operadic composition of in-splits, it is enough to check that there is a decomposition
β 2 = σ 1 ○ ⋯ ○ σ p ∈ WD W V .
Since the iterated operadic composition on the right has no delay nodes, it remains to check that its supplier assignment s is equal to s β 2 (4.3.3.2).
On W out = Z out ⊆ Dm σ 1 the supplier assignment s is the composition of p identity functions, hence the identity function. On
V in = X in = p ∐ i=1 s −1 β (z i ) ⊆ Dm σp the supplier assignment s sends elements in each s −1 β (z j ) to z j ∈ W in , so it is equal to s β 2 .
Motivation 4.3.10. Next, to show that β 3 is either a colored unit or an iterated operadic composition of out-splits, we first need a lemma that says that the following wiring diagram is generated by out-splits.
X x Y in y 1 y k ⋮ X out ∖ {x}
The following observation is the out-split analogue of Lemma 4.3.8. Lemma 4.3.11. Suppose:
• X, Y ∈ Box S such that X in = Y in .
• There exist x ∈ X out and distinct elements y 1 , . . . , y k ∈ Y out with k ≥ 1 such that
Y out = X out ∖ {x} ∐ {y 1 , . . . , y k } and v(x) = v(y i ) ∈ S for 1 ≤ i ≤ k. • σ ∈ WD Y X
is a wiring diagram with no delay nodes and with supplier assignment
Dm σ = Y out ∐ X in = [X out ∖ {x}] ∐ {y 1 , . . . , y k } ∐ X in sσ Sp σ = Y in ∐ X out = X in ∐ X out
given by
s σ (z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x if z = y 1 , . . . , y k ; z if z ∈ [X out ∖ {x}] ∐ X in .
Then:
(1) σ is the Y-colored unit if k = 1;
(2) σ is an iterated operadic composition of (k − 1) out-splits if k ≥ 2.
Proof. If k = 1, then s σ is the identity function, so σ is a colored unit.
The assertion for k ≥ 2 is proved by induction. If k = 2, then σ is by definition the out-split σ Y,y 1 ,y 2 (Def. 3.1.7).
Suppose k ≥ 3. We will factor σ into two wiring diagrams as depicted in the picture
X x Y in y k y 1 ⋮ y k−1 X out ∖ {x} w σ = σ 1 ○ σ 2
in which the intermediate gray box will be called W below. The inner wiring diagram σ 2 will be an out-split, and the outer wiring diagram σ 1 will be an iterated operadic composition of (k − 2) out-splits. To define such a decomposition, we will need the following definitions.
(1) Define the box W ∈ Box S with W in = X in = Y in and W out = X out ∖ {x} ∐ {w, y k } for some w = y k such that v(w) = v(y k ) ∈ S. In particular, we have Y out = W out ∖ {w} ∐ {y 1 , . . . , y k−1 }.
(4.3.11.1)
(2) Define the wiring diagram σ 1 ∈ WD Y W with no delay nodes and with supplier assignment
Dm σ 1 = Y out ∐ W in = [X out ∖ {x}] ∐ {y 1 , . . . , y k } ∐ Y in s σ 1 Sp σ 1 = Y in ∐ W out = Y in ∐ [X out ∖ {x}] ∐ {w, y k } given by s σ 1 (z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ w if z = y 1 , . . . , y k−1 ; z otherwise.
(3) Define the wiring diagram σ 2 ∈ WD W X with no delay nodes and with supplier assignment
Dm σ 2 = W out ∐ X in = [X out ∖ {x}] ∐ {w, y k } ∐ W in s σ 2 Sp σ 2 = W in ∐ X out
given by
s σ 2 (z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x if z = w, y k ; z otherwise.
Then σ 1 ○ σ 2 ∈ WD Y X is a wiring diagram with no delay nodes. To see that it is equal to σ, it suffices to check that the supplier assignment of σ 1 ○ σ 2 is equal to s σ . This is true by a direct inspection of s σ 1 and s σ 2 .
By (4.3.11.1) the induction hypothesis applies to σ 1 , which says that it is an iterated operadic composition of (k − 2) out-splits. By definition σ 2 is the out-split σ W,w,y k . Combined with the previous paragraph, it follows that σ = σ 1 ○ σ 2 is the iterated operadic composition of (k − 1) out-splits, finishing the induction.
Finally, we consider β 3 . Proof. Recall that the wiring diagram β 3 ∈ WD V X has no delay nodes and has supplier assignment (4.3.3.3)
Dm β 3 = V out ∐ X in = Z out ∐ X in s β 3 Sp β 3 = V in ∐ X out = X in ∐ X out
the coproduct of the identity function on X in and s β ∶ Z out G G X out . Since β has no internal wasted wires, by Lemma 4.3.4 the map s β ∶ Z out G G X out is surjective.
Write X out = {x 1 , . . . , x r }, so Z out = ∐ r i=1 s −1 β (x i ) with each s −1 β (x i ) non-empty.
If r = 0, then X out = ∅ = Z out , and β 3 is a colored unit.
Suppose r > 0. We will decompose β 3 as in the picture:
X x 1 ⋮ xr V in = X in s −1 β (x 1 ) s −1 β (xr) ⋮ ⋱ ⋮ V out = Z out β 3 = σ 1 ○ ⋯ ○ σ r
The outermost gray box will be called V 1 , and the innermost gray box will be called V r−1 below. To define such a decomposition, we first need some definitions.
(1) For 0 ≤ j ≤ r define the box V j ∈ Box S with V in j = V in = X in and V out j = {x 1 , . . . , x j } ∅ if j = 0 ∐ r ∐ i=j+1 s −1 β (x i ) ∅ if j = r . Note that V out 0 = Z out = V out , so V 0 = V. Also, V out r = X out , so V r = X. (2) For 1 ≤ j ≤ r define the wiring diagram σ j ∈ WD V j−1 V j
with no delay nodes and with supplier assignment
Dm σ j = V out j−1 ∐ V in j = {x 1 , . . . , x j−1 } ∐ r ∐ i=j s −1 β (x i ) ∐ X in sσ j Sp σ j = V in j−1 ∐ V out j = X in ∐ {x 1 , . . . , x j } ∐ r ∐ i=j+1 s −1 β (x i )
given by
s σ j (z) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x j if z ∈ s −1 β (x j ); z otherwise.
The iterated operadic composition
σ 1 ○ ⋯ ○ σ r ∈ WD V 0 Vr = WD V X
has no delay nodes. To see that it is equal to β 3 , it remains to check that the former's supplier assignment s is equal to s β 3 .
• On X in ⊆ Dm σ r the supplier assignment s is the composition of r identity functions, hence the identity function.
• On V out = Z out ⊆ Dm σ 1 the supplier assignment s sends elements in each
s −1 β (x i ) to x i .
So s is equal to s β 3 .
By Lemma 4.3.11 each σ j for 1 ≤ j ≤ r is either a colored unit or an iterated operadic composition of out-splits. Therefore, β 3 = σ 1 ○ ⋯ ○ σ r is either a colored unit or an iterated operadic composition of out-splits.
Summary of Chapter 4
(1) A wiring diagram with no input boxes and no delay nodes is generated by the empty wiring diagram and a finite number of 1-wasted wires.
(2) Every wiring diagram ψ has a decomposition
ψ = π 1 ○ β 1 ○ β 2 ○ β 3 ○ ϕ in which:
• π 1 is generated by 1-loops;
• β 1 is generated by 1-wasted wires;
• β 2 is generated by in-splits;
• β 3 is generated by out-splits;
• ϕ is either the empty wiring diagram or is generated by 2-cells and 1-delay nodes.
Finite Presentation
Fix a class S, with respect to which the Box S -colored operad WD of wiring diagrams is defined (Theorem 2.3.11). The main purpose of this chapter is to establish finite presentations for the operad WD of wiring diagrams and its variants WD • and WD 0 . For the operad WD, our finite presentation means the following two statements.
(1) The 8 generating wiring diagrams (Def. 3.1.9) generate the operad WD. This means that every wiring diagram can be expressed as a finite iterated operadic composition involving only generating wiring diagrams.
(2) If a wiring diagram can be operadically generated by the generating wiring diagrams in two different ways, then there exists a finite sequence of elementary equivalences from the first iterated operadic composition to the other one. An elementary equivalence is induced by either an elementary relation (Def. 3.3.30) or an operad associativity/unity axiom for the generating wiring diagrams.
In Chapter 6 we will use these finite presentations to describe algebras over the operads WD, WD • , and WD 0 in terms of finitely many generating structure maps and generating axioms corresponding to the generating wiring diagrams and elementary relations. In Section 6.3 we will use the finite presentation for WD-algebras to study the propagator algebra. In Section 6.7 we will use the finite presentation for WD 0 -algebras to study the algebra of open dynamical systems.
In Section 5.1 we establish the first part of the finite presentation theorem for WD by showing that every wiring diagram has a stratified presentation (Theorem 5.1.11). A stratified presentation (Def. 5.1.9) is a highly structured iterated operadic composition of the generating wiring diagrams. The proof of the second part of the finite presentation theorem also requires the use of stratified presentations.
87
In Section 5.2 we establish the second part of the finite presentation theorem for WD. We show that any two presentations of the same wiring diagram in terms of generating wiring diagrams are connected by a finite sequence of elementary equivalences (Theorem 5.2.11).
In Section 5.3 we establish a finite presentation for the operad WD • of wiring diagrams without delay nodes, which we call normal wiring diagrams. Normal wiring diagrams appeared in Spivak's study of mode-dependent networks and dynamical systems [Spi15,Spi15b].
In Section 5.4 we restrict further and establish a finite presentation for the operad WD 0 of wiring diagrams without delay nodes and whose supplier assignments are bijections. We call them strict wiring diagrams. They appeared in [VSL15]. We will use strict wiring diagrams in Section 6.7 to study the algebra of open dynamical systems.
Stratified Presentation
In this section, we define a stratified presentation and show that every wiring diagram has a stratified presentation (Theorem 5.1.11). We also need the concept of a simplex to discuss generators and relations in the operad WD of wiring diagrams.
Motivation 5.1.1. In plain language, a simplex is a finite parenthesized word whose alphabets are generating wiring diagrams, in which each pair of parentheses has a well defined associated ○ i -composition. In particular, a simplex has a well defined operadic composition. As we have seen in Chapter 3, it is often possible to express a wiring diagram as an operadic composition of generating wiring diagrams in multiple ways. In other words, a wiring diagram can have many different simplex presentations. We now start to develop the necessary language to say precisely that any two such simplex presentations of the same wiring diagram are equivalent in some way.
Definition 5.1.2. Suppose n ≥ 1. An n-simplex Ψ and its composition Ψ ∈ WD are defined inductively as follows.
(1) A 1-simplex is a generating wiring diagram (Def. 3.1.9) ψ. Its composition ψ is defined as ψ itself.
(2) Suppose n ≥ 2 and that k-simplices for 1 ≤ k ≤ n − 1 and their compositions in WD are already defined. An n-simplex is a tuple
Ψ = ψ, i, φ consisting of • an integer i ≥ 1,
• a p-simplex ψ for some p ≥ 1, and • a q-simplex φ for some q ≥ 1 such that:
(i) p + q = n;
(ii) the operadic composition
Ψ def == ψ ○ i φ (5.1.2.1) is defined in WD (Def. 2.3.4). The wiring diagram Ψ in (5.1.2.1) is the composition of Ψ.
A simplex in WD is an m-simplex in WD for some m ≥ 1. We say that a simplex Ψ is a presentation of the wiring diagram Ψ .
Notation 5.1.3. To simplify notations, we will sometimes use the right side of (5.1.2.1) to denote a simplex. To simplify notations even further, we may even just list the generating wiring diagrams (ψ 1 , . . . , ψ n ) in a simplex in the order in which they appear in the composition (5.1.2.1), omitting all the pairs of parentheses and the operadic compositions from the notations. In the next three Examples, every ψ i denotes a generating wiring diagram, and we will use Notation 5.1.3.
Example 5.1.5. A 2-simplex has the form ψ 1 , i, ψ 2 , which we abbreviate to ψ 1 ○ i ψ 2 , for some integer i ≥ 1. For instance, suppose d ∈ S, and X is the box with X in = {d} = X out . Suppose Y is an arbitrary box. Then there is a 2-simplex
θ X,Y , 1, δ d in which θ X,Y is a 2-cell (Def. 3.1.4) and δ d is a 1-delay node (Def. 3.1.2). Its compo- sition θ X,Y ○ 1 δ d is the wiring diagram d Y θ X,Y ○ 1 δ d in WD X∐Y Y with one delay node.
Example 5.1.6. A 3-simplex is an iterated operadic composition in WD of the form
(ψ 1 ○ i ψ 2 ) ○ j ψ 3 or ψ 1 ○ i (ψ 2 ○ j ψ 3 )
for some integers i, j ≥ 1. Once again these are really abbreviations for the 3simplices
(ψ 1 , i, ψ 2 ) , j, ψ 3 or ψ 1 , i, (ψ 2 , j, ψ 3 ) .
For instance, continuing the example above, suppose λ X∐Y,d ∈ WD Y X∐Y is a 1-loop (Def. 3.1.5). Then there is a 3-simplex
λ X∐Y,d , 1, θ X,Y , 1, δ d whose composition λ X∐Y,d ○ 1 (θ X,Y ○ 1 δ d ) is the wiring diagram d Y in WD Y Y with one delay node. Example 5.1.7. A 4-simplex is an iterated operadic composition in WD of the form (ψ 1 ○ i ψ 2 ) ○ j ψ 3 ○ k ψ 4 , ψ 1 ○ i (ψ 2 ○ j ψ 3 ) ○ k ψ 4 , (ψ 1 ○ i ψ 2 ) ○ j (ψ 3 ○ k ψ 4 ), ψ 1 ○ i (ψ 2 ○ j ψ 3 ) ○ k ψ 4 , or ψ 1 ○ i ψ 2 ○ j (ψ 3 ○ k ψ 4 )
for some integers i, j, k ≥ 1. For instance, continuing the previous example, suppose
Y out = {y} and Z is a box such that Z out = {z, z ′ } with v(z) = v(z ′ ) = v(y) and that Z (z = z ′ ) = Y. Suppose σ Z,z,z ′ ∈ WD Z Y is an out-split (Def. 3.1.7). Then there is a 4-simplex σ Z,z,z ′ , 1, λ X∐Y,d , 1, (θ X,Y , 1, δ d ) whose composition σ Z,z,z ′ ○ 1 λ X∐Y,d ○ 1 (θ X,Y ○ 1 δ d ) is the wiring diagram d y z z ′ Y in WD Z Y with one delay node.
In Section 5.2 we will show that any two presentations of the same wiring diagram are equivalent in a certain way. For this purpose, we will need a more structured kind of presentation.
Motivation 5.1.8. If we think of a simplex as a parenthesized word whose alphabets are generating wiring diagrams, then the stratified simplex in the next definition is a word where the same alphabets must occur in a consecutive string. For example, all the 1-loops must occur together as a string (λ 1 , . . . , λ n ). Furthermore, we can even insist that these strings for different types of generating wiring diagrams occur in a specific order, with name changes and 1-loops at the top and with 1-delay nodes and 2-cells at the bottom.
Definition 5.1.9. A stratified simplex in WD is a simplex in WD (Def. 5.1.2) of one of the following two forms, where Notation 5.1.3 is used:
(1) ω, ǫ , where:
• ω is a possibly empty string of 1-wasted wires (Def. 3.1.8);
• ǫ is the empty wiring diagram (Def. 3.1.1).
(2) τ, λ, ω, σ * , σ * , θ, δ , where: • τ is a name change (Def. 3.1.3);
• λ is a possibly empty string of 1-loops (Def. 3.1.5);
• ω is a possibly empty string of 1-wasted wires;
• σ * is a possibly empty string of in-splits (Def. 3.1.6);
• σ * is a possibly empty string of out-splits (Def. 3.1.7);
• θ is a possibly empty string of 2-cells (Def. 3.1.4);
• δ is a possibly empty string of 1-delay nodes (Def. 3.1.2).
We call these stratified simplices of type (1) and of type (2), respectively. If Ψ is a stratified simplex, then we call it a stratified presentation of the wiring diagram Ψ .
Remark 5.1.10. Stratified simplices of type (1) and of type (2) are mutually exclusive. Indeed, the composition of a stratified simplex of type (1) has no input boxes and no delay nodes. On the other hand, the composition of a stratified simplex of type (2) either has at least one input box or at least one delay node or both.
Using the decompositions in the previous Chapter, we now observe that the generating wiring diagrams generate the operad WD of wiring diagrams in a highly structured way.
Theorem 5.1.11. Every wiring diagram has a stratified presentation (Def. 5.1.9).
Proof. Suppose ψ ∈ WD Y X is a general wiring diagram as in Assumption 4.1.1 with input boxes X = (X 1 , . . . , X N ) and delay nodes DN ψ = {d 1 , . . . , d r }. Recall Notation 3.3.1 for (iterated) ○ 1 .
If N = r = 0, then ψ has a stratified presentation of type (1) by Lemma 4.1.3.
Next suppose N + r ≥ 1. We use the decomposition ψ = α ○ ϕ (4.1.7.1) and show that ψ has a stratified presentation of type (2) using the following observations.
(1) If N + r = 1, then ϕ is either a colored unit, which can be ignored in a simplex by Lemma 2.3.6, or a 1-delay node by Lemma 4.1.8.
(2) If N + r ≥ 2, then ϕ has a stratified presentation θ, δ consisting of 2-cells and 1-delay nodes by Lemma 4.1.10 and by the equivalence between γ and the ○ i -compositions (2.1.12.1).
(3) By definition α (4.1.6.1) has one input box and no delay nodes. There is a decomposition α = π 1 ○ π 2 by Lemma 4.2.6. The outer wiring diagram π 1 is either a colored unit or has a stratified presentation (λ) consisting of 1-loops by Lemma 4.2.7.
(4) Furthermore, by Lemma 4.3.5 there is a decomposition π 2 = β 1 ○ β 2 ○ β 3 in which:
• β 1 is a colored unit or has a stratified presentation (ω) consisting of 1-wasted wires by Lemma 4.3.6. • β 2 is a colored unit or has a stratified presentation (σ * ) consisting of in-splits by Lemma 4.3.9. • β 3 is a colored unit or has a stratified presentation (σ * ) consisting of out-splits by Lemma 4.3.12.
Using the decomposition
ψ = π 1 ○ β 1 ○ β 2 ○ β 3 ○ ϕ
together with Convention 4.2.4, we obtain the desired stratified presentation of type (2) for ψ when N + r ≥ 1.
Finite Presentation for Wiring Diagrams
The purpose of this section is to establish the second part of the finite presentation theorem for the operad WD. In what follows, we will regard each operad associativity or unity axiom as an equality. We remind the reader of Notation 5.1.3 regarding simplices in WD.
Motivation 5.2.1.
Recall that a simplex is essentially a parenthesized word whose alphabets are generating wiring diagrams. In the next definition, we develop the precise concept through which one simplex presentation of a wiring diagram may be replaced by another. We only allow replacement of strings within a simplex corresponding to either one of the 28 elementary relations or an operad associativity/unity axiom. When such a replacement within a simplex is possible, we say that the two simplices are elementarily equivalent. (1) A subsimplex of Ψ is a simplex in WD defined inductively as follows.
• If Ψ is a 1-simplex, then a subsimplex of Ψ is Ψ itself.
• Suppose n ≥ 2 and Ψ = ψ, i, φ for some i ≥ 1, p-simplex ψ, and q- (3) Suppose Φ is another simplex in WD. Then Ψ and Φ are said to be equivalent if their compositions are equal; i.e., Ψ = Φ ∈ WD.
simplex φ with p + q = n. Then a subsimplex of Ψ is -a subsimplex of ψ, -a subsimplex of φ, or -Ψ itself. If Ψ ′ is a subsimplex of Ψ, then we write Ψ ′ ⊆ Ψ.
(4) Suppose:
• Ψ ′ ⊆ Ψ is an elementary subsimplex corresponding to one side of R, which is either an elementary relation or an operad associativity/unity axiom for the generating wiring diagrams. • Ψ" is the simplex given by the other side of R.
• Ψ 1 is the simplex obtained from Ψ by replacing the subsimplex Ψ ′ by Ψ". We say that Ψ and Ψ 1 are elementarily equivalent. Note that elementarily equivalent simplices are also equivalent.
(5) If Ψ and Φ are elementarily equivalent, we write Ψ ∼ Φ and call this an elementary equivalence.
(6) Suppose Ψ 0 , . . . , Ψ r are simplices for some r ≥ 1 and that there exist elementary equivalences
Ψ 0 ∼ Ψ 1 ∼ ⋯ ∼ Ψ r .
Then we say that Ψ 0 and Ψ r are connected by a finite sequence of elementary equivalences. Note that in this case Ψ 0 and Ψ r are equivalent.
Remark 5.2.3.
In the definition of an elementary subsimplex and an elementary equivalence, we did not use the operad equivariance axiom (2.1.10.7). The reason is that the associativity and commutativity properties of 2-cells-namely, the elementary relations (3.3.9.1) and (3.3.10.1)-are enough to guarantee the operad equivariance axiom involving only the generating wiring diagrams.
Example 5.2.4. In a 3-simplex (ψ 1 ○ i ψ 2 ) ○ j ψ 3 , both (ψ 1 , ψ 2 ) = ψ 1 ○ i ψ 2 and (ψ 1 , ψ 2 , ψ 3 ) = (ψ 1 ○ i ψ 2 ) ○ j ψ 3
are subsimplices. However, (ψ 2 , ψ 3 ) is not a subsimplex.
Example 5.2.5. A given wiring diagram may have many different equivalent presentations. For example, suppose X ∈ Box S . Then the 1-simplex consisting of the X-colored unit 1 X (2.3.2.1) is elementarily equivalent to:
(1) the 2-simplex θ X,∅ ○ 2 ǫ by (3.3.8.1);
(2) the 3-simplex (λ Z,x ) ○ (ω Z,x 1 ) ○ σ Y,x 1 ,x 2 by (3.3.20.1); (3) the 2-simplex σ Y,x,y ○ ω Y,y by (3.3.25.1).
Any two of these three simplices are connected by a finite sequence of elementary equivalences. Note that elementarily equivalent simplices may have different lengths.
Example 5.2.6. Suppose:
• θ X,Y ∈ WD X∐Y X,Y is a 2-cell (Def. 3.1.4). • θ V,W ∈ WD X V,W is a 2-cell with X = V ∐ W ∈ Box S . • σ T,t 1 ,t 2 ∈ WD Y T is an in-split (Def. 3.1.6).
Then the 3-simplices
θ X,Y ○ 1 θ V,W ○ 3 σ T,t 1 ,t 2 and θ X,Y ○ 2 σ T,t 1 ,t 2 ○ 1 θ V,V W T X in X out Y in Y out
can be created from θ X,Y by substituting in the two gray boxes in either order.
Convention 5.2.7.
In what follows, to simplify the presentation, elementary equivalences corresponding to an operad associativity/unity axiom-(2.1.10.3), (2.1.10.4), (2.1.10.5), or (2.1.10.6)-for the generating wiring diagrams will often be applied tacitly wherever necessary. For instance, an elementary equivalence given by replacing one of the 3-simplices in Example 5.2.6 by the other one will often not be mentioned explicitly.
Our next goal is to show that any two equivalent simplices are connected by a finite sequence of elementary equivalences. In other words, with respect to the generating wiring diagrams, the 28 elementary relations and the operad associativity/unity axioms-(2.1.10.3), (2.1.10.4), (2.1.10.5), and (2.1.10.6)-for the generating wiring diagrams generate all the relations in WD. During the first reading, the reader may wish to skip the proofs of the following three Lemmas.
The first step is to show that every simplex is connected to a stratified simplex in the following sense. Therefore, after a finite sequence of elementary equivalences, we may assume that there is at most one name change in Ψ, which is the left-most entry. If there are further elementary equivalences later that create name changes, we will perform the same procedure without explicitly mentioning it.
The empty wiring diagram ǫ ∈ WD ∅ (Def. 3.1.1) and the 1-delay nodes δ d ∈ WD d (Def. 3.1.2) have no input boxes, so no operadic composition of the forms ǫ ○ i − or δ d ○ i − can be defined. Therefore, after a finite sequence of elementary equivalences corresponding to the horizontal associativity axiom (2.1.10.3), we may assume that Ψ has the form τ, Ψ 1 , ǫ, δ in which:
• τ is a name change;
• all the 1-delay nodes δ are at the right-most entries;
• all the empty wiring diagrams ǫ are just to their left;
• Ψ 1 is either empty or is a subsimplex involving 2-cells (Def. 3.1.4), 1-loops (Def. 3.1.5), in-splits (Def. 3.1.6), out-splits (Def. 3.1.7), and 1-wasted wires (Def. 3.1.8).
Next we use the elementary relations (3.3.11.1)-(3.3.14.1) to move all the 2-cells in Ψ to just the left of ǫ. Then we use the elementary relations (3.3.16.1)-(3.3.18.1) to move all the remaining 1-loops to just the right of the name change τ. After that, we use the elementary relations (3.3.24.1) and (3.3.28.1) to move all the 1-wasted wires to just the right of the 1-loops. Then we use the elementary relation (3.3.23.1) to move all the in-splits to just the right of the 1-wasted wires. So after a finite sequence of elementary equivalences, we may assume that the simplex Ψ has the form τ, λ, ω, σ * , σ * , θ, ǫ, δ .
(5.2.8.1)
If the string ǫ of empty wiring diagrams is empty, then we are done because this is now a stratified simplex of type (2).
So suppose the string ǫ in (5.2.8.1) is non-empty. Using finitely many elementary equivalences corresponding to the elementary relations (3.3.8.1)-(3.3.10.1), we may cancel all the unnecessary empty wiring diagrams in (5.2.8.1). If there are no empty wiring diagrams left after the cancellation, then we have a stratified simplex of type (2).
Suppose that, after the cancellation in the previous paragraph, the resulting string ǫ is still non-empty. Then it must contain a single empty wiring diagram ǫ, and there are no 2-cells θ and no 1-delay nodes δ in the resulting simplex Ψ. Since the output box of ǫ is the empty box, the current simplex Ψ cannot have any 1-loops λ, in-splits σ * , or out-splits σ * . Therefore, in this case the simplex (5.2.8.1) has the form
τ, ω, ǫ . (5.2.8.2)
There are now two cases. First suppose the string ω in (5.2.8.2) is empty. Since the output box of ǫ is the empty box, in the simplex (τ, ǫ) the name change τ must be the colored unit of the empty box. So by the left unity axiom (2.1.10.5), the simplex (1 ∅ , ǫ) is elementarily equivalent to the simplex (ǫ), which is a stratified simplex of type (1).
Next suppose the string ω in (5.2.8.2) is non-empty. Using finitely many elementary equivalences corresponding to the elementary relation (3.3.7.1), the simplex (5.2.8.2) is connected to a simplex of the form
ω, τ, ǫ (5.2.8.3)
with τ ○ ǫ as one of the operadic compositions. As in the previous case, the composition τ ○ ǫ forces τ to be the colored unit of the empty box. So the simplex (ω,
1 ∅ , ǫ) in (5.2.8.3) is elementarily equivalent to the simplex (ω, ǫ), which is a stratified sim- plex of type (1).
The next step is to show that equivalent stratified simplices are connected. We begin with stratified simplices of type (1). Proof. Suppose Ψ 1 = ω 1 , ǫ and Ψ 2 = ω 2 , ǫ are equivalent stratified simplices of type (1) with common composition ψ. Then ψ has no input boxes and no delay nodes, and its output box contains only external wasted wires as in Lemma 4.1.3. Each 1-wasted wire in each Ψ i creates one external wasted wire in ψ. So the 1wasted wire strings ω 1 and ω 2 have the same length. It follows that the simplices Ψ 1 and Ψ 2 are connected by a finite sequence of elementary equivalences corresponding to the elementary relation (3.3.29.1) and the vertical associativity axiom (2.1.10.4). • other elementary relations that move the generating wiring diagrams around the simplices, we may assume that there are no unnecessary generating wiring diagrams in these stratified simplices. Here unnecessary refers to either a colored unit or generating wiring diagrams whose (iterated) operadic composition is a colored unit.
The name change τ 1 in Ψ 1 has output box Y and input box uniquely determined by ψ, and the same is true for the name change τ 2 in Ψ 2 . It follows that τ 1 is equal to τ 2 . So we may assume that there are no name changes in the two stratified simplices Ψ i .
The string of delay nodes δ i in each simplex Ψ i represents the set of delay nodes in ψ. Therefore, the two Ψ i without their strings of delay nodes are also equivalent. Moreover, if these simplices without delay nodes are connected by a finite sequence of elementary equivalences, then so are the two Ψ i themselves by the horizontal associativity axiom (2.1.10.3). So we may assume that the wiring diagram ψ and the two simplices Ψ i have no delay nodes. At this stage, each stratified simplex Ψ i has the form
λ i , ω i , σ i * , σ * i , θ i .
The composition of the string of 2-cells θ i in each simplex Ψ i has the same input boxes as ψ. So using finitely many elementary equivalences corresponding to the elementary relations (3.3.9.1) and (3.3.10.1), we may assume that the wiring diagram ψ has only one input box and that the simplices Ψ i have no 2-cells. At this stage, each stratified simplex Ψ i has the form
λ i , ω i , σ i * , σ * i .
Observe that for each i ∈ {1, 2}, the string of 1-wasted wires ω i in the simplex Ψ i corresponds to precisely the set ψ w − ∐ ψ w + of external and internal wasted wires in the wiring diagram ψ (Def. 2.2.13). Here an internal wasted wire in ψ is created by applying a 1-loop to a 1-wasted wire as in (3.2.3.1). Therefore, using finitely many elementary equivalences corresponding to the elementary relations (3.3.15.1) and (3.3.18.1), we may assume that the wiring diagram ψ and the two simplices Ψ i have no 1-wasted wires. At this stage, each stratified simplex Ψ i has the form
λ i , σ i * , σ * i .
Using finitely many elementary equivalences corresponding to the elementary relation (3.3.19.1), we may assume that each loop element in the wiring diagram ψ (Def. 4.2.1) corresponds to precisely one 1-loop λ in each simplex Ψ i . At this stage, the 1-loops in each simplex Ψ i are in bijection with the loop elements in ψ. Moreover, the two stratified subsimplices σ 1 * , σ * 1 ⊆ Ψ 1 and σ 2 * , σ * 2 ⊆ Ψ 2 are equivalent. Therefore, using finitely many elementary equivalences corresponding to the elementary relation (3.3.15.1), we may assume that the wiring diagram ψ has no loop elements and that the simplices Ψ i have no 1-loops. So each stratified simplex Ψ i now has the form We are now ready for the finite presentation theorem for wiring diagrams. It describes the wiring diagram operad WD (Theorem 2.3.11) in terms of finitely many generators and finitely many relations.
σ i * , σ * i .
Finite Presentation for Normal Wiring Diagrams
In this section, we establish a finite presentation theorem for the operad of wiring diagrams without delay nodes. (1) A wiring diagram is said to be normal if its set of delay nodes is empty.
(2) The collection of normal wiring diagrams is denoted by WD • . If we want to emphasize S, then we will write WD S • . Example 5.3.2. Among the 8 generating wiring diagrams (section 3.1):
(1) A 1-delay node δ d (Def. 3.1.2) is not normal. (2) The empty wiring diagram ǫ (Def. 3.1.1), a name change τ X,Y (Def. 3.1.3), a 2-cell θ X,Y (Def. 3.1.4), a 1-loop λ X,x (Def. 3.1.5), an in-split σ X,x 1 ,x 2 (Def.
3.1.6), an out-split σ Y,y 1 ,y 2 (Def. 3.1.7), and a 1-wasted wire ω Y,y (Def. 3.1.8) are normal.
In particular, there is a proper inclusion WD • ⊊ WD. Furthermore, the 1-internal wasted wire ω X,x (Def. 3.2.1) is normal. (1) ϕ (4.1.5.1) is not normal, unless r = 0.
(2) ψ in (4.1.3.1), α (4.1.6.1), π 1 (4.2.5.1), π 2 (4. Proof. We can reuse the proof of Theorem 2.3.11-that WD is a Box S -colored operadas long as we know that the relevant structure is well-defined in WD • .
The collection WD • is closed under the equivariant structure map (2.3.1.1). Furthermore, each colored unit
1 Y (2.3.2.1) is in WD • .
Suppose both ϕ and ψ are normal wiring diagrams such that ϕ ○ i ψ ∈ WD is defined. Then ϕ ○ i ψ is also normal because
DN ϕ○ i ψ = DN ϕ ∐ DN ψ = ∅.
Therefore, Lemmas 2.3.6, 2.3.8, and 2.3.10 all apply to WD • to show that it is an operad.
Our next objective is to obtain a version of the finite presentation theorem for WD • . For this purpose, we will use the following definitions.
Definition 5.3.6. Consider the operad WD • of normal wiring diagrams.
(1) A normal generating wiring diagram is a generating wiring diagram (Def.
3.1.9) except for 1-delay nodes δ d (Def. 3.1.2).
(2) A normal simplex is defined as in Def. 5.1.2 using normal generating wiring diagrams and WD • in place of WD.
(3) A normal stratified simplex and a normal stratified presentation are defined as in Def. 5.1.9 with WD • in place of WD, except that a normal stratified simplex of type (2) has the form τ, λ, ω, σ * , σ * , θ .
(4) All of Def. 5.2.2 is repeated with normal generating wiring diagrams and WD • in place of WD.
The following result is the finite presentation theorem for normal wiring diagrams. So in the WD • versions of these Lemmas, we simply ignore all the delay nodes in the original proofs.
Finite Presentation for Strict Wiring Diagrams
In this section, we establish a finite presentation theorem for the operad of strict wiring diagrams. Such wiring diagrams are used in [VSL15] to study open dynamical systems.
Definition 5.4.1. Fix a class S.
(1) A wiring diagram (Def. 2.2.15) is said to be strict if (i) it is normal (Def. 5.3.1) and
(ii) its supplier assignment is a bijection.
(2) The collection of strict wiring diagrams is denoted by WD 0 . If we want to emphasize S, then we will write WD S 0 . Remark 5.4.2. What we call a strict wiring diagram is simply called a wiring diagram in [VSL15] (Def. 3.5). In [VSL15] (Remark 2.7) S is a set of representatives of isomorphism classes of second-countable smooth manifolds. The noninstantaneity requirement (2.2.13.2) in this case is called the no passing wires requirement in [VSL15]. As noted in [VSL15] (Remark 3.6), strictness implies the non-existence of external wasted wires, internal wasted wires (Def. 2.2.13), and split wires, i.e., multiple (at least two) demand wires having the same supply wire. So strict wiring diagrams are much simpler than a general wiring diagram.
Example 5.4.3. Among the 8 generating wiring diagrams (section 3.1):
(1) A 1-delay node δ d (Def. 3.1.2) is not normal (Def. 5.3.1), hence also not strict. (2) The empty wiring diagram ǫ (Def. 3.1.1), a name change τ X,Y (Def. 3.1.3), a 2-cell θ X,Y (Def. 3.1.4), and a 1-loop λ X,x (Def. 3.1.5) are strict.
(3) An in-split σ X,x 1 ,x 2 (Def. 3.1.6), an out-split σ Y,y 1 ,y 2 (Def. 3.1.7), and a 1wasted wire ω Y,y (Def. 3.1.8) are normal but not strict.
In particular, there are proper inclusions WD 0 ⊊ WD • ⊊ WD. Furthermore, the 1-internal wasted wire ω X,x (Def. 3.2.1) is normal but not strict. (2) The other 18 are normal but not strict. (1) ψ in (4.1.3.1) is normal but not strict.
(2) α (4.1.6.1), π 2 (4.2.5.2), β 1 (4.3.3.1), β 2 (4.3.3.2), and β 3 (4.3.3.3) are normal but not strict in general.
(3) π 1 (4. Proof. The argument is essentially identical to the proof of Prop. 5.3.5 with a minor modification. Suppose both ϕ and ψ are strict wiring diagrams such that ϕ ○ i ψ ∈ WD is defined. Then ϕ ○ i ψ is also strict. Indeed, we already know that it is normal. Next, one can check directly from the definition of the supplier assignment s ϕ○ i ψ (2.3.4.1) that it is a bijection because, in all cases, it is defined as a composition of the bijections s ϕ and s ψ . Therefore, Lemmas 2.3.6, 2.3.8, and 2.3.10 all apply to WD 0 to show that it is an operad.
Our next objective is to obtain a version of the finite presentation theorem for WD 0 . For this purpose, we will use the following definitions. (1) A strict generating wiring diagram means the empty wiring diagram ǫ (Def.
3.1.1), a name change τ X,Y (Def. 3.1.3), a 2-cell θ X,Y (Def. 3.1.4), or a 1-loop λ X,x (Def. 3.1.5).
(2) A strict simplex is defined as in Def. 5.1.2 using strict generating wiring diagrams and WD 0 in place of WD.
(3) A strict stratified simplex is a stratified simplex (Def. 5.1.9) of the form (ǫ) or τ, λ, θ .
(4) If Ψ is a strict stratified simplex, then we call it a strict stratified presentation of the strict wiring diagram Ψ . The following result is the finite presentation theorem for strict wiring diagrams.
Theorem 5.4.8. Consider the operad WD 0 of strict wiring diagrams.
(1) Every strict wiring diagram has a strict stratified presentation.
(2) Every strict wiring diagram can be obtained from finitely many strict generating wiring diagrams via iterated operadic compositions (Def. 2.1.10).
(3) Any two equivalent strict simplices are connected by a finite sequence of strict elementary equivalences.
Proof. As in the proof of Theorem 5.3.7, for statement (1), we reuse the proof of Theorem 5.1.11 while assuming the wiring diagram ψ is strict. In this case, ψ is either the empty wiring diagram ǫ or has a decomposition (using (4.1.7.1) and (4.2.6.1)) ψ = π 1 ○ π 2 ○ ϕ in which π 2 (4.2.5.2) is a name change. The desired strict stratified presentation follows from Convention 4.2.4 and the facts that:
• π 1 is either a colored unit or has a stratified presentation (λ) by Lemma 4.2.7;
• ϕ is either a colored unit or has a stratified presentation (θ) by Lemma 4.1.10.
Statement (2) is a special case of statement (1).
For statement (3) we use the strict versions of the proofs of Lemma 5.2.8, Lemma 5.2.9, and Lemma 5.2.10. The key observation is that, in this case, only strict generating wiring diagrams and strict elementary equivalences are used in these proofs.
Summary of Chapter 5
(1) A simplex in WD is a finite non-empty parenthesized word of generating wiring diagrams in which each pair of parentheses is equipped with an operadic ○ i -composition.
(2) A stratified simplex in WD is a simplex of one of the following two forms.
• ω, ǫ
• τ, λ, ω, σ * , σ * , θ, δ
(3) Every wiring diagram has a stratified presentation.
(4) Two simplices are elementarily equivalent if one can be obtained from the other by replacing a subsimplex Ψ ′ by an equivalent simplex Ψ ′′ such that Ψ ′ = Ψ ′′ is either one of the twenty-eight elementary relations in WD or an operad associativity/unity axiom involving only the eight generating wiring diagrams.
(5) Any two simplex presentations of a given wiring diagram are connected by a finite sequence of elementary equivalences.
(6) A normal wiring diagram is a wiring diagram with no delay nodes.
(7) The operad WD • of normal wiring diagrams satisfies a finite presentation theorem involving the seven normal generating wiring diagrams and the twenty-eight elementary relations.
(8) A strict wiring diagram is a wiring diagram with no delay nodes and whose supplier assignment is a bijection.
(9) The operad WD 0 of strict wiring diagrams satisfies a finite presentation theorem involving the four strict generating wiring diagrams and the eight strict elementary relations.
Chapter 6
Finite Presentation for Algebras over Wiring Diagrams
The main purpose of this chapter is to provide finite presentations for algebras over the operads WD (Theorem 2.3.11), WD • (Prop. 5.3.5), and WD 0 (Prop. 5.4.6). The advantage of such a finite presentation is that sometimes the general structure map of an operad algebra can be a bit difficult to write down and understand. On the other hand, our generating structure maps and generating axioms are all fairly easy to write down and understand, as we will illustrate with examples in Sections 6.3, 6.5, and 6.7.
In Section 6.1 we recall the basics of algebras over an operad.
In Section 6.2 we first define a WD-algebra in terms of 8 generating structure maps and 28 generating axioms corresponding to the generating wiring diagrams (Def. 3.1.9) and the elementary relations (Def. 3.3.30), respectively. Then we observe that this finite presentation for a WD-algebra is in fact equivalent to the general definition of a WD-algebra (Theorem 6.2.2). This is an application of the finite presentation theorem for the operad WD (Theorem 5.2.11).
In Section 6.3 we provide a finite presentation for the WD-algebra called the propagator algebra. In its original form, the propagator algebra was the main example in [RS13].
In Section 6.4 we observe that algebras over the operad WD • of normal wiring diagrams have a similar finite presentation with 7 generating structure maps and 105 28 generating axioms. In Section 6.5 we provide a finite presentation for the WD •algebra called the algebra of discrete systems. In its original form, this algebra was one of the main examples in [Spi15b].
In Section 6.6 we observe that algebras over the operad WD 0 of strict wiring diagrams admit a finite presentation with 4 generating structure maps and 8 generating axioms. In Section 6.7 we provide a finite presentation for the WD 0 -algebra called the algebra of open dynamical systems. In its original form, this algebra was one of the main examples in [VSL15].
Operad Algebras
Let us first recall the definition of an algebra over an operad. The following definition can be found in [Yau16] (Def. 13.2.3). In its original 1-colored topological form, it was first given in [May72]. Fix a class S as before.
Motivation 6.1.1. One can think of an algebra over an operad as a generalization of a module over a ring. Given a ring R, a left R-module M is equipped with structure maps r ∶ M G G M for each element r ∈ R that satisfy some axioms. In particular, these structure maps are associative in the sense that
(r 1 r 2 )(m) = r 1 (r 2 m)
for r 1 , r 2 ∈ R and m ∈ M. Furthermore, the multiplicative unit 1 R of R acts as the identity map, so 1 R (m) = m. For algebras over an operad, there is also an equivariance axiom because operads can model operations with multiple inputs. (1) For each c ∈ S, A is equipped with a class A c called the c-colored entry of A.
(2) For each d ∈ S, c = (c 1 , . . . , c n ) ∈ Prof(S), and ζ ∈ O d c , A is equipped with a structure map A c def == n ∏ i=1 A c i µ ζ G G A d (6.1.2.1)
in which an empty product, for the case n = 0, means the one-point set { * }.
This data is required to satisfy the following associativity, unity, and equivariance axioms.
Associativity: Suppose d c ∈ Prof(S) × S is as above with n ≥ 1, ζ 0 ∈ O d c , ζ i ∈ O c i b i for each 1 ≤ i ≤ n with b i ∈ Prof(S), and b = (b 1 , . . . , b n ) as in (2.1.3.2). Write ζ = γ ζ 0 ; ζ 1 , . . . , ζ n ∈ O d b . Then the diagram A b µ ζ G G = A d A b 1 × ⋯ × A b n ∏ µ ζ i G G A c 1 × ⋯ × A cn = A c µ ζ 0 y y (6.1.2.2)
is commutative.
Unity: For each c ∈ S, the structure map
A c µ 1c G G A c (6.1.2.3) is the identity map, where 1 c ∈ O c c is the c-colored unit of O. Equivariance: Suppose ζ ∈ O d c as in (6.1.2.1), σ ∈ Σ n , and ζ σ ∈ O d cσ is the image of ζ under the right action (2.1.3.1). Then the diagram A c µ ζ σ −1 G G A cσ µ ζ σ A d = G G A d (6.1.2.4)
is commutative. Here the top σ −1 permutes the factors of A c from the left, and cσ = c σ(1) , . . . , c σ(n) .
To simplify the notations, we will sometimes denote an O-algebra by just A and denote the structure map µ ζ by ζ.
Just as operads can be equivalently expressed in terms of the ○ i -compositions (Prop. 2.1.12), so can operad algebras.
Associativity: Suppose:
• d ∈ S, c = (c 1 , . . . , c n ) ∈ Prof(S) with n ≥ 1, and 1 ≤ i ≤ n;
• b ∈ Prof(S) and c ○ i b as in (2.1.10.2); • ζ ∈ O d c , ξ ∈ O c i b , and ζ ○ i ξ ∈ O d c○ i b . Then the diagram A c○ i b = µ ζ○ i ξ G G A d A (c 1 ,...,c i−1 ) × A b × A (c i+1 ,...,cn) (Id,µ ξ ,Id) G G A (c 1 ,...,c i−1 ) × A c i × A (c i+1 ,...,cn) = A c µ ζ y y (6.1.3.1)
is commutative. Notation 6.1.4. To simplify the notations, we will sometimes denote the structure map µ ζ by ζ. We will also write the composition in the diagram (6.1.3.1) as µ ζ ○ i µ ξ , called the ○ i -composition of µ ζ and µ ξ . So this associativity axiom states that
µ ζ○ i ξ = µ ζ ○ i µ ξ .
In other words, the structure map of the ○ i -composition ζ ○ i ξ is the ○ i -composition of the structure maps corresponding to ζ and ξ.
Using the associativity and the unity axioms in Def. Remark 6.1.5. In Def. 6.1.2 and Def. 6.1.3, each entry of an operad algebra has no structure beyond being a class. We could have just as easily defined operad algebras in a symmetric monoidal category, which is in fact the setting in [Yau16] (Def. 13.2.3). One simply replaces the product with the symmetric monoidal product and the one-point set with the monoidal unit. However, to keep the presentation simple, we chose to define operad algebras whose entries are just classes. This is sufficient for the main examples in Sections 6.3, 6.5, and 6.7. (1) For the associative operad As, an As-algebra with an underlying set is exactly a monoid.
(2) For the commutative operad Com, a Com-algebra with an underlying set is exactly a monoid whose multiplication is commutative.
Example 6.1.8 (Traffic Spaces and Probability Spaces). This is a continuation of Example 2.1.8, where we discussed the graph operation operad GrOp. Here we consider GrOp-algebras in the category of complex vector spaces; see Remark 6.1.5. In particular, a GrOp-algebra A has an underlying complex vector space, and all the structure maps are linear maps, with tensor products playing the roles of products. The graph operation • ∈ GrOp 0 , consisting of a single vertex and no edges, yields an element in A, also denoted by •. For a general graph operation G ∈ GrOp n , the corresponding structure map A ⊗n G G A is also denoted by G. We will write δ ∈ GrOp 1 for the graph operation consisting of a single vertex and a loop. a pair (A, ϕ) in which A is a GrOp-algebra and ϕ ∶ A G G C is a linear functional such that the following two conditions are satisfied.
A traffic space [Mal16] is
Unity and Diagonality: ϕ(•) = 1 and ϕ = ϕ ○ δ.
Input-Independence:
For each graph operation G ∈ GrOp n , the graph operation δ(G) ∈ GrOp n is obtained from G by identifying its input and output. Suppose G ′ is a graph operation obtained from δ(G) by choosing a different vertex as the input/output. Then ϕ ○ δ(G) = ϕ ○ G ′ .
For example, suppose G ∈ GrOp 4 is the graph operation on the left: Then δ(G) is the graph operation in the middle, and the graph operation on the right is an example of a G ′ .
Traffic spaces play an important role in (non-commutative) probability theory. Indeed, a non-commutative probability space, also known as a quantum probability space, is a pair (A, ϕ) in which:
(1) A is a unital C-algebra.
(2) ϕ ∶ A G G C is a unital linear functional.
A * -probability space is a non-commutative probability space (A, ϕ) in which:
(1) A is equipped with an anti-linear involution * such that (ab) * = b * a * for all a, b ∈ A.
(2) ϕ satisfies the positivity condition that ϕ(a * a) ≥ 0 for all a ∈ A.
Then a commutative * -probability space is an example of a traffic space, since the product of n elements in A is well-defined and is independent of the order of those elements. It is, furthermore, a * -algebra in the following sense. For each graph operation G ∈ GrOp n , its transpose G t is the graph operation obtained from G by reversing the direction of each edge and swapping the input and the output. Then G(a 1 ⊗ ⋯ ⊗ a n ) * = G t (a * 1 ⊗ ⋯ ⊗ a * n ) for all a 1 , . . . , a n ∈ A.
Algebras over the Operad of Wiring Diagrams
The purpose of this section is to provide a finite presentation for WD-algebras. We begin by defining a WD-algebra in terms of the generating wiring diagrams and the elementary relations. Immediately afterwards we will establish its equivalence with Def. 6.1.3 when O = WD. Recall that WD is a Box S -colored operad (Theorem 2.3.11).
Definition 6.2.1. A WD-algebra A consists of the following data. For each X ∈ Box S , A is equipped with a class A X called the X-colored entry of A. It is equipped with the following 8 generating structure maps corresponding to the generating wiring diagrams (Def. 3.1.9).
(1) Corresponding to the empty wiring diagram ǫ ∈ WD ∅ (Def. 3.1.1), it has a structure map * i.e., a chosen element in A d .
ǫ G G A ∅ ,(6.
(3) Corresponding to each name change τ X,Y ∈ WD Y X (Def. 3.1.3), it has a structure map
A X τ X,Y G G A Y (6.2.1.3)
that is, furthermore, the identity map if τ X,X is the colored unit 1 X (2.3.2.1).
(4) Corresponding to each 2-cell θ X,Y ∈ WD X∐Y X,Y (Def. 3.1.4), it has a structure map
A X × A Y θ X,Y G G A X∐Y .(A X σ X,x 1 ,x 2 G G A Y .
(6.2.1.6) (7) Corresponding to each out-split σ Y,y 1 ,y 2 ∈ WD Y X (Def. 3.1.7), it has a structure map
A X σ Y,y 1 ,y 2 G G A Y . (6.2.1.7) (8) Corresponding to each 1-wasted wire ω Y,y ∈ WD Y X (Def. 3.1.8), it has a structure map A X ω Y,y G G A Y .
(6.2.1.8)
The following 28 diagrams, called the generating axioms, which correspond to the elementary relations (Def. 3.3.30), are required to be commutative.
(1) In the setting of (3.3.2.1), the diagram
A X τ X,Y G G τ X,Z 4 4 ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ A Y τ Y,Z A Z (6.2.1.9)
is commutative.
(2) In the setting of (3.3.3.1), the diagram
A X × A Y θ X,Y G G (τ X,X ′ ,τ Y,Y ′ ) A X∐Y τ X∐Y,X ′ ∐Y ′ A X ′ × A Y ′ θ X ′ ,Y ′ G G A X ′ ∐Y ′ (6.2.1.10)
is commutative.
(3) In the setting of (3.3.4.1), the diagram
A X λ X,x G G τ X,Y A X∖x τ X∖x,Y∖y A Y λ Y,y G G A Y∖y (6.2.1.11)
is commutative.
(4) In the setting of (3.3.5.1), the diagram
A X σ X,x 1 ,x 2 G G τ X,Y A X ′ τ X ′ ,Y ′ A Y σ Y,y 1 ,y 2 G G A Y ′ is commutative.
(5) In the setting of (3.3.6.1), the diagram
A X ′ σ X,x 1 ,x 2 G G τ X ′ ,Y ′ A X τ X,Y A Y ′ σ Y,y 1 ,y 2 G G A Y is commutative.
(6) In the setting of (3.3.7.1), the diagram
A X ′ ω X,x G G τ X ′ ,Y ′ A X τ X,Y A Y ′ ω Y,y G G A Y
is commutative.
(7) In the setting of (3.3.8.1), the diagram
A X × * (Id,ǫ) G G = A X × A ∅ θ X,∅ A X = G G A X∐∅ (6.2.1.12)
is commutative.
(8) In the setting of (3.3.9.1), the diagram
A X × A Y × A Z (Id,θ Y,Z ) G G (θ X,Y ,Id) A X × A Y∐Z θ X,Y∐Z A X∐Y × A Z θ X∐Y,Z G G A X∐Y∐Z (6.2.1.13)
is commutative.
(9) In the setting of (3.3.10.1), the diagram
A Y × A X permute G G θ Y,X A X × A Y θ X,Y A Y∐X = G G A X∐Y (6.2.1.14)
is commutative.
(10) In the setting of (3.3.11.1), the diagram
A X × A Y θ X,Y G G (λ X,x ,Id) A X∐Y λ X∐Y,x A X∖x × A Y θ X∖x,Y G G A (X∐Y)∖{x} (6.2.1.15)
is commutative.
(11) In the setting of (3.3.12.1), the diagram
A X × A Y θ X,Y G G (σ X,x 1 ,x 2 ,Id) A X∐Y σ X∐Y,x 1 ,x 2 A X ′ × A Y θ X ′ ,Y G G A X ′ ∐Y
is commutative.
(12) In the setting of (3.3.13.1), the diagram
A X ′ × A Y θ X ′ ,Y G G (σ X,x 1 ,x 2 ,Id) A X ′ ∐Y σ X∐Y,x 1 ,x 2 A X × A Y θ X,Y G G A X∐Y is commutative.
(13) In the setting of (3.3.14.1), the diagram
A X ′ × A Y θ X ′ ,Y G G (ω X,x 0 ,Id) A X ′ ∐Y ω X∐Y,x 0 A X × A Y θ X,Y G G A X∐Y
is commutative.
(14) In the setting of (3.3.15.1), the diagram
A X λ X,x 2 G G λ X,x 1 A X∖x 2 λ X∖x 2 ,x 1 A X∖x 1 λ X∖x 1 ,x 2 G G A X∖x (6.2.1.16)
is commutative.
(15) In the setting of (3.3.16.1), the diagram
A X λ X,x G G σ X,x 1 ,x 2 A X∖x σ X∖x,x 1 ,x 2 A X ′ λ X ′ ,x G G A X ′ ∖x
is commutative.
(16) In the setting of (3.3.17.1), the diagram
A X ′ σ X,x 1 ,x 2 G G λ X ′ ,x A X λ X,x A X ′ ∖x σ X∖x,x 1 ,x 2 G G A X∖x is commutative.
(17) In the setting of (3.3.18.1), the diagram
A X ′ ω X,x 0 G G λ X ′ ,x A X λ X,x A X ′ ∖x ω X∖x,x 0 G G A X∖x is commutative.
(18) In the setting of (3.3.19.1), the diagram
A X σ Y,x 1 ,x 2 G G σ X,x 1 ,x 2 A Y λ Y,x(1) G G A Y∖x(1) λ Y∖x(1),x(2) A X ′ λ X ′ ,x G G A X * (6.2.1.17)
is commutative.
(19) In the setting of (3.3.20.1), the diagram
A X = G G σ Y,x 1 ,x 2 A X A Y ω Z,x 1 G G A Z λ Z,x y y is commutative.
(20) In the setting of (3.3.21.1), the diagram
A X σ X,x 2 ,x 3 G G σ X,x 1 ,x 2 A X 23 σ X 23 ,x 1 ,x 23 A X 12 σ X 12 ,x 12 ,x 3 G G A Y is commutative.(21)
In the setting of (3.3.22.1), the diagram
A X σ X,x 3 ,x 4 G G σ X,x 1 ,x 2 A X 34 σ X 34 ,x 1 ,x 2 A X 12 σ X 12 ,x 3 ,x 4 G G A Y is commutative.
(22) In the setting of (3.3.23.1), the diagram
A X σ Z,z 1 ,z 2 G G σ X,z 1 ,z 2 A Z σ Z,z 1 ,z 2 A W σ Y,z 1 ,z 2 G G A Y is commutative.
(23) In the setting of (3.3.24.1), the diagram
A X ω Z,z G G σ X,z 1 ,z 2 A Z σ Z,z 1 ,z 2 A W ω Y,z G G A Y is commutative.
(24) In the setting of (3.3.25.1), the diagram
A X ω Y,y G G = 4 4 ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ A Y σ Y,x,y A X is commutative.
(25) In the setting of (3.3.26.1), the diagram
A X σ Y 23 ,y 1 ,y 23 G G σ Y 12 ,y 12 ,y 3 A Y 23 σ Y,y 2 ,y 3 A Y 12 σ Y,y 1 ,y 2 G G A Y is commutative.
(26) In the setting of (3.3.27.1), the diagram
A X σ Y 34 ,y 1 ,y 2 G G σ Y 12 ,y 3 ,y 4 A Y 34 σ Y,y 3 ,y 4 A Y 12 σ Y,y 1 ,y 2 G G A Y is commutative.
(27) In the setting of (3.3.28.1), the diagram
A X ω W,y G G σ Z,y 1 ,y 2 A W σ Y,y 1 ,y 2 A Z ω Y,y G G A Y is commutative.
(28) In the setting of (3.3.29.1), the diagram
A X ω Y 2 ,y 1 G G ω Y 1 ,y 2 A Y 2 ω Y,y 2 A Y 1 ω Y,y 1 G G A Y is commutative.
This finishes the definition of a WD-algebra.
At this moment we have two definitions of a WD-algebra.
(1) On the one hand, in Def. 6.1.3 with O = WD, a WD-algebra has a structure map µ ζ (6.1.2.1) for each wiring diagram ζ. This structure map satisfies the associativity axiom (6.1.3.1) for a general operadic composition in WD, together with the unity and the equivariance axioms in Def. 6.1.2.
(2) On the other hand, in Def. 6.2.1 a WD-algebra has 8 generating structure maps and satisfies 28 generating axioms.
We now observe that these two definitions are equivalent, so WD-algebras indeed have a finite presentation as in Def. 6.2.1. The generating axiom (6.2.1.14) is a special case of the equivariance diagram (6.1.2.4), so it is commutative. Each of the other 27 generating axioms corresponds to an elementary relation that describes two different ways to construct the same wiring diagram as an iterated operadic composition of generating wiring diagrams. Each such generating axiom asserts that the two corresponding compositions of generating structure maps-defined using the composition in the diagram (6.1.3.1)are equal. The associativity axiom (6.1.3.1) of (A, µ) applied twice guarantees that two such compositions are indeed equal.
Conversely, suppose A is a WD-algebra in the sense of Def. 6.2.1, so it has eight generating structure maps that satisfy 28 generating axioms. We must show that it is a WD-algebra in the sense of Def. 6.1.3 For a wiring diagram ψ ∈ WD with a presentation Ψ (Def. 5.1.2), we define its structure map µ ψ (6.1.2.1) inductively as follows.
(1) If Ψ is a 1-simplex, then Ψ = (ψ), and ψ is a generating wiring diagram by the definition of a simplex. In this case, we define µ ψ as the corresponding generating structure map (6.2.1.1)-(6.2.1.8) of A.
(2) Inductively, suppose Ψ is an n-simplex for some n ≥ 2, so Ψ = (Φ, i, Θ) for some i ≥ 1, p-simplex Φ, and q-simplex Θ with p + q = n. Since 1 ≤ p, q < n, by the induction hypothesis, the structure maps µ Φ and µ Θ are already defined. Then we define the structure map
µ ψ = µ Φ ○ i µ Θ (6.2.2.1)
as in Notation 6.1.4.
By Theorem 5.1.11 every wiring diagram has a stratified presentation, hence a presentation. To see that the structure map µ ψ as above is well-defined, we need to show that the map µ ψ is independent of the choice of a presentation Ψ. Any two presentations of a wiring diagram are by definition equivalent simplices. By Theorem 5.2.11(2) (= the relations part of the finite presentation theorem for WD), any two equivalent simplices are either equal or are connected by a finite sequence of elementary equivalences. Therefore, it suffices to show that every elementary equivalence in WD yields a commutative diagram involving the generating structure maps of A, where ○ i is interpreted as in Notation 6.1.4. Recall from Def. 5.2.2 that an elementary equivalence comes from either an elementary relation or an operad associativity/unity axiom for the generating wiring diagrams.
It follows from a direct inspection that the operad associativity and unity axioms- By definition each of the 28 generating axioms of A corresponds to an elementary relation (Def. 3.3.30) and is a commutative diagram. Therefore, the structure map µ ψ for each wiring diagram ψ is well-defined.
It remains to check that the structure map µ satisfies the required unity, equivariance, and associativity axioms. The unity axiom (6.1.2.3) holds because it is part of the assumption on the generating structure map corresponding to a name change (6.2.1.3).
The associativity axiom (6.1.3.1) holds because the structure map µ ψ is defined above (6.2.2.1) by requiring that the diagram (6.1.3.1) be commutative.
For the equivariance axiom (6.1.2.4), first note that it is enough to check it when the wiring diagram in questioned is an iterated operadic composition of 2-cells. This is because 2-cells are the only binary generating wiring diagrams (Remark 3.1.10). All other generating wiring diagrams are either 0-ary or unary, for which equivariance is trivial.
So now suppose ζ in the equivariance axiom (6.1.2.4) is an iterated operadic composition of 2-cells. If ζ is a 2-cell and the permutation σ is the transposition (1 2) ∈ Σ 2 , then the equivariance axiom (6.1.2.4) is true by the generating axiom (6.2.1.14). The general case now follows from this special case using:
• the generating axiom (6.2.1.13) corresponding to the associativity property of 2-cells (3.3.9.1);
• the operad associativity axioms (2.1.10.3) and (2.1.10.4) when applied to 2-cells;
• the fact that the transpositions (i, i + 1) for 1 ≤ i ≤ n − 1 generate the symmetric group Σ n .
So (A, µ) is a WD-algebra in the sense of Def. 6.1.3.
Finite Presentation for the Propagator Algebra
As an illustration of Theorem 6.2.2, in this section we provide a finite presentation for the WD-algebra called the propagator algebra that was first introduced in [RS13] (Section 3). This finite presentation is about the structure maps, not the underlying sets. As explained in [RS13] (Section 3.4), the propagator algebra can be used, for example, to provide an iterative description of the Fibonacci sequence. In contrast to the original definition in [RS13], we will define the propagator algebra using finitely many generating structure maps and axioms-8 generating structure maps and 28 generating axioms to be exact. Since our generating structure maps are rather simple, our description of the propagator algebra is less involved than the original definition and verification in [RS13]. The equivalence between the two definitions of the propagator algebra is explained in Remark 6.3.23. We will often omit the subscript and just write ∂.
(2) A k-historical propagator from T to U is a function f ∶ Prof(T) G G Prof(U) such that: (i) f (t) = t + k for all t ∈ Prof(T); (ii) If t ∈ Prof(T) has length t ≥ 1, then ∂ U f (t) = f (∂ T t). (6.3.2.2)
The condition (6.3.2.2) is called historicity.
(3) The set of k-historical propagators from T to U is denoted by Hist k (T, U).
(4) A historical propagator from T to U is an m-historical propagator from T to U for some m ≥ 0. in which the right side starts with k entries of the base point * , is a k-historical propagator, called the k-moment delay function in [RS13].
Before we can define the propagator algebra, we first need to define its entries.
Definition 6.3.4. Suppose X = (X in , X out ) ∈ Box S (Def. 2.2.7).
(1) Define the pointed sets
X in v = x∈X in v(x) and X out v = x∈X out v(x) (6.3.4.1)
in which each v(x), a pointed set, is the value of x (Def. 2.2.6) and an empty product means the one-point set.
(2) Define the set
P X = Hist 1 X in v , X out v (6.3.4.2) of 1-historical propagators of type X.
So a 1-historical propagator of type X is a function that takes each X in v -profile to an X out v -profile whose length is one higher than before and that satisfies historicity (6.3.2.2). Example 6.3.5. Suppose (N, 1) is the pointed set of non-negative integers with base point 1. Consider the box X with X in and X out both equal to the one-point set * with value v( * ) = (N, 1).
X N N
A 1-historical propagator of type X is a function that takes each finite sequence of non-negative integers to a sequence of non-negative integers whose length is one higher than before and that satisfies historicity. For example:
(1) The 1-moment delay function in Example 6.3.3, given by is a 1-historical propagator of type X, denoted "Σ" in [RS13]. This 1historical propagator takes a sequence of non-negative integers to the sequence whose ith entry is the sum of the first i − 1 entries of the given sequence. For instance, we have f (1, 5, 6) = (0, 1, 6, 12).
(
Y N N N
A 1-historical propagator of type Y is a function that takes each finite sequence of ordered pairs of non-negative integers to a sequence of non-negative integers whose length is one higher than before and that satisfies historicity. For example: A 1-historical propagator of type Z is a function that takes each finite sequence of ordered triples of non-negative integers to a sequence of ordered pairs of nonnegative integers whose length is one higher than before and that satisfies historicity. For example, the function ℓ ∶ Prof (N × N × N) G G Prof (N × N) given by
ℓ (m 1 , m ′ 1 , m ′′ 1 ), . . . , (m n , m ′ n , m ′′ n ) = (1, 1), (m 1 , m ′ 1 + m ′′ 1 ), . . . , (m n , m ′ n + m ′′ n )
is a 1-historical propagator of type Z. For instance, we have ℓ (3, 1, 4), (7, 2, 9), (8, 5, 10) = (1, 1), (3, 5), (7, 11), (8, 15) .
We will come back to this example several times below.
The generating structure map of the propagator algebra associated to a 1-loop requires a few notations in its definition. So here we define this map first. The reader should keep in mind that the following definition as well as all the proofs in this section involve simple inductions on the length of some profiles.
Definition 6.3.8. Suppose X ∈ Box S , x − ∈ X in , and x + ∈ X out such that v(x − ) = v(x + ) as pointed sets. The box X ∖ x ∈ Box S is obtained from X by removing x = {x ± }. For t ∈ Prof X out v
, we will write:
(i) t x+ ∈ Prof v(x + ) for the profile obtained from t by taking only the v(x + )entry;
(ii) t ∖x+ ∈ Prof (X ∖ x) out v for the profile obtained from t by removing the v(x + )entry.
Suppose g ∈ P X (6.3.4.2). Define two functions
Prof (X ∖ x) in v λg G G Prof (X ∖ x) out v (6.3.8.1) and Prof (X ∖ x) in v Gg G G Prof v(x + ) (6.3.8.2)
with the properties (λg)(?) = ? + 1 = G g (?) (6.3.8.3) inductively as follows.
(i) For the empty profile, define
(λg)(∅) = g(∅) ∖x+ ∈ Prof (X ∖ x) out v G g (∅) = g(∅) x+ ∈ Prof v(x + ) . (6.3.8.4)
In each definition in (6.3.8.4), the first ∅ is the empty (X ∖ x) in v -profile, and the second ∅, to which g applies, is the empty X in v -profile. The profile g(∅) has length 1 because g ∈ Hist 1 X in v , X out v . So both (λg)(∅) and G g (∅) have length 1.
(ii) Inductively, suppose w ∈ Prof (X ∖ x) in v has length n ≥ 1. Define (λg)(w) = g w, G g (∂w) ∖x+ ∈ Prof (X ∖ x) out v G g (w) = g w, G g (∂w) x+ ∈ Prof v(x + ) . (6.3.8.5)
Here ∂ is the truncation (6.3.2.1), so the profile
G g (∂w) ∈ Prof v(x + ) = Prof v(x − )
is already defined and has length n by the induction hypothesis. In each definition in (6.3.8.5), w, G g (∂w) ∈ Prof X in v has length n, so its image under g has length n + 1. Therefore, both (λg)(w) and G g (w) have length n + 1.
We say that λg and G g are defined with respect to x = {x ± }.
Example 6.3.9. This is a continuation of Example 6.3.7, where the box Z has Z in =
{z 1 , z 2 , z 3 }, Z out = {z 1 , z 2 }, and all v(−) = (N, 1). For z 1 ∈ Z in and z 1 ∈ Z out , suppose Z ∖ z is the box obtained from Z by removing z = {z 1 , z 1 }. So we have (Z ∖ z) in v = v(z 2 ) × v(z 3 ) = N × N; (Z ∖ z) out v = v(z 2 ) = N. Z z 3 z 2 z 1 z 2 z 1 (Z ∖ z) in (Z ∖ z) out λ Z,z ∈ WD Z∖z Z
For the 1-historical propagator ℓ of type Z defined as
ℓ (m 1 , m ′ 1 , m ′′ 1 ), . . . , (m n , m ′ n , m ′′ n ) = (1, 1), (m 1 , m ′ 1 + m ′′ 1 ), . . . , (m n , m ′ n + m ′′ n ) , the functions Prof(N × N) λℓ G G Prof(N)
and (1, 1, . . . , 1), in which on the right side of G ℓ (m) there are n + 1 copies of 1. In particular, λℓ = h, the 1-historical propagator of type Y in Example 6.3.6. In this example, both λℓ and G ℓ are 1-historical propagators. This is not an accident, as we show in the next result.
Prof(N × N) G ℓ G G Prof(N)
The following observation will use Def. 6.3.2 about historical propagators. Lemma 6.3.10. In the context of Def. 6.3.8 with g ∈ P X , the following statements hold.
(1) G g ∈ Hist 1 (X ∖ x) in v , v(x + ) . (2) λg ∈ Hist 1 (X ∖ x) in v , (X ∖ x) out v = P X∖x .
Proof. In this proof, we will abbreviate G g to G. For statement (1), we are trying to show that G is a 1-historical propagator from (X ∖ x) in v to v(x + ). In view of the property (6.3.8.3), it remains to check historicity (6.3.2.2) for G, which we will do by induction. Suppose w ∈ Prof (X ∖ x) in v has length ≥ 1. If w = 1, then ∂w = ∅. Using the definitions (6.3.8.4) and (6.3.8.5) we have:
∂G(w) = ∂g w, G(∅) x+ = g ∂ (w, g(∅) x+ ) x+ by historicity of g = g(∅) x+ = G(∅).
Inductively, suppose w ≥ 2. Using the definition (6.3.8.5), we have: This finishes the proof of statement (1).
Statement (2) is proved by essentially the same argument as above, except that, in view of the definitions (6.3.8.4) and (6.3.8.5), the various right-most subscripts x + are replaced by ∖x + .
We are now ready to define the propagator algebra in terms of finitely many generating structure maps and generating axioms as in Def. 6.2.1. Most of the generating structure maps below are easily seen to be well-defined. The only exception is the generating structure map associated to a 1-loop, which we dealt with in Lemma 6.3.10 above. Definition 6.3.11. Define the propagator algebra P as the WD-algebra, in the sense of Def. 6.2.1, with X-colored entry
P X = Hist 1 X in v , X out v
as in (6.3.4.2) for each X ∈ Box S . Its 8 generating structure maps are defined as follows.
(1) Corresponding to the empty wiring diagram ǫ ∈ WD ∅ (Def (2) Corresponding to each 1-delay node δ d ∈ WD d (Def. 3.1.2) with d a pointed set, the structure map
{ * } δ d G G P d = Hist 1 d, d (6.3.11.2) sends * to the function (δ d * )(t) = * , t for each t ∈ Prof(d).
Here the * on the right is the base point in d. In other words, δ d * is the 1-moment delay function in Example 6.3.3.
(3) Corresponding to each name change τ X,Y ∈ WD Y X (Def. 3.1.3), the structure map
Hist 1 X in v , X out v = P X τ X,Y G G P Y = Hist 1 Y in v , Y out v (6.3.11.3)
is the identity map. Here we are using the fact that, if x ∈ X in ∐ X out and y ∈ Y in ∐ Y out corresponds to x under τ X,Y , then v(x) = v(y) as pointed sets.
(4) Corresponding to each 2-cell θ X,Y ∈ WD X∐Y X,Y (Def. 3.1.4), the structure map
P X × P Y = Hist 1 X in v , X out v × Hist 1 Y in v , Y out v θ X,Y P X∐Y = Hist 1 (X ∐ Y) in v , (X ∐ Y) out v = Hist 1 X in v × Y in v , X out v × Y out v (6.3.11.4) is given by θ X,Y ( f X , f Y ) = f X × f Y for f X ∈ Hist 1 X in v , X out v and f Y ∈ Hist 1 Y in v , Y out v .
(5) Corresponding to each 1-loop λ X,x ∈ WD X∖x X (Def. 3.1.5), the structure map P X λ X,x G G P X∖x (6.3.11.5) sends each g ∈ P X to λg ∈ P X∖x (6.3.8.1), which is well-defined by Lemma 6.3.10.
(6) Corresponding to each in-split σ X,x 1 ,x 2 ∈ WD Y X (Def. 3.1.6), the structure map
P X σ X,x 1 ,x 2 G G P Y (6.3.11.6)
is given by (σ X,x 1 ,x 2 g) (y) = g(πy)
for g ∈ P X and y ∈ Prof(Y in v ). Here πy ∈ Prof(X in v ) is the same as y except that its v(x 1 )-entry and v(x 2 )-entry are both given by the v(x 12 )-entry of y.
(7) Corresponding to each out-split σ Y,y 1 ,y 2 ∈ WD Y X (Def. 3.1.7), the structure map
P X σ Y,y 1 ,y 2 G G P Y (6.3.11.7)
is given by σ Y,y 1 ,y 2 g (y) = πg(y)
for g ∈ P X and y ∈ Prof(Y in v ) = Prof(X in v ). Here πg(y) ∈ Prof(Y out v )
is the same as g(y) ∈ Prof(X out v ) except that its v(y 1 )-entry and v(y 2 )-entry are both given by the v(y 12 )-entry of g(y).
(8) Corresponding to each 1-wasted wire ω Y,y ∈ WD Y X (Def. 3.1.8), the structure map
P X ω Y,y G G P Y (6.3.11.8)
is given by ω Y,y g (t) = g t ∖y for g ∈ P X and t ∈ Prof(Y in v ). Here t ∖y ∈ Prof(X in v ) is obtained from t by removing the v(y)-entry.
This finishes the definition of the propagator algebra P.
The following four examples continue Examples 6.3.7 and 6.3.9, where Z is the box with Z in = {z 1 , z 2 , z 3 }, Z out = {z 1 , z 2 }, and all v(−) = (N, 1). The 1-historical propagator ℓ ∈ P Z = Hist 1 (N 3 , N 2 ) is defined as
ℓ (m 1 , m ′ 1 , m ′′ 1 ), . . . , (m n , m ′ n , m ′′ n ) = (1, 1), (m 1 , m ′ 1 + m ′′ 1 ), . . . , (m n , m ′ n + m ′′ n )
. Let us consider the image of ℓ under some of the structure maps of P.
Z z 1 z 1 Y in Y out λ Z,z in WD Y Z , the structure map P Z = Hist 1 (N 3 , N 2 ) λ Z,z G G Hist 1 (N 2 , N) = P Y in (6.3.11.5), and m = (m 1 , m ′ 1 ), . . . , (m n , m ′ n ) ∈ Prof(Y in v ) = Prof(N 2 )
. As observed in Example 6.3.9 we have
(λ Z,z ℓ)(m) = h(m) = 1, m 1 + m ′ 1 , . . . , m n + m ′ n in Prof(Y out v ) = Prof(N),
where h ∈ P Y is the 1-historical propagator in Example 6.3.6. For instance, we have (λ Z,z ℓ) (2, 5), (4, 9), (3, 7) = (1, 7, 13, 10).
Example 6.3.13. Suppose W is the box with W in = Z in (z 1 = z 2 ) = {w, z 3 }, and W out = Z out . For the in-split (Def. 3.1.6) Z z 1 z 2 z 3 W in w W out σ Z,z 1 ,z 2 in WD W Z , the structure map P Z = Hist 1 (N 3 , N 2 ) σ Z,z 1 ,z 2 G G Hist 1 (N 2 , N 2 ) = P W
in (6.3.11.6) applied to the 1-historical propagator ℓ ∈ P Z is given as follows.
For
m = (m 1 , m ′ 1 ), . . . , (m n , m ′ n ) ∈ Prof(W in v ) = Prof(N 2 ), we have σ Z,z 1 ,z 2 ℓ (m) = ℓ(πm) = ℓ (m 1 , m 1 , m ′ 1 ), . . . , (m n , m n , m ′ n ) = (1, 1), (m 1 , m 1 + m ′ 1 ), . . . , (m n , m n + m ′ n ) in Prof(W out v ) = Prof(N 2 )
. For instance, we have σ Z,z 1 ,z 2 ℓ (2, 5), (4, 9), (3, 7) = (1, 1), (2, 7), (4, 13), (3, 10) .
Example 6.3.14. Suppose V is a box with V out = {v, v ′ , z 2 } such that V out (v = v ′ ) = Z out , and V in = Z in . For the out-split (Def. 3.1.7) Z z 1 z 2 V in V out v v ′ σ V,v,v ′ in WD V Z , the structure map P Z = Hist 1 (N 3 , N 2 ) σ V,v,v ′ G G Hist 1 (N 3 , N 3 ) = P V
in (6.3.11.7) applied to the 1-historical propagator ℓ ∈ P Z is given as follows.
For
m = (m 1 , m ′ 1 , m ′′ 1 ), . . . , (m n , m ′ n , m ′′ n ) ∈ Prof(V in v ) = Prof(N 3 ), we have σ V,v,v ′ ℓ (m) = πℓ(m) = π (1, 1), (m 1 , m ′ 1 + m ′′ 1 ), . . . , (m n , m ′ n + m ′′ n ) = (1, 1, 1), (m 1 , m 1 , m ′ 1 + m ′′ 1 ), . . . , (m n , m n , m ′ n + m ′′ n ) in Prof(V out v ) = Prof(N 3 ). For instance, we have σ V,v,v ′ ℓ (2, 5, 1)
, (4, 9, 10), (3, 7, 6) = (1, 1, 1), (2, 2, 6), (4, 4, 19), (3, 3, 13) .
Example 6.3.15. Suppose U is a box such that U ∖ u = Z for some u ∈ U in with v(u) = (N, 1). For the 1-wasted wire (Def. 3.1.8) Z U in u U out ω U,u in WD U Z , the structure map P Z = Hist 1 (N 3 , N 2 ) ω U,u G G Hist 1 (N 4 , N 2 ) = P U
in (6.3.11.8) applied to ℓ ∈ P Z is given as follows. For
m = (m 1 i , m 2 i , m 3 i , m 4 i ) n i=1 ∈ Prof(U in v ) = Prof v(u) × v(z 1 ) × v(z 2 ) × v(z 3 ) = Prof(N 4 ), we have ω U,u ℓ (m) = ℓ (m ∖u ) = ℓ (m 2 i , m 3 i , m 4 i ) n i=1 = (1, 1), (m 2 1 , m 3 1 + m 4 1 ), . . . , (m 2 n , m 3 n + m 4 n )
in Prof(U out v ) = Prof(N 2 ). For instance, we have ω U,u ℓ (2, 5, 1, 7), (4, 9, 10, 2), (3, 7, 6, 5) = (1, 1), (5, 8), (9, 12), (7, 11) .
The following observation is the finite presentation theorem for the propagator algebra.
Theorem 6.3.16. The propagator algebra in Def. 6.3.11 is actually a WD-algebra in the sense of Def. 6.2.1, hence also in the sense of Def. 6.1.3 by Theorem 6.2.2.
Proof. We need to check the 28 generating axioms in Def. 6.2.1. The 8 generating structure maps are all rather simple functions except for λ X,x (6.3.11.5). The only generating axioms that are not obvious are the ones that involve a composition of two such generating structure maps, namely (6.2.1.16) and (6.2.1.17). These two generating axioms are verified in Lemma 6.3.20 and Lemma 6.3.22 below.
In preparation for Lemma 6.3.20, we will need a few definitions and notations. Recall that the generating axiom (6.2.1.16) is really the WD-algebra manifestation of Prop. 3.3.15. The next definition is essentially the double-loop version of Def. 6.3.8. Definition 6.3.17. Suppose:
• X ∈ Box S , x 1 − = x 2 − ∈ X in , and x 1 + = x 2 + ∈ X out such that v(x i + ) = v(x i − ) as pointed sets for each i ∈ {1, 2}. • X ∖ x 1 , X ∖ x 2 , and X ∖ x ∈ Box S are obtained from X by removing x 1 = {x 1 ± }, x 2 = {x 2 ± }, and x = {x 1 ± , x 2 ± }, respectively. Suppose t ∈ Prof X out v . (i) Write v(x + ) = v(x 1 + ) × v(x 2 + ) and v(x − ) = v(x 1 − ) × v(x 2 − ). (ii) Write t x+ ∈ Prof v(x + )
for the profile obtained from t by taking only the v(x + )-entries.
(iii) For i ∈ {1, 2}, write t x i + ∈ Prof v(x i + ) for the profile obtained from t by taking only the v(x i + )-entry. (iv) The profile z x i + is also defined as long as
z ∈ Prof u∈T v(u) for some subset {x 1 + , x 2 + } ⊆ T ⊆ X out . (v) Write t ∖x+ ∈ Prof (X ∖ x) out v
for the profile obtained from t by removing the v(x + )-entries.
(vi) For i ∈ {1, 2}, write t ∖x i + ∈ Prof (X ∖ x i ) out v
for the profile obtained from t by removing the v(x i + )-entry.
Suppose g ∈ P X . Define two functions
Prof (X ∖ x) in v λ 2 g G G Prof (X ∖ x) out v (6.3.17.1)
and
Prof (X ∖ x) in v G 2 g G G Prof v(x + ) (6.3.17.2)
with the properties (λ 2 g)(?) = ?
+ 1 = G 2 g (?) (6.3.17.3)
inductively as follows.
(i) For the empty profile, define
(λ 2 g)(∅) = g(∅) ∖x+ ∈ Prof (X ∖ x) out v G 2 g (∅) = g(∅) x+ ∈ Prof v(x + ) . (6.3.17.4)
In each definition in (6.3.17.4), the first ∅ is the empty (X ∖ x) in v -profile, and the second ∅, to which g applies, is the empty X in v -profile. The profile g(∅) has length 1 because g ∈ Hist 1
X in v , X out v . So both (λ 2 g)(∅) and G 2 g (∅) have length 1. (ii) Inductively, suppose w ∈ Prof (X ∖ x) in v has length n ≥ 1. Define (λ 2 g)(w) = g w, G 2 g (∂w) ∖x+ ∈ Prof (X ∖ x) out v G 2 g (w) = g w, G 2 g (∂w) x+ ∈ Prof v(x + ) .
(6.3.17.5)
Here ∂ is the truncation (6.3.2.1), so the profile
G 2 g (∂w) ∈ Prof v(x + ) = Prof v(x − )
is already defined and has length n by the induction hypothesis. In each definition in (6.3.17.5), w, G 2 g (∂w) ∈ Prof X in v has length n, so its image under g has length n + 1. Therefore, both (λ 2 g)(w) and G 2 g (w) have length n + 1. Example 6.3.18. This is a continuation of Examples 6.3.7 and 6.3.9, where Z is the box with Z in = {z 1 , z 2 , z 3 }, Z out = {z 1 , z 2 }, and all v(−) = (N, 1). The 1-historical propagator ℓ ∈ P Z = Hist 1 (N 3 , N 2 ) is defined as
ℓ (m 1 , m ′ 1 , m ′′ 1 ), . . . , (m n , m ′ n , m ′′ n ) = (1, 1), (m 1 , m ′ 1 + m ′′ 1 ), . . . , (m n , m ′ n + m ′′ n ) . With w 1 = {z 1 , z 1 } and w 2 = {z 2 , z 2 }, suppose Z ∖ w is the box obtained from Z by removing {z 1 , z 2 , z 1 , z 2 }. Then (Z ∖ w) in v = v(z 3 ) = N and (Z ∖ w) out v = * ,
where * here is the one-point set (= empty product).
z 1 z 1 z 2 z 2 z 3 (Z ∖ w) in (Z ∖ w) out λ 2 ∈ WD Z∖w Z
The functions
Prof(N) The generating axiom (6.2.1.16) for the propagator algebra P claims that the diagram
P X λ X,x 2 G G λ X,x 1 P X∖x 2 λ X∖x 2 ,x 1 P X∖x 1 λ X∖x 1 ,x 2 G G P X∖x (6.3.18.1)
is commutative. We will consider the top-right composition, so let us use the abbreviations λ 2 = λ X,x 2 and λ 1 = λ X∖x 2 ,x 1 . (6.3.18.2)
For the proof of Lemma 6.3.20, we will need the following observation. It provides an explicit formula for the function G 2 g (6.3.17.2) in terms of G g (defined with respect to x 2 ) and G λ 2 g (6.3.8.2) (defined with respect to x 1 ) for each g ∈ P X . Lemma 6.3.19. In the context of Def. 6.3.17, suppose g ∈ P X and w ∈ Prof (X ∖ x) in v with length at least 1. Then the following equalities hold.
G 2 g (∂w) x 1 + = G λ 2 g (∂w) ∈ Prof v(x 1 + ) G 2 g (∂w) x 2 + = G g ∂w, ∂G λ 2 g (∂w) ∈ Prof v(x 2 + ) (6.3.19.1)
In the above equalities:
(1) ∂ is the truncation (6.3.2.1).
(2) λ 2 g ∈ P X∖x 2 by Lemma 6.3.10(2).
(
3) The function
Prof
(X ∖ x) in v G λ 2 g G G Prof v(x 1 + ) is defined with respect to x 1 = {x 1 + , x 1 − } (6.3.8.2). (4) The function Prof (X ∖ x 2 ) in v Gg G G Prof v(x 2 + ) is defined with respect to x 2 = {x 2 + , x 2 − }.
Proof. The proof of (6.3.19.1) is by induction on w ≥ 1. If w = 1, then ∂w = ∅. So by (6.3.8.4) and (6.3.17.4) we have
G λ 2 g (∅) = (λ 2 g)(∅) x 1 + = g(∅) ∖x 2 + x 1 + = g(∅) x 1 + = G 2 g (∅) x 1 + . Likewise, we have G g (∅) = g(∅) x 2 + = G 2 g (∅) x 2 + .
This proves the initial case w = 1.
For the induction step, suppose w ≥ 2. For the first equality in (6.3.19.1) we have:
G λ 2 g (∂w) = (λ 2 g) ∂w, G λ 2 g (∂ 2 w) x 1 + by (6.3.8.5) = g ∂w, G λ 2 g (∂ 2 w), G g ∂ 2 w, ∂G λ 2 g (∂ 2 w) x 1 + by (6.3.8.5) = g ∂w, G 2 g (∂ 2 w) x 1 + , G 2 g (∂ 2 w) x 2 + x 1 + by induction hypothesis = G 2 g (∂w) x 1 +
by (6.3.17.5).
In the second equality above, we used the fact that g(⋯) ∖x 2
+ x 1 + = g(⋯) x 1 + .
For the second equality in (6.3.19.1) we have:
G g ∂w, ∂G λ 2 g (∂w)
= G g ∂w, G λ 2 g (∂ 2 w) by Lemma 6.3.10(1) Proof. We will use the abbreviations (6.3.18.2). By the symmetry between x 1 = {x 1 ± } and x 2 = {x 2 ± }, it suffices to show that λ 1 λ 2 g (w) = (λ 2 g)(w) (6.3.20.1)
= g ∂w, G λ 2 g (∂ 2 w), G g ∂ 2 w, ∂G λ 2 g (∂ 2 w) x 2 + by (6.3.8.5) = g ∂w, G 2 g (∂ 2 w) x 1 + , G 2 g (∂ 2 w) x 2 + x 2 + by induction hypothesis = G 2 g(
for g ∈ P X and w ∈ Prof (X ∖ x) in v , where λ 2 g is defined in (6.3.17.1). We prove (6.3.20.1) by induction on the length w .
If w = 0, then by (6.3.8.4) and (6.3.17.4) the left side of (6.3.20.1) is:
λ 2 g)(∅) ∖x 1 + = g(∅) ∖x 2 + ∖x 1 + = g(∅) ∖x+ = (λ 2 g)(∅).
For the induction step, suppose w ≥ 1. Then the left side of (6.3.20.1) is:
(λ 2 g) w, G λ 2 g (∂w) ∖x 1 + by (6.3.8.5) = g w, G λ 2 g (∂w), G g ∂w, ∂G λ 2 g (∂w) ∖x+ by (6.3.8.5) = g w, G 2 g (∂w) x 1 + , G 2 g (∂w) x 2 + ∖x+ by (6.3.19.1) = (λ 2 g)(w) by (6.3.17.5).
This finishes the induction.
The proof of the generating axiom (6.2.1.17) in Lemma 6.3.22 below will use the following observation. Recall that the generating axiom (6.2.1.17) is really a WD-algebra manifestation of the generating relation (3.3.19.1).
Lemma 6.3.21. In the context of Prop. 3.3.19, recall that
X = Y (x 1 = x 2 ) , X ′ = X (x 1 = x 2 )
, and X * = X ∖ {x 12 , x 1 , x 2 }.
Consider the maps:
P X σ• = σ X,x 1 ,x 2 σ • = σ Y,x 1 ,x 2 G G P Y P X ′
Then for g ∈ P X and w ∈ Prof X * in v with length ≥ 1, the equality
G 2 σ • g (∂w) = G σ•g (∂w), G σ• g (∂w) (6.3.21.1)
holds. Here:
(1) σ • g ∈ P Y and σ • g ∈ P X ′ .
(2) The function
Prof X * in v G 2 σ • g G G Prof v(x 1 ) × v(x 2 )
is defined as in (6.3.17.2), starting with the box Y and the wires x 1 = x 2 ∈ Y out and x 1 = x 2 ∈ Y in .
(3) The function
Prof X * in v Gσ • g G G Prof v(x 12 )
is defined as in (6.3.8.2), starting with the box X ′ and the wires x 12 ∈ X ′in and x 12 ∈ X ′out .
Proof. The proof is by induction on the length of w. If w = 1, the ∂w = ∅. So we have:
G 2 σ • g (∅) = (σ • g)(∅) {x 1 ,x 2 } by (6.3.17.4) = g(∅) x 12 , g(∅)
x 12 by (6.3.11.7)
= (σ • g)(∅) x 12 , (σ • g)(∅) x 12 by (6.3.11.6) = G σ•g (∅), G σ•g (∅) by (6.3.8.4)
For the induction step, suppose w ≥ 2. Then we have:
G 2 σ • g (∂w) = (σ • g) ∂w, G 2 σ • g (∂ 2 w) {x 1 ,x 2 } by (6.3.17.5) = (σ • g) ∂w, G σ•g (∂ 2 w), G σ•g (∂ 2 w) {x 1 ,x 2 } by induction hypothesis = g ∂w, G σ•g (∂ 2 w), G σ• g (∂ 2
w) x 12 , same by (6.3.11.7)
= (σ • g) ∂w, G σ•g (∂ 2 w) x 12 , same by (6.3.11.6) = G σ•g (∂w), G σ•g (∂w) by (6.3.8.5)
This finishes the induction.
P X σ• = σ X,x 1 ,x 2 σ • = σ Y,x 1 ,x 2 G G P Y λ (1) = λ Y,x(1) G G P Y∖x(1) λ (2) = λ Y∖x(1),x(2) P X ′ λ = λ X ′ ,x G G P X *
is commutative.
Algebras over the Operad of Normal Wiring Diagrams
Proof. For g ∈ P X and w ∈ Prof X * in v , we will prove the desired equality (λσ • g)(w) = λ (2)
λ (1) σ • g (w) ∈ Prof X * out v (6.3.22.1)
by induction on the length of w. If w = 0, then both sides of (6.3.22.1) are equal to g(∅) ∖x 12 .
For the induction step, suppose w ≥ 1. Then we have:
(λσ • g)(w) = (σ • g) w, G σ•g (∂w)
∖x 12 by (6.3.8.5) = g w, G σ•g (∂w), G σ•g (∂w)
∖x 12 by (6.3.11.6) = g w, G 2 σ • g (∂w) ∖x 12 by (6.3.21.1)
= (σ • g) w, G 2 σ • g (∂w)
∖{x 1 ,x 2 } by (6.3.11.7)
= λ (2) λ (1) σ • g (w) by (6.3.17.5) and (6.3.20.1)
This finishes the induction.
Remark 6.3.23.
To see that our definition of the propagator algebra P in Def. 6.3.11 agrees with the one in [RS13] (Section 3), recall that our version of the propagator algebra is based on Def. 6.2.1. On the other hand, the propagator algebra in [RS13] is based on Def. 6.1.2, which is equivalent to Def. 6.1.3. A direct inspection of [RS13] (Announcement 3.3.3 and Eq. (17)) reveals that their structure map of P, when applied to the generating wiring diagrams (section 3.1), reduces to our 8 generating structure maps in Def. 6.3.11. Theorem 6.2.2 then guarantees that the two definitions are equivalent.
Algebras over the Operad of Normal Wiring Diagrams
The purpose of this section is to provide a finite presentation for algebras over the Box S -colored operad WD • of normal wiring diagrams (Prop. 5.3.5). We begin by defining these algebras in terms of finitely many generators and relations. Recall from Def. 5.3.6 that a normal generating wiring diagram is a generating wiring diagram that is not a 1-delay node δ d .
Definition 6.4.1. A WD • -algebra A consists of the following data.
(1) For each X ∈ Box S , A is equipped with a class A X called the X-colored entry of A.
(2) It is equipped with the 7 generating structure maps in Def. 6.2.1 corresponding to the normal generating wiring diagrams.
This data is required to satisfy the same 28 generating axioms in Def. 6.2.1.
The next observation is the WD • version of Theorem 6.2.2. It guarantees that the two existing definitions of a WD • -algebra are equivalent. The first one (Def. 6.1.3) is in terms of a general structure map satisfying an associativity axiom for a general operadic composition. The other one (Def. 6.4.1) is in terms of 7 generating structure maps and 28 generating axioms regarding the normal generating wiring diagram. Therefore, algebras over WD • have a finite presentation. Proof. Simply restrict the proof of Theorem 6.2.2 to normal (generating) wiring diagrams. Instead of Theorem 5.1.11, here we use Theorem 5.3.7 for the existence of a presentation involving only normal generating wiring diagrams.
Remark 6.4.3. In [Spi15b] (Def. 4.1-4.4) several closely related WD • -algebras were defined, although they appeared in the language of symmetric monoidal categories. By Theorem 6.4.2 each of these WD • -algebras has a finite presentation with 7 generating structure maps and 28 generating axioms as in Def. 6.4.1. In Section 6.5 we will discuss one of these WD • -algebras and its finite presentation. In Section 6.7 we will discuss a similar algebra of open dynamical systems over the operad WD 0 of strict wiring diagrams. Essentially the same formalism applies to the other WD • -algebras in [Spi15b].
Finite Presentation for the Algebra of Discrete Systems
The purpose of this section is to provide a finite presentation for the algebra of discrete systems introduced in (1) a set T, called the state set;
(2) a function f rd ∶ T G G B, called the readout function;
(3) a function f up ∶ A × T G G T, called the update function.
Definition 6.5.3. Suppose X = (X in , X out ) ∈ Box S is a box. An X-discrete system is an (X in v , X out v )-discrete system, where X in v = x∈X in v(x) and X out v = x∈X out v(x) ∈ Set
as in (6.3.4.1) (but with sets instead of pointed sets). In other words, an X-discrete system is a triple (T, f rd , f up ) such that T is a set and that
T f rd G G X out v and X in v × T f up G G T are functions.
The collection of all X-discrete systems is denoted by DS(X).
Example 6.5.4. If X = ∅ ∈ Box S is the empty box, then X in v = X out v = * by convention. A readout function f rd ∶ T G G * gives no information, and * × T ≅ T. So
DS(∅) = (T, f up ) ∶ T ∈ Set, f up ∶ T G G T a function . (6.5.4.1)
In particular, the collection DS(∅) is not a set but a proper class. This example explains why in Def. 6.1.2 we defined an entry of an operad algebra to be a class and not a set.
Example 6.5.5. Suppose X is the box with
X in = {x 1 , x 2 }, X out = {x 1 , x 2 }, and values v(x 1 ) = {a 1 , b 1 }, v(x 2 ) = {a 2 , b 2 }, v(x 1 ) = {a 1 , b 1 }, and v(x 2 ) = {a 2 , b 2 }. x 1 x 2 x 1 x 2 An X-discrete system is an (X in v , X out v )-discrete system, where X in v = v(x 1 ) × v(x 2 ) = {a 1 , b 1 } × {a 2 , b 2 }; X out v = v(x 1 ) × v(x 2 ) = {a 1 , b 1 } × {a 2 , b 2 }. Suppose T = {1, 2} is the state set. There is an X-discrete system (T, f rd , f up ) with the readout function f rd ∶ T G G X out v and update function f up ∶ X in v × T G G T de- fined as follows. f rd (1) = (a 1 , a 2 ) f up (a 1 , a 2 ), 1 = 1 f up (a 1 , a 2 ), 2 = 1 f rd (2) = (b 1 , a 2 ) f up (a 1 , b 2 ), 1 = 2 f up (a 1 , b 2 ), 2 = 1 f up (b 1 , a 2 ), 1 = 2 f up (b 1 , a 2 ), 2 = 2 f up (b 1 , b 2 ), 1 = 1 f up (b 1 , b 2 ), 2 = 2
Visually it can also be represented by the transition diagram:
state = 1 readout = (a 1 , a 2 ) state = 2 readout = (b 1 , a 2 )
For example, the arrow labeled (b 1 , a 2 ) from the box for state 1 to the box for state 2 represents the value f up (b 1 , a 2 ), 1 = 2, and likewise for the other arrows.
We now define the algebra of discrete systems in terms of 7 very simple generating structure maps. Definition 6.5.6. The algebra of discrete systems is the WD • -algebra DS in the sense of Def. 6.4.1 defined as follows. For each X ∈ Box S , the X-colored entry is the class DS(X) of X-discrete systems in Def. 6.5.3
The 7 generating structure maps-as in Def. 6.2.1 but without δ d -are defined as follows.
(1) Corresponding to the empty wiring diagram ǫ ∈ WD • ∅ (Def. 3.1.1), the chosen element in DS(∅) (6.5.4.1) is the pair ( * , Id) with:
• the one-point set * as its state set;
• the identity map as its update function.
(2) Corresponding to each name change τ X,
Y ∈ WD • Y X (Def. 3.1.3), the struc- ture map DS(X) τ X,Y = G G DS(Y)
is the identity map, using the fact that
X in v = Y in v and X out v = Y out v .
(3) Corresponding to a 2-cell θ X,Y ∈ WD • X∐Y X,Y (Def. 3.1.4), it has the structure map
DS(X) × DS(Y) θ X,Y (T X , f rd X , f up X ), (T Y , f rd Y , f up Y ) ❴ DS(X ∐ Y) T X × T Y , f rd X × f rd Y , f up X × f up Y using the fact that (X ∐ Y) in v = X in v × Y in v and (X ∐ Y) out v = X out v × Y out v . (4) Corresponding to a 1-loop λ X,x ∈ WD • X∖x X (Def. 3.1.5) with x = x − , x + ∈ X in × X out (so v(x + ) = v(x − )), it has the structure map DS(X) λ X,x G G DS(X ∖ x) T, f rd , f up ✤ G G T, f rd ∖x , f up ∖x in which, for (y, t) ∈ (X ∖ x) in v × T: f rd (t) = f rd (t) ∖x+ , f rd (t) x+ ∈ X out v = (X ∖ x) out v × v(x + ); f rd ∖x (t) = f rd (t) ∖x+ ∈ (X ∖ x) out v ; f up ∖x y, t = f up y, f rd (t) x+ , t .
(5) Corresponding to an in-split σ X,
x 1 ,x 2 ∈ WD • Y X (Def. 3.1.6) with v(x 1 ) = v(x 2 ) and Y = X (x 1 = x 2 ), it has the structure map DS(X) σ X,x 1 ,x 2 G G DS(Y) T, f rd , f up ✤ G G T, f rd , σ • f up in which the update function is σ • f up y, t = f up σ • y, t for (y, t) ∈ Y in v × T.
Here σ • y ∈ X in v is obtained from y by using the v(x 12 )entry of y in both the v(x 1 )-entry and the v(x 2 )-entry, where x 12 ∈ Y in is the identified element of x 1 and x 2 .
(6) Corresponding to an out-split σ Y,y 1 ,
y 2 ∈ WD • Y X (Def. 3.1.7) with v(y 1 ) = v(y 2 ) and X = Y (y 1 = y 2 ), it has the structure map DS(X) σ Y,y 1 ,y 2 G G DS(Y) T, f rd , f up ✤ G G T, σ • f rd , f up in which the readout function is σ • f rd (t) = σ • f rd (t) ∈ Y out v for t ∈ T.
Here σ • f rd (t) is obtained from f rd (t) ∈ X out v by using its v(y 12 )entry in both the v(y 1 )-entry and the v(y 2 )-entry, where y 12 ∈ X out is the identified element of y 1 and y 2 .
(7) Corresponding to a 1-wasted wire ω Y,y ∈ WD • Y X (Def. 3.1.8) with y ∈ Y in and X = Y ∖ y, it has the structure map
DS(X) ω Y,y G G DS(Y) T, f rd , f up ✤ G G T, f rd , ω f up in which the update function is ω f up (z, t) = f up z ∖y , t for (z, t) ∈ Y in v × T.
Here z ∖y ∈ X in v is obtained from z by removing the v(y)-entry.
This finishes the definition of the WD • -algebra of discrete systems. Remark 6.5.7. In [Spi15b] (Example 2.7) the image of θ X,Y is called the parallel composition. The structure map λ X,x corresponding to a 1-loop was discussed in [Spi15b] (Example 2.9). The structure maps σ X,x 1 ,x 2 and σ Y,y 1 ,y 2 corresponding to an in-split and an out-split were discussed in [Spi15b] (Example 2.8).
The following four examples refer to the X-discrete system (T, f rd , f up ) in Example 6.5.5, where
X in = {x 1 , x 2 }, X out = {x 1 , x 2 }, v(x i ) = {a i , b i }, and v(x i ) = {a i , b i } for i = 1, 2.
Let us consider the effects of some of the structure maps in Def. 6.5.6 on (T, f rd , f up ) ∈ DS(X). Example 6.5.8. Suppose a 1 = a 1 and b 1
= b 1 , so v(x 1 ) = {a 1 , b 1 } = v(x 1 ). Suppose X ∖ x is the box obtained from X by removing x = {x 1 , x 1 }, so (X ∖ x) in v = v(x 2 ) and (X ∖ x) out v = v(x 2 ). Consider the 1-loop (Def. 3.1.5) X x 1 x 1 (X ∖ x) in (X ∖ x) out λ X,x in WD • X∖x X .
Then
λ X,x (T, f rd , f up ) = (T, f rd ∖x , f up ∖x ) ∈ DS(X ∖ x)
is the (X ∖ x)-discrete system with update and readout functions
(X ∖ x) in v × T = {a 2 , b 2 } × T f up ∖x G G T f rd ∖x G G {a 2 , b 2 } = (X ∖ x) out v .
Its transition diagram is:
state = 1 readout = a 2 state = 2 readout = a 2 a 2 b 2 a 2 b 2 For instance, we have f up ∖x (a 2 , 1) = f up (a 2 , f rd (1) x 1 ), 1 = f up (a 1 , a 2 ), 1 = 1,
which explains the loop at state 1 labeled a 2 . Similar calculation yields the rest of the update function and the readout function.
Example 6.5.9. Suppose a 1 = a 2 and b 1
= b 2 , so v(x 1 ) = {a 1 , b 1 } = v(x 2 ). Suppose Y is the box X (x 1 = x 2 ), so Y in v = v(x 1 ) and Y out v = X out v . Consider the in-split (Def. 3.1.6) X x 1 x 2 Y in y Y out σ X,x 1 ,x 2 in WD • Y X . Then σ X,x 1 ,x 2 (T, f rd , f up ) = (T, f rd , σ • f up ) ∈ DS(Y)
is the Y-discrete system with update and readout functions
Y in v × T = {a 1 , b 1 } × T σ• f up G G T f rd G G Y out v = {a 1 , b 1 } × {a 2 , b 2 }.
Its transition diagram is:
state = 1 readout = (a 1 , a 2 ) state = 2 readout = (b 1 , a 2 ) a 1 b 1 b 1 a 1 For instance, we have (σ • f up )(a 1 , 2) = f up (a 1 , a 2 ), 2 = 1,
which explains the arrow from state 2 to state 1 labeled a 1 . Similar calculation yields the rest of the update function.
Example 6.5.10. Suppose Z is a box with z = z ′ in Z out such that v(z) = v(z ′ ) = v(x 1 ) and that Z (z = z ′ ) = X. So Z in v = X in v and Z out v = v(x 1 ) × v(x 1 ) × v(x 2 ). Consider the out-split (Def. 3.1.7) X x 1 Z in Z out z z ′ σ Z,z,z ′ in WD • Z X . Then σ Z,z,z ′ (T, f rd , f up ) = (T, σ • f rd , f up ) ∈ DS(Z)
is the Z-discrete system with update and readout functions
Z in v × T = X in v × T f up G G T σ • f rd G G Z out v .
Its transition diagram is:
state = 1 readout = (a 1 , a 1 , a 2 ) state = 2 readout = (b 1 , b 1 , a 2 ) (a 1 , a 2 ) (b 1 , b 2 ) (a 1 , b 2 ) (b 1 , a 2 ) (b 1 , a 2 ) (b 1 , b 2 ) (a 1 , a 2 ) (a 1 , b 2 )
This transition diagram is the same as that of (T, f rd , f up ), except for the values of the readout function at states 1 and 2.
Example 6.5.11. Suppose W is a box such that W ∖ w = X for some w ∈ W in . So
W in v = v(w) × X in v and W out v = X out v . Consider the 1-wasted wire (Def. 3.1.8) X W in w W out ω W,w in WD • W X . Then ω W,w (T, f rd , f up ) = (T, f rd , ω f up ) ∈ DS(W)
is the W-discrete system with update and readout functions
W in v × T = v(w) × X in v × T ω f up G G T f rd G G W out v .
Its transition diagram is:
state = 1 readout = (a 1 , a 2 ) state = 2 readout = (b 1 , a 2 ) (s, a 1 , a 2 ) (s, b 1 , b 2 ) (s, a 1 , b 2 ) (s, b 1 , a 2 ) (s, b 1 , a 2 ) (s, b 1 , b 2 ) (s, a 1 , a 2 ) (s, a 1 , b 2 )
Here each double arrow ⇒ represents the set of arrows as s runs through v(w). For instance, for each s ∈ v(w) we have
(ω f up ) (s, a 1 , b 2 ), 1 = f up (a 1 , b 2 ), 1 = 2,
which explains the double arrow from state 1 to state 2 labeled (s, a 1 , b 2 ). Similar calculation yields the rest of the update function.
The following observation ensures that DS is a well-defined WD • -algebra, i.e., that it satisfies the generating axioms. Theorem 6.5.12. The algebra of discrete systems DS in Def. 6.5.6 is actually a WD •algebra in the sense of Def. 6.4.1, hence also in the sense of Def. 6.1.3 by Theorem 6.4.2.
Proof. We must check that DS satisfies the 28 generating axioms in Def. 6.2.1, which are all trivial to check. For example, the generating axiom (6.2.1.17) says that, in the setting of (3.3.19.1) with X * = X ∖ {x 12 , x 1 ,
x 2 } and v(x 12 ) = v(x 1 ) = v(x 2 ), the diagram DS(X) σ Y,x 1 ,x 2 G G σ X,x 1 ,x 2 DS(Y) λ Y,x(1) G G DS(Y ∖ x(1)) λ Y∖x(1),x(2) DS(X ′ ) λ X ′ ,x G G DS(X * )
is commutative. When applied to a typical element (T, f rd , f up ) ∈ DS(X), a simple direct inspection reveals that both compositions in the above diagram yield (T, g rd , g up ) ∈ DS(X * ), in which
g rd (t) = f rd (t) ∖x 12 ; g up y, t = f up y, f rd (t) x 12 , f rd (t) x 12 , t Here for (y, t) ∈ X * in v × T, we have f rd (t) = f rd (t) ∖x 12 , f rd (t) x 12 ∈ X out v = X * out v × v(x 12 ); y, f rd (t) x 12 , f rd (t) x 12 ∈ X * in v × v(x 1 ) × v(x 2 ) = X in v .
The other generating axioms are checked similarly.
Remark 6.5.13. Our definition of the algebra of discrete systems DS actually agrees with the one in [Spi15b] (Example 2.7 and Def. 4.9). To see this, note that Spivak's definition is essentially based on Def. 6.1.2, except that it is stated in terms of symmetric monoidal categories. Spivak's structure map of DS, when applied to the 7 normal generating wiring diagrams (Def. 5.3.6(1)), agrees with ours in Def. 6.5.6. So Theorems 6.4.2 and 6.5.12 imply that the two definitions of DS-namely, the one in [Spi15b] and our Def. 6.5.6-are equivalent.
Algebras over the Operad of Strict Wiring Diagrams
The purpose of this section is to provide a finite presentation for algebras over the Box S -colored operad WD 0 of strict wiring diagrams (Prop. 5.4.6). We begin by defining these algebras in terms of finitely many generators and relations. Recall from Def. 5.4.7) that:
(1) The strict generating wiring diagrams are the empty wiring diagram ǫ, a name change τ X,Y , a 2-cell θ X,Y , and a 1-loop λ X,x .
(2) The strict elementary relations are the 8 elementary relations that involve only strict generating wiring diagrams on both sides. Definition 6.6.1. A WD 0 -algebra A consists of the following data.
(1) For each X ∈ Box S , A is equipped with a class A X called the X-colored entry of A.
(2) It is equipped with the 4 generating structure maps in Def. Proof. Simply restrict the proof of Theorem 6.2.2 to strict (generating) wiring diagrams. Instead of Theorem 5.1.11, here we use Theorem 5.4.8 for the existence of a presentation involving only strict generating wiring diagrams.
Finite Presentation for the Algebra of Open Dynamical Systems
The purpose of this section is to provide a finite presentation for the algebra of open dynamical systems introduced in [VSL15]. In [VSL15] the algebra of open dynamical systems G was defined and verified using essentially Def. 6.1.2 but in the form of symmetric monoidal categories and monoidal functors. Our definition of G in Def. 6.7.7 is based on Def. 6.6.1, which involves four relatively simple generating structure maps. Our verification that G is actually a WD 0 -algebra in Theorem 6.7.9 boils down to verifying the generating axiom (6.2.1.16) for a double-loop. This is a simple exercise involving the definition of the generating structure map corresponding to a 1-loop (6.7.7.4). The equivalence between the two definitions of the algebra of open dynamical systems is explained in Remark 6.7.10.
Let us first recall the setting of [VSL15]. For the definitions of the basic objects in differential geometry that appear below, the reader may consult, for example, [Hel78] (Ch.1). Assumption 6.7.1. Throughout this section:
(1) S is a chosen set of representatives of isomorphism classes of second-countable smooth manifolds, henceforth called manifolds.
(2) The operad of strict wiring diagrams WD 0 (Prop. 5.4.6) is defined using this choice of S.
(3) All the maps between manifolds are smooth maps.
(4) For a manifold M, denote by π ∶ TM G G M the projection map of the tangent bundle.
x I = (x m ) m∈I ∈ I v = m∈I v(m) x ∖I = (x m ) m∈M∖I ∈ (M ∖ I) v = m∈M∖I v(m). (6.7.2.2)
Definition 6.7.
3. An open dynamical system, or ods for short, is a tuple M, U in , U out , f consisting of:
(1) manifolds M, U in , and U out ;
(2) a pair of maps f = ( f in , f out ),
M × U in f in G G project 7 7 ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ TM π M f out M U out
such that the left diagram is commutative.
Next is [VSL15] (Def. 4.2).
Definition 6.7.4. For each X = (X in , X out ) ∈ Box S , define the class
G X = (M, f ) ∶ M ∈ Fin S , (M v , X in v , X out v , f ) is an ods (6.7.4.1)
in which M v , X in v , and X out v are as in (6.7.2.1). Example 6.7.5. For the empty box ∅ = (∅, ∅) ∈ Box S and an S-finite set M, to say that
(M v , ∅ v = { * }, ∅ v = { * }, f ) is an open dynamical system means that f is a pair of maps f = ( f in , f out ), M v = M v × { * } f in G G Id 9 9 ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ TM v π M v f out M v { * }
such that the left diagram is commutative. Since f out gives no information, f = f in is equivalent to a vector field on M v . So
G ∅ = (M, f ) ∶ M ∈ Fin S , f is a vector field on M v . (6.7.5.1) Example 6.7.6. Suppose W is the box with W in = {w 1 , w 2 }, W out = {w 1 , w 2 }, and all v(−) = R, so W in v = W out v = R 2 .
Suppose M is the one-point set with value R. There is an element (M, f ) ∈ G W whose structure maps
R × R 2 f in G G π 1 8 8 ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ TM = R 2 π 1 R f out R R 2
are given by
f in (x, y, z) = (x, ax + by + cz) and f out (x) = (dx, e x )
for any choice of fixed parameters a, b, c, d ∈ R.
We now define the algebra of open dynamical systems in terms of 4 generating structure maps. Definition 6.7.7. The algebra of open dynamical systems is the WD 0 -algebra G in the sense of Def. 6.6.1 defined as follows. For each box X ∈ Box S , the X-colored entry is G X in (6.7.4.1).
The 4 generating structure maps (Def. 6.2.1) are defined as follows.
(1) Corresponding to the empty wiring diagram ǫ ∈ WD 0 ∅ (Def. 3.1.1), the structure map * ǫ G G G ∅ (6.7.7.1) sends * to (∅, * ) ∈ G ∅ (6.7.5.1). Here ∅ ∈ Fin S is the empty set, in which ∅ v = { * }, and in the second entry * is the trivial vector field.
(2) Corresponding to a name change τ X,Y ∈ WD 0 Y X (Def. 3.1.3), the structure map
G X τ X,Y = G G G Y (6.7.7.2)
is the identity map, using the fact that
X in v = Y in v and X out v = Y out v .
(3) Corresponding to a 2-cell θ X,Y ∈ WD 0 X∐Y X,Y (Def. 3.1.4), it has the structure map
G X × G Y θ X,Y (M X , f X ), (M Y , f Y ) ❴ G X∐Y M X ∐ M Y , f X × f Y (6.7.7.3) in which M X ∐ M Y is the coproduct in Fin S (Def. 2.2.6). (4) Corresponding to a 1-loop λ X,x ∈ WD 0 X∖x X (Def. 3.1.5) with x = x − , x + ∈ X in × X out , it has the structure map G X λ X,x G G G X∖x (M, f ) ✤ G G M, f ∖x . (6.7.7.4) The maps f ∖x = f in ∖x , f out ∖x are defined as f in ∖x m, y = f in m, f out (m) x+ , y ∈ TM v f out ∖x (m) = f out (m) ∖x+ ∈ (X ∖ x) out v for m ∈ M v and y ∈ (X ∖ x) in v . Recalling that f out (m) ∈ X out v , the elements f out (m) x+ ∈ v(x + ) = v(x − ) and f out (m) ∖x+ ∈ (X ∖ x) outw = {w 1 , w 1 }, so (W ∖ w) in v = (W ∖ w) out v = R. Consider the 1-loop (Def. 3.1.5) W w 1 w 1 (W ∖ w) in (W ∖ w) out λ W,w in WD 0 W∖w W . Then λ W,w (M, f ) = (M, f ∖w ) ∈ G W∖w has structure maps R × R f in ∖w G G π 1 8 8 ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ TM = R 2 π 1 R f out ∖w R R given by f out ∖w (x) = f out (x) ∖w 1 = (dx, e x ) ∖w 1 = e x ; f in ∖w (x, y) = f in x, ( f out (x) w 1 , y) = f in (x, dx, y) = x, (a + bd)x + cy .
The following observation ensures that G is a well-defined WD 0 -algebra, i.e., that it satisfies the generating axioms.
Theorem 6.7.9. The algebra of open dynamical systems G in Def. 6.7.7 is actually a WD 0algebra in the sense of Def. 6.6.1, hence also in the sense of Def. 6.1.3 by Theorem 6.6.2.
Proof. We must check the 8 generating axioms corresponding to the strict elementary relations listed in Def. 6.6.1. All of them follow from a quick inspection of the definitions except for (6.2.1.16). This generating axiom says that, in the setting of the elementary relation (3.3.15.1) corresponding to a double-loop, the diagram
G X λ X,x 2 G G λ X,x 1 G X∖x 2 λ X∖x 2 ,x 1 G X∖x 1 λ X∖x 1 ,x 2 G G G X∖x (6.7.9.1) is commutative.
To prove (6.7.9.1), suppose (M, f ) ∈ G X . First define the element (M, f ∖x ) ∈ G X∖x with the maps f ∖x = ( f in ∖x , f out ∖x ),
M v × (X ∖ x) in v f in ∖x G G project 9 9 ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ TM v π M v f out ∖x M v (X ∖ x) out v defined as follows. Given m ∈ M v and y ∈ (X ∖ x) in v = y∈X in ∖{x 1 − ,x 2 − } v(y) we define f in ∖x m, y = f in m; f out (m) {x 1 + ,x 2 + } , y ∈ TM v f out ∖x (m) = f out (m) ∖{x 1 + ,x 2 + } ∈ (X ∖ x) out v .
Recalling that f out (m) ∈ X out v , the elements
f out (m) {x 1 + ,x 2 + } ∈ v(x 1 + ) × v(x 2 + ) = v(x 1 − ) × v(x 2
− ) and f out (m) ∖{x 1 + ,x 2 + } are as in (6.7.2.2). Now it follows from a direct inspection using the definition (6.7.7.4) that both composites in (6.7.9.1), when applied to (M, f ), yields (M, f ∖x ) ∈ G X∖x . This proves the generating axiom (6.7.9.1) for G.
Remark 6.7.10. Our definition of the algebra of open dynamical systems G actually agrees with the one in [VSL15] (Def. 4.4 and 4.5). To see this, note that among the four generating structure maps in Def. 6.7.7:
• ǫ (6.7.7.1), τ X,Y (6.7.7.2), and λ X,x (6.7.7.4) agree with [VSL15] (Def. 4.4);
• θ X,Y (6.7.7.3) agrees with [VSL15] (Def. 4.5).
Theorem 6.6.2 then implies that the two definitions of G-namely, the one in [VSL15] and our Def. 6.7.7-are equivalent.
Summary of Chapter 6
(1) For an S-colored operad O, an O-algebra A consists of a class A c for each c ∈ S and a structure map
A c 1 × ⋯ × A cn ζ G G A d for each ζ ∈ O d c 1 ,.
..,cn that satisfies the associativity, unity, and equivariance axioms.
(2) Each WD-algebra can be described using eight generating structure maps that satisfy twenty-eight generating axioms.
(3) The propagator algebra is a WD-algebra.
(4) Each WD • -algebra can be described using seven generating structure maps that satisfy twenty-eight generating axioms.
(5) The algebra of discrete systems is a WD • -algebra.
(6) Each WD 0 -algebra can be described using four generating structure maps that satisfy eight generating axioms.
(7)
The algebra of open dynamical systems is a WD 0 -algebra.
Part 2
Undirected Wiring Diagrams
The main purpose of this part is to describe the combinatorial structure of the operad UWD of undirected wiring diagrams. The main result is a finite presentation theorem that describes the operad UWD in terms of 6 operadic generators and 17 generating relations.
The operad UWD of undirected wiring diagrams is recalled in Chapter 7. Operadic generators and generating relations for the operad UWD are presented in Chapter 8. Various decompositions of undirected wiring diagrams are given in Chapter 9. Using the results in Chapters 8 and 9, the finite presentation theorem for the operad UWD is proved in Chapter 10. In Chapter 11 we prove the corresponding finite presentation theorem for UWD-algebras and discuss the main example of the relational algebra. This finite presentation theorem for algebras describes each UWD-algebra in terms of finitely many generating structure maps and relations among these maps. Also given in this Chapter is a partial verification of a conjecture of Spivak about the rigidity of the relational algebra.
Reading Guide. The reader who already knows about pushouts of finite sets may skip Section 7.2. In Section 7.3, where we define the operad structure on undirected wiring diagrams, the reader may wish to skip the proofs of Lemmas 7.3.11 and 7.3.13 and just study the accompanying pictures. Chapter 7
Undirected Wiring Diagrams
The purposes of this chapter are (1) to recall the definition of an undirected wiring diagram (Def. 7.1.4) from [Spi13];
(2) to give a proof that the collection of undirected wiring diagrams forms an operad (Theorem 7.3.14).
There is a subtlety regarding the definition and the operadic composition of undirected wiring diagrams; see Remark 7.1.5(4) and Example 7.3.8. Many more examples of undirected wiring diagrams and their operadic composition will be given in the next chapter.
Fix a class S for this chapter.
Defining Undirected Wiring Diagrams
In this section, we recall the definition of an undirected wiring diagram. Recall from Def. 2.2.6 that an S-finite set is a pair (X, v) with X a finite set and v ∶ X G G S a function, called the value assignment. Maps between S-finite sets are functions compatible with the value assignments. The category of S-finite sets is denoted by Fin S . As in earlier chapters, if there is no danger of confusion, then we write an S-finite set (X, v) simply as X. The number of elements in a finite set T is denoted by T .
As in Section 2.2 we first define undirected prewiring diagrams. Undirected wiring diagrams are then defined as the appropriate equivalence classes.
153
Motivation 7.1.1. Before we define an undirected wiring diagram precisely, let us first provide some motivation for it. An undirected wiring diagram is a picture similar to this:
y 1 y 2 y 3 y 4 y 5 y 6 Y x 1 x 2 x 3
x 4 x 5 x 6
x 2
x 1 c 1 c 2 c 3 c 4 c 5 c 6 c 7
There are two input boxes X 1 = {x 1 , . . . , x 6 } and X 2 = {x 1 , x 2 }, an output box Y = {y 1 , . . . , y 6 }, and seven cables {c 1 , . . . , c 7 }. In general, there can be any finite numbers of input boxes and cables, and each input/output box is a finite set. For each element z in the input boxes and the output box, we need to specify a cable c to which z is connected. For instance, x 4 , x 5 , and x 2 are connected to the cable c 6 .
Depending on the given undirected wiring diagrams, there are four possible kinds of cables. First, a cable may only be connected to elements in the inside boxes, such as c 6 and c 7 in the picture above. Second, a cable may only be connected to elements in the outside box, such as c 1 and c 5 above. Third, a cable may be connected to elements in both the inside boxes and the outside box, such as c 2 and c 3 above. Finally, there may be standalone cables that are not connected to anything in the inside and the outside boxes, such as c 4 above. Such distinction among the set of cables will be important later when we discuss the finite presentation for the operad of undirected wiring diagrams. We will come back to this picture in Example 7.1.7 below.
The following definition is a slight generalization of [Spi13] (Examples 2.1.7 and 4.1.1); see Remark 7.1.5.
Definition 7.1.2. Suppose S is a class. An undirected S-prewiring diagram is a tuple
ϕ = X, Y, C, f , g (7.1.2.1)
consisting of the following data.
(1) Y ∈ Fin S , called the output box of ϕ. An element in Y is called an output wire for ϕ.
(2) X = (X 1 , . . . , X n ) is a Fin S -profile for some n ≥ 0 (Def. 2.1.1); i.e., each X i ∈ Fin S .
• We call X i the ith input box of ϕ.
• An element in each X i is called an input wire for ϕ.
• Denote by X = ∐ n i=1 X i ∈ Fin S the coproduct.
• Each element in X ∐ Y is called a wire.
(3) C ∈ Fin S , called the set of cables of ϕ. Each element in C is called a cable.
(4) f and g are maps in the diagram, called a cospan
X 1 ∐ ⋯ ∐ X n = X f G G C Y g o o (7.1.2.2) in Fin S .
• f is called the input soldering function and g the output soldering function. • If f (x) = c, then we say that x is soldered to c and that c is soldered to x via f . If g(y) = c, then we say that y is soldered to c and that c is soldered to y via g. • If c ∈ C is a cable and if m = f −1 (c) and n = g −1 (c) , then c is called an (m, n)-cable.
• A (0, 0)-cable is also called a wasted cable. In other words, a wasted cable is a cable that is in neither the image of f nor the image of g.
Given Y and X, we will denote ϕ as either the tuple (C, f , g) or the cospan (7.1.2.2). The cables tell us how to wire the input wires and the output wires together. So the names of the cables should not matter. This is made precise in the following definition.
Definition 7.1.4. Suppose ϕ = (X, Y, C, f , g) and ϕ ′ = (X, Y, C ′ , f ′ , g ′ ) are two undirected S-prewiring diagrams with the same output box Y and input boxes X.
(1) An equivalence h ∶ ϕ G G ϕ ′ is an isomorphism h ∶ C G G C ′ ∈ Fin S such that the diagram X f G G f ′ 3 3 ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ C h ≅ Y g o o g ′ } } ④ ④ ④ ④ ④ ④ ④ ④ C ′ in Fin S is commutative.
(2) We say that ϕ and ϕ ′ are equivalent if there exists an equivalence ϕ G G ϕ ′ .
(3) An undirected S-wiring diagram is an equivalence class of undirected Sprewiring diagrams. If S is clear from the context, we will drop S and just say undirected wiring diagram.
(4) The class of undirected S-wiring diagrams with output box Y and input boxes X = (X 1 , . . . , X n ) is denoted by
UWD Y X or UWD Y X 1 ,...,Xn . (7.1.4.1)
The class of all undirected S-wiring diagrams is denoted by UWD. If we want to emphasize the class S, we will write UWD S . 3.1) Fong also used cospans, but did not insist that they be jointly surjective in any way.
Convention 7.1.6. To simplify the presentation, we usually suppress the difference between an undirected prewiring diagram and an undirected wiring diagram. Each undirected wiring diagram ϕ = (X, Y, C, f , g) has a unique representative in which:
• each cable is an element in S;
• the value assignment v ∶ C G G S sends each cable to itself.
Unless otherwise specified, we will always use this representative of an undirected wiring diagram.
Example 7.1.7. Suppose S is any class. Consider the undirected wiring diagram ϕ ∈ UWD Y X 1 ,X 2 defined as follows.
• The input boxes are
X 1 = {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } and X 2 = {x 1 , x 2 } ∈ Fin S .
• The output box is Y = {y 1 , y 2 , y 3 , y 4 , y 5 , y 6 } ∈ Fin S .
• The set of cables is C = {c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , c 7 } ∈ Fin S .
Their value assignments satisfied the following conditions:
• v(c 1 ) = v(y 1 ) = v(y 2 ) ∈ S. • v(c 2 ) = v(x 1 ) = v(y 3 ) ∈ S. • v(c 3 ) = v(x 2 ) = v(x 3 ) = v(x 1 ) = v(y 4 ) = v(y 5 ) ∈ S.
• v(c 4 ) ∈ S is arbitrary.
•
v(c 5 ) = v(y 6 ) ∈ S. • v(c 6 ) = v(x 4 ) = v(x 5 ) = v(x 2 ) ∈ S. • v(c 7 ) = v(x 6 ) ∈ S.
The input and the output soldering functions
X 1 ∐ X 2 f G G C Y g o o
are defined as follows:
• c 1 = g(y 1 ) = g(y 2 ) is a (0, 2)-cable. • c 2 = f (x 1 ) = g(y 3 ) is a (1, 1)-cable. • c 3 = f (x 2 ) = f (x 3 ) = f (x 1 ) = g(y 4 ) = g(y 5 ) is a (3, 2)-cable.
• c 4 is a (0, 0)-cable, i.e., a wasted cable.
• c 5 = g(y 6 ) is a (0, 1)-cable. a (3, 0)-cable.
• c 6 = f (x 4 ) = f (x 5 ) = f (x 2 ) is
• c 7 = f (x 6 ) is a (1, 0)-cable.
Graphically we represent this undirected wiring diagram ϕ ∈ UWD Y X 1 ,X 2 as follows.
y 1 y 2 y 3 y 4 y 5 y 6 Y x 1 x 2 x 3 x 4 x 5 x 6 x 2 x 1 c 1 c 2 c 3 c 4 c 5 c 6 c 7 (7.1.7.1)
The input boxes X 1 = {x 1 , . . . , x 6 } and X 2 = {x 1 , x 2 } are drawn as the smaller boxes inside. The output box Y = {y 1 , . . . , y 6 } is drawn as the outer rectangle. Each element within each box is drawn along the boundary, either just inside (as in X 1 and X 2 ) or just outside (as in Y). Note that no orientation is attached to the sides of these squares and rectangles. For example, we could have drawn y 6 ∈ Y on the bottom side of the outer rectangle. Each cable c i ∈ C is drawn as a small gray disk, which is not to be confused with a delay node in a wiring diagram (such as d in (2.2.18.1)).
The soldering functions f and g tell us how to connect the wires in X = X 1 ∐ X 2 and Y to the cables. We will revisit this example in Example 9.1.2 below.
Pushouts
The operadic composition on the collection of undirected wiring diagrams UWD (Def. 7.1.4) involves the basic categorical concept of a pushout, which we recall in this section. The reader may consult [Awo10] (5.6) and [Mac98] (p.65-66) for more discussion of pushouts. Roughly speaking, a pushout is a way of summing two objects with some identification. We will only use the following definition when the category is Fin S (Def. 2.2.6), so the reader may simply take the category C below to be Fin S and objects and maps to be those in Fin S .
Y X f o o g G G Z (7.2.1.1)
is a diagram in C. Then a pushout of this diagram is a tuple (W, α, β) consisting of an object W ∈ C and maps α ∶ Y G G W and β ∶ Z G G W in C such that the following two conditions hold.
(1) The square
X f g G G Z β Y α G G W in C is commutative, i.e., α f = βg. (2) Suppose (W ′ , α ′ , β ′ ) is another such tuple, i.e., α ′ f = β ′ g. Then there exists a unique map h ∶ W G G W ′ in C such that the diagram X f g G G Z β β ′ Y α G G α ′ G G W h ❊ ❊ ❊ ❊ 4 4 ❊ ❊ ❊ ❊ W ′ (7.2.1.2)
in C is commutative, i.e., α ′ = hα and β ′ = hβ.
If a pushout exists, then by definition it is unique up to unique isomorphisms. In a general category, a pushout may not exist for a diagram of the form (7.
W = Y ∐ Z f (x) = g(x) ∶ x ∈ X (7.2.2.1) taken in Fin S .
Proof. The maps α ∶ Y G G W and β ∶ Z G G W are the obvious maps, each being the inclusion into Y ∐ Z followed by the quotient map to W. Then the tuple (W, α, β) has the required universal property of a pushout in Fin S . Example 7.2.3. A commutative square with opposite identity maps is a pushout square. In other words, a pushout of the diagram
X X = o o g G G Z
in any category is given by the commutative square
X = g G G Z = X g G G Z
as can be checked by a direct inspection.
Operad Structure
Fix a class S. In this section we define the Fin S -colored operad structure (Def. 2.1.10) on the collection of undirected wiring diagrams UWD (Def. 7.1.4). When S is either { * } or the collection of sets, this operad structure on UWD was first introduced in [Spi13] using the structure map γ (2.1.3.2).
Definition 7.3.1 (Equivariance in UWD). Suppose Y ∈ Fin S , X = (X 1 , . . . , X n ) is a Fin S -profile of length n, and σ ∈ Σ n is a permutation. Define the function
UWD Y X 1 ,...,Xn = UWD Y X σ ≅ G G UWD Y Xσ = UWD Y X σ(1) ,...,X σ(n) (7.3.1.1) by sending ϕ = (C, f , g) ∈ UWD Y X to ϕ = (C, f , g) ∈ UWD Y Xσ , using the fact that ∐ n i=1 X i = ∐ n i=1 X σ(i) .
In other words, the equivariant structure in UWD simply relabels the input boxes.
Next we define the colored units in UWD. The Y-colored unit in UWD, for Y ∈ Fin S , may be depicted as follows.
1 Y = Y = G G Y Y = o o ∈ UWD Y Y . (7.3.2.1)
Motivation 7.3.3. Next we define the ○ i -composition in UWD. The operadic composition ϕ ○ i ψ can be visualized in the following picture.
ϕ ⋯ X 1 X i X n ψ ○ i (7.3.3.1)
Intuitively, to form the operadic composition ϕ ○ i ψ in UWD, we replace the ith input box X i in ϕ by the input boxes in ψ. The set of cables in ψ is added to the set of cables in ϕ, with appropriate identification from the input and the output soldering functions in ϕ and ψ.
The following notation will be useful in the definition of the ○ i -composition.
Notation 7.3.4. Suppose X = (X 1 , . . . , X n ) is a Fin S -profile. (1) Write X = X 1 ∐ ⋯ ∐ X n ∈ Fin S .
(2) For integers i and j, define
X [i,j] = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ X i ∐ ⋯ ∐ X j if 1 ≤ i ≤ j ≤ n; ∅ otherwise. (7.3.4.1)
Definition 7.3.5 (○ i -Composition in UWD). Suppose:
• ϕ = X fϕ G G C ϕ Y gϕ o o ∈ UWD Y X with X = (X 1 , .
. . , X n ), n ≥ 1, and 1 ≤ i ≤ n;
• ψ = W f ψ G G C ψ X i g ψ o o ∈ UWD X i W with W = m ≥ 0.
Define the undirected wiring diagram
ϕ ○ i ψ = (C, f , g) ∈ UWD Y X○ i W as the cospan Y gϕ g X 1 ∐ ⋯ ∐ X n fϕ G G (Id,g ψ ,Id) pushout C ϕ X [1,i−1] ∐ W ∐ X [i+1,n] (Id, f ψ ,Id) G G f S S X [1,i−1] ∐ C ψ ∐ X [i+1,n] G G C (7.3.5.1)
in Fin S . Here:
• The square is a pushout in Fin S , which always exists by Prop. 7.2.2.
• The S-finite sets W, X [1,i−1] , and X [i+1,n] are as in Notation 7.3.4.
• The Fin S -profile
X ○ i W = X 1 , . . . , X i−1 , W, X i+1 , . . . , X n
is as in (2.1.10.2).
• The map f is the bottom horizontal composition, and the map g is the right vertical composition.
In the following observation, we describe the undirected wiring diagram ϕ ○ i ψ more explicitly. Proposition 7.3.6. Consider the diagram (7.3.5.1).
(1) A choice of a pushout C is the quotient
C = C ϕ ∐ C ψ f ϕ (x) = g ψ (x) ∶ x ∈ X i (7.3.6.1)
in Fin S . The following statements use this choice of C.
(2) The maps C ϕ G G C and C ψ G G C are the obvious maps, each being the inclusion into C ϕ ∐ C ψ followed by the quotient map to C.
(3) In the map f and in the horizontal unnamed map, for j = i, the map X j G G C is the composition
X j fϕ G G C ϕ G G C .
(4) On W the map f is the composition
W f ψ G G C ψ G G C .
Proof. A direct inspection shows that the first three statements indeed describe a pushout of the square in the diagram (7.3.5.1). The last assertion follows from the definition of f as the bottom horizontal composition.
Remark 7.3.7. Consider Def. 7.3.5.
(1) Pushouts are unique up to unique isomorphisms. Since undirected wiring diagrams are defined as equivalence classes of undirected prewiring diagrams (Def. 7.1.4), the undirected wiring diagram ϕ ○ i ψ is well-defined.
(2) In [Spi13] S was taken to be either the one-point set or the collection of sets. The operadic composition in UWD was defined in terms of the operadic composition γ (2.1.3.2). By Prop. 2.1.12 the two descriptions-i.e., the one in [Spi13] and Def. 7.3.5-are equivalent.
Recall from Def. 7.1.2 that a wasted cable in an undirected wiring diagram is a cable that is not in the image of the input and the output soldering functions. Example 7.3.8. In this example, we observe that wasted cables can be created by the ○ i -composition, even if the original undirected wiring diagrams have no wasted cables. Suppose:
• S = { * } and X = {x 1 , x 2 } ∈ Fin is a two-element set. • ϕ = X G G { * } ∅ o o ∈ UWD ∅ X . • ψ = ∅ G G X X = o o ∈ UWD X ∅ .
Note that neither ϕ nor ψ has a wasted cable. On the other hand, the undirected wiring diagram ϕ ○ 1 ψ ∈ UWD ∅ ∅ is the cospan
∅ ~X = G G { * } ∅ G G S S X G G { * }
in Fin, in which the square is a pushout by Example 7.2.3. The following picture gives a visualization of this ○ 1 -composition.
x 1 x 2 * ψ ϕ = * ϕ ○ 1 ψ
In particular, the unique cable in ϕ ○ 1 ψ is a wasted cable. This example illustrates that the ○ i -composition of two undirected wiring diagrams without wasted cables may have wasted cables, which we mentioned in Remark 7.1.5. We will revisit this example in Section 8.3 and Example 10.1.5 below.
We now prove that the collection UWD of undirected wiring diagrams is a Fin Scolored operad in the sense of Def. 2.1.10. Proof. The equivariance axiom holds because the equivariant structure (7.3.1) simply relabels the input boxes. The unity axioms follow from the definitions of the colored units in UWD (7.3.2.1) and Example 7.2.3. Motivation 7.3.10. The horizontal associativity axiom in UWD may be visualized as follows.
ψ ζ ϕ ⋯ Y i Y j Y k (ϕ ○ j ζ) ○ i ψ ⋯ ψ ζ Y k
To keep the picture simple, we omitted drawing the wires and the cables. Note that this is basically the undirected version of the picture (2.3.7.1). Proof. Suppose:
• ϕ = C ϕ , f ϕ , g ϕ ∈ UWD Z Y with Y = n ≥ 2 and 1 ≤ i < j ≤ n; • ψ = C ψ , f ψ , g ψ ∈ UWD Y i W with W = l; • ζ = C ζ , f ζ , g ζ ∈ UWD Y j X with X = m.
We must show that
ϕ ○ j ζ ○ i ψ = (ϕ ○ i ψ) ○ j−1+l ζ ∈ UWD Z (Y○ j X)○ i W . (7.3.11.1)
Consider the undirected wiring diagram
(C, f , g) ∈ UWD Z (Y○ j X)○ i W
given by the cospan
Y [1,i−1] ∐ W ∐ Y [i+1,j−1] ∐ X ∐ Y [j+1,n] f G G C Z g o o
in Fin S , where Notation 7.3.4 was used. In this cospan:
• The set of cables is the quotient
C = C ϕ ∐ C ψ ∐ C ζ f ϕ (y i ) = g ψ (y i ), f ϕ (y j ) = g ζ (y j ) ∶ y i ∈ Y i , y j ∈ Y j .
• The output soldering function g is the composition Z gϕ G G C ϕ G G C .
• For the input soldering function f :
-The restriction to Y k is the composition Y k fϕ G G C ϕ G G C for k = i, j. -The restriction to W is the composition W f ψ G G C ψ G G C . -The restriction to X is the composition X f ζ G G C ζ G G C .
Using the description of ○ i in Prop. 7.3.6, a direct inspection reveals that both sides of (7.3.11.1) are equal to (C, f , g).
Motivation 7.3.12. The vertical associativity axiom in UWD may be visualized as follows.
○ j ○ i ζ ψ ⋯ X j X k ϕ ⋯ Y i Y l ϕ ○ i (ψ ○ j ζ) ⋯ ⋯ ⋯ ζ X k Y l
As before, to keep the picture simple, we did not draw the wires and the cables. Note that this is basically the undirected version of the picture (2.3.9.1). Proof. Suppose:
• ϕ = C ϕ , f ϕ , g ϕ ∈ UWD Z Y with Y = n ≥ 1 and 1 ≤ i ≤ n; • ψ = C ψ , f ψ , g ψ ∈ UWD Y i X with X = m ≥ 1 and 1 ≤ j ≤ m; • ζ = C ζ , f ζ , g ζ ∈ UWD X j W with W = l.
We must show that
(ϕ ○ i ψ) ○ i−1+j ζ = ϕ ○ i ψ ○ j ζ ∈ UWD Z (Y○ i X)○ i−1+j W . (7.3.13.1)
Consider the undirected wiring diagram
(C, f , g) ∈ UWD Z (Y○ i X)○ i−1+j W
given by the cospan
Y [1,i−1] ∐ X [1,j−1] ∐ W ∐ X [j+1,m] ∐ Y [i+1,n] f G G C Z g o o
in Fin S , where Notation 7.3.4 was used. In this cospan:
• The set of cables is the quotient
C = C ϕ ∐ C ψ ∐ C ζ f ϕ (y) = g ψ (y), f ψ (x) = g ζ (x) ∶ y ∈ Y i , x ∈ X j .
• The output soldering function g is the composition
Z gϕ G G C ϕ G G C .
• For the input soldering function f :
-The restriction to Y l is the composition Y l fϕ G G C ϕ G G C for l = i. -The restriction to X k is the composition X k f ψ G G C ψ G G C for k = j. -The restriction to W is the composition W f ζ G G C ζ G G C .
Using the description of ○ i in Prop. 7.3.6, a direct inspection reveals that both sides of (7.3.13.1) are equal to (C, f , g). (2) If S = Set, the collection of sets, then our Fin Set -colored operad UWD is called the operad of typed wiring diagrams in [Spi13] (Example 4.1.1).
We defined the operad UWD in terms of the ○ i -compositions (Def. 7.3.5). The following observation expresses the operad UWD in terms of the operadic composition γ (2.1.3.2). In [Spi13] the operad structure on undirected wiring diagrams was actually defined in terms of γ. Proposition 7.3.16. Suppose: . . . , X n ) for some n ≥ 1 and X = X 1 ∐ ⋯ ∐ X n .
• ϕ = (C ϕ , f ϕ , g ϕ ) ∈ UWD Y X with X = (X 1 ,• For each 1 ≤ i ≤ n, ψ i = (C i , f i , g i ) ∈ UWD X i W i with W i = W i,1 , . . . , W i,k i for some k i ≥ 0. • W = (W 1 , . . . , W n ), W i = W i,1 ∐ ⋯ ∐ W i,k i , and W = ∐ 1≤i≤n W i . Then γ ϕ; ψ 1 , . . . , ψ n = (C, f , g) ∈ UWD Y W is given by the cospan Y gϕ g X fϕ G G ∐g i pushout C ϕ W ∐ f i G G f W W C 1 ∐ ⋯ ∐ C n G G C (7.3.16.1)
in Fin S , in which the square is a pushout. In this diagram:
(1) C is the quotient
C = C ϕ ∐ C 1 ∐ ⋯ ∐ C n f ϕ (x) = g i (x) ∶ x ∈ X i , 1 ≤ i ≤ n in Fin S .
(2) The maps C ϕ G G C and C i G G C are the obvious maps, each being an inclusion followed by a quotient map to C.
(3) The restriction of f to W i is the composition of
f i ∶ W i G G C i and C i G G C.
Proof. This follows from (i) the correspondence (2.1.12.1) between γ and the ○ icompositions and (ii) the description of ○ i given in Proposition 7.3.6.
Summary of Chapter 7
(1) An undirected S-wiring diagram has a finite number of input boxes, an output box, an S-finite set of cables, an input soldering function, and an output soldering function.
(2) For each class S, the collection of S-wiring diagrams UWD is a Fin S -colored operad.
Fix a class S, and consider the Fin S -colored operad UWD of undirected wiring diagrams (Theorem 7.3.14). The purpose of this chapter is to describe a finite number of undirected wiring diagrams that we will later show to be sufficient to describe the entire operad UWD. One may also regard this chapter as consisting of a long list of examples of undirected wiring diagrams.
In Section 8.1 we describe 6 undirected wiring diagrams, called the generating undirected wiring diagrams. Later we will show that they generate the operad UWD of undirected wiring diagrams. This means that every undirected wiring diagram can be obtained from finitely many generating undirected wiring diagrams via iterated operadic compositions. For now one may think of the generating undirected wiring diagrams as examples of undirected wiring diagrams.
In Section 8.2 we describe 17 elementary relations among the generating undirected wiring diagrams. Later we will show that these elementary relations together with the operad associativity and unity axioms-
Generating Undirected Wiring Diagrams
Recall the definition of an undirected wiring diagram (Def. 7.1.4). In this section, we introduce 6 undirected wiring diagrams, called the generating undirected wiring diagrams. They will be used in later chapters to give a finite presentation for the operad UWD of undirected wiring diagrams. The undirected wiring diagrams in this section all have directed analogues in Section 3.1.
The following undirected wiring diagram is an undirected analogue of the empty wiring diagram (Def. 3.1.1).
ǫ = ∅ G G ∅ ∅ o o ∈ UWD ∅ ,
where ∅ is the empty S-finite set. Note that the empty cell is a 0-ary element in UWD.
Next we define the undirected wiring diagram with no input boxes and whose unique cable is a (0, 1)-cable. This is an undirected analogue of a 1-wasted wire (Def. 3.1.8). To simplify the typography, we will often write x for the one-point set {x}.
Definition 8.1.2. Suppose * ∈ Fin S is a one-element S-finite set. Define the 1-output wire
ω * = ∅ G G * * o o ∈ UWD * .
Note that a 1-output wire is a 0-ary element in UWD.
Next we define the undirected wiring diagram Y X ⋮ with 1 input box and whose cables are all (1, 1)-cables. This is an undirected analogue of a name change (Def. 3.1.3).
Definition 8.1.3. Suppose f ∶ X G G Y ∈ Fin S is a bijection. Define the undirected name change τ f = X f ≅ G G Y Y = o o ∈ UWD Y X .
If the bijection f is clear from the context, then we write τ f as τ X,Y or just τ. If there is no danger of confusion, then we will call τ f a name change.
Next we define the undirected wiring diagram
X ∐ Y X ⋮ Y ⋮
with 2 input boxes and whose cables are all (1, 1)-cables. This is an undirected analogue of a 2-cell (Def. 3.1.4).
Definition 8.1.4. Suppose X, Y ∈ Fin S and X ∐ Y is their coproduct. Define the undirected 2-cell θ (X,Y) = X ∐ Y = G G X ∐ Y X ∐ Y = o o ∈ UWD X∐Y X,Y .
If there is no danger of confusion, then we will call it a 2-cell.
Next we define the undirected wiring diagram
X ∖ x ± x− x+ X ⋮
with a (2, 0)-cable, all other cables being (1, 1)-cables. This is an undirected analogue of a 1-loop (Def. 3.1.5).
Definition 8.1.5. Suppose:
• X ∈ Fin S , and x + , x − ∈ X are two distinct elements with v(x + ) = v(x − ) ∈ S.
• X ∖ x ± ∈ Fin S is obtained from X by removing x + and x − .
• X (x + = x − ) ∈ Fin S is the quotient of X with x + and x − identified.
Define the loop
λ (X,x±) = X projection G G X (x+=x−) X ∖ x ± inclusion o o ∈ UWD X∖x± X .
Next we define the undirected wiring diagram (1, 2)-cable, all other cables being (1, 1)-cables. This is an undirected analogue of an out-split (Def. 3.1.7).
x 1 x 2 X X ′ ⋮ x with a
Definition 8.1.6. Suppose:
• X ∈ Fin S , and x 1 , x 2 are two distinct elements in X with v(x 1 ) = v(x 2 ) ∈ S. • X ′ ∈ Fin S , and x ∈ X ′ such that v(x) = v(x 1 ) and that X ′ ∖ {x} = X ∖ {x 1 , x 2 }.
Define the split
σ (X,x 1 ,x 2 ) = X ′ = G G X ′ X o o ∈ UWD X X ′
in which the output soldering function X G G X ′ sends x 1 , x 2 ∈ X to x ∈ X ′ and is the identity function on X ∖ {x 1 , x 2 }.
Definition 8.1.7. The 6 undirected wiring diagrams in Def. 8.1.1-8.1.6 will be referred to as generating undirected wiring diagrams. If the context is clear, we will simply call them generators.
Remark 8.1.8. Among the generating undirected wiring diagrams:
(1) None has a wasted cable (Def. 7.1.2). As we will see in Section 8.3, wasted cables can be created by the generators.
(2) The empty cell ǫ (Def. 8.1.1) and a 1-output wire ω * (Def. 8.1.2) are 0-ary elements in UWD.
Elementary Relations
The purpose of this section is to introduce 17 elementary relations among the generating undirected wiring diagrams (Def. 8.1.7). Each elementary relation is proved using Prop. 7.3.6 and Example 7.2.3 and by a simple inspection of the relevant definitions of the generating undirected wiring diagrams and operadic compositions. Each proof is similar to Example 7.3.8 and the proofs of Lemma 7.3.11 and Lemma 7.3.13. Therefore, we will omit the proofs, providing a picture instead in most cases. Some, but not all, of the following relations have directed analogues in Section 3.3.
Recall the operadic composition in the Fin S -colored operad UWD (Def. 7.3.5) and Notation 3.3.1 for (iterated) ○ 1 . The first five relations are about name changes (Def. 8.1.3). The first one says that two consecutive name changes can be composed down into one name change.
Proposition 8.2.1. Suppose:
• f ∶ X G G Y and g ∶ Y G G Z ∈ Fin S are bijections. • τ f ∈ UWD Y X , τ g ∈ UWD Z Y ,
and τ g f ∈ UWD Z X are the corresponding name changes.
Then
τ g ○ τ f = τ g f ∈ UWD Z X . (8.2.1.1)
The next relation says that a name change of a 1-output wire (Def. 8.1.2) is again a 1-output wire. • ω x ∈ UWD X and ω y ∈ UWD Y are the corresponding 1-output wires.
• τ X,Y ∈ UWD Y X is the name change corresponding to the bijection X G G Y ∈ Fin S . Then τ X,Y ○ ω x = ω y ∈ UWD Y . (8.2.2.1)
The next relation says that name changes inside a 2-cell (Def. 8.1.4) can be rewritten as a name change of a 2-cell.
Proposition 8.2.3.
Suppose:
• f 1 ∶ X 1 G G Y 1 and f 2 ∶ X 2 G G Y 2 ∈ Fin S are bijections. • f 1 ∐ f 2 ∶ X 1 ∐ X 2 G G Y 1 ∐ Y 2 ∈ Fin S is their coproduct. • τ f 1 ∈ UWD Y 1 X 1 , τ f 2 ∈ UWD Y 2 X 2 , and τ f 1 ∐ f 2 ∈ UWD Y 1 ∐Y 2 X 1 ∐X 2 are the corresponding name changes. • θ (X 1 ,X 2 ) ∈ UWD X 1 ∐X 2 X 1 ,X 2 and θ (Y 1 ,Y 2 ) ∈ UWD Y 1 ∐Y 2 Y 1 ,Y 2 are 2-cells.
Then
θ (Y 1 ,Y 2 ) ○ 1 τ f 1 ○ 2 τ f 2 = τ f 1 ∐ f 2 ○ θ (X 1 ,X 2 ) ∈ UWD Y 1 ∐Y 2 X 1 ,X 2 . (8.2.3.1)
The next relation says that a name change inside a loop (Def. 8.1.5) can be rewritten as a name change of a loop.
Proposition 8.2.4. Suppose:
• X ∈ Fin S , and x + , x − ∈ X are two distinct elements with v(x + ) = v(x − ) ∈ S. • f ∶ X G G Y ∈ Fin S is a bijection with y + = f (x + ) and y − = f (x − ).
• X ∖ x ± ∈ Fin S and Y ∖ y ± ∈ Fin S are obtained from X and Y by removing the indicated elements.
• f ′ ∶ X ∖ x ± G G Y ∖ y ± is the corresponding bijection. • τ f ∈ UWD Y X and τ f ′ ∈ UWD Y∖y±
X∖x± are name changes.
• λ (X,x±) ∈ UWD X∖x± X and λ (Y,y±) ∈ UWD Y∖y± Y are loops.
Then
λ (Y,y±) ○ τ f = τ f ′ ○ λ (X,x±) ∈ UWD Y∖y± X . (8.2.4.1)
The next relation says that a name change inside a split (Def. 8.1.6) can be rewritten as a name change of a split.
Proposition 8.2.5. Suppose:
• X ∈ Fin S , and x 1 , x 2 are two distinct elements in X with v(x 1 ) = v(x 2 ) ∈ S. • X ′ ∈ Fin S , and x ∈ X ′ such that v(x) = v(x 1 ) and that X ′ ∖ {x} = X ∖ {x 1 , x 2 }. • f ∶ X G G Y ∈ Fin S is a bijection with y 1 = f (x 1 ) and y 2 = f (x 2 ). • Y ′ ∈ Fin S , and y ∈ Y ′ such that v(y) = v(y 1 ) ∈ S and that Y ′ ∖ {y} = Y ∖ {y 1 , y 2 }. • f ′ ∶ X ′ G G Y ′ ∈ Fin S
is a bijection such that f ′ (x) = y and that its restriction to
X ′ ∖ {x} = X ∖ {x 1 , x 2 } is equal to that of f . • τ f ∈ UWD Y X and τ f ′ ∈ UWD Y ′ X ′ are name changes. • σ (X,x 1 ,x 2 ) ∈ UWD X X ′ and σ (Y,y 1 ,y 2 ) ∈ UWD Y Y ′ are splits.
Then
σ (Y,y 1 ,y 2 ) ○ τ f ′ = τ f ○ σ (X,x 1 ,x 2 ) ∈ UWD Y X ′ . (8.2.5.1)
The following two relations involve 1-output wires in somewhat subtle ways. The next relation says that the undirected wiring diagram X ⋮ x with a (1, 0)-cable, all other cables being (1, 1)-cables, can be obtained from the generators as either one of the following two (iterated) operadic compositions.
X ⋮ Y x x y W = W Y X ⋮ x x y
As in Section 7.3, the gray boxes here indicate an operadic composition.
• On the left, a 1-output wire ω y is substituted into a 2-cell θ (X,y) , which is then substituted into a loop λ (Y,x,y) .
• On the right, a split σ (Y,x,y) is substituted into a loop λ (Y,x,y) .
Proposition 8.2.6. Suppose:
• Y ∈ Fin S , and x, y ∈ Y are distinct elements with v(x) = v(y) ∈ S.
• X = Y ∖ y ∈ Fin S is obtained from Y by removing y.
• ω y ∈ UWD y is the 1-output wire for y.
• θ (X,y) ∈ UWD Y X,y is a 2-cell. • λ (Y,x,y) ∈ UWD W Y is a loop, where W = Y ∖ {x, y} = X ∖ x. • σ (Y,x,y) ∈ UWD Y X is a split. Then λ (Y,x,y) ○ θ (X,y) ○ 2 ω y = λ (Y,x,y) ○ σ (Y,x,y) ∈ UWD W X . (8.2.6.1)
The next relation says that the X-colored unit (Def. 7.3.2) can be obtained from the generators as the following iterated operadic composition.
X W Y X ⋮ x x w y x x
More precisely, it says that the X-colored unit 1 X can be obtained by substituting a 1-output wire ω y into a 2-cell θ (X,y) , then into a split σ (W,x,w) , and then into a loop λ (W,w,y) .
Proposition 8.2.7. Suppose:
• W ∈ Fin S , and w, x, y are distinct elements in W with v(w) = v(x) = v(y) ∈ S.
• Y = W ∖ w ∈ Fin S is obtained from W by removing w.
• X = Y ∖ y ∈ Fin S is obtained from Y by removing y.
• ω y ∈ UWD y is the 1-output wire for y.
• θ (X,y) ∈ UWD Y X,y is a 2-cell. • σ (W,x,w) ∈ UWD W Y is a split. • λ (W,w,y) ∈ UWD X W is a loop.
Then λ (W,w,y) ○ σ (W,x,w) ○ θ (X,y) ○ 2 ω y = 1 X ∈ UWD X X , (8.2.7.1)
in which 1 X is the X-colored unit (Def. 7.3.2).
The next five relations are about 2-cells (Def. 8.1.4). The following relation is the unity property of 2-cells.
Proposition 8.2.8. Suppose: the empty cell (Def. 8.1.1).
• θ (X,∅) ∈ UWD X X,∅ is a 2-cell. • ǫ ∈ UWD ∅ isThen θ (X,∅) ○ 2 ǫ = 1 X ∈ UWD X X . (8.2.8.1)
The next relation is the associativity property of 2-cells. It gives two different ways to construct the following undirected wiring diagram using two 2-cells. Then
θ (X∐Y,Z) ○ 1 θ (X,Y) = θ (X,Y∐Z) ○ 2 θ (Y,Z) ∈ UWD X∐Y∐Z X,Y,Z . (8.2.9.1)
The next relation is the commutativity property of 2-cells. It uses the equivariant structure (7.3.1) in UWD.
Proposition 8.2.10. Suppose:
• θ X,Y ∈ UWD X∐Y X,Y is a 2-cell. • (1 2) ∈ Σ 2 is the non-trivial permutation.
Then
θ (X,Y) (1 2) = θ (Y,X) ∈ UWD Y∐X Y,X . (8.2.10.1)
The next relation is the commutativity property between a 2-cell and a loop. It gives two different ways to construct the following undirected wiring diagram.
X ∐ Y ′ X ⋯ Y ⋯ y+ y−
Proposition 8.2.11. Suppose:
• Y ∈ Fin S , and y + ,
y − are distinct elements in Y with v(y + ) = v(y − ) ∈ S.
• Y ′ = Y ∖ y ± ∈ Fin S is obtained from Y by removing y + and y − .
• λ (Y,y±) ∈ UWD Y ′ Y is a loop. • θ (X,Y) ∈ UWD X∐Y X,Y and θ (X,Y ′ ) ∈ UWD X∐Y ′ X,Y ′ are 2-cells for some X ∈ Fin S . • λ (X∐Y,y±) ∈ UWD X∐Y ′ X∐Y is a loop.
Then
θ (X,Y ′ ) ○ 2 λ (Y,y±) = λ (X∐Y,y±) ○ θ (X,Y) ∈ UWD X∐Y ′ X,Y . (8.2.11.1)
The next relation is the commutativity between a 2-cell and a split. It gives two different ways to construct the following undirected wiring diagram.
X ∐ Y X ⋯ Y ′ ⋯ y y 1 y 2
Proposition 8.2.12. Suppose:
• Y ∈ Fin S , and y 1 , y 2 are distinct elements in Y with v(y 1 ) = v(y 2 ) ∈ S.
• Y ′ ∈ Fin S , and y ∈ Y ′ such that v(y) = v(y 1 ) and that
Y ′ ∖ {y} = Y ∖ {y 1 , y 2 }. • σ (Y,y 1 ,y 2 ) ∈ UWD Y Y ′ is a split. • θ (X,Y) ∈ UWD X∐Y X,Y and θ (X,Y ′ ) ∈ UWD X∐Y ′ X,Y ′ are 2-cells for some X ∈ Fin S . • σ (X∐Y,y 1 ,y 2 ) ∈ UWD X∐Y X∐Y ′ is a split.
Then
θ (X,Y) ○ 2 σ (Y,y 1 ,y 2 ) = σ (X∐Y,y 1 ,y 2 ) ○ θ (X,Y ′ ) ∈ UWD X∐Y X,Y ′ . (8.2.12.1)
The following four relations are about splits. The next relation is the commutativity property of splits. It gives two different ways to construct the following undirected wiring diagram using two splits. • X ∈ Fin S , and y 1 , y 2 , z 1 , z 2 are distinct elements such that v(y 1 ) = v(y 2 ) and v(z 1 ) = v(z 2 ) ∈ S.
• X ′ ∈ Fin S , and y and z are distinct elements in X ′ such that
-v(y) = v(y 1 ) and v(z) = v(z 1 ); -X ′ ∖ {y, z} = X ∖ {y 1 , y 2 , z 1 , z 2 }. • Y = X ′ ∖ {y} ∐ {y 1 , y 2 } and Z = X ′ ∖ {z} ∐ {z 1 , z 2 } ∈ Fin S • σ (X,y 1 ,y 2 ) ∈ UWD X Z and σ (Z,z 1 ,z 2 ) ∈ UWD Z X ′ are splits. • σ (X,z 1 ,z 2 ) ∈ UWD X Y and σ (Y,y 1 ,y 2 ) ∈ UWD Y X ′ are splits. Then σ (X,y 1 ,y 2 ) ○ σ (Z,z 1 ,z 2 ) = σ (X,z 1 ,z 2 ) ○ σ (Y,y 1 ,y 2 ) ∈ UWD X X ′ . (8.2.13.1)
The next relation is the associativity property of splits. It gives two different ways to construct the following undirected wiring diagram using two splits.
Y X ⋮ x y 1 y 2 y 3
Proposition 8.2.14. Suppose:
• Y ∈ Fin S , and y 1 , y 2 , y 3 are distinct elements in Y with v(y 1 ) = v(y 2 ) = v(y 3 ) ∈ S.
• X ∈ Fin S , and x ∈ X such that -v(x) = v(y 1 ); -X ∖ {x} = Y ∖ {y 1 , y 2 , y 3 }.
• Y 1 = Y (y 1 = y 2 ) ∈ Fin S is the quotient of Y with y 1 and y 2 identified, called y 12 ∈ Y 1 .
• Y 2 = Y (y 2 = y 3 ) ∈ Fin S is the quotient of Y with y 2 and y 3 identified, called y 23 ∈ Y 2 .
• σ (Y,y 1 ,y 2 ) ∈ UWD Y Y 1 and σ (Y 1 ,y 12 ,y 3 ) ∈ UWD Y 1 X are splits. • σ (Y,y 2 ,y 3 ) ∈ UWD Y Y 2 and σ (Y 2 ,y 1 ,y 23 ) ∈ UWD Y 2 X are splits.
Then σ (Y,y 1 ,y 2 ) ○ σ (Y 1 ,y 12 ,y 3 ) = σ (Y,y 2 ,y 3 ) ○ σ (Y 2 ,y 1 ,y 23 ) ∈ UWD Y X .
(8.2.14.1)
The next relation is the commutativity property between a split and a loop. It gives two different ways to construct the following undirected wiring diagram using a split and a loop. • Y ∈ Fin S , and y 1 and y 2 are distinct elements in Y with v(y 1 ) = v(y 2 ) ∈ S.
• X ∈ Fin S , and x, x + , x − are distinct elements in X such that -v(x) = v(y 1 ); -v(x + ) = v(x − ); -X ∖ {x, x + , x − } = Y ∖ {y 1 , y 2 }. • Y ′ = X ∖ {x} ∐ {y 1 , y 2 } and X ′ = X ∖ {x + , x − }.
• σ (Y ′ ,y 1 ,y 2 ) ∈ UWD Y ′ X and σ (Y,y 1 ,y 2 ) ∈ UWD Y X ′ are splits.
• λ (Y ′ ,x±) ∈ UWD Y Y ′ and λ (X,x±) ∈ UWD X ′ X are loops. Then λ (Y ′ ,x±) ○ σ (Y ′ ,y 1 ,y 2 ) = σ (Y,y 1 ,y 2 ) ○ λ (X,x±) ∈ UWD Y X . (8.2.15.1)
The next relation says that the undirected wiring diagram or in the counterpart in which x + and x − are switched. In (8.2.16.1) below, this picture corresponds to the left side, and its counterpart corresponds to the right side.
Proposition 8.2.16. Suppose:
• X ∈ Fin S , and x + and x − are distinct elements in X with v(x + ) = v(x − ) ∈ S.
• Y ∈ Fin S , and y ∈ Y such that v(y) = v(x + ) and that X ∖ x ± = Y ∖ y.
• W = X ∐ y = Y ∐ x ± ∈ Fin S . • σ (W,y,x+) ∈ UWD W X and σ (W,y,x−) ∈ UWD W X are splits. • λ (W,x±) ∈ UWD Y W is a loop.
Then
λ (W,x±) ○ σ (W,y,x+) = λ (W,x±) ○ σ (W,y,x−) ∈ UWD Y X . (8.2.16.1)
The final relation is the commutativity property of loops. It gives two different ways to construct the following undirected wiring diagram using two loops.
Y X ⋯ x 1 x 2 x 3 x 4
Proposition 8.2.17. Suppose:
• X ∈ Fin S , and x 1 , x 2 , x 3 , x 4 are distinct elements in X with v(x 1 ) = v(x 2 ) and v(x 3 ) = v(x 4 ) ∈ S. • W = X ∖ {x 1 , x 2 }, Z = X ∖ {x 3 , x 4 }, and Y = X ∖ {x 1 , x 2 , x 3 , x 4 } ∈ Fin S . • λ (W,x 3 ,x 4 ) ∈ UWD Y W and λ (X,x 1 ,x 2 ) ∈ UWD W X are loops. • λ (Z,x 1 ,x 2 ) ∈ UWD Y Z and λ (X,x 3 ,x 4 ) ∈ UWD Z X are loops.
Then
λ (W,x 3 ,x 4 ) ○ λ (X,x 1 ,x 2 ) = λ (Z,x 1 ,x 2 ) ○ λ (X,x 3 ,x 4 ) ∈ UWD Y X .(8
Wasted Cables
The
(1) ϕ = X G G * ∅ o o ∈ UWD ∅ X is the loop λ (X,x 1 ,x 2 ) . (2) ψ = ∅ G G X X = o o ∈ UWD X ∅ is the iterated operadic composition ψ = θ (∅,X) ○ 2 θ (x 1 ,x 2 ) ○ 2 ω x 1 ○ 2 ω x 2
involving two 2-cells and two 1-output wires.
So the composition
ϕ ○ ψ = ∅ G G * ∅ o o
∈ UWD ∅ ∅ , which is depicted as = and has one wasted cable, is the iterated operadic composition
λ (X,x 1 ,x 2 ) ○ θ (∅,X) ○ 2 θ (x 1 ,x 2 ) ○ 2 ω x 1 ○ 2 ω x 2 (8.3.1.1)
involving 5 generators.
Example 8.3.2. As a variation of the previous example, consider any box Y ∈ Fin S and the undirected wiring diagram with one wasted cable
ζ Y = Y inclusion G G Y ∐ * Y inclusion o o ∈ UWD Y Y . It is depicted as follows. Y Y ⋮
This undirected wiring diagram can be created by replacing the empty box ∅ in the 2-cell θ (∅,X) by Y and the loop λ (X,x 1 ,x 2 ) by the loop λ (Y∐X,x 1 ,x 2 ) in (8.3.1.1) above. The resulting operadic composition
ζ Y = λ (Y∐X,x 1 ,x 2 ) ○ θ (Y,X) ○ 2 θ (x 1 ,x 2 ) ○ 2 ω x 1 ○ 2 ω x 2 ∈ UWD Y Y (8.3.2.1)
involves 5 generators: one loop, two 2-cells, and two 1-output wires. It corresponds to the following picture.
Y Y ⋮ x 1 x 2 The intermediate gray box is Y ∐ X = Y ∐ {x 1 , x 2 }.
Roughly speaking, the operadic composition (8.3.2.1) says that a wasted cable can be created by applying a loop to two 1-output wires. Additional wasted cables can similarly be created using more 2-cells, 1-output wires, and loops. Example 8.3.3. The undirected wiring diagram ζ Y in Example 8.3.2 can also be created as in the following picture.
Y Y ⋮ x 1 x 1 x 2
The inner gray box is Y ∐ x 1 , and the outer gray box is Y ∐ X. In terms of the generators, the above picture is realized as the operadic composition
ζ Y = λ (Y∐X,x 1 ,x 2 ) ○ σ (Y∐X,x 1 ,x 2 ) ○ θ (Y,x 1 ) ○ 2 ω x 1 ∈ UWD Y Y . (8.3.3.1)
It involves one loop, one split, one 2-cell, and one 1-output wire. Roughly speaking, the operadic composition (8.3.3.1) says that a wasted cable can be created by applying a loop to a split that is attached to a 1-output wire. 1) and (8.3.3.1). These two decompositions of ζ Y are actually connected as follows.
λ (Y∐X,x 1 ,x 2 ) ○ θ (Y,X) ○ 2 θ (x 1 ,x 2 ) ○ 2 ω x 1 ○ 2 ω x 2 by (8.3.2.1) = λ (Y∐X,x 1 ,x 2 ) ○ θ (Y∐x 1 ,x 2 ) ○ 1 θ (Y,x 1 ) ○ 2 ω x 1 ○ 2 ω x 2 by elem. rel. (8.2.9.1) = λ (Y∐X,x 1 ,x 2 ) ○ θ (Y∐x 1 ,x 2 ) ○ 1 θ (Y,x 1 ) ○ 2 ω x 1 ○ 2 ω x 2 by vertical ass. (2.1.10.4) = λ (Y∐X,x 1 ,x 2 ) ○ θ (Y∐x 1 ,x 2 ) ○ 2 ω x 2 ○ 1 θ (Y,x 1 ) ○ 2 ω x 1 by horizontal ass. (2.1.10.3) = λ (Y∐X,x 1 ,x 2 ) ○ θ (Y∐x 1 ,x 2 ) ○ 2 ω x 2 ○ θ (Y,x 1 ) ○ 2 ω x 1 by vertical ass. (2.1.10.4) = λ (Y∐X,x 1 ,x 2 ) ○ σ (Y∐X,x 1 ,x 2 ) ○ θ (Y,x 1 ) ○ 2 ω x
Summary of Chapter 8
(1) There are six generating undirected wiring diagrams.
(2) There are seventeen elementary relations in UWD.
(3) Wasted cables can arise from operadic composition of undirected wiring diagrams with no wasted cables.
Chapter 9
Decomposition of Undirected Wiring Diagrams
This chapter is the undirected analogue of Chapter 4. As part of the finite presentation theorem for the operad UWD of undirected wiring diagrams (Theorem 7.3.14), in Theorem 10.1.12 we will observe that each undirected wiring diagram has a highly structured decomposition in terms of generators (Def. 8.1.7), called a stratified presentation (Def. 10.1.8). Stratified presentations are also needed to establish the second part of the finite presentation theorem for the operad UWD regarding relations (Theorem 10.2.7). The purpose of this chapter is to provide all the steps needed to establish the existence of a stratified presentation for each undirected wiring diagram. We remind the reader about Notation 3.3.1 for (iterated) operadic compositions.
Fix a class S, with respect to which the operad UWD of undirected wiring diagrams (Def. 7.3.14) is defined.
A Motivating Example
Before we establish the desired decomposition of a general undirected wiring diagram, in this section we consider an elaborate example that will serve as a guide and motivation for the construction later in this chapter for the general case. The point of this decomposition is to break the complexity of a general undirected wiring diagram into several stratified pieces, each of which is easy to understand and visualize.
185
The following notations regarding subsets of cables will be used frequently in this chapter. Recall that an (m, n)-cable is a cable to which exactly m input wires and exactly n output wires are soldered (Def. 7.1.2).
Notation 9.1.1. Suppose ψ = (C ψ , f ψ , g ψ ) is an undirected wiring diagram and m, n ≥ 0. Define:
• C (m,n) ψ ⊆ C ψ as the subset of (m, n)-cables. In particular, C (0,0) ψ is the set of wasted cables.
• C (≥m,n) ψ ⊆ C ψ as the subset of (l, n)-cables with l ≥ m.
• C (m,≥n) ψ ⊆ C ψ as the subset of (m, k)-cables with k ≥ n.
• C (≥m,≥n) ψ ⊆ C ψ as the subset of (j, k)-cables with j ≥ m and k ≥ n.
• C ≥3 ψ ⊆ C ψ as the subset of (k, l)-cables with k, l ≥ 1 and k + l ≥ 3.
A cable in C (≥m,n) ψ is called an (≥ m, n)-cable, and similarly for cables in the other subsets defined above.
As in the case of wiring diagrams (see Convention 4.2.4), name changes (Def. 8.1.3) are easy to deal with. Therefore, in the following example, to keep the presentation simple, we will ignore name changes.
Example 9.1.2. Consider ϕ = C ϕ , f ϕ , g ϕ ∈ UWD Y X 1 ,X 2 in (7.1.7.1), which is visualized as
y 1 y 2 y 3 y 4 y 5 y 6 Y x 1 x 2 x 3 x 4 x 5 x 6 x 2 x 1 c 1 c 2 c 3 c 4 c 5
c 6 c 7 with X 1 = {x 1 , . . . , x 6 } and X 2 = {x 1 , x 2 }. We can decompose it as
ϕ = ϕ 1 ○ ϕ 2 (9.1.2.1)
as indicated in the following picture.
Y y 1 y 2 y 3 y 4 y 5 y 6 Z x 1 x 2 x 3
x 4 x 5 x 6
x 2
x 1 c 1 x 1 x 2 x 3 x 1 c 4+ c 4− c 5 x 2 x 4 x 5 x 6 c 7 c 1 c 2 c 3 c 4 c 5 c 6 c 7
As before the intermediate gray box Z indicates that an operadic composition occurs along it. The box Z is defined as
Z = X ∐ {c 4+ , c 4− } ∐ {c 1 , c 5 } ∐ {c 7 } ∈ Fin S
in which:
• X = X 1 ∐ X 2 .
• c 4+ and c 4− are two copies of the wasted cable c 4 in ϕ, so {c 4+ ,
c 4− } = C (0,0) ϕ ∐ C (0,0) ϕ . • {c 1 , c 5 } = C (0,≥1) ϕ . • {c 7 } = C (1,0) ϕ .
So we may also write Z as In the decomposition (9.1.2.1) of ϕ, the inside undirected wiring diagram is the cospan
Z = X ∐ C (0,0) ϕ,± ∐ C (0,≥1) ϕ ∐ C (1,0) ϕ (9.1.2.2) in which C (0,0) ϕ,± = C (0,0) ϕ ∐ C (0,ϕ 2 = X inclusion G G Z Z = o o ∈ UWD Z X 1 ,X 2 .
Note that in ϕ 2 :
• All the input wires-i.e., those in X-are soldered to (1, 1)-cables.
• All other cables-i.e., those in C
(0,0) ϕ,± ∐ C (0,≥1) ϕ ∐ C (1,0) ϕ -are (0, 1)-cables.
• There are no wasted cables.
As we will see later in (9.3.8.1), such an undirected wiring diagram can be decomposed into 2-cells (Def. 8.1.4) and 1-output wires (Def. 8.1.2). For example, this ϕ 2 needs:
• five 1-output wires, exactly as many as the number of (0, 1)-cables;
• six 2-cells, where 6 is the number of input boxes plus the number of (0, 1)cables minus 1.
The outside undirected wiring diagram in the decomposition ϕ = ϕ 1 ○ ϕ 2 is the cospan
ϕ 1 = Z ( fϕ,ι) G G C ϕ Y gϕ o o ∈ UWD Y Z .
Here:
• f ϕ ∶ X G G C ϕ is the input soldering function of ϕ. • ι ∶ C (0,0) ϕ,± ∐ C (0,≥1) ϕ ∐ C (1,0) ϕ G G C ϕ is the inclusion map on each coproduct summand. • g ϕ ∶ Y G G C ϕ is the output soldering function of ϕ.
• Every cable is an (m, n)-cable with m ≥ 1 and n ≥ 0. In other words, every cable in ϕ 1 is soldered to some input wires, so in particular there are no wasted cables in ϕ 1 .
• There are also no (1, 0)-cables, but there are (≥ 2, 0)-cables.
As we will see later, such an undirected wiring diagram can be decomposed into loops (Def. 8.1.5) and splits (Def. 8.1.6). In the case of ϕ 1 , which is the undirected wiring diagram
Y y 1 y 2 y 3 y 4 y 5 y 6 Z x 1 x 2 x 3 x 4 x 5 x 6 x 2 x 1 c 1 c 4+ c 4− c 5 c 7 c 1 c 2 c 3 c 4 c 5
c 6 c 7 this further decomposition
ϕ 1 = φ 1 ○ φ 2 (9.1.2.3)
can be visualized as follows.
Z x 1 x 2 x 3 x 4 x 5 x 6 x 2 x 1 c 1 c 4+ c 4− c 5 c 7 W Y y 1 y 2 y 3
y 4 y 5 y 6 (9.1.
2.4)
In this decomposition ϕ 1 = φ 1 ○ φ 2 , the inner undirected wiring diagram is the cospan
φ 2 = Z = G G Z W g φ 2 o o ∈ UWD W Z
with g φ 2 surjective. So every cable in φ 2 is a (1, n)-cable for some n ≥ 1 . As we will see later, such an undirected wiring diagram is generated by splits (Def. 8.1.6). For example, this φ 2 is the iterated operadic composition of 5 splits-one for the cable soldered to c 1 , one for the cable soldered to x 5 , and three for the cable soldered to x 2 .
The outer undirected wiring diagram in the decomposition (9.1.2.3) is the cospan
φ 1 = W G G C φ 1 Y o o ∈ UWD Y W
in which every cable is either a (1, 1)-cable or a (2, 0)-cable. We will show later that such an undirected wiring diagram is generated by loops (Def. 8.1.5). For example, this φ 1 is the iterated operadic composition of 6 loops, where 6 is the number of (2, 0)-cables in φ 1 .
In summary, we decompose ϕ ∈ UWD Y X 1 ,X 2 as the iterated operadic composition (9.1.2.5)
ϕ = ϕ 1 ○ ϕ 2 = φ 1 ○ φ 2 ○ ϕ 2 = λ, . . . ,
Here λ, σ, θ, and ω denote a loop, a split, a 2-cell, and a 1-output wire, respectively. This decomposition in terms of the generators is called a stratified presentation (Def. 10.1.8). In the next few sections, we will establish all the steps needed to obtain a stratified presentation for a general undirected wiring diagram.
Factoring Undirected Wiring Diagrams
In this section, using Example 9.1.2 as a guide and motivation, we establish a decomposition of a general undirected wiring diagram into two simpler undirected wiring diagrams (Theorem 9.2.3). This is the general version of the decomposition (9.1.2.1) above. Each undirected wiring diagram in this decomposition will be decomposed further, eventually leading to the desired stratified presentation.
Assumption 9.2.1.
Suppose
ψ = X f ψ G G C ψ Y g ψ o o ∈ UWD Y X (9.2.1.1)
is a general undirected wiring diagram with:
• output box Y ∈ Fin S and input boxes X = (X 1 , . . . , X N ) for some N ≥ 0;
• X = X 1 ∐ ⋯ ∐ X N ∈ Fin S .
Recall Notation 9.1.1 for certain subsets of cables. The undirected wiring diagrams ψ 1 and ψ 2 in the next definition are the general versions of ϕ 1 and ϕ 2 in the decomposition (9.1.2.1) above.
Definition 9.2.2. Suppose ψ = C ψ , f ψ , g ψ ∈ UWD Y
X is a general undirected wiring diagram as in (9.2.1.1) with X = (X 1 , . . . , X N ). (2) Define the undirected wiring diagram
ψ 1 = Z ( f ψ ,ι) G G C ψ Y g ψ o o ∈ UWD Y Z (9.2.2.2) in which C (0,0) ψ,± ∐ C (0,≥1) ψ ∐ C (1,0) ψ ι G G C ψ
is the inclusion map on each coproduct summand.
(3) Define the undirected wiring diagram
ψ 2 = X inclusion G G Z Z = o o ∈ UWD Z X .(ψ = ψ 1 ○ ψ 2 ∈ UWD Y X . (9.2.3.1)
Proof. By the definition of ○ = ○ 1 (Def. 7.3.5), the operadic composition ψ 1 ○ ψ 2 is given by the cospan
Y g ψ g ψ Ð Ð Z ( f ψ ,ι) G G = C ψ = X inclusion G G f ψ S S Z ( f ψ ,ι) G G C ψ
in Fin S . The square is a pushout by Example 7.2.3. This cospan is equal to ψ.
Example 9.2.4. If ψ = ǫ ∈ UWD ∅ is the empty cell (Def. 8.1.1), then:
• ψ 1 = 1 ∅ , the ∅-colored unit (Def. 7.3.2) with ∅ ∈ Fin S the empty box;
• ψ 2 = ǫ.
So in this case the decomposition (9.2.3.1) simply says ǫ = 1 ∅ ○ ǫ.
Remark 9.2.5. In the decomposition (9.2.3.1), both ψ 1 and ψ 2 are simpler than ψ for the following reasons.
(1) ψ 1 has the same set of cables C ψ and the same output soldering function g ψ as ψ. Furthermore, its input soldering function ( f ψ , ι) includes the input soldering function f ψ of ψ. However, every cable in ψ 1 is soldered to at least one input wire (i.e., C (0,≥0)
ψ 1 = ∅), whereas C (0,≥0)
ψ may be non-empty. In particular, ψ 1 has no wasted cables, even though ψ may have some. Furthermore, ψ 1 has only one input box Z, while ψ has N ≥ 0 input boxes.
(2) ψ 2 has the same input boxes X as ψ. However, it is, in general, much simpler than ψ and ψ 1 because its cables are either (1, 1)-cables or (0, 1)cables. In particular, ψ 2 also has no wasted cables.
(3) Neither ψ 1 nor ψ 2 has any (1, 0)-cables, even though ψ may have some.
The Inner Undirected Wiring Diagram
The purpose of this section is to analyze the undirected wiring diagram ψ 2 in the decomposition (9.2.3.1). The undirected wiring diagram ψ 1 will be studied in the next few sections. We begin with the following observation regarding iterated operadic compositions of 2-cells (Def. 8.1.4).
... X 1 ⋯ X n ⋯
can be generated by 2-cells.
Proposition 9.3.2. Suppose n ≥ 2, X i ∈ Fin S for 1 ≤ i ≤ n, and X = ∐ n i=1 X i . Then the undirected wiring diagram
Θ = X = G G X X = o o ∈ UWD X X 1 ,...,Xn (9.3.2.1)
is:
• a 2-cell if n = 2;
• an iterated operadic composition
Θ = θ 1 ○ 2 θ 2 ○ 3 ⋯ ○ n−1 θ n−1 with each θ j a 2-cell if n ≥ 3.
Proof. This is proved by induction on n ≥ 2. The initial case simply says that Θ is the 2-cell θ (X 1 ,X 2 ) by Def. 8.1.4. Suppose n ≥ 3. By the definition of ○ n−1 (Def. 7.3.5) and Example 7.2.3, we may decompose Θ as
Θ = Θ 1 ○ n−1 θ (X n−1 ,Xn) in which Θ 1 = X = G G X X = o o
∈ UWD X X 1 ,...,X n−2 ,X n−1 ∐Xn and θ (X n−1 ,Xn) ∈ UWD X n−1 ∐Xn X n−1 ,Xn is a 2-cell. Since the induction hypothesis applies to Θ 1 , the proof is finished.
Example 9.3.3. In the previous Proposition:
(1) If n = 3, then Θ decomposes as
Θ = θ (X 1 ,X 2 ∐X 3 ) ○ 2 θ (X 2 ,X 3 )
into two 2-cells.
(2) If n = 4, then Θ decomposes as
Θ = θ (X 1 ,X 2 ∐X 3 ∐X 4 ) ○ 2 θ (X 2 ,X 3 ∐X 4 ) ○ 3 θ (X 3 ,X 4 )
into three 2-cells.
Notation 9.3.4. In the context of (9.2.2.1), write:
• C ′ ψ = C (0,0) ψ,± ∐ C (0,≥1) ψ ∐ C (1,0) ψ ∈ Fin S , so Z = X ∐ C ′ ψ . • p = C ′ ψ .
The following observation covers the marginal cases for ψ 2 .
Lemma 9.3.5. For
ψ 2 = X inclusion G G Z Z = o o ∈ UWD Z X in (9.2.2.3):
(1) If N = p = 0, then ψ 2 is the empty cell ǫ (Def. 8.1.1).
(2) If N = 0 and p = 1, then ψ 2 is a 1-output wire (Def. 8.1.2).
(3) If N = 1 and p = 0, then ψ 2 is the X 1 -colored unit (Def. 7.3.2).
Proof. Since X = (X 1 , . . . , X N ) and X = X 1 ∐ ⋯ ∐ X N , all three statements follow immediately from the definition of ψ 2 .
The next observation covers the other cases for ψ 2 . Recall C ′ ψ in Notation 9.3.4.
Motivation 9.3.6. The following result says that an undirected wiring diagram of the form ... • N, p ≥ 1, and C ′ ψ = {c 1 , . . . , c p };
• ω j = ∅ G G c j c j = o o ∈ UWD c j is the 1-output wire for c j (Def. 8.1.2) for 1 ≤ j ≤ p.
Then there is a decomposition
ψ 2 = Θ ○ N+1 ω 1 ⋯ ○ N+1 ω p ∈ UWD Z X (9.3.7.1)
in which every pair of parentheses starts on the left and
Θ = Z = G G Z Z = o o ∈ UWD Z X 1 ,...,X N ,c 1 ,...,cp .
Proof. The right side of (9.3.7.1) is a well-defined element in Z X . By the correspondence between the ○ i -compositions and γ (2.1.12.1) in the operad UWD, the right side of (9.3.7.1) can be rewritten as
ψ ′ 2 = γ Θ; 1 X 1 , . . . , 1 X N , ω 1 , . . . , ω p . Since Z = X ∐ {c 1 , . . . , c p }, by Prop. 7.3.16 the cospan for ψ ′ 2 is Z = Z = = G G Z = X inclusion G G Z = G G Z in Fin S .
This is equal to the cospan that defines ψ 2 .
The following observation says that, if N, p ≥ 1, then ψ 2 is generated by 2-cells and 1-output wires.
Corollary 9.3.8. Suppose ψ 2 ∈ UWD Z X in (9.2.2.3) has N = X , p = C ′ ψ ≥ 1.
Then there is a decomposition
ψ 2 = θ 1 ○ 2 θ 2 ○ 3 ⋯ ○ N+p−1 θ N+p−1 ○ N+1 ω 1 ⋯ ○ N+1 ω p (9.3.8.1)
with:
• each θ i a 2-cell;
• each ω j a 1-output wire;
• each pair of parentheses starting on the left.
Proof. This is true by the decomposition (9.3.7.1) above and Prop. 9.3.2 with n = N + p ≥ 2, applied to Θ.
The Outer Undirected Wiring Diagram
The purpose of this section is to establish a decomposition for the undirected wiring diagram ψ 1 (9.2.2.2) that appeared in (9.2.3.1). This is the general version of the decomposition (9.1.2.3), so the reader may wish to refer back there for specific examples of the constructions below. Each of the constituent undirected wiring diagrams in this decomposition will be studied further in later sections. The goal is to decompose ψ 1 into two undirected wiring diagrams in which the outer one, called φ 1 below, is generated by loops (Def. 8.1.5), while the inner one, called φ 2 below, is generated by splits (Def. 8.1.6).
Recall Notation 9.1.1 for certain subsets of cables. Also recall from Remark 9.2.5 that ψ 1 ∈ UWD Y Z has neither (0, ≥ 0)-cables nor (1, 0)-cables. So ψ 1 satisfies the hypotheses of the next definition.
Definition 9.4.1. Suppose ϕ = C ϕ , f ϕ , g ϕ ∈ UWD B
A is an undirected wiring diagram with
• one input box A and
• C (0,≥0) ϕ = ∅ = C (1,0) ϕ .
We will write f ϕ and g ϕ as f and g, respectively.
(1) For each cable c ∈ C
(≥3,0) ϕ ∐ C ≥3 ϕ , choose a wire a c ∈ f −1 c ⊆ A, where f −1 c = f −1 ({c}) is the set of f -preimages of c.
(2) Define
W = B ∐ f −1 C (2,0) ϕ ∐ ∐ c∈C (≥3,0) ϕ ∐C ≥3 ϕ f −1 c ∖ a c ± ∈ Fin S (9.4.1.1) in which f −1 c ∖ a c ± = f −1 c ∖ a c + ∐ f −1 c ∖ a c −
is the coproduct of two copies of f −1 c ∖ a c . This W is the general version of the W in the example (9.1.2.4).
(3) Define
V = B ∐ C (2,0) ϕ ∐ ∐ c∈C (≥3,0) ϕ ∐C ≥3 ϕ f −1 c ∖ a c ∈ Fin S . (9.4.1.2)
This V is the general version of the set of cables between W and Y in the example (9.1.2.4).
(4) Define
φ 1 = W f 1 G G V B g 1 inclusion o o ∈ UWD B W (9.4.1.3)
in which the restrictions of f 1 to the coproduct summands of W are defined as follows.
• f 1 ∶ B G G B is the identity map. • f 1 ∶ f −1 C (2,0) ϕ G G C (2,0) ϕ is the map f . • f 1 ∶ f −1 c ∖ a c ± G G f −1 c ∖ a c is the fold map for each c ∈ C (≥3,0) ϕ ∐ C ≥3
ϕ . That is, the restriction of f 1 to each of f −1 c ∖ a c + and f −1 c ∖ a c − is the identity map. This φ 1 is the general version of that in the example (9.1.2.4).
(5) Define
φ 2 = A f 2 = G G A W g 2 o o ∈ UWD W A (9.4.1.4)
as follows. We will use the equality
B = g −1 C (1,≥1) ϕ ∐ g −1 C (≥2,≥1) ϕ which is true because C (0,≥0) ϕ = ∅ = C (1,0) ϕ . For w ∈ W = g −1 C (1,≥1) ϕ ∐ g −1 C (≥2,≥1) ϕ ∐ f −1 C (2,0) ϕ ∐ ∐ c∈C (≥3,0) ϕ ∐C ≥3 ϕ f −1 c ∖ a c ± ∈ Fin S (9.4.1.5) define g 2 (w) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ f −1 g(w) ∈ A if w ∈ g −1 C (1,≥1) ϕ ; a g(w) ∈ f −1 g(w) ⊆ A if w ∈ g −1 C (≥2,≥1) ϕ ; w if w ∈ f −1 C (2,0) ϕ or w ∈ f −1 c ∖ a c + ; a c ∈ f −1 c ⊆ A if w ∈ f −1 c ∖ a c − .
In the first line of this definition, we used the fact that each cable in C (2) The output soldering function g 2 of φ 2 is surjective because of the assumption C Proof. It suffices to check that the operadic composition φ 1 ○ φ 2 ∈ UWD B A is given by the cospan
B g 1 =inclusion g v v W = B ∐ f −1 C (2,0) ϕ ∐ ∐ f −1 c ∖ ac ± g 2 f 1 (Id, f ,fold) G G V = B ∐ C (2,0) ϕ ∐ ∐ f −1 c ∖ ac g 3 =(g,incl., f ) A f 2 = G G f R R A f G G Cϕ
in Fin S . Here the two coproducts ∐ are both indexed by all c ∈ C (≥3,0) ϕ ∐ C ≥3 ϕ . By the definition of ○ = ○ 1 (7.3.5.1), we just need to check that the rectangle is a pushout (Def. 7.2.1) in Fin S . It follows from direct inspection of each coproduct summand of W in (9.4.1.5) that the rectangle is commutative.
Next, suppose given a solid-arrow commutative diagram
W g 2 f 1 G G V g 3 β A α G G f G G C ϕ h 3 3 U
in Fin S . We must show that there exists a unique map h that makes the diagram commutative. Recall that
C ϕ = C (1,1) ϕ ∐ C (2,0) ϕ ∐ C (≥3,0) ϕ ∐ C ≥3 ϕ because C (0,≥0) ϕ = ∅ = C (1,0) ϕ . Define h ∶ C ϕ G G U as h(c) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ α f −1 (c) if c ∈ C (1,1) ϕ ; β(c) if c ∈ C (2,0) ϕ ⊆ V; α(a c ) if c ∈ C (≥3,0) ϕ ∐ C ≥3 ϕ .
One checks by direct inspection that (i) h f = α and hg 3 = β and that (ii) h is the only such map.
As we mentioned just before Def. 9.4.1, the decomposition (9.4.3.1) applies to ψ 1 ∈ UWD Y Z defined in (9.2.2.2). In the next two sections, we will show that, up to name changes, φ 1 is generated by loops (Prop. 9.6.2), and φ 2 is generated by splits (Prop. 9.5.3).
Iterated Splits
The purpose of this section is to show that φ 2 (9.4.1.4) is either a name change or is generated by splits. First let us adopt the following convention, which is the undirected version of Convention 4.2.4. with f ρ = Id A and g ρ surjective.
(
1) If A = ∅, then ρ = 1 ∅ ∈ UWD ∅ ∅ (Def. 7.3.2). (2) Suppose A = ∅.
(i) If g ρ is a bijection, then ρ is a name change τ A,B .
(ii) Otherwise, ρ is an iterated operadic composition of splits (Def. 8.1.6).
Proof.
We will write f ρ and g ρ as f and g, respectively. If A = ∅, then ρ is the cospan
( ∅ G G ∅ ∅ o o )
, which is the ∅-colored unit in UWD. Suppose A = ∅. If g is a bijection, then by definition g is the name change τ g −1 .
So suppose g is surjective but is not a bijection. We must show that ρ is an iterated operadic composition of splits. The first step is to decompose ρ in such a way that each constituent undirected wiring diagram creates one group of output
wires g −1 a i . Decompose A as A = A 1 ∐ A 2 ∈ Fin S in which • A 1 = a ∈ A ∶ g −1 a = 1 ; • A 2 = a ∈ A ∶ g −1 a ≥ 2 = {a 1 , .
. . , a n }.
By assumption A 2 = ∅. To decompose ρ we will use the following intermediate boxes. For each 1 ≤ i ≤ n + 1, define
D i = A 1 ∐ ∐ 1≤k<i g −1 a k ∅ if i = 1 ∐ {a i , . . . , a n } ∅ if i = n + 1 ∈ Fin S .
Note that D 1 = A and D n+1 ≅ B.
For 1 ≤ i ≤ n define ρ i = D i f i = G G D i D i+1 g i o o ∈ UWD D i+1 D i (9.5.3.2) in which, for d ∈ D i+1 , g i (d) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a i if d ∈ g −1 a i , d otherwise.
A direct inspection using Example 7.2.3 and Prop. 7.3.6 shows that, up to a name change, there is a decomposition
ρ = ρ n ○ ⋯ ○ ρ 1 ∈ UWD B A . (9.5.3.3)
When n = 2, this decomposition is depicted in the following picture.
A a 1 a 2 ⋮ g −1 a 1 ⋮ g −1 a 2 ⋮
To finish the proof, it suffices to show that each ρ i is an iterated operadic composition of splits. We will prove this assertion in the next result.
π = D ∐ b = G G D ∐ b D ∐ g −1 b g o o ∈ UWD D∐g −1 b D∐b (9.5.5.1)
such that
• g D = Id D and
• p = g −1 b ≥ 2.
Then π is an iterated operadic composition of p − 1 splits.
Proof. Write g −1 b = {b 1 , . . . , b p }. If p = 2 then π is the split σ (D∐{b 1 ,b 2 },b 1 ,b 2 ) by definition.
Suppose p ≥ 3. Then there is a decomposition of π into p − 1 splits as [2,p] . (9.5.5.2)
π = σ D∐{b 1 ,...,bp},b p−1 ,bp ○ ⋯ ○ σ D∐{b 1 ,b [2,p] },b 1 ,b
Here each b [j,p] means the wires b l for j ≤ l ≤ p are identified into one element. Starting from the right, the jth split in the above decomposition of π, namely [j,p] in Fin S . Here the output soldering function sends b j and b [j+1,p] to b [j,p] and is the identity function everywhere else.
σ D∐ b 1 ,...,b j ,b [j+1,p] ,b j ,b [j+1,p] ∈ UWD D∐{b 1 ,...,b j ,b [j+1,p] } D∐{b 1 ,...,b j−1 ,b [j,p] } , is the cospan D ∐ b 1 , . . . , b j , b [j+1,p] D ∐ b 1 , . . . , b j−1 , b [j,p] = G G D ∐ b 1 , . . . , b j−1 , b
Example 9.5.6. For each 1 ≤ i ≤ n the undirected wiring diagram ρ i ∈ UWD D i+1 D i in (9.5.3.2) is of the form π (9.5.5.1) with
D = A 1 ∐ ∐ 1≤k<i g −1 a k ∐ {a i+1 , .
. . , a n } and b = a i . Therefore, ρ in (9.5.3.3) is an iterated operadic composition of splits.
Example 9.5.7. In (9.5.5.2) above:
(1) If p = 3, then π decomposes into two splits as [2,3] .
π = σ (D∐{b 1 ,b 2 ,b 3 },b 2 ,b 3 ) ○ σ D∐{b 1 ,b [2,3] },b 1 ,b
(2) If p = 4, then π decomposes into three splits as [2,4] .
π = σ (D∐{b 1 ,b 2 ,b 3 ,b 4 },b 3 ,b 4 ) ○ σ D∐{b 1 ,b 2 ,b [3,4] },b 2 ,b [3,4] ○ σ D∐{b 1 ,b [2,4] },b 1 ,b
Iterated Loops
The purpose of this section is to show that φ 1 (9.4.1.3) is either a name change or is generated by loops (Def. 8.1.5). Recall Convention 9.5.1 regarding name changes. Also recall from Remark 9.4.2 that in φ 1 (9.4.1.3) each cable is either a (1, 1)-cable or a (2, 0)-cable. Therefore, the following result applies to φ 1 .
Motivation 9.6.1. The following result says that an undirected wiring diagram of the form ⋮ ⋯ can be generated by loops.
Proposition 9.6.2. Suppose A, B, C ∈ Fin S and
ξ = A f G G C B g inclusion o o ∈ UWD B A
in which each cable is either a (1, 1)-cable or a (2, 0)-cable. Suppose ξ has q (2, 0)-cables. Then:
(1) ξ is a name change if q = 0.
(2) ξ is the iterated operadic composition of q loops (Def. 8.1.5) if q ≥ 1.
Proof. Since ξ only has (1, 1)-cables and (2, 0)-cables, up to a name change we may write it as the cospan
ξ = B ∐ T ± f (Id,fold) G G B ∐ T B g inclusion o o
with:
• T the set of (2, 0)-cables in ξ;
• f −1 T = T ± = T + ∐ T − the coproduct of two copies of T.
If q = 0 (i.e., T = ∅), then ξ is the B-colored unit (Def. 7.3.2).
If q = 1 with T = {t} and T ± = {t ± }, then ξ is the loop λ (B∐{t±},t±) by definition.
Suppose q ≥ 2. We may write T = {t 1 , . . . , t q } and T ± = t 1 ± , . . . , t q ± . The picture
t 1 + t 1 − t 2 + t 2 −
⋮ depicts a decomposition of ξ into two loops when q = 2. A direct inspection shows that there is a decomposition of ξ into q loops as
ξ = λ B∐t q ± ,t q ± ○ ⋯ ○ λ B∐{t 1 ± ,...,t q ± },t 1 ± .
(9.6.2.1)
Starting from the right, the jth loop in the above decomposition of ξ, namely
λ B∐{t j ± ,...,t q ± },t j ± ∈ UWD B∐{t j+1 ± ,...,t q ± } B∐{t j ± ,...,t q ± } , is the cospan B ∐ t j+1 ± , . . . , t q ± inclusion B ∐ t j ± , . . . , t q ± t j ± ↦t j G G B ∐ t j+1 ± , . . . , t q ± ∐ t j in Fin S .
Here the input soldering function sends t j ± to t j and is the identity function everywhere else.
Example 9.6.3. In (9.6.2.1) above:
(1) If q = 2, then ξ decomposes into two loops as
ξ = λ B∐t 2 ± ,t 2 ± ○ λ B∐{t 1 ± ,t 2 ± },t 1 ± .
(2) If q = 3, then ξ decomposes into three loops as
ξ = λ B∐t 3 ± ,t 3 ± ○ λ B∐{t 2 ± ,t 3 ± },t 2 ± ○ λ B∐{t 1 ± ,t 2 ± ,t 3 ± },t 1 ± .
Summary of Chapter 9
Every undirected wiring diagram ϕ has a decomposition ϕ = φ 1 ○ φ 2 ○ ϕ 2 in which:
• φ 1 is generated by loops;
• φ 2 is generated by splits;
• ϕ 2 is generated by 2-cells and 1-output wires.
Chapter 10
Finite Presentation for Undirected Wiring Diagrams
Fix a class S, with respect to which the Fin S -colored operad UWD of undirected wiring diagrams is defined (Theorem 7.3.14). The main purpose of this chapter is to establish a finite presentation for the operad UWD; see Theorem 10.2.7. This means the following two statements. This finite presentation theorem for UWD is the undirected analogue of the finite presentation theorem for WD (Theorem 5.2.11). As in the directed case, this result leads to a finite presentation theorem for UWD-algebras, which we will discuss in Chapter 11.
We will continue to use Notation 3.3.1 for (iterated) operadic compositions.
203
Stratified Presentation
In this section, we define a stratified presentation in UWD and show that every undirected wiring diagram has a stratified presentation (Theorem 10.1.12). The following definition is the undirected analogue of Def. 5.1.2.
Motivation 10.1.1. A simplex below is a finite parenthesized word whose alphabets are generating undirected wiring diagrams, in which each pair of parentheses has a well defined associated ○ i -composition. In particular, a simplex has a well defined operadic composition. As we have seen in Chapter 8, it is often possible to express an undirected wiring diagram as an operadic composition of generating undirected wiring diagrams in multiple ways. In other words, an undirected wiring diagram can have many different simplex presentations. We now start to develop the necessary language to say precisely that any two such simplex presentations of the same undirected wiring diagram are equivalent in some way.
Definition 10.1.2. Suppose n ≥ 1. An n-simplex Ψ and its composition Ψ ∈ UWD are defined inductively as follows.
(1) A 1-simplex is a generator (Def. 8.1.7) ψ. Its composition ψ is defined as ψ itself.
(2) Suppose n ≥ 2 and that k-simplices for 1 ≤ k ≤ n − 1 and their compositions in UWD are already defined. An n-simplex in UWD is a tuple Ψ = ψ, i, φ consisting of • an integer i ≥ 1, • a p-simplex ψ for some p ≥ 1, and • a q-simplex φ for some q ≥ 1 such that:
(i) p + q = n;
(ii) the operadic composition
Ψ def == ψ ○ i φ (10.1.2.1)
is defined in UWD (Def. 7.3.5). The undirected wiring diagram Ψ in (10.1.2.1) is the composition of Ψ.
A simplex in UWD is an m-simplex in UWD for some m ≥ 1. We say that a simplex Ψ is a presentation of the undirected wiring diagram Ψ . have the same composition.
(2) The elementary relation (8.2.7.1) says that the 4-simplex λ (W,w,y) ○ σ (W,x,w) ○ θ (X,y) ○ 2 ω y and the 1-simplex (1 X ) have the same composition. In other words, the former has composition 1 X .
Example 10.1.5. In (8.3.1.1) we considered the 5-simplex
λ (X,x 1 ,x 2 ) ○ θ (∅,X) ○ 2 θ (x 1 ,x 2 ) ○ 2 ω x 1 ○ 2 ω x 2 ,
whose composition has a wasted cable.
Example 10.1.6. In (8.3.2.1) we considered the 5-simplex
λ (Y∐X,x 1 ,x 2 ) ○ θ (Y,X) ○ 2 θ (x 1 ,x 2 ) ○ 2 ω x 1 ○ 2 ω x 2 ,
whose composition also has a wasted cable.
Motivation 10.1.7. If we think of a simplex as a parenthesized word whose alphabets are generating undirected wiring diagrams, then the stratified simplex in the next definition is a word where the same alphabets must occur in a consecutive string. For example, all the loops must occur together as a string (λ 1 , . . . , λ n ). Furthermore, we can even insist that these strings for different types of generating undirected wiring diagrams occur in a specific order, with name changes and loops at the top, followed by splits, 2-cells, and 1-output wires at the bottom. (1) (ǫ), where ǫ ∈ UWD ∅ is the empty cell (Def. 8.1.1).
(2) τ, λ, σ, θ, ω , where:
• τ is a name change (Def. 8.1.3);
• λ is a possibly empty string of loops (Def. 8.1.5);
• σ is a possibly empty string of splits (Def. 8.1.6);
• θ is a possibly empty string of 2-cells (Def. 8.1.4);
• ω is a possibly empty string of 1-output wires (Def. 8.1.2).
We call these stratified simplices of type (1) and of type (2), respectively. If Ψ is a stratified simplex in UWD, then we call it a stratified presentation of the undirected wiring diagram Ψ .
Remark 10.1.9. The composition of a stratified simplex of type (2) cannot be the empty cell ǫ. Indeed, if the composition of a stratified simplex of type (2) is in UWD ∅ , then 1-output wires must be involved. So the composition must have at least one cable, and it cannot be the empty cell. We now observe that the generators generate the operad UWD of undirected wiring diagrams in a highly structured way.
Theorem 10.1.12. Every undirected wiring diagram admits a stratified presentation.
Proof. Suppose ψ is an undirected wiring diagram. If ψ is the empty cell ǫ (Def. 8.1.1), then (ǫ) is a stratified presentation of ψ. So let us now assume that ψ is not the empty cell. We will show that it admits a stratified presentation of type (2). We remind the reader of Convention 9.5.1.
Combining (9.2.3.1) and (9.4.3.1), there is a decomposition
ψ = φ 1 ○ φ 2 ○ ψ 2 .
In this decomposition:
• The undirected wiring diagram ψ 2 either is a colored unit or has a stratified presentation (θ, ω) by Lemma 9.3.5 and Corollary 9.3.8.
• The undirected wiring diagram φ 2 either is a name change or has a stratified presentation (σ) by Prop. 9.5.3.
• The undirected wiring diagram φ 1 either is a name change or has a stratified presentation (λ) by Prop. 9.6.2.
By the above three statements, ψ has a stratified presentation of type (2).
Elementary Equivalences
The purpose of this section is to establish the second part of the finite presentation theorem for the operad UWD of undirected wiring diagrams. Recall (1) A subsimplex of Ψ is a simplex in UWD defined inductively as follows.
• If Ψ is a 1-simplex, then a subsimplex of Ψ is Ψ itself.
• Suppose n ≥ 2 and Ψ = ψ, i, φ for some i ≥ 1, p-simplex ψ, and qsimplex φ with p + q = n. Then a subsimplex of Ψ is a subsimplex of ψ, -a subsimplex of φ, or -Ψ itself. If Ψ ′ is a subsimplex of Ψ, then we write Ψ ′ ⊆ Ψ. (3) Suppose Φ is another simplex in UWD. Then Ψ and Φ are said to be equivalent if their compositions are equal; i.e., Ψ = Φ ∈ UWD.
(4) Suppose:
• Ψ ′ ⊆ Ψ is an elementary subsimplex corresponding to one side of R, which is either an elementary relation or an operad associativity/unity axiom for the generators in UWD. • Ψ" is the simplex given by the other side of R.
• Ψ 1 is the simplex obtained from Ψ by replacing the subsimplex Ψ ′ by Ψ". We say that Ψ and Ψ 1 are elementarily equivalent. Note that elementarily equivalent simplices are also equivalent.
(5) If Ψ and Φ are elementarily equivalent, we write Ψ ∼ Φ and call this an elementary equivalence.
(6) Suppose Ψ 0 , . . . , Ψ r are simplices in UWD for some r ≥ 1 and that there exist elementary equivalences
Ψ 0 ∼ Ψ 1 ∼ ⋯ ∼ Ψ r .
Then we say that Ψ 0 and Ψ r are connected by a finite sequence of elementary equivalences. Note that in this case Ψ 0 and Ψ r are equivalent. 1) repeatedly to compose them down into one name change. Therefore, after a finite sequence of elementary equivalences, we may assume that there is at most one name change in Ψ, which is the left-most entry. If there are further elementary equivalences later that create name changes, we will perform the same procedure without explicitly mentioning it.
The empty cell ǫ ∈ UWD ∅ (Def. 8.1.1) and a 1-output wire ω * ∈ UWD * (Def. 8.1.2) have no input boxes, so no operadic composition of the forms ǫ ○ i − or ω * ○ i − can be defined. Therefore, after a finite sequence of elementary equivalences corresponding to the operad horizontal associativity axiom (2.1.10.3), we may assume that Ψ has the form τ, Ψ 1 , ǫ, ω .
Here:
• τ is a name change;
• all the 1-output wires ω are at the right-most entries;
• all the empty cells ǫ are just to their left;
• Ψ 1 is either empty or is a subsimplex involving 2-cells (Def. 8.1.4), loops (Def. 8.1.5), and splits (Def. 8.1.6).
Next Suppose that, after the cancellation in the previous paragraph, the resulting string ǫ of empty cells is still non-empty. Then it must contain a single empty cell ǫ, and there are no 2-cells θ and no 1-output wires ω in the resulting simplex Ψ. Since the output box of ǫ is the empty box, the current simplex Ψ cannot have any loops λ or splits σ. Therefore, in this case the simplex (10.2.5.1) has the form τ, ǫ .
Since the output box of ǫ is the empty box ∅, this name change τ must be the colored unit 1 ∅ . So by the operad left unity axiom (2.1.10.5), the simplex 1 ∅ , ǫ is elementarily equivalent to (ǫ), which is a stratified simplex of type (1). The next step is to show that equivalent stratified simplices are connected. The following observation is the undirected analogue of Lemmas 5.2.9 and 5.2.10. Proof. Suppose Ψ 1 and Ψ 2 are equivalent stratified simplices in UWD. By Remark 10.1.9 Ψ 1 and Ψ 2 are both of type (1) or both of type (2). If they are both of type (1), then they are both equal to (ǫ) by definition. So let us now assume that Ψ 1 and Ψ 2 are distinct but equivalent stratified simplices of type (2) in UWD. The rest of the proof is similar to that of Lemma 5.2.10 and consists of a series of reductions.
Write ψ ∈ UWD Y X for the common composition of Ψ 1 and Ψ 2 . Using elementary equivalences corresponding to we may assume that there are no unnecessary generators in these stratified simplices. Here unnecessary refers to either a colored unit or generators whose (iterated) operadic composition is a colored unit. If ψ itself is a name change τ, then at this stage both Ψ 1 and Ψ 2 are equal to the 1-simplex (τ). So let us now assume that ψ is not a name change.
The name change τ 1 in Ψ 1 has output box Y, and the same is true for the name change τ 2 in Ψ 2 . We may actually assume that the input boxes of τ 1 and of τ 2 are also equal, provided that we change the output boxes of other generators in Ψ 2 accordingly by a name change if necessary. Such changes correspond to elementary equivalences coming from the operad unity and associativity axioms and the elementary relation (8.2.1.1). Using finitely many elementary equivalences, we may therefore assume that τ 1 is equal to τ 2 . So we may as well assume that there are no name changes in the two stratified simplices Ψ i . At this stage, each stratified simplex Ψ i has the form
λ i , σ i , θ i , ω i .
Using finitely many elementary equivalences corresponding to the elementary relation (8.2.6.1) and other elementary relations that move the generators around, we may assume that in the simplices Ψ 1 and Ψ 2 all the (1, 0)-cables in ψ are created as in the left side of (8.2.6.1), i.e., as λ, θ, ω rather than as λ, σ . In plain language, this means that every (1, 0)-cable in ψ is created in both Ψ i by applying a loop to a 1-output wire and some other wire, rather than by applying a loop to a split.
Similarly, using Example 8.3.4 and finitely many elementary equivalences, we may further assume that every wasted cable in ψ is created in both simplices Ψ i as λ, θ, θ, ω, ω as in (8.3.2.1), rather than as λ, σ, θ, ω as in (8.3.3.1). In plain language, this means that every wasted cable in ψ is created in both Ψ i by applying a loop to two 1-output wires, as in the first picture after (8.3.2.1).
Recall that each 1-output wire ω is a 0-ary element in UWD, while loops and splits are unary. Therefore, by the operad associativity axioms (2.1.10.3) and (2.1.10.4), we may assume that in each simplex Ψ i , the right portion θ i , ω i is a subsimplex. By the above reductions, at this stage the two simplices θ 1 , ω 1 and θ 2 , ω 2 are uniquely determined by ψ and have the same composition. In fact, each simplex θ i , ω i has composition ψ 2 (9.2.2.3). Using finitely many elementary equivalences corresponding to the elementary relations (8.2.9.1) and (8.2.10.1) and the operad associativity axioms, we may now assume that the simplices θ 1 , ω 1 and θ 2 , ω 2 are equal.
Similarly, at this stage the simplices λ 1 , σ 1 and λ 2 , σ 2 both have composition ψ 1 (9.2.2.2). Using finitely many elementary equivalences corresponding to the elementary relations (8.2.13.1)-(8.2.17.1) and the operad vertical associativity axiom (2.1.10.4), we may now assume that the simplices λ 1 , σ 1 and λ 2 , σ 2 are equal.
We are now ready for the finite presentation theorem for undirected wiring diagrams. It describes the operad UWD of undirected wiring diagrams (Theorem 7.3.14) in terms of finitely many generators and finitely many relations.
Summary of Chapter 10
(1) A simplex in UWD is a finite non-empty parenthesized word of generating undirected wiring diagrams in which each pair of parentheses is equipped with an operadic ○ i -composition.
(2) A stratified simplex in UWD is a simplex of one of the following two forms.
• (ǫ) • τ, λ, σ, θ, ω
(3) Every undirected wiring diagram has a stratified presentation.
(4) Two simplices in UWD are elementarily equivalent if one can be obtained from the other by replacing a subsimplex Ψ ′ by an equivalent simplex Ψ ′′ such that Ψ ′ = Ψ ′′ is either one of the seventeen elementary relations in UWD or an operad associativity/unity axiom involving only the six generating undirected wiring diagrams.
(5) Any two simplex presentations of a given undirected wiring diagram are connected by a finite sequence of elementary equivalences.
Algebras of Undirected Wiring Diagrams
The main purpose of this chapter is to provide a finite presentation theorem for algebras over the operad UWD of undirected wiring diagrams (Theorem 7.3.14).
As in the case of wiring diagrams (Theorem 6.2.2), this finite presentation for UWDalgebras is a consequence of the finite presentation for the operad UWD (Theorem 10.2.7). This finite presentation allows us to reduce the understanding of a UWDalgebra to just a few basic structure maps and a small number of easy-to-check axioms. We will illustrate this point further with the relational algebra of a set and the typed relational algebra.
In Section 11.1 we first define a UWD-algebra in terms of 6 generating structure maps and 17 generating axioms corresponding to the generators (Def. 8.1.7) and the elementary relations (Def. 8.2.18) in UWD. Then we observe that this finite presentation for a UWD-algebra is in fact equivalent to the general definition (Def. 6.1.3) of a UWD-algebra (Theorem 11.1.2). This is an application of the finite presentation theorem for the operad UWD (Theorem 10.2.7).
In Section 11.2 we provide, for each set A, a finite presentation for the UWDalgebra called the relational algebra of A (Theorem 11.2.5). In its original form, the relational algebra was the main algebra example in [Spi13]. In [Spi13] Spivak pointed out that the relational algebra of a set and its variant called the typed relational algebra, to be discussed in Section 11.4, have applications in digital circuits, machine learning, and database theory.
In Section 11.3 we prove a rigidity property of the relational algebra of a set (Theorem 11.3.3). It says that a given map of sets is a bijection precisely when it 213 induces a map between the two relational algebras. The motivation for this rigidity property comes from [Spi13] Section 3, where several examples suggest that there are very few interesting maps out of the relational algebra of a set. In fact, in [Spi13] Conjecture 3.1.6, Spivak conjectured that the relational algebra of any set is quotient-free. Although our rigidity result does not prove Spivak's conjecture in its full generality, it adds further evidence that the conjecture should be true.
In Section 11.4 we consider a generalization of the relational algebra of a set, called the typed relational algebra. We observe that, similar to the relational algebra of a set, the typed relational algebra has a finite presentation (Theorem 11.4.7). In its original form, the typed relational algebra was first defined in [Spi13] Section 4.
Finite Presentation for Algebras
The main purpose of this section is to prove a finite presentation theorem for UWDalgebras. We first define a UWD-algebra in terms of a finite number of generators and relations. Immediately afterwards we will show that this definition agrees with the general definition of an operad algebra (Def. 6.1.3) when the operad is UWD.
Fix a class S, with respect to which the Fin S -colored operad UWD of undirected wiring diagrams is defined (Theorem 7.3.14).
Definition 11.1.1. A UWD-algebra A consists of the following data. For each X ∈ Fin S , A is equipped with a set A X called the X-colored entry of A. Moreover, it is equipped with the following 6 generating structure maps corresponding to the generators in UWD (Def. 8.1.7).
(1) Corresponding to the empty cell ǫ ∈ UWD ∅ (Def. 8.1.1), it has a structure map * ǫ G G A ∅ , (11.1.1.1)
i.e., a chosen element in A ∅ .
(2) Corresponding to each 1-output wire ω * ∈ UWD * (Def. 8.1.2), it has a structure map * ω * G G A * , (11.1.1.2)
i.e., a chosen element in A * .
(3) Corresponding to each name change τ f ∈ UWD Y X (Def. 8.1.3), it has a structure map
A X τ f G G A Y (11.1.1.3)
that is, furthermore, the identity map if f is the identity map on X.
(4) Corresponding to each 2-cell θ (X,Y) ∈ UWD X∐Y X,Y (Def. 8.1.4), it has a structure map
A X × A Y θ (X,Y) G G A X∐Y .
(11.1.1.4) (5) Corresponding to each loop λ (X,x±) ∈ UWD X∖x± X (Def. 8.1.5), it has a structure map
A X λ (X,x ± ) G G A X∖x± .
(11.1.1.5) (6) Corresponding to each split σ (X,x 1 ,x 2 ) ∈ UWD X X ′ (Def. 8.1.6), it has a structure map
A X ′ σ (X,x 1 ,x 2 ) G G A X .
(11.1.1.6)
The following 17 diagrams, called the generating axioms, which correspond to the elementary relations in UWD (Def. 8.2.18), are required to be commutative.
(1) In the setting of (8.2.1.1), the diagram
A X τ f G G τ g f 4 4 ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ A Y τg A Z is commutative.
(2) In the setting of (8.2.2.1), the diagram *
ωx G G ωy 3 3 ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ A x τx,y A y is commutative.
(3) In the setting of (8.2.3.1), the diagram
A X 1 × A X 2 (τ f 1 ,τ f 2 ) θ (X 1 ,X 2 ) G G A X 1 ∐X 2 τ f 1 ∐ f 2 A Y 1 × A Y 2 θ (Y 1 ,Y 2 ) G G A Y 1 ∐Y 2
is commutative.
(4) In the setting of (8.2.4.1), the diagram
A X τ f λ (X,x ± ) G G A X∖x± τ f ′ A Y λ (Y,y ± ) G G A Y∖y± is commutative.
(5) In the setting of (8.2.5.1), the diagram
A X ′ τ f ′ σ (X,x 1 ,x 2 ) G G A X τ f A Y ′ σ (Y,y 1 ,y 2 ) G G A Y is commutative.
(6) In the setting of (8.2.6.1), the diagram
A X × * ≅ A X (Id,ωy) σ (Y,x,y) G G A Y λ (Y,x,y) A X × A y θ (X,y) G G A Y λ (Y,x,y) G G A W (11.1.1.7)
is commutative.
(7) In the setting of (8.2.7.1), the diagram
A X ≅ A X × * Id (Id,ωy) G G A X × A y θ (X,y) G G A Y σ (W,x,w) A X A W λ (W,w,y) o o is commutative.
(8) In the setting of (8.2.8.1), the diagram
A X Id ≅ G G A X × * (Id,ǫ) A X A X × A ∅ θ (X,∅) o o
is commutative.
(9) In the setting of (8.2.9.1), the diagram
A X × A Y × A Z (Id,θ (Y,Z) ) G G (θ (X,Y) ,Id) A X × A Y∐Z θ (X,Y∐Z) A X∐Y × A Z θ (X∐Y,Z) G G A X∐Y∐Z (11.1.1.8)
is commutative.
(10) In the setting of (8.2.10.1), the diagram
A Y × A X permute G G θ (Y,X) A X × A Y θ (X,Y) A Y∐X = G G A X∐Y (11.1.1.9)
is commutative.
(11) In the setting of (8.2.11.1), the diagram
A X × A Y Id,λ (Y,y ± ) θ (X,Y) G G A X∐Y λ (X∐Y,y ± ) A X × A Y ′ θ (X,Y ′ ) G G A X∐Y ′ is commutative.
(12) In the setting of (8.2.12.1), the diagram
A X × A Y ′ Id,σ (Y,y 1 ,y 2 ) θ (X,Y ′ ) G G A X∐Y ′ σ (X∐Y,y 1 ,y 2 ) A X × A Y θ (X,Y) G G A X∐Y is commutative.
(13) In the setting of (8.2.13.1), the diagram
A X ′ σ (Z,z 1 ,z 2 ) σ (Y,y 1 ,y 2 ) G G A Y σ (X,z 1 ,z 2 ) A Z σ (X,y 1 ,y 2 ) G G A X is commutative.
(14) In the setting of (8.2.14.1), the diagram
A X σ (Y 1 ,y 12 ,y 3 ) σ (Y 2 ,y 1 ,y 23 ) G G A Y 2 σ (Y,y 2 ,y 3 ) A Y 1 σ (Y,y 1 ,y 2 ) G G A Y is commutative.
(15) In the setting of (8.2.15.1), the diagram
A X σ (Y ′ ,y 1 ,y 2 ) λ (X,x ± ) G G A X ′ σ (Y,y 1 ,y 2 ) A Y ′ λ (Y ′ ,x ± ) G G A Y is commutative.
(16) In the setting of (8.2.16.1), the diagram
A X σ (W,y,x + ) σ (W,y,x − ) G G A W λ (W,x ± ) A W λ (W,x ± ) G G A Y is commutative.
(17) In the setting of (8.2.17.1), the diagram
A X λ (X,x 1 ,x 2 ) λ (X,x 3 ,x 4 ) G G A Z λ (Z,x 1 ,x 2 ) A W λ (W,x 3 ,x 4 ) G G A Y is commutative.
This finishes the definition of a UWD-algebra.
At this moment we have two definitions of a UWD-algebra.
• On the one hand, in Def. 6.1.3 with O = UWD, a UWD-algebra has a structure map µ ζ (6.1.2.1) for each undirected wiring diagram ζ ∈ UWD. This structure map satisfies the associativity axiom (6.1.3.1) for a general operadic composition in UWD, together with the unity and the equivariance axioms in Def. 6.1.2.
• On the other hand, in Def. 11.1.1 a UWD-algebra has 6 generating structure maps and satisfies 17 generating axioms.
We now observe that these two definitions are equivalent, so UWD-algebras indeed have a finite presentation as in Def. 11.1.1. The generating axiom (11.1.1.9) is a special case of the equivariance diagram (6.1.2.4), so it is commutative. Each of the other 16 generating axioms corresponds to an elementary relation in UWD that describes two different ways to construct the same undirected wiring diagram as an iterated operadic composition of generators in UWD. Each such generating axiom asserts that the two corresponding compositions of generating structure maps-defined using the composition in the diagram (6.1.3.1)-are equal. The associativity axiom (6.1.3.1) of (A, µ) applied twice guarantees that two such compositions are indeed equal.
Conversely, suppose A is a UWD-algebra in the sense of Def. 11.1.1, so it has 6 generating structure maps that satisfy 17 generating axioms. We must show that it is a UWD-algebra in the sense of Def. 6.1.3 For an undirected wiring diagram ψ ∈ UWD with a presentation Ψ (Def. 10.1.2), we define its structure map µ ψ (6.1.2.1) inductively as follows.
(1) If Ψ is a 1-simplex, then Ψ = (ψ), and ψ is a generator in UWD by definition.
In this case, we define µ ψ as the corresponding generating structure map (11.1.1.1)-(11.1.1.6) of A.
(2) Inductively, suppose Ψ is an n-simplex for some n ≥ 2, so Ψ = (Φ, i, Θ) for some i ≥ 1, p-simplex Φ, and q-simplex Θ with p + q = n. Since 1 ≤ p, q < n, by the induction hypothesis, the structure maps µ Φ and µ Θ are already defined. Then we define the structure map
µ ψ = µ Φ ○ i µ Θ (11.1.2.1)
as in Notation 6.1.4.
By Theorem 10.1.12 every undirected wiring diagram has a stratified presentation, hence a presentation. To see that the structure map µ ψ as above is welldefined, we need to show that the map µ ψ is independent of the choice of a presentation Ψ. Any two presentations of an undirected wiring diagram are by definition equivalent simplices. By Theorem 10.2.7(2) (= the relations part of the finite presentation theorem for UWD), any two equivalent simplices in UWD are either equal or are connected by a finite sequence of elementary equivalences. Therefore, it suffices to show that every elementary equivalence in UWD yields a commutative diagram involving the generating structure maps of A, where ○ i is interpreted as in Notation 6.1.4. Recall from Def. 10.2.1 that an elementary equivalence comes from either an elementary relation in UWD or an operad associativity/unity axiom for the generators in UWD.
It follows from a direct inspection that the operad associativity and unity axioms- By definition each of the 17 generating axioms of A corresponds to an elementary relation (Def. 8.2.18) and is a commutative diagram. Therefore, the structure map µ ψ for each wiring diagram ψ is well-defined.
It remains to check that the structure map µ satisfies the required unity, equivariance, and associativity axioms. The unity axiom (6.1.2.3) holds because it is part of the assumption on the generating structure map corresponding to a name change (11.1.1.3).
The associativity axiom (6.1.3.1) holds because the structure map µ ψ is defined above (11.1.2.1) by requiring that the diagram (6.1.3.1) be commutative.
For the equivariance axiom (6.1.2.4), first note that it is enough to check it when the undirected wiring diagram in questioned is an iterated operadic composition of 2-cells. This is because 2-cells are the only binary generators in UWD (Remark 8.1.8). All other generators are either 0-ary or unary, for which equivariance is trivial.
So now suppose ζ in the equivariance axiom (6.1.2.4) is an iterated operadic composition of 2-cells. If ζ is a 2-cell and the permutation σ is the transposition (1 2) ∈ Σ 2 , then the equivariance axiom (6.1.2.4) is true by the generating axiom (11.1.1.9). The general case now follows from this special case using:
• the generating axiom (11. 1
Finite Presentation for the Relational Algebra
The purpose of this section is to provide an illustration of Theorem 11.1.2, the finite presentation theorem of UWD-algebras, using the relational algebra. This algebra was originally introduced as the main algebra example in [Spi13] using Def. 6.1.2. We will describe the relational algebra in terms of 6 generating structure maps and 17 generating axioms. First we need some notations.
Definition 11.2.1. Suppose A and X are sets.
(1) A X = Set(X, A) is the set of functions X G G A.
(2) An element u ∈ A X is called an X-vector in A, and u(x) ∈ A is called the x-entry of u for x ∈ X.
(3) (X) = T ⊆ X is the set of subsets of X, called the power set of X.
(4) Rel A (X) = A X is the power set of the set A X of X-vectors in A.
Example 11.2.2. Suppose A, B, X, Y are sets and * is a one-element set.
(1) Since A ∅ = * , it follows that (3) There is a canonical bijection
A X × A Y ≅ A X∐Y . (11.2.2.3) (4) Suppose h ∶ X G G Y is a function. (i) There is an induced map h * ∶ A Y G G A X sending each map Y G G A to the composition of X G G Y G G A; i.e., h * is pre-composition with h. (ii) Likewise, there is an induced map Rel A (Y) h * G G Rel A (X) (11.2.2.4)
given by pre-composition with h.
(5) Suppose p ∶ A G G B is a function. (i) There is an induced map p * ∶ A X G G B X sending each map X G G A to the composition X G G A G G B; i.e., p * is post-composition with p. (ii) Likewise, there is an induced map Rel A (X) p * G G Rel B (X) (11.2.2.5)
given by post-composition with p.
Assumption 11.2.3. Throughout this section, S is the one-element set. So Fin S = Fin is the collection of finite sets, and UWD is a Fin-colored operad (Theorem 7.3.14).
We now define the relational algebra in terms of finitely many generating structure maps and generating axioms.
Definition 11.2.4. Suppose A is a set. The relational algebra of A, denoted Rel A , is the UWD-algebra in the sense of Def. 11.1.1 defined as follows. For each finite set X, the X-colored entry is
Rel A (X) = A X
as in Def. 11.2.1. Its 6 generating structure maps are defined as follows.
(1) For the empty cell ǫ ∈ UWD ∅ (Def. 8.1.1), the chosen element in Rel A (∅) = {∅, * } (11.2.2.1) is * .
(2) For a 1-output wire ω * ∈ UWD * (Def. 8.1.2), the chosen element in Rel A ( * ) = (A) (11.2.2.2) is A.
(3) For a bijection f ∶ X G G Y ∈ Fin and the name change τ f ∈ UWD Y X (Def. 8.1.3), the generating structure map is the pre-composition map (11.2.2.4)
Rel A (X) τ f = ( f −1 ) * G G Rel A (Y) .
In other words, each X-vector in A is reindexed as a Y-vector in A using the bijection f .
(4) For a 2-cell θ (X,Y) ∈ UWD X∐Y X,Y (Def. 8.1.4), the generating structure map is defined using the bijection (11.2.2.3) as follows.
Rel A (X) × Rel A (Y) θ (X,Y) U ⊆ A X , V ⊆ A Y ❴ Rel A (X ∐ Y) U × V ⊆ A X × A Y ≅ A X∐Y
In other words, concatenate every given X-vector with every given Yvector to form an (X ∐ Y)-vector.
(5) For a loop λ (X,x±) ∈ UWD X∖x± X (Def. 8.1.5), the generating structure map is defined as
Rel A (X) λ (X,x ± ) U ⊆ A X ❴ Rel A (X ∖ x ± ) u ∖x± ∶ u ∈ U, u(x + ) = u(x − ) ⊆ A X∖x± (11.2.4.1)
in which u ∖x± is the composition
X ∖ x ± inclusion G G X u G G A .
In other words, for a given subset of X-vectors in A, look for those whose x + -entry and x − -entry are equal, and then delete these two entries to form a subset of (X ∖ x ± )-vectors.
(6) For a split σ (X,x 1 ,x 2 ) ∈ UWD X X ′ (Def. 8.1.6), denote by p ∶ X G G X ′ the projection map that sends both x 1 , x 2 ∈ X to x ∈ X ′ and is the identity function everywhere else. The generating structure map is defined as the pre-composition map (11.2.2.4)
Rel A (X ′ ) σ (X,x 1 ,x 2 ) = p * G G Rel A (X) .
In other words, for a given subset of X ′ -vectors in A, use the x-entry for both the x 1 -entry and the x 2 -entry to form a subset of X-vectors.
The following observation is the finite presentation theorem for the relational algebra of a set. Proof. We need to check that Rel A satisfies the 17 generating axioms in Def. 11.1.1. Due to the simplicity of the 6 generating structure maps, all the generating axioms can be checked by a direct inspection. For example, the generating axiom (11.1.1.7), which corresponds to the elementary relation (8.2.6.1), is the assertion that the diagram
Rel A (X) × * ≅ Rel A (X) (Id,ωy) σ (Y,x,y) G G Rel A (Y) λ (Y,x,y) Rel A (X) × Rel A (y) θ (X,y) G G Rel A (Y) λ (Y,x,y) G G Rel A (W) is commutative, where X = Y ∖ y and W = Y ∖ {x, y} = X ∖ x. For each element U ⊆ A X ∈ Rel A (X),
one can check that both compositions in the above diagram send U to the element
W inclusion G G X u G G A ∶ u ∈ U ⊆ A W in Rel A (W
Spivak's Conjecture: Rigidity of the Relational Algebra
The purpose of this section is to partially verify a conjecture in [Spi13] (Conjecture 3.1.6) that states that the relational algebra Rel A in Def. 11.2.4 is quotient-free. To state this conjecture, we first need the definition of a map between operad algebras. Recall from Section 6.1 the definition of an operad algebra. that is compatible with the structure maps in the following sense.
f * ∶ Rel A G G Rel B is a map of UWD-algebras. This means that the diagram n ∏ i=1 Rel A (X i ) µ A ζ ∏ f X i G G n ∏ i=1 Rel B (X i ) µ B ζ Rel A (Y) f Y G G Rel B (Y) (11.3.3.1)
is commutative for each ζ ∈ UWD Y X with X = (X 1 , . . . , X n ). We will consider two special cases, which will show that f is surjective and injective.
(1) To show that f is surjective, consider a 1-output wire ω * ∈ UWD * (Def. 8.1.2). In this case, the commutative diagram (11.3.3.1) becomes the diagram * (2) To show that f is injective, we argue by contradiction. So suppose there exist distinct elements a + , a − ∈ A such that f (a + ) = f (a − ) ∈ B. We will show that this assumption leads to a contradiction. Consider the loop λ (A,a±) ∈ UWD A∖a± A (Def. 8.1.5). In this case, the commutative diagram (11.3.3.1) becomes the diagram
ω A * = G G * ω B * Rel A ( * ) = (A) f * G G (B) = Rel B ( * ) in which ω A * ( * ) = A ∈ (A)Rel A (A) λ (A,a ± ) f * G G Rel B (A) λ (A,a ± ) Rel A (A ∖ a ± ) f * G G Rel B (A ∖ a ± ) (11.3.3.2)
in which both horizontal maps f * are post-composition with f . Consider the element
U = A = G G A ∈ Rel A (A) = A A ,
which is the single-element set consisting of the identity map of A. We will show that the two compositions in (11.3.3.2) do not agree at U.
Algebras of Undirected Wiring Diagrams
Assumption 11.4.1. Throughout this section, S is the collection of sets, so UWD is the Fin Set -colored operad of undirected wiring diagrams (Theorem 7.3.14).
Definition 11.4.2. Suppose (X, v) ∈ Fin Set (Def. 2.2.6), so X is a finite set and v ∶ X G G Set assigns to each x ∈ X a set v(x).
(1) Define the set
X v = x∈X v(x) (11.4.2.1)
in which an empty product, for the case X = ∅, means the one-point set.
(2) An element in X v is also called an X-vector.
(3) For x ∈ X, the x-entry of an X-vector u is denoted by u x .
As before, we will omit writing v in (X, v) if there is no danger of confusion.
f ∶ (X, v) G G (Y, v) ∈ Fin Set induces a map Y v = ∏ y∈Y v(y) fv G G ∏ x∈X v(x) = X v . (11.4.4.1)
For (a y ) y∈Y ∈ Y v with each a y ∈ v(y), its image is defined as
f v (a y ) y∈Y = a f (x) x∈X using the fact that a f (x) ∈ v( f (x)) = v(x)
. If A is a set and v(y) = v(x) = A for all y ∈ Y and x ∈ X, then f v is the pre-composition map in Example 11.2.2. In what follows, a map induced by the map f v will often be denoted by f v as well.
We now define the typed relational algebra in terms of finitely many generating structure maps and generating axioms.
Definition 11.4.5. The typed relational algebra Rel is the UWD-algebra in the sense of Def. 11.1.1 defined as follows. For each (X, v) ∈ Fin Set , the X-colored entry is
Rel(X) = (X v )
with X v as in (11.4.2.1) and (−) the power set (Def. 11.2.1). Its 6 generating structure maps are defined as follows.
(1) For the empty cell ǫ ∈ UWD ∅ (Def. 8.1.1), the chosen element in
Rel(∅) = (∅ v ) = ( * ) = {∅, * } is * .
(2) For a 1-output wire ω x ∈ UWD x (Def. 8.1.2), the chosen element in
Rel(x) = (v(x)) is v(x).
(3) For a bijection f ∶ X G G Y ∈ Fin Set and the name change τ f ∈ UWD Y X (Def. 8.1.3), the generating structure map is the bijection .1.4), the generating structure map is defined as follows.
Rel(X) = (X v ) τ f = f −1 v G G (Y v ) = Rel(Y) induced by f −1 v as in (11.4.4.1). (4) For a 2-cell θ (X,Y) ∈ UWD X∐Y X,Y (Def. 8Rel(X) × Rel(Y) θ (X,Y) U ⊆ X v , V ⊆ Y v ❴ Rel(X ∐ Y) U × V ⊆ X v × Y v ≅ (X ∐ Y) v
(5) For a loop λ (X,x±) ∈ UWD X∖x± X (Def. 8.1.5), the generating structure map is defined as
Rel(X) λ (X,x ± ) U ⊆ X v ❴ Rel (X ∖ x ± ) u ∖x± ∶ u ∈ U, u x+ = u x− ⊆ (X ∖ x ± ) v (11.4.5.1)
in which u ∖x± is obtained from u by deleting the x ± -entries. In other words, if ι ∶ (X ∖ x ± ) G G X is the inclusion map, then
u ∖x± = ι v (u),
provided u x+ = u x− , where ι v is as in (11.4.4.1).
(6) For a split σ (X,x 1 ,x 2 ) ∈ UWD X X ′ (Def. 8.1.6), denote by p ∶ X G G X ′ the projection map that sends both x 1 , x 2 ∈ X to x ∈ X ′ and is the identity function everywhere else. The generating structure map is the map
Rel(X ′ ) σ (X,x 1 ,x 2 ) = pv G G Rel(X)
induced by p v in (11.4.4.1). In other words, σ (X,x 1 ,x 2 ) takes each X ′ -vector u in any given subset of X ′ -vectors to the X-vector whose x 1 -entry and x 2entry are both equal to the x-entry of u, and all other entries remain the same.
Example 11.4.6. Suppose A is a set and (X, v) ∈ Fin Set such that v(x) = A for all x ∈ X. Then The following observation is the finite presentation theorem for the typed relational algebra.
Rel(X) = (X v ) = A X = Rel A (X),
Theorem 11.4.7. The typed relational algebra Rel in Def. 11.4.5 is actually a UWDalgebra in the sense of Def. 11.1.1, hence also in the sense of Def. 6.1.3 by Theorem 11.1.2.
Proof. As in the proof of Theorem 11.2.5 for the relational algebra of A, we just need to check that Rel satisfies the 17 generating axioms in Def. 11.1.1. Due to the simplicity of the 6 generating structure maps, all the generating axioms can be checked by a direct inspection. For example, the generating axiom (11.1.1.7), which corresponds to the elementary relation (8.2.6.1), is the assertion that the diagram
Rel(X) × * ≅ Rel(X) (Id,ωy) σ (Y,x,y) G G Rel(Y) λ (Y,x,y) Rel(X) × Rel(y) θ (X,y) G G Rel(Y) λ (Y,x,y) G G Rel(W) is commutative, where X = Y ∖ y and W = Y ∖ {x, y} = X ∖ x. For each element U ⊆ X v = v(x) ∈ Rel(X),
one can check that both compositions in the above diagram send U to the element
ι v (u) ∶ u ∈ U ⊆ W v
in Rel(W), where ι ∶ W G G X is the inclusion map and ι v is as in (11.4.4.1). The other 16 generating axioms are checked similarly.
Remark 11.4.8. To see that our relational algebra Rel in Def. 11.4.5 agrees with the one in [Spi13] Section 4, note that the latter is based on Def. 6.1.2, which is equivalent to Def. 6.1.3. A direct inspection of [Spi13] Lemma 4.1.2 reveals that Spivak's structure map of Rel, when applied to the 6 generators in UWD (Def. 8.1.7), reduces to our generating structure maps in Def. 11.4.5. Theorem 11.4.7 then guarantees that the two definitions of the relational algebra Rel-i.e., our Def. 11.4.5 and [Spi13] Section 4-are equivalent.
Summary of Chapter 11
(1) Each UWD-algebra can be described using six generating structure maps and seventeen generating axioms.
(2) The (typed) relational algebra is a UWD-algebra.
(3) Spivak's Conjecture holds when restricted to relational algebras.
then there are corresponding maps of operads for each of the four operads of (undirected) wiring diagrams.
Section 12.2 contains the first main result Theorem 12.2.4 of this chapter. It says that there is a map of operads χ ∶ WD • G G UWD defined by forgetting directions.
In Section 12.3 we provide a series of examples to illustrate the operad map χ.
Section 12.4 contains the second main result Theorem 12.4.1 of this chapter. This result identifies precisely the image of the operad map χ ∶ WD • G G UWD as consisting of the undirected wiring diagrams with no wasted cables and no (0, ≥ 2)cables.
In Section 12.5 we consider the restriction of the operad map χ to the operad WD 0 of strict wiring diagrams. Recall that a strict wiring diagram is a wiring diagram with no delay nodes and whose supplier assignment is a bijection. In Theorem 12.5.1 we will show that the image of the operad map WD 0 G G UWD consists of precisely the undirected wiring diagrams with only (1, 1)-cables and (2, 0)cables.
Operad Maps
In this section, we define an operad map and record some obvious maps among the various operads of (undirected) wiring diagrams (Prop. 12.1.7 and 12.1.9). Recall from Def. 2.1.10 the definition of an S-colored operad.
Definition 12.1.1. Suppose O is an S-colored operad and P is a T-colored operad. A map of operads, also called an operad map, f ∶ O G G P consists of a pair of maps ( f 0 , f 1 ) as follows.
(1) f 0 ∶ S G G T, called the color map.
(2) For each d ∈ S and c = (c 1 , . . . , c n ) ∈ Prof(S) with n ≥ 0, it has a map, called an entry map,
O d c f 1 G G P f d f c in which f d = f 0 d ∈ T and f c = ( f 0 c 1 , .
. . , f 0 c n ) ∈ Prof(T).
We will usually write both f 0 and f 1 as f . These maps are required to preserve the operad structure in the sense that the following three conditions hold.
A f c 1 × ⋯ × A f cn = A f c 1 × ⋯ × A f cn µ f ζ = µ f ζ G G A f d = A f d .
Here µ ? is the P-algebra structure map of A, and f ζ ∈ P f d f c 1 ,..., f cn . Since an operad map is assumed to preserve all the operad structure, a direct inspection reveals that A f is indeed an O-algebra.
We say that the O-algebra A f is induced along f .
In the next two observations, we record some obvious operad maps among the various operads of (undirected) wiring diagrams.
Recall the Box S -colored operad WD of wiring diagrams (Theorem 2.3.11), the Box S -colored operad WD • of normal wiring diagrams (Prop. 5.3.5), and the Box Scolored operad WD 0 of strict wiring diagrams (Prop. 5.4.6). Remember that a normal wiring diagram is a wiring diagram without delay nodes, and a strict wiring diagram is a normal wiring diagram whose supplier assignment is a bijection. Also recall that WD S means the operad of S-wiring diagrams, and the symbol S is suppressed from the notation WD S unless we need to emphasize it. Similar remarks apply to WD S 0 and WD S • . Proposition 12.1.7. Given a map f ∶ S G G T of classes, there exists an induced commutative diagram
WD S 0 G G f * WD S • G G f * WD S f * WD T 0 G G WD T • G G WD T (12.1.7.1)
of maps of operads in which:
• the horizontal maps are operad inclusions;
• the vertical maps are induced by f on value assignments.
Proof. In each row of the diagram (12.1.7.1), the maps on colors (i.e., either Box S or Box T ) are the identity map. For a fixed class, a strict wiring diagram is by definition also a normal wiring diagram, which by definition is also a wiring diagram. In each of WD 0 and WD • , the operad structure-i.e., the equivariant structure, the colored units, and the operad composition-is defined as that in WD (Def. 2.3.1-2.3.4). So the horizontal entrywise inclusions actually define operad inclusions.
For the vertical maps in the diagram (12.1.7.1), first note that f induces maps Fin S G G Fin T and Box S G G Box T that are the identity map on the underlying finite sets. On value assignments, these maps are post-composition with f . A direct inspection reveals that, using these two maps, every (strict/normal) S-wiring diagram is sent to a (strict/normal) T-wiring diagram. Moreover, all the operad structure (Def. 2.3.1-2.3.4) is preserved by these maps. So the vertical entrywise defined maps in the diagram (12.1.7.1) are maps of operads. The commutativity of the diagram is immediate from the definitions of the operad maps.
Example 12.1.8. By Example 12.1.6 and Prop. 12.1.7, every WD-algebra (Def. 6.2.1) restricts to a WD • -algebra (Def. 6.4.1), and every WD • -algebra restricts to a WD 0algebra (Def. 6.6.1). For example, the propagator algebra (Def. 6.3.11), which is a WD-algebra, restricts to a WD • -algebra and also to a WD 0 -algebra.
Recall that UWD S is the Fin S -colored operad of undirected S-wiring diagrams (Theorem 7.3.14), and the symbol S in UWD S is dropped unless we need to emphasize S. Essentially the same as in Prop. 12.1.7, we have the following operad maps for undirected wiring diagrams. Then the inclusion * G G Set induces an operad inclusion UWD * G G UWD Set . In [Spi13] Example 2.1.7, UWD * is denoted by S and is called the operad of singlytyped wiring diagrams. In [Spi13] Example 4.1.1, UWD Set is denoted by T and is called the operad of typed wiring diagrams.
Normal to Undirected Wiring Diagrams
Fix a class S for the rest of this chapter, with respect to which the operad WD • of normal wiring diagrams (Prop. 5.3.5) and the operad UWD of undirected wiring diagrams (Theorem 7.3.14) are defined. Recall that a normal wiring diagram is a wiring diagram without delay nodes. The purpose of this section is to construct a map of operads WD • G G UWD given by forgetting directions (Theorem 12.2.4). The existence of such a map of operads was hinted at in the discussion in [RS13] Section 4.1.
Example 12.2.1. To motivate the definition of the map of operads WD • G G UWD to be defined below, consider the following normal wiring diagram.
ϕ ∈ WD • Y X 1 ,X 2 X 1 X 2 y 1 y 2 y 1 y 2 y 3 Here X in 1 = x in 11 , x in 12 , x in 13 , X out 1 = x out 11 , x out 12 , X in 2 = x in 2 , and X out 2 = x out 2 .
The supplier assignment of ϕ (Def. 2.2.13), s ∶ Dm ϕ G G Sp ϕ , is given by
• y 1 = s x in 12 = s x in 2 ; • x out 11 = s x in 11 = s(y 1 ); • x out 2 = s(y 2 ) = s(y 3 ) = s x in 13 .
Note that y 2 ∈ Y in is an external wasted wire, and x out 12 ∈ X out 1 is an internal wasted wire.
A natural way to make ϕ into an undirected wiring diagram is to forget the directions of all the arrows. For instance, we send Y = Y in , Y out = {y 1 , y 2 }, {y 1 , y 2 , y 3 } ∈ Box S to Y = Y in ∐ Y out = y 1 , y 2 , y 1 , y 2 , y 3 ∈ Fin S , and similarly we send X i ∈ Box S to X i = X in i ∐ X out i ∈ Fin S for i = 1, 2. So we have
X 1 = x in 11 , x in 12 , x in 13 , x out 11 , x out 12 and X 2 = x in 2 , x out 2 .
Inserting cables at appropriate places, we obtain the following undirected wiring diagram.
ϕ ∈ UWD Y X 1 ,X 2 X 1 X 2 y 1yDm ϕ○ i ψ = Y out ∐ ∐ j =i X in j ∐ W in s ϕ○ i ψ G G Y in ∐ ∐ j =i X out j ∐ W out =χ (ϕ ○ i ψ) ∈ UWD Y Z is the cospan Y = Y in ∐ Y out Id ∐s ϕ○ i ψ ∐ j =i X j ∐ ∐ k W k s ϕ○ i ψ ,Id G G Sp ϕ○ i ψ = Y in ∐ ∐ j =i X out j ∐ W out (12.2.4.3)
in Fin S . Here:
• The input soldering function is made up of the identity map on ∐ j =i X out j ∐ W out ;
the supplier assignment
s ϕ○ i ψ ∶ ∐ j =i X in j ∐ W in G G Sp ϕ○ i ψ .
• The output soldering function is the coproduct of the identity map on Y in ; -the supplier assignment
s ϕ○ i ψ ∶ Y out G G ∐ j =i X out j ∐ W out .
On the other hand, by (12.2.2.3) and Def. 7.3.5, the ○ i -composition
χϕ ○ i χψ ∈ UWD
in Fin S . We must show that there exists a unique dotted map η that makes the diagram commutative. By a direct inspection the only possible candidate for η is given by the restrictions • The relational algebra of a set A (Def. 11.2.4 with S = * ) induces a WD •algebra along the operad map χ.
Y in ∐ ∐ j =i X out j η = β G G V and W out η = α G G V .
• The typed relational algebra (Def. 11.4.5 with S = Set) also induces a WD •algebra along the operad map χ.
Examples of the Operad Map
The purpose of this section is to provide concrete examples of the map of operads χ ∶ WD • G G UWD in Theorem 12.2.4. Recall that the map χ was defined in Def. 12.2.2. Similar to Section 8.2, all the assertions in this section are checked by a direct inspection of the normal and undirected wiring diagrams involved. So we will omit the proofs.
First we consider the normal generating wiring diagrams (Def. 5.3.6) that are sent by χ to generators in UWD (Def. 8.1.7). .1.7), the image χσ Y,y 1 ,y 2 ∈ UWD Y X is the split σ (Y,y 1 ,y 2 ) (Def. 8.1.6).
X = X in ∐ X out f in ∐( f out ) −1 G G Y in ∐ Y out = Y ∈ Fin S induced by f .
Next we consider an in-split and a 1-wasted wire. They are not sent by the map of operads χ to generators in UWD. So we will express their χ-images as operadic compositions of the generators in UWD. By Theorem 10.1.12 this is always possible.
Example 12.3.6. For an in-split σ X,
x 1 ,x 2 ∈ WD • Y X (Def. 3.1.6) with Y = X (x 1 =x 2 ) , sup- pose Z = Y ∐ x + 1 , x − 1 ∈ Fin S in which v(x + 1 ) = v(x − 1 ) = v(x 1 ) ∈ S. Identify Z (x 12 = x + 1 ) = X ∐ x + 1 , x − 1 (x 1 = x 2 = x + 1 ) = X via x + 1 ↦ x 1 and x − 1 ↦ x 2 . Then we have χσ X,x 1 ,x 2 = λ Z,x ± 1 ○ σ Z,x 12 ,x + 1 ∈ UWD Y X (12.3.6.1)
in which:
• λ Z,x ± 1 ∈ UWD Y Z is a loop (Def. 8.1.5); • σ Z,x 12 ,x + 1 ∈ UWD Z X is a split (Def. 8.1.6).
Observe that the right side of (12.3.6.1) also appeared in the elementary relation (8.2.16.1) in UWD. The equality (12.3.6.1) may be visualized as the following picture.
X x 1 x 2 x 12 σ X,x 1 ,x 2 ↝ χ X x 1 x 2 x 12 ⋮ χσ X,x 1 ,x 2 = x 1 x 2 Z λ Z,x ± 1 ○ σ Z,x 12 ,x + 1 x 12 x + 1 x − 1 ⋮
On the right side, the intermediate gray box is Z. In χσ X,x 1 ,x 2 we drew all of X ∖ {x 1 , x 2 } on the right side of the box to make the picture easier to read. It has a (2, 1)-cable, and all other cables are (1, 1)-cables.
Next we consider the χ-image of a 1-wasted wire.
Example 12.3.7. For a 1-wasted wire ω Y,y ∈ WD • Y X (Def. 3.1.8) with X = Y ∖ y, we have χω Y,y = θ (X,y) ○ 2 ω y ∈ UWD Y X (12.3.7.1)
in which:
• θ (X,y) ∈ UWD X∐y X,y is a 2-cell (Def. 8.1.4); • ω y ∈ UWD y is a 1-output wire (Def. 8.1.2).
Observe that the right side of (12. Here:
• θ (∅,y) ∈ UWD y ∅,y is a 2-cell (Def. 8.1.4). • ǫ ∈ UWD ∅ is the empty cell (Def. 8.1.1).
• ω y,y ∈ WD • y ∅ is the 1-wasted wire ω Y,y (Def. 3.1.8) with Y out = ∅ and Y in = y.
• ǫ ∈ WD • ∅ is the empty wiring diagram (Def. 3.1.1).
Note that in the first two lines of (12.3.8.1), the operadic compositions are in the operad UWD. On the other hand, in the last line of (12.3.8.1) the operadic composition is in the operad WD • .
Example 12.3.9. By Examples 12.3.1-12.3.5 and 12.3.8, all 6 types of generators in UWD (Def. 8.1.7) are in the image of the operad map χ ∶ WD • G G UWD. However, one must be careful that this does not imply that every undirected wiring diagram is in the image of χ. We will make precise the image of the operad map χ in Theorem 12.4.1 below.
Next we consider a 1-internal wasted wire, which by Prop. 3.2.3 can be generated by a 1-loop and a 1-wasted wire. in which:
• Z = Y ∐ x ± ∈ Fin S with v(x + ) = v(x − ) = v(x) ∈ S;
• λ (Z,x±) ∈ UWD Y Z is a loop (Def. 8.1.5);
• σ (Z,x±) ∈ UWD Z X is a split (Def. 8.1.6).
Observe that the right side of (12.3.10.1) also appeared in the elementary relation (8.2.6.1). The equality (12.3.10.1) may be visualized as the following picture.
X Y x ω X,x ↝ χ χω X,x X x ⋮ = Y λ (Z,x±) ○ σ (Z,x±) X ⋮
x x+
x− On the right side, the gray box is Z. In χω X,x , all of X ∖ x = Y is drawn on the left side. It has a (1, 0)-cable, and all other cables are (1, 1)-cables.
Example 12.3.11. Consider the wiring diagram π ∈ WD • Y X in Example 4.2.3. Then χπ ∈ UWD Y X is the right side of the following picture.
X π ↝ χ X χπ
So χπ has a (0, 1)-cable, a (2, 1)-cable, a (3, 1)-cable, a (1, 1)-cable, and a (1, 0)cable.
Example 12.3.12. Consider the wiring diagram π 2 ∈ WD • Z X in Example 4.3.2. Then χπ 2 ∈ UWD Z X is the right side of the following picture.
X π 2 ↝ χ X χπ 2
So χπ 2 has two (2, 1)-cables, two (0, 1)-cables, a (1, 2)-cable, and two (1, 1)-cables.
Image of the Operad Map
The purpose of this section is to give an explicit description of the image of the map of operads χ ∶ WD • G G UWD in Theorem 12.2.4. Recall that the map χ was defined in Def. 12.2.2. Also recall the notations and terminology in Notation 9.1.1 regarding subsets of cables. Proof. The color map χ 0 is surjective because, for each X ∈ Fin S , we have (∅, X) ∈ Box S and χ(∅, X) = X.
For the second assertion, we will prove the required inclusions in both directions. First suppose ψ ∈ WD • Y X 1 ,...,Xn for some n ≥ 0. Recall that χψ = ψ ∈ UWD Y X 1 ,...,Xn is the cospan (12.2.2.3)
Y = Y in ∐ Y out Id Y in ∐s ψ Y out n ∐ i=1 X i = X in ∐ X out s ψ X in ,Id X out G G Sp ψ = Y in ∐ X out
in Fin S . To see that ψ has no wasted cables (i.e., (0, 0)-cables) and no (0, ≥ 2)-cables, suppose c ∈ Sp ψ is not in the image of the input soldering function s ψ X in , Id X out . We must show that c is a (0, 1)-cable in ψ. Since c is not in the image of Id X out , we have c ∈ Y in . By the non-instantaneity requirement (2.2.13.2), we also have c ∈ s ψ (Y out ). Therefore, c is in the image of the output soldering function Id Y in ∐s ψ Y out exactly once, so c is a (0, 1)-cable in ψ.
To improve readability, the other half of the second assertion-i.e., that every undirected wiring diagram with no wasted cables and no (0, ≥ 2)-cables is in the image of the operad map χ-will be proved in Proposition 12.4.10 below.
First we consider the special case when there are no (0, ≥ 0)-cables. (1) For each cable c ∈ C ϕ , pick a wire
u c ∈ f −1 ϕ (c) ⊆ N ∐ j=1 U j (12.4.2.2)
in the f ϕ -preimage of c. This is possible because the assumption C (0,≥0) ϕ = ∅ means exactly that f ϕ is surjective. We will use the canonical bijection
u c ∶ c ∈ C ϕ fϕ ≅ G G C ϕ ∈ Fin S (12.4.2.3)
below.
(2) For each 1 ≤ j ≤ N define a box X j ∈ Box S as X out j = u c ∈ U j ∶ c ∈ C ϕ ⊆ U j and X in j = U j ∖ X out j .
Note that we have X j = U j ,
X out = N ∐ j=1 X out j = u c ∶ c ∈ C ϕ ≅ C ϕ , and X in = N ∐ j=1 U j ∖ u c ∶ c ∈ C ϕ .
(3) Define Y = (∅, V) ∈ Box S , so Y = V.
(4) Using the bijection (12.4.2.3), define ψ ∈ WD • Y X 1 ,...,X N whose supplier assignment So X 1 has 2 inputs and 4 outputs, and X 2 has 2 inputs and no outputs. Note that according to Convention 2.2.10 we should draw inputs on the left and outputs on the right. However, we drew ψ to resemble ϕ to make the construction easier to understand.
Dm ψ = Y out ∐ X in = V ∐ ⎡ ⎢ ⎢ ⎢ ⎣ N ∐ j=1 U j ∖ u c ∶ c ∈ C ϕ ⎤ ⎥ ⎥ ⎥ ⎦ s ψ = (gϕ, fϕ) Sp ψ = Y in ∐ X out ≅ C ϕ(V = Y in ∐ Y out Id∅ ∐s ψ N ∐ j=1 U j = X in ∐ X out (s ψ ,Id) G G Sp ψ = Y in ∐ X out ≅ C ϕ in
Next we consider the general case where there may be (0, 1)-cables in ϕ. The idea is to decompose ϕ as ϕ 1 ○ 1 ϕ 0 such that the following statements hold.
• ϕ 0 satisfies the hypotheses of Lemma 12.4.3, so it has no (0, ≥ 0)-cables.
• ϕ 1 contains all the (0, 1)-cables in ϕ; its other cables are all (1, 1)-cables.
• Each of ϕ 0 and ϕ 1 can be lifted back to WD • in such a way that the lifted wiring diagrams are operadically composable in WD • .
The fact that χ ∶ WD • G G UWD is an operad map will then show that ϕ has a χpreimage. • Define which has only 1 input box V 0 .
ϕ 1 = V 0 inclusion G G V 0 ∐ V 1 = V V = o o ∈ UWD V V 0 ,(12.• Define Y 0 = (∅, V 0 ) ∈ Box S , so Y 0 = V 0 ∈ Fin S . • Define Y = (V 1 , V 0 ) ∈ Box S , so Y = V ∈ Fin S .
First we observe that the two undirected wiring diagrams in the previous definition give a decomposition of ϕ.
Lemma 12.4.7. In the context of Def. 12.4.6, there is a decomposition ϕ = ϕ 1 ○ ϕ 0 .
Proof. Since ϕ is the cospan
V = gϕ } } V 0 gϕ inclusion G G V 0 ∐ V 1 gϕ N ∐ j=1 U j fϕ G G fϕ S S Im( f ϕ ) inclusion G G C ϕ = Im( f ϕ ) ∐ C (0,1) ϕ
in Fin S , by the definition of ○ = ○ 1 in UWD (Def. 7.3.5) it is enough to check that the square is a pushout. Since g ϕ ∶ V 1 ≅ C (0,1) ϕ is a bijection, a direct inspection reveals that the square is a pushout. Proof. Define ψ 1 ∈ WD • Y Y 0 whose supplier assignment with ϕ 0 ∈ UWD V 0 U 1 ,...,U N and ϕ 1 ∈ UWD V V 0 as in Def. 12.4.6. Moreover, ϕ 0 satisfies the hypotheses of Lemma 12.4.3 (i.e., that its input soldering function is surjective). So there exists ψ 0 ∈ WD • Then one choice of a χ-preimage ψ ∈ WD • , as constructed in Prop. 12.4.10, is the following normal wiring diagram. Y X 1 X 2 Note that the two (0, 1)-cables in ϕ are lifted to external wasted wires in ψ. To construct a χ 0preimage of ϕ, first note that there is a decomposition
Dm ψ 1 = Y out ∐ Y in 0 = V 0 s ψ 1 = inclusion Sp ψ 1 = Y in ∐ Y out 0 = V 1 ∐ V 0 = V(
Map from Strict to Undirected Wiring Diagrams
U = U 1 ∐ U 2 + ∐ U 2 −
such that the following statements hold.
• U 1 = u ∈ U ∶ f ϕ u ∈ C (1,1) ϕ , so there are bijections
U 1 fϕ ≅ G G C (1,1) ϕ V gϕ ≅ o o .
(12.5.3.1) • For each c ∈ C (2,0) ϕ , there exist unique u + = u c ∈ U 2 + and u − ∈ U 2 − such that f ϕ (u ± ) = c. The correspondence u + ↔ u − defines a bijection U 2 + ≅ U 2 − .
• U 2 + = u c ∈ f −1 ϕ (c) ∶ c ∈ C
In (12.4.2.4) we already defined ψ ∈ WD • Y X 1 ,...,X N with supplier assignment
Dm ψ = Y out ∐ X in = V ∐ U ∖ u c ∶ c ∈ C ϕ s ψ = (gϕ, fϕ) Sp ψ = Y in ∐ X out ≅ C ϕ = C (1,1) ϕ ∐ C (2,0)
ϕ such that χψ = ϕ by Lemma 12.4.3. So it is enough to show that ψ is a strict wiring diagram. Since ψ ∈ WD • has no delay nodes, it suffices to show that its supplier assignment s ψ is a bijection. First note that the map g ϕ ∶ V G G C (1,1) ϕ (12.5.3.1) is a bijection.
It remains to show that the map U ∖ u c ∶ c ∈ C ϕ fϕ G G C (2,0) ϕ is also a bijection. We have
U ∖ u c ∶ c ∈ C ϕ = U 1 ∐ U 2 + ∐ U 2 − ∖ u c ∶ c ∈ C (1,1) ϕ ∐ C (2,0) ϕ = U 2 + ∐ U 2 − ∖ u c ∶ c ∈ C (2,0) ϕ = U 2 − . Since f ϕ ∶ U 2 − G G C (2,0)
ϕ is a bijection, the proof is complete.
The proof of Theorem 12.5.1 is complete.
Example 12.5.4. The following generators in UWD (Def. 8.1.7) are in the image of the operad map χ 0 ∶ WD 0 G G UWD.
• the empty cell ǫ ∈ UWD ∅ (Def. 8.1.1);
In Section 13.1 we prove that an operad map ρ ∶ WD G G UWD exists and that it extends the existing operad map χ ∶ WD • G G UWD. The construction of the operad map ρ is motivated in Example 13.1.1, where we discuss how delay nodes should be sent to undirected wiring diagrams.
In Section 13.2 we provide a series of examples, all containing delay nodes, to further illustrate the operad map ρ ∶ WD G G UWD.
In Section 13.3 we prove that the operad map ρ ∶ WD G G UWD is surjective. This section ends with Example 13.3.4, which provides a detailed illustration of how to lift an undirected wiring diagram back to a wiring diagram.
Wiring Diagrams to Undirected Wiring Diagrams
The purpose of this section is to construct an operad map ρ ∶ WD G G UWD that extends the operad map χ ∶ WD • G G UWD in Theorem 12.2.4. Since normal wiring diagrams are wiring diagrams with no delay nodes, to construct the operad map ρ, we need to decide how to map the delay nodes to undirected wiring diagrams.
Example 13.1.1. Before we define the desired operad map ρ ∶ WD G G UWD, let us consider a motivating example that explains what happens to delay nodes. In the following picture, the wiring diagram ϕ ∈ WD Y on the left is sent to the undirected wiring diagram ϕ ∈ UWD Y on the right.
ϕ ∈ WD Y d 1 d 2 y d 3 ↝ ρ ϕ ∈ UWD Y
In ϕ there are 3 delay nodes and no input boxes. In ϕ there are 3 cables and no input boxes. As in the operad map χ ∶ WD • G G UWD, every supply wire {y, d 1 , d 2 , d 3 } in ϕ yields a cable in ϕ. However, since a delay node is both a demand wire and a supply wire, we need to identify the cables corresponding to a delay node d and its supply wire s(d).
• In ϕ the top delay node d 1 supplies only itself, so the identification is trivial. It yields a wasted cable in ϕ.
• The middle delay node d 2 is supplied by the global input y, so their cables are identified, yielding a (0, 2)-cable in ϕ.
• The bottom delay node d 3 supplies itself and three global outputs, so the identification is trivial. Its cable is a (0, 3)-cable in ϕ.
For a general wiring diagram, this identification is defined in (13.1.2.2) below.
Observe that in ϕ, there are a wasted cable, a (0, 2)-cable, and a (0, 3)-cable, none of which is possible in the image of χ ∶ WD • G G UWD by Theorem 12.4.1. In fact, this example suggests that the desired operad map ρ ∶ WD G G UWD is surjective because wasted cables and (0, ≥ 2)-cables are now realizable by carefully chosen delay nodes. We will prove in Theorem 13.3.3 that this is indeed the case.
We now define the operad map ρ ∶ WD G G UWD that extends the operad map χ ∶ WD • G G UWD in Theorem 12.2.4. Recall the color map χ 0 ∶ Box S G G Fin S in (12.2.2.1) with
χ 0 Y = Y = Y in ∐ Y out ∈ Fin S for each Y = (Y in , Y out ) ∈ Box S .Y = Y in ∐ Y out Id Y in ∐s ψ Y out | | Sp ψ = Y in ∐ X out ∐ DN ψ quotient X in ∐ X out s ψ X in ,Id X out G G I I Sp ψ quotient G G C ρψ = Sp ψ d = s ψ d ∶ d ∈ DN ψ (13.1.2.2)
in Fin S . Here:
• X out = ∐ n i=1 X out i and X in = ∐ n i=1 X in i . • The map Dm ψ = Y out ∐ X in ∐ DN ψ s ψ G G Y in ∐ X out ∐ DN ψ = Sp ψ
is the supplier assignment for ψ (Def. 2.2.13).
• s ψ Y out ∶ Y out G G X out ∐ DN ψ by the non-instantaneity requirement (2.2.13.2).
• C ρψ in the lower right corner is the quotient set obtained from Sp ψ by identifying d and s ψ d for each delay node d ∈ DN ψ .
Remark 13.1.3. In Def. 13.1.2 suppose ψ ∈ WD • Y X 1 ,...,Xn , i.e., DN ψ = ∅. Then C ρψ = Sp ψ = Y in ∐ X out , so ρ 1 ψ in (13.1.2.2) is equal to χ 1 ψ in (12.2.2.3). In other words, when applied to normal wiring diagrams, the entry maps ρ 1 and χ 1 are the same. So Def. 13.1.2 is indeed an extension of Def. 12.2.2 to all wiring diagrams. Proof. This proof is similar to that of Theorem 12.2.4. The difference here is that we now need to take into account the delay nodes.
We will write both ρ 0 and ρ 1 as ρ. We must check that ρ preserves the operad structure in the sense of Def. 12.1.1. In both WD (2.3.1.1) and UWD (7.3.1.1), the equivariant structure is given by permuting the labels of the input boxes. So ρ preserves equivariance in the sense of (12.1.1.1). Likewise, it follows immediately from the definitions of the colored units in WD (2.3.2.1) and UWD (7.3.2.1) that they are preserved by ρ in the sense of (12.1.1.2).
To check that ρ preserves operadic composition in the sense of (12.1.1.3), suppose ϕ ∈ WD Y X 1 ,...,Xn with n ≥ 1, 1 ≤ i ≤ n, and ψ ∈ WD X i W 1 ,...,Wm with m ≥ 0. We must show that ρ ϕ ○ i ψ = ρϕ ○ i ρψ ∈ UWD To prove (13.1.4.2), on the one hand, by Def. 2.3.4 ϕ ○ i ψ ∈ WD Y X○ i W has supplier assignment
Dm ϕ○ i ψ = Y out ∐ ∐ j =i X in j ∐ W in ∐ DN ϕ ∐ DN ψ s ϕ○ i ψ Sp ϕ○ i ψ = Y in ∐ ∐ j =i X out j ∐ W out ∐ DN ϕ ∐ DN) ρ (ϕ ○ i ψ) ∈ UWD Y Z is the cospan Y = Y in ∐ Y out Id Y in ,s ϕ○ i ψ Y out Z in ∐ Z out s ϕ○ i ψ Z in ,Id Z out G G C ρ(ϕ○ i ψ) = Sp ϕ○ i ψ = Y in ∐ Z out ∐ DN ϕ ∐ DN ψ d = s ϕ○ i ψ d ∶ d ∈ DN ϕ ∐ DN ψ (13.1.4.3)
in Fin S . In (13.1.4.3):
• Z = X ○ i W ∈ Box S , so Z in = ∐ j =i X in j ∐ W in and Z out = ∐ j =i X out j ∐ W out ∈ Fin S .
(13.1.4.4)
• The input soldering function is induced by the identity map on Z out ; -the supplier assignment s ϕ○ i ψ ∶ Z in G G Sp ϕ○ i ψ .
• The output soldering function is induced by the identity map on Y in ; -the supplier assignment s ϕ○ i ψ ∶ Y out G G Z out ∐ DN ϕ ∐ DN ψ .
In (13.1.4.3) and in what follows, to simplify the notation, we use the same symbol to denote a map and a map induced by it.
On the other hand, by (13.1.2.2) and Def. 7.3.5, the ○ i -composition
ρϕ ○ i ρψ ∈ UWD Y Z is the cospan Y = Y in ∐ Y out Id Y in ,s ϕ Y out
Surjectivity of the Operad Map
The reader is reminded of Notation 9.1.1 regarding subsets of cables. The purpose of this section is to show that the operad map ρ ∶ WD G G UWD in Theorem 13.1.4 is surjective. Our strategy is similar to the proof of Lemma 12.4.3, except that here the input soldering function may not be surjective. Cables not in the image of the input soldering function are (0, ≥ 0)-cables. Wasted cables (i.e., (0, 0)-cables) and (0, ≥ 2)-cables are realized using delay nodes, similar to the delay nodes d 1 and d 3 in Example 13.1.1. Moreover, (0, 1)-cables are realized using external wasted wires, similar to y in the picture (12.3.7.2).
Given an undirected wiring diagram, we now define a wiring diagram that will be shown to be a ρ-preimage. Below we will use the map χ 0 = ρ 0 ∶ Box S G G Fin S (12.2.2.1), usually denoted by χ 0 Y = Y = Y in ∐ Y out . The following definition is the general version of Def. 12.4.2 in the sense that now we do not require the input soldering function to be surjective. A detailed example of the following definition will be given in Example 13.3.4. and its inverse below.
(2) For each 1 ≤ j ≤ N define a box X j = (X in j , X out j ) ∈ Box S as X out j = u c ∈ U j ∶ c ∈ C (≥1,≥0) ϕ ⊆ U j and X in j = U j ∖ X out j . (13.3.1.4) Note that we have X j = X in j ∐ X out j = U j ;
X out = N ∐ j=1 X out j = u c ∶ c ∈ C (≥1,≥0) ϕ ≅ C (≥1,≥0) ϕ ; X in = N ∐ j=1 X in j = U ∖ u c ∶ c ∈ C (≥1,≥0) ϕ . (3) Define a box Y = (Y in , Y out ) ∈ Box S as Y in = g −1 ϕ C (0,1) ϕ ; Y out = g −1 ϕ C (≥1,≥0) ϕ ∐ g −1 ϕ C (0,≥2) ϕ .
(13.3.1.5)
Note that Y = Y in ∐ Y out = V.
(4) Define ψ ∈ WD Y X 1 ,...,X N with delay nodes DN ψ = C Recall the sets
Dm ψ = Y out ∐ X in ∐ DN ψ and Sp ψ = Y in ∐ X out ∐ DN ψ
of demand wires and of supply wires. The supplier assignment for ψ (2) The definition (13.3.1.4) of each box X j means that:
Dm ψ = g −1 ϕ C (≥1,≥0) ϕ ∐ g −1 ϕ C (0,≥2) ϕ ∐ U ∖ {u c } ∐ C (0,0) ϕ ∐ C (0,≥2)
• For each cable c ∈ C (≥1,≥0) ϕ = Im( f ϕ ), one wire in U = ∐ N j=1 U j , namely u c , soldered to c is made into an output wire in ψ.
• All other wires in U are made into input wires in ψ.
(3) The definition (13.3.1.5) of the box Y means that:
• Elements in V that are soldered to (0, 1)-cables in ϕ are made into global input wires in ψ. • All other elements in V are made into global output wires in ψ. (1) ρ is surjective on colors.
(2) In the context of Def. 13.3.1, we have that ρψ = ϕ. In particular, the operad map ρ is surjective on entries as well.
Proof. The color map of ρ is ρ 0 = χ 0 ∶ Box S G G Fin S (12.2.2.1), which is surjective by Theorem 12.4.1(1).
For the second assertion, the undirected wiring diagram ρψ ∈ UWD Y X 1 ,...,Xn = UWD V U 1 ,...,U N is by definition the cospan in (13.1.2.2). Since every delay node in ψ (13.3.1.6) is supplied by itself, the set of cables in ρψ is C ρψ = Sp ψ , the set of supply wires in ψ. Furthermore, there is a bijection By the definition of s ψ (13.3.1.7), ρψ ∈ UWD V U 1 ,...,U N is the cospan (13.1.2.2)
C ρψ = Sp ψ = g −1 ϕ C (0,1) ϕ ∐ {u c } ∐ CV = Y = Y in g −1 ϕ C (0,1) ϕ ∐ Y out g −1 ϕ C (≥1,≥0) ϕ ∐ g −1 ϕ C (0,≥2) ϕ Id Y in ∐s ψ Y out = Id ∐ f −1 ϕ gϕ∐gϕ U = U ∖ {u c } X in ∐ {u c } X out s ψ X in ,Id X out = f −1 ϕ fϕ,Id G G C ρψ = g −1 ϕ C (0,1) ϕ ∐ {u c } ∐ C (0,0) ϕ ∐ C (0,≥2)
ϕ in which f −1 ϕ is the inverse of the bijection (13.3.1.3). Combining this cospan for ρψ with the bijection (13.3.3.1), there is a commutative diagram Next, for each cable c ∈ C (≥1,≥0) ϕ , we are supposed to choose an f ϕ -preimage u c ∈ U = U 1 ∐ U 2 . We may choose, for example, u c 2 = u 1 , u c 3 = u 2 , u c 6 = u 5 , and u c 7 = u 6 , all in U 1 . With such choices, the boxes X 1 and X 2 ∈ Box S (13.3.1.4) are X 1 = (X in 1 , X out 1 ) = {u 3 , u 4 }, {u 1 , u 2 , u 5 , u 6 } ;
V = g −1 ϕ C (0,1) ϕ ∐ g −1 ϕ C (≥1,≥0) ϕ ∐ g −1 ϕ C (0,≥2) ϕ Id ∐ f −1 ϕ gϕ∐gϕ gϕ Õ Õ C ρψ = g −1 ϕ C (0,1) ϕ ∐ {u c } ∐ C (0,0) ϕ ∐ C (0,X 2 = (X in 2 , X out 2 ) = {u 1 , u 2 }, ∅ .
The box Y ∈ Box S (13.3.1.5) is
(Y in , Y out ) = g −1 ϕ C (0,1) ϕ , g −1 ϕ C (≥1,≥0) ϕ ∐ g −1 ϕ C (0,≥2) ϕ = {v 6 }, {v 1 , v 2 , v 3 , v 4 , v 5 } .
The set of delay nodes of ψ ∈ WD Y X 1 ,X 2 (13.3.1.6) is The supplier assignment for ψ (13.3.1.7) is the function
Dm ψ = Y out ∐ X in ∐ DN ψ = {v 1 , v 2 , v 3 , v 4 , v 5 } ∐ {u 3 , u 4 , u 1 , u 2 } ∐ {c 1 , c 4 } s ψ Sp ψ = Y in ∐ X out ∐ DN ψ = {v 6 } ∐ {u 1 , u 2 , u 5 , u 6 } ∐ {c 1 , c 4 }
given by The supply wire v 6 is an external wasted wire, and u 6 is an internal wasted wire in ψ. We may draw ψ ∈ WD Y X 1 ,X 2 as follows.
s ψ (v 1 ) = s ψ (v 2 ) = c 1 , s ψ (v 3 ) = u 1 , s ψ (v 4 ) = s ψ (v 5 ) = u 2 ,Y v 6 c 4 c 1 v 2 v 1 X 1 u 2 u 3 u 1 v 5 v 4 u 6 u 1 v 3 u 5 u 4 u 2 X 2
By Theorem 13.3.3(2) or a direct inspection, the map ρ ∶ WD G G UWD sends ψ to ϕ.
Note that ψ ∈ WD Y X 1 ,X 2 is certainly not the only ρ-preimage of ϕ ∈ UWD V U 1 ,U 2 . For example, the wiring diagram
Z ψ ′ ∈ WD Z X 1 ,X 2 v 6 c 4 c 1 v 1 v 2 X 1 u 2 u 3 u 1 v 5 v 4 u 6 u 1 v 3 u 5 u 4 u 2 X 2
also satisfies ρψ ′ = ϕ. Here the output box is Z = (Z in , Z out ) = {v 1 , v 6 }, {v 2 , v 3 , v 4 , v 5 } ∈ Box S , and s ψ ′ (c 1 ) = v 1 . Everything else is the same as in ψ.
Index
set of (≥ m, ≥ n)-cables Introduction 1.1. What are Wiring Diagrams?
by Prof(S) the class of finite ordered sequences of elements in S.
When n ≥ 1, the composition f ○ (g 1 , . . . , g n ) corresponds to the 2-level tree:f ... g 1 ...
Definition 2 .1. 3 .
23Suppose S is a class. An S-colored operad (O, 1, γ) consists of the following data.(1) For any d ∈ S and c ∈ Prof(S) with length n ≥ 0, O is equipped with a class O d c = O d c 1 ,...,cn called the entry of O with input profile c and output color d. An element in O d c is called an n-ary element in O. (2) For d c ∈ Prof(S) × S as above and a permutation σ ∈ Σ n , O is equipped with right action or the symmetric group action, in which cσ = (c σ(1) , . . . , c σ(n) )
Prof(S) × S as above with n ≥ 1, suppose b 1 , . . . , b n ∈ Prof(S) and b = (b 1 , . . . , b n ) ∈ Prof(S) is their concatenation. Then O is equipped with operadic composition. For y ∈ O d c and x i ∈ O c i b i for 1 ≤ i ≤ n, the image of the operadic composition is written as γ y; x 1 , . . . , x n ∈ O d b .
Definition 2.1.10. Suppose S is a class. An S-colored operad (O, 1, ○) consists of the following data.
Lemma 2. 3 . 10 .
310The ○ i -composition in Def. 2.3.4 satisfies the vertical associativity axiom (2.1.10.4).
Definition 3 .1. 1 .
31Define the empty wiring diagram ǫ ∈ WD ∅ with:
( 2 )
2the output box d = ({d}, {d}), in which both d's have values d ∈ S;
Notation 3.3. 1 .
1Suppose O is an S-colored operad (Def. 2.1.10), and T is a set.
( 1 )
1If ϕ ∈ O d c where the input profile (c) has length 1 and if φ ∈ O c b , then we write
Proposition 3 .3. 4 .
34Suppose:
seven relations are about 2-cells (Def. 3.1.4). The following relation says that substituting the empty wiring diagram (Def. 3.1.1) into a 2-cell yields a colored unit (2.3.2.1). Proposition 3.3.8. Suppose:
Proposition 3.3.13. Suppose:
Proposition 3.3.20. Suppose:
Proposition 3. 3 . 21 .
321Suppose:
the intermediate gray box is called Y below.
Proposition 3.3. 25 .
25Suppose:
Proposition 3.3.26. Suppose:
Proposition 3.3. 27 .
27Suppose:
Proposition 3.3. 29 .
29Suppose:
. 30 .
30The 28 relations (3.3.2.1)-(3.3.29.1) are called elementary relations.
. 2 .
2The first observation is about the marginal case where ψ has no input boxes and no delay nodes. So it looks like this with finitely many, possibly zero, external wasted wires. In any case, it can be written in terms of the empty wiring diagram and finitely many 1-wasted wires.
Lemma 4 .1. 3 .
43Suppose N = r = 0 in ψ;
Lemma 4 .1. 7 .
47Given a wiring diagram ψ (4.1.1.1), there is a decomposition
Lemma 4.1. 8 .
8Consider the wiring diagram ϕ (4.1.5.1).
( 1 )
1If N = r = 0, then ϕ = ǫ, the empty wiring diagram (Def. 3.1.1).(2) If (N, r) = (1, 0), then ϕ = 1 X 1 , the colored unit of X 1(Def. 2.3.2).
( 3 )
3If (N, r) = (0, 1), then ϕ = δ d 1 ,a 1-delay node (Def. 3.1.2).
Lemma 4.1.10. Suppose N + r ≥ 2 in the wiring diagram ϕ ∈ WD X ′ X (4.1.5.1). Then it admits a decomposition
Definition 4 .2. 1 .
41Suppose π ∈ WD Y X is a wiring diagram with one input box X and no delay nodes.
name changes can always be rewritten on the outside (i.e., left side) of an iterated operadic composition in WD. Moreover, using the elementary relation (3.3.2.1), an iteration of name changes can be composed down into just one name change. To simplify the presentation, in what follows these elementary relations regarding name changes are automatically applied wherever necessary.
Lemma 4.3.1. Consider the wiring diagram π 2 ∈ WD Z X in Def. 4.2.5.
Lemma 4 .3. 4 .
44In the context of Def. 4.3.3:
Lemma 4.3.9. The wiring diagram β 2 ∈ WD W V in Def. 4.3.3 is either a colored unit or an iterated operadic composition of in-splits.
Lemma 4.3.12. In the context of Def. 4.3.3, suppose β has no internal wasted wires (such as π 2 (4.2.5.2) by Lemma 4.3.1). Then β 3 ∈ WD V X is either a colored unit or an iterated operadic composition of out-splits.
Remark 5.1.4. In Def. 5.1.2 we could have made the definition for a general operad O other than WD, using a specified collection of elements in O in place of the generating wiring diagrams. Such a definition would be useful in discussing generators and relations in a general operad O.
First we define precisely what it means for two presentations of the same wiring diagram to be related to each other. Recall the 28 elementary relations (Def. 3.3.30) and the definition of a colored operad (Def. 2.1.10).
Definition 5 .2. 2 .
52Suppose Ψ is an n-simplex in WD as in Def. 5.1.2.
( 2 )
2An elementary subsimplex Ψ ′ of Ψ is a subsimplex of one of two forms: (i) Ψ ′ is one side (either left or right) of a specified elementary relation (Def. 3.3.30). (ii) Ψ ′ is one side (either left or right) of a specified operad associativity or unity axiom-(2.1.10.3), (2.1.10.4), (2.1.10.5), or (2.1.10.6)-involving only the generating wiring diagrams (Def. 3.1.9).
Lemma 5.2.8. Every simplex is either a stratified simplex or is connected to an equivalent stratified simplex by a finite sequence of elementary equivalences (Def. 5.2.2). Proof. Using Notation 5.1.3 suppose Ψ = (ψ 1 , . . . , ψ n ) is a simplex with composition Ψ = ψ ∈ WD Y X as in Assumption 4.1.1. So ψ has input boxes X = (X 1 , . . . , X N ) and delay nodes DN ψ = {d 1 , . . . , d r }. Suppose Ψ is not a stratified simplex. We will show that Ψ is connected to an equivalent stratified simplex by a finite sequence of elementary equivalences. Using the five elementary relations (3.3.3.1)-(3.3.7.1), first we move all the name changes (Def. 3.1.3) in Ψ, if there are any, to the left. Then we use the elementary relation (3.3.2.1) repeatedly to compose them down into one name change.
Lemma 5.2.9. Any two equivalent stratified simplices of type (1) are either equal or are connected by a finite sequence of elementary equivalences (Def. 5.2.2).
Lemma 5.2.10. Any two equivalent stratified simplices of type (2) are either equal or are connected by a finite sequence of elementary equivalences (Def. 5.2.2). Proof. The proof consists of a series of reductions. Suppose Ψ 1 and Ψ 2 are distinct but equivalent stratified simplices of type (2) with common composition ψ ∈ WD Y X . Using elementary equivalences corresponding to • the operad unity axioms (2.1.10.5) and (2.1.10.6), • the elementary relations (3.3.8.1), (3.3.20.1), and (3.3.25.1) regarding colored units, and
The two stratified subsimplices σ i * ⊆ Ψ i of in-splits for i ∈ {1, 2} are also equivalent. They are connected by a finite sequence of elementary equivalences corresponding to the elementary relations (3.3.21.1) and (3.3.22.1). Likewise, the stratified subsimplices σ * i ⊆ Ψ i of out-splits for i ∈ {1, 2} are connected by a finite sequence of elementary equivalences corresponding to the elementary relations (3.3.26.1) and (3.3.27.1). So the two simplices Ψ i are also connected by a finite sequence of elementary equivalences.
Example 5 .3. 3 .
53All the wiring diagrams that appear in the 28 elementary relations (section 3.3) are normal. Example 5.3.4. Among the wiring diagrams in Chapter 4:
involve only strict wiring diagrams on both sides. Indeed, both (3.3.20.1) and (3.3.25.1) involve a 1-wasted wire, which is not strict. Example 5.4.5. Among the wiring diagrams in Chapter 4:
Definition 5.4.7. Consider the operad WD 0 of strict wiring diagrams.
( 5 )
5A strict elementary relation means one of the 8 elementary relations that involve only strict wiring diagrams on both sides, namely, of Def. 5.2.2 is repeated with WD 0 in place of WD using strict generating wiring diagrams, strict simplices, and strict elementary relations. The resulting notions are called strict elementary equivalences, and so forth.
Definition 6.1.3. Suppose (O, 1, ○) is an S-colored operad as in Def. 2.1.10. An Oalgebra (A, µ) is defined exactly as in Def. 6.1.2 except that the associativity axiom (6.1.2.2) is replaced by the following axiom.
2.1.3 and Def. 2.1.10, it is not hard to check that Def. 6.1.2 and Def. 6.1.3 are in fact equivalent. A proof of this equivalence can be found in [Yau16] (Prop. 16.7.8 and Exercises 10 and 11 in Chapter 16).
Example 6.1.6 (Monoid Modules). Suppose (A, µ, 1) is a monoid (Example 2.1.6). Recall that a left A-module is a set M equipped with a structure map a ∶ M G G M for each a ∈ A that is associative and unital. Associativity means (ab)m = a(bm) for a, b ∈ A and m ∈ M. Unity means 1m = m for m ∈ M. If we regard A as a 1colored operad O concentrated in arity 1 as in Example 2.1.6, then left A-modules yield O-algebras in the sense of Def. 6.1.2. The only slight difference between a left A-module and an O-algebra is that the former has an underlying set, while the latter is allowed to have an underlying class.Example 6.1.7 (Associative and Commutative Monoids). This is a continuation of Example 2.1.7.
(2.1.10.3), (2.1.10.4), (2.1.10.5), and (2.1.10.6)-for the generating wiring diagrams yield commutative diagrams involving the generating structure maps of A. In fact, the diagrams involving the horizontal and the vertical associativity axioms (2.1.10.3) and (2.1.10.4) are commutative because composition of functions is associative. The diagrams for the two unity axioms (2.1.10.5) and (2.1.10.6) are commutative because the generating structure map for a colored unit (6.2.1.3) is required to be the identity map.
Assumption 6.3. 1 .
1Throughout this section, S denotes the class of pointed sets, with respect to which S-boxes (Def. 2.2.7) and the operad WD are defined (Theorem 2.3.11). In a pointed set, the base point is denoted by * . Recall the concept and notations regarding profiles from Def. 2.1.1. Let us first recall a few definitions from [RS13] (section 3). Definition 6.3.2. Suppose T and U are pointed sets and k ≥ 0. (1) Define the truncation ∂ T ∶ Prof ≥1 (T) G G Prof(T) as the function ∂ T (t 1 , . . . , t n ) = (t 1 , . . . , t n−1 ). (6.3.2.1)
Example 6.3. 3 .
3Given a pointed set T and an integer k ≥ 0, the function D k ∶ Prof(T) G G Prof(T) defined as D k (t 1 , . . . , t n ) = ( * , . . . , * , t 1 , . . . , t n ),
D 1 (m 1 ,
1. . . , m n ) = (1, m 1 , . . . , m n ) for m i ∈ N, is a 1-historical propagator of type X. For instance, we have D 1 (1, 5, 6) = (1, 1, 5, 6).
( 2 )
2The function f ∶ Prof(N) G G Prof(N) defined as f (m 1 , . . . , m n ) = 0, m 1 , m 1 + m 2 , . . . ,
3 )
3The function g ∶ Prof(N) G G Prof(N) defined as g(m 1 , . . . , m n ) = 1, m 1 , m 1 m 2 , . . . , historical propagator of type X. This 1-historical propagator takes a sequence of non-negative integers to the sequence whose ith entry is the product of the first i − 1 entries of the given sequence. For instance, we have g(1, 5, 6) = (1, 1, 5, 30).Example 6.3.6. Consider the box Y with Y in = {y 1 , y 2 }, Y out = {y}, and values v(y 1 ) = v(y 2 ) = v(y) = (N, 1).
( 1 )
1The function h ∶ Prof(N × N) G G Prof(N) given by h (m 1 , m ′ 1 ), . . . , (m n , m ′ n ) = 1, m 1 + m ′ 1 , . . . , m n + m ′ n is a 1-historical propagator of type Y, denoted " + " in [RS13]. For instance, we have h (1, 4), (5, 2), (6, 8) = (1, 5, 7, 14). (2) The function j ∶ Prof(N × N) G G Prof(N) given by j (m 1 , m ′ 1 ), . . . , (m n , m ′ n ) = 1, m 1 m ′ 1 , . . . , m n m ′ n is a 1-historical propagator of type Y. For instance, we have j (1, 4), (5, 2), (6, 8) = (1, 4, 10, 48). Example 6.3.7. Consider the box Z with Z in = {z 1 , z 2 , z 3 }, Z out = {z 1 , z 2 }, and all v(−) = (N,
are given as follows. Supposem = (m 1 , m ′ 1 ), . . . , (m n , m ′ n ) ∈ Prof(N × N) is an(N × N)-profile of length n ≥ 0. Then a simple induction shows that (λℓ)(m) = 1, m 1 + m ′ 1 , . . . , m n + m ′ n , G ℓ (m) =
∂G(w) = ∂g w, G(∂w) x+ = g ∂w, ∂G(∂w) x+ by historicity of g = g ∂w, G(∂∂w) x+ by the induction hypothesis on G = G(∂w).
G
G P ∅ = Hist 1 { * }, { * } = { * } (6.3.11.1) sends * to the unique function (ǫ * ) * , . . . , * m = * , . . . , * m+1 .
Example 6.3.12. Suppose Y is the box obtained from Z by removing z = {z 1 , z 1 }. Consider the 1-loop (Def. 3.1.5)
G
G Prof(N × N) in (6.3.17.1) and (6.3.17.2) are given as follows. For m = (m 1 , . . . , m n ) ∈ Prof(N), a simple induction shows that (λ 2 ℓ)(m) = ( * , . . . , * ), (G 2 ℓ )(m) = (1, 1), (1, 1 + m 1 ), (1, 1 + m 1 + m 2 ), . . . , (1, 1 + m 1 + ⋯ + m n ) , where on the right side of (λ 2 ℓ)(m) there are n + 1 copies of * . For instance, we have (G 2 ℓ )(6, 3, 2, 9) = (1, 1), (1, 7), (1, 10), (1, 12), (1, 21) .
Theorem 6 .4. 2 .
62For the operad WD • of normal wiring diagrams (Prop. 5.3.5), Def. 6.1.3 with O = WD • and Def. 6.4.1 of a WD • -algebra are equivalent.
[Spi15b] (Definition 4.9). Let us first recall some definitions from [Spi15b] (Sections 2.1 and 4.1). Assumption 6.5.1. Throughout this section, S denotes the class of sets. So Box S = Box Set , and WD • is the Box Set -colored operad of normal wiring diagrams (Prop. 5.3.5). Definition 6.5.2. Suppose A and B are sets. An (A, B)-discrete system is a triple (T, f rd , f up ) consisting of:
Theorem 6 .6. 2 .
62For the operad WD 0 of strict wiring diagrams (Prop. 5.4.6), Def. 6.1.3 with O = WD 0 and Def. 6.6.1 of a WD 0 -algebra are equivalent.
Definition 6 .7. 2 .)
62Suppose (M, v) ∈ Fin S is an S-finite set (Def. For a subset I ⊆ M and x = (x m ) m∈M ∈ M v , define
the definition of the WD 0 -algebra of open dynamical systems.Example 6.7.8. This is a continuation of Example 6.7.6, where W is the box with W in = {w 1 , w 2 } and W out = {w 1 , w 2 } and M is the one-point set, all with v(−) = R. Suppose W ∖ w is the box obtained from W by removing
Section 8.3 is not technically needed in later sections. However, it contains several illuminating examples about how wasted cables can be created from undirected wiring diagrams without wasted cables. So we urge the reader not to skip these examples.The decompositions in Chapter 9 are illustrated with a detailed example in Section 9.1. The reader may read that section and skip the rest of the Chapter during the first reading.In Section 10.2, after the initial definitions and examples, the reader may skip the proofs of Lemmas 10.2.5 and 10.2.6 and go straight to Theorem 10.2.7, the finite presentation theorem for undirected wiring diagrams.
Remark 7 .1. 3 .
73In Def. 2.2.7 an S-box is an element in Fin S × Fin S . In the context of undirected (pre)wiring diagrams, the name box refers to an element in Fin S , such as the output box or one of the input boxes. The context itself should make it clear what box means.
Definition 7.3.2 (Units in UWD). For each Y ∈ Fin S , the Y-colored unit is defined as the undirected wiring diagram
Lemma 7.3. 9 .
9The ○ i -composition in Def. 7.3.5 satisfies the left unity axiom (2.1.10.5), the right unity axiom (2.1.10.6), and the equivariance axiom (2.1.10.7).
(2.1.10.3), (2.1.10.4), (2.1.10.5), and (2.1.10.6)-for the generating undirected wiring diagrams generate all the relations in the operad UWD of undirected wiring diagrams. In other words, suppose an arbitrary undirected wiring diagram can be built in two ways using the generating undirected wiring diagrams. Then there exists a finite sequence of steps connecting them in which each step is given by one of the 17 elementary relations or an operad associativity/unity axiom for the generating undirected wiring diagrams. For now one may think of the elementary relations as examples of the operadic composition in the operad UWD.
( 3 )
3A name change τ (Def. 8.1.3), a loop λ (X,x±) (Def. 8.1.5), and a split σ (X,x 1 ,x 2 ) (Def. 8.1.6) are unary elements in UWD.(4) A 2-cell θ (X,Y) (Def. 8.1.4) is a binary element in UWD.
Proposition 8 .2. 2 .
82Suppose:• X = {x} and Y = {y} are two 1-element S-finite sets with v(x) = v(y) ∈ S.
. 9 .
9Suppose: • θ (X∐Y,Z) ∈ UWD X∐Y∐Z X∐Y,Z and θ (X,Y) ∈ UWD X∐Y X,Y are 2-cells. • θ (X,Y∐Z) ∈ UWD X∐Y∐Z X,Y∐Z and θ (Y,Z) ∈ UWD Y∐Z Y,Z are 2-cells.
purpose of this section is to consider several examples of how the generators in the operad UWD can create wasted cables (Def. 7.1.2). Example 8.3.4 provides an illustration of some of the elementary relations in UWD. The examples in this section provide a good warm-up exercise for the discussion in Chapter 10 about stratified presentations and elementary equivalences. Recall from Remark 8.1.7 that none of the generators has a wasted cable. Example 8.3.1. In the context of Example 7.3.8 with X = {x 1 , x 2 }:
Example 8 .3. 4 .
84As an illustration of using the elementary relations in UWD, recall the undirected wiring diagram ζ Y ∈ UWD Y Y in Examples 8.3.2 and 8.3.3. It can be generated by the generators as either one of the two iterated operadic compositions (8.3.2.
coproduct of two copies of the set of wasted cables C
∈
UWD Z X in (9.2.2.3), suppose:
unique f -preimage in A. In the second line, a ? was defined earlier in the current definition, using the fact C(≥2,≥1) ϕ ⊆ C ≥3ϕ . This φ 2 is the general version of that in the example (9.1.2.4).Remark 9.4.2.Consider the previous definition.
( 1 )
1The input soldering function f 1 of φ 1 is surjective. Furthermore, all the cables in φ 1 are either (1, 1)-cables (namely, those cables in B ⊆ V) or (2, 0)cables (namely, those in V ∖ B).
. 3 .
3In the context of Def. 9.4.1, there is a decompositionϕ = φ 1 ○ φ 2 .(9.4.3.1)
. 3 .
3always be rewritten on the outside (i.e., left side) of an iterated operadic composition in UWD. Moreover, using the elementary relation (8.2.1.1), an iteration of name changes can be composed down into just one name change. To simplify the presentation, in what follows these elementary relations regarding name changes are automatically applied wherever necessary. With this in mind, in the sequel we will mostly not mention name changes.Recall from Remark 9.4.2 that in φ 2 (9.4.1.4), the input soldering function is the identity function and the output soldering function is surjective. So the following Proposition applies to φ 2 .Motivation 9.5.2. The following result says that an undirected wiring diagram Suppose A, B ∈ Fin S , and
Motivation 9 .5. 4 ..5. 5 .
945The following result says that an undirected wiring diagram of Suppose D ∈ Fin S , D ∋ b ∈ S, and
( 1 )
1The 6 generating undirected wiring diagrams (Def. 8.1.7) generate the operad UWD. This means that every undirected wiring diagram can be expressed as a finite iterated operadic composition involving only the 6 generators. See Theorem 10.1.12. (2) If an undirected wiring diagram can be operadically generated by the generators in two different ways, then there exists a finite sequence of elementary equivalences (Def. 10.2.1) from the first iterated operadic composition to the other one. See Theorem 10.2.7. An elementary equivalence is induced by either an elementary relation in UWD (Def. 8.2.18) or an operad associativity/unity axiom for the generators.
Remark 10.1.3. To simplify the presentation, as in Notation 5.1.3, we will sometimes use either• the right side of (10.1.2.1) or • even just the list of generators (ψ 1 , . . . , ψ n ) in a simplex in the order in which they appear in (10.1.2.1) to denote a simplex in UWD. Example 10.1.4. Elementary relations in UWD (Def. 8.2.18) provide a large source of simplices in UWD. In fact, each side, either left or right, of each elementary relation in UWD is a simplex. For example:(1) The elementary relation (8.2.6.1) says that the 3-simplex λ (Y,x,y) ○ θ (X,y) ○ 2 ω y and the 2-simplex λ (Y,x,y) ○ σ (Y,x,y)
Definition 10.1. 8 .
8A stratified simplex in UWD is a simplex in UWD (Def. 10.1.2) of one of the following two forms:
( 2 )
2An elementary subsimplex Ψ ′ of Ψ is a subsimplex of one of two forms: (i) Ψ ′ is one side (either left or right) of a specified elementary relation in UWD (Def. 8.2.18). (ii) Ψ ′ is one side (either left or right) of a specified operad associativity or unity axiom-(2.1.10.3), (2.1.10.4), (2.1.10.5), or (2.1.10.6)-involving only the generators in UWD (Def. 8.1.7).
Convention 10.2.4. As in Convention 5.2.7, to simplify the presentation, elementary equivalences corresponding to an operad associativity/unity axiom-(2.1.10.3), (2.1.10.4), (2.1.10.5), or (2.1.10.6)-for the generators in UWD (Def. 8.1.7) will often be applied tacitly wherever necessary. The goal of this section is to show that any two equivalent simplices in UWD are connected by a finite sequence of elementary equivalences. The first step is to show that every simplex in UWD is connected to a stratified simplex (Def. 10.1.8) in the following sense. The following observation is the undirected analogue of Lemma 5.2.8. Lemma 10.2.5. Every simplex in UWD is either a stratified simplex (Def. 10.1.8) or is connected to an equivalent stratified simplex by a finite sequence of elementary equivalences (Def. 10.2.1). Proof. Suppose Ψ is a simplex in UWD that is not a stratified simplex. Using the three elementary relations (8.2.3.1)-(8.2.5.1), first we move all the name changes (Def. 3.1.3) in Ψ, if there are any, to the left. Then we use the elementary relation (8.2.1.
we use the elementary relations (8.2.9.1)-(8.2.12.1) to move all the 2-cells in Ψ to just the left of ǫ. Then we use the elementary relations (8.2.15.1) and (8.2.17.1) to move all the remaining loops to just the right of the name change τ. So after a finite sequence of elementary equivalences, we may assume that the simplex Ψ has the form τ, λ, σ, θ, ǫ, ω . (10.2.5.1) If the string ǫ of empty cells is empty, then we are done because this is now a stratified simplex of type (2). So suppose the string ǫ of empty cells in (10.2.5.1) is non-empty. Using finitely many elementary equivalences corresponding to the elementary relations (8.2.8.1)-(8.2.10.1), we may cancel all the unnecessary empty cells in (10.2.5.1). If there are no empty cells left after the cancellation, then we have a stratified simplex of type (2).
Lemma 10.2.6. Any two equivalent stratified simplices (Def. 10.1.8) in UWD are either equal or are connected by a finite sequence of elementary equivalences (Def. 10.2.1).
•
the operad unity axioms (2.1.10.5) and (2.1.10.6), • the elementary relations (8.2.7.1) and (8.2.8.1) regarding colored units, and • other elementary relations in UWD that move the generators around the stratified simplices,
(2.1.10.3), (2.1.10.4), (2.1.10.5), and (2.1.10.6)-for the generators yield commutative diagrams involving the generating structure maps of A. In fact, the diagrams involving the horizontal and the vertical associativity axioms (2.1.10.3) and (2.1.10.4) are commutative because composition of functions is associative. The diagrams for the two unity axioms (2.1.10.5) and (2.1.10.6) are commutative because the generating structure map for a colored unit (11.1.1.3) is required to be the identity map.
Rel A (∅) = ( * ) = {∅, * }. (11.2.2.1) (2) Since A * = A, it follows that Rel A ( * ) = (A), (11.2.2.2)the power set of A.
Definition 11.3.1. Suppose O is an S-colored operad, and (A, µ A ) and (B, µ B ) are O-algebras. A map of O-algebras f ∶ A G G B consists of a collection of maps A c fc G G B c ∶ c ∈ S
and ω B * ( * ) = B ∈ (B) by Def. 11.2.4. The bottom horizontal map f * is post-composition with f , so it sends each subset U ⊆ A to its image f (U) ⊆ B. Therefore, the commutativity of the above diagram forces f (A) = f * (A) = B, so f is surjective.
( 1 )
1For each c ∈ S, A f is equipped with the c-colored entry A f c = A f c .(2) For each d c 1 ,...,cn ∈ Prof(S) × S and ζ ∈ O d c 1 ,...,cn , A f is equipped with the structure map
Proposition 12.1.9. Given a map f ∶ S G G T of classes, there exists an induced map of operads UWD S G G UWD T . Example 12.1.10. Suppose S = * , a one-point set, and T = Set, the collection of sets.
Example 12.3.10. For a 1-internal wasted wire ω X,x ∈ WD • Y X (Def. 3.2.1) with Y = X ∖ x, we have χω X,x = λ (Z,x±) ○ σ (Z,x±)
Theorem 12.4.1. Consider the operad map χ ∶ WD • G G UWD in Theorem 12.2.4. Then:(1) The color map χ 0 ∶ Box S G G Fin S (12.2.2.1) is surjective.(2) The image of the entry map χ 1 ∶ WD • G G UWD (12.2.2.2) consists of precisely the undirected wiring diagrams with no wasted cables and no (0, ≥ 2)-cables.
Fin S . Using the bijection (12.4.2.3) and the definition of s ψ (12.4.2.4), it follows that this cospan is equivalent to the given cospan (12.4.2.1) in the sense of Def. 7.1.4. So they define the same undirected wiring diagram, i.e., χψ = ϕ. Example 12.4.4. Consider ϕ in (12.4.2.1) with N = 0. Then ∐ j U j = ∅. Since f ϕ is sujective, it follows that C ϕ = V = ∅. So ϕ is the empty cell ǫ ∈ UWD ∅ (Def. 8.1.1).The construction (12.4.2.4) above yields ψ = ǫ ∈ WD • ∅ , the empty wiring diagram. So the conclusion χǫ = ǫ agrees with Example 12.3.1.Example 12.4.5. Suppose ϕ with C choice of a χ-preimage ψ ∈ WD • , as constructed in (12.4.2.4), is the following normal wiring diagram.
∈
By assumption there is a decompositionC ϕ = Im( f ϕ ) ∐ C (0,1) ϕ in which Im( f ϕ ) is the image of f ϕ . • Define V 0 = g −1 ϕ Im( f ϕ ) and V 1 = g V = V 0 ∐ V 1 , and there is a bijection g ϕ ∶ V 1 ≅ C UWD V 0 U 1 ,...,U N , (12.4.6.3)in which the input soldering function is surjective, i.e., C
Lemma 12.4.8. For ϕ 1 ∈ UWD V V 0 in (12.4.6.4), there exists ψ 1 ∈ WD • Y Y 0 such that χψ 1 = ϕ 1 .
inclusion map. Then it follows from the definition of χ (12.2.2.3) that χψ 1 = ϕ 1 . Remark 12.4.9. By Lemma 4.3.6 ψ 1 in (12.4.8.1) is an iterated operadic composition of V 1 1-wasted wires (Def. 3.1.8). Since V 1 ≅ g .4.6.2), this means that the (0, 1)-cables in ϕ are lifted to external wasted wires in ψ 1 . Proposition 12.4.10. Every undirected wiring diagram ϕ with C is in the image of the operad map χ ∶ WD • G G UWD.Proof. Suppose ϕ ∈ UWD V U 1 ,...,U N with C
Y 0 X
01 ,...,X N such that χψ 0 = ϕ 0 . (12.4.10.2) With ψ 1 ∈ WD • Y Y 0 as in Lemma 12.4.8, we have χ ψ 1 ○ ψ 0 = χψ 1 ○ χψ 0 by Theorem 12.2.4 = ϕ 1 ○ ϕ 0 by Lemma 12.4.8 and (12.4.10.2) = ϕ by (12.4.10.1).This proves that ϕ is in the image of χ.Proposition 12.4.10 finishes the proof of Theorem 12.4.1.Example 12.4.11. This is an extension of Example 12.4.5. Suppose ϕ with C
a fixed class S, recall the Box S -colored operad of strict wiring diagrams WD 0 (Prop. 5.4.6). As pointed out in Example 12.1.4, we can compose the operad map χ ∶ WD • G G UWD in Theorem 12.2.4 with the operad inclusion WD 0 G G WD • in Prop. 12.1.7 to obtain an operad map of this section is to identify precisely the image of this operad map.has only (1, 1)-cables and (2, 0)-cables, i.e., C ϕ = C
Definition 13.1.2. Fix a class S. For each Y X 1 ,...,Xn ∈ Prof (Box S ) × Box S with n ≥ 0. For ψ ∈ WD Y X 1 ,...,Xn , its image ρ 1 ψ = ψ ∈ UWD Y X 1 ,...,Xnis the cospan
Theorem 13.1.4. The maps ρ 0 = χ 0 (12.2.2.1) and ρ 1 (13.1.2.1) define a map of operads WD ρ G G UWD . (13.1.4.1)
Z
= (X ○ i W) = X 1 , . . . , X i−1 , W 1 , . . . , W m , X i+1 , . . . , X n ∈ Prof (Fin S )as in (2.1.10.2), X = (X 1 , . . . , X n ), and W = (W 1 , . . . , W m ).
∈ f −1 ϕ (c) ⊆ U.We will use the bijection{u c } = u c ∶ c ∈ C
is the inverse of the bijection (13.3.1.3). Remark 13.3.2. Consider Def. 13.3.1.
(i.e., f ϕ is surjective), then Def. 13.3.1 reduces to Def. 12.4.2.
( 4 )
4The supplier assignment s ψ in (13.3.1.7) satisfies the non-instantaneity requirement (2.2.13.2) because Y in = g from the image of s ψ , which is {u c } ∐ C the set of external wasted wires in ψ.(5) Each delay node (13.3.1.6) is supplied by itself, i.e., d = s ψ d for each d ∈ DN ψ . Theorem 13.3.3. Consider the operad map ρ ∶ WD G G UWD in Theorem 13.1.4.
f ϕ is the bijection(13.3.1.3).
Uin
= U ∖ {u c } ∐ {u c } Fin S . In this diagram, the outer cospan is ϕ (13.3.1.1). Therefore, by Def. 7.1.4 we have proved ϕ = ρψ. Example 13.3.4. This is an illustration of Def. 13.3.1 and Theorem 13.3.3. Consider the undirected wiring diagram ϕ ∈ UWD V U V = {v 1 , . . . , v 6 }, U 1 = {u 1 , . . . , u 6 }, U 2 = {u 1 , u 2 }, and C ϕ = {c 1 , . . . , c 7 }.
=
Im( f ϕ ) = {c 2 , c 3 , c 6 , c 7 }.
=
{c 1 , c 4 }.
s ψ (u 3 ) = s ψ (u 1 ) = u 2 , s ψ (u 4 ) = s ψ (u 2 ) = u 5 , s ψ (c 1 ) = c 1 ,and s ψ (c 4 ) = c 4 .
Convention 2.2.8.From now on, whenever Fin S or Box S is used, we always assume that the class S is non-empty. box is called a signed finite set. It is a slight generalization of what appears in[RS13, VSL15]. In[RS13] S is the class of pointed sets, where an S-box is called a black box. In[VSL15] S is a set of representatives of isomorphism classes of second-countable smooth manifolds and smooth maps, or more generally the class of objects in a category with finite products.Convention 2.2.10.For the purpose of visualization, an S-box X will be drawn as follows.Remark 2.2.9. In [Spi15] (Def. 3.1) an S-X
X in
X out
or
X
(2.2.10.1)
Remark 2.2.17. Def. 2.2.15 of an S-wiring diagram is a slight generalization of the one given in[RS13], where S was taken to be the class of pointed sets. Note that when S is a proper class (e.g., the class of pointed sets), the collection WDY
X
(2.2.15.1) is also a proper class, not a set. This is the reason why in Def. 2.1.3 we
allow O
d
c to be a class, in contrast to what was stated in [RS13] (Def. 2.1.2B).
Example 2.2.18. Suppose S is a non-empty class and:
Proof. We need to check that the supplier assignment for ϕ ○ i ψ satisfies the noninstantaneity requirement (2.2.13.2). So suppose y ∈ Y out . We must show that s ϕ○ i ψ (y) ∈ Y in . By (2.3.4.2) we haveLemma 2.3.5. Def. 2.3.4 indeed defines a wiring diagram ϕ ○ i ψ in WD Y
X○ i W .
Lemma 2.3.6. The ○ i -composition in Def. 2.3.4 satisfies the left unity axiom (2.1.10.5), the right unity axiom (2.1.10.6), and the equivariance axiom (2.1.10.7).Proof. This follows from a direct inspection of the definitions of the equivariant structure (2.3.1.1) and the colored units in WD (2.3.2.1).Next we show that WD satisfies the associativity axioms (2.1.10.3) and (2.1.10.4). The reader may wish to skip the proofs of the following two Lemmas and simply look at the pictures during the first reading.Motivation 2.3.7. For the horizontal associativity axiom (2.1.10.3), one should keep in mind the following picture for the iterated operadic composition
we already know that both sides are well-defined wiring diagrams in the indicated entry of WD. Moreover, both sides have DN ϕ ∐ DN ψ ∐ DN ζ ∈ Fin S as the set of delay nodes. So it remains to show that their supplier assignments are equal.2.3.8.1)
By Lemma 2.3.5 Note that both sides in (2.3.8.1) have demand wires
we already know that both sides are well-defined wiring diagrams in the indicated entry of WD. Moreover, both sides have DN ϕ ∐ DN ψ ∐ DN ζ ∈ Fin S as the set of delay nodes. So it remains to show that their supplier assignments are equal.1+j W .
(2.3.10.1)
By Lemma 2.3.5 Note that both sides in (2.3.10.1) have demand wires
Remark 3.1.10. Among the generating wiring diagrams:(1) A 1-delay node δ d (Def. 3.1.2) is the only wiring diagram that has a delay node.Definition 3.1.9. The eight wiring diagrams in Def. 3.1.1-3.1.8 will be referred to
as generating wiring diagrams.
.10.1) both have delay nodes {d 1 , . . . , d r }, it remains to check that the supplier assignment of the right side is equal to s ϕ = Id. This follows from a direct inspection because (i) colored units (Def. 2.3.2), 1-delay nodes (Def. 3.1.2), and 2-cells (Def. 3.1.4) all have identity supplier assignments and because (ii) γ (2.1.12.1) is an iteration of various ○ i (Def. 2.3.4).
W are elementarily equivalent by the horizontal associativity axiom (2.1.10.3). This elementary equivalence expresses the fact that the wiring diagram
Theorem 5.2.11. Consider the operad WD of wiring diagrams. (1) Every wiring diagram can be obtained from finitely many generating wiring diagrams (Def. 3.1.9) via iterated operadic compositions (Def. 2.1.10). Proof. The first statement is a special case of Theorem 5.1.11. The second statement is a combination of Remark 5.1.10, Lemma 5.2.8 twice, Lemma 5.2.9, and Lemma 5.2.10.(2) Any two equivalent simplices are either equal or are connected by a finite sequence
of elementary equivalences (Def. 5.2.2).
Such wiring diagrams are used in [Spi15, Spi15b] to study mode-dependent networks and dynamical systems. Recall Def. 2.2.13, Def. 2.2.15, and Convention 2.2.16 regarding wiring diagrams. Definition 5.3.1. Fix a class S.
WD • is a Box S -colored operad, called the operad of normal wiring diagrams.2.5.2), β 1 (4.3.3.1), β 2 (4.3.3.2),
and β 3 (4.3.3.3) are normal.
Proposition 5.3.5. With respect to
• the equivariant structure in Def. 2.3.1,
• the colored units in Def. 2.3.2, and
• the ○ i -compositions in Def. 2.3.4,
Theorem 5.3.7. Consider the operad WD • of normal wiring diagrams.(1) Every normal wiring diagram has a normal stratified presentation.(2) Every normal wiring diagram can be obtained from finitely many normal generating wiring diagrams via iterated operadic compositions (Def. 2.1.10).Proof. For statement (1), we reuse the proof of Theorem 5.1.11 while assuming r = 0. Statement (2) is a special case of statement (1).For statement (3) we reuse the proof of Theorem 5.2.11(2). In other words, we simply reuse the proofs of Lemma 5.2.8, Lemma 5.2.9, and Lemma 5.2.10 while assuming r = 0 throughout. The key observation is that, for normal simplices, elementary equivalences as in Def. 5.2.2 involve either:•elementary relations (Def. 3.3.30), none of which involves delay nodes, or • an operad associativity or unity axiom-(2.1.10.3), (2.1.10.4), (2.1.10.5), or (2.1.10.6)-for the normal generating wiring diagrams.(3) Any two equivalent normal simplices are connected by a finite sequence of ele-
mentary equivalences (Def. 5.2.2).
2.5.1) is a strict wiring diagram by Lemma 4.2.7. Proposition 5.4.6. With respect to • the equivariant structure in Def. 2.3.1, • the colored units in Def. 2.3.2, and • the ○ i -compositions in Def. 2.3.4, WD 0 is a Box S -colored operad, called the operad of strict wiring diagrams.
Definition 6.1.2. Suppose (O, 1, γ) is an S-colored operad as in Def. 2.1.3. An Oalgebra is a pair (A, µ) consisting of the following data.
bles can have wasted cables. In other words, while individual undirected wiring diagrams may have no wasted cables, the operadic composition can actually create wasted cables. So there is no such thing as the operad of undirected wiring diagrams without wasted cables. In[Fon15] (Def.Remark 7.1.5. Consider Def. 7.1.2 and 7.1.4.
(1) If S = { * }, a one-point set, then what we call an undirected { * }-wiring
diagram is called a singly-typed wiring diagram in [Spi13] (Example 2.1.7).
(2) If S = Set, the collection of sets, then what we call an undirected Set-wiring
diagram is called a typed wiring diagram in [Spi13] (Example 4.1.1).
(3) Cospans (7.1.2.2) are also used in other work about processes and net-
works. For example, cospans in a category, rather than just Fin S , are used
in [Fon15]. That setting is then used in [BF15, BFP16] to study passive
linear networks and Markov processes.
(4) In the book [Spi14] p.464 (but not in [Spi13] Example 2.1.7), Spivak's defi-
nition of an undirected { * }-wiring diagram is slightly different from ours.
More precisely, Spivak insisted that the maps ( f , g) in the cospan (7.1.2.2)
be jointly surjective, meaning that there are no wasted cables. However,
undirected wiring diagrams whose structure maps ( f , g) are jointly sur-
jectivity are not closed under the operad structure in UWD, to be defined
in Section 7.3. In Example 7.3.8 and Section 8.3 we will illustrate that the
operadic composition of undirected wiring diagrams without wasted ca-
Definition 7.2.1. Suppose C is a category (Def. 2.2.2), and
2.1.1). Even if it exists, it may be difficult to describe. Luckily, for S-finite sets (Def. 2.2.6), pushouts always exist and are easy to describe, as the following observation shows. Proposition 7.2.2. In the category Fin S of S-finite sets, each diagram of the form (7.2.1.1) has a pushout given by the quotient
If S = { * }, a one-point set, then our Fin-colored operad UWD is called the operad of singly-typed wiring diagrams in [Spi13] (Example 2.1.7).Theorem 7.3.14. For any class S, when equipped with the structure in Def. 7.3.1-7.3.5,
UWD in Def. 7.1.4 is a Fin S -colored operad, called the operad of undirected wiring dia-
grams.
Proof. In view of Def. 2.1.10, this follows from Lemmas 7.3.9, 7.3.11, and 7.3.13.
Example 7.3.15. Consider Theorem 7.3.14.
(1)
Definition 8.1.1. Define the empty cell
Theorem 10.2.7. Consider the operad UWD of undirected wiring diagrams.(1) Every undirected wiring diagram can be obtained from finitely many generators
(Def. 8.1.7) via iterated operadic compositions (Def. 2.1.10).
(2) Any two equivalent simplices in UWD are either equal or are connected by a finite
sequence of elementary equivalences (Def. 10.2.1).
Proof. The first statement is a special case of Theorem 10.1.12. The second state-
ment is a combination of Lemma 10.2.5 twice and Lemma 10.2.6.
.1.8) corresponding to the associativity property of 2-cells (8.2.9.1);• the fact that the transpositions (i, i + 1) for 1 ≤ i ≤ n − 1 generate the symmetric group Σ n .So (A, µ) is a UWD-algebra in the sense of Def. 6.1.3.Remark 11.1.3. The proofs of the finite presentation theorems 11.1.2 and 6.2.2 for UWD-algebras and WD-algebras are almost identical. In fact, it is not difficult to formulate and prove a more general result that has both of these finite presentation theorems as special cases. Such a result would say that, if an operad O has a finite presentation (i.e., specific finite sets of generators and generating relations expressed in terms of simplices and elementary equivalences similar to Def. 10.1.2 and 10.2.1), then O-algebras have a corresponding finite presentation. We purposely chose not to present the material this way in order to avoid the higher level of abstraction that is unnecessary for actual applications of (undirected) wiring diagrams. Although the context is slightly different, the formulation and proof of such a finite presentation theorem for O-algebras can be extracted from the Strong Biased Definition Theorem in[YJ15] page 193.• the operad associativity axioms (2.1.10.3) and (2.1.10.4) when applied to
2-cells;
Theorem 11.2.5. For each set A, the relational algebra Rel A in Def. 11.2.4 is actually a UWD-algebra in the sense of Def. 11.1.1, hence also in the sense of Def. 6.1.3 by Theorem 11.1.2.
). The other 16 generating axioms are checked similarly. Remark 11.2.6. To see that our relational algebra Rel A in Def. 11.2.4 agrees with the one in [Spi13] Example 2.2.10, note that the latter is based on Def. 6.1.2, which is equivalent to Def. 6.1.3. A direct inspection of [Spi13] Eq. (11) reveals that Spivak's structure map of Rel A , when applied to the 6 generators in UWD (Def. 8.1.7), reduces to our generating structure maps in Def. 11.2.4. Theorem 11.2.5 then guarantees that the two definitions of the relational algebra Rel A -i.e., our Def. 11.2.4 and [Spi13] Example 2.2.10-are equivalent.
Example 11.4.3. Suppose A is a set and (X, v) ∈ Fin Set such that v(x) = A for all x ∈ X. Then X v = x∈X A = A X as in Def. 11.2.1Example 11.4.4. Each map
the X-colored entry of the relational algebra of A (Def. 11.2.4). Furthermore, if in Def. 11.4.5 all the value assignments v take the constant value A, then the 6 generating structure maps of Rel reduce to those of Rel A in Def. 11.2.4.
Example 12.1.6. Suppose f ∶ O G G P is an operad map as in Def. 12.1.1 and A = {A t } t∈T is a P-algebra as in Def. 6.1.2 or Def. 6.1.3. Then there is an induced Oalgebra A f defined by the following data.
Remark 12.2.3. Consider the output soldering function of ψ in (12.2.2.3). Due to the non-instantaneity requirement (2.2.13.2), the restriction of the supplier assignments ψ to Y out is a map Y out G G X out .Proof. As before we will write both χ 0 andχ 1 as χ. We must check that χ preserves the operad structure in the sense of Def. 12.1.1. In both WD • (2.3.1.1) and UWD (7.3.1.1), the equivariant structure is given by permuting the labels of the input boxes. So χ preserves equivariance in the sense of (12.1.1.1). Likewise, it follows immediately from the definitions of the colored units in WD • (2.3.2.1) and UWD (7.3.2.1) that they are preserved by χ in the sense of (12.1.1.2).To check that χ preserves operadic composition in the sense of (12.1.1.3), suppose ϕ ∈ WD • Y X 1 ,...,Xn with n ≥ 1, 1 ≤ i ≤ n, and ψ ∈ WD • X i W 1 ,...,Wm with m ≥ 0. We must show that χ ϕ ○ i ψ = χϕ ○ i χψ ∈ UWD Z = (X ○ i W) = X 1 , . . . , X i−1 , W 1 , . . . , W m , X i+1 , . . . , X n ∈ Prof (Fin S )as in (2.1.10.2), X = (X 1 , . . . , X n ), and W = (W 1 , . . . , W m ). prove (12.2.4.2), on the one hand, by Def. 2.3.4 ϕ ○ i ψ ∈ WD •2
y 1
y 2
y 3
Theorem 12.2.4. The maps χ 0 (12.2.2.1) and χ 1 (12.2.2.2) define a map of operads
WD •
χ
G G UWD .
(12.2.4.1)
Y
Z
(12.2.4.2)
in which
To Y
X○ i W has supplier
assignment
Sp ϕ○ i ψ that is given by s ϕ , s ψ s ϕ , s ψ , s ϕ s ψ , or s ψ s ϕ s ψ according to (2.3.4.2) and (2.3.4.3). HereW in =
m
∐
k=1
W in
k
and
W out =
m
∐
k=1
W out
k
∈ Fin S .
So by (12.2.2.3)
With this definition of η, it remains to check the equalities η Id ∐s ψ X out Both of these equalities can be checked by a direct inspection. This finishes the proof that the square (12.2.4.5) is a pushout and, therefore, that χ preserves operadic composition (12.2.4.2).Example 12.2.5. By Example 12.1.6 and Theorem 12.2.4, every UWD-algebra (Def. 11.1.1) induces a WD • -algebra (Def. 6.4.1) along the operad map χ ∶ WD • G G UWD. For example:i
= β
and
ηh = α.
(12.2.4.6)
Example 12.3.1. For the empty wiring diagram ǫ ∈ WD • ∅ (Def. 3.1.1), the image χǫ ∈ UWD ∅ is the empty cell (Def. 8.1.1). Example 12.3.2. For a name change τ f ∈ WD • Y X (Def. 3.1.3), the image χτ f ∈ UWD Y X is the name change τ f (Def. 8.1.3) corresponding to the bijection
Example 12.3.3. For a 2-cellθ X,Y ∈ WD • X∐Y X,Y (Def. 3.1.4), the image χθ X,Y ∈ UWD X∐Y X,Y is the 2-cell θ (X,Y) (Def. 8.1.4). Example 12.3.4. For a 1-loop λ X,x ∈ WD • X∖x X (Def. 3.1.5), the image χλ X,x ∈ UWDis the loop λ (X,x±) (Def. 8.1.5).Example 12.3.5. For an out-split σ Y,y 1 ,y 2 ∈ WD •X∖x±
X
Y
X (Def. 3
In χω Y,y we drew all of X on the right side to make the picture easier to read. It has a (0, 1)-cable, and all other cables are (1, 1)-cables.Example 12.3.8. Each 1-output wire is in the image of χ. Indeed, for a 1-output wire ω y ∈ UWD y (Def. 8.1.2), we have ω y = θ (∅,y) ○ 2 ω y ○ ǫ = χω y,y ○ χǫ by Examples 12.3.7 and 12.3.1= χ ω y,y ○ ǫ .3.7.1) also appeared in the elementary relations
(8.2.6.1) and (8.2.7.1) and the example (8.3.3.1). The equality (12.3.7.1) may be visu-
alized as the following picture.
X
y
ω Y,y
↝
χ
X
y
⋮
χω Y,y
(12.3.7.2)
(12.3.8.1)
12.4.2.4) is• g ϕ when restricted to V;• f ϕ when restricted to ∐ j U j ∖ {u c }.Lemma 12.4.3. In the context of Def. 12.4.2, we haveχψ = ϕ ∈ UWD Y X 1 ,...,X N = UWD V U 1 ,...,U N .Proof. By definition χψ is the cospan (12.2.2.3)
ψ that is given by s ϕ , s ψ s ϕ , s ψ , s ϕ s ψ , or s ψ s ϕ s ψ according to (2.3.4.2) and (2.3.4.3). Here ∈ Fin S . So by (13.1.2.2W in =
m
∐
k=1
W in
k
and
W out =
m
∐
k=1
W out
k
1 -
1delay node, 40 1-internal wasted wire, 45 1-loop, 41 1-output wire, 170 1-wasted wire, 44 2-cell, 41 algebra of discrete systems, 138 algebra of open dynamical systems, 146 associative law, 22 associative operad, 18 associativity of 2-cells, 49 associativity of a category, 24 associativity of an algebra, 107 associativity of an operad, 15, 20 associativity of in-splits, 56 associativity of out-splits, 59 associativity of splits, 178 associativity of undirected 2-cells, 176 colored entries of an algebra, 106 colored operad, 15, 20 colored unit, 15 commutative law, 22 commutative operad, 18 commutativity of 1-loops, 52 commutativity of 1-wasted wires, 60 commutativity of 2-cells, 50 commutativity of in-splits, 56 commutativity of loops, 180 commutativity of out-splits, 59 commutativity of splits, 178 commutativity of undirected 2-cells, 176 ○ i -composition, 20 ○ i -composition in UWD, 161 ○ i -composition in WD, 31 ○ i -composition of structure maps, 108 composition, 20 composition of a simplex, 88 composition of a simplex in UWD, 204 concatenation, 15 conncted graph, 18 coproduct of boxes, 26 coproduct of finite sets, 25 cospan, 155 cut, 22 edge substitution, 19 elementarily equivalent, 93 elementarily equivalent in UWD, 207 elementary equivalence, 93 elementary equivalence in UWD, 207 elementary relations, 61 elementary relations in UWD, 180 elementary subsimplex, 93 elementary subsimplex in UWD, 207 empty cell, 170 empty profile, 14 empty wiring diagram, 40 endomorphism operad, 17 entry map, 236 entry of an operad, 15 ǫ, 40, 170 equivalence of prewiring diagrams, 28 equivalent simplices, 93 equivalent simplices in UWD, 207 equivariance in UWD, 160, 163 equivariance in WD, 30, 32 equivariance of an algebra, 107 equivariance of an operad, 16, 22 external wasted wire, 28 externally supplied element, 69 Fin, 25 Fin S , 25 finite presentation for the propagator algebra, 129 finite presentation for the algebra of discrete systems, 143 finite presentation for the algebra of open dynamical systems, 148 finite presentation for the relational algebra, 223 finite presentation for the typed relational algebra, 230 finite presentation for UWD, 211 finite presentation for UWD-algebras, 219 finite presentation for WD, 98 finite presentation for WD-algebras, 117 finite presentation for WD•, 100 finite presentation for WD•-algebras, 136 finite presentation for WD 0 , 103 finite presentation for WD 0 -algebras, 144 generating undirected wiring diagrams, 172 generating wiring diagrams, 44 generators, 172 global input, 27 global output, 27 graph, 18 graph groupoids, 5 graph operation, 18 historical propagator, 119 historical propagators of type X, 120 historicity, 119 horizontal associativity, 21 horizontal associativity in UWD, 164 horizontal associativity in WD, 33 in-split, 42 induced operad algebra, 238 initial operad, 237 input, 25 input box, 27, 154 input profile, 15 input soldering function, 155 internal input, 28 internal output, 28 internal wasted wire, 28 internally supplied element, 69 isomorphism of operad algebras, 225 iterated loops, 201 iterated splits, 198 λ (X,x±) , 171 λ X,x , 41 left unity, 16, 21 length of a profile, 14 loop, 18, 171 loop element, 69 manifolds, 144 map of operad algebras, 224 map of operads, 236 (m, n)-cable, 155, 186 (≥ m, n)-cable, 186 (m, ≥ n)-cable, 186 (≥ m, ≥ n)-cable, 186 mode-dependent networks, 1 module over a monoid, 108 moment delay, 120 monoid, 17 monoid with multiple objects, 24 monoidal category, 4 morphism, 23 multicategory, 22, 280 multimaps, 22 n-ary element, 15 name change, 40 no passing wires, 101 non-commutative probability space, 109 non-instantaneity requirement, 28 normal generating wiring diagram, 100 normal simplex, 100 normal stratified presentation, 100 normal stratified simplex, 100 normal wiring diagram, 99 object, 23 ω * , 170 ω X,x , 45 ω Y,y , 44 open dynamical system, 145 operad, 15, 20 operad algebra, 106 operad algebra induced along a map, 238 operad inclusion, 237 operad map, 236 operad map composition, 237 operadic composition, 15, 20 out-split, 43 output, 25 output box, 27 output color, 15 output soldering function, 155 readout function, 136 relational algebra of a set, 222 right action, 15 right permutation, 15 right unity, 16, 21 S-box, 25 S-finite set, 25 σ X,x 1 ,x 2 , 42 σ (X,x 1 ,x 2 ) , 172 σ Y,y 1 ,y 2 , 43 signal-flow graphs, 5 signed finite set, 25 simplex, 88 simplex in UWD, 204 singly-typed wiring diagram, 156 soldered to, 155 Sp, 27 Spivak's Conjecture, 225 split, 172 state set, 136 stratified presentation, 91, 206 stratified presentations exist, 91 stratified presentations exist in UWD, 206 stratified simplex, 91 stratified simplex in UWD, 205 strict elementary equivalences, 103 strict elementary relation, 103 strict generating wiring diagram, 102 strict simplex, 102 strict stratified presentation, 103 strict stratified simplex, 103 strict wiring diagram, 101 string diagrams, 5 structure map of an algebra, 106 sub-operad, 237 subsimplex, 92 subsimplex in UWD, 207 substitution, 22 supplier, 28 supplier assignment, 28 supply, 27 supply wire, 28 symmetric group, 15 symmetric group action, 15 symmetric operad, 17 tangent bundle, 145 τ X,Y , 40, 170 terminal operad, 237 θ (X,Y) , 171 θ X,Y , 41 top equivariance, 17 traced monoidal category, 279 traffic space, 109 transition diagram, 137 truncation, 119 typed relational algebra, 228 typed wiring diagram, 156 underline notation, 17 undirected 2-cell, 171 undirected name change, 170 undirected prewiring diagram, 154 undirected wiring diagram, 155 units in UWD, 160 units in WD, 30 unity in UWD, 163 unity in WD, 32 Index unity of 2-cells, 49 unity of a category, 24 unity of an algebra, 107 unity of an operad, 16, 21 update function, 136 UWD, 156 UWD is an operad, 166 UWD-algebra, 214 value assignment, 25 vector field, 146 vertical associativity, 21 vertical associativity in UWD, 165 vertical associativity in WD, 35 vertical notation, 17 wasted cable, 155 WD, 29 WD is an operad, 36 WD-algebra, 110 WD•, 99 WD• is an operad, 99 WD•-algebra, 135 WD 0 , 101 WD 0 is an operad, 102 WD 0 -algebra, 144 wheeled prop, 5, 281 wiring diagram, 29black box, 25
block permutation, 17
block sum, 17
bottom equivariance, 17
box, 25, 155
cable, 155
category, 23
category of finite sets, 25
class, 13
coherence, 4
color map, 236
decomposition, 63, 190
decomposition of unary wiring diagrams, 71,
76
delay node, 27
δ d , 40
demand, 27
discrete system, 136
Dm, 27
DN, 27
289
290
Index
double-loop, 52
dynamical systems, 1
Pentagon Axiom, 5
power set, 221
presentation, 89, 204
prewiring diagram, 27
profile, 14
prop, 280
propagator algebra, 124
props, 5
pushout, 158
. Wiring Diagrams
Proof. Since σ has no delay nodes, if k = 1, then s σ = Id. So σ is a colored unit.Suppose k ≥ 2. We will prove that σ is an iterated operadic composition of
(a 1 , a 2 )(b 1 , b 2 ) (a 1 , b 2 ) (b 1 , a 2 ) (b 1 , a 2 ) (b 1 , b 2 ) (a 1 , a 2 ) (a 1 , b 2 )
Motivation 9.3.1. The following result says that an undirected wiring diagram of the form
Compatibility with Structure Maps: For each d ∈ S, c = (c 1 , . . . , c n ) ∈ Prof(S), and ζ ∈ O d c , the diagramis commutative.Furthermore, we call f an isomorphism if there exists a map of O-algebras g ∶ B G G A such that g f = Id A and f g = Id B . If such a map g exists, then it is necessarily unique.The following conjecture regarding the relational algebra is[Spi13]Conjecture 3.1.6. Since we are talking about the relational algebra, here S is a one-element set, and UWD is a Fin-colored operad.Conjecture 11.3.2. Suppose:• A is a set, and Rel A is the relational algebra of A in Def. 11.2.4.Then at least one of the following two statements holds.(1) f is an isomorphism.(2) B X is a one-element set for each finite set X.Roughly speaking this conjecture states that there are no interesting maps out of any relational algebra. In the following observation, we will verify Conjecture 11.3.2 in the special case when the map Rel A G G B is induced by a map of sets out of A.Theorem 11.3.3. Suppose f ∶ AG G B is a map of sets. Then the following statements are equivalent.(1) f is a bijection of sets.(2) The post-composition maps f * ∶ Rel A(X)G G Rel B (X) (11.2.2.5), with X running through all the finite sets, form an isomorphism of UWD-algebras.(3) The post-composition maps f * ∶ Rel A(X)G G Rel B(X), with X running through all the finite sets, form a map of UWD-algebras.ProofOn the other hand, applying the top horizontal map f * to U yieldswhich is the single-element set consisting of the map f . Since f (a + ) = f (a − ), applying the right vertical map now yields the single-element setwhich contains one element. So λ (A,0,1) f * = f * λ (A,0,1) , and the diagram (11.3.3.2) in this case is not commutative. In particular, this non-injective function f does not induce a map of UWD-algebras Rel A G G Rel B .Finite Presentation for the Typed Relational AlgebraThe relational algebra Rel A in Def. 11.2.4 has a fixed set A as the set of potential values in each coordinate in an X-vector (Def. 11.2.1). There is a more general version of the relational algebra, called the typed relational algebra, in which each coordinate in an X-vector has its own set of potential values. The typed relational algebra was first introduced in [Spi13] (Section 4) using Def. 6.1.2. In this section, we observe that the typed relational algebra also has a finite presentation, similar to the one for the relational algebra in Theorem 11.2.5.Part 3Maps Between Operads of Wiring DiagramsSo far we have considered four operads constructed from wiring diagrams and undirected wiring diagrams:(1) the Box S -colored operad of wiring diagrams WD (Theorem 2.3.11);(2) the Box S -colored operad of normal wiring diagrams WD • (Prop. 5.3.5);(3) the Box S -colored operad of strict wiring diagrams WD 0 (Prop. 5.4.6);(4) the Fin S -colored operad of undirected wiring diagrams UWD (Theorem 7.3.14).The purpose of this part is to study maps between these operads. We will show that there is a commutative diagramof operad maps, in which the horizontal maps are operad inclusions. All the operad maps in this diagram except ρ are discussed in Chapter 12. The operad map ρ is discussed in Chapter 13. The existence of such operad maps implies that:• Every UWD-algebra induces a WD-algebra along ρ.• Every WD-algebra restricts to a WD • -algebra.• Every WD • -algebra restricts to a WD 0 -algebra.For each of the three operad maps that end at UWD, we will compute precisely the image. An undirected wiring diagram is in the image of • χ 0 if and only if its cables are either (1, 1)-cables or (2, 0)-cables;• χ if and only if it has no (0, 0)-cables and no (0, ≥ 2)-cables.Furthermore, the operad map ρ ∶ WD G G UWD is surjective, so every undirected wiring diagram is the ρ-image of some wiring diagram. Delay nodes play a crucial role in the surjectivity of the operad map ρ.Reading Guide. The reader who already knows about operad maps may skip most of Section 12.1 and go straight to Propositions 12.1.7 and 12.1.9. Instead of the proof of Theorem 12.2.4 about the existence of the operad map χ, the reader may wish to concentrate on the motivating Example 12.2.1. Likewise, the proofs of Theorems 12.4.1 and 12.5.1 about the images of the operad maps χ and χ 0 may be skipped during the first reading.Instead of the proof of Theorem 13.1.4 on the existence of the operad map ρ, the reader may wish to concentrate on the motivating Example 13.1.1. Likewise, beforeChapter 12Map from Normal to Undirected Wiring DiagramsThis chapter has three main purposes.(1) We show that there exists an operad map χ ∶ WD • G G UWD from the operad WD • of normal wiring diagrams to the operad UWD of undirected wiring diagrams. See Theorem 12.2.4. Recall that a normal wiring diagram is a wiring diagram with no delay nodes. Intuitively, the map χ is given by forgetting directions.(2) We compute precisely the image of the operad map χ ∶ WD • G G UWD in Theorem 12.4.1. An undirected wiring diagram is in the image of χ if and only if it has no wasted cables and no (0, ≥ 2)-cables.(3) We consider the restriction χ 0 ∶ WD 0 G G UWD of the operad map χ ∶ WD • G G UWD to the operad WD 0 of strict wiring diagrams and compute precisely its image in Theorem 12.5.1. An undirected wiring diagram is in the image of χ 0 if and only if its cables are either (1, 1)-cables or (2, 0)cables.In Theorem 13.1.4 we will extend the operad map χ ∶ WD • G G UWD to an operad map ρ ∶ WD G G UWD defined for all wiring diagrams. Furthermore, we will show in Theorem 13.3.3 that the operad map ρ ∶ WD G G UWD is surjective.In Section 12.1 we define a map of operads and observe that there are inclusions of operads WD 0 G G WD • G G WD. Furthermore, if the underlying class S changes,Preservation of Equivariance:For each d c ∈ Prof(S) × S as above and permutation σ ∈ Σ n , the diagramPreservation of Colored Units:For each c ∈ S,Preservation of Operadic Composition: For eachis commutative. Call the two cables on the left, from top to bottom, c 1 and c 2 and the three cables on the right, also from top to bottom, c 3 , c 4 , and c 5 . Then on the left side c 1 is a (2, 1)cable, and c 2 is a (0, 1)-cable. On the right side, c 3 , c 4 , and c 5 are a (2, 1)-cable, a (1, 0)-cable, and a (2, 2)-cable (Def. 7.1.2), respectively.Note that the set of cables {c 1 , . . . , c 5 } in ϕ is in canonical bijection with the set of supply wires in ϕ (Def. 2.2.13), namelyWith this identification, the input and output soldering functions of ϕ are completely determined by the identity map on Sp ϕ and the supplier assignment of ϕ. We will make this precise in (12.2.2.3) below.With the previous example as motivation, we now define the map of operads WD • G G UWD. Recall the definitions of normal wiring diagrams (Def. 2.2.13 and 5.3.1) and of undirected wiring diagrams (Def. 7.1.2 and 7.1.4).Definition 12.2.2. Fix a class S.(1) Define the map χ 0 ∶ Box S G G Fin S by(2) For eachin Fin S . Here:is the supplier assignment for ψ (Def. 2.2.13).Theorem 12.5.1. The image of the operad map χ 0 ∶ WD 0 G G UWD consists of precisely the undirected wiring diagrams whose cables are either (1, 1)-cables or (2, 0)-cables.Proof. To make the argument easier to read, we will prove the two required inclusions in Lemmas 12.5.2 and 12.5.3 below. Proof. Suppose ψ ∈ WD 0 Y X 1 ,...,Xn , so it has no delay nodes and its supplier assignmentin Fin S . By definition (12.2.2.3), χ 0 ψ ∈ UWD Y X 1 ,...,Xn is the following cospan.Observe that:Since there are no other cables, this finishes the proof.Lemma 12.5.3. If ϕ ∈ UWD has only (1, 1)-cables and (2, 0)-cables, then it is in the image of the operad map χ 0 ∶ WD 0 G G UWD.Proof. Supposeare not in the image of χ 0 .Example 12.5.5. In the following picture, the strict wiring diagram on the left is sent by χ 0 ∶ WD 0 G G UWD to the undirected wiring diagram on the right.On the right, there are two (2, 0)-cables, and the other cables are (1, 1)-cables.Example 12.5.6. In the following picture, the strict wiring diagram on the left is sent by χ 0 ∶ WD 0 G G UWD to the undirected wiring diagram on the right.On the right, there are two (2, 0)-cables and three (1, 1)-cables.Summary of Chapter 12(1) There is an operad map χ ∶ WD • G G UWD given by forgetting directions whose image consists of precisely the undirected wiring diagrams with no wasted cables and no (0, ≥ 2)-cables.(2) The restriction of χ to WD 0 is an operad map χ 0 ∶ WD 0 G G UWD. Its image consists of precisely the undirected wiring diagrams whose cables are either (1, 1)-cables or (2, 0)-cables.Chapter 13Map from Wiring Diagrams to Undirected Wiring DiagramsThis chapter has two main purposes.(1) We extend the operad map χ ∶ WD • G G UWD in Theorem 12.2.4, defined for normal wiring diagrams (i.e., those without delay nodes), to an operad map ρ ∶ WD G G UWD that is defined for all wiring diagrams. See Theorem 13.1.4.(2) Furthermore, we will show that the operad map ρ ∶ WD G G UWD is surjective; see Theorem 13.3.3. In other words, every undirected wiring diagram is the ρ-image of some wiring diagram.We remind the reader that the image of the operad map χ ∶ WD • G G UWD was identified in Theorem 12.4.1. It consists of precisely those undirected wiring diagrams with no wasted cables and no (0, ≥ 2)-cables.At first glance, the existence of the operad map ρ ∶ WD G G UWD is not obvious because a general wiring diagram has delay nodes, but undirected wiring diagrams have no obvious analogues of delay nodes. In fact, for this reason Rupel and Spivak ([RS13] 4.1) expressed doubt that there exists an operad map from WD to UWD. Our main results in this chapter, Theorems 13.1.4 and 13.3.3, show that not only is there an operad map ρ ∶ WD G G UWD, but also it is surjective. As we will see in (13.3.1.6), delay nodes play a critical role in realizing wasted cables and (0, ≥ 2)cables in undirected wiring diagrams.In δ d the delay node is supplied by the unique global input, so in ρδ d there is only one cable. In ρδ d the output box {d, d} ∈ Fin S is the S-finite set with two copies of the element d ∈ S. There are no input boxes in ρδ d , and the only cable in it is a (0, 2)-cable.Example 13.2.2. In the following picture, the wiring diagram ϕ ∈ WD Y X with one delay node d is sent by ρ to the undirected wiring diagram on the right.Indeed, the set of supply wires in ϕ isSince the delay node d is supplied by the global input y 1 , by definition (13.1.2.2) the set of cables of ρϕ isTherefore, in ρϕ the cable represented by y 1 is a (2, 3)-cable. It is soldered to: y 1 , the input of X supplied by y 1 in ϕ, and the two global output wires and the input wire of X supplied by d in ϕ. The cable represented by y 2 is a (0, 1)-cable. The other two cables are a (1, 0)-cable and a (2, 1)-cable.Example 13.2.3. In the following picture, the wiring diagram ϕ ∈ WD Y X with one delay node d is sent by ρ to the undirected wiring diagram on the right.In ϕ the delay node is supplied by itself, so the set of cables in ρϕ isIn ρϕ the cable corresponding to d is a (1, 2)-cable. The other two cables are both (1, 1)-cables.Example 13.2.4. In the following picture, the wiring diagram ϕ ∈ WD Y X with one delay node d is sent by ρ to the undirected wiring diagram on the right.In ϕ the delay node d is supplied by the unique output wire x of X, so their corresponding cables are identified in ρϕ. This cable is a (2, 2)-cable. The other cable is a (1, 1)-cable.PropsWhile an operad models operations with multiple inputs and one output, a propshort for product and permutation-models operations with multiple inputs and multiple outputs. A typical example of a prop is the collection of functions Map(X m , X n ) for a set X with m, n ≥ 0. Using the kind of pictures in Motivation 2.1.2, a function f ∶ X m G G X n may be depicted as follows. The articles[Mar08,Val12]provide surveys of these compositional structures. The book[YJ15]is a comprehensive foundation of this subject.
. ∈ Uwd Y X, Def. 8.1. 3∈ UWD Y X (Def. 8.1.3);
. X , Y ∈ Uwd X∐y X, Y (Def. 8.1.4)• a 2-cell θ X,Y ∈ UWD X∐Y X,Y (Def. 8.1.4);
extends to an operad map ρ ∶ WD G G UWD that sends each delay node to a cable. G G Uwd, G G UWD extends to an operad map ρ ∶ WD G G UWD that sends each delay node to a cable.
Give a detailed proof for each elementary relation. These proofs are similar to those for Lemma 2.3.8, Lemma 2.3.10, and Prop. 3.2.3Give a detailed proof for each elementary relation. These proofs are similar to those for Lemma 2.3.8, Lemma 2.3.10, and Prop. 3.2.3.
For each elementary relation, draw a picture that depicts the operadic compositions. similar to those just before Prop. 3.3.9 and Prop. 3.3.11, if one was not givenFor each elementary relation, draw a picture that depicts the operadic compositions, similar to those just before Prop. 3.3.9 and Prop. 3.3.11, if one was not given.
Write down precisely the wiring diagrams π, π 1 , and π 2 , including their systems in [VSL15], when applied to the strict generating wiring diagrams-namely. ǫ, τ X,Y , θ X,Y , and λ X,x -Write down precisely the wiring diagrams π, π 1 , and π 2 , including their systems in [VSL15], when applied to the strict generating wiring diagrams-namely, ǫ, τ X,Y , θ X,Y , and λ X,x -
for categories, operads, props, and their applitheory, of which [Mac98] is the most advanced and [Awo10, Lei14, Rie16] are more basic. The basic concepts of categories, functors, and natural transformations were all introduced in the founding article. for categories, operads, props, and their appli- theory, of which [Mac98] is the most ad- vanced and [Awo10, Lei14, Rie16] are more basic. The basic concepts of categories, functors, and natural transformations were all introduced in the founding article
Category Theory, 2nd. S Awodey, Oxford Logic Guides. OxfordOxford Univ. Press52S. Awodey, Category Theory, 2nd. ed., Oxford Logic Guides 52, Ox- ford Univ. Press, Oxford, 2010.
M Barr, C Wells, Category Theory for Computing Science. Prentice-HallM. Barr and C. Wells, Category Theory for Computing Science, Prentice- Hall , 1990.
General theory of natural equivalences. S Eilenberg, S Maclane, Trans. Amer. Math. Soc. 58S. Eilenberg and S. MacLane, General theory of natural equivalences, Trans. Amer. Math. Soc. 58 (1945), 231-294.
Traced monoidal categories. A Joyal, R Street, D Verity, Mathematical Proceedings of the Cambridge Philosophical Society. 119A. Joyal, R. Street, and D. Verity, Traced monoidal categories, Mathe- matical Proceedings of the Cambridge Philosophical Society 119 (1996), 447-468.
F W Lawvere, S H Schanuel, Conceptual Mathematics: A first introduction to categories. Cambridge2nd ed.F.W. Lawvere and S.H. Schanuel, Conceptual Mathematics: A first in- troduction to categories, 2nd ed., Cambridge, 2009.
Basic Category Theory, Cambridge Studies in Adv. T Leinster, MathT. Leinster, Basic Category Theory, Cambridge Studies in Adv. Math.
Categories for the working mathematician. S Mac Lane, Grad. Texts in Math. 5Springer-Verlag2nd ed.S. Mac Lane, Categories for the working mathematician, Grad. Texts in Math. 5, 2nd ed., Springer-Verlag, New York, 1998.
Basic Category Theory for Computer Scientists. B C Pierce, MIT PressB.C. Pierce, Basic Category Theory for Computer Scientists, MIT Press,1991.
E , Category Theory in Context. Dover, New YorkE. Riehl, Category Theory in Context, Dover, New York, 2016.
Category Theory for the Sciences. D I Spivak, MIT PressD.I. Spivak, Category Theory for the Sciences, MIT Press, 2014.
. R F C Walters, CambridgeCategories and Computer ScienceR.F.C. Walters, Categories and Computer Science, Cambridge, 1991.
G M Kelly, On the operads of J.P. May. 13G.M. Kelly, On the operads of J.P. May, Reprints in Theory Appl. Categ. 13 (2005), 1-13.
Deductive systems and categories. II. Standard constructions and closed categories. J Lambek, Category Theory, Homology Theory and their Applications, I (Battelle Inst. Conf., Seattle, Wash. BerlinSpringer1J. Lambek, Deductive systems and categories. II. Standard construc- tions and closed categories, in: 1969 Category Theory, Homology Theory and their Applications, I (Battelle Inst. Conf., Seattle, Wash., 1968, Vol. 1) p.76-122, Springer, Berlin, 1969.
Operads in Algebra. M Markl, S Shnider, J Stasheff, Topology and Physics, Math. Surveys and Monographs. 96Amer. Math. SocM. Markl, S. Shnider, and J. Stasheff, Operads in Algebra, Topology and Physics, Math. Surveys and Monographs 96, Amer. Math. Soc., Providence, 2002.
The geometry of iterated loop spaces. J P May, Lecture Notes in Math. 271Springer-VerlagJ.P. May, The geometry of iterated loop spaces, Lecture Notes in Math. 271, Springer-Verlag, New York, 1972.
Definitions: operads, algebras and modules. J P May, Contemp. Math. 202J.P. May, Definitions: operads, algebras and modules, Contemp. Math. 202, p.1-7, 1997.
Algebraic Operads, Grund. der math. J.-L Loday, B Vallette, Wiss. 346Springer-VerlagJ.-L. Loday and B. Vallette, Algebraic Operads, Grund. der math. Wiss. 346, Springer-Verlag, New York, 2012.
Colored Operads, Graduate Studies in Math. D Yau, Amer. Math. Soc170Providence, RID. Yau, Colored Operads, Graduate Studies in Math. 170, Amer. Math. Soc., Providence, RI, 2016.
Categorical algebra. S Mac Lane, Bull. Amer. Math. Soc. 71S. Mac Lane, Categorical algebra, Bull. Amer. Math. Soc. 71 (1965), 40-106.
Operads and PROPs, Handbook of Algebra 5. M , ElsevierM. Markl, Operads and PROPs, Handbook of Algebra 5, p.87-140, El- sevier, 2008.
Wheeled PROPs, graph complexes and the master equation. M Markl, S Merkulov, S Shadrin, J. Pure Appl. Algebra. 213M. Markl, S. Merkulov, and S. Shadrin, Wheeled PROPs, graph com- plexes and the master equation, J. Pure Appl. Algebra 213 (2009), 496-535.
B Vallette, arXiv:1202.3245Algebra + Homotopy = Operad, preprint. B. Vallette, Algebra + Homotopy = Operad, preprint, arXiv:1202.3245.
of categories, operads, and props in the sciences, including dynamical systems, computer science, engineering, network theory, linguistics. D Yau, M W Johnson, Math. Surveys and Monographs. 203biology. and machine learningD. Yau and M.W. Johnson, A Foundation for PROPs, Algebras, and Mod- ules, Math. Surveys and Monographs 203, Amer. Math. Soc., Providence, RI, 2015 of categories, operads, and props in the sciences, including dynamical systems, computer science, engineering, network theory, lin- guistics, biology, neuroscience, and machine learning.
Monads: Some Linguistic Applications. A Asudeh, A. Asudeh, Monads: Some Linguistic Applications, available at http://users.ox.ac.uk/∼cpgl0036/handouts/asudeh-se-lfg13.pdf.
. J C Baez, J Erbele, arXiv:1405.6881Categories in control. J.C. Baez and J. Erbele, Categories in control, arXiv:1405.6881.
J C Baez, B Fong, arXiv:1504.05625A compositional framework for passing linear networks. J.C. Baez and B. Fong, A compositional framework for passing linear networks, arXiv:1504.05625.
A compositional framework for Markov processes. J C Baez, B Fong, B S Pollard, J. Math. Phys. 573pJ.C. Baez, B. Fong, and B.S. Pollard, A compositional framework for Markov processes, J. Math. Phys. 57, No. 3, 033301, 30 p. (2016).
Operads and the tree of life. J Baez, N Otter, J. Baez and N. Otter, Operads and the tree of life, available at http://math.ucr.edu/home/baez/tree of life/tree of life.pdf.
Logic and Computation: A Rosetta Stone. J C Baez, M Stay, New Structures for Physics. Bob CoeckeBerlinSpringer813Physics, TopologyJ.C. Baez and M. Stay, Physics, Topology, Logic and Computation: A Rosetta Stone, in: New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics vol. 813, Springer, Berlin, 2011, pp. 95-174.
A categorical semantics of signal flow graphs. F Bonchi, P Sobociński, F Zanasi, Lecture Notes in Comp. Sci. 8704SpringerF. Bonchi, P. Sobociński, and F. Zanasi, A categorical semantics of sig- nal flow graphs, pp. 435-450, Lecture Notes in Comp. Sci. 8704, Springer, 2014.
Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience. R Brown, T Porter, Proceedings of the International Conference on Theoretical Neurobiology. Nandini Singhthe International Conference on Theoretical NeurobiologyDelhi; National Brain Research Centre1R. Brown and T. Porter, Category Theory and Higher Dimensional Alge- bra: potential descriptive tools in neuroscience, in: Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1 (2003) 80-92.
Quantum picturalism. B Coecke, Contemporary Physics. 51B. Coecke, Quantum picturalism, Contemporary Physics 51 (2010), 59- 83.
C Heunen, M Sadrzadeh, E Grefenstette, Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse. OxfordC. Heunen, M. Sadrzadeh, and E. Grefenstette, ed., Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse, Oxford, 2013.
O Iordache, Self-Evolvable Systems: Machine Learning in Social Media. BerlinSpringer-VerlagO. Iordache, Self-Evolvable Systems: Machine Learning in Social Media, Springer-Verlag, Berlin, 2012.
. M A Méndez, Set Operads in Combinatorics and Computer Science. SpringerM.A. Méndez, Set Operads in Combinatorics and Computer Science, Springer, New York, 2015.
D , D I Spivak, arXiv:1307.6894The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes. D. Rupel and D.I. Spivak, The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes, arXiv:1307.6894.
The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits. D I Spivak, arXiv:1305.0297D.I. Spivak, The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits, arXiv:1305.0297.
Nesting of dynamic systems and mode-dependent networks. D I Spivak, arXiv:1502.07380D.I. Spivak, Nesting of dynamic systems and mode-dependent net- works, arXiv:1502.07380.
The steady states of coupled dynamical systems compose according to matrix arithmetic. D I Spivak, arXiv:1512.00802D.I. Spivak, The steady states of coupled dynamical systems compose according to matrix arithmetic, arXiv:1512.00802.
D I Spivak, P Schultz, D Rupel, arXiv:1508.01069String diagrams for traced and compact categories are oriented 1-cobordisms. D.I. Spivak, P. Schultz, and D. Rupel, String diagrams for traced and compact categories are oriented 1-cobordisms, arXiv:1508.01069.
Algebras of open dynamical systems on the operad of wiring diagrams. D Vagner, D I Spivak, E Lerman, Theory Appl. Categ. 30D. Vagner, D.I. Spivak, and E. Lerman, Algebras of open dynamical systems on the operad of wiring diagrams, Theory Appl. Categ. 30 (2015), 1793- 1822.
Monads: Some Linguistic Applications. A Asudeh, A. Asudeh, Monads: Some Linguistic Applications, available at http://users.ox.ac.uk/∼cpgl0036/handouts/asudeh-se-lfg13.pdf. 282
Oxford Logic Guides 52. S Awodey, Oxford Univ. Press158279OxfordCategory TheoryS. Awodey, Category Theory, 2nd. ed., Oxford Logic Guides 52, Oxford Univ. Press, Oxford, 2010. 23, 158, 279
Introduction to Commutative Algebra. M F Atiyah, I G Macdonald, Addison-WesleyMassachusettsM.F. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra, Addison- Wesley, Massachusetts, 1969. 4
. J C Baez, J Erbele, arXiv:1405.6881Categories in control. 5282J.C. Baez and J. Erbele, Categories in control, arXiv:1405.6881. 5, 282
J C Baez, B Fong, arXiv:1504.05625A compositional framework for passing linear networks. 156282J.C. Baez and B. Fong, A compositional framework for passing linear networks, arXiv:1504.05625. 156, 282
A compositional framework for Markov processes. J C Baez, B Fong, B S Pollard, J. Math. Phys. 573282J.C. Baez, B. Fong, and B.S. Pollard, A compositional framework for Markov processes, J. Math. Phys. 57, No. 3, 033301, 30 p. (2016). 156, 282
Operads and the tree of life. J Baez, N Otter, J. Baez and N. Otter, Operads and the tree of life, available at http://math.ucr.edu/home/baez/tree of life/tree of life.pdf. 282
Logic and Computation: A Rosetta Stone. J C Baez, M Stay, New Structures for Physics. Bob CoeckeBerlinSpringer813Physics, TopologyJ.C. Baez and M. Stay, Physics, Topology, Logic and Computation: A Rosetta Stone, in: New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics vol. 813, Springer, Berlin, 2011, pp. 95-174. 282
M Barr, C Wells, Category Theory for Computing Science. Prentice-Hall279M. Barr and C. Wells, Category Theory for Computing Science, Prentice-Hall , 1990. 279
A categorical semantics of signal flow graphs. F Bonchi, P Sobociński, F Zanasi, Lecture Notes in Comp. Sci. 8704282SpringerF. Bonchi, P. Sobociński, and F. Zanasi, A categorical semantics of signal flow graphs, pp. 435-450, Lecture Notes in Comp. Sci. 8704, Springer, 2014. 5, 282
Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience. R Brown, T Porter, Proceedings of the International Conference on Theoretical Neurobiology. Nandini Singhthe International Conference on Theoretical NeurobiologyDelhi1National Brain Research CentreR. Brown and T. Porter, Category Theory and Higher Dimensional Algebra: poten- tial descriptive tools in neuroscience, in: Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1 (2003) 80-92. 282
Quantum picturalism. B Coecke, Contemporary Physics. 51B. Coecke, Quantum picturalism, Contemporary Physics 51 (2010), 59-83. 282
General theory of natural equivalences. S Eilenberg, S Maclane, Trans. Amer. Math. Soc. 58S. Eilenberg and S. MacLane, General theory of natural equivalences, Trans. Amer. Math. Soc. 58 (1945), 231-294. 279
Decorated cospans. B Fong, Theory Appl. Categ. 30B. Fong, Decorated cospans, Theory Appl. Categ. 30 (2015), 1096-1120. 156
Koszul duality for operads. V Ginzburg, M Kapranov, Duke Math. J. 765V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), 203-272. 5
Naive Set Theory, Undergrad. Texts in Math. P R Halmos, SpringerNew YorkP.R. Halmos, Naive Set Theory, Undergrad. Texts in Math., Springer, New York, 1974. 13
S Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces. San DiegoAcademic Press144S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, San Diego, 1978. 144
C Heunen, M Sadrzadeh, E Grefenstette, Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse. Oxford282C. Heunen, M. Sadrzadeh, and E. Grefenstette, ed., Quantum Physics and Linguistics: A Compositional, Diagrammatic Discourse, Oxford, 2013. 282
O Iordache, Self-Evolvable Systems, Machine Learning in Social Media. BerlinSpringer-Verlag282O. Iordache, Self-Evolvable Systems: Machine Learning in Social Media, Springer- Verlag, Berlin, 2012. 282
On homotopy invariance for algebras over colored PROPs. M W Johnson, D Yau, J. Homotopy and Related Structures. 4M.W. Johnson and D. Yau, On homotopy invariance for algebras over colored PROPs, J. Homotopy and Related Structures 4 (2009), 275-315. 17
Traced monoidal categories. A Joyal, R Street, D Verity, Mathematical Proceedings of the Cambridge Philosophical Society. 119279A. Joyal, R. Street, and D. Verity, Traced monoidal categories, Mathematical Proceedings of the Cambridge Philosophical Society 119 (1996), 447-468. 5, 279
G M Kelly, On the operads of J.P. May. 13280G.M. Kelly, On the operads of J.P. May, Reprints in Theory Appl. Categ. 13 (2005), 1-13. 17, 280
Deductive systems and categories. II. Standard constructions and closed categories. J Lambek, Category Theory, Homology Theory and their Applications, I (Battelle Inst. Conf., Seattle, Wash. BerlinSpringer1280J. Lambek, Deductive systems and categories. II. Standard constructions and closed cat- egories, in: 1969 Category Theory, Homology Theory and their Applications, I (Battelle Inst. Conf., Seattle, Wash., 1968, Vol. 1) p.76-122, Springer, Berlin, 1969. 2, 22, 280
F W Lawvere, S H Schanuel, Conceptual Mathematics: A first introduction to categories. Cambridge2792nd ed.F.W. Lawvere and S.H. Schanuel, Conceptual Mathematics: A first introduction to cate- gories, 2nd ed., Cambridge, 2009. 279
Basic Category Theory, Cambridge Studies in Adv. T Leinster, Math. 14323280Cambridge Univ. PressT. Leinster, Basic Category Theory, Cambridge Studies in Adv. Math. 143, Cambridge Univ. Press, Cambridge, 2014. 23, 279, 280
Algebraic Operads, Grund. der math. J.-L Loday, B Vallette, Wiss. 346280Springer-VerlagJ.-L. Loday and B. Vallette, Algebraic Operads, Grund. der math. Wiss. 346, Springer- Verlag, New York, 2012. 280
. S Mac Lane, Natural Associativity and Commutativity. 494Rice Univ. StudiesS. Mac Lane, Natural Associativity and Commutativity, Rice Univ. Studies 49, 28-46. 4
Categorical algebra. S Mac Lane, Bull. Amer. Math. Soc. 71S. Mac Lane, Categorical algebra, Bull. Amer. Math. Soc. 71 (1965), 40-106. 281
Categories for the working mathematician. S Mac Lane, Grad. Texts in Math. 5280Springer-Verlag2nd ed.S. Mac Lane, Categories for the working mathematician, Grad. Texts in Math. 5, 2nd ed., Springer-Verlag, New York, 1998. 4, 23, 158, 279, 280
C Male, arXiv:1114.4662v6Traffics distributions and independence: the permutation invariant matrices and the notions of independence. 18109C. Male, Traffics distributions and independence: the permutation invariant matrices and the notions of independence, arXiv:1114.4662v6. 18, 109
Models for operads. M , 1471-1500. 19Comm. Algebra. 24M. Markl, Models for operads, Comm. Algebra 24 (1996), 1471-1500. 19
Operads and PROPs, Handbook of Algebra 5. M , Elsevier22281M. Markl, Operads and PROPs, Handbook of Algebra 5, p.87-140, Elsevier, 2008. 22, 281
Wheeled PROPs, graph complexes and the master equation. M Markl, S Merkulov, S Shadrin, J. Pure Appl. Algebra. 213M. Markl, S. Merkulov, and S. Shadrin, Wheeled PROPs, graph complexes and the mas- ter equation, J. Pure Appl. Algebra 213 (2009), 496-535. 281
Operads in Algebra. M Markl, S Shnider, J Stasheff, Topology and Physics, Math. Surveys and Monographs. 96280Amer. Math. SocM. Markl, S. Shnider, and J. Stasheff, Operads in Algebra, Topology and Physics, Math. Surveys and Monographs 96, Amer. Math. Soc., Providence, 2002. 17, 280
The geometry of iterated loop spaces. J P May, Lecture Notes in Math. 271280Springer-VerlagJ.P. May, The geometry of iterated loop spaces, Lecture Notes in Math. 271, Springer- Verlag, New York, 1972. 2, 17, 106, 280
Definitions: operads, algebras and modules. J P May, Contemp. Math. 202280J.P. May, Definitions: operads, algebras and modules, Contemp. Math. 202, p.1-7, 1997. 2, 17, 280
. M A Méndez, Set Operads in Combinatorics and Computer Science. 282SpringerM.A. Méndez, Set Operads in Combinatorics and Computer Science, Springer, New York, 2015. 282
Basic Category Theory for Computer Scientists. B C Pierce, MIT Press280B.C. Pierce, Basic Category Theory for Computer Scientists, MIT Press,1991. 280
C C Pinter, A Book of Set Theory. New YorkDoverC.C. Pinter, A Book of Set Theory, Dover, New York, 2014. 13
E , Category Theory in Context. Dover, New York279280E. Riehl, Category Theory in Context, Dover, New York, 2016. 279, 280
Advanced Modern Algebra. J J Rotman, Prentice HallNew JerseyJ.J. Rotman, Advanced Modern Algebra, Prentice Hall, New Jersey, 2002. 4
D , D I Spivak, arXiv:1307.6894The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes. 276282D. Rupel and D.I. Spivak, The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes, arXiv:1307.6894. 1, 2, 4, 6, 9, 13, 25, 29, 36, 105, 119, 120, 121, 135, 239, 259, 275, 276, 282
D I Spivak, arXiv:1305.0297The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits. 239282D.I. Spivak, The operad of wiring diagrams: formalizing a graphical language for databases, recursion, and plug-and-play circuits, arXiv:1305.0297. 1, 2, 3, 8, 153, 154, 156, 160, 162, 166, 213, 214, 221, 224, 225, 227, 230, 239, 282
Category Theory for the Sciences. D I Spivak, MIT Press156280D.I. Spivak, Category Theory for the Sciences, MIT Press, 2014. 1, 8, 156, 280
Nesting of dynamic systems and mode-dependent networks. D I Spivak, arXiv:1502.0738099283D.I. Spivak, Nesting of dynamic systems and mode-dependent networks, arXiv:1502.07380. 1, 2, 25, 88, 99, 283
The steady states of coupled dynamical systems compose according to matrix arithmetic. D I Spivak, arXiv:1512.00802143283D.I. Spivak, The steady states of coupled dynamical systems compose according to ma- trix arithmetic, arXiv:1512.00802. 1, 2, 88, 99, 106, 136, 140, 143, 283
D I Spivak, P Schultz, D Rupel, arXiv:1508.01069String diagrams for traced and compact categories are oriented 1-cobordisms. 5283D.I. Spivak, P. Schultz, and D. Rupel, String diagrams for traced and compact categories are oriented 1-cobordisms, arXiv:1508.01069. 5, 283
Algebras of open dynamical systems on the operad of wiring diagrams. D Vagner, D I Spivak, E Lerman, Theory Appl. Categ. 30283D. Vagner, D.I. Spivak, and E. Lerman, Algebras of open dynamical systems on the operad of wiring diagrams, Theory Appl. Categ. 30 (2015), 1793-1822. 1, 2, 3, 25, 88, 101, 106, 144, 145, 149, 277, 283
B Vallette, arXiv:1202.3245.281Algebra + Homotopy = Operad, preprint. B. Vallette, Algebra + Homotopy = Operad, preprint, arXiv:1202.3245. 281
R F C Walters, Categories and Computer Science. CambridgeCambridge Univ. Press280R.F.C. Walters, Categories and Computer Science, Cambridge Univ. Press, Cambridge, 1991. 280
Colored Operads, Graduate Studies in Math. D Yau, Amer. Math. Soc. 170280D. Yau, Colored Operads, Graduate Studies in Math. 170, Amer. Math. Soc., Providence, RI, 2016. 2, 15, 17, 22, 106, 108, 276, 280
A Foundation for PROPs, Algebras, and Modules. D Yau, M W Johnson, Math. Surveys and Monographs. 203281Amer. Math. SocD. Yau and M.W. Johnson, A Foundation for PROPs, Algebras, and Modules, Math. Surveys and Monographs 203, Amer. Math. Soc., Providence, RI, 2015 2, 5, 221, 281
| [] |
[
"A computationally universal phase of quantum matter",
"A computationally universal phase of quantum matter"
] | [
"Robert Raussendorf \nDepartment of Physics and Astronomy\nUniversity of British Columbia\nVancouverBCCanada\n\nStewart Blusson Quantum Matter Institute\nUniversity of British Columbia\nVancouverBCCanada\n",
"Cihan Okay \nDepartment of Physics and Astronomy\nUniversity of British Columbia\nVancouverBCCanada\n\nStewart Blusson Quantum Matter Institute\nUniversity of British Columbia\nVancouverBCCanada\n",
"Dong-Sheng Wang \nDepartment of Physics and Astronomy\nUniversity of British Columbia\nVancouverBCCanada\n\nStewart Blusson Quantum Matter Institute\nUniversity of British Columbia\nVancouverBCCanada\n",
"David T Stephen \nMax-Planck-Institut für Quantenoptik\nHans-Kopfermann-Straße 185748GarchingGermany\n",
"Hendrik Poulsen Nautrup \nInstitut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstr. 21aA-6020InnsbruckAustria\n"
] | [
"Department of Physics and Astronomy\nUniversity of British Columbia\nVancouverBCCanada",
"Stewart Blusson Quantum Matter Institute\nUniversity of British Columbia\nVancouverBCCanada",
"Department of Physics and Astronomy\nUniversity of British Columbia\nVancouverBCCanada",
"Stewart Blusson Quantum Matter Institute\nUniversity of British Columbia\nVancouverBCCanada",
"Department of Physics and Astronomy\nUniversity of British Columbia\nVancouverBCCanada",
"Stewart Blusson Quantum Matter Institute\nUniversity of British Columbia\nVancouverBCCanada",
"Max-Planck-Institut für Quantenoptik\nHans-Kopfermann-Straße 185748GarchingGermany",
"Institut für Theoretische Physik\nUniversität Innsbruck\nTechnikerstr. 21aA-6020InnsbruckAustria"
] | [] | We provide the first example of a symmetry protected quantum phase that has universal computational power. This two-dimensional phase is protected by one-dimensional line-like symmetries that can be understood in terms of local symmetries of a tensor network. These local symmetries imply that every ground state in the phase is a universal resource for measurement based quantum computation.Z Zvirtual quantum register circuit model time X X X arXiv:1803.00095v2 [quant-ph] | 10.1103/physrevlett.122.090501 | [
"https://arxiv.org/pdf/1803.00095v2.pdf"
] | 89,620,487 | 1803.00095 | a83c3f875299b8d7be201aba5dec3a0077338066 |
A computationally universal phase of quantum matter
5 Mar 2019
Robert Raussendorf
Department of Physics and Astronomy
University of British Columbia
VancouverBCCanada
Stewart Blusson Quantum Matter Institute
University of British Columbia
VancouverBCCanada
Cihan Okay
Department of Physics and Astronomy
University of British Columbia
VancouverBCCanada
Stewart Blusson Quantum Matter Institute
University of British Columbia
VancouverBCCanada
Dong-Sheng Wang
Department of Physics and Astronomy
University of British Columbia
VancouverBCCanada
Stewart Blusson Quantum Matter Institute
University of British Columbia
VancouverBCCanada
David T Stephen
Max-Planck-Institut für Quantenoptik
Hans-Kopfermann-Straße 185748GarchingGermany
Hendrik Poulsen Nautrup
Institut für Theoretische Physik
Universität Innsbruck
Technikerstr. 21aA-6020InnsbruckAustria
A computationally universal phase of quantum matter
5 Mar 2019(Dated: March 6, 2019)
We provide the first example of a symmetry protected quantum phase that has universal computational power. This two-dimensional phase is protected by one-dimensional line-like symmetries that can be understood in terms of local symmetries of a tensor network. These local symmetries imply that every ground state in the phase is a universal resource for measurement based quantum computation.Z Zvirtual quantum register circuit model time X X X arXiv:1803.00095v2 [quant-ph]
PACS numbers: 03.67.Mn, 03.65.Ud, 03.67.Ac
In the presence of symmetry, quantum phases of matter can have computational power. This was first conjectured in [1]- [3], and has been proven [4]- [6] or numerically supported [7], [8] in several instances. The important property is that the computational power is uniform. It does not depend on the precise choice of the state within the phase, and is thus a property of the phase itself. In this way, phases of quantum matter acquire a computational characterization and computational value.
The quantum computational power of physical phases is utilized by measurement based quantum computation (MBQC) [9], where the process of computation is driven by local measurements on an initial entangled state. Here, we consider initial states that originate from symmetry protected topological (SPT) phases [10]- [12].
Proofs of the existence of such "computational phases of quantum matter" have so far been confined to spatial dimension one. After it was shown that computational wire-the ability to shuttle quantum information from one end of a spin chain to the other-is a property of certain SPT phases [3], the first phase permitting quantum computations on a single logical qubit was described in [4]. In fact, uniform computational power is ubiquitous in one-dimensional SPT phases [5], [6].
Computationally, physical phases in dimension 2 and higher are more interesting than in dimension 1. The reason is that, in MBQC, one spatial dimension plays the role of circuit model time. Therefore, MBQC in dimension D corresponds to the circuit model in dimension D − 1, and universal MBQC is possible only in D ≥ 2.
Yet, to date, the evidence for quantum computational phases of matter is much more scant for D ≥ 2 than for D = 1. Numerical evidence exists for deformed Affleck-Kennedy-Lieb-Tasaki Hamiltonians on the honeycomb lattice [7,8,13]. In addition, extended regions of constant computational power have also been observed in SPT phases with Z 2 -symmetry [14].
Numerous computationally universal resources states for MBQC have been constructed [15]- [18] using the tools of group cohomology that also form the basis for the clas-sification of SPT order [11], [12]. From the starting point of these special states, it remains open what happens to the computational power as one probes deeper into the SPT phases surrounding them.
For the cluster phase in D = 2, a symmetry protected phase that contains the cluster state, it was shown analytically that universal computational power persists throughout a finite region around the cluster state [22].
Here, we prove the existence of a computationally universal phase of quantum matter in spatial dimension two. As in [22], the phase we consider is protected by onedimensional line-like symmetries, generalizing the conventional notion of symmetry protected topological order defined by global on-site symmetries. As in the case of global symmetries, these line symmetries can be built from the local symmetries of a tensor network which persist throughout the phase. Using this, we establish that computational universality persists throughout the entire phase. The backbone of the computational scheme is symmetry protected correlations in a virtual quantum register, see Symmetry protected quantum correlations enable uniform computational power throughout the 2D cluster phase. The long-range symmetry shown is composed of the symmetries of local PEPS tensors. Setting and result. We consider a two-dimensional (2D) simple square spin lattice which is, for simplicity of boundary conditions, embedded in a torus of a small circumference n and a large circumference nN , with n, N ∈ N, N n and n even. Its Hamiltonian is invariant under all lattice translations and the symmetries
U c,+ = nN −1 x=0 X x,c+x , U c,− = nN −1 x=0 X x,c−x ,(1)
for all c ∈ Z n . Therein, X ≡ σ x , and the addition in the second index of X is mod n. A graphical rendering of these symmetries is provided in Fig. 2a. These symmetries were previously considered in [22]. We consider phases in which the ground state is unique, and thus shares the symmetries. As the Hamiltonian is varied while respecting the symmetries Eq. (1), the respective ground states arrange into phases. The central object of interest is the "2D cluster phase", i.e. the physical phase which respects the symmetries Eq. (1) and which contains the 2D cluster state.
The main result of this paper is the following.
Theorem 1 For a spin-1/2 lattice on a torus with circumferences n and N n, where N n and n even, all ground states in the cluster phase, except a possible set of measure zero, are universal resources for measurementbased quantum computation on n/2 logical qubits.
To facilitate the proof of Theorem 1, we introduce the notion of "cluster-like" states |Φ . For a square grid in dimension 2, we represent states as projected entangled pair states (PEPS) with local tensors A Φ , such that contracting virtual legs on a torus as in Fig. 1 describes the wave function of |Φ . The "cluster-like" states are those whose PEPS tensors have the symmetries
= = = = Z Z I I Z X I I X Z X Z X I I I Z X I I I (2)
Therein, red (blue) legs indicate Pauli operators X (Z). This notation means that, for example, acting on the physical leg (diagonally directed) of the PEPS tensor with a Pauli X is equivalent to a corresponding action of Pauli operators on the virtual legs. The reason for calling states satisfying Eq. (2) "cluster-like" is that if we add Z X = I to those symmetries, then we obtain cluster states as the only solution of the joint symmetry constraints.
The proof of Theorem 1 splits into two parts. First we show that all states in the 2D cluster phase are clusterlike, and then demonstrate that cluster-likeness implies universal computational power.
A 2D physical phase of cluster-like states. Here we prove the following result.
Proposition 1 Every ground state |Φ in the 2D cluster phase has a description in terms of a local tensor A Φ that has the symmetries of Eq. (2).
Our starting point is the characterization of SPT phases in terms of symmetric quantum circuits. A symmetric quantum circuit is a sequence of unitary gates U = l i=1 U i where each gate U i is invariant under the symmetry group G of Eq. (1), [U i , U (g)] = 0, for all g ∈ G. In a local such circuit, each gate U i acts only on a bounded number of qubits [19]. We then have the following result [11], Lemma 1 Symmetric gapped ground states in the same SPT phase are connected by symmetric local quantum circuits of constant depth.
To prove Proposition 1, we analyze the structure of the symmetry-respecting gates U Φ,i of the circuit U Φ mapping the cluster state |C to a given state |Φ in the cluster phase, |Φ = U Φ |C . Writing
U Φ,i = j d j P j , with d j ∈ C, ∀j,(3)
only symmetry-respecting n-qubit Pauli operators P j , P j ∈ P n , can appear on the r.h.s. The generators of such Pauli operators are displayed in Fig. 2 (b), (c). Furthermore, the operators shown in Fig. 2 . Now expanding the entire circuit U Φ into a sum of Pauli operators, every Pauli operator in this expansion is also a product of X k and Star-operators. We further observe that, by the form of the cluster state stabilizer,
Star k |C = X k |C ,(4)
for all lattice sites k. Using relation Eq. (4), all star operators in the expansion of U Φ can be eliminated. We thereby obtain a transformation T Φ that satisfies the relation T Φ |C = U Φ |C = |Φ , and is composed of Pauli-X operators only,
T Φ = j c j X(j).(5)
Therein, X(j) := k (X k ) j k an X-type Pauli operator with support on the n × nN torus, i.e., j is a binary vector with n 2 N components.
Proof of Proposition 1. To illustrate the idea of the proof, we first discuss the special case where the map T Φ is a tensor product of local factors, T Φ = k t Φ,k . Then, to obtain a local tensor A Φ representing |Φ , we apply T Φ site-wise to the local tensor C representing the cluster state. Graphically,
= A C Φ t Φ .
Since by Eq. (5) t Φ is a linear combination of I and X, it commutes with X. Hence, the symmetries Eq. (2) of the cluster state tensors C are also symmetries of the tensors A Φ representing |Φ . Now turning to the general case, the action of T Φ on |C results in local tensors A Φ of the form
C = A Φ B Φ a b c d ,(6)
where the "junk tensor" B Φ [3] forms a tensor network representation of the map T Φ , and emerges as a consequence of the non-locality of the map T Φ . It inherits from T Φ the property that on the physical leg of C (pointing upwards) it acts as I or X, depending on the state of the virtual links a, .., d (for details, see the SM, Section I B). The junk tensor B Φ thus commutes with the action of the local Pauli X-operator,
= B Φ X B Φ X .
In result, the symmetries Eq. (2) hold for all tensors A Φ describing a state |Φ in the cluster phase.
Cluster symmetries and computation. We now show that the symmetries Eq. (2) of PEPS tensors imply MBQC universality of the corresponding quantum state. This proceeds in two steps. We establish (i) computational wire, i.e. the ability to shuttle quantum information across the torus, and (ii) a universal set of quantum gates.
(i) Computational wire. We now map to a quasi-1D setting by grouping spins into blocks of size n × n. If we block n × n copies of the tensor A Φ , as in Fig. 1, we obtain the block tensor A Φ which forms a matrix product state (MPS) representation of the quasi-1D system. Contracting the physical legs of this tensor with local X eigenstates labelled by the n 2 -component binary vector i gives the tensor component A Φ (i). We can now use the symmetries in Eq. (2) to constrain these tensor components:
Lemma 2 Consider a torus of size n × nN , and n ∈ 2N. For all ground states |Φ in the 2D cluster phase, the corresponding block tensors A Φ (i) satisfy
A Φ (i) = C(i) ⊗ B Φ (i).(7)
The logical tensors C(i) are constant throughout the phase, and
C(i) ∈ P n , ∀i.(8)
Lemma 2 establishes the primitive of computational wire, similar to Theorem 1 in [3]. The Hilbert space on which the tensor components A Φ (i) act is the so-called virtual space, which decomposes into "logical subsystem" and "junk subsystem" [3]. Upon measurement in the X-basis of all spins in a block, the logical subsystem is acted on by the operators C(i), which are uniform across the cluster phase. Conversely, the operators B Φ (i) acting on the junk space vary uncontrollably across the phase. Thus, to achieve computation, the logical subspace is used to encode and process information. The operators C(i) become the usual outcome-dependent byproduct operators of MBQC. They are of computational use, as described below under "quantum gates". Two points are worth noting, one technical, one physical. (i) With Lemma 2, we have mapped the original two-dimensional system to an effectively one-dimensional system composed of blocks. A wealth of techniques established for 1D SPT order thereby becomes available [3]- [6], [10]- [12]. (ii) The blocking notwithstanding, the basis {|i } in which Eq. (8) holds is local at the level of individual spins, not only at block level. (It is the local X-eigenbasis.) Since MBQC uses 1-spin local measurements, we require this stronger notion of locality.
Finally, we explain why Lemma 2 is a consequence of the symmetries of the local tensors A Φ in the cluster phase. The local symmetries Eq. (2) can be combined in such a way that they map Pauli operators on the virtual logical register one column farther to the right,
, X Z X = = X l X X Z X l -1 l +1 l l l ,(9)
for all l. (The tensor factors "I" for the action of the symmetries on the junk systems have been omitted). Iterating these propagation relations n times (n is the circumference of the torus), we find that, upon measurement of the physical qubits in the local X-basis, each virtual local Pauli operator Z is mapped onto itself up to sign. See Fig. 1 for illustration (n = 6 is shown). The same is true for Pauli operators X, cf. Fig. 4 in the SM. Thus, every virtual Pauli operator is mapped to itself up to sign, after one clock cycle of duration n. Therefore, the action of A Φ on the logical subsystem is indeed by Pauli operators, as stated by Lemma 2. As a technical remark, we note that the following construction requires that Lemma 2 holds also when A Φ is put into the so-called canonical MPS form [23]. Details of this condition, as well as the proof of its veracity, are given in the SM, Section III A.
(ii) Quantum gates. The subsequent construction significantly differs from the standard mapping to the circuit model [9]. Specifically, the technique of "cutting out coupled wires" by local Z-measurements is not available throughout the cluster phase, and is therefore replaced.
As a first step, we observe that the byproduct operators C(i) are of the form
C(i) ∼ k∈K C[k] i k ,(10)
where "∼" is equality up to phase, K is the n × n block of spins, and i k the measurement outcome at location k. Eq. (10) means that every site k in the block has its own byproduct operator C[k]. This is known to hold for the cluster state [9], and by Lemma 2 it extends to the entire cluster phase.
Next, we find the precise form of the byproduct operators C[k] for certain sites k ∈ K. Namely, for the sites k = (1, l), (2, l) and (n, l) in the first, second and last column of each block, the operators C[k] are
C[(1, l)] = Z l , C[(2, l)] = Z l−1 X l Z l+1 , C[(n, l)] = X l .(11)
They can be understood as follows. For the last column in the block, n, the operator C[n, l] is the standard byproduct operator for cluster states. By Lemma 2 it holds in the entire cluster phase. (See the SM, Sec. III B for the result in canonical form.) The C[r, l] for earlier columns r are also X-operators, inserted at position (r, l). They are then propagated forward to the right boundary of the block using Eq. (9), resulting in Eq. (11).
If the resource is a 2D cluster state, the special state in the phase of interest, then onsite measurements in the X/Y -plane of the Bloch sphere are universal [24]. Because of the product form of the byproduct operators Eq. (10), every local measurement implements one logical gate. Suppose the measurement at site k is in the basis spanned by |0,
α k = cos(α) |0 k − i sin(α) |1 k , |1, α k = −i sin(α) |0 k + cos(α) |1 k , with |0 , |1 refer- ring to eigenstates of X. The resulting gate is U α (i k ) = k i k , α|0 k I + k i k , α|1 k C[k], hence U α (i k ) = C[k] i k exp(iα C[k]).
Here, the operators C[k] of Eq. (8) become a computational tool, as it specifies the unitary gate implemented. The outcome-dependent byproduct operator can be compensated for by classical side-processing and adaptive measurement bases [9]. With Eq. (11), the gate set
U = {e iα Z l−1 X l Z l+1 , e iα Z l , e iα X l , ∀α ∈ R}.(12)
can be realized. U is a universal set [27]; also see Section IV B of the SM. When moving away from the cluster state into the cluster phase, non-trivial tensors B Φ appear, and measurement in a local basis away from the symmetry-respecting X-basis becomes non-trivial. If unaccounted for, the logical subsystem becomes entangled with the junk subsystem through such measurement [3], which introduces decoherence into the logical processing. However, this undesirable effect can be prevented by the techniques of [6]. By virtue of Lemma 2, we mapped to a quasi-1D setting to which we can apply Theorem 2 of [6]. (The essentials of [6] are reviewed in Section IV A of the SM.) In result, the universal gate set U can be implemented in the whole cluster phase, not only on the cluster state.
To summarize, the argument for computational universality of the 2D cluster phase splits into two parts. First, we have shown in Proposition 1 that all ground states in the 2D cluster phase are cluster-like, i.e., they satisfy the symmetry constraints Eq. (2). Second, by mapping to a quasi one-dimensional system we showed that the symmetries Eq. (2) lead to universal computational power. Taken together, these two results yield Theorem 1.
Conclusion.
We have described the first symmetry protected topological phase in which every ground state (up to a possible set of measure zero) has universal power for measurement based quantum computation. Our phase is protected by symmetries acting on a lower dimensional subsystem, and it is associated with a set of local symmetries of tensor networks, see Eq. (2). These symmetries are sufficient to guarantee computational universality of the corresponding tensor network. What implications these symmetries have on the physics of this phase and others like it remains an interesting question.
As for the implications on the computational side, we ask: Can the computational power of quantum phases of matter be classified? In the spirit of this question, we conclude with three more specific ones: (i) How broadly can the present construction be generalized? (ii) The line-like symmetries we consider are neither global symmetries, which are typically used to define SPT phases, nor are they local like in a lattice gauge theory. Indeed, they are more closely related to the "higher-form" symmetries considered in [20], [21], [26] which act on lower dimensional submanifolds of the whole lattice. Is this type of symmetry necessary for a computationally universal phase, or can other structurally different symmetries lead to similar results? (iii) As one-dimensional computational phases [5], [6] build on symmetry protected computational wire [3], the present construction builds on a symmetry protected quantum cellular automaton. In particular, Eq. (9) defines the transition function of a quantum cellular automaton. Quantum cellular automata have been classified [28]- [31]. What is the relation between this classification and computational phases of quantum matter? 1) [m] and whose support fits into a skewed square of horizontal and vertical extension smaller than n is equal to a product of star operators.
To prove Lemma 5 we need a further result.
Lemma 6 Any product of Z-operators within a k × k region of a skewed square lattice can be moved to the boundary of that region by multiplying with star operators.
Proof of Lemma 6. W.l.o.g., consider one of the two sub-lattices (even or odd). The proof is by induction. (I) The statement is true for a skewed square L 2 of size 2 × 2. We now prove that (II) If the statement is true for the skewed square L k−1 , then it is also true for L k . W.l.o.g., assume that Z-operators are only located on the boundary of L k−1 . Each such Z can be moved into a corner of L k by multiplying with star operators, leaving a trail of Zs in the boundary of L k ; see Fig. 5a.
Proof of Lemma 5. W.l.o.g., consider one of the two sub-lattices. By Lemma 6, the Z-operators can all be moved to the boundary of that region by multiplying with star operators. Since the initial operator Z(v) is invariant under the transformations Eq. (1) [m], and the star operators are invariant, so is the resulting operator Z(w). Now consider a tensor factor Z i in Z(w). Since Z(w) is symmetric under the transformations (1) [m] and these symmetries act on diagonals, it exists in conjunction with three additional local operators Z j , Z k , Z l ; see Fig. 5b for the labeling. The two diagonals α, β that intersect in i only intersect again on the torus at a horizontal and vertical distance of n/2 (see Fig. 2b). Since by assumption the support of Z(w) is contained within a skewed square of vertical and horizontal extension smaller than n, α and β do not intersect twice within such a square. Therefore, the locations i, j, k, l are all distinct, and their product can be removed by a product of star operators. The procedure is iterated until no local Pauli operators Z remain.
Structure and symmetry of the tensors AΦ
Here we prove Eq. (6) [m], with the additional symmetry property B Φ X = XB Φ . We already sketched the argument in the main text, and now provide additional detail.
We denote the 5-legged PEPS tensors of a cluster state (one physical, four virtual legs) by C; also see [25]. The corresponding tensors for the other states |Φ = U Φ |C in the cluster phase are denoted by A Φ . We prove that the tensor A Φ is invariant under the symmetries Eq. (2) [m], by constructing it explicitly.
With each transformation T Φ we associate a network of "junk" tensors that has the same geometry as the tensor network for |C . That is, at each lattice site resides a tensor B Φ with four virtual legs and two physical legs (input and output). The Hilbert space associated to each virtual leg is spanned by the vectors |j, a, b , where a ∈ [1, nN ] and b ∈ [1, n] are integers and j is the same index appearing in Eq. (5) [m], which we rewrite here for convenience:
T Φ = j c j X(j).(13)
Each pair a, b naturally corresponds to a site in the 2D lattice, which we denote k a,b . We define the tensor B Φ by the following non-zero components, a,b j,a,b j,a,b+1 j,a+1,
B Φ j,b = c 1 n 2 N j j k X a,b .(14)
We now need to establish two properties of the tensor B Φ , namely (i) The transformation T Φ corresponds to the tensor network composed of B Φ , with all virtual legs pairwise contracted, and (ii) The tensor A Φ representing a state |Φ satisfies the symmetries of Eq. (2) [m].
(i) First note that T Φ is invariant under all lattice translations: for every j, all operators that can be obtained from X(j) via lattice translations appear with the same coefficient c j in Eq. (13). Then we can rewrite,
T Φ = 1 n 2 N j c j [X(j) + trans.],(15)
where trans. indicates all possible lattice translations of X(j), which may or may not be distinct from X(j). In order to obtain Eq. (15) from Eq. (14) We contract the network of tensors Eq. (14) in two steps. Consider first fixing the index j to the same value j * on each link. Then, summing over the indices a, b on each link produces the term c j * [X(j * ) + trans.]. Note that this sum is greatly simplified because, in order to get a non-zero contraction, fixing the indices a, b on one link also fixes them on all other links. Now we sum over j * to get T Φ as defined in Eq. (13), up to a constant factor n 2 N . We emphasize that the same tensor B Φ is placed on each lattice site, so the tensor network is translationally invariant.
(ii) The tensor A Φ representing the state |Φ is constructed as A Φ = B Φ C. Graphically,
C = A Φ B Φ .
With the form Eq. (14) of the tensors B Φ , those tensors commute with the action of a local Pauli X-operator on the physical legs,
= B Φ X B Φ X .
With this relation, and since the symmetries Eq. (2) [m] hold for the cluster state tensor C, we have,
= C B Φ = C B Φ A Φ = A Φ X X I Z I X I Z I X X X Z Z = C B Φ X X X X Z Z .
Thus, the tensors A Φ describing the state |Φ share the first of the symmetries in Eq. (2) [m] with the cluster state tensors C. The same holds for the remaining symmetries of Eq. (2) [m], and the proof is analogous.
CORRELATIONS IN VIRTUAL SPACE
As an additional guide to the proof of Lemma 2 in the main text, a graphical representation of an X-type correlation in virtual space is provided in Fig. 6.
CANONICAL FORM FOR THE QUASI-1D BLOCK TENSORS AΦ
We briefly expand on the importance of the canonical form for computation. Given a quantum state, there are infinitely many ways to represent it as an MPS. Using this freedom, one can always choose an MPS tensor such that it is in the canonical form [23]. The canonical form of an MPS is a powerful tool that allows one to relate mathematical properties of the MPS tensor to physical properties of the state. For example, an MPS has a finite correlation length if and only if a certain linear map (to be introduced in the next section) has a unique fixed point. We require this uniqueness for the computational primitive of oblivious wire, upon which all non-trivial gates are based; see [6] and the next section. It is guaranteed it in the canonical form since every state in the cluster phase has a finite correlation length.
In the main text and Section 4, we construct a PEPS representation A Φ of a state |Φ in the cluster phase by stacking a tensor network representation of the circuit U Φ on top of the cluster state PEPS. While this is legitimate PEPS representation, it may not lead to an MPS in canonical form upon blocking into the tensor A Φ . Hence, we need to go through the extra step of showing that our results also hold in canonical form in order to use its properties when constructing a computational scheme.
Proof of Lemma 2 in canonical form
Here we use the quasi-1D picture to show that Lemma 2 [m] holds even when our block MPS tensor A Φ is put into canonical form.
Proof of Lemma 2 in canonical form. The proof proceeds by considering the quasi-1D SPT phase containing the 2D cluster state protected by the groupG of conelike symmetries generated by the symmetries shown in Fig. (1) [m] and Fig. (6), and their translates. For every state in this phase, Eqs. (7) [m] and (8) [m] are guaranteed to hold for the canonical MPS representation of that state, as stated in Theorem 1 of Ref. [3]. To complete the proof, we need only show that every state in the cluster phase is also in this quasi-1D phase.
Consider a state |Φ = U Φ |C in the cluster phase. The circuit U Φ is defined to be symmetric under the line symmetries defining the cluster phase. It turns out that this guarantees that it is also symmetric under the cone symmetries defining the groupG, which is a strictly larger group than the group G formed by the line symmetries. This is because U Φ is a local circuit of constant depth with a 2D notion of locality.
In the proof of Lemma 1 [m] we identified three types of Pauli operators that commute with the stripe symmetries. While the local operators shown in Fig. 2 (b) [m] commute with all elements ofG, non-local operators like that in Fig. 2 (c) [m] do not. But, because of the local structure of U Φ , the non-local operators will not appear in its expansion (Eq. (3) [m]). So every U Φ which commutes with the stripe symmetries will also commute with all elements ofG. The circuit U Φ is local in the 2D sense, so it also satisfies the weaker notion of 1D block locality. Hence, |Φ is connected to the cluster state via a local circuit of constant depth which commutes withG, and it is thus in the quasi-1D SPT phase protected byG.
Operators C(i) in canonical form Now that we have shown that Eqs. (7) [m] and (8) [m] hold in the canonical form, we would like to identify the precise form of the operators C(i) therein. In particular, we must confirm that the relations given in Eq. (11) [m] still hold in canonical form, as we didn't establish this of Eq. (9) [m] which was used to derive Eq. (11) [m]. We again use Theorem 1 of Ref. [3], which gives an explicit recipe for determining C(i).
Denote by u(g) the representation ofG given by the cone-like symmetry operators in Figs. 1 [m] and 6. Depending on the binary vector i, the action of u(g) on A Φ (i) yields a sign χ i (g) = ±1 according to u(g)|i = χ i (g)|i . Then we have [3]
C(i)V (g) = χ i (g) V (g)C(i), ∀g ∈G.(16)
Therein, V (g) are the Pauli operators acting in the virtual space associated to u(g) as in Figs We restate Eq. (11) [m] here for later use in Section 4, now valid also in canonical form,
C[(1, l)] = Z l ,(17a)C[(n, l)] = X l ,(17b)C[(2, l)] = Z l−1 X l Z l+1 .(17c)
COMPUTATION IN SPT PHASES
Review of techniques for quantum computation in SPT phases
Here we review the basic techniques for computing in 1D SPT phases from [6]. The proof of Theorem 1, which is provided in the next section, substantially relies on these techniques. This section is only intended as an intuitive guide to the main techniques employed in [6]. It does not replace [6] as technical background.
In the 1D scenario discussed here, we deal with a chain of N -level spins, without any substructure. We discuss two techniques for MBQC in SPT phases, namely "oblivious wire", and "logical gates by averaging over measurement outcomes".
Oblivious wire. The purpose of oblivious wire is to decouple the logical subsystem of the virtual space from the junk subsystem. An elementary segment L of oblivious wire is implemented by (i) measuring a number m of successive spins in the symmetry protected basis, (ii) propagating the logical byproduct operator C(i) forward to the end of the computation, and (iii) forgetting the measurement outcomes.
The resulting action on the virtual space is I ⊗ L, with
L(ρ) = N −1 i=0 B Φ (i)ρB Φ (i) † .
Note that the tensors B Φ are location-independent, cf. Section 4. Here we finally use the property of translationinvariance of the physical setup.
Since the ground state anywhere in the SPT phase has finite correlation length, the MPS tensors B Φ (i) are injective [3], [22], and L therefore has a unique fixed point. Note that these implications hold only for tensors in the canonical form, hence why we require the proof in Appendix B. Denote by λ 1 the second-largest eigenvalue of L, and ξ := −1/ ln λ 1 the corresponding correlation length. Then, for m ξ,
L m (ρ) → ν ρ ρ fix ,
where ρ fix is the unique fixed point of L and ν ρ ∈ R + . Oblivious wire applied to a state τ defined on the entire virtual space decouples the logical from the junk part,
L m (τ ) → σ ⊗ ρ fix .
Logical gates. Small rotations ∼ dα are implemented by measuring slightly off the symmetry protected basis, followed by oblivious wire. The reason for implementing only small rotation angles between two pieces of oblivious wire is that unitarity on the logical subspace is violated at second order in dα [6].
To realize a logical unitary, we measure in the basis
|i = |i + e iδ dα |j , |j = |j − e iδ dα |i ,(18)
and |r = |r for r = i, j. Therein, {|i , i = 0, .., N − 1} is the symmetry-respecting basis. Then, the logical operation
U (δ, dα) = exp idα e −iδ ν ji C − e iδ ν * ji C † i(19)
is implemented, for small angles dα. Therein,
C = C(i) −1 C(j),(20)
for 0 ≤ i, j = N − 1 and i = j, and the constant ν ij ∈ C is given by
lim m→∞ L m (B Φ (i)ρ fix B Φ (j) † ) = ν ij ρ fix .
U (δ, dα) is implemented after accounting for the byproduct operator through forward propagation, and subsequent averaging over the measurement outcomes. Finite rotation angles are accumulated by repetition.
The constants ν ij ∈ C affect the gate operation and vary across the phase. They need to be measured prior to computation. By translation invariance they are the same for all sites.
Finally, a weak measurement of the observable C is performed by measuring in the basis |i + |j √ 2 , |i − |j √ 2 , |r , ∀r = i, j
A near-projective measurement of C is achieved by repeating the above procedure many times. For details and the changeover between logical unitary and logical measurement see [6].
Proof of Theorem 1
The proof of Theorem 1 is similar to that of Theorem 2 of [6], see Sections III -VI therein. There is one additional complication, however. In the present setting there are two distinct notions of locality, namely "block locality" on which the effective 1D SPT order hinges, and "single-site locality" on which MBQC hinges. The latter is more stringent: only single site measurements are available to MBQC, not block-local ones. It needs to be shown that quantum computational universality persists under this restriction of measurement bases.
For concreteness, in the 1D case we could choose to mix any pair of basis states |i and |j in Eq. (18), to construct a computationally useful measurement basis. In the present 2D case, the corresponding linear combinations of block states |i and |j are constrained by the requirement of single-site locality.
Proof of Theorem 1. The proof proceeds in 3 steps, which address unitary gates, universality, and measurement and initialization.
(i) Unitary gates. Consider all but one qubit k in a given block being measured in the symmetry protected X-basis, while qubit k is measured in the basis B(δ, dα) spanned by |+ = |+ + e iδ dα|− , |− = |− − e iδ dα|+ . (21) These measurements are followed by oblivious wire.
In the following, we establish that (1) single site measurements in the basis B(δ, dα) are sufficient to implement logical quantum gates, and (2) the resulting gate set is universal.
(1) We consider the conditional logical transformation implemented given certain measurement outcomes. Denote i
We are interested in particular in the operators C ii (k) which result from measuring qubit k of the given block in a symmetry-breaking local basis of Eq. (21). From Eq. (10) [m] it follows that
FIG. 1. Symmetry protected quantum correlations enable uniform computational power throughout the 2D cluster phase. The long-range symmetry shown is composed of the symmetries of local PEPS tensors.
FIG. 2 .
2(a) Stripe-like symmetry of Eq. (1). All translates are also symmetries. (b)-(c) The generators of Pauli operators that commute with the symmetries Eq. (1). (b) The local operators X k and Star l , for all sites k, l. (c) Geometrically non-local operators Zi ⊗Zj. The locations i, j are consecutive intersections of the supports α, β of two symmetries.
This work is supported by NSERC (CO, DSW, DTS, RR), Cifar (RR), the Stewart Blusson Quantum Matter Institute (CO and DSW), and the Austrian Science Fund FWF within the DK-ALM: W1259-N27 (HPN). DSW thanks Z.C. Gu for discussions. notation: Numbered equations from the main text are labeled with a suffix "[m]". For example, Eq. (3) from the main text is referenced in this Supplementary Material as "Eq. (3) [m]". The numbering of figures and lemmas is continued from the article. PROOF OF PROPOSITION 1 To complete the proof of Proposition 1 [m] we need to show (a) that all bounded range Pauli operators commuting with the symmetries Eq. (1) [m] are products of local X-operators and Z-type star operators, and (b) that any state |Φ in the cluster phase can be expressed in terms of local tensors A Φ of the form Eq. (6) [m]. The respective arguments are provided in Sections 4 and 4 below.
For a torus of size n×nN , any tensor product of Pauli Z-operators that is symmetric under the transformations Eq. (
Illustration for Lemma 6. Relocating Pauli operators Z to the boundary of the skewed square by multiplying with star operators. (b) Illustration for Lemma 5. By the symmetries Eq. (1) [m], every Pauli operator Zi in the boundary of a small skewed square region has 3 three distinct partners.
FIG. 6 .
6Symmetry-protected quantum correlations complementary to the symmetries in Fig. 1. The red (blue) lines represent the action of Pauli operators X (Z) on the virtual legs of PEPS tensors, such that matching pairs cancel. Symmetries displayed in both figures and their vertical translates generate a symmetry groupG = Z n 2 × Z n 2 . The boundary conditions in the vertical direction are periodic.
FIG. 7 .
7(k) := i + e k mod 2, for any given site k under consideration. Therein, e k is the n 2 -component binary vector with an entry 1 in position k, and 0 everywhere else. T i∪i (k) is the conditional logical transformation applied if either the outcome i or i (k) was obtained, and Illustration for the proof of Theorem 1. (a) Measurement patterns for unitary gates and measurements. (b) Simplified pattern for initialization by measurement.p i ∪ i (k) is the probability for obtaining the outcome i or i (k) . In analogy with Eq. (20) for the 1D case, we define the operators C ij := C(i) −1 C(j).
and hence they appear in the canonical form of the tensor A Φ , as desired.. 1 [m] and 6.
Writing C(i) ∼ k∈K C[k] i k as in the main text, Eq. (16)
can be used to determine C[k]. Then, one can straight-
forwardly verify that the operators given in Eq. (11) [m]
satisfy Eq. (16),
We thus find the following form for the logical operation T i∪i (k) resulting from the measurement in the basis Eq.(21), up to linear order in dα,(23) Therein, the overall byproduct operator C(i) has been separated out. The constants ν ii (k) are given by the relationwith ρ fix the unique fixed point state of the junk subsystem. The uniqueness of ρ fix is guaranteed by Lemma 2 in canonical form, as proved in Section 4.As in[6], logical gates are implemented by "forgetting" the measurement outcomes i after accounting for the corresponding byproduct operator. That is, we implement the weighted probabilistic average of the gates T i∪i (k) described in Eq.(23). These gates are. These gates are indeed unitary up to linear order in dα. The angle dα therefore has to be chosen small. Finite rotation angles are accumulated by repetition.At special points in the phase, reachable by fine-tuning, the conversion rate of measurement angle to rotation angle is exactly zero. For those ground states we cannot establish universal computational power. Hence the restriction to "all ground states except a possible set of measure zero" in Theorem 1.(2) Now we make special choices for the site k. First, for k = (1, l), with Eq. (17a), for a suitable angle δ the gate action is U (1,l) (dα) = exp (2idα |ν[(1, l)]|Z l ) , i.e., an infinitesimal rotation about the Z l -axis is performed.Second, choosing k = (n, l), leads to gates U (n,l) = exp(2i|ν[(n, l)]|dαX l ) due to Eq. (17b). Finally, choosing k = (2, l) leads to gatesin accordance with Eqs.(24)and (17c). Again, finite rotation angles are accumulated by repetition.(ii) Universality. We show that the local X and Zrotations, and the entangling gates of Eq. (25) form a universal set for n/2 qubits. After initialization, all logical qubits are in a Z-eigenstates. Through the available local rotations, they can be rotated into the logical state |+ , up to the action of byproduct operators in P n . Thereafter, the state of the odd-numbered qubits remains unchanged throughout. Only the local rotations for even-numbered qubits, and the rotations Eq. (25) for odd k are subsequently used. Because the odd-numbered qubits are frozen to |+ , the latter gates become U (2,l) ∼ = exp(iβZ l−1 Z l+1 ), for l even.Together these gates are universal for the even-numbered qubits.(iii) Measurement and initialization. With Eq. (17a), for weak measurements of a logical qubit l in the Z-basis, the measurement pattern ofFig. 7a, with O = Z, is performed. For a near-projective measurement of the logical subsystem of the virtual space, this pattern is repeated on a large number of consecutive blocks of physical qubits.Initialization can be performed by measurement. To simplify the procedure, all logical qubits can be measured in the Z-basis simultaneously by the measurement pattern inFig. 7b.
* These authors contributed equally to this work. * These authors contributed equally to this work
. A C Doherty, S D Bartlett, Phys. Rev. Lett. 10320506A.C. Doherty, S.D. Bartlett, Phys. Rev. Lett. 103, 020506 (2009).
. A Miyake, Phys. Rev. Lett. 10540501A. Miyake, Phys. Rev. Lett. 105, 040501 (2010).
. D V Else, I Schwarz, S D Bartlett, A C Doherty, Phys. Rev. Lett. 108240505D.V. Else, I. Schwarz, S.D. Bartlett and A.C. Doherty, Phys. Rev. Lett. 108, 240505 (2012).
. J Miller, A Miyake, Phys. Rev. Lett. 114120506J. Miller and A. Miyake, Phys. Rev. Lett. 114, 120506 (2015).
. D T Stephen, D.-S Wang, A Prakash, T.-C Wei, R Raussendorf, Phys. Rev. Lett. 11910504D.T. Stephen, D.-S. Wang, A. Prakash, T.-C. Wei, R. Raussendorf, Phys. Rev. Lett. 119, 010504 (2017).
. R Raussendorf, D.-S Wang, A Prakash, T.-C Wei, D T Stephen, Phys. Rev. A. 9612302R. Raussendorf, D.-S. Wang, A. Prakash, T.-C. Wei, D.T. Stephen, Phys. Rev. A 96, 012302, (2017).
. A S Darmawan, G K Brennen, S D Bartlett, New J. Phys. 1413023A.S. Darmawan, G.K. Brennen, S.D. Bartlett, New J. Phys. 14, 013023 (2012).
. C.-Y Huang, M A Wagner, T.-C Wei, arXiv:1605.08417C.-Y. Huang, M.A. Wagner, and T.-C. Wei, arXiv:1605.08417.
. R Raussendorf, H.-J Briegel, Phys. Rev. Lett. 865188R. Raussendorf and H.-J. Briegel, Phys. Rev. Lett. 86, 5188 (2001).
. Z C Gu, X G Wen, Phys. Rev. B. 80155131Z.C. Gu and X.G. Wen, Phys. Rev. B 80, 155131 (2009).
. X Chen, Z C Gu, X G Wen, Phys. Rev. B. 82155138X. Chen, Z.C. Gu, and X.G. Wen, Phys. Rev. B 82, 155138 (2010).
. X Chen, Z C Gu, Z X Liu, X G Wen, Phys. Rev. B. 87155114X. Chen, Z.C. Gu, Z.X. Liu, X.G. Wen, Phys. Rev. B 87, 155114 (2013).
. H Niggemann, A Klümper, J Zittartz, Z. Phys. B. 104103H. Niggemann, A. Klümper, and J. Zittartz, Z. Phys. B 104, 103 (1997).
. T.-C Wei, C.-Y Huang, Phys. Rev. A. 9632317T.-C. Wei and C.-Y. Huang, Phys. Rev. A 96, 032317 (2017).
. H Poulsen Nautrup, T C Wei, Phys. Rev. A. 9252309H. Poulsen Nautrup and T.C. Wei, Phys. Rev. A 92, 052309 (2015).
. J Miller, A Miyake, NPJ Quantum Information. 216036J. Miller and A. Miyake, NPJ Quantum Information 2, 16036 (2016).
. Y Chen, A Prakash, T C Wei, Phys. Rev. A. 9722305Y. Chen, A. Prakash, T.C. Wei, Phys. Rev. A 97, 022305 (2018).
. J Miller, A Miyake, arXiv:1612.08135J. Miller and A. Miyake, arXiv:1612.08135.
. Y Huang, X Chen, Phys. Rev. B. 91195143Y. Huang, X. Chen, Phys. Rev. B 91, 195143 (2015).
. B Yoshida, Phys. Rev. B. 93155131B. Yoshida, Phys. Rev. B 93, 155131 (2016).
. S Roberts, B Yoshida, A Kubica, S D Bartlett, Phys. Rev. A. 9622306S. Roberts, B. Yoshida, A. Kubica, S.D. Bartlett, Phys. Rev. A 96, 022306 (2017).
. D V Else, S D Bartlett, A C Doherty, New J. Phys. 14113016D.V. Else, S.D. Bartlett, A.C. Doherty, New J. Phys. 14, 113016 (2012).
. D Perez-Garcia, F Verstraete, M M Wolf, J.I. Cirac, Quant. Inf. Comput. 7401D. Perez-Garcia, F. Verstraete, M.M. Wolf, J.I. Cirac, Quant. Inf. Comput. 7, 401 (2007).
. A Mantri, T F Demarie, J F Fitzsimons, arXiv:1607.- 00758A. Mantri, T.F. Demarie, J.F. Fitzsimons, arXiv:1607.- 00758.
. D Gross, J Eisert, Phys. Rev. Lett. 98220503D. Gross and J. Eisert, Phys. Rev. Lett. 98, 220503 (2007).
In: Algebra, Geometry, and Physics in the 21st Century. A Kapustin, R Thorngren, Progress in Mathematics. 324Birkhäuser VerlagA. Kapustin, R. Thorngren, In: Algebra, Geometry, and Physics in the 21st Century. Progress in Mathematics 324. Birkhäuser Verlag, Cham, Switzerland, 177-202.
. R Raussendorf, Phys. Rev. 7252301R. Raussendorf, Phys. Rev. 72, 052301 (2005).
. D M Schlingemann, H Vogts, R F Werner, J. Math. Phys. 49112104D.M. Schlingemann, H. Vogts and R.F. Werner, J. Math. Phys. 49, 112104 (2008).
. D Gross, V Nesme, H Vogts, R F Werner, Commun. Math. Phys. 310419D. Gross, V. Nesme, H. Vogts, R.F. Werner, Commun. Math. Phys. 310, 419 (2012).
. C Cedzich, T Geib, F A Grünbaum, C Stahl, L Velazquez, A H Werner, R F Werner, Ann. Henri Poincaré. 19325C. Cedzich, T. Geib, F.A. Grünbaum, C. Stahl, L. Ve- lazquez, A.H. Werner, R.F. Werner, Ann. Henri Poincaré 19, 325 (2018).
. J I Cirac, D Perez-Garcia, N Schuch, F Verstraete, J. Stat. Mech. 83105J.I. Cirac, D. Perez-Garcia, N. Schuch, F. Verstraete, J. Stat. Mech. 083105 (2017).
| [] |
[
"Coexistence and Transition between Shear Zones in Slow Granular Flows",
"Coexistence and Transition between Shear Zones in Slow Granular Flows"
] | [
"Robabeh Moosavi \nDepartment of Physics\nInstitute for Advanced Studies in Basic Sciences\n45137-66731ZanjanIran\n",
"M Reza Shaebani \nComputational Physics Group\nUniversity of Duisburg\nD-47048Essen, DuisburgGermany\n\nDepartment of Theoretical Physics\nSaarland University\nD-66041SaarbrückenGermany\n",
"Maniya Maleki \nDepartment of Physics\nInstitute for Advanced Studies in Basic Sciences\n45137-66731ZanjanIran\n",
"János Török \nDepartment of Theoretical Physics\nBudapest University of Technology and Economics\nH-1111BudapestHungary\n",
"Dietrich E Wolf \nComputational Physics Group\nUniversity of Duisburg\nD-47048Essen, DuisburgGermany\n",
"Wolfgang Losert \nDepartment of Physics\nUniversity of Maryland\nCollege Park, MarylandUSA\n"
] | [
"Department of Physics\nInstitute for Advanced Studies in Basic Sciences\n45137-66731ZanjanIran",
"Computational Physics Group\nUniversity of Duisburg\nD-47048Essen, DuisburgGermany",
"Department of Theoretical Physics\nSaarland University\nD-66041SaarbrückenGermany",
"Department of Physics\nInstitute for Advanced Studies in Basic Sciences\n45137-66731ZanjanIran",
"Department of Theoretical Physics\nBudapest University of Technology and Economics\nH-1111BudapestHungary",
"Computational Physics Group\nUniversity of Duisburg\nD-47048Essen, DuisburgGermany",
"Department of Physics\nUniversity of Maryland\nCollege Park, MarylandUSA"
] | [] | We report experiments on slow granular flows in a split-bottom Couette cell that show novel strain localization features. Nontrivial flow profiles have been observed which are shown to be the consequence of simultaneous formation of shear zones in the bulk and at the boundaries. The fluctuating band model based on a minimization principle can be fitted to the experiments over a large variation of morphology and filling height with one single fit parameter, the relative friction coefficient µ rel between wall and bulk. The possibility of multiple shear zone formation is controlled by µ rel . Moreover, we observe that the symmetry of an initial state, with coexisting shear zones at both side walls, breaks spontaneously below a threshold value of the shear velocity. A dynamical transition between two asymmetric flow states happens over a characteristic time scale which depends on the shear strength.The intriguing rheology of granular materials has been widely studied over the years for its fundamental scientific interest and industrial and geophysical importance[1][2][3]. Shear banding is a widespread phenomenon in slow flows of complex materials, ranging from foams [4] and emulsions [5] to colloids [6] and granular matter[2,[7][8][9][10][11][12][13][14][15][16][17][18][19]. A clear understanding of how the strain is localized and the material yields is crucial in order to develop a consistent continuum theory at low inertial numbers, which is currently an important open issue[10,20,21].Slowly sheared granular materials usually develop narrow shear zones, often localized near a boundary, e.g., in avalanches [7], geological faults[8], and Couette flows[9][10][11][12], to mention a few. The characteristic length scale of the flow gradient is independent of the shear rate, ranges up to few particle diameters, and depends on particle shape and properties[9,10]. An important question we address is whether the formation of boundary-localized shear zones is intrinsic to granular matter or whether it can be prevented or controlled by suitable boundary conditions. Note that wide shear zones in granular bulk flow have been created, in a modified split-bottom Couette cell[13][14][15]. The emerging flow profiles were found to have shear zones tens of particle diameters wide. The wide shear zones were found to obey a number of scaling laws, with a transition from a shear zone near the surface at low filling heights to a closed cupola shape at high filling heights. It has not been clear, so far, whether or under what conditions the coexistence of these wide shear zones with the boundary-localized ones is possible, and what happens to the universality of the flow profiles when dealing with more complex boundary conditions.In slow flows, i.e. the state with rate-independent stresses, one expects that the steady-state flow pattern remains stable. One of the major findings of the present study is that the above concept does not work at shear velocities below a critical value.In this letter we report on the experimental and numerical study of complex shear zone formation in a Couette cell geometry in which the split at the bottom is located at the outer cylinder (seeFig. 1). We show that the surface flow patterns can be explained by the linear combination of three distinct shear zones. Their existence is explained by a model based on an optimization principle which was already applied to shear zone formation in granular materials[16][17][18][19]. The relative magnitude of the bulk and wall effective friction coefficients turns out to be the key control parameter which determines the possibility of simultaneous formation of shear zones in the system and, hence, the overall shape of the flow profiles. | 10.1103/physrevlett.111.148301 | [
"https://arxiv.org/pdf/1309.5294v1.pdf"
] | 795,459 | 1309.5294 | 88c9ff1a5bd09e9693c5cb5b8b69c600d3f297e6 |
Coexistence and Transition between Shear Zones in Slow Granular Flows
20 Sep 2013
Robabeh Moosavi
Department of Physics
Institute for Advanced Studies in Basic Sciences
45137-66731ZanjanIran
M Reza Shaebani
Computational Physics Group
University of Duisburg
D-47048Essen, DuisburgGermany
Department of Theoretical Physics
Saarland University
D-66041SaarbrückenGermany
Maniya Maleki
Department of Physics
Institute for Advanced Studies in Basic Sciences
45137-66731ZanjanIran
János Török
Department of Theoretical Physics
Budapest University of Technology and Economics
H-1111BudapestHungary
Dietrich E Wolf
Computational Physics Group
University of Duisburg
D-47048Essen, DuisburgGermany
Wolfgang Losert
Department of Physics
University of Maryland
College Park, MarylandUSA
Coexistence and Transition between Shear Zones in Slow Granular Flows
20 Sep 2013(Dated: May 22, 2014)numbers: 4757Gc4570Mg8350Ax8380Fg
We report experiments on slow granular flows in a split-bottom Couette cell that show novel strain localization features. Nontrivial flow profiles have been observed which are shown to be the consequence of simultaneous formation of shear zones in the bulk and at the boundaries. The fluctuating band model based on a minimization principle can be fitted to the experiments over a large variation of morphology and filling height with one single fit parameter, the relative friction coefficient µ rel between wall and bulk. The possibility of multiple shear zone formation is controlled by µ rel . Moreover, we observe that the symmetry of an initial state, with coexisting shear zones at both side walls, breaks spontaneously below a threshold value of the shear velocity. A dynamical transition between two asymmetric flow states happens over a characteristic time scale which depends on the shear strength.The intriguing rheology of granular materials has been widely studied over the years for its fundamental scientific interest and industrial and geophysical importance[1][2][3]. Shear banding is a widespread phenomenon in slow flows of complex materials, ranging from foams [4] and emulsions [5] to colloids [6] and granular matter[2,[7][8][9][10][11][12][13][14][15][16][17][18][19]. A clear understanding of how the strain is localized and the material yields is crucial in order to develop a consistent continuum theory at low inertial numbers, which is currently an important open issue[10,20,21].Slowly sheared granular materials usually develop narrow shear zones, often localized near a boundary, e.g., in avalanches [7], geological faults[8], and Couette flows[9][10][11][12], to mention a few. The characteristic length scale of the flow gradient is independent of the shear rate, ranges up to few particle diameters, and depends on particle shape and properties[9,10]. An important question we address is whether the formation of boundary-localized shear zones is intrinsic to granular matter or whether it can be prevented or controlled by suitable boundary conditions. Note that wide shear zones in granular bulk flow have been created, in a modified split-bottom Couette cell[13][14][15]. The emerging flow profiles were found to have shear zones tens of particle diameters wide. The wide shear zones were found to obey a number of scaling laws, with a transition from a shear zone near the surface at low filling heights to a closed cupola shape at high filling heights. It has not been clear, so far, whether or under what conditions the coexistence of these wide shear zones with the boundary-localized ones is possible, and what happens to the universality of the flow profiles when dealing with more complex boundary conditions.In slow flows, i.e. the state with rate-independent stresses, one expects that the steady-state flow pattern remains stable. One of the major findings of the present study is that the above concept does not work at shear velocities below a critical value.In this letter we report on the experimental and numerical study of complex shear zone formation in a Couette cell geometry in which the split at the bottom is located at the outer cylinder (seeFig. 1). We show that the surface flow patterns can be explained by the linear combination of three distinct shear zones. Their existence is explained by a model based on an optimization principle which was already applied to shear zone formation in granular materials[16][17][18][19]. The relative magnitude of the bulk and wall effective friction coefficients turns out to be the key control parameter which determines the possibility of simultaneous formation of shear zones in the system and, hence, the overall shape of the flow profiles.
We report experiments on slow granular flows in a split-bottom Couette cell that show novel strain localization features. Nontrivial flow profiles have been observed which are shown to be the consequence of simultaneous formation of shear zones in the bulk and at the boundaries. The fluctuating band model based on a minimization principle can be fitted to the experiments over a large variation of morphology and filling height with one single fit parameter, the relative friction coefficient µ rel between wall and bulk. The possibility of multiple shear zone formation is controlled by µ rel . Moreover, we observe that the symmetry of an initial state, with coexisting shear zones at both side walls, breaks spontaneously below a threshold value of the shear velocity. A dynamical transition between two asymmetric flow states happens over a characteristic time scale which depends on the shear strength. The intriguing rheology of granular materials has been widely studied over the years for its fundamental scientific interest and industrial and geophysical importance [1][2][3]. Shear banding is a widespread phenomenon in slow flows of complex materials, ranging from foams [4] and emulsions [5] to colloids [6] and granular matter [2,[7][8][9][10][11][12][13][14][15][16][17][18][19]. A clear understanding of how the strain is localized and the material yields is crucial in order to develop a consistent continuum theory at low inertial numbers, which is currently an important open issue [10,20,21].
Slowly sheared granular materials usually develop narrow shear zones, often localized near a boundary, e.g., in avalanches [7], geological faults [8], and Couette flows [9][10][11][12], to mention a few. The characteristic length scale of the flow gradient is independent of the shear rate, ranges up to few particle diameters, and depends on particle shape and properties [9,10]. An important question we address is whether the formation of boundary-localized shear zones is intrinsic to granular matter or whether it can be prevented or controlled by suitable boundary conditions. Note that wide shear zones in granular bulk flow have been created, in a modified split-bottom Couette cell [13][14][15]. The emerging flow profiles were found to have shear zones tens of particle diameters wide. The wide shear zones were found to obey a number of scaling laws, with a transition from a shear zone near the surface at low filling heights to a closed cupola shape at high filling heights. It has not been clear, so far, whether or under what conditions the coexistence of these wide shear zones with the boundary-localized ones is possible, and what happens to the universality of the flow profiles when dealing with more complex boundary conditions.
In slow flows, i.e. the state with rate-independent stresses, one expects that the steady-state flow pattern remains stable. One of the major findings of the present study is that the above concept does not work at shear velocities below a critical value.
In this letter we report on the experimental and numerical study of complex shear zone formation in a Couette cell geometry in which the split at the bottom is located at the outer cylinder (see Fig. 1). We show that the surface flow patterns can be explained by the linear combination of three distinct shear zones. Their existence is explained by a model based on an optimization principle which was already applied to shear zone formation in granular materials [16][17][18][19]. The relative magnitude of the bulk and wall effective friction coefficients turns out to be the key control parameter which determines the possibility of simultaneous formation of shear zones in the system and, hence, the overall shape of the flow profiles. More interestingly, upon decreasing the driving strength below a critical value Ω c , we observe a dynamical transition between boundary-localized shear zones. Setup -The experimental setup is shown in Fig. 1, with the inner and outer radius, R i and R o , respectively. The bottom plate and the inner cylinder of the apparatus rotate while the outer wall remains at rest. The cylindrical gap between the moving and standing parts has a size (<400 µm) much smaller than the typical grain size, so that no particle can escape. The apparatus was filled up to height H with spherical glass or steel beads of average diameter 0.5, 1, 2, or 3 mm with size polydispersity of about 15%. A layer of grains is glued to the bottom and side walls to obtain rough boundaries. The size polydispersity ensures that the flow profiles near the walls are not influenced by the ordering of grains [9,22]. While the bulk and boundary beads are always chosen of the same material, their size ratio δ=d wall /d bulk was varied in order to investigate the impact of the relative boundary roughness η which is defined by the normalized penetration of the flowing particles into the rough surface as η=1+δ− 1+2δ−δ 2 /3 [23,24]. For smooth walls η=0.
Velocity profiles -The inner cylinder and the comoving bottom plate are rotated at angular velocity Ω. To avoid rate-dependent stresses [25], a gear is used to decrease the rotating shaft speed down to the range 0.05 rad/s< Ω <0.15 rad/s where the steady-state velocities are proportional to Ω. Here we show results for Ω=0.15 rad/s. The resulting surface flow is monitored from above using a fast CCD camera with pixel resolution 70 µm at a frame rate of 60 s −1 . The average angular velocity ω(r) at the free top surface is obtained by means of particle image velocimetry method, which determines the average angular cross-correlation function in terms of the radial coordinate r for temporally separated frames. After the flow reaches a steady state (generally in a few seconds), we measure ω(r) at the free surface as a function of r. The flow is wall-localized for very shallow (H→0) and deep (3(R o −R i )<H) layers, with exponentially decaying strain rates. However, a rich variety of surface flow patterns can be observed in the middle range of H [see Figs. 1(c) and 2(a)]; The profile shapes strongly depend on the choice of H, η, and material properties. The basic question is how does the system adopt a stationary velocity profile.
Variational approach -To provide physical insight into what determines the flow profile shape, we use a variational minimization procedure [26]. This method has been successfully applied to predict the closed cupola forms of shearing regions in deep granular beds [14][15][16][17] and the refraction of shear zones in layered granular materials [18,19,27]. Dry granular materials are best described by the Mohr-Coulomb theory, which limits the shear stress divided by the normal stress by the effective friction coefficient µ eff of the material. Once the stress ratio exceeds µ eff , the material fails and a shear band forms. Due to cylindrical symmetry the whole system can be described by a two dimensional radial cut. The resulting shear band must be compatible with the boundary conditions and it should be the one which fails under the least torque or equivalently under the least dissipation rate. The last criterion can be formulated as
H 0 µ eff r(h) 2 (H − h) 1 + (dr/dh) 2 dh = min,(1)
where we search for the r(h) function, i.e. the shear band position in the bulk of material at a given height h. Here we used hydrostatic pressure since Janssen-effect plays no role due to the constant agitation of the driving [28]. The above plastic event (i.e. the instantaneous shear band) modifies the structure of the material in its vicinity. Hence, due to local fluctuations, another shear band can be optimal in the next instance. This is thus a self organized process, where the shear band appears as a global optimum which itself modifies the medium in which the optimization is carried out. This is incorporated in a kinetic elastoplastic theory [20] which takes such self organization into account. However, it is impossible to solve the model for the geometry of our problem, therefore, we use a fluctuating band model. The details of this model can be found in [17], here we reiterate only the main points: The two dimensional cut is coarse grained (coarse graining length can be as small as the particle diameter) into small mesoscopic cells which are characterized by a local effective friction coefficient. The friction coefficient is different in the bulk µ bulk eff and at the wall µ wall eff due to differences in texture. The actual strength of a particular cell in the bulk (at the wall) is chosen randomly from the interval [0, µ bulk eff ] ([0, µ wall eff ]). An instantaneous shear band is chosen by minimizing Eq. (1). In the scope of this model, the width of this shear band is considered to be only one cell wide. Once the shear band is found, the local strength along it and in its neighborhood (next neighbor sites) are updated randomly. Shear profiles are obtained by an ensemble average over instantaneous slips. We note that: (i) The actual probability distribution of µ eff [29] is not important in itself; The central limit theorem ensures that only its average and variance play a role in the integral of Eq. (1). (ii) The model has other parameters which are fixed by the geometry using the coarse graining length of a particle diameter size. The only free parameter we can vary for a given test is the ratio of the friction coefficients of wall and bulk µ rel =µ wall eff /µ bulk eff . The numerical velocity profiles obtained by tuning the single free parameter µ rel match remarkably with the experimental data, as shown in Figs. 1(c) and 2(a), given the fact that the boundary roughness is nonuniform, and size polidispersity would also influence the mechanical properties [30]. For a given set of bulk and wall particle sizes, the corresponding values of µ rel at different filling heights are obtained from the best fit to the experimental data with Eq. (1) within 7% error, showing that µ rel is roughly invariant with H [ Fig. 2(b)]. The constant nature of µ rel indicates that the fluctuating band model captures the right physics behind the effect of the walls because µ rel is fixed by the material size and type on the wall and in the bulk, therefore, it should be the same for all filling heights for a given set of materials.
When looking at different strain rate profiles, up to three maxima can be observed, one in the middle and two at the boundaries. In the geometry of our setup, these are indeed the only feasible choices of shear zones which minimize the rate of energy dissipation. The competition between these types of minimal paths gives rise to a rich shear zone phase diagram. Roughly speaking, the energy dissipation along the shear zone at the outer cylinder is proportional to µ wall eff R 2 o H 2 , while the cost of the path which sticks to the bottom plate and then to the inner cylinder grows with µ wall
eff (R 2 i H 2 + 2 3 (R 3 o −R 3 i )H).
Hence, one expects that the inner shear zone wins the race only above H∼80mm. Assuming that the middle shear zone with the center position R w is the universal wide zone reported in [13][14][15], it should follow a path in the bulk of material which is given by [16]
h = H − r 1 − R o r [1 − (H/R o ) α ] 1/α ,(2)
and the total dissipation along the broad shear zone is equal to 2πµ bulk ist) and the trajectory which sticks to the outer cylinder shows that the former becomes favorable only for µ rel 0.8 [ Fig. 2(c)]. After detailed calculations, Fig. 3 summarizes the results of the formation and coexistence of shear zones in a phase diagram in the (µ rel , H) space. The numerical diagram reveals that the H-dependence of the surface profile shape has a nontrivial dependence on µ rel . This has been confirmed by the experimental results, obtained for the accessible values of (µ rel , H). The model numerically reproduces the experiment well without providing an explicit analytical expression for the velocity profiles. In the following we address whether a functional form can be proposed, based on the combination of possible basic ingredients: wall-localized shear zones with exponential flow profiles [9][10][11][12] and wide shear zones with Gaussian velocity gradient profiles [13,17]. We find that both all experimental and numerical profiles are well fitted by a superposition of a Gaussian and two exponential curves [solid lines in Figs. 1(c) and 2(a)]:
dω(r) dr = a i exp[−b i (r−R i )] + a o exp[−b o (R o −r)] + a w √ π ξ exp[−(x − R w ) 2 /ξ 2 ].(3)
The contribution of different terms evolves with H in such a way that confirms the validity of the numerical phase diagram. Also, the universality of the wide shear zone is preserved, i.e. the evolution of the width ξ and the center position R w of the wide zone follows, respec- tively, H 2/3 and R o (1−(H/R o ) α ), compatible with prior work [13,15]. The exponent α, however, ranges between 1.4−2.5. The discrepancy can be attributed to the relatively large d bulk compared to the system size. We find that our additional parameters, the characteristic lengths of the exponential decays b i and b o , are influenced by the particle size and type. They evolve with the filling height in the following way: For a given experimental setting, b o scales with H as exp(−λ o H) [ Fig. 4(a)]. The decay constant λ o grows weakly with increasing η, meaning that the larger roughness is accompanied by the faster suppression of the outer shear zone with increasing the filling height. The exponent b i shows a saturation behavior with H [ Fig. 4(b)], with the following empirical scaling relation
b i (H)/b ∞ i ∼ 1 + tanh H − H o 2w ,(4)
with H o and w being the center and width of the hyperbolic tangent. The saturation value b ∞ i decays exponentially with η for a given material (not shown). In short, the surface flow pattern is a linear combination of a few basic elements, each of which satisfies simple scaling laws.
We also determine the relation between µ rel and the boundary roughness η. As illustrated in Fig. 4(c), a clear dependency on the material type can be observed. One expects that µ rel saturates towards µ ∞ rel =1 at η→∞, since the bulk particles fill the holes and smoothen the boundary roughness so that they practically roll over each other. The behavior at η→0 depends on material type and particle size. We attribute the particle size dependence to the roughness caused by the uneven gluing.
Instability at low shear velocities -All the experimental results reported so far were obtained in the rate independent regime, 0.05 rad/s<Ω<0.15 rad/s, where the flow profiles rapidly reach to their final steady-state shapes. Let us consider a case with two coexisting shear zones at both side walls, obtained at η=0.36 and by adjusting the height of steel bead layer to H≃80 mm. We observe an anomalous behavior, a spontaneous symmetry breaking of the flow profile, as the shear velocity is decreased below Ω c ∼5×10 −3 rad/s (see Fig. 5). The system is found in either of the two asymmetric flow states with strain localization at only one boundary. A dynamical transition between the two states takes place over a characteristic time scale, which decays to zero at Ω→Ω c (Ω<Ω c ). A similar asymmetric shear zone has been recently reported in experiments on colloidal glasses [31] (although with permanent rather than transient behavior), and in numerical simulations of plane shear flow [23]. Based on the analysis of velocity fluctuations, a plausible scenario is that the agitations induced by the external driving at shear velocities lower than Ω c are not strong enough to trigger shear zones at both walls. Thus the system is trapped in one of two minimal states. The shear rate plays the role of a kind of "temperature", enabling the system to visit both minimal states. When the system is sheared slower than Ω c , it freezes in one of the shear zone locations for a long time. As the shear velocity approaches Ω c , the switching happens more frequently, and the transition time goes to zero. Note that the velocity profiles in the small Ω state can be also recovered from the fluctuating band model when averaging over a long time window.
In conclusion, the possibility of multiple shear zones and the transitions between two of the most thoroughly investigated kinds of shear flow behaviors in dry granular materials is studied through careful comparison of experiment and modeling. We describe those aspects of the microstructure that are translated to the global rheology, and verify that the formation of localized boundary shear zones is not an intrinsic property of granular matter. One can adjust the relative strength of bulk and boundary shear zones by tuning the relative effective friction. Tuning it via the boundary conditions and material properties it is possible to either enhance or minimize boundary shear zones.
Our study may also be used as a template for a practical tool to measure the effective friction coefficient of the material from surface flow patterns. Our finding, that the minimization of energy dissipation governs the intriguing behavior, is a major step forward towards understanding the mechanisms of shear localization in granular materials which is an outstanding challenge in physics of complex flows, geophysics, and industry. The observed instabilities at low shear velocities deserves further detailed studies to uncover the underlying physics. The results of plane shear flows [23] suggest that the rotational degrees of freedom of particles play a crucial role in facilitating the dynamical transitions between the optimum states.
We would like to thank T. Unger and Z. Shojaaee for helpful discussions. R.M. and M.M. acknowledge the support of this project by the Institute for Advanced Studies in Basic Sciences (IASBS) Research Council under grant No. G2010IASBS136, and M.R.S. and D.E.W. by DGF Grant No. Wo577/8-1 within the priority program "Particles in Contact". WL acknowledges support from DTRA grant 10DTRA1077.
PACS numbers: 47.57.Gc, 45.70.Mg, 83.50.Ax, 83.80.Fg
FIG. 1 :
1(color online). (a) The experimental setup and its top view (inset). (b) Schematic of the side view. Ri=57mm, Ro=99mm. (c) Velocity profiles at different values of H and d bulk (in mm), and η. Symbols are experimental data, dashed curves are obtained from the variational approach Eq. (1), and solid curves are the fits with Eq. (3). The gray color denotes the localized profiles at the two extreme limits of H.
FIG. 2 :
2(color online). (a) Radial dependence of the strain rate. Same symbols and lines as in Fig. 1(c). (b) µ rel obtained from the best fit of Eq. (1) to the data at different H. The horizontal solid lines indicate the mean values and the dashed lines are the best linear fits. (c) The rate of energy dissipation χ, scaled by the maximum dissipation of the bulk profile χ ref , versus H. The dissipation of the wide shear zone (dashed line) is compared to that of the localized shear zone at the outer cylinder at different values of µ rel (solid lines).
RwFIG. 3 :
3(H−h)r 2 1 + (dh/dr) 2 dr. The exponent α is introduced after Eq. (3). A comparison between this trajectory (within the range of H it may ex-(color online). Phase diagram of shear zone coexistence for d bulk =2 mm. The squares, circles, and stars denote the shear zone at the outer and inner cylinders and in the bulk, respectively. The gray shaded regions denote the values of (µ rel , H) for which the experiments are performed. Insets: Typical velocity fields (sketched with arrows) and the corresponding strain rates (red curves) (both corrected for the radial dependence).
FIG. 4 :
4(color online). (a),(b) Evolution of the decay exponents bo and bi with H. The lines indicate exponential fits (a), and fits given by Eq.(4) (b). (c) µ rel (averaged over all filling heights) vs. the wall roughness η. The curves are exponential saturation fits as guides to the eye.
FIG. 5 :
5(color online). The surface flow profile at Ω=0.1 rad/s (solid curve), and two typical profiles at Ω=4×10 −3 rad/s (dashed curves), and the corresponding strain rates (inset).
. Y Forterre, O Pouliquen, Annu. Rev. Fluid Mech. 401Y. Forterre and O. Pouliquen, Annu. Rev. Fluid Mech. 40, 1 (2008);
. I S Aranson, L S Tsimring, Rev. Mod. Phys. 78641I. S. Aranson and L. S. Tsimring, Rev. Mod. Phys. 78, 641 (2006).
. P Schall, M Van Hecke, Annu. Rev. Fluid Mech. 4267P. Schall and M. van Hecke, Annu. Rev. Fluid Mech. 42, 67 (2010).
R Nedderman, Statics and Kinematics of Granular Materials. CambridgeCambridge University PressR. Nedderman, Statics and Kinematics of Granular Ma- terials (Cambridge University Press, Cambridge, 1992).
. R Höhler, S Cohen-Addad, J. Phys. Condens. Matter. 171041R. Höhler and S. Cohen-Addad, J. Phys. Condens. Mat- ter 17, R1041 (2005);
. J Lauridsen, G Chanan, M Dennin, Phys. Rev. Lett. 9318303J. Lauridsen, G. Chanan, and M. Dennin, Phys. Rev. Lett. 93, 018303 (2004);
. G Katgert, M E Möbius, M Van Hecke, Phys. Rev. Lett. 10158301G. Katgert, M. E. Möbius, and M. van Hecke, Phys. Rev. Lett. 101, 058301 (2008).
. L Bécu, S Manneville, A Colin, Phys. Rev. Lett. 96138302L. Bécu, S. Manneville, and A. Colin, Phys. Rev. Lett. 96, 138302 (2006);
. N D Denkov, Phys. Rev. Lett. 100138301N. D. Denkov et al., Phys. Rev. Lett. 100, 138301 (2008).
. J K G Dhont, Faraday discuss. 123157J. K. G. Dhont et al., Faraday discuss. 123, 157 (2003);
. I Cohen, Phys. Rev. Lett. 97215502I. Cohen et al., Phys. Rev. Lett. 97, 215502 (2006);
. R Isa, W C K Besseling, Poon, Phys. Rev. Lett. 98198305Isa, R. Besseling, and W. C. K. Poon, Phys. Rev. Lett. 98, 198305 (2007).
. A Daerr, S Douady, Nature. 399241A. Daerr and S. Douady, Nature 399, 241 (1999);
. T Börzsönyi, T C Halsey, R E Ecke, Phys. Rev. Lett. 94208001T. Börzsönyi, T. C. Halsey, and R. E. Ecke, Phys. Rev. Lett. 94, 208001 (2005).
. D R Scott, Nature. 381592D. R. Scott, Nature 381, 592 (1996);
. G Gudehus, K Nubel, Géotechnique. 54187G. Gudehus and K. Nubel, Géotechnique 54, 187 (2004).
. D M Mueth, Nature. 406385D. M. Mueth et al., Nature 406, 385 (2000).
. L Bocquet, Phys. Rev. E. 6511307L. Bocquet et al., Phys. Rev. E 65, 011307 (2001).
. W Losert, Phys. Rev. Lett. 851428W. Losert et al., Phys. Rev. Lett. 85, 1428 (2000).
. M , Eur. Phys. J. E. 11325M. Latzel et al., Eur. Phys. J. E 11, 325 (2003).
. D Fenistein, M Van Hecke, Nature. 425256D. Fenistein and M. van Hecke, Nature 425, 256 (2003).
. X Cheng, Phys. Rev. Lett. 9638001X. Cheng et al., Phys. Rev. Lett. 96, 038001 (2006).
. D Fenistein, J W Van De Meent, M Van Hecke, Phys. Rev. Lett. 96118001D. Fenistein, J. W. van de Meent, and M. van Hecke, Phys. Rev. Lett. 96, 118001 (2006).
. T Unger, Phys. Rev. Lett. 92214301T. Unger et al., Phys. Rev. Lett. 92, 214301 (2004).
. J Török, Phys. Rev. E. 7511305J. Török et al., Phys. Rev. E 75, 011305 (2007).
. T Unger, Phys. Rev. Lett. 9818301T. Unger, Phys. Rev. Lett. 98, 018301 (2007).
. T Börzsönyi, T Unger, B Szabó, Phys. Rev. E. 8060302T. Börzsönyi, T. Unger, and B. Szabó, Phys. Rev. E 80, 060302(R) (2009).
. L Bocquet, A Colin, A Ajdari, Phys. Rev. Lett. 10336001L. Bocquet, A. Colin and A. Ajdari, Phys. Rev. Lett. 103, 036001 (2009).
. V Mansard, A Colin, Soft Matter. 84025V. Mansard and A. Colin, Soft Matter 8, 4025 (2012).
. G Chambon, Phys. Rev. E. 6811304G. Chambon et al., Phys. Rev. E 68, 011304 (2003).
. Z Shojaaee, Phys. Rev. E. 8611302Z. Shojaaee et al., Phys. Rev. E 86, 011302 (2012).
. G Koval, Granular Matter. 13525G. Koval et al., Granular Matter 13, 525 (2011).
. R R Hartley, R P Behringer, Nature. 421928R. R. Hartley and R. P. Behringer, Nature 421, 928 (2003).
. L Onsager, Phys. Rev. 37405L. Onsager, Phys. Rev. 37, 405 (1931);
. Phys. Rev. 382265Phys. Rev. 38, 2265 (1931);
. R Baker, M Garber, Geotechnique. 28395R. Baker and M. Garber, Geotechnique 28, 395 (1978);
. I Einav, M Randolph, Geotechnique. 56501I. Einav and M. Randolph, Geotechnique 56, 501 (2006).
. H A Knudsen, J Bergli, Phys. Rev. Lett. 103108301H. A. Knudsen and J. Bergli, Phys. Rev. Lett. 103, 108301 (2009).
. K , Phys. Rev. Lett. 10478302K. Nichol et al., Phys. Rev. Lett. 104, 078302 (2010).
. M R Shaebani, T Unger, J Kertész, Phys. Rev. E. 7811308M. R. Shaebani, T. Unger, and J. Kertész, Phys. Rev. E 78, 011308 (2008).
. M R Shaebani, M Madadi, S Luding, D E Wolf, Phys. Rev. E. 8511301M. R. Shaebani, M. Madadi, S. Luding, and D. E. Wolf, Phys. Rev. E 85, 011301 (2012).
. R Besseling, Phys. Rev. Lett. 105268301R. Besseling et al., Phys. Rev. Lett. 105, 268301 (2010).
| [] |
[
"NONLINEAR MAXWELL-SCHRÖDINGER SYSTEM AND QUANTUM MAGNETO-HYDRODYNAMICS IN 3D",
"NONLINEAR MAXWELL-SCHRÖDINGER SYSTEM AND QUANTUM MAGNETO-HYDRODYNAMICS IN 3D"
] | [
"Paolo Antonelli ",
"Michele D'amico ",
"Pierangelo Marcati "
] | [] | [] | Motivated by some models arising in quantum plasma dynamics, in this paper we study the Maxwell-Schrödinger system with a power-type nonlinearity. We show the local well-posedness in H 2 (R 3 ) × H 3/2 (R 3 ) and the global existence of finite energy weak solutions, these results are then applied to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems.2000 Mathematics Subject Classification. 35Q40, 35Q35 (Primary); 76Y05, 82D10. | null | [
"https://arxiv.org/pdf/1702.00751v1.pdf"
] | 119,139,840 | 1702.00751 | a65fbc74fbe6ed1a51ef3478363cec9e646fa485 |
NONLINEAR MAXWELL-SCHRÖDINGER SYSTEM AND QUANTUM MAGNETO-HYDRODYNAMICS IN 3D
2 Feb 2017
Paolo Antonelli
Michele D'amico
Pierangelo Marcati
NONLINEAR MAXWELL-SCHRÖDINGER SYSTEM AND QUANTUM MAGNETO-HYDRODYNAMICS IN 3D
2 Feb 2017
Motivated by some models arising in quantum plasma dynamics, in this paper we study the Maxwell-Schrödinger system with a power-type nonlinearity. We show the local well-posedness in H 2 (R 3 ) × H 3/2 (R 3 ) and the global existence of finite energy weak solutions, these results are then applied to the analysis of finite energy weak solutions for Quantum Magnetohydrodynamic systems.2000 Mathematics Subject Classification. 35Q40, 35Q35 (Primary); 76Y05, 82D10.
Introduction
In this paper we investigate the existence of local and global in time solutions to the following 3-D nonlinear Maxwell-Schrödinger system (1) i∂ t u = − 1 2 ∆ A u + φu + |u| 2(γ−1) u A = PJ(u, A) with the initial data
u(0) = u 0 , A(0) = A 0 , ∂ t A(0) = A 1 .
Here all the physical constants are normalized to 1, ∆ A = (∇ − iA) 2 denotes the magnetic Laplacian, φ = φ(ρ) = (−∆) −1 ρ, with ρ := |u| 2 , represents the Hartreetype electrostatic potential, while the power nonlinearity describes the self-consistent interaction potential. J(u, A) = Im(ū(∇ − iA)u) is the electric current density and P = I − ∇ div ∆ −1 denotes the Leray-Helmholtz projection operator onto divergence free vector fields. The Maxwell-Schrödinger system
(2) i∂ t u = − 1 2 ∆ A u + φu − ∆φ − ∂ t div A = ρ A + ∇(∂ t φ + div A) = J,
is used in the literature to describe the dynamics of a charged non-relativistic quantum particle, subject to its self-generated (classical) electro-magnetic field, see for instance [32,8]. In particular the Maxwell-Schrödinger system (2) can be seen as a classical approximation to the quantum field equations for an electro-dynamical non-relativistic many body system. It is well known to be invariant under the gauge transformation
(3) (u, A, φ) → (u ′ , A ′ , φ ′ ) = (e iλ u, A + ∇λ, φ − ∂ t λ),
therefore for our convenience we can decide to work in the Coulomb gauge, namely by assuming div A = 0. Consequently under this gauge the system (2) takes the form
i∂ t u = − 1 2 ∆ A u + φu A = PJ(u, A).
It is straightforward to verify that also power-type nonlinearities of the previous form are gauge invariant. The Maxwell-Schrödinger system (2) has been widely studied in the mathematical literature in the various choice of gauges, for instance among the first mathematical treatments we mention [26,40], where the authors studied the local and global well-posedness in high regularity spaces by means of the Lorentz gauge. The global existence of finite energy weak solutions has been investigated in [15], by using the method of vanishing viscosity. However the uniqueness and the global well-posedness of the finite energy weak solutions is not easily achievable with this approach. In [27,28], by using the semigroup associated to the magnetic Laplacian following Kato's theory [19,20] and hence by means of a fixed point argument, the authors obtained global well-posedness with higher order Sobolev regularity.
More recently a global well-posedness result in the energy space has been proven in [6] by using the analysis of a short time wave packet parametrix for the magnetic Schrödinger equation and the related linear, bilinear, and trilinear estimates. Therefore strong H 1 solutions to (2) are obtained as the unique strong limit of H 2 solutions. Moreover in the same paper the authors obtained a continuous dependence on initial data in the energy space. The asymptotic behavior and the long-range scattering of solutions to (2) has been studied for instance in [11,12,33] (see also the references therein). The global well-posedness in the space of energy for the 2D Maxwell-Schrödinger system in Lorentz gauge has been investigated by [41] . In the present paper we focus on the Cauchy problem for the Maxwell-Schrödinger system with a power-type nonlinearity; our interest in this problem is motivated by the possibility to develop a general theory for quantum fluids in presence of self-induced electromagnetic interacting fields. The related Quantum Magneto-Hydrodynamic (QMHD) systems, with a nontrivial pressure tensor, arise in the description of quantum plasmas, for example in astrophysics, where magnetic fields and quantum effects are non negligible, see [16,17,34,35] and the references therein. The hydrodynamic equations describing a bipolar gas of ions and electrons can be recovered from the Maxwell-Schrödinger system (1) by applying the Madelung transforms as done in [1], where the authors studied a general class of quantum fluids in the non-magnetic case. We refer to the Section 5 for a more detailed discussion concerning the connection between QMHD and the Maxwell-Schrödinger system (1). We state in the sequel the two main results of this paper. The first one regards the local well-posedness theory for (1) in H 2 (R 3 ) × H 3 2 (R 3 ). More precisely let us denote by
X := (u 0 , A 0 , A 1 ) ∈ H 2 (R 3 ) × H 3/2 (R 3 ) × H 1/2 (R 3 ) s.t. div A 0 = div A 1 = 0 .
. Theorem 1.1 (Local wellposedness). Let γ > 3 2 . For all (u 0 , A 0 , A 1 ) ∈ X there exists a (maximal) time 0 < T max ≤ ∞ and a unique (maximal) solution (u, A) to (1) such that
• u ∈ C([0, T max ); H 2 (R 3 )),
• A ∈ C([0, T max ); H • Let Γ(u 0 , A 0 , A 1 ) = (u(·), A(·), ∂ t A(·)), then Γ ∈ C(X; C([0, T ]; X)), for any 0 < T < T max The following blowup alternative holds: either T max = ∞ or T max < ∞ and
lim t↑Tmax u(t) H 2 (R 3 ) = ∞, lim t↑Tmax A(t) H 3 2 (R 3 ) = ∞, lim t↑Tmax ∂ t A(t) H 1 2 (R 3 ) = ∞ .
Our proof plays on the the construction of the evolution operator associated to the magnetic Laplacian, based on Kato's approach [19,20], then we perform a fixed point argument to approximate the solutions to the Maxwell-Schrödinger system by the classical Picard iteration. Differently from [28], in our case the solutions obtained by this method cannot be extended globally in time, indeed the power-type nonlinearity does not lead to a Gronwall type inequality capable to bound the higher order norms of the solution at any time, see also [30] for a similar problem. To circumvent this difficulty we regularize the system (1) by making use of the so-called Yosida approximations of the identity, hence we are able to get the global well-posedness for the approximating system in H 2 (R 3 ) × H 3/2 (R 3 ). Moreover, by using the uniform bounds provided by the higher order energy, defined by the norm of X, we prove the existence of a finite energy weak solution to (1), in the sense defined in [15]. This is established by the following theorem.
Theorem 1.2 (Global Weak Solutions). Let 1 < γ < 3, (u 0 , A 0 , A 1 ) ∈ X, then there exists, globally in time, a finite energy weak solution (u, A) to (1), such that u ∈ L ∞ (R + ; H 1 (R 3 )), A ∈ L ∞ (R + ; H 1 (R 3 )) ∩ W 1,∞ (R + ; L 2 (R 3 )).
Remark 1.3. The same results can be obtained in a straightforward way, by using the previous results on the Coulomb gauge, in any other admissible gauge. Remark 1.4. It is possible to include a Hartree (nonlocal) nonlinear potential of the form (| · | −α * |u| 2 )u, with 0 < α < 3. It can be dealt in the same fashion as for power nonlinearities.
The final goal of this paper is to develop a suitable theory for the QMHD system (39). The major obstacle in this direction, which is also the major difference with respect to the usual usual QHD theory, regards the possibility to give sense to the nonlinear term related to the Lorenz force. For sake of simplicity we will consider the case without the nonlinear potential. Let us recall the definition of the macroscopic hydrodynamic variables via the so-called Madelung transformations, namely
ρ := |u| 2 J := Re(ū(−i∇ − A)u)
From the Maxwell equations we have
E = −∂ t A − ∇φ B = ∇ ∧ A F L := ρE + J ∧ B,
where E, B, F L , φ denote the Electric field, the Magnetic field, the Lorenz force and the (scalar) electrostatic potential, respectively. The fields equations are supplemented by the involution of the magnetic field and in the Coulomb gauge by the Poisson equation (here all the physical constants are normalized to one), namely
div B = 0, div E = −∆φ = ρ
The usual energy estimates on the Maxwell-Schrödinger system (1), as we will see in the Section 5, lead to
J √ ρ ∈ L ∞ t L 2 x , ∇ √ ρ ∈ L ∞ t L 2 x , J ∈ L ∞ t L 3/2 x , B ∈ L ∞ t L 2 x , ∇ ∧ J ∈ L ∞ t L 1 x ∩ L ∞ t W −1,3/2 x
. Unfortunately these bounds are not sufficient to apply the compensated compactness of Tartar [24,25,37] and in particular the argument in the Lecture 40 of [38], indeed J / ∈ L 2 x and B / ∈ L 3 x (the boundedness in at least one of these norms would be sufficient). Therefore the analysis of the Lorenz force for finite energy solutions needs still to be better understood. In [4],the authors investigate the weak stability of the Lorentz force by a detailed frequency analysis, in the case of incompressible dynamics (where J ∈ L 2 ). In [6] the authors obtain a global wellposedness result, in the sense that finite energy strong solutions are the unique limit of H 2 regular solutions, but however these solutions do not allow to treat the Lorenz force term. The results of [27,28], obtained without the nonlinear potential, include global well-posedness in higher order Sobolev spaces which combined with the methods of [1,2] allows instead to analyze the pressureless QMHD case. The additional difficulty introduced by the power nonlinearity in the Maxwell-Schrödinger system (1) in 3-D, namely a nonlinear pressure term in the QMHD system, can't be easily managed. Usually the proof of higher order well-posedness for the NLS, combines higher order energy estimates with the use of sharp Strichartz estimates. However Strichartz estimates of the same type are not, to our knowledge, available for the system (1), while a brute force higher order energy estimate would end up in a superlinear Gronwall inequality and hence into an upper bound which blows up in finite time. Our theory deals with the presence of a hydrodynamic pressure and it will provide a local well-posedness of QMHD in the higher regularity framework.
The paper is organized as follows. In Section 2 we collect some estimates which will be used afterwards and we study the evolution operator associated to the linear magnetic Schrödinger equation. In Section 3 we prove Theorem 1.1. In Section 4 we introduce an approximating system to (1) for which we show global existence of solutions and then we pass to the limit, proving Theorem 1.2. Finally, in Section 5 we discuss about the application of our main results to the existence theory for the QMHD system.
Notation and Preliminaries
In this Section we introduce the notation and we review some preliminary results we are going to use throughout the paper. Let A, B be two quantities, we say A B if A ≤ CB for some constant C > 0. We denote by L p (R 3 ) the usual Lebesgue spaces, H s,p (R 3 ) are the Sobolev spaces defined throught the norms f H s,p := (1 − ∆) s/2 f L p . For a given reflexive Banach space X we let C([0, T ]; X ) (resp. C 1 ([0, T ]; X ))) denote the space of continuous (resp. differentiable) maps [0, T ] → X . Analogously, L p (0, T ; X ) is the space of functions whose
Bochner integral f L p (0,T ;X ) := T 0 f (t) X dt 1/p is finite. Lemma 2.1 (Generalized Kato-Ponce inequality). Suppose 1 < p < ∞, s ≥ 0, α ≥ 0, β ≥ 0 and 1 p = 1 pi + 1 qi with i = 1, 2, 1 < q 1 ≤ ∞, 1 < p 2 ≤ ∞. Setting Λ s = (I − ∆) s 2 we have Λ s (f 1 f 2 ) L p (R 3 ) Λ s+α (f 1 ) L p 1 (R 3 ) Λ −α (f 2 ) L q 1 (R 3 ) + Λ −β (f 1 ) L p 2 (R 3 ) Λ s+β (f 2 ) L q 2 (R 3 )
Proof. Those estimates are generalization of Kato-Ponce commutator estimates, for a proof of this Lemma see for example Theorem 1.4 in [22].
Lemma 2.2. Let p, q be such that 1 ≤ q < 3 2 < p ≤ ∞, then (4) (−∆) −1 f L ∞ f θ L p f 1−θ L q ,
where θ ∈ (0, 1) is given by θ = (q ′ −3)p ′ 3(q ′ −p ′ ) . Furthermore, the following estimates hold
(−∆) −1 (f 1 f 2 )f 3 L 2 (R 3 ) f 1 L 2 (R 3 ) f 2 L 3 (R 3 ) f 3 L 3 (R 3 ) (5) (−∆) −1 |f | 2 L ∞ (R 3 ) f 2 L 2 (R 3 ) + f 2 L ∞ (R 3 )(6)
Proof. Let R > 0, then we have
4π((−∆) −1 f )(x) = 1 |y| f (x − y) dx = |y|<R 1 |y| f (x − y) dx + |y|≥R 1 |y| f (x − y) dx,
then by Hölder's inequality we have
(−∆) −1 f L ∞ |y|<R |y| −p ′ dy 1/p ′ f L p + |y|≥R |y| −q ′ dy 1/q ′ f L q .
The two integrals on the right hand side are finite by the assumptions on p, q. By optimizing the above inequality in R we then get (4). To prove (5) we apply Hölder and Hardy-Littlewood-Sobolev inequality to get
(−∆) −1 (f 1 f 2 )f 3 L 2 ≤ (−∆) −1 (f 1 f 2 ) L 6 f 3 L 3 f 1 f 2 L 6/5 f 3 L 3 .
Using again Hölder inequality for f 1 f 2 L we get (5).
Inequality (6) follows from (4) by choosing p = ∞, q = 1 and by applying Young's inequality.
Next Lemma will be useful to estimate the Hartree term in the fixed point argument in Section 3.
Lemma 2.3. Let u ∈ H 2 (R 3 ), then (7) (−∆) −1 (|u| 2 )u H 2 u 2 H 3/4 u H 2 . Proof. We have (−∆) −1 (|u| 2 )u H 2 (−∆) −1 (1 − ∆)(|u| 2 )u L 2 + (−∆) −1 (|u| 2 )(1 − ∆)u L 2 (−∆) −1 (1 − ∆)|u| 2 L 6 u L 3 + (−∆) −1 |u| 2 L ∞ (1 − ∆)u L 2 .
By the Hardy-Littlewood-Sobolev inequality we have
(−∆) −1 (1 − ∆)|u| 2 L 6 (1 − ∆)|u| 2 L 6/5 u L 3 (1 − ∆)u L 2 ,
where the last inequality follows from Lemma 2.1. On the other hand, by using (4), with p, q sufficiently close to 3 2 , and Sobolev embedding we see that
(−∆) −1 |u| 2 L ∞ u 2 H 1 2 +ε . Consequently, (−∆) −1 (|u| 2 )u H 2 u 2 H 1 2 +ε u H 2 . Lemma 2.4. Let A ∈ H 1 (R 3 ) and u ∈ H 2 (R 3 )
. Then the following estimates hold:
(∇ − iA)u H 1 (R 3 ) (1 + A H 1 (R 3 ) ) u H 2 (R 3 ) ,(8)PJ(u, A) H 1 2 (R 3 ) u H 1 (R 3 ) u H 2 (R 3 ) + A H 1 (R 3 ) u 2 H 2 (R 3 ) ,(9)∆ A u L 2 u H 2 + A 4 H 1 u L 2 ,(10)u H 2 ∆ A u L 2 + A 4 H 1 u L 2 ,(11)(∇ + iA)u L 6 u H 2 + A 4 H 1 u L 2 .(12)
Proof. We begin with the proof of (8). By using Lemma 2.1 we have
(∇ − iA)u H 1 (R 3 ) ≤ ∇u H 1 (R 3 ) + Au H 1 (R 3 ) u H 2 (R 3 ) + A H 1 (R 3 ) u L ∞ (R 3 ) + A L 6 (R 3 ) u H 1,3 (R 3 ) u H 2 (R 3 ) + A H 1 (R 3 ) u H 2 (R 3 ) + A H 1 (R 3 ) u H 3 2 (R 3 ) ,
where in the last inequality we used the Sobolev embedding theorem. Thus (8) is proved. We now consider (9); by Lemma 2.1,
u∇u H 1 2 (R 3 ) u H 1 2 ,3 (R 3 ) ∇u L 6 (R 3 ) + u L 6 (R 3 ) ∇u H 1 2 ,3 (R 3 ) u H 1 2 ,3 (R 3 ) ∇u H 1 (R 3 ) + u H 1 (R 3 ) ∇u H 1 2 ,3 (R 3 ) u H 1 (R 3 ) u H 2 (R 3 )
and
A|u| 2 H 1 2 (R 3 ) A H 1 2 (R 3 ) u 2 L ∞ (R 3 ) + A L 6 |u| 2 H 1 2 ,3 (R 3 ) A H 1 (R 3 ) u 2 H 2 (R 3 )
. By adding the two estimates above we then obtain
PJ H 1/2 J H 1/2 u H 1 u H 2 + A H 1 u 2 H 2 . For (10) we have ∆ A u L 2 u H 2 + A · ∇u L 2 + |A| 2 u L 2 u H 2 + A H 1 u H 3/2 + A 2 H 1 u H 1 u H 2 + A H 1 u 1/4 L 2 u 3/4 H 2 + A 2 H 1 u 1/2 L 2 u 1/2 H 2 .
By using Young's inequality we obtain (10). Estimate (11) is proved in an analogous way. Finally, for (12) we have
(∇ − iA)u L 6 ∇(∇ − iA)u L 2 ∆ A u L 2 + A(∇ − iA) L 2 ∆ A u L 2 + A H 1 u H 3/2 + A 2 H 1
u H 1 and proceed as for the previous estimates.
Let us now state the Strichartz estimates for the wave equation we are going to use in our paper. For a proof see for example [10,36] and references therein.
I × R 3 → C be a Schwartz solution to the wave equation B = F with initial data B(0) = B 0 , ∂ t B(0) = B 1 .
Then the following estimate holds
B L q t L r x (I×R 3 ) + B CtḢ s x (I×R 3 ) + ∂ t B CtḢ s−1 x (I×R 3 ) B 0 Ḣs (R 3 ) + B 1 Ḣs−1 + F Lq ′ t Lr ′ x (I×R 3 )
whenever s ≥ 0, 2 ≤ q,q ≤ ∞ and 2 ≤ r,r < ∞ obey the scaling condition
1 q + 3 r = 3 2 − s = 1 q ′ + 3
r ′ − 2 and the wave admissibility condition
1 q + 1 r , 1 q + 1 r ≤ 1 2
As a consequence we also obtain the following energy estimate.
Lemma 2.6. Let s ∈ R, B 0 ∈ H s (R 3 ), B 1 ∈ H s−1 (R 3 ) and F ∈ L 1 ([0, T ]; H s−1 (R 3 )), T > 0, then B ∈ C([0, T ]; H s (R 3 )) ∩ H s−1 ([0, T ]; H s−1 (R 3 ))
defined as in previous Lemma satisfies
B CtH s x ([0,T ]×R 3 ) + ∂ t B CtH s−1 ([0,T ]×R 3 ) (1 + T )( B 0 H s (R 3 ) + B 1 H s−1 (R 3 ) + F L 1 t H s−1 x ([0,T ]×R 3 ) ).(13)
We conclude this Section by recalling some results concerning the Schrödinger propagator associated to the magnetic Laplacian ∆ A . More precisely, let A be a given, time dependent, divergence-free vector field, we then consider the following initial value problem (14) i∂
t u = − 1 2 ∆ A u u(s) = f,
and we study the properties of its solution. (14). Moreover, it holds
Proposition 2.7. Let 0 < T < ∞ and let us assume that A ∈ C([0, T ]; H 1 (R 3 )), ∂ t A ∈ L 1 ([0, T ]; L 3 (R 3 )). Then there exists a unique u ∈ C([0, T ]; H 2 (R 3 ))∩C 1 ([0, T ]; L 2 (R 3 )) solution to(15) u L ∞ (0,T ;H 2 (R 3 )) f H 2 1 + A 4 L ∞ t H 1 x e ∂tA L 1 t L 3
x .
Proposition 2.8. Let A ∈ L ∞ ([0, T ]; H 1 (R 3 ))∩W 1,1 ([0, T ]; L 3 ), f ∈ L 1 ([0, T ]; H −2 (R 3 )) and let v ∈ C([0, T ]; L 2 (R 3 )) ∩ W 1,1 ([0, T ]; H −2 (R 3 )) be solution to i∂ t v = − 1 2 ∆ A v + f Then for every t 0 ∈ [0, T ], v(t) = U A (t, t 0 ) − i t t0 U A (t, s)f (s)ds
Proof. See [28] for a proof of Propositions 2.7 and 2.8.
From Proposition 2.7 we can then define the propagator U A (t, s) associated to (14),
i.e. U A (t, s)f = u(t),
where u is the solution in Proposition 2.7, and U A satisfies the following properties:
• U A (t, s)H 2 ⊂ H 2 , for any t, x ∈ [0, T ]; • U A (t, t) = I; • U A (t 1 , t 2 )U A (t 2 , t 3 ) = U A (t 1 , t 3 ), for any t 1 , t 2 , t 3 ∈ [0, T ].
Moreover, by (15) we have
K 2 := sup t,s∈[0,T ] U A (t, s) H 2 →H 2 ≤ 1 + A 4 L ∞ t H 1 e ∂tA L 1 t L 3 x .
From the unitarity of U A (t, s) in L 2 , U A (t, s)f L 2 = f L 2 , and by interpolation, we can then infer
sup t,s∈[0,T ] U A (t, s)f H s →H s < ∞, ∀ s ∈ [0, 2].
Local well-posedness
In this section we are going to prove the local well-posedness result stated in Theorem 1.1 by using a fixed point argument. We split the proof in two parts: in Proposition 3.1 we are going to show the existence and uniqueness of a local solution by means of a fixed point argument, then Proposition 3.2 will be about the continuous dependence of the solution on the initial data.
Proposition 3.1. Let γ > 3 2 . For all (u 0 , A 0 , A 1 ) ∈ X there exists T max > 0 and a unique maximal solution (u, A) to (1) such that u ∈ C([0, T max ); H 2 (R 3 )), A ∈ C([0, T max ); H 3 2 (R 3 ) ∩ C 1 ([0, T max ); H 1 2 (R 3 )), div A = 0. Moreover the blowup alter- native holds true.
Proof. First of all, let us define the space
X T := {(u, A) s.t. u ∈ C([0, T ], H 2 (R 3 )), A ∈ C([0, T ], H 3 2 (R 3 )) ∩ C 1 ([0, T ], H 1 2 (R 3 )), div A = 0, u L ∞ t H 2 x (R 3 ) ≤ R 1 , A L ∞ t H 3 2 (R 3 ) + ∂ t A L ∞ t H 1 2 x (R 3 ) ≤ R 2 } ,(16)
where R 1 , R 2 , T > 0 will be chosen later. It is straightforward to see that X T , endowed with the distance
(17) d((u 1 , A 1 ), (u 2 , A 2 )) = max{ u 1 − u 2 L ∞ t L 2 x (R 3 ) , A 1 − A 2 L 4 t L 4 x (R 3 ) } , is a complete metric space. We also define (18) (u, A) XT := u L ∞ t H 2 x ([0,T ]×R 3 ) + A L ∞ t H 3/2 x ([0,T ]×R 3 ) + ∂ t A L ∞ t H 1/2 x ([0,T ]×R 3 ) . Let (u 0 , A 0 , A 1 ) ∈ X, we define the map Φ : X T → X T , (u, B) = Φ(u, A), (u, A) ∈ X T , where (19) u(t) = U A (t, 0)u 0 − i t 0 U A (t, s)(φu + |u| 2(γ−1) u)(s)ds and B(t) = cos(t √ −∆)A 0 + sin(t √ −∆) √ −∆ A 1 + t 0 sin(t √ −∆) √ −∆ PJ(u, A)(s)ds
Let us first show that Φ maps X T into itself. By (15) we have that for any s ∈ [0, T ],
U A (t, s)f H 2 ≤ f H 2 1 + A 4 L ∞ t H 1 x e ∂tA L 1 t L 3 x and since ∂ t A L 1 t L 3 x T ∂ t A L ∞ t H 1/2 x , we have U A (t, s)f H 2 ≤ C(1 + R 4 2 )e T R2 f H 2 .
Let us consider the nonlinear terms in (19). Since γ > 3 2 the function z → |z| 2(γ−1) z is C 2 (C; C), then by the Sobolev embedding H 2 ֒→ L ∞ and by Lemma 2.1 we have
|u| 2(γ−1) u L ∞ t H 2 x u 2γ−1 L ∞ t H 2 x R 2γ−1 1 .
Furthermore, from (7) we have
φu L ∞ t H 2 x u 2 L ∞ t H 3/4 x u L ∞ t H 2 x R 3 1 .
so that by putting everything together, we obtain
u L ∞ t H 2 x ≤ C 1 (1 + R 4 2 ) exp(T R 2 ) u 0 H 2 + T R 2γ−1 1 + T R 3 1 .
On the other hand, by using the Strichartz estimates for the wave equation stated in Lemma 2.5 we have
B L ∞ t H 3/2 x + ∂ t B L ∞ t H 1/2 x (1 + T ) A 0 H 3/2 + A 1 H 1/2 + PJ L 1 t H 1/2 x . By (9) we have PJ L ∞ t H 1/2 x R 2 1 (1 + R 1 ), so that B L ∞ t H 3/2 x + ∂ t B L ∞ t H 1/2 x ≤ C 2 (1 + T ) A 0 H 3/2 + A 1 H 1/2 + T R 2 1 (1 + R 1 )
. Let us now choose R 1 , R 2 , T ; without loss of generality we can assume that T < 1. Let
R 2 := 4C 2 A 0 H 3/2 + A 1 H 1/2 R 1 := 2C 1 (1 + R 4 2 )e R2 u 0 H 2 Then u L ∞ t H 2 x (R 3 ) ≤ R 1 2 + C 1 (1 + R 4 2 )T e R2 R 1 (R 2(γ−1) 1 + R 2 1 ) B L ∞ t H 3 2 x (R 3 ) + ∂ t B L ∞ t H 1 2 x (R 3 ) ≤ R 2 2 + 2C 2 R 2 1 (1 + R 2 )T Now by choosing T such that max C 1 (1 + R 4 2 )e R2 (R 2(γ−1) 1 + R 2 1 )T, 2C 2 R 2 1 (1 + R 2 ) R 2 T < 1 2 ,
we see that Φ maps X T into itself We now prove that, possibly choosing a smaller value for T > 0, the map Φ is indeed a contraction on X T . Let us define
(u 1 , B 1 ) = Φ(u 1 , A 1 ) (u 2 , B 2 ) = Φ(u 2 , A 2 ) .
By writing the difference of the equations for u 1 and u 2 we write
(u 1 − u 2 )(t) = −i t 0 U A (t, s)F (s) ds,
where F is given by
F = 2i(A 1 − A 2 ) · ∇u 2 + (|A 1 | 2 − |A 2 | 2 )u 2 + (φ(|u 1 | 2 ) − φ(|u 2 | 2 ))u 2 + φ(|u 1 | 2 )(u 1 − u 2 ) + |u 1 | 2(γ−1) u 1 − |u 2 | 2(γ−1) u 2 =: 5 j=1 F j .(20)
Hence we have
(21) u 1 − u 2 L ∞ t L 2 x j F j L 1 t L 2 x .
We now estimate term by term; by using Hölder's inequality and Sobolev embedding we have
F 1 L 1 t L 2 x T 3/4 ∇u 2 L ∞ t H 1 x A 1 − A 2 L 4 t,x , F 2 L 1
For the term F 4 we use (6) and Sobolev embedding to get
F 4 L 1 t L 2 x T u 1 2 L ∞ t H 2 x u 1 − u 2 L ∞ t L 2
x . The last term is estimated by
F 5 L 2 (R 3 ) ( u 1 2(γ−1) L ∞ + u 2 2(γ−1) L ∞ ) u − u ′ L 2 (R 3 ) R 2(γ−1) 1 u 1 − u 2 L 2 (R 3 ) ,
where we used the following inequality
||u 1 | 2(γ−1) u 1 − |u 2 | 2(γ−2) u 2 | (|u 1 | 2(γ−1) + |u 2 | 2(γ−1) )|u 1 − u 2 |
By putting everything together in (21), and by using Hölder's inequality in time, we obtain
(22) u 1 − u 2 L ∞ t L 2 x (T 3/4 + T )C(R 1 , R 2 )d((u 1 , A 1 ), (u 2 , A 2 )).
Analogously, for B 1 , B 2 we write
(23) (B 1 − B 2 )(t) = t 0 sin((t − s) √ −∆) √ −∆ G(s) ds, where G = 3 j=1 G j is given by: G = P Im{(u 1 − u 2 )(∇ − iA 1 )u 1 − iu 1 u 2 (A 1 − A 2 ) − (u 1 − u 2 )(∇ + iA 2 )u 2 } .
Here we have used the fact that P(u 2 ∇(u 1 − u 2 )) = −P((u 1 − u 2 )∇u 2 ). Using the Strichartz estimates in Lemma 2.5 with q = r =q =r = 4, we get
(24) B 1 − B 2 L 4 t,x ≤ G L 4 3 t,x
We estimate the three terms in G. The terms G 1 and G 3 are treated similarly, by Sobolev embedding and by using (8) we have
G 1 L 4/3 t,x + G 3 L 4/3 t,x T 3/4 1 + A 1 L ∞ t H 1 x + A 2 L ∞ t H 1 x u 1 L ∞ t H 2 x + u 2 L ∞ t H 2 x u 1 − u 2 L ∞ t L 2
x . By using Hölder's inequality, G 2 is bounded by
G 2 L 4/3 t,x T 1/2 u 1 L ∞ t H 1 x u 2 L ∞ t H 1 x A 1 − A 2 L 4
t,x . Resuming, by estimating the terms in (24) we obtain
(25) A 1 − A 2 L 4 t,x (T 1/2 + T 3/4 )C(R 1 , R 2 )d((u 1 , A 1 ), (u 2 , A 2 )).
By summing up (22) and (25), we finally get
d((u 1 , B 1 ), (u 2 , B 2 )) ≤ (T 1/2 + T )C(R 1 , R 2 )d((u 1 , A 1 ), (u 2 , A 2 )).
Thus, if T > 0 is chosen sufficiently small, then Φ is a contraction. This proves that for any initial data (u 0 , A 0 , A 1 ) ∈ X, there exists a unique local solution (u,
A) to (1) such that u ∈ C([0, T ]; H 2 (R 3 )), A ∈ C([0, T ]; H 3/2 (R 3 )) ∩ C 1 ([0, T ]; H 1/2 (R 3 )).
By a standard argument it is straightforward to show that it may be extended to a maximal solution (u,
A), with u ∈ C([0, T max ); H 2 (R 3 )), A ∈ C([0, T max ); H 3/2 (R 3 )) ∩ C 1 ([0, T max ); H 1/2 (R 3 )
) and that the blow-up alternative holds true, namely if T max < ∞ then we have
lim t→T − max ( u(t) H 2 + A(t) H 3/2 + ∂ t A(t) H 1/2 ) = ∞.
Next Proposition states the continuous dependence of solution on the initial data. Its proof goes through a series of technical lemmas and it follows this strategy: first we prove the continuous dependence for more regular solutions, then by an approximation argument we prove the general result for solutions (u, A) ∈ X. This will finish the proof of Theorem 1.1. In the remaining part of the Section we state the Proposition and the Lemmas needed to prove the continuous dependence for regular solutions. Then we show how to extend it to arbitrary solutions (u, A) ∈ X. The proofs of Lemmas 3.3, 3.4 and 3.5 will be given in the Appendix. For the lemmas we consider two different solutions (u, A), (u ′ , A ′ ) defined from two sets of initial data (u 0 , A 0 ,
A 1 ), (u ′ 0 , A ′ 0 , A ′ 1 ) ∈ X such that (u 0 , A 0 , A 1 ) X , (u ′ 0 , A ′ 0 , A ′ 1 ) X ≤ R.
Moreover, we are also going to exploit the uniform bounds given by the total energy of system (1),
(26) E(t) = 1 2 |(∇ − iA)u| 2 + 1 2 |∂ t A| 2 + 1 2 |∇A| 2 + 1 2 |∇φ| 2 + 1 γ |u| 2γ dx.
It is straightforward to see that it is conserved along the flow of solutions to (1)
, thus if (u, A), resp. (u ′ , A ′ ), is the solutions emanated from (u 0 , A 0 , A 1 ), resp. (u ′ 0 , A ′ 0 , A ′ 1 )
, then we may consider E > 0 such that
E(t), E ′ (t) ≤ E,
where E(t), resp. E ′ (t), is the total energy associated to (u, A), resp. (u ′ , A ′ ).
u − u ′ L ∞ t H 2 x ∂ t (u − u ′ )(0) L 2 + T ∂ t u ′ L ∞ t H 2 x A − A ′ L ∞ t H 1/2 x + u − u ′ L ∞ t L 2 x + T (u, A) − (u ′ , A ′ ) XT ,
where the constant depends only on R, E defined as above.
Lemma 3.4. Let (u, A), (u ′ , A ′ ) be solutions to (1) defined as above, then we have
u − u ′ L ∞ t L 2 x + A − A ′ L ∞ t H 1/2 x (u 0 , A 0 , A 1 ) − (u ′ 0 , A ′ 0 , A ′ 1 ) L 2 ×H 1/2 ×H −1/2 ,where
the constant depends only on R, E defined as above.
Lemma 3.5. We have ∂ t u L ∞ t H 2 x u 0 H 4 + A 0 H 5/2 + A 1 H 3/2 ,
where the constant depends only on T, R, E.
Proof of Proposition 3.2. By combining the above Lemmas, it is possible to show the continuous dependence for solutions whose initial data are (u 0 , A 0 , A 1 ) ∈ H 4 × H 5/2 × H 3/2 . Indeed Lemmas 3.3, 3.4 and 3.5 imply the following estimate
(u, A) − (u ′ , A ′ ) XT (u 0 , A 0 , A 1 ) − (u ′ 0 , A ′ 0 , A ′ 1 ) X + T ( u ′ 0 H 4 + A ′ 0 H 5/2 + A ′ 1 H 3/2 ) (u 0 , A 0 , A 1 ) − (u ′ 0 , A ′ 0 , A ′ 1 ) L 2 ×H 1/2 ×H −1/2 + T (u, A) − (u ′ , A ′ ) XT .
A straightforward bootstrap argument yields to (27) (
u, A) − (u ′ , A ′ ) XT (u 0 , A 0 , A 1 ) − (u ′ 0 , A ′ 0 , A ′ 1 ) X + T ( u ′ 0 H 4 + A ′ 0 H 5/2 + A ′ 1 H 3/2 ) (u 0 , A 0 , A 1 ) − (u ′ 0 , A ′ 0 , A ′ 1 ) L 2 ×H 1/2 ×H −1/2 .
Let us now consider general initial data (u 0 , A 0 , A 1 ) ∈ X and let us consider a mollified
η δ (x) = δ −3 η(x/δ), δ > 0, where η ∈ C ∞ c (R 3 ) is a smooth, radial function with η = 1. We define u δ 0 = η δ * u 0 , A δ 2 0 = η δ 2 * A 0 , A δ 2 1 = η δ 2 * A 1 .
It is straightforward to check that this definition implies
u 0 H 4 + A δ 2 0 H 5/2 + A δ 2 1 H 3/2 δ −2 ( u 0 H 2 + A 0 H 3/2 + A 1 H 1/2 ) and that u 0 − u δ L 2 + A 0 − A δ 2 0 H 1/2 + A 1 − A δ 2 1 H −1/2 = 0(δ 2 )
. By using (27) above we then infer (28) (
u, A) − (u δ , A δ 2 ) XT (u 0 , A 0 , A 1 ) − (u δ 0 , A δ 2 0 , A δ 2 1 ) X + T u δ 0 H 4 + A δ 2 0 H 5/2 + A δ 2 1 H 3/2 (u 0 , A 0 , A 1 ) − (u δ 0 , A δ 2 0 , A δ 2 1 ) L 2 ×H 1/2 ×H −1/2 (u 0 , A 0 , A 1 ) − (u δ 0 , A δ 2 0 , A δ 2 1 ) X + T O(δ 2 )o(δ −2 ).
Consequently we have that (u δ , A δ 2 ) converges to (u, A) in X T as δ → 0. Let now {(u 0,n , A 0,n , A 1,n } ⊂ X be a sequence converging to (u 0 , A 0 , A 1 ) ∈ X. We want to prove that the solutions (u n , A n ) emanated from (u 0,n , A 0,n , A 1,n ) converge to (u, A) in X T . To do this, we regularize the initial data by considering (u δ 0,n , A δ 2 0,n , A δ 2 1,n ). From (28) we know that {(u δ n , A δ 2 n )} converges to (u , A n ) in X T , as δ → 0, where (u n , A n ) is the solution to (1) with initial data (u 0,n , A 0,n , A 1,n ). On the other hand, {(u δ 0,n , A δ 2 0,n , A δ 2 1,n )} generate regular solutions, so that by (27) we have that {(u δ n , A δ 2 n )} converges to (u δ , A δ 2 ) in X T , for n → ∞. The triangular inequality then yields the convergence of (u n , A n ) to (u, A) in X T .
Global existence
In the previous Section we proved the local well-posedness of (1) in H 2 × H 3/2 . However, the presence of the power-type nonlinearity in (1) prevents us to obtain a global bound for (u(t), A(t), ∂ t A(t)) X . This is different, for example, from what can be proven in [22]. Indeed, while in the case of Hartree nonlinearity it is possible to use (7) which is linear in the higher order norm, in the case of the power-type nonlinearity one has
|u| 2(γ−1) u H 2 (R 3 ) u 2(γ−1) L ∞ (R 3 ) u H 2 (R 3 )
, which requires to bound u in H s (R 3 ), with s > 3 2 . Therefore it follows that the related Gronwall type inequality becomes superlinear in the higher order norm, hence it blows up in finite time.
Our strategy to investigate global in time existence will be based on the regularization of the nonlinear terms, provided by the classical Yosida approximations of the identity.
We then consider the following approximating system
(29) iu ε t = − 1 2 ∆à ε u ε + φ ε u ε + N ε (u ε ) A ε =J ε PJ ε u ε (0) =u 0 , A ε (0) = A 0 , ∂ t A ε (0) = A 1 , where J ε = (I −ε∆) −1 ,à ε = J ε A ε , N ε (u ε ) = J ε |J ε u ε | 2(γ−1) J ε u ε , J ε = J(u ε , A ε ), φ ε = φ(|u ε | 2 )
and we denote ∇Ãε = ∇ − ià ε . The total energy of this approximating system is given by
(30) E = R 3 |∇Ãε u ε | 2 + 1 2 |∇φ ε | 2 + 1 2 |∇A ε | 2 + 1 2 |∂ t A ε | 2 + 1 γ |J ε u ε | 2γ dx
which is conserved along the flow of solutions. A local well-posedness result, analogous to Theorem 1.1, can be proved for the system (29) in a straightforward way.
Proposition 4.1. For all (u 0 , A 0 , A 1 ) ∈ X, there exists T ε max > 0 and a unique maximal solution (u ε , A ε ) to (29) such that u ε ∈ C([0, T ε max ); H 2 (R 3 )), A ε ∈ C([0, T ε max ); H 3/2 (R 3 ))∩ C 1 ([0, T ε max ); H 1/2 (R 3 )
) and the usual blow-up alternative holds true. Moreover, the solution depends continuously on the initial data.
Proof. We only remark here that the local well-posedness result for system (29) holds for any γ ∈ (1, ∞), while in Theorem 1.1 we restrict the range to γ ∈ ( 3 2 , ∞). Indeed, because of the Yosida regularisation, we have
N ε (u ε ) H 2 |J ε u ε | 2(γ−1) J ε u ε L 2 u ε 2γ−1 H 2 .
The regularisation of the nonlinear terms yields indeed the global existence of solutions.
Proposition 4.2. The solution obtained in Proposition (4.1) exists globally in time,
namely (u ε (t), A ε (t), ∂ t A ε (t)) X is finite for any t ∈ R.
The proof of Proposition 4.2 is based on the following Lemma 4.3. Let ε > 0, then for every t ∈ R,
(31) u ε (t) H 2 ≤ C( u 0 L 2 , E)e t ∂tA ǫ L ∞ t H 1/2 x .
Proof. By (11) we have
u ε H 2 ∆à ε u ε L 2 + A ε 4 H 1 u ε L 2 ≤C( u 0 L 2 , E) ∆à ε u ε L 2 , therefore it is convenient to estimate the norm ∆Ãεu ε L 2 instead of u H 2 (R 3 )
. By a standard energy method it follows that
d dt ( (∆à ε u ε )(t) L 2 ) ≤ ∆à ε (φ ε u ε ) L 2 + ∆à ε N ε (u ε ) L 2 + [∂ t , ∆à ε ]u ε L 2 .
The first term can be estimated by using (10) and (7),
∆Ãε(φ ε u ε ) L 2 φ ε u ε H 2 + A ε 4 H 1 φ ε u ε L 2 u ε 2 H 3/4 ∆Ãεu ε L 2 + A ε 4 H 1 u ε L 2 ≤C( u 0 L 2 , E) ∆à ε u ε L 2 .
The nonlinear term N ε (u ε ) can be controlled by exploting the regularization given by
J ε ∆ÃεN ε (u ε ) L 2 N ε (u ε ) H 2 + A ε 4 H 1 N ε (u ε ) L 2 |J ε u ε | 2(γ−1) J ε u ε L 2 + A ε 4 H 1 N ε (u ε ) L 2 u ε 2(γ−1) H 1 u ε H 2 + A ε 4 H 1 u ε L 2 ,
where the last inequality follows from Sobolev embedding. The commutator [∂ t , ∆à ε ]u ε = 2∂ tà ε (∇ + ià ε )u ε can be estimated by using the Hölder's inequality and the Sobolev embedding
∂ tÃ ε · (∇ + ià ε )u ε L 2 ≤ ∂ tà ε L 3 (∇ + ià ε )u ε L 6 ∂ t A ε H 1/2 ∆à ε u ε L 2 + A ε 4 H 1 u ε L 2 .
By summing up the previous three terms
d dt ( (∆Ãεu ε )(t) L 2 ) ≤ C( u 0 L 2 , E) ∂ t A ε H 1/2 ∆Ãεu ε L 2 ,
hence (31).
Proof of Proposition 4.2.
In order to get a bound on the H 2 norm of the approximating solution u ǫ , by Lemma 4.3 it is sufficient to control ∂ t A ε
L ∞ t H 1/2 x .
Using the energy estimate for the wave equation
A ε L ∞ t H 3/2 x + ∂ t A ε L ∞ t H 1/2 x C(T ) A 0 H 3/2 + A 1 H 1/2 + J ε PJ ε L ∞ t H 1/2 x ,
and, by exploiting the Yosida regularization, we get
J ε PJ ε L ∞ t H 1/2 x PJ ε L ∞ t H −1/2 x J ε L ∞ t L 3/2 x ≤ C(E).
It follows that A ε (t) H 3/2 + ∂ t A ε (t) H 1/2 is uniformly bounded on compact time intervals and consequently by (31) also u ε (t) H 2 is finite. Hence, by the blow-up alternative, the solution (u ε , A ε ) to (29) exists globally in time.
Now we conclude the proof of Theorem 1.2 by showing that (u ε , A ε ) converges to a solution to (1), as ε → 0. This will conclude the proof of Theorem (1.2). The conservation of mass and energy yields the following a priori bounds (32) u ε
L ∞ t H 1 x (R×R 3 ) ≤ C, A ε L ∞ t H 1 x (R×R 3 ) ≤ C, ∂ t A ε L ∞ t L 2 x (R×R 3 ) ≤ C , which imply that, up to subsequences, there exist u ∈ L ∞ t H 1 x , A ∈ L ∞ t H 1 x ∩ W 1,∞ t L 2 x , such that u ε * ⇀ u in L ∞ t H 1 x (R × R 3 ) (33) A ε * ⇀ A in L ∞ t H 1 x (R × R 3 ) (34) ∂ t A ε * ⇀ ∂ t A in L ∞ t L 2 x (R × R 3 )(35)
Proposition 4.4. The weak limit (u, A) in (33), (34) is a finite energy weak solution to the Cauchy problem (1), with initial datum (u 0 , A 0 , A 1 ).
Proof. Let us consider u ε , by using equation (29) and the a priori bounds given by the energy we have {∂ t u ε } is uniformly bounded in L ∞ (R; H −1 (R 3 )). Hence, by using the Aubin-Lions lemma and from the assumption 1 < γ < 3 we may infer (36) u
ε → u in L 4 loc (R × R 3 ) ∩ L 2γ loc (R × R 3 ) .
This also implies that |u ε | 2 ⇀ |u| 2 in L 2 t L 6/5
x , and consequently from Hardy-Littlewood-Sobolev we obtain
(37) (−∆) −1 (|u ε | 2 ) ⇀ (−∆) −1 (|u| 2 ), in L 2 t L 6 x .
Analogously for A ε , the a priori bounds yield
(38) A ε → A in L 4 loc (R × R 3 ) .
We are now able to show the convergence for the nonlinear terms
K ε (u ε , A ε ), N ε (u ε ), J ε PJ ε , where K ε (u ε ,à ε ) = iÃ ε · ∇u ε + 1 2 |à ε | 2 u ε + φ(u ε )u ε ,
Indeed, by using the convergences (33)-(38) we may conclude
K ε (u ε ,Ã ε ) ⇀ K(u, A) in L 4 3 loc (R × R 3 ) , PJ(u ε ,Ã ε ) ⇀ PJ(u, A) in L 4 3 loc (R × R 3 ) , N ε (u ε ) ⇀ N (u), in L 2γ 2γ−1 loc (R × R 3 ).
It remains to see that the initial condition is satisfied. We have that
∂ t A ∈ L ∞ t L 2 x (R× R 3 ) and ∂ 2 t A, ∂ t u ∈ L ∞ t H −1 x (R × R 3 )
, and consequently (u, A, ∂ t A) ∈ C(R; H −1 × L 2 × H −1 ). Moreover, the energy bounds imply (u, A, ∂ t A) ∈ L ∞ (R; H 1 × H 1 × L 2 ) and hence we may also infer the weak continuity (u, A, ∂ t A) ∈ C w (R; H 1 × H 1 × L 2 ). Since A ε ∈ L 2 (0, T ; H 1 (R 3 )) and ∂ t A ε ∈ L 2 (0, T ; L 2 (R 3 )), integrating by parts we have
T 0 A ε (t)∂ t f (t) + ∂ t A ε (t)f (t), ϕ H 1 ,H −1 ds = − A 0 , ϕ
for every ϕ ∈ L 2 (R 3 ) and all f ∈ C ∞ (R) with f (0) = 1 and f (T ) = 0. As ε → 0 we obtain
T 0 {A(t)∂ t f (t) + ∂ t A(t)h(t)} dt = −A 0 in L 2 (R 3 ), which implies A| t=0 = A 0 . Now we have T 0 ∂ t A ε ∂ t f (t) + {∆A ε − PJ(u ε ,Ã ε )}f (t), η = A 1 , η ,
and as ε → 0, we find
T 0 {∂ t A(t)∂ t f (t) + ∂ 2 t A(t)f (t)} = A 1 in H −1 (R 3 )
, which gives us ∂ t A| t=0 = A 1 . Applying the same argument to u ε we deduce that u| t=0 = u 0 .
Quantum Magnetohydrodynamics
Our last Section is devoted to point out the relation between the nonlinear Maxwell-Schrödinger system (1) and quantum magnetohydrodynamic (QMHD) models. Such hydrodynamic systems have been introduced in the physics literature, motivated by various applications to semiconductor devices, dense astrophysical plasmas (e.g. in white dwarfs), or laser plasmas [16,17,34,35]. As a simplification, let us consider a one-spiecies charged quantum fluid with self-generated electromagnetic fields. The dynamics is described by the following system
(39) ∂ t ρ + div J = 0 ∂ t J + div J ⊗ J ρ + ∇P (ρ) = ρE + J ∧ B + 1 2 ρ∇ ∆ √ ρ √ ρ ,
where ρ denotes the charge density and J the current density of the quantum fluid.
Here all the constants are normalized to one. The pressure term P (ρ) is assumed to be isentropic of the form P (ρ) = γ−1 γ ρ γ , 1 < γ < 3. The last term in the equation for the current density can be written in different ways
(40) 1 2 ρ∇ ∆ √ ρ √ ρ = 1 4 ∇∆ρ − div(∇ √ ρ ⊗ ∇ √ ρ) = 1 4 div(ρ∇ 2 log ρ).
and it can be seen as a self-consistent quantum potential (the so called Bohm potential) or as a quantum correction to the stress tensor. Mathematically speaking, this is a third order nonlinear dispersive term. The hydrodynamical system above is complemented by the Maxwell equations for the electromagnetic fields E and B
(41) div E = ρ, ∇ ∧ E = −∂ t B div B = 0, ∇ ∧ B = J + ∂ t E.
In recent years a global existence theory of finite energy weak solutions for a class a quantum hydrodynamic systems has been established by the first and third author of this paper in [1,2,3]. By means of a polar factorization techinque it is possible to define the hydrodynamic quantities by considering the Madelung transform of a wave function solution to a nonlinear Schrödinger equation. In this way the definition of the velocity field in the nodal regions is no longer needed. We also mention in the H 2 case the construction given in [9]. Furthermore it could be interesting to consider also confining potentials as in [5], generated by external magnetic fields. The aim of this Section is to show the existence of a finite energy weak solution to (39)-(41) by taking advantage of our results on the system (1).
Definition 5.1. Let ρ 0 , J 0 , E 0 , B 0 ∈ L 1 loc (R 3 )
, then a finite energy weak solution to system (39)- (41)
in the space-time slab [0, T ) × R 3 is given by a quadruple ( √ ρ, Λ, φ, A) such that (1) √ ρ ∈ L ∞ ([0, T ); H 1 (R 3 )), Λ ∈ L ∞ ([0, T ); L 2 (R 3 )), φ ∈ L ∞ ([0, T ); H 1 (R 3 )), A ∈ L ∞ ([0, T ); H 1 (R 3 )) ∩ W 1,∞ ([0, T ); L 2 (R 3 )); (2) ρ := ( √ ρ) 2 , J := √ ρΛ, E := −∂ t A − ∇φ, B := ∇ ∧ A; (3) J ∈ L 2 ([0, T ); L 2 loc (R 3 )); (4) ∀ η ∈ C ∞ c ([0, T ) × R 3 ), T 0 R 3 ρ∂ t η + J · ∇η dxdt + R 3 ρ 0 (x)η(0, x) dx = 0; (5) ∀ζ ∈ C ∞ c ([0, T ) × R 3 ; R 3 ), T 0 R 3 J · ∂ t ζ + Λ ⊗ Λ : ∇ζ + P (ρ) div ζ + ρE · ζ + (J ∧ B) · ζ + ∇ √ ρ ⊗ ∇ √ ρ : ∇ζ + 1 4 ρ∆ div ζ dxdt + R 3 J 0 (x) · ζ(0, X) dx = 0; (6) E, B satisfy (41) in [0, T ) × R 3
in the sense of distributions; (7) (finite energy) the total mass and energy defined by
(42) M (t) := R 3 ρ(t, x) dx,(43)E(t) = R 3 1 2 |∇ √ ρ| 2 + 1 2 |Λ| 2 + f (ρ) + 1 2 |∂ t A| 2 + 1 2 |∇A| 2 + 1 2 |∇φ| 2 dx
respectively, are finite for every t ∈ [0, T ). Here f (ρ) = 1 γ ρ γ .
Proposition 5.2. Let (ρ 0 , J 0 , B 0 , E 0 ) be such that ρ 0 := |u 0 | 2 , J 0 := Re(ū 0 (−i∇ − A 0 )u 0 ), B 0 := ∇∧A 0 , E 0 := −A 1 −∇φ 0 , φ 0 := (−∆) −1 |u 0 | 2 for some (u 0 , A 0 , A 1 ) ∈ X, then there exists T max > 0 such that ( √ ρ, Λ, φ, A) is a finite energy weak solution to (39)- (41) with initial data (ρ 0 , J 0 , B 0 , E 0 ) in the space-time slab [0, T max ) × R 3 . Moreover, the energy is conserved for all t ∈ [0, T max ).
To prove this Proposition we are going to use a polar factorization argument, in analogy with the electrostatic case treated in [1,2]. Given any complex valued fuction u ∈ H 1 (R 3 ), we may define the set of its polar factors as
P (u) := {ϕ ∈ L ∞ (R 3 ) : ϕ L ∞ ≤ 1, u = √ ρϕ a.e. in R 3 },
where √ ρ := |u|. Thus, for any ϕ ∈ P (u), we have |ϕ| = 1 √ ρ dx a.e. in R 3 and ϕ is uniquely defined √ ρ dx a.e. in R 3 . Clearly the polar factor is not uniquely defined in the nodal regions, i.e. in the set {ρ = 0}. In the following Lemma we exploit the polar factorization of a given wave function ψ in order to define the hydrodynamical quantities associated to ψ. This approach overcomes the WKB ansatz in the finite energy framework and allows to define the hydrodynamical quantities almost everywhere in the space, without passing through the construction of the velocity field, which is not uniquely defined in the nodal region. Furthermore, we show how this definition which uses the polar factorization is stable in H 1 (R 3 ).
Lemma 5.3. Let u ∈ H 1 (R 3 ), A ∈ L 3 (R 3 )
, and let √ ρ := |u|, ϕ ∈ P (u). Let us define
Λ := Re(φ(−i∇ − A)u) ∈ L 2 (R 3 ), then we have • √ ρ ∈ H 1 (R 3 )
and ∇ √ ρ = Re(φ∇u); • the following identity holds a.e. in R 3 ,
(44) Re{(−i∇ − A)u ⊗ (−i∇ − A)u} = ∇ √ ρ ⊗ ∇ √ ρ + Λ ⊗ Λ. Moreover, let {u n } ⊂ H 1 (R 3 ), {A n } ⊂ L 3 (R 3 )
be such that u n converges strongly to u in H 1 and A n converges strongly to A in L 3 , then we have
∇ √ ρ n → ∇ √ ρ, Λ n → Λ, in L 2 (R 3 ),
where √ ρ n := |u n |, Λ n := Re(φ n (−i∇ − A n )u n ).
Proof. Let u ∈ H 1 (R 3 ) and let us consider a sequence of smooth functions converging to u,
{u n } ⊂ C ∞ c (R 3 ), u n → u in H 1 (R 3 ).
For each u n we may define
ϕ n (x) := u n (x) |u n (x)| if u n (x) = 0 0 if u n (x) = 0.
The ϕ n 's are clearly polar factors for the wave functions u n . Since ϕ n L ∞ ≤ 1, then (up to subsequences) there exists ϕ ∈ L ∞ (R 3 ) such that
(45) ϕ n * ⇀ ϕ, L ∞ (R d ).
It is easy to check that ϕ is indeed a polar factor for u. Since {u n } ⊂ C ∞ c (R 3 ), we have ∇ √ ρ n = Re(φ n ∇u n ), a.e. in R 3 .
It follows from the convergence above
∇ √ ρ n → ∇ √ ρ, L 2 (R 3 ) Re(φ n ∇u n ) ⇀ Re(φ∇u), L 2 (R 3 ),
thus ∇ √ ρ = Re(φ∇u) in L 2 (R 3 ) and consequently the equality holds a.e. in R 3 . It should be noted that here we have ∇ √ ρ = Re(φ∇u), where ϕ is the weak− * limit in (45). However the identity above for ∇ √ ρ does not depend on the choice of ϕ. Indeed, by Theorem 6.19 in [23] we have ∇u = 0 for almost every x ∈ u −1 ({0}) and, on the other hand, ϕ is uniquely determined on {x ∈ R 3 : |u(x)| > 0} almost everywhere. Consequently, for any ϕ 1 , ϕ 2 ∈ P (u), we have Re(φ 1 ∇u) = Re(φ 2 ∇u) = ∇ √ ρ. The same argument applies for Λ := Re(φ(−∇ − A)u). Let us now prove the identity (44). Recall that we have |ϕ| = 1 √ ρ dx a.e. in R 3 , hence again by invoking Theorem 6.19 in [23] we have
Re{(−i∇ − A)u ⊗ (−i∇ − A)u} = Re ϕ(−i∇ − A)u ⊗ (φ(−i∇ − A)u) = Re{ϕ(−i∇ − A)u} ⊗ Re{φ(−i∇ − A)u} − Im{ϕ(−i∇ − A)u} ⊗ Im{φ(−i∇ − A)u} =Λ ⊗ Λ + ∇ √ ρ ⊗ ∇ √ ρ,
a.e. in R 3 . Furthermore, by taking the trace on both sides of the above equality we furthermore obtain
(46) |(−i∇ − A)u| 2 = |∇ √ ρ| 2 + |Λ| 2 .
For the second part of the Lemma, let us consider a sequence {u n } ⊂ H 1 strongly converging to u ∈ H 1 and vector fields {A n } ⊂ L 3 strongly converging to A ∈ L 3 . As before it is straightforward to show that
Re(φ n ∇u n ) ⇀ Re(φ∇u), L 2 Re(φ n (−i∇ − A n )u n ) ⇀ Re(φ(−i∇ − A)u), L 2 .
Moreover, from (46), the strong convergence of u n and the weak convergence for ∇ √ ρ n , Λ n , we obtain
(−i∇ − A)u 2 L 2 = ∇ √ ρ 2 L 2 + Λ 2 L 2 ≤ lim inf n→∞ ∇ √ ρ n 2 L 2 + Λ n 2 L 2 = lim n→∞ (−i∇ − A n )u n 2 L 2 = (−i∇ − A)u 2 L 2 .
Hence, we obtain ∇ √ ρ n L 2 → ∇ √ ρ L 2 and Λ n L 2 → Λ L 2 . Consequently, from the weak convergence in L 2 and the convergence of the L 2 norms we may infer the strong convergence ∇ √ ρ n → ∇ √ ρ, Λ n → Λ, in L 2 (R 3 ).
In view of Lemma 5.3 we can now prove Proposition 5.2. Let (u 0 , A 0 , A 1 ) ∈ X be given, then by our main Theorem 1.1 there exists a unique solution (u, A) to (1)
in [0, T max ) × R 3 such that u ∈ C([0, T max ); H 2 (R 3 )), A ∈ C([0, T max ); H 3/2 (R 3 )) ∩ C 1 ([0, T max ); H 1/2 (R 3 )
). Let us now define √ ρ := |u|, Λ := Re(φ(−i∇ + A)u), where ϕ is a polar factor for u, and let φ := (−∆) −1 ρ. By differentiating ρ with respect to time we have
∂ t ρ =2 Re ū − i 2 (−i∇ − A) 2 u − iφu − i|u| 2(γ−1) u = Im ū(−i∇ − A) 2 u = Im −i div ū(−i∇ − A)u + (−i∇ − A)u · (−i∇ − A)u = − div (Re(ū(−i∇ − A)u)) .
Hence by defining J = Re (ū(−i∇ − A)u) = √ ρΛ we obtain the continuity equation for
ρ ∂ t ρ + div J = 0.
Now let us differentiate J with respect to time,
∂ t J = Re i 2 (−i∇ − A) 2 u + iφū + i|u| 2(γ−1)ū (−i∇ − A)u + Re ū(−i∇ − A) − i 2 (−i∇ − A) 2 u − iφu − i|u| 2(γ−1) u − ρ∂ t A = 1 2 Im ū(−i∇ − A) (−i∇ − A) 2 u − (−i∇ − A) 2 u(−i∇ − A)u + Re ū(φ + |u| 2(γ−1) )∇u −ū∇ φu + |u| 2(γ−1) u − ρ∂ t A.
Now the last line equals ρ∇φ − ρ∇ρ γ−1 − ρ∂ t A = ρ(−∂ t A − ∇φ) + ∇P (ρ), where P (ρ) = γ−1 γ ρ γ . After some tedious but rather straightforward calculations we may see that
1 2 Im ū(−i∇ − A) (−i∇ − A) 2 u − (−i∇ − A) 2 u(−i∇ − A)u = 1 4 ∇∆ρ − div Re (−i∇ − A)u ⊗ (−i∇ − A)u + J ∧ (∇ ∧ A).
By putting everything together we then obtain
∂ t J + div Re{(−i∇ − A)u ⊗ (−i∇ − A)u} + ∇P (ρ) = ρ(−∂ t A − ∇φ) + J ∧ (∇ ∧ A) + 1 4 ∇∆ρ.
We now use the polar factorization Lemma to infer that
Re{(−i∇ − A)u ⊗ (−i∇ − A)u} = ∇ √ ρ ⊗ ∇ √ ρ + Λ ⊗ Λ
and consequently we get
∂ t J + div(Λ ⊗ Λ) + ∇P (ρ) = ρE + J ∧ B + 1 4 ∇∆ρ − div(∇ √ ρ ⊗ ∇ √ ρ).
By recalling identity (40) we see that this is the equation for the current density in the QMHD system (39). The above calculations are rigorous only when (u, A) are sufficiently regular, however for solutions to (1)
Appendix -Continuous dependence
In this Appendix we are going to prove the Lemmas 3.3, 3.4, 3.5 used to show the continuous dependence stated in Proposition 3.2. We consider two initial data
(u 0 , A 0 , A 1 ), (u ′ 0 , A ′ 0 , A ′ 1 ) ∈ X such that (u 0 , A 0 , A 1 ) X , (u ′ 0 , A ′ 0 , A ′ 1 ) X ≤ R,
and whose energies, defined as in (26), satisfy
E(t), E ′ (t) ≤ E.
All throughout this Appendix we are going to denote by (u, A), (u ′ , A ′ ) the solutions to (1) emanated from (u 0 , A 0 , A 1 ), (u ′ 0 , A ′ 0 , A ′ 1 ) ∈ X, respectively. First of all we are going to prove Lemma 3.3; we will split it into two steps, see the two Lemmas 6.1 and 6.2 below.
Lemma 6.1. We have u − u ′ L ∞ t H 2 x (R 3 ) R,E ∂ t (u − u ′ ) L ∞ t L 2 x (R 3 ) + u − u ′ L ∞ t L 2 x (R 3 ) + A − A ′ L ∞ t H 1 2 x (R 3 ) .(47)
Proof. Let us consider the equation for the difference u − u ′ ; we have
i∂ t (u − u ′ ) = −∆(u − u ′ ) + 2iA · ∇(u − u ′ ) + |A| 2 (u − u ′ ) + F where F = 2i(A − A ′ ) · ∇u ′ + (|A| 2 − |A ′ | 2 )u ′ + (φ(|u| 2 ) − φ(|u ′ | 2 ))u ′ + φ(|u| 2 )(u − u ′ ) + |u| 2(γ−1) u − |u ′ | 2(γ−1) u ′ .
This implies
∆(u − u ′ ) L 2 (R 3 ) ≤ ∂ t (u − u ′ ) L 2 (R 3 ) + A · ∇(u − u ′ ) L 2 (R 3 ) + |A| 2 (u − u ′ ) L 2 (R 3 ) + F L 2 (R 3 )
From Hölder's inequality and Sobolev embedding theorem we have
A · ∇(u − u ′ ) L 2 (R 3 ) ≤ A L 6 (R 3 ) ∇(u − u ′ ) L 3 (R 3 ) ∇A L 2 (R 3 ) u − u ′ H 3 2 E u − u ′ H 3 2 (R 3 ) E u − u ′ 1 4 L 2 (R 3 ) u − u ′ 3 4 H 2 (R 3 ) E C(ε) u − u ′ L 2 (R 3 ) + ε u − u ′ H 2 (R 3 )
, where we do not consider the explicit dependence of the constants on R and E. Similarly we have
|A| 2 (u − u ′ ) L 2 (R 3 ) A 2 L 6 u − u ′ H 1 (R 3 ) E C(ε) u − u ′ L 2 (R 3 ) + ε u − u ′ H 2 (R 3 )
We can deal with F as already done previously, getting
F L 2 (R 3 ) R,E A − A ′ H 1 2 (R 3 ) + u − u ′ L 2 (R 3 )
Finally, putting all togheter the previous inequality, we have
u − u ′ L ∞ t H 2 x (R 3 ) ≤ u − u ′ L ∞ t L 2 x (R 3 ) + ∆(u − u ′ ) L ∞ t L 2 x (R 3 ) R,E C(ε) u − u ′ L ∞ t L 2 x (R 3 ) + ∂ t (u − u ′ ) L ∞ t L 2 x (R 3 ) + A − A ′ L ∞ t H 1 2 x (R 3 ) + ε u − u ′ L ∞ t H 2 x (R 3 )
. Now, by choosing ε sufficently small, we get (47).
We note that in the same way we can prove that
(48) ∂ t (u − u ′ ) L ∞ t L 2 x (R 3 ) R,E u − u ′ L ∞ t H 2 x (R 3 ) + A − A ′ L ∞ t H 1 2 x (R 3 )
In order to estimate the term
∂ t (u − u ′ ) L ∞ t L 2 x (R 3 )
we use next lemma. Lemma 6.2. The following inequality holds:
∂ t u − ∂ t u ′ L ∞ t L 2 x (R 3 ) R,E ∂ t (u − u ′ )(0) L 2 (R 3 ) + T ∂ t u ′ L ∞ t H 2 x (R 3 ) A − A ′ L ∞ t H 1 2 x (R 3 ) + u − u ′ L ∞ t L 2 x (R 3 ) + T (u − u ′ , A − A ′ , ∂ t A − ∂ t A ′ ) X(49)
Proof. We start by differentiating in time the equation
i∂ t u = −∆ A u + φ(u)u + |u| 2(γ−1) u.
We then get
i∂ 2 t u = −∆ A ∂ t u + φ(u)∂ t u + (2i∂ t A(∇ − iA) + ∂ t φ)u + ∂ t (|u| 2(γ−1) u)
Writing the corresponding equation for ∂ 2 t u ′ and taking the difference with the previous one we get
(50) i∂ 2 t (u − u ′ ) = −∆ A (∂ t u − ∂ t u ′ ) + F , where F is given by F = 2i(A − A ′ ) ∇ − i 2 (A + A ′ ) + (φ − φ ′ ) ∂ t u ′ + φ(∂ t u − ∂ t u ′ ) + (2i∂ t A(∇ − iA) + ∂ t φ)(u − u ′ ) + ∂ t (|u| 2(γ−1) u − |u ′ | 2(γ−1) u ′ ) + (2i∂ t (A − A ′ )(∇ − iA) − 2i(A − A ′ )∂ t A ′ + ∂ t (φ − φ ′ ))u ′ .(51)
Using the unitarity in L 2 (R 3 ) of U A (t, s) we get
(52) ∂ t (u − u ′ )(t) L 2 (R 3 ) ≤ ∂ t (u − u ′ )(0) L 2 + t 0 F (s) L 2 (R 3 ) ds .
We estimate the inhomogenous term F , we have
2i(A − A ′ )(∇ − i 2 (A + A ′ )) + (φ − φ ′ ) ∂ t u ′ L 2 (R 3 ) R,E u − u ′ L 2 (R 3 ) + A − A ′ H 1 2 (R 3 ) ∂ t u ′ H 2 (R 3 )
This inequality follows from
(A − A ′ ) ∇ − i 2 (A + A ′ ) ∂ t u ′ L 2 (R 3 ) ≤ A − A ′ L 3 (R 3 ) ∇ − i 2 (A + A ′ ) ∂ t u ′ L 6 (R 3 ) A − A ′ H 1 2 (R 3 ) ∇∂ t u ′ H 1 (R 3 ) + A + A ′ L 6 (R 3 ) ∂ t u L ∞ (R 3 ) A − A ′ H 1 2 (R 3 ) ∂ t u H 2 (R 3 ) 1 + ∇A L 2 (R 3 ) + ∇A ′ L 2 (R 3 ) E A − A ′ H 1 2 (R 3 ) ∂ t u ′ H 2 (R 3 ) and (φ − φ ′ )∂ t u ′ L 2 (R 3 ) ≤ ∆ −1 ((u − u ′ )u + (u − u ′ )u ′ )∂ t u ′ L 2 (R 3 ) u − u ′ L 2 (R 3 ) u L 3 (R 3 ) ∂ t u L 3 (R 3 ) + (u − u ′ ) L 2 (R 3 ) u ′ L 3 (R 3 ) ∂ t u L 3 (R 3 ) R u − u ′ L 2 (R 3 ) ∂ t u ′ H 2 (R 3 )
where we used Hölder inequality, the Sobolev embeddings (5). Furthermore, from (6) we may infer
H 1 (R 3 ) ֒→ L 6 (R 3 ), H 1 2 (R 3 ) ֒→ L 3 (R 3 ) andφ(∂ t u − ∂ t u ′ ) L 2 (R 3 ) R ∂ t u − ∂ t u ′ L 2 (R 3 ) . Again, 2i∂ t A(∇ − iA)(u − u ′ ) L 2 (R 3 ) ∂ t A L 3 (R 3 ) (∇ − iA)(u − u ′ ) L 6 R,E u − u ′ H 2 (R 3 )
and, by using (8) and (5),
∂ t φ(u − u ′ ) L 2 (R 3 ) (∆ −1 (2 Re(u∂ t u)))(u − u ′ ) L 2 (R 3 ) ∂ t u L 2 (R 3 ) u L 3 (R 3 ) u − u ′ H 2 (R 3 ) R u − u ′ H 2 (R 3 ) . Observe that one has ∂ t (|u| 2(γ−1) u) = γ|u| 2(γ−1) ∂ t u + (γ − 1)|u| 2(γ−2) u 2 ∂ t u, therefore it follows ∂ t (|u| 2(γ−1) u − |u ′ | 2(γ−1) u ′ ) = γ∂ t u(|u| 2(γ−1) − |u ′ | 2(γ−1) ) + γ|u ′ | 2(γ−1) ∂ t (u − u ′ ) + (γ − 1)∂ t u(|u| 2(γ−2) u 2 − |u ′ | 2(γ−2) u ′2 ) + (γ − 1)|u ′ | 2(γ−2) u 2 ∂ t (u − u ′ )
We then have
∂ t (|u| 2 u − |u ′ | 2 u ′ ) L 2 (R 3 ) R ∂ t (u − u ′ ) L 2 (R 3 ) + u − u ′ H 2 (R 3 ) ,
where we used the following two inequalities
|z| 2(γ−1) − |z ′ | 2(γ−1) |z| 2γ−3 + |z ′ | 2γ−3 |z − z ′ | |z| 2(γ−2) z 2 − |z ′ | 2(γ−2) z ′2 |z| 2γ−3 + |z ′ | 2γ−3 |z − z ′ | .
For the last term, with similar computations, we have
∂ t (A − A ′ )(∇ − iA)u ′ L 2 (R 3 ) R,E ∂ t (A − A ′ ) H 1 2 (R 3 ) ∂ t A ′ (A − A ′ )u ′ L 2 (R 3 ) ∂ t A ′ L 3 (R 3 ) A − A ′ L 6 (R 3 ) u ′ L ∞ (R 3 ) R A − A ′ H 3 2 (R 3 ) (∂ t φ − ∂ t φ ′ )u ′ L 2 (R 3 ) R ∂ t u − ∂ t u ′ L 2 (R 3 ) + u − u ′ H 2 (R 3 )
. By putting everything together, we obtain
∂ t u − ∂ t u ′ L ∞ t L 2 x (R 3 ) R,E ∂ t (u − u ′ )(0) L 2 (R 3 ) + T ∂ t u − ∂ t u ′ L ∞ t L 2 x (R 3 ) + T ∂ t u ′ L ∞ t H 2 x (R 3 ) A − A ′ L ∞ t H 1 2 x (R 3 ) + u − u ′ L ∞ t L 2 x (R 3 ) + T u − u ′ L ∞ t H 2 x (R 3 ) + A − A ′ L ∞ t H 3 2 x (R 3 ) + ∂ t (A − A ′ ) L ∞ t H 1 2 x (R 3 )
, which gives (49), by using (48) for the term
∂ t (u − u ′ ) L ∞ t L 2 x (R 3 )
in the righthand side of the previous inequality.
By putting together the two previous Lemmas we then have Lemma 3.3. Now we are going to estimate the term A − A ′
L ∞ t H 1 2 x (R 3 ) + u − u ′ L ∞ t L 2 x (R 3 ) .
Lemma 6.3. Let (u, A), (u ′ , A) be as in previous lemmas, then
A − A ′ L ∞ t H 1 2 (R 3 ) + u − u ′ L ∞ t L 2 (R 3 ) R,E (u 0 − u ′ 0 , A 0 − A ′ 0 , A 1 − A ′ 1 ) L 2 (R 3 )×H 1 2 (R 3 )×H − 1 2 (R 3 )(53)
Proof. Writing the difference equation for A and A ′ we get
(A − A ′ ) = G , with G = P Im{(u − u ′ )(∇ − iA)u − iuu ′ (A − A ′ ) − (u − u ′ )(∇ + iA ′ )u ′ }
where we used the fact that P(u ′ ∇(u − u ′ )) = −P((u − u ′ )∇u ′ ). By applying the energy estimate (13) we get
A − A ′ L ∞ t H 1 2 x (R 3 ) (1 + T ) (A 0 − A ′ 0 , A 1 − A ′ 1 ) H 1 2 (R 3 )×H − 1 2 (R 3 ) + (1 + T ) G L 1 t H − 1 2 x (R 3 )
Using the embedding L
2 (R 3 ) ֒→ H − 1 2 (R 3 ) we have (u − u ′ )(∇ − iA)u L 3 2 (R 3 ) ≤ u − u ′ L 2 (R 3 ) (∇ − iA)u L 6 (R 3 ) u − u ′ L 2 (R 3 ) ∇u H 1 (R 3 ) + Au L 6 (R 3 ) u − u ′ L 2 (R 3 ) u H 2 (R 3 ) 1 + ∇A L 2 (R 3 ) R,E u − u ′ L 2 (R 3 ) . Analogously (u − u ′ )(∇ + iA)u ′ L 3 2 (R 3 ) R,E u − u ′ L 2 (R 3 ) and uu ′ (A − A ′ ) L 3 2 (R 3 ) R A − A ′ H 1 2 (R 3 )
In a similar way, using the difference of the equations for u and u ′ we get
u − u ′ L ∞ t L 2 (R 3 ) R,E u 0 − u ′ 0 L 2 (R 3 ) + T A − A ′ L ∞ t H 1 2 (R 3 ) + u − u ′ L ∞ t L 2 (R 3 )
Putting all togheter, taking T sufficiently small, we get (53). Now, using (53) in (49), we get
∂ t u − ∂ t u ′ L ∞ t L 2 (R 3 ) ∂ t (u − u ′ )(0) L 2 (R 3 ) + T ∂ t u ′ L ∞ t H 2 (R 3 ) (u 0 − u ′ 0 , A 0 − A ′ 0 , A 1 − A ′ 1 ) X 0, 1 2 + T (u − u ′ , A − A ′ , ∂ t A − ∂ t A ′ ) X(54)
On the other hand, by analogous arguments, we have the following estimate for the Maxwell part
A − A ′ L ∞ t H 3 2 (R 3 ) + ∂ t A − ∂ t A ′ L ∞ t H 1 2 (R 3 ) (A 0 − A ′ 0 , A 1 − A ′ 1 ) H 3 2 (R 3 )×H 1 2 (R 3 ) + T (u − u ′ , A − A ′ , ∂ t A − ∂ t A ′ ) X(55)
In order to get the estimate for (u − u ′ , A − A ′ , ∂ t A − ∂ t A ′ ) X we put togheter (47), choosing a sufficiently small T , (54) and (55) to get
(u − u ′ , A − A ′ , ∂ t A − ∂ t A ′ ) X (u 0 − u ′ 0 , A 0 − A ′ 0 , A 1 − A ′ 1 ) X + ( ∂ t u ′ L ∞ t H 2 (R 3 ) + 1) (u 0 − u ′ 0 , A 0 − A ′ 0 , A 1 − A ′ 1 ) X 0, 1 2(56)
where we applied (48) to the term ∂ t (u − u ′ )(0) L 2 (R 3 ) .
Finally we are going to estimate the term ∂ t u ′ L ∞ t H 2 (R 3 ) . Lemma 6.4. The following estimate holds:
(57) ∂ t u L ∞ t H 2 (R 3 ) ≤ ∂ 2 t u L 2 (R 3 ) + C(E, R)
Proof. From the equation
i∂ 2 tt u = −∆∂ t u + 2iA · ∇∂ t u + |A| 2 ∂ t u + 2i∂ t A · ∇u + 2A · ∂ t Au + ∂ t φu + φ∂ t u + ∂ t (|u| 2(γ−1) u)
we can estimate ∂ t u H 2 (R 3 ) . Indeed
∂ t u H 2 (R 3 ) ≤ ∂ t u L 2 (R 3 ) + ∆∂ t u L 2 (R 3 ) ≤ C(R) + ∆∂ t u L 2 (R 3 )
So we have ∆∂ t u L 2 (R 3 ) ≤ ∂ 2 tt u L 2 (R 3 ) + A · ∇∂ t u L 2 (R 3 ) + |A| 2 ∂ t u L 2 (R 3 ) + ∂ t A · ∇u L 2 (R 3 ) + A · ∂ t Au L 2 (R 3 ) + ∂ t φu + φ∂ t u L 2 (R 3 ) + ∂ t (|u| 2(γ−1) u) L 2 (R 3 )
We begin with the estimate of the right-hand side of the previous inequality.
A · ∇∂ t u L 2 (R 3 ) A L 6 (R 3 ) ∇∂ t u L 3 (R 3 ) ∇A L 2 (R 3 ) ∂ t u H 3 2 (R 3 ) √ E ∂ t u 1 4 L 2 (R 3 ) ∂ t u 3 4 H 2 (R 3 ) √ E(C(ε) ∂ t u L 2 (R 3 ) + ε ∂ t u H 2 (R 3 ) ) C(E, R) + C(R)ε ∂ t u H 2 (R 3 )
In the same way
|A| 2 ∂ t u L 2 (R 3 ) A 2 L 6 (R 3 ) ∂ t u L 6 (R 3 ) ∇A 2 L 2 (R 3 ) ∂ t u H 1 (R 3 ) (1 + ε ∂ t u H 2 (R 3 ) )
The other terms are all bounded by C(R); for instance
∂ t A∇u H 2 (R 3 ) ∂ t A H 1 2 (R 3 ) u H 2 (R 3 ) ≤ C(R) or ∂ t (|u| 2(γ−1) u) L 2 (R 3 ) |u| 2(γ−1) ∂ t u L 2 (R 3 ) u 2(γ−1) L ∞ ∂ t u L 2 (R 3 ) ≤ C(R)
We can deal with the remaining terms analogously. Finally we get
∂ t u H 2 (R 3 ) ∂ 2 tt u L 2 (R 3 ) + C(E, R) + C(R)ε ∂ t u H 2 (R 3 )
which gives (57) for sufficiently small ε.
To complete the estimates we have to deal with ∂ 2 t u L ∞ t L 2 (R 3 ) . We write the equation for the time derivative ∂ 2 t u i∂ 3 t u = −∆u + 2iA · ∇∂ tt u + |A| 2 ∂ tt u + G where G = 4i∂ t A · ∇∂ t u + 4A · ∂ t A∂ t u + 2i∂ 2 t A(∇u − iAu) + 2(∂ t A) 2 u + 2∂ t φ∂ t u + ∂ 2 t φu + ∂ 2 t uφ + ∂ 2 t (|u| 2(γ−1) u) Using Duhamel's representation in Lemma (2.8) we have
∂ tt u L ∞ t L 2 (R 3 ) ∂ tt u(0) L 2 (R 3 ) + T G L ∞ t L 2 (R 3 )
Proceeding as before we finally get
∂ 2 t u L 2 (R 3 ) ∂ 2 t u(0) L 2 (R 3 ) + T C(R, E) ∂ t u L ∞ t H 2 (R 3 ) + A L ∞ t H 5 2 (R 3 ) + ∂ t A L ∞ t H 3 2 (R 3 )
We estimate the right-hand side of the previous inequality. We have
A L ∞ t H 5 2 (R 3 ) + ∂ t A L ∞ t H 3 2 (R 3 ) (1 + T ) (A 0 , A 1 ) H 5 2 (R 3 )×H 3 2 (R 3 ) + T (1 + T ) J L ∞ t H 3 2 (R 3 )
For the term with J, proceeding as in (9), we have u∇u H
3 2
3(R 3 ) ∩ C 1 ([0, T max ); H 1 2 (R 3 )), div A = 0
Lemma 2. 5 (
5Strichartz estimates for the wave equation). Let I be a time interval, and let B :
Proposition 3. 2 (
2Continuous dependence on the initial data). Let 0 < T < T max , then the mapping(u 0 , A 0 , A 1 ) → (u, A, ∂ t A), where (u,A)is the solution to (1) is continuous as a mapping from X to C([0, T ]; X).
Lemma 3. 3 .
3Let (u, A), (u ′ , A ′ ) be solutions to (1) defined as above, then we have
considered in Theorem 1.1 they can be rigorsouly justified in the weak sense, namely in the sense of Definition 5.1 by regularising the initial data and by exploiting the continuous dependence showed in Proposition 3.2 and the H 1 −stability of the polar factorization stated in Lemma 5.3. It only remains to prove that E, B satisfy the Maxwell equations, but this comes in a straightforward way from the wave equation in (1) and the definitions E = −∂ t A − ∇φ, B = ∇ ∧ A. Finally we remark that for solutions (u, A) to (1) considered in Theorem 1.1 the total energy (26) is conserved. Again by using Lemma 5.3 we see that the energy in (26) equals the one defined in (43) this equals the energy defined in (43). This concludes the proof of Proposition 5.2.
3 2
3(R 3 ) C(R) u H 4 (R 3 )
t L 2 x T 3/4 u 2 L ∞ t,x A 1 L ∞ t H 1 x + A 2 L ∞ t H 1 x A 1 − A 2 L 4 t,xBy using (5), the third term is estimated byF 3 L 1 t L 2 x T u 1 L ∞ t H 1 x + u 2 L ∞ t H 1 x u 2 L ∞ t H 1 x u 1 − u 2 L ∞ t L 2 x .
On the finite energy weak solutions to a system in Quantum Fluid Dynamics. P Antonelli, P Marcati, Comm. Math. Phys. 287P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in Quantum Fluid Dynamics, Comm. Math. Phys. 287 (2009), no 2, 657-686.
The Quantum Hydrodynamics system in two space dimensions. P Antonelli, P Marcati, Arch. Rat. Mech. Anal. 203P. Antonelli and P. Marcati, The Quantum Hydrodynamics system in two space dimensions, Arch. Rat. Mech. Anal. 203 (2012), 499-527.
Some results on systems for quantum fluids, accepted Cont. P Antonelli, P Marcati, MathP. Antonelli, P. Marcati, Some results on systems for quantum fluids, accepted Cont. Math.
Derivation of the magnetohydrodynamic system from Navier-Stokes-Maxwell systems. D Arsénio, S Ibrahim, N Masmoudi, Arch. Rat. Mech. Anal. 2163D. Arsénio, S. Ibrahim, N. Masmoudi, Derivation of the magnetohydrodynamic system from Navier-Stokes-Maxwell systems, Arch. Rat. Mech. Anal. 216, no. 3 (2015), 767-812.
Scattering for Nonlinear Schrodinger Equation Under Partial Harmonic Confinement Comm. P Antonelli, R Carles, J D Silva, Math.Phys. 3341P. Antonelli, R. Carles, J.D. Silva, Scattering for Nonlinear Schrodinger Equation Under Partial Harmonic Confinement Comm.Math.Phys. 334 (2015), no.1, 367-396.
Global well-posedness in the energy space for the Maxwell-Schrödinger system. I Bejenaru, D Tataru, Comm. Math. Phys. 2881I. Bejenaru, D. Tataru, Global well-posedness in the energy space for the Maxwell-Schrödinger system, Comm. Math. Phys. 288 (2009), no. 1, 145-198.
On a method for studying abstract evolution equations in the hyperbolic case. G Da Prato, M Iannelli, Comm. PDEs. 16G. Da Prato, M. Iannelli, On a method for studying abstract evolution equations in the hyperbolic case, Comm. PDEs 1 (1976), no. 6, 585-608.
The Schrödinger equation in a classical context: a seminar on superconductivity. R P Feynman, R B Leighton, M Sands, The Feynman lectures on physics. Reading, Mass; LondonAddison-Wesley Publishing Co., IncIII Quantum MechanicsR.P. Feynman, R.B. Leighton and M. Sands, The Schrödinger equation in a classical context: a seminar on superconductivity (Chapter 21) The Feynman lectures on physics,Vol. III Quantum Mechanics, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1995.
Quantum hydrodynamics, Wigner transforms and the classical limit. I Gasser, P A Markowich, Asymptot. Anal. 142I. Gasser, P.A. Markowich, Quantum hydrodynamics, Wigner transforms and the classical limit Asymptot. Anal. 14 (1997), no. 2, 97-116.
Generalized Strichartz inequalities for the wave equation. J Ginibre, G Velo, J. Funct. Anal. 1331J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal. 133, no. 1 (1995), 50-68.
Long range scattering and modified wave operators for the Maxwell-Schrödinger system I. The case of vanishing asymptotic magnetic field. J Ginibre, G Velo, Comm. Math. Phys. 2363J. Ginibre, G. Velo, Long range scattering and modified wave operators for the Maxwell- Schrödinger system I. The case of vanishing asymptotic magnetic field, Comm. Math. Phys, 236 (2003), no. 3, 395-448.
Long range scattering and modified wave operators for the Maxwell-Schrödinger system II. The general case. J Ginibre, G Velo, Ann. Henri Poincaré. 85J. Ginibre, G. Velo, Long range scattering and modified wave operators for the Maxwell- Schrödinger system II. The general case, Ann. Henri Poincaré 8 (2007). no. 5, 917-994.
Uniqueness at infinity in time for the Maxwell-Schrödinger system with arbitrarily large asymptotic data. J Ginibre, G Velo, Port. Math. 654J. Ginibre, G. Velo, Uniqueness at infinity in time for the Maxwell-Schrödinger system with arbitrarily large asymptotic data, Port. Math. 65 (2004), no. 4, 509-534.
Long range scattering for the Maxwell-Schrödinger system with arbitrarily large asymptotic data. J Ginibre, G Velo, Hokkaido Math. J. 374J. Ginibre, G. Velo, Long range scattering for the Maxwell-Schrödinger system with arbitrarily large asymptotic data, Hokkaido Math. J. 37 (2008), no. 4, 795-811.
Global finite-energy solutions to the Maxwell-Schrödinger system. Y Guo, K Nakamitsu, W Strauss, Comm. Math. Phys. 170Y. Guo, K. Nakamitsu, W. Strauss, Global finite-energy solutions to the Maxwell-Schrödinger system, Comm. Math. Phys. 170 (1995), 181-196.
A magnetohydrodynamic model for quantum plasmas. F Haas, Phys. Plasmas. 1262117F. Haas, A magnetohydrodynamic model for quantum plasmas, Phys. Plasmas 12 (2005), 062117.
Quantum plasmas: An hydrodynamic approach. F Haas, SpringerNew YorkF. Haas, Quantum plasmas: An hydrodynamic approach, New York: Springer.
Existence and uniqueness of the solution to the modified Schrödinger map. J Kato, Math. Res. Lett. 122-3J. Kato, Existence and uniqueness of the solution to the modified Schrödinger map, Math. Res. Lett 12 (2005), no. 2-3, 171-186.
Linear evolution equations of "hyperbolic" type. T Kato, J. Fac. Sci. Univ. Tokyo Sect. I. 17T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), 241-258.
Linear evolution equations of "hyperbolic" type, II. T Kato, J¿ Math. Soc. Japan. 25T. Kato, Linear evolution equations of "hyperbolic" type, II, J¿ Math. Soc. Japan 25 (1973), 648-666.
. M Keel, T Tao, Endpoint Strichartz Estimates. Amer. J. Math. 120M. Keel, T. Tao, Endpoint Strichartz Estimates. Amer. J. Math. 120 (1998), 955-980.
Exact smoothing properties of Schrödinger semigroups. M A Kon, A Gulisashvili, Amer. J. Math. 1886M.A. Kon, A. Gulisashvili, Exact smoothing properties of Schrödinger semigroups, Amer. J. Math. 188, no. 6 (1996), 1215-1248.
. E Lieb, M Loss, Graduate Studies in Mathematics. 14AMSAnalysisE. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, AMS, 2001.
Compacité par compensation. F Murat, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 54F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5, no. 3 (1978), 489-507.
Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. F Murat, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 84F. Murat, Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8, no. 1 (1981), 69-102.
The Cauchy problem for the coupled Maxwell-Schrödinger equations. K Nakamitsu, M Tsutsumi, J. Math. Phys. 27K. Nakamitsu, M. Tsutsumi, The Cauchy problem for the coupled Maxwell-Schrödinger equations, J. Math. Phys. 27 (1986), 211-216.
Local well-posedness for the Maxwell-Schrödinger equation. M Nakamura, T Wada, Math. Ann. 3323M. Nakamura, T. Wada, Local well-posedness for the Maxwell-Schrödinger equation, Math. Ann. 332 (2005), no. 3, 565-604.
Global existence and uniqueness of solutions to the Maxwell-Schrödinger equations. M Nakamura, T Wada, Comm. Math. Phys. 276M. Nakamura, T. Wada, Global existence and uniqueness of solutions to the Maxwell-Schrödinger equations, Comm. Math. Phys. 276 (2007), 315-339.
Semigroups of linear operators and applications to partial differential equations. A Pazy, Appl. Math. Sciences. 44Springer-VerlagA. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sciences, 44, Springer-Verlag, New York, 1983.
Existence of a unique local body solutions to the many-body Maxwell-Schrödinger initial value problem. K Petersen, arXiv:1402.3680v1preprint archived asK. Petersen, Existence of a unique local body solutions to the many-body Maxwell-Schrödinger initial value problem, preprint archived as arXiv:1402.3680v1.
Methods of modern mathematical physics II: Fourier analysis, selfadjointness. M Reed, B Simon, ElsevierM. Reed, B. Simon, Methods of modern mathematical physics II: Fourier analysis, self- adjointness, Elsevier.
L I Schiff, Quantum Mechanics. New-YorkMcGraw-Hill2nd edL.I. Schiff, Quantum Mechanics, 2nd ed. New-York: McGraw-Hill, 1955.
Modified wave operators for Maxwell-Schrödinger equations in three space dimensions. A Shimoura, Ann. Henri Poincaré. 4A. Shimoura, Modified wave operators for Maxwell-Schrödinger equations in three space dimen- sions, Ann. Henri Poincaré 4 (2003), 661-683.
Nonlinear aspects of quantum plasma physics. P K Shukla, B Eliasson, Phys. Usp. 53P.K. Shukla, B. Eliasson, Nonlinear aspects of quantum plasma physics, Phys. Usp. 53 (2010), 51-76.
Novel attractive force between ions in quantum plasmas. P K Shukla, B Eliasson, Phys. Rev. Lett. 108165007P.K. Shukla, B. Eliasson, Novel attractive force between ions in quantum plasmas, Phys. Rev. Lett. 108 (2012), 165007.
Nonlinear dispersive equations: local and global analysis, CBMS regional conference series in mathematics. T Tao, T. Tao, Nonlinear dispersive equations: local and global analysis, CBMS regional conference series in mathematics, 2006.
Compensated compactness and applications to partial differential equations. L Tartar, Nonlinear analysis and mechanics: Heriot-Watt Symposium. Boston, Mass.-LondonPitmanIVL. Tartar, Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV. Research Notes in Mathematics, vol. 39, Pitman, Boston, Mass.-London, 136-212, 1979.
An introduction to Navier-Stokes equation and oceanography Lecture Notes of the Unione Matematica Italiana. L Tartar, Springer-Verlag1Berlin; UMI, BolognaL.Tartar, An introduction to Navier-Stokes equation and oceanography Lecture Notes of the Unione Matematica Italiana, vol. 1, Springer-Verlag, Berlin; UMI, Bologna, 2006
Global existence and asymptotic behavior of solutions for the Maxwell-Schrödinger equations in three space dimensions. Y Tsutsumi, Comm. Math. Phys. 151Y. Tsutsumi, Global existence and asymptotic behavior of solutions for the Maxwell-Schrödinger equations in three space dimensions, Comm. Math. Phys. 151 (1993), 543-576.
Global existence of solutions to the Cauchy problem for coupled Maxwell-Schrödinger equations in two space dimensions. M Tsutsumi, K Nakamitsu, Physical mathematics and nonlinear partial differential equations. J.H. Lightbourne, S.M. RankinNew YorkMarcel DekkerM. Tsutsumi, K. Nakamitsu, Global existence of solutions to the Cauchy problem for coupled Maxwell-Schrödinger equations in two space dimensions, in: J.H. Lightbourne, S.M. Rankin (eds.) Physical mathematics and nonlinear partial differential equations, New York: Marcel Dekker (1985), 139-155.
Smoothing effects for Schrödinger equations with electro-magnetic potentials and applications to the Maxwell-Schrödinger equations. T Wada, J. Funct. Anal. 263T. Wada, Smoothing effects for Schrödinger equations with electro-magnetic potentials and ap- plications to the Maxwell-Schrödinger equations, J. Funct. Anal. 263 (2012), 1-24.
| [] |
[
"A constructive account of the Kan-Quillen model structure and of Kan's Ex ∞ functor",
"A constructive account of the Kan-Quillen model structure and of Kan's Ex ∞ functor"
] | [
"Simon Henry "
] | [] | [] | We give a fully constructive proof that there is a proper cartesian ωcombinatorial model structure on the category of simplicial sets, whose generating cofibrations and trivial cofibrations are the usual boundary inclusion and horn inclusion. The main difference with classical mathematics is that constructively not all monomorphisms are cofibrations (only those satisfying some decidability conditions) and not every object is cofibrant.The proof relies on three main ingredients: First, our construction of a weak model categories on simplicial sets, then the interplay with the semi-simplicial versions of this weak model structure and finally, the use of Kan Ex ∞ -functor, and more precisely of S.Moss' direct proof that the natural map X → Ex ∞ X is an anodyne morphism, which we show is constructive when X is cofibrant. | null | [
"https://arxiv.org/pdf/1905.06160v1.pdf"
] | 155,091,716 | 1905.06160 | 7df48bb22689a33779d317586b10e255e8b7a50d |
A constructive account of the Kan-Quillen model structure and of Kan's Ex ∞ functor
15 May 2019
Simon Henry
A constructive account of the Kan-Quillen model structure and of Kan's Ex ∞ functor
15 May 2019
We give a fully constructive proof that there is a proper cartesian ωcombinatorial model structure on the category of simplicial sets, whose generating cofibrations and trivial cofibrations are the usual boundary inclusion and horn inclusion. The main difference with classical mathematics is that constructively not all monomorphisms are cofibrations (only those satisfying some decidability conditions) and not every object is cofibrant.The proof relies on three main ingredients: First, our construction of a weak model categories on simplicial sets, then the interplay with the semi-simplicial versions of this weak model structure and finally, the use of Kan Ex ∞ -functor, and more precisely of S.Moss' direct proof that the natural map X → Ex ∞ X is an anodyne morphism, which we show is constructive when X is cofibrant.
As we do not assume the axiom of choice, one needs to precise some details regarding theorem 1.1: a "structure of fibration" (resp. trivial fibration) on a map f is the choice of a solution to each lifting problem of a horn inclusion (resp. boundary inclusion) against f . No uniformity condition is required on these lift. A fibration (resp. trivial fibration) is a morphism which admits at least one structure of fibration (resp. trivial fibration), but the choice of the structure is considered irrelevant.
More generally, we will follow the convention that (unless exceptionally stated otherwise) every statement of the form ∀a, ∃b should be interpreted as the existence of a function that given "a" produces a "b". In particular, when one says that a morphism has the lifting property against some set of arrow it means that one has a function that produces a solution to each lifting problem. We will use the convention constantly in the present paper, i.e. every time we say that "there exists" some x, we mean that one specific x has been chosen for each possible value of the parameters involved in the statement.
As fibrations and trivial fibrations are defined by the right lifting property against a small set of morphisms between finitely presented objects, it is very easy to apply a constructive version of the small object argument to show that one has two weak factorization systems, which will be called as follows:
1.2 Definition.
• The weak factorization system cofibrantly generated by the boundary inclusion ∂∆[n] ֒→ ∆[n] is called "cofibrations/trivial fibrations.
• The weak factorization system cofibrantly generated by the horn inclusion Λ k [n] ֒→ ∆[n] is called "Anodyne morphisms/Kan fibrations".
We have discussed the constructive validity of the small object argument in appendix D of [6], though there are probably other references doing this. Note that anodyne morphisms will in the end be the trivial cofibrations, and Kan fibrations will be what we have called fibrations in the statement of the main theorem 1.1, but this will be one of the last result we will prove. In the meantime we will distinguish between Kan fibrations and "strong fibrations" and between anodyne morphisms and "trivial cofibrations" (these two other concept being defined in 2.2.3). Simplicial sets whose map to the terminal simplicial set is a Kan fibration will be called either Kan complexes, or fibrant simplicial sets.
1.3 Remark. Before going any further, we should pause here to insist on a very important remark: one of the key difference between what we are doing in the present paper and the usual construction of the Kan-Quillen model structure in classical mathematics is that the cofibrations are no longer exactly the monomorphisms. It can be shown, see for example proposition 5.1.4 in [6], that the class of cofibrations generated by the boundary inclusion, i.e. the class of arrow which have the left lifting property against all trivial fibration is exactly the class of cofibrations described in the statement of theorem 1.1. In particular one has:
Not every simplicial set is cofibrant ! A simplicial set X is cofibrant if and only if it is decidable whether a cell of X is degenerate or not.
This introduces some changes compared to the classical situation, for example the left properness of the model structure on simplicial set is no longer automatic, and the assumption that certain objects needs to be cofibrant tends to appears in a lot of results. Compare for example 3.3.4, 3.3.5 or 3.4.1 to their classical counterparts.
One can also show the classical Eilenberg-Zilber lemma, asserting that a cell x ∈ X([n]) can be written uniquely as σ * y for σ a degeneracy and y a nondegenerated cells holds if and only if X is cofibrant. A general constructive version of the Eilenberg-Zilber lemma can be found as Lemma 5.1.2 in [6] and does implies that the statement above holds for cofibrant simplicial sets. The converse (that the validity of the Eilenberg-Zilber lemma implies cofibrancy of X) is immediate from the decidability of equality between morphisms of the category ∆: if a cell is written σ * y with y non-degenerate one can decide if it is degenerate or not depending on if σ is the the identity (an isomorphism) or not.
The general structure of the proof of this theorem (and in fact of the paper) is as follows:
• In subsection 2.1 we review the existence of a "weak model structure" on simplicial sets and semi-simplicial sets from [6], which is our starting point.
• In subsection 2.2, more precisely in theorem 2.2.8, we will (up to a technical detail, see the remark 1.4 below) extend this in a model structure on the category of simplicial sets with cofibrations (and trivial fibration) as specified above, but we will not show that trivial cofibrations are the same as anodyne morphisms, or equivalently that the fibrations (called "strong fibrations") are the Kan fibrations. This part is based on the use of semi-simplicial sets.
• Left properness of this model structure follow also from semi-simplicial techniques (see proposition 2.2.9).
• The overall goal 1 of section 3 is to introduce Kan's Ex ∞ -functor. This is done following the work of S.Moss from [12], which can be made constructive at the cost of only minor modification. This will allows us to show that the fibrations of the model structure above are exactly the Kan fibrations (proposition 3.5.1) and to prove the right properness of this model structure (proposition 3.5.2), as well as to fix a small gap in constructiveness of subsection 2.2 (see the remark below).
1.4 Remark. The gap we are referring too in this last point is that in subsection 2.2, the "strong fibrations" (i.e. the fibrations of the model structure on simplicial sets) are defined as the map having the right lifting property against all cofibrations which are equivalences. It is unclear if they can be defined by a lifting property against a small set and hence if trivial cofibration/strong cofibration do form a weak factorization system as a model category structure should require. In proposition 2.2.7 we give a formal argument that shows it is the case, but it is unlikely that this argument can be made constructive. What definitely solve the problem constructively is the proof in 3.5.1 that this factorization is actually just the "anodyne morphisms/Kan fibrations" factorization, but this require all the material of section 3. This being said, the reader should note that even before section 3, it holds constructively that the anodyne/Kan fibration of an arrow with fibrant target is a "trivial cofibration/strong fibration" factorization (because of the third point of lemma 2.2.6). Hence it holds constructively, even without the results of section 3, that any arrow with fibrant target admit such a factorization, i.e. one already has something similar to a (right 2 ) semi-model category without invoking the properties of Kan Ex ∞ functor.
1.5 Remark. The fact that we need to invoke the good properties of Kan's Ex ∞ functor to show that the class of fibration is indeed the class of Kan fibrations of course remind us of D-C.Cisinski's approach to the construction of Kan-Quillen model structure in [3]. We do not really know how deep are the similarities between our proof and D-C.Cisinski's proof. Our initial plan on this problem was actually to try to see if this approach of Cisinski can be made constructive or not. While we definitely do not exclude this is the case, it seemed to represent a considerably harder task than what we have achieved here. One of the problem is that Cisinski's theory relies heavily on a set theoretical argument similar to the one we mention in the proof of 2.2.7, whose constructiveness seems unlikely. The other problem being simply that Cisinski approach, while very elegant, relies on a considerable amount of machinery whose constructivity would have to be carefully checked.
1.6 Remark. Finally, I only said that "constructive" meant something like internal logic of an elementary topos with a natural number object for simplicity, but everything is actually completely predicative for some, relatively strong, sense of this word. I believe that everything can be formalized within the internal logic of an "Arithmetic universe", i.e. a pretopos with parametrized list objects (see for example [10] ). Such a formalization of course requires some modification: for example it wouldn't make sense to say that a morphisms "is a fibration" in the sense that "there exists a structure of fibrations on the morphisms" as the set of all "structure of fibration" on a given morphism cannot be defined, but it would make sense to consider a morphism endowed with a structure of fibration, and to show that given such a pair one can perform some construction.
Though working in such framework in an explicitly way forces to be extremely careful about a huge number of details and makes everything considerably more complicated, and would make the paper considerably more complicated. For this reason we will not do it explicitly. It seems to me that this is typically the sort of thing that should be done with a proof assistant. There is one part of this claim that I have not checked carefully: Whether such a weak framework is sufficient to use the case of the small object argument that we need, i.e. construct the cofibration/trivial fibration and the anodyne/Kan fibrations factorization systems (generated respectively by boundary inclusion and horn inclusion) on simplicial set and semi-simplicial sets, though it seems reasonable that a complicated encoding with list object can achieve this. More precisely this should follow from the fact that the initial model theorem for partial horn theories of Vickers and Palmgren in [13] is believed to be provable internally in an arithmetic universe, and the factorization obtained from R.Garner's version of the small object argument (from [5]) are constructed as certain initial structure that can be described using partial horn logic.
1.7 Remark. In a joint paper with Nicola Gambino ([4]), we will show that this Quillen model structure on simplicial sets admit all the necessary structure to interpret homotopy type theory, with type and context being interpreted as bifibrant objects. This was the main motivation for the present paper and the two papers have been written in close connection. I would also like to thanks Nicola Gambino for the helpful comments he made about earlier version of the present paper. [6], which is the starting point of the present paper, is the construction of a "weak model structure" on the category of simplicial sets where fibrations (between fibrant objects) and cofibrations (between cofibrant objects) are as specified above. More explicitly this means that there is a class of maps called "equivalences 3 " in the category of simplicial sets that are either fibrant or cofibrant (in the sense above) such that:
• Weak equivalences (between objects that are either fibrants or cofibrant) contains isomorphisms, are stable under composition and satisfies 2-outof-3 (and the stronger 2-out-of-6 property).
• A cofibration between cofibrant objects is a weak equivalence if and only if it has the left lifting properties against all fibrations between fibrant objects (such a map is called a trivial cofibration).
• A fibrations between fibrant objects is a trivial fibrations if and only if it is a weak equivalence 4 .
• The localization of the category of fibrant or cofibrant objects at the weak equivalences can be described as the category of fibrant and cofibrant objects with homotopy classes of maps between them. Where the homotopy relation is defined as usually, using equivalently a path object or a cylinder object. This localization is called the homotopy category.
• The weak equivalences are exactly the morphisms that are invertible in the homotopy category (which proves the first point immediately).
One can deduce from this various characterization of weak equivalences: for example, a map from a cofibrant object to a fibrant object is a weak equivalence if and only if it can be factored as a trivial cofibration followed by a trivial fibration. Note that at this point it does not makes sense to ask whether a map X → Y is a weak equivalence if one of X or Y is neither fibrant nor cofibrant.
2.1.2.
In [6, theorem 5.5.6] we also showed that a similar "weak model structure" exists on the category of semi-simplicial sets. Semi-simplicial sets are "simplicial sets without degeneracies", i.e. collection of sets X 0 , . . . , X n , . . . with "faces maps" satisfying the same relations as the face maps of a simplicial sets. Equivalently they are presheaves on the category ∆ + of finite non-empty ordinals and injective order preserving maps between them. The generating cofibrations in the category of semi-simplicial sets are the semisimplicial boundary inclusion:
∂∆ + [n] ֒→ ∆ + [n],
where ∂∆ + [n] and ∆ + [n] respectively denotes the semi-simplicial subset of nondegenerated cells in ∆[n] and ∂∆ [n]. Note that the ∆ + [n] also corresponds to the representable semi-simplicial sets, so that a morphism ∆ + [n] → X is the same as an n-cell of X and a morphism ∂∆ + [n] → X is the data of a collection of n cells of dimension n − 1 with compatible boundary exactly as simplicial morphisms from ∂∆[n] to a simplicial sets X. In particular a morphism f : X → Y of simplicial sets is a trivial fibration if and only if its image by the forgetful functor to semi-simplicial sets is a trivial fibration (in the sense that it has the right lifting property against the generating cofibration). As there is no degeneracies anymore in ∆ + the description of cofibrations simplifies to just "levelwise complemented monomorphisms" i.e. the class of monomorphism f : X → Y such that for each n, and for each y ∈ Y ([n]) it is decidable whether y ∈ X([n]) or not (this is also discussed in [6, theorem 5.5.6]). In particular, every object is cofibrant. Similarly, a morphism of semi-simplicial sets is said to be a Kan fibration when it has the lifting property against the semi-simplicial version of the horn inclusion Λ k
+ [n] ֒→ ∆ + [n], where Λ k + [n]
and ∆ + [n] respectively denotes respectively the semi-simplicial sets of non-degenerate cells in Λ k [n] and ∆[n]). As above a simplicial morphisms between simplicial sets is a Kan fibration if and only if its image by the forgetful functor to simplicial set is a Kan fibration of semisimplicial sets. In this weak model structure on semi-simplicial sets, the cofibration are as described above, the fibrant objects are the semi-simplicial Kan complexes and the fibrations and trivial fibration between fibrant object are the Kan fibrations and trivial fibrations. The big difference with the model structure on simplicial sets is that as every semi-simplicial set is cofibrant, the classes of weak equivalences is defined between arbitrary objects of the category. Note that we do not claim that every trivial cofibration (i.e. cofibration which is an equivalence) is an anodyne morphism (i.e. a retract of a transfinite composition of pushout of coproducts of semi-simplicial horn inclusion) : the anodyne morphism have the left lifting property against all Kan fibrations, the trivial cofibration only against Kan fibration between Kan complexes.
Remark.
Note that it is well known, even classically, that this model structure cannot be a Quillen model structure. As every object is cofibrant, it can be seen by a combinatorial argument that, at least classically, it is a "right semi-model structure" in the sense of [2] The forgetful functor from simplicial sets to semi-simplicial sets is very well behaved: we showed in [6, theorem 5.5.6] that it is both a left and right Quillen equivalence, and we will prove in 2.2.2 that it preserves and detect weak equivalences without any assumption of fibrancy/cofibrancy. As all object in ∆ + are cofibrant, this will allow to remove some assumption of cofibrancy in various places.
Sketch of proof of 2.1.1. We finish this section by presenting the main steps of the argument given in [6] of the existence of the weak model structure on simplicial sets, i.e. all the claims made in 2.1.1. The details of this can be found in [6], but we hope the following summary will be of help to the reader. The proof for semi-simplicial sets is quite similar. The first (and essentially only) important technical step is the proof of the so-called "pushout-product" or "corner-product" conditions for the simplicial generating cofibrations and trivial cofibrations. This follows from a completely constructive results of Joyal (theorem 3.2.2 of [8]), in [6] it corresponds to lemma 5.2.2 (and how it is used in the proof of theorem 5.2.1 in 5.2.3). In the present paper we also reproduce a different proof of this claim as 3.2.6, which is due to S.Moss (see [12, 2.12]). From the corner-product conditions one deduces formally 5 all the usual property of stability of cofibrations, anodyne morphisms, fibrations, and trivial fibrations under product and exponential expected in a cartesian model category (see proposition 3.2.6 and the remark directly below it). This allows to construct nicely behaved cylinder objects as ∆[1] × X and path objects as X ∆ [1] , whose legs are appropriately (trivial) (co)fibrations as soon as X is (co)fibrant. More generally, one can construct relative path object for any fibration X ։ Y and relative cylinder object for any cofibration A ֒→ Y . Having such relative cylinders and path objects is the definition of weak model structure that we gave in section 2 of [6]. The precise observation that one get a weak model structure from such a tensor product satisfying the corner-product condition is essentially the construction done in section 3 of [6], summarized by theorem 3.2 there.
Then all the claims made in 2.1.1 follows from the general theory of weak model structure developed in section 2.1 and 2.2 of [6]. We sketch the general strategy here, though at this point we recommend looking directly at subsection 2.1 and 2.2 of [6] which are mostly self contained. One uses these cylinders and path objects to define the homotopy relation between maps from a cofibrant object to a fibrant objects. Using the lifting property one show that the homotopy relation with respect to any cylinder object is equivalent to the homotopy relation with respect to any path object and that these define an equivalence relation compatible to pre-composition and post-composition. The proof is essentially the same as in a full Quillen model structure: the definition of weak model structure is exactly tailored so that the usual proof of these claims can be applied. This allows to give a first definition of the homotopy category as the category whose objects are the fibrant-cofibrant objects and the maps are the homotopy class of maps. One then proves formally that this homotopy category is equivalent to various localization (see theorem 2.2.6 in [6]), the last one being the localization of the category of simplicial sets that are either fibrant or cofibrant at all trivial cofibration with cofibrant domain and all trivial fibration with fibrant target. One can then defines weak equivalences as the arrow that are invertible in this localization, and one automatically have 2-out-of-6 and all the other good properties of weak equivalences. The fact that trivial fibration (with fibrant domain) are exactly the fibration that are equivalence is a little harder and use again the property of the relative path objects (see proposition 2.2.9 in [6]), and similarly for cofibrations.
The simplicial model structure
To obtain that simplicial sets form a full Quillen model structure we first need to extend the meaning of "equivalences" so that it makes sense also for arrows between objects that are neither fibrant nor cofibrant. We will do this by exploiting the forgetful functor from the category of simplicial sets to the category ∆ + of semi-simplicial sets. As in the category of semi-simplicial sets every object is cofibrant the notion of weak equivalence there is defined for arbitrary arrows, and we will show it is reasonable to define equivalences of simplicial sets as arrow that are equivalences of the underlying semi-simplicial sets. We start by the following observation:
2.2.1 Lemma.
• If f : X → Y is an anodyne morphism in ∆, then its image in ∆ + is also anodyne, and in particular is an equivalence.
• Let f : X ։ Y be a trivial fibration in ∆. Then the image of f in ∆ + is an equivalence.
Note that in the second case, it is obvious that f is a trivial fibration in ∆ + , but this is not enough to deduce that is is an equivalence in general, unless its target is fibrant, as ∆ + only has a weak model structure.
Proof.
• This is corollary 5.5.15.(ii) of [6].
• We first assume that X is cofibrant. In this case one can construct a strong cylinder object for X using the cartesian structure of simplicial sets:
X X ֒→ ∆[1] × X ∼ → X
with the two maps X ֒→ ∆[1] × X being anodyne morphisms (this follows from the fact that X is cofibrant and the corner-product conditions). Because of the previous point this produces a strong cylinder object for the underlying semi-simplicial set of X in the category of semi-simplicial sets.
In ∆ + , every object is cofibrant, and the arrow f : X → Y is still a trivial fibration, so one can find some dotted lifting for the following two squares in ∆ + :
∅ X Y Y s X X X ∆[1] × X X Y (IdX ,sf ) h
In particular, s is a section of f , i.e. f s = Id Y , and h an homotopy between Id X and sf . Hence s is an inverse of f in the homotopy category of ∆ + , which makes f an equivalence in ∆ + .
In the general case (when we do not assume that X is cofibrant), one take a cofibrant replacement (with a trivial cofibration) X c ∼ ։ X and the result above applies to both the trivial fibration X c ∼ ։ X and the composite trivial fibration X c ∼ ։ Y . By 2-out-of-3 for weak equivalences in ∆ + this implies that the map X ∼ ։ Y is indeed an equivalence in ∆ + .
Proposition.
For a morphism f : X → Y between simplicial sets that are either fibrant or cofibrant the following are equivalent:
• f is an equivalence for the weak model structure in ∆.
• The image of f in ∆ + is an equivalence for the weak model structure on
∆ + Proof. If Y is cofibrant, then one can take a fibrant replacement Y ∼ ֒→ Y f . The map Y ∼ ֒→ Y f
is an equivalence both in ∆ and ∆ + , so in both category f is an equivalence if and only if the composite X → Y f is an equivalence, so it is enough to prove the result when Y is fibrant. A similar argument using a cofibrant replacement allows to assume that X is cofibrant. Assuming both X cofibrant and Y fibrant, one factors f as an anodyne morphism (with cofibrant domain) followed by a Kan fibration (with fibrant target). The anodyne morphism is an equivalence in both categories, hence (in both category) f is an equivalence if and only if the Kan fibration part is a trivial fibration. But for a map in ∆, being a trivial fibration in ∆ and in ∆ + are the exact same condition (the lifting property only involves face operation, no degeneracies).
This last proposition makes the following definition very reasonable:
Definition.
• An arrow in ∆ is said to be an equivalence if its image by the forgetful functor to ∆ + is an equivalence for the semi-simplicial version of the Kan-Quillen weak model structure mentioned in 2.1.2.
• A trivial cofibration is a cofibration which is also an equivalence.
• A strong fibration is an arrow that has the right lifting property against all trivial cofibrations.
We remind that the reader, that we will prove in 3.5.1 that these notion of strong fibrations and trivial cofibrations are equivalent to the usual notion of Kan fibrations and anodyne morphisms.
Remark. With this definition it is immediate that:
• Isomorphisms are equivalences, and equivalences are stable under composition, satisfies the 2-out-of-3 and even the 2-out-of-6 properties.
• Anodyne morphisms are trivial cofibrations. Indeed they are cofibrations by definition and they are equivalences in the sense of definition 2.2.3 by the first point of lemma 2.2.1.
• As a consequence, strong fibrations are Kan fibrations.
• Trivial fibrations, defined by the right lifting property against boundary inclusion, are both strong fibrations because they have the right lifting property against all cofibrations, and equivalence because of lemma 2.2.1.
• A Kan fibration (or strong fibrations) with fibrant target is a trivial fibrations if and only if it is an equivalence (this follows from proposition 2.2.2 and the fact that this fact holds in weak model categories).
Maybe it is a good point to recall the following very classical lemma that we will use constantly in this paper:
2.2.5 Lemma.
Assume that a map f is factored as f = pi. If i has the left lifting property against f , then f is a retract of p. If p has the right lifting property against f then f is a retract of i.
Proof. We only prove the first half of the claim, the second is just the dual statement. One form a morphism h as the dotted diagonal filler in first square below (obtained by the lifting property of i against f ), which can then be used to form a retract diagram:
A A B C i f p h A B A C C C f i p h f 2.2.6
Lemma.
(i) A cofibration is a trivial cofibration if and only if it has the left lifting property against all Kan fibrations between Kan complexes. (ii) An arrow whose target is a Kan complex is a trivial cofibration if and only if it is anodyne. (iii) An arrow whose target is a Kan complex is a strong fibration if and only if it is a Kan fibration. (iv) A map is a trivial fibration if and only if it is a strong fibration and an equivalence.
Because of the third point it is equivalent for a simplicial set X that X → 1 is a a Kan fibration (i.e. X is a Kan complex) and that X → 1 is a strong fibration. One will simply say that X is fibrant.
Proof.
(i) Let f : A ֒→ B be a cofibration that is also an equivalence, and we consider a lifting problem of f against a Kan fibration between Kan complexes:
A X B Y f v p u
In the special case where both u and v are equivalences, then by 2-out-of-3, the map p is also an equivalence. As it is a Kan fibration between Kan complexes it is also a trivial fibration, and hence the lifting problem has a solution because f is a cofibration. We will now show that one can bring back the general case to this situation:
One can factor u as an anodyne morphism followed by a Kan fibration: B ∼ ֒→ Y ′ ։ Y and complete the diagram above by forming the pullback
P = Y ′ × Y X : A P X B Y ′ Y f v ′ ∼
The map v ′ can be factorized as an anodyne morphism followed by a Kan fibration:
A P ′ P X B Y ′ Y f ∼ ∼
The case treated above, where the two horizontal maps are equivalences, allows to produce a dotted diagonal lifting of the form:
A P ′ P X B Y ′ Y f ∼ ∼
and this concludes the proof in the general case.
Conversely, assume i : A ֒→ B is a cofibration that has the left lifting property against all Kan fibrations between Kan complexes. One needs to show that i is an equivalence. By taking an anodyne morphism B
∼ ֒→ B f to a fibrant objects the composite A ֒→ B f still has the announced lifting property so one can freely assume that B is fibrant in order to show that i is an equivalence. Under that assumption one factors i as an anodyne morphism followed by a Kan fibration, the Kan fibration has a fibrant target so it has the right lifting property against i. Hence by the retract lemma 2.2.5, i is a retract of the anodyne part of the factorization, hence anodyne itself and hence is an equivalence.
(ii) This second observation follows from last part of the proof of (i) where we explicitly showed that a trivial cofibration with fibrant target is anodyne.
(iii) We have mentioned already that strong fibrations are Kan fibrations, and (i) shows that Kan fibrations between Kan complexes are strong fibrations.
(iv) Trivial fibration have the right lifting property against all cofibrations, in particular against trivial cofibration hence they are strong fibration, and lemma 2.2.1 shows they are equivalences. For the other direction, the proof is essentially the dual the proof of (i). Let p be a strong fibration that is also a weak equivalence, and consider a lifting problem of p against a cofibration:
A X B Y p
By factoring the map A → X into a cofibration A → A ′ followed by a trivial fibrations and taking the pushout of A ֒→ B along this map A → A ′ one reduces the problem to the case where the top map is an equivalence. One can then factor the bottom map as a cofibration followed a trivial fibration:
A A ′ X B B ′ Y ′ Y ∼ p ∼
where the dotted arrow exists because the composed cofibration A ֒→ Y ′ is a weak equivalence by the 2-out-of-3 properties, and hence has the left lifting property against p. This provides a dotted filling for the initial square.
In order to conclude that one has a Model structure on simplicial sets, one needs one more proposition.
Proposition. Any morphism can be factored as a trivial cofibration followed by a strong fibration.
Again, we will show in 3.5.1 that this factorization system is actually the same as the anodyne/Kan fibrations factorizations system, i.e. that trivial cofibration are the same anodyne morphisms and that strong fibration are the same as Kan fibrations. Note that at this point it is immediate that anodyne morphism are trivial cofibrations, and hence that fibrations are Kan fibrations.
Proof. We will give two proof of this claim. The first one follows from [11], more precisely its theorem 3.2, which is not known to be constructive but allows to give a simple and direct proof of the present proposition. In order to fix the issue with constructivity one gives a second, considerably less direct proof: as mentioned above in 3.5.1 we will prove independently of the present proposition that trivial cofibrations are the same as anodyne morphisms, hence showing that the weak factorization mentioned in the proposition exists and is simply the anodyne-Kan fibration weak factorization system (whose existence follows from the small object arguments). We still give the first proof as we believe it is interesting on its own as it allows to construct the model structure on simplicial sets without needing to invoke Kan Ex ∞ -functor. Theorem 3.2 of [11] claims that the 2-category of presentable categories endowed with a class of cellular morphisms generated by a set of morphisms is closed under pseudo-pullback, and that these pullback are constructed explicitly: the underlying category is the pullback of categories, and the class of cellular morphisms are the morphisms whose image in each component are in the specified classes. We apply this to the following square:
P (Kan-Cplx, TrivFib) ( ∆, Cof ) (Kan-Cplx, All arrows)
Where "Cof" denotes the class of cofibration in ∆ which is generated by a set. Kan-Cplx denotes the category of "algebraic Kan complexes", i.e. simplicial set endowed with chosen lifting against horn inclusion and of morphisms compatible to these choices of lifting. The functor ∆ → Kan-Cplx send any simplicial set to the "free algebraic Kan complexes it generates",i.e. the left adjoint to the forgetful functor from algebraic Kan complex to simplicial set, or equivalently the functor sending a simplicial set to its canonical fibrant replacement as produced by R.Garner version of the small object argument. The class TrivFib is the left class of the weak factorization on Kan-Cplx cofibrantly generated by the image of the horn inclusion in ∆. The right class of the weak factorization system are hence exactly the morphism whose image by the forgetful functor to ∆ are Kan fibrations. It follows that the morphism in ∆ which are sent to "trivial cofibrations" in Kan-Cplx are exactly the arrows that have the left lifting property against all Kan fibration between Kan complexes. Hence in this case the pullback is the category of simplicial sets with as set of cellular morphisms the map that are both cofibrations and have the left lifting property against Kan fibration between Kan complexes, i.e. the "trivial cofibrations" as defined above, hence this class of arrow is generated by a set, and hence by the small object argument it is one half of a weak factorization system.
Theorem.
There is a model structure on the category of simplicial sets such that:
• The equivalences are as defined in 2.2.3.
• The cofibrations and trivial fibrations are the same as in theorem 1.1.
• Proof. Given a pushout square in the category of simplicial sets:
A C B D ∼ f
Then as the forgetful functor to semi-simplicial sets preserves all colimits, this square is again a pushout in the category of semi-simplicial sets. In this category every object is cofibrant, and pushout along a cofibration between cofibrant objects is a left Quillen functor hence preserves equivalences between cofibrant objects, hence f is an equivalence in the category of semi-simplicial sets, and hence is an equivalence in ∆ by definition ( 2.2.3).
Kan Ex ∞ -functors
The goal of this section is to introduce Kan's Ex and Ex ∞ functors and to use them in subsection 3.5 to prove the remaining claim concerning the simplicial model structure. Most of the results here were (in their classical form) originally proved by Kan in [9] (often with quite different proof that the ones we will provide here), but we will mostly follow the approach of S.Moss in [12] which we will make constructive by only adding some details. Subsection 3.1 is a preliminary section that is of some independent interest but which will have only a very marginal role in the paper: it will only be used to prove some decidability conditions (more precisely lemma 3.4.3, which will be an easy consequence of 3.1.8 and proposition 3.1.10). As such it can be easily ignored by the reader. Subsection 3.2 review the notion of "P-structure" introduced by S.Moss, which is mostly a language to talk more conveniently about "Strongly anodyne morphisms", i.e. transfinite composition of pushout of horn inclusion. This is a key tool to structure the proof of the main results of section 3.4. Subsection 3.3 introduce Kan's barycentric subdivision functor Sd, its right adjoint Ex and Kan's Ex ∞ functor and proves some of their basic properties. This is very classical material that we reproduce here just for completeness and to discuss some constructive aspect. Subsection 3.4 reproduces (with some modification to make it constructive) S.Moss' proof in [12] that the natural transformation X → Ex ∞ X is an anodyne extension. Constructively this only works when X is cofibrant. We also noted that S.Moss prove can be used to obtain a result which apparently was not known even classically: for any morphisms f : X → Y (with X cofibrant) the natural morphisms:
X → Ex ∞ X × Ex ∞ Y Y
is anodyne. This was known when Y is terminal, or when X → Y is a fibration, and we will actually only use it in these two special cases. Finally subsection 3.5 uses the properties of this functor to conclude that all Kan fibrations are strong fibrations (proposition 3.5.1) and that the model structure on simplicial sets is indeed right proper (proposition 3.5.2).
Degeneracy quotient and questions of decidability
In this section we establish some general results about a notion of "degeneracy quotient" that we will introduce. While the notion might have some interest on its own in other context its only use in the present paper is to prove some decidability results, which will follow from lemma 3.1.8 below. In fact, the only uses of this section in the present paper is in the proof of the decidability conditions of lemma 3.4.3. Proposition 3.1.11 is not useful for the present paper, but will serve in some future work, in particular in [4] and it was more natural to include its proof here.
3.1.1 Definition. A morphism f : X → Y between simplicial sets is said to be degeneracy detecting if:
∀x ∈ X, f (x) is degenerated ⇒ x is degenerated
Of course the converse implication is true for any simplicial map, so one has that x is degenerated if and only if f (x) is degenerated. One says that a cell x ∈ X n is σ-degenerated for some degeneracy σ :
[n] → [m] if x = σ * y for some y. 3.1.2 Lemma. Let σ : [n] → [m]
be any degeneracy and x ∈ X n any cell. The following are equivalent: and x = s * y = σ * j * y is indeed σ-degenerated. If now σd is non-injective for some section d of s, then y = d * x is degenerated by assumptions, hence one can write x = s ′ * y ′ for y ′ of lower dimension than x and start the argument above again, an induction on the dimension concludes the proof. One easily see it is also a necessary condition.
(i) x is σ-degenerated. (ii) For all face map i : [k] → [n] such that the composite σi is non-injective, the cell i * x is degenerated. Proof. If x = σ * y then for any such i, i * x = (σi) * y which is degenerated if σi is non-injective, so (i) ⇒ (ii
Proof. One needs to show that, under the assumption of the lemma, for any two elements i, j ∈ [n] if σi = σj then si = sj. If si = sj, then there is a section d of s such that dsi = i and dsj = j hence σj = (σd)(sj) and σi = (σd)(si), so the injectivity of σd implies that σi = σj. As equality in [n] is decidable one can take the contraposite and concludes the proof.
3.1.4
Proposition. Let f : X → Y be a map between simplicial sets, then the followings conditions are equivalents:
(i) f is degeneracy detecting.
(ii) If f (x) is σ-degenerated for some degeneracy σ then x is σ-degenerated as well.
(iii) f has the (unique) right lifting property against all the degeneracy map
∆[n] → ∆[m].
Proof. (ii) clearly implies the (i) and the converse is immediate from lemma 3.1.2. The lifting in (iii) is automatically unique as degeneracy are epimorphisms in the presheaf category and this lifting property is a reformulation of (ii).
Given a simplicial set X, x ∈ X([n]) and σ : [n] → [m] a degeneracy, one defines X[(x, σ)] as the pushout:
∆[n] X ∆[m] X[(x, σ)] σ x X[(x, σ)] is the universal for map X → Y making x "σ-degenerated", i.e. given a morphism f : X → Y , it factors as X → X[(x, σ)] if and only if f (x) = σ * y for some y ∈ Y ([m])
, and such a factorization is unique when it exists. More generally, given a collection (x i ∈ X([n i ])) i∈I and σ i :
[n i ] → [m i ] one can define an object X[(x i , σ i )]
as the pushout of a coproduct of degeneracy maps, which has the following universal property: a morphism f :
X → Y factors (uniquely) through X → X[(x i , σ i )] → Y if and only if for all i ∈ I, f (x i ) is σ i -degenerated.
Definition.
A morphisms is said to be a degeneracy quotient if it is obtain as X → X[(x i , σ i )] for some collection of x i ∈ X([n i ]) and σ i : [n i ] ։ [m i ] as above.
3.1.6 Proposition. Degeneracy quotient and degeneracy detecting map form an orthogonal factorization system. More precisely, for any morphisms f : X → Y its factorization is obtained as:
X → X[(x i , σ i )] → Y where (x i , σ i ) is the collection of all x i and σ i such that f (x i ) is σ i -degenerated.
Note that this is essentially nothing more than the small object argument, though it is notable that in this case it converges in a single step.
Proof. It is clear from the universal property of X[(x i , σ i )] that one has a factorization as in the lemma, and the first map is by definition a degeneracy quotient.
The map X[(x i , σ i )] → Y is degeneracy detecting: given x ∈ X[(x i , σ i )], it is the image of a x 0 ∈ X, if the image of x is degenerated in Y one has f (x 0 ) = σ * y, hence (x 0 , σ) appears in the definition of X[(x i , σ i )]
, which forces the image of x 0 , i.e. x, to be degenerated. The orthogonality of the two class is relatively immediate as well: given a lifting problem:
X A X[(x i , σ i ) B
where the right map is degeneracy detecting, then a diagonal filling exists if and only the image of the x i in A satisfies the appropriate degeneracy conditions. As their images in B satisfies them because of the existence of the square, and as the map A → B is degeneracy detecting, this is immediate.
The following is more or less a reformulation of what is a degeneracy quotient that will be convenient:
∀a ∈ A([n]) p(a) is degenerated ⇒ f (a) is degenerated. (D)
Note that if such a factorization exists then condition (D) holds without any assumption on p, so that if p is a degeneracy quotient then a factorization exists if and only condition (D) holds.
Proof. It follows from 3.1.2, that condition (D) is equivalent to: This observation has a quite interesting consequence that will be extremely useful to us, and in fact is the unique reason why we are interested in degeneracy quotient in the present paper: Proof. One can use condition (D) of lemma 3.1.7 to test whether such a diagonal lift exists. As B is finite and decidable, degeneracy in B is decidable. So for each cell a ∈ A it is decidable if " p(a) is degenerated ⇒ f (a) is degenerated" as both side of the implication are decidable. Moreover this condition is automatically valid for all degenerated cells of A, so it is necessary to test it only on a finite number of cells to know whether f factors through p, which makes the validity of condition (D) decidable and hence the existence of a diagonal lift decidable.
∀a ∈ A([n]) p(a) is σ-degenerated ⇒ f (a) is σ-degenerated.
The following lemma is obvious, but will be a convenient a technical tools to show that certain maps are degeneracy quotient:
3.1.9 Lemma. Let p : A → B be an epimorphism. One considers the equivalence relation ∼ p on A generated by:
• If p(a) is σ-degenerated, then a ∼ p σ * t * a for any section t of σ.
• ∼ p is compatible with all the faces and degeneracy maps of A.
Then p is a degeneracy quotient if and only if any two a, a ′ ∈ A such that pa = pa ′ one has a ∼ p a ′ .
Note that for any morphisms, a ∼ p a ′ ⇒ pa = pa ′ . 3.1.10 Proposition. Let P be a poset with an idempotent order preserving endomorphism π satisfying either ∀x, πx x or ∀x, πx x. Let Q = πP . Then the morphisms between the simplicial nerve:
Proof. One easily see that ∼ p is exactly the simplicial equivalence relation by which one needs to quotient
A to obtain A[(a i , σ i )] where (a i , σ i ) is the family of all a i such that p(a i ) is σ i degenerated in B.N (P ) → N (Q)
induced by π : P → Q is a degeneracy quotient.
Proof. We assume that πx x. The other case follows by simply reversing the order relation on P and on all objects of the category ∆. Let p 0 p 1 · · · p n be an element of N (P ) n and assumes that p 0 , . . . , p i−1 ∈ Q, then one forms p 0 p 1 · · · p i−1 πp i p i · · · p n It is an element of N (P ) n+1 whose image in Q is degenerated as σ i * (πp 0 · · · πp n ). This implies that in N (P ):
(p 0 · · · p n ) ∼ (p 0 · · · p i−1 πp i p i+1 · · · p n ) Hence using this for all i from 0 to n, one obtains that for any sequence p 0 · · · p n all the (πp 0 · · · πp i−1 p i · · · p n ) for i = 0, . . . , n + 1 are equivalent. In particular any sequence is equivalent to its image by π and finally any two sequences whose image in N (Q) are the same are equivalent.
We finish with a proposition that will only be useful in future work ([4]):
Proposition. The class of degeneracy quotient is stable under pullback.
Proof. First we show that given a pullback of the form:
P ∆[n] ∆[k] ∆[m] φ σ f
where σ is a degeneracy map, the map φ is a degeneracy quotient. This is proved using proposition 3.1.10. Indeed in such a pullback P is nerve of the pullback of posets, that we will also denote P (because the nerve functor commutes to pullback). We will show that the map P → . This is still an element of P , π ′ (i, j) (i, j) it is idempotent, and its image identifies naturally with [k]. Hence φ : P → ∆[k] is indeed a degeneracy quotient by proposition 3.1.10. We now show that given any pullback of the form:
P ∆[n] X ∆[m] φ σ f
for a degeneracy σ, the map φ is a degeneracy quotient. Indeed, one write:
X = Colim ∆[k]→X ∆[k]
Given a x : ∆[k] → X one write P x the pullback:
P x P ∆[n] ∆[k] X ∆[m] φx φ σ f
All map φ x are degeneracy quotient by the first part of the proof. As the category of simplicial sets is a topos, colimits are universal, one has the morphism φ is the colimit of the arrows φ x (in the category of arrows). As the class of degeneracy quotient is the left class of an orthogonal factorization system, the colimit φ is also a degeneracy quotient. To give an explicit argument: given a lifting problem of φ against a degeneracy detecting map one can construct for each x a lifting:
P x P A ∆[k] X B φx φ
By uniqueness of the lifts, they will all be compatible and produces a morphisms from the colimits to A making the square commutes. Finally we can prove the claim in the proposition. Given a morphism f :
X → Y any degeneracy map ∆[n] → ∆[m] over Y (i.e with δ[m] → Y )
is sent by the pullback functor ∆ /Y → ∆ /X to a degeneracy quotient. send degeneracy quotient to degeneracy quotient. But a general degeneracy quotient is pushout of coproduct of degeneracy map, this coproduct and pushout are preserved by the pullback functor (because the category of simplicial sets is cartesian closed), and coproduct of pushout of degeneracy are degeneracy quotient so this concludes the proof.
P-structures
This section recalls the notion of P -structure introduced in [12] with some minor modification to make it more suitable to the constructive context. A "Pstructure" on a morphism f : A → B is essentially a recipe for constructing it as an iterated pushout of (coproduct of) horn inclusion Λ i These two cells are connected by F = d i P . So if A ֒→ B is constructed by iterating such pushout, then one can partition the non-degenerate cells of B that are not in A into "type I" and "type II" and there should be a bijection which associate to any type II cell the type I cell that is added by the same pushout. The formal definition look like this:
3.2.1
Definition. Let f : A → B be a cofibration of simplicial sets. A Pstructure on f is the data of:
• A (decidable) partition of the set of non-degenerate cells of B which are not in A into:
B I B II
called respectively type I cells and type II cells.
• A bijection P : B II ∼ → B I .
Such that:
1. For all x ∈ B II , dim(P x) = dim(x) + 1 2. For all x ∈ B II , there is a unique i such that d i (P x) = x.
Every cell of B II has finite P -height (see definition 3.2.2 and lemma 3.2.3 below).
In [12], the last condition was formulated as a well-foundness condition. Wellfoundness is a tricky notion constructively so we prefer to avoid it. It should be clear to the reader that the condition we will now explain is equivalent to well-foundness if one assumes classical logic, or if one has a nice enough notion of well-foundness constructively. Intuitively this last condition just assert that the "recipe" given by the P -structure to construct B from A as an iterated pushout of horn inclusion is indeed well-founded, i.e. can be executed. We will formulate it by introducing for each cell b ∈ B a set:
Ant(b)
of "antecedent of b" which corresponds to the set of cells that needs to be constructed before b in the process described by P . In [12] the well-foundness condition is essentially that the order relation generated by b ′ ∈ Ant(b) is wellfounded. As each Ant(b) is a finite set this is equivalent to the fact that for each b there is an integer k such that when iterating Ant(b) more than k times one has only cells in A. This is this second definition that we will use in our constructive context.
More precisely: Given a cell b ∈ B II and let i be the unique integer such that d i P x = x, one defines the set Ant(b) of antecedent of b as: Similarly, if b = P b ′ is type I, one defines:
Ant 0 (b) = {d j P (b)|j = i}Ant(b) = Ant(b ′ )
Finally, if b ∈ A:
Ant(b) = ∅
and if b is not in A but degenerated, then
Ant(b) = Ant(b ′ ) where b ′ is the unique non-degenerate cell such that b = σ * b ′ .
One also defines Ant II (b) to be the set of non-degenerate type II cell in Ant 0 (b). Note that in all cases Ant(b) and Ant 0 (b) are Kurawtowski-finite 6 sets, and as the subset of type II cell is decidable, Ant II (b) is also Kurawtowski-finite. One defines Ant k (b) and Ant k II (b) by:
Ant 1 (b) = Ant(b) Ant k (b) = c∈Antb Ant k−1 c Ant 1 II (b) = Ant II (b) Ant k II (b) = c∈AntIIb Ant k−1 II c
Note that when applied to a non-degenerate type II cell b ∈ B, all elements of Ant II (b) (and hence of Ant k II (b) as well) are non-degenerate type II cells of the same dimension as b.
Definition.
• One says that b has finite P -height if there exists an integer k such that:
Ant k (b) = ∅
• One says that b has finite weak P -height if there is an integer k such that:
Ant k II (b) = ∅ Note that for each given k and b ∈ B, as the sets Ant k (b) and Ant k II (b) are Kuratowski-finite it is decidable whether or not Ant k (b) and Ant k II (b) are empty. In particular, assuming b has finite (weak) P -height there is smallest integer k, called the (weak) P -height of b, such that Ant k (II) (b) = ∅. But in general it might not be decidable whether b has finite (weak) P -height or not.
3.2.3
Lemma. Let f : A ֒→ B with a P -structure satisfying all the conditions of definition 3.2.1 but the last. Then the following are equivalent:
• Every b ∈ B has finite P -height.
• Every non-degenerate type II cell b ∈ B II has finite weak P -height.
Proof. It is clear that Ant k II (b) ⊂ Ant k (b) hence the first condition implies the second. Conversely, assume that every b ∈ B has finite weak P -height. We will prove by double induction on both the dimension and the weak P -height that all cells of B have finite P -height. First we assume that all cell of dimension < n have finite P -height. Cells of A have P -height zero. All cells of B of dimension n that are either degenerate or of type I satisfies Ant(b) = Ant(b ′ ) for some b ′ of dimension strictly less than n, hence for b ′ of finite P -height by the induction assumption. As Ant k (b) = Ant k (b ′ ) this implies that b has finite P -height as well. It remains to show that all non-degenerate n-cell of type II in B have finite P -height. We do that by induction on their weak P -height. Indeed for a general type II cell b, Ant(b) is constituted of:
• Degenerate or type I cell, that are already known to have finite P -height.
• Faces of cell in Ant 0 (b) which are hence of dimension < n and hence known to be of finite P -height.
• Non-degenerate type II cells that are hence elements of Ant II (b), but
∅ = Ant k II (b) = c∈AntIIb Ant k−1 II c
hence all c ∈ Ant II b have weak P -height at most k − 1, and hence they all have finite P -height by induction.
So all elements of Ant(b) have finite P -height, let m be the maximum of all these P -height, one has that:
Ant m+1 (b) = c∈Ant(b) Ant m (b) = ∅
Lemma.
A cofibration with a P -structure is anodyne. More precisely it is a ω-transfinite composition of pushout of coproduct of horn inclusions.
A map will be called "strongly anodyne" if it admits a P -structure.
Proof. Let A ֒→ B be a cofibration with a P -structure. Let B k ⊂ B be the subset of B of cell of P -height at most k. One has B 0 = A, and B k is a sub-simplicial set. Indeed, for every cell b ∈ B all faces of b appears in Ant(b) or are such that Ant(d i b) = Ant(b) and all degeneracies of b satisfies Ant(σ * b) = Ant(b), hence they all have P -height at most k. Let U be the set of non-degenerate type II cell of B of P -height exactly k. For each u ∈ U , let i u be the unique integer such that d iu P (u) = u.
Then the corresponding map ∆[n]
P u → B k send Λ iu [n] to B k−1 and both u and P u are in B k − B k−1 . Hence taking the pushout:
Λ iu [n] ∆[n] B k−1 R
produces the simplicial set R ⊂ B k whose cells are all those of B k−1 , u and P u and all their degeneracy. Taking the pushout by the coproduct of all these horn inclusions for all u ∈ U gives B k−1 → B k .
Hence B = B k is a ω-transfinite composition of the maps B k → B k+1 which are all pushout of coproduct horn inclusion.
Classically one also has the converse: any transfinite composition of pushouts of coproduct horn inclusion has a canonical P -structure. Constructively this sort of statement is somehow problematic, mostly because the general notion of "transfinite composition" require a notion of ordinal to be formulated appropriately, but it works perfectly fine if one restrict to ω-composition:
3.2.5 Proposition. The class of strongly anodyne morphism contains all horn inclusion and is stable under pushout and ω-transfinite 7 composition. Any morphism can be factored as a strongly anodyne morphisms followed by a Kan fibration, and any anodyne morphism is a retract of a strongly anodyne morphism.
Proof. Horn inclusion have a trivial P -structure with one cell of type I and one cell of type II. It is easy to see that coproduct, pushout and transfinite composition of strongly anodyne map have P -structure induced by the P -structure we start from, for example if A ֒→ B has a P -structure, then C → B A C has a P -structure where a cell in B A C is type I or II if and only if it is type I or II for the P -structure on A ֒→ B and the map P is the same as the one on B, and similarly for coproduct and transfinite composition. It follows that the factorization of the map as an anodyne followed by a Kan fibration obtained by the small object argument is a strongly anodyne morphism as it is constructed as a ω-transfinite composition of pushout of coproduct of horn inclusion. Finally any anodyne morphism j can be factored as a strongly anodyne morphism followed by a Kan fibration, and the usual retract lemma (2.2.5) shows that j is a retract of the strongly anodyne part of the factorization.
We finish this section by mentioning a very important example where this machinery applies, mostly to serve as an example to show how it can be used.
Given two morphisms f : A → B and g : X → Y between simplicial sets one define as usual f × g the "corner-product" of f and g as the morphism:
f × g : (A × Y ) A×X (B × X) → B × Y
One then has the following well known proposition, which we have referred to in the introduction as the corner-product conditions, and which is a key point in establishing the existence of the weak model structure on simplicial sets. It also corresponds to the fact the model structure on simplicial sets that we are constructing is cartesian.
3.2.6
Proposition. If i and j are cofibrations, then i × j is a cofibration as well. Is one of them is anodyne then i × j is also anodyne.
As usual (following for example the appendix of [7]) this implies the dual condition, that if i : A → B is a cofibration and p : Y → X is a fibration, then
the map [B, Y ] → [B, X] × [A,X] [A, Y ]
is a fibration (the brackets denotes the cartesian exponential in simplicial sets), and it is a trivial fibration as soon as either i is anodyne or p is a trivial fibration.
Proof. By usual abstract manipulation (see for example the appendix of [7]) it is sufficient to show it when i and j are generating cofibrations/generating anodyne map. If i and j are generating cofibrations it is very easy to check that i × j is a cofibration as defined in the statement of our main theorem 1. for some k, m, then i × j is anodyne. This is done by constructing an explicit P -structure on i × j.
The first direct proof of this claim that we know of is in [8] (theorem 3.2.2), here we follow the proof of S.Moss' in 2.12 of [12] to show how P-structures works.
We only treat the case k < m for simplicity, by reversing the order relation on can treat the case k > 0 similarly, which in particular cover the case k = m. One says that a cell is type II if either it skip the k th row by going directly from (a, k − 1) to (a + 1, k + 1), in which case one define P x by adding the intermediate step (a, k − 1), (a + 1, k), (a + 1, k + 1) , or if the last point where the k th row is reached, is (a, k) followed by (a + 1, k + 1) in which case P x is defined by inserting the intermediate step: (a, k), (a, k + 1), (a + 1, k + 1). It is an easy exercise to check that this defines a P -structure. (Ex X) n = Hom(Sd ∆[n], X)
Kan Ex and Sd functors
Sd X = Colim ∆[n]→X Sd ∆[n]
The barycentric subdivision construction has a nice expression not just for the ∆[n], but also for all objects which are in the image of the functor ∆ + → ∆, indeed:
3.3.1 Proposition. The composite:
∆ + → ∆ Sd → ∆
Is the functor sending a semi-simplicial set X to N (∆ + /X).
One can note that as the category ∆ + /X is directed, the nerve N (∆ + /X) is itself the image of the semi-simplicial set of its non-degenerate cells. We won't make any use of this remark though.
Proof. This functors X → N (∆ + /X) preserves colimit, because it can be rewritten as:
N (∆ + /X) k = F :[k]→∆+ X(F (k))
which is levelwise a coproduct of colimits preserving functors. Hence we are comparing to colimits preserving functor, so it is enough to show they are isomorphic when restricted to representable. But ∆ + /[n] ≃ K[n] functorially on map of ∆ + so this concludes the proof. ). It can then be checked completely explicitly that this is a (strongly) anodyne morphisms, see Proposition 2.14 of [12] for an explicit description of a P -structure.
There is a natural transformation:
Sd ∆[n] → ∆[n]
Which is induced by the order preserving function:
max : K[n] → [n]
sending each (decidable) subset of [n] to its maximal element. By Kan extension, this gives us natural transformations:
Sd m → Id Id n → Ex
One can hence define a sequences of functors:
X Ex X Ex 2 X . . . Ex k X . . . Ex ∞ X nx nEx X n Ex 2 X n Ex k−1 X n Ex k X
with Ex ∞ the colimit.
Lemma.
For each k, n, there is a (dotted) arrow Ψ k n making the following triangle commutes.
Sd 2 Λ k [n] Sd Λ k [n] Sd 2 ∆[n] Sd(m Λ k [n] ) Ψ k n
Proof. The proof given in [3] as proposition 2.1.39 is purely combinatorial and constructive.
Corollary.
For every cofibrant simplicial set X, Ex ∞ X is a Kan complex.
The proof that follows essentially comes from [3]. If one does not assume that X is cofibrant it still applies to proves that X has the "existential" right lifting property against horn inclusion, but it does not seems possible to give a uniform choice of solution to all lifting problems without this assumption. Without such a uniform choice of lifting against horn inclusion one cannot construct solution to lifting problems against more complicated anodyne morphism that involves an infinite number of pushout of horn inclusion, unless we assume the axiom of choice.
Proof. Lemma 3.3.3 allows to show that given any solid diagram as below, there is a dotted filling:
Λ k [n] Ex X ∆[n] Ex 2 X nEx X
Indeed, through the adjunction the map Λ k [n] → Ex X corresponds to an arrow Sd Λ k [n] → X, which due to lemma 3.3.3 can be extended in:
Sd 2 Λ k [n] Sd Λ k [n] X Sd 2 ∆[n] Sd m Λ k [n] Sd 2 ψ k n
The resulting map Sd 2 ∆[n] → X corresponds to a map ∆[n] → Ex 2 X which has exactly the right property to make the square above commutes. Now by smallness of Λ k [n], any map Λ k [n] → Ex ∞ X factors in Ex k X, the observation above produces a canonical filling in ∆[n] → Ex k+1 X. The choice of the filling, seen as taking values in Ex ∞ X, in general depends on k, but if one further assume that X is cofibrant, than by lemma 3.4.3, the maps Ex k X → Ex k+1 X are all level wise decidable inclusion, so there is a smallest k such that the map Λ k [n] → Ex ∞ X factors into Ex k X and this produces a canonical solution to the lifting problem.
3.3.5
Proposition. If f : X → Y is a fibration (resp. a trivial fibration) with X and Y cofibrant then
Ex ∞ f : Ex ∞ X → Ex ∞ Y is
also a fibration (resp. a trivial fibration).
Similarly to what happen with corollary 3.3.4, without the assumption that X and Y are cofibrant it is only possible to obtain the "existential" form of the lifting property and no canonical choice of lifting.
Proof. Given a lifting problem:
Λ k [n] Ex ∞ X ∆[n] Ex ∞ Y ∼
There is an i such that it factors into:
Λ k [n] Ex i X Ex ∞ X ∆[n] Ex i Y Ex ∞ Y ∼
Moreover, assuming X and Y are cofibrant, lemma 3.4.3 shows that Ex i X ⊂ Ex i+1 X are levelwise decidable inclusion, so (by finiteness of Λ k [n] and ∆[n]) the set of i such that a factorization as above exists is decidable, and hence there is a smallest such i. Proposition 3.3.2 shows that Ex i f is a fibration, so the first square has a diagonal lifting and this concludes the proof.
S.Moss' proof that X → Ex X is anodyne
Let f : X → Y be a simplicial morphisms. One has a square:
X Y Ex ∞ X Ex ∞ Y
Our goal in this section is to show that when X is cofibrant the induced map:
X → Ex ∞ X × Ex ∞ Y Y isX → Ex X × Ex Y Y
is strongly anodyne.
The proof will be concluded in 3.4.5, essentially, we will construct an explicit P -structure on this map. This construction is mostly due to S.Moss in [12]. In addition to the dependency in Y , the main new contributions of this paper in this section is to show that assuming X is cofibrant one can show that sufficiently many decidability conditions can be proved to make S.Moss' argument constructive. In order to do that properly one needs to completely reproduce his argument. Following, [12]
j k n {i} = {i} if i k {0, . . . , i} if i > k r k n {i} = {i} if i k {0, . . . , i − 1} if i = k + 1 {i − 1} if i > k + 1
Both extended to non-singleton elements as binary join preserving maps. These functions satisfies a certain number of equations, we list here those that we will need, they are all due to S.Moss.
Lemma.
j k n j h n = j h n j k n = j h n 0 h k n (1)
Id ∆[n] = r k n • Sd ∂ k+1 n+1 0 k n (2) j k n r k n = (Sd σ k n )j k n+1 0 k n (3) j h n r k n = j h n (Sd σ k n ) 0 h < k n (4) r k n j h n+1 = j h n r k n 0 h k n (5) r k n (Sd ∂ i+1 n+1 ) = (Sd ∂ i n )r k n−1 0 k < i n(6)
j k n r k n r k n+1 = j k n r k n (Sd σ k+1 n+1 ) 0 k n (7)
j k n+1 (Sd ∂ h n+1 )j k n = j k n+1 (Sd ∂ h n+1 ) 0 k n and 0 h n + 1 (8) j k n r k n (Sd ∂ i n+1 )j k−1 n = j k n r k n (Sd ∂ i n+1 ) 0 i k n (9) (Sd σ h n )j k n+1 r k n+1 = j k−1 n r k−1 n (Sd σ h n+1 ) 0 h < k n + 1 (10) (Sd σ h n )j k n+1 r k n+1 = j k n r k n (Sd σ h+1 n+1 ) 0 k h n(11)
Proof. All the functions involved are nerve of join preserving maps between the K[n], so it is enough to check the relations at the level of posets and when function are evaluated at {i}, where one has explicit formula for all of them.
As functions between the Sd ∆[n], j k n and r k n automatically acts one the cells of Ex X. One denotes this action by x → xj k n and x → xr k n which is compatible to the identification of cells of Ex X with functions Sd ∆[n] → x. By equation (1), the j k n are an increasing family of commuting projection whose image defines a series of subsets: X n = J 0 n ⊂ J 1 n ⊂ . . . J n n = (Ex X) n where the identifications with (Ex X) n and X n comes from the fact that j n n is the identity, and j
Ex Y (X) = Ex X × Ex Y Y An n-cell in Ex Y is a morphism Sd ∆[n] → X whose image in Y factors through the map Sd ∆[n] → ∆[n]
. I.e. it is an n-cell of x ∈ (Ex X) n which satisfies: (1) and (5), Ex Y X is stable under the action of j k n and r k n . Before going any further, one needs to state some decidability conditions:
f xj 0 n = f x Note that because of relation
3.4.3 Lemma. If X is a cofibrant simplicial set, then:
1. The inclusion X ⊂ Ex Y X is levelwise decidable. 2. Ex Y X is cofibrant and X → Ex Y X is a cofibration. 7. If x ∈ J k n − J k−1 n
then for all i, with k + 1 < i n + 1, d i (P x) is either of type I or degenerated. 8. A non-degenerated cell x in (Ex Y X) n − X n is type I if and only P x is degenerated.
Proof.
1. d k+1 P x is xr k n (Sd ∂ k+1 ) which is equal to x by equation (2).
2. Let k is the smallest value such that xj k n = x, i.e.P x = xr k n . Equation (5) gives xr k n j k n+1 = xj k n r k n = xr k n . Hence P x ∈ J k n+1 , in particular x ∈ J h n ⇒ k h ⇒ P x ∈ J h n+1 . Conversely, if P x ∈ J k n+1 then:
xj k n = (P x)(Sd ∂ h+1 )j k n (as x = d h+1 P x) = (P x)j k n+1 (Sd ∂ h+1 )j k n ( as P x ∈ J k n+1 ) = (P x)j k n+1 (Sd ∂ h+1 ) (by equation (8)) = x ( P x ∈ J k n+1 and x = d h+1 P x) Hence x ∈ J k n .
3. xr k n = xj k−1 n r k n is degenerated because of equation (4) 4. Let k such that x ∈ J k n − J k−1 n , then P x = xr k n = xj k n r k n and P x ∈ J k n+1 − J k−1 n+1 because of point (2), hence P 2 x = xr k n r k n+1 = xj k n r k n r k n+1 which is degenerated because of equation (7). (10) and (11) show that if x is degenerated then P x is degenerated. If x ∈ X, i.e. x ∈ J 0 n then P x = xr 0 n but r 0 n = Sd σ 0 so P x is degenerated.
Equation
It follows that if x is of type I, then x = yr k n with y ∈ J k n if y ∈ J k−1 n then x is degenerated because of point (3) hence P x is degenerated because of the first part of the present point, if y / ∈ J k−1 n then x = P y and hence P x is degenerated because of point (4). 6. This follows immediately from equation (9) as d i (P x) = xj k n r k n (Sd ∂ i ).
7. For k + 1 < i n + 1 on has:
j k n r k n (Sd ∂ i n+1 ) = j k n (Sd ∂ i−1 n )r k n−1 by equation (6) = j k n (Sd ∂ i−1 n )j k n−1 r k n−1
by equation (8) This equations shows that for x ∈ J k n , d i P x is of the form yr k n−1 for y ∈ J k n−1 , namely y = x(Sd ∂ i−1 )j k n−1 , hence, if d i P x is non-degenerated, it is of type I. 8. We have shown in 5 that if x is type I then P x is degenerated. Conversely let x be a non-degenerated cell such that P x is degenerated. Let k be such that x ∈ J k n − J k−1 n . One has x = d k+1 P x by point 1 of the lemma, hence d k+1 P x is non-degenerated, which means that P x can only be σ k -degenerated or σ k+1 -degenerated (otherwise d k+1 P X would also be degenerated). If P x is σ k -degenerated then d k P x = d k+1 P x = x, but by point 6 of the lemma d k P x ∈ J k−1 n so this is impossible. If P x is σ k+1degenerated then d k+2 P x = d k+1 P x = x hence point 7 shows that x is of type I.
3.4.5.
We are now ready to prove proposition 3.4.1:
Proof. The goal is to show that the type I cell and the operation P we have defined satisfies the condition of 3.2.1, so that the map is anodyne because of 3.2.4. Point (8) of lemma 3.4.4 (combined with lemma 3.4.3) shows that being a type I cell is decidable. So one can indeed defines type II cells as the cells that are not of type I (and non-degenerate nor in the domain) and get a partition of the non-degenerate cells. It also follows from point (8) that if x is a type II cell then P x is a non-degenerate cell, and it is type I (either by definition or because of point (4) ). Finally, point (2) show that P preserve the k such that x ∈ J k n , as X ⊂ Ex Y X corresponds to J 0 n it shows that P never send cell to cell in X. So P restricts into a function from type II cells to type I cells.
We now show that it is a bijection: If x is a type I cell than it can be written as yr k n with y ∈ J k n . By point (3) of lemma 3.4.4, if y ∈ J k−1 n , then x = yr k n is degenerated, hence y / ∈ J k−1 n and hence x = P y. By point (5) of lemma 3.4.4 if y is degenerated or type I then x = P y is degenerated, hence y is a type II cell. This proves the surjectivity of P . If x is a type II cell and y = P x, then x = d k+1 P x (because of point 1 of lemma 3.4.4) where k can be characterized as the unique integer such that y ∈ J k n+1 − J k−1 n+1 (because of point 2 of lemma 3.4.4). Hence P is injective on type II cell and this concludes the proof that P is a bijection between nondegenerated type II cells and non-degenerated type I cells.
Finally if x is a non-degenerate type II cell, and let k such that x ∈ J k n − J k−1 n . Point (1) of lemma 3.4.4 shows that d k+1 (P x) = x, while point (6) and (7) shows that for all i = k + 1, d i P x is either in J k−1 n , type I or degenerated, hence always distinct from x. So there is indeed a unique i such that d i P x = x, and it is k + 1. It remains to proves the "well-foundness" or "finite height" condition. It follows from point (6) and (7) of lemma 3.4.4 that given x ∈ J k n −J k−1 n a non-degenerate type II cell, Ant II (x) ⊂ J k−1 n . In particular, any cell x ∈ J k n has weak P -height at most k, hence by lemma 3.2.3 this shows that every cell has finite P -height and hence concludes the proof.
3.4.6
Corollary. For any f : X → Y with X cofibrant, the morphism:
X → Ex ∞ X × Ex ∞ Y Y
Is strongly anodyne.
Proof. Consider Ex k X × Ex k Y Y → Y and apply the functor Ex Y to it. One obtains:
Ex Y Ex k X × Ex k Y Y = Ex Ex k X × Ex k Y Y × Ex Y Y = Ex k+1 X × Ex k+1 Y Ex Y × Ex Y Y
In the last terms the map from the term Ex k+1 X × Ex k+1 Y Ex Y to Ex Y used in the fiber product is just the second projection, so the fiber product simplifies to:
Ex Y Ex k X × Ex k Y Y = Ex k+1 X × Ex k+1 Y Y
And the natural map Ex k X × Ex k Y Y → Ex Y Ex k X × Ex k Y Y corresponds through this identification to just:
n Ex k X × n Ex k Y Id Y : Ex k X × Ex k Y Y → Ex k+1 X × Ex k+1 Y Y
It follows by induction that the sequence of maps:
X → Ex X × Ex Y Y → · · · → Ex k X × Ex k Y Y → Ex k+1 X × Ex k+1 Y Y → . . .
are all strong anodyne maps (and all these objects are cofibrant), and the map X → Ex ∞ X × Ex ∞ Y Y is their transfinite composite (this last claim can either be observed very explicitly, or formally by commutation of directed colimits with finite limits). The proof given here, at least the case of a Kan fibration between cofibrant object, is essentially the proof proposition 2.1.41 of [3].
Applications
Proof. We start with the first half: we observed in 2.2.4 that strong fibrations are Kan fibrations. So we only need to show that any Kan fibration is a strong fibration. We first show this claim for p : A ։ B a Kan fibration between cofibrant object. One has that Ex ∞ (f ) is a Kan fibration (by 3.3.2) between fibrant objects (because of 3.3.4), hence it is a strong fibration (by lemma 2.2.6.(iii)), in particular any pullback of Ex ∞ (f ) is also a strong fibration. This gives a factorization of p:
A Ex ∞ (A) × Ex ∞ (B) B Ex ∞ A B Ex ∞ B ∼ p Ex ∞ p
in an anodyne map (by corollary 3.4.6) followed by strong fibration as a pullback of the strong fibration Ex ∞ (p). So p is a retract of the strong fibration part by the retract lemma (2.2.5) and hence is itself a strong fibration. We now move to the case of a general Kan fibration. We first show that a Kan fibration that is also an equivalence is a trivial fibration. Let p : X → Y be such a Kan fibration and weak equivalence, one needs to show that it has the right lifting property against all boundary inclusion: ∂∆[n] ֒→ ∆[n], consider such a lifting problem:
∂∆[n] X ∆[n] Y f
One first factors the map ∆[n] → Y as a cofibration followed by a trivial fibration and we form a pullback of f along the fibration part to get a diagram:
∂∆[n] P X ∆[n] Z Y u ∼ f ′ f
∼ By 2-out-of-3 the new fibration f ′ is again a weak equivalence, but note that now the object Z is cofibrant. One can further factor u in a cofibration followed by a trivial fibration:
∂∆[n] K P X ∆[n] Z Y f ′′ ∼ ∼ f ′ f ∼
f ′′ is a Kan fibration between cofibrant objects, hence is a strong fibration by the first part of the proof, moreover it is an equivalence hence it is a trivial fibration by the last point of lemma 2.2.6, and hence it has the right lifting property against the boundary inclusion which show that the morphism f is a trivial fibration as well. One can then concludes the proof by the same argument as used in the proof of the first part of lemma 2.2.6: Given a lifting problem of a trivial cofibration against a Kan fibration one can, using appropriate factorization, reduce to the case where the top and bottom map of the lifting square are weak equivalences, in which case the Kan fibration is a weak equivalence by 2-out-of-3 and hence is a trivial fibration by the claim we just made, and hence has the right lifting property against all cofibration which concludes the proof. For the second half of the proposition, given a trivial cofibration j one factors it as an anodyne morphisms followed by a Kan fibration. By the first half of the proof the Kan fibration is a strong fibration and hence has the right lifting property against j. It immediately follows from the retract lemma 2.2.5 that j is a retract of the anodyne morphism and hence is anodyne it self. Proof. We start with the case where all the objects in the pullback are cofibrant. This implies that the pullback itself is cofibrant because it is a subobject of the product which is cofibrant because of the cartesianess of the model structure 3.2.6, and the explicit description of cofibrant objects in terms of decidability of degeneratness of cell, immediately shows that a subobject of a cofibrant simplicial sets is cofibrant. In this case, the result follows immediately from an application of Kan's Ex ∞ functor: It preserves the pullback square (because it is a right adjoint), it send each object to a fibrant object, when all the object are fibrant the result is true in any (weak) model category (a clearly constructive argument, valid in weak model category is given as corollary 2.4.4 in [6]), and it detect equivalences between cofibrant objects because the morphism X → Ex ∞ X is anodyne (hence an equivalence) for X cofibrant. It appears that having right properness when all the objects are cofibrant is sufficient to deduce the general case by taking cofibrant replacement of all the objects involved in the appropriate order: Given a pullback P = B × A C one constructs cofibrant replacement of B c ∼ ։ B, . . . which still form a diagram such that the comparison maps B c × A c C c → B× A C is again a trivial fibrations. This is achieved by constructing first A c and then defining B c and C c respectively as cofibrant replacement of the pullbacks B × A A c and C × A A c . Assuming moreover that B ։ A is a fibration one also obtains this way that B c ։ A c is a fibration. Once this is done one deduces immediately the result in the general case from the result for the cofibrant replacement.
An epimorphism of simplicial set p : A → B is a degeneracy quotient if and only if for any map f : A → X, the map f factors through p if and only if the following condition holds:
of f through p is always unique as p is an epimorphism, so saying that f factors through p if and only if condition (D') (or (D) ) holds is equivalent to saying that B (endowed with the map p : A → B) has the universal property of A[(a i , σ i )] where (a i , σ i ) are all the pairs of a i ∈ A([n]) such that p(a i ) is σ i -degenerated. Hence this indeed holds if and only if A → B is a degeneracy quotient, as because of proposition 3.1.6, any degeneracy quotient p : A → B is isomorphic to A → A[(a i , σ i )] where (a i , σ i ) are all the pairs of a i ∈ A([n]) such that p(a i ) is σ i -degenerated.
By the second half of proposition 3.1.6, the map p is a degeneracy quotient if and only if the second maps in the factorization A → A[(a i , σ i )] → B is an isomorphism, which happens if and only if the relation ∼ p is equivalent to p(a) = p(a ′ ). We continue with a proposition that will be convenient to get examples of degeneracy quotient (see for example the proof of lemma 3.4.3 for examples).
[k] is of the form of proposition 3.1.10. The map σ : [n] ։ [m] is of this form, with the section [n] → [m] sending each i ∈ [m] to the smallest element of the fiber, this gives an order preserving idempotent π : [n] → [n] such that πx x. This induce an idempotent on P sending a pair (i, j) (with i ∈ [k], j ∈ [n]) to π ′ (i, j) = (i, πj)
[n] ֒→ ∆[n]. The general idea of this definition is that in such an iterated pushout cells are added by pairs: each pushout by a horn inclusion Λ i [n] → ∆[n] adds exactly two non-degenerate cells: (I) The cell P corresponding to the identity of ∆[n]. (II) The cell F corresponding to the the i-th face ∂ i [n] : ∆[n − 1] → ∆[n].
And one defines Ant(b) as Ant 0 (b) together with all (iterated) faces of cells appearing in Ant 0 (b).
A i-cell of ∆[n] × ∆[m] is an order preserving function [i] → [n] × [m]. It is non-degenerate if and only if it is an injective function. The domain D of i × j is:∆[n] × Λ k [m] ∂∆[n]×Λ k [m] (∂∆[n] × ∆[m]) = ∆[n] × Λ k [m] (∂∆[n] × ∆[m])It corresponds to the morphisms [i] → [n] × [m] such that either they skip a column or they skip a row other than k, where we consider that [n] = {0, . . . , n} numbers the column of [n] × [m] and [m] = {0, . . . , k, . . . , m} numbers the row. So the only non-degenerate cell of ∆[n] × ∆[m] that are not in D are injection [i] → [n] × [m]whose first projection takes all possible value, and whose second projection takes all possible values except maybe k.
the barycentric subdivision functor ∆ → ∆: ∆[n] → Sd ∆[n] := N K([n]) Where K([n]) denotes the set of finite non-empty decidable subsets of [n]. Functoriality in [n] is given by direct image of subsets on K[n]). This extend to an adjunction: Sd : ∆ ⇆ ∆ : Ex with:
Sd preserves cofibrations and anodyne morphisms, Ex preserves fibrations and trivial fibrations. Proof. It is enough to check that the image of the generating cofibrations and generating anodyne maps by Sd are cofibrations and anodyne respectively. In both case one can use proposition 3.3.1 to computes Sd on the generators as they are image of semi-simplicial maps. This makes the results immediate for cofibrations: Sd ∂∆[n] → Sd ∆[n] is the morphism N (K[n] − {[n]}) → N (K[n]) which is clearly a levelwise complemented monomorphisms between finite decidable, hence cofibrant, simplicial sets. For anodyne: Sd Λ i [n] → Sd ∆[n] is the morphisms N (K[n] − {[n], [n] − {i}}) → N (K[n]
Kan fibration are the same as the strong fibrations of definition 2.2.3. Dually, the trivial cofibrations of definition 2.2.3 are the same as anodyne morphisms.
The model structure of 2.2.8 is right proper, i.e. the pullback of a weak equivalence along a fibration is again a fibration.
The boundary inclusion map is denotes ∂ n or ∂[n] : ∂∆[n] → ∆[n], the i-th face maps is denoted ∂ i [n] or ∂ i n or just ∂ i : ∆[n − 1] → ∆[n], for the map corresponding to the order preserving injection from [n − 1] to [n] which only skip i. The degeneracy ∆[n + 1] → ∆[n] that hits i twice is denoted σ i . Given a simplicial or semi-simplicial sets X, the image of a cell x ∈ X n be the i-th face map is denoted d i x. Constructing the model structure 2.1 Review of the weak model structures 2.1.1. One of the achievement of1.8 Notation. ∆ and ∆ + denotes the category of finite non-empty ordinal, re-
spectively with non-decreasing map and non-decreasing injection between them.
∆ is the category of simplicial sets and ∆ + is the category of semi-simplicial sets
(see 2.1.2). One denotes by ∆[n] and ∆ + [n] the representable simplicial and
semi-simplicial sets corresponding to the ordinal [n] = {0, . . . , n}. Our usual no-
tation for the boundary of the n-simplex and its k-th horn, both for simplicial
and semi-simplicial versions are: ∂∆[n] Λ k [n] ∂∆ + [n] Λ k
+ [n]
2
). But for example the codiagonal map ∆ + [0] ∆ + [0] → ∆ + [0], where ∆ + [0] denotes the representable semi-simplicial sets by the ordinal [0] = {0} is easily seen to have the lifting property of trivial fibrations (there is no higher cells to lift ! ) while it is clearly not a weak equivalence.
The fibrations are the strong fibration of definition 2.2.3.Proof. We have two weak factorization systems, trivial cofibrations have been defined as the cofibrations that are equivalences, and it was shown in 2.2.6 that trivial fibrations are the (strong) fibrations that are equivalences. Equivalences are stable by composition, satisfies 2-out-of-6 and contains isomorphisms by definition, so this concludes the proof.2.2.9 Proposition. The model structure of theorem 2.2.8 is left proper, i.e.
the pushout of weak equivalence along a cofibration is a weak equivalence.
). Conversely, let x satisfying (ii). If σ is the identity the result is trivial. If σ is not injective, then x is in particular degenerated, i.e. there exist a nontrivial degeneracy s : [n] → [k] such that x = s * y. Note that y = d * x for d : [k] → [n] any section of s. If for all section d of s, σd is injective, then lemma 3.1.3 below shows that s factors as jσ for some degeneracy j : [m] → [k]
3.1.3 Lemma. Let σ : [n] → [m] and s : [n] → [k] be two degeneracy, assume that for all d : [k] → [n] a section of s, σd is injective, then there exists a (unique) j : [m] → [k] such that s = jσ.
one introduces two functions between the Sd ∆[n]. Let j k n : Sd ∆[n] → Sd ∆[n] and r k n : Sd ∆[n + 1] → Sd ∆[n] be the maps defined at the level of posets by:
0 n : K[n] → K[n] has image isomorphic to [n], with j 0 n : K[n] → [n] being the "Max" function used in the definition of the natural transformation Sd ∆[n] → ∆[n]. We define:
We will give a more detailed account of its contents at the beginning of this section.
not exactly though: in the standard terminology a right semi-model structure concern a weakening of the cofibration/trivial fibration factorization to arrows with fibrant target, where here it is the other weak factorization system which is concerned.
In most of the literature this are called weak equivalence, though we can't think of any reasons to keep the adjective "weak" other than history, so we will simply drop it.4 Here we use the fact that that trivial fibrations are characterized by a lifting property against cofibration between cofibrant objects, which might not be the case in a general weak model category.
using the so-called "Joyal-Tierney calculus" presented in the appendix of[7], though this types of manipulation were known before, maybe in a less elegant or general way.
A set X is said to be Kuratowski-finite if ∃n, ∃x 1 , . . . , xn ∈ X, ∀x ∈ X, x = x 1 or . . . or x = xn.
Here the restriction to "ω" is only to avoid the discussion of what is an ordinal constructively.
It appears that because of point 2 of lemma 3.4.4 and the fact that r 0 n is the same as Sd σ 0 it is actually a consequence from the rest of the definition that type I cells are not in X.
3. The sets J n k ⊂ (Ex Y X) n are decidable.Proof. All these decidability problems corresponds to the decidability of a factorization of a map Sd ∆[n] → X through some epimorphism Sd ∆[n] → K. In all this case we will show that the corresponding epimorphism is a degeneracy quotient using lemma 3.1.10 and conclude about the decidability using lemma 3. We can now give the definition of the P -structure on X ֒→ Ex Y X.• Type I cells are the non-degenerated cells v ∈ Ex Y (X) which are not 8 in X and can be written as yr k n with y ∈ J k n ⊂ Ex Y X. • Point 8 of lemma 3.4.4 will prove that being type I is decidable. Type II cells are just the cells that are not of type I (and which are non-degenerated and not in X).• For any cell x one defines P x as xr k n where k is the smallest integer such that x ∈ J k n , i.e. x ∈ J k n − J k−1 n . Lemma 3.4.3 shows that the J k n are decidable so there is indeed such a smaller integer k.In order to show that being type I is decidable and that P defined this way defines a bijection from type II cells to type I cells, one needs a few technical lemma that we have regrouped in:2.x ∈ J k n if and only if P x ∈ J k n+1 3. If x ∈ J k−1 n then xr k n is degenerate.4. P 2 x is always degenerated.5. If x is degenerated or type I or in X, then P x is degenerated.6. If x ∈ J k n − J k−1 n then for all i k d i (P x) ∈ J k−1 n .
Notes on constructive set theory. P Aczel, M Rathjen, P. Aczel and M. Rathjen. Notes on constructive set theory. Available from http://www1.maths.leeds.ac.uk/~rathjen/book.pdf, 2010.
On left and right model categories and left and right Bousfield localizations. Clark Barwick, Homology, Homotopy and Applications. 122Clark Barwick. On left and right model categories and left and right Bous- field localizations. Homology, Homotopy and Applications, 12(2):245-320, 2010.
Les préfaisceaux comme modèles des types d'homotopie. Société mathématique de France. Denis-Charles Cisinski, Denis-Charles Cisinski. Les préfaisceaux comme modèles des types d'homotopie. Société mathématique de France, 2006.
Towards a constructive simplicial model of Univalent Foundations. Nicola Gambino, Simon Henry, ArXiv. To appears onNicola Gambino and Simon Henry. Towards a constructive simplicial model of Univalent Foundations. To appears on ArXiv, 2019.
Understanding the small object argument. Richard Garner, 17Applied categorical structuresRichard Garner. Understanding the small object argument. Applied cate- gorical structures, 17(3):247-285, 2009.
Weak model categories in constructive and classical mathematics. Simon Henry, ArXiv:1807.02650v2ArXiv preprintSimon Henry. Weak model categories in constructive and classical mathe- matics. ArXiv preprint, ArXiv:1807.02650v2, 2018.
Quasi-categories vs segal spaces. ArXiv preprint math/0607820. André Joyal, Myles Tierney, André Joyal and Myles Tierney. Quasi-categories vs segal spaces. ArXiv preprint math/0607820, 2006.
Notes on simplicial homotopy theory. André Joyal, Myles Tierney, PreprintAndré Joyal and Myles Tierney. Notes on simplicial homotopy theory. Preprint, 2008.
On css complexes. M Daniel, Kan, American Journal of Mathematics. 793Daniel M Kan. On css complexes. American Journal of Mathematics, 79(3):449-476, 1957.
An induction principle for consequence in arithmetic universes. Maria Emilia Maietti, Steven Vickers, Journal of Pure and Applied Algebra. 2168-9Maria Emilia Maietti and Steven Vickers. An induction principle for con- sequence in arithmetic universes. Journal of Pure and Applied Algebra, 216(8-9):2049-2067, 2012.
Cellular categories. Michael Makkai, Jiří Rosický, Journal of Pure and Applied Algebra. 2189Michael Makkai and Jiří Rosický. Cellular categories. Journal of Pure and Applied Algebra, 218(9):1652-1664, 2014.
Another approach to the kan-quillen model structure. Sean Moss, arXiv:1506.04887arXiv preprintSean Moss. Another approach to the kan-quillen model structure. arXiv preprint arXiv:1506.04887, 2015.
Partial horn logic and cartesian categories. Erik Palmgren, J Steven, Vickers, Annals of Pure and Applied Logic. 1453Erik Palmgren and Steven J Vickers. Partial horn logic and cartesian categories. Annals of Pure and Applied Logic, 145(3):314-353, 2007.
| [] |
[
"A new characterization of Conrad's property for group orderings, with applications",
"A new characterization of Conrad's property for group orderings, with applications"
] | [
"Andrés Navas ",
"Cristóbal Rivas ",
"Adam Clay "
] | [] | [] | We provide a pure algebraic version of the dynamical characterization of Conrad's property given in[10]. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof of a theorem first established by Linnell: an orderable group having infinitely many orderings has uncountably many. This proof is achieved by extending to uncountable orderable groups a result of the first author about orderings which may be approximated by their conjugates. This last result is illustrated by an example of an exotic ordering on the free group given by the third author in the Appendix. * Funded by the PBCT/Conicyt Research Network on Low Dimensional Dynamics. | 10.2140/agt.2009.9.2079 | [
"https://arxiv.org/pdf/0901.0880v2.pdf"
] | 5,117,846 | 0901.0880 | 32c8ece7d7eb40e36ff9cbac3705548bc5f8dd13 |
A new characterization of Conrad's property for group orderings, with applications
18 Jan 2009
Andrés Navas
Cristóbal Rivas
Adam Clay
A new characterization of Conrad's property for group orderings, with applications
18 Jan 2009
We provide a pure algebraic version of the dynamical characterization of Conrad's property given in[10]. This approach allows dealing with general group actions on totally ordered spaces. As an application, we give a new and somehow constructive proof of a theorem first established by Linnell: an orderable group having infinitely many orderings has uncountably many. This proof is achieved by extending to uncountable orderable groups a result of the first author about orderings which may be approximated by their conjugates. This last result is illustrated by an example of an exotic ordering on the free group given by the third author in the Appendix. * Funded by the PBCT/Conicyt Research Network on Low Dimensional Dynamics.
Introduction
In recent years, relevant progress has been made in the theory of (left) orderable groups. This has been achieved mainly by means of the use of a recently introduced mathematical object, namely the space of group orderings (see for instance [4,6,8,11]). This space may be endowed with a natural topology (roughly, two orderings are close if they coincide over large finite sets), and the study of this topological structure should reveal some algebraic features of the underlying group. In [10] it was realized that, for this study, the classical Conrad property for group orderings becomes relevant. Bringing ideas and techniques from the theory of codimension-one foliations, the 'dynamical' insight of this property was revealed. Unfortunately, many proofs of [10] are difficult to read for people with a pure algebraic view of orderable groups. More importantly, some of the results therein do not cover the case of uncountable groups. Indeed, the dynamical analysis of group orderings is done via the so-called 'dynamical realization' of orderable groups as groups of homeomorphisms of the line, which is not available for general uncountable orderable groups.
Motivated by this, we develop here an algebraic counterpart of (part of) the analysis of [10]. We begin by giving a new characterization of the Conrad property that is purely algebraic, although it has a dynamical flavor (c.f., Theorem 1.4). This leads naturally to the notion of Conradian actions on totally ordered spaces. A relevant example concerns the action of an ordered group on the space of cosets with respect to a convex subgroup. In this setting, we define the notion of Conradian extension (c.f., Example 1.10), and we generalize Conrad's classical theorem on the 'level' structure of groups admitting Conradian orderings (c.f., Theorem 1.13, Corollary 1.14).
A relevant concept introduced in [10] is the Conradian soul, which corresponds to the maximal subgroup of an ordered group that is convex and restricted to which the ordering is Conradian. In [10], a more geometrical view of this notion was given in the case of countable groups. Here we provide an analogous algebraic description which applies to general (possibly uncountable) ordered groups (c.f., Theorem 2.1).
The Conradian soul was introduced as a main tool for dealing with the problem of approximating a group ordering by its conjugates. For instance, it was shown in [10] that if the Conradian soul of an ordering on a non-trivial countable group is trivial, then this ordering is an accumulation point of its set of conjugates. The extension of this result to uncountable orderable groups appears here as Theorem 2.7. We point out that an independent proof using completely different ideas was given by Adam Clay in [2].
Based on the work of Linnell [6], it was shown in [10] that if an ordering on a group is isolated in the corresponding space of orderings, then its Conradian soul is 'almost trivial', in the sense that it has only finitely many orderings. It is then natural to deal with ordered groups (Γ, ) for which the Conradian soul C (Γ) is non-trivial but has only finitely many orderings. If is not Conradian, then to each of the orderings on C (Γ) corresponds an ordering on Γ (roughly, the new orderings on Γ are obtained by changing the original one on C (Γ) but preserving the set of elements bigger than the identity outside). As it was proved in [10], at least one of these orderings on Γ is an accumulation point of its set of conjugates provided that Γ is countable. Here we extend this result to the case of uncountable groups (c.f., Corollary 2.10).
The property of being approximated by its conjugates does not hold for all of the finitely many orderings on Γ obtained by the preceding construction. A remarkable example illustrating this fact, namely the Dubrovina-Dubrovin ordering DD on braid groups B n [4], was extensively studied from this point of view in [10]. In the Appendix, Adam Clay provides a different kind of example, namely an 'exotic' ordering C on the free group F 2 . As it is the case of DD , the Conradian soul of C is isomorphic to Z, and C is not an accumulation point of the set of its conjugates. (This answers by the negative a question suggested in [10,Remark 4.11].) The main difference between DD and C lies on the fact that DD is an isolated point of the (uncountable) space of orderings of B n , while C is non-isolated in the (also uncountable) space of orderings of F 2 . (Actually, the space of orderings of F 2 is homeomorphic to the Cantor set [7,10].)
As a final application of our methods, we give a new proof of a theorem first established by Linnell [6]: if a group has infinitely many orderings, then it has uncountably many. Linnell's proof uses an argument from General Topology for reducing the general case to that of Conradian orderings for which prior arguments by Zenkov [12] apply. To deal with the non Conradian case, we use our machinery on Conradian souls. Note that this was already done in [10] for countable groups: Theorem 3.1 here corresponds to the extension to the case of uncountable groups.
1 Crossings and Conradian orderings 1.1 An equivalent for Conrad's property Let be an ordering on a group Γ, that is, a total order relation which is invariant by left multiplication. Recall that is said to be Conradian if for all f ≻ 1 and all g ≻ 1 (for short, for all positive elements f, g) there exists n ∈ N such that f g n ≻ g. (See however Remark 1.5.) A subgroup Γ 0 of Γ is -Conradian if the restriction of to it is a Conradian ordering.
A crossing for the ordered group (Γ, ) is a 5-uple (f, g, u, v, w) of elements in Γ such that:
-u ≺ w ≺ v, -g n u ≺ v and f n v ≻ u for every n ∈ N, -there exist M, N in N so that f N v ≺ w ≺ g M u.
Remark 1.1. It follows from the third condition that neither f nor g can be equal to the identity.
Remark 1.2. If (f, g, u, v, w) is a crossing, then the inequalities f n v ≻ u and g n u ≺ v actually hold for every integer n. Indeed, we necessarily have f v ≺ v, since in the other case we would have
v ≻ w ≻ f N v ≻ f N −1 v ≻ . . . ≻ f v ≻ v, which is absurd. Therefore, for n > 0, f −n v ≻ f n−1 v ≻ . . . f −1 v ≻ v ≻ u.
The inequality g −n u ≺ v for n > 0 may be checked similarly.
Remark 1.3. The reason of the use of different type of letter for the elements f, g and u, v, w will become clear in §1.2. Somehow, u, v, w should be thought of as 'reference points' instead of genuine group elements (see Figure 1).
Figure 1: A crossing
u v w f N v g M u • • • • • f g
The next result is the natural analogue of both Propositions 3.16 and 3.19 of [10] in our setting. Proof. Suppose that is not Conradian, and let f, g be positive elements so that f g n ≺ g for every n ∈ N. We claim that (f, g, u, v, w) is a crossing for (Γ, ) for the choice u = 1, and v = f −1 g, and w = g 2 . Indeed:
-From f g 2 ≺ g one obtains g 2 ≺ f −1 g, and since g ≻ 1, this gives 1 ≺ g 2 ≺ f −1 g, that is, u ≺ w ≺ v.
-From f g n ≺ g one gets g n ≺ f −1 g, that is, g n u ≺ v (for every n ∈ N); on the other hand, since both f, g are positive, we have f n−1 g ≻ 1, and thus f n (f −1 g) ≻ 1, that is, f n v ≻ u (for every n ∈ N).
-The relation f (f −1 g) = g ≺ g 2 may be read as f N v ≺ w for N = 1; on the other hand, the relation
g 2 ≺ g 3 is w ≺ g M u for M = 3.
Conversely, assume that (f, g, u, v, w) is a crossing for (Γ, ) so that f N v ≺ w ≺ g M u (with M, N in N). We will prove that is not Conradian by showing that, for h = g M f N andh = g M , both elements w −1 hw and w −1h w are positive, but
(w −1 hw)(w −1h w) n ≺ w −1h w for all n ∈ N.
To show this, first note that gw ≻ w. Indeed, if not then we would have
w ≺ g N u ≺ g N w ≺ g N −1 w ≺ . . . ≺ gw ≺ w,
which is absurd. Clearly, the inequality gw ≻ w implies
g M w ≻ g M −1 w ≻ . . . ≻ gw ≻ w, and hence w −1h w = w −1 g M w ≻ 1.(1)
Moreover,
hw = g M f N w ≻ g M f N f N v = g M f 2N v ≻ g M u ≻ w.
and hence w −1 hw ≻ 1.
Now note that, for every n ∈ N,
hh n w = hg M n w ≺ hg M n g M u = hg M n+M u ≺ hv = g M f N v ≺ g M w =hw.
After multiplying by the left by w −1 , the last inequality becomes
(w −1 hw)(w −1h w) n = w −1 hh n w ≺ w −1h w,
as we wanted to check. Together with (1) and (2), this shows that is not Conradian.
Remark 1.5. A fact that will be not used in this work is that, for every Conradian group ordering , one actually has f g 2 ≻ g for all positive elements f, g (i.e., one can take 'n = 2' in the original definition). The proof given in [10,Proposition 3.7] uses the fact that, if f, g are positive elements for which f g 2 ≺ g, then letting h = f g one has f h n ≺ h for all n ∈ N. This is illustrated by Figure 2. Notice that, as shown below, in this situation (f, f g, 1, f g, g) is a crossing for M = N
1 g 2 • • f f g f g g f g g • • • • • f g Remark 1.6.
The second condition in the definition of crossing may seem difficult to handle. A more 'robust' property is that of reinforced crossing, which is a 5-uple (f, g, u, v, w) of elements in an ordered group (Γ, ) such that:
-u ≺ w ≺ v, -f u ≻ u and g(v) ≺ v, -there exist M, N in N so that f N v ≺ w ≺ g M u.
. Figure 3: A reinforced crossing
u v w f N v g M u • • • • • f g One easily checks that a reinforced crossing is a crossing. Conversely, if (f, g, u, v, w) is a crossing, then (f N g M , g M f N , f N w, g M w, w) is a reinforced crossing (here, M, N in N are such that f N v ≺ w ≺ g M u). Indeed, from the properties of crossing one gets f N g M (g M w) ≺ f N v ≺ w and g M f N (f N w) ≻ g M u ≻ w. Moreover, f N g M (f N w) ≻ f N g M u ≻ f N w and g M f N (g M w) ≺ g M f N v ≺ g M w.
Remark 1.7. The dynamical characterization of Conrad's property may serve as inspiration for introducing other relevant properties for group orderings. (Compare [10, Question 3.22].) For instance, one can say that a 6-uple (f, g, u, v, w 1 , w 2 ) of elements in an ordered group (Γ, ) is a (reinforced) double crossing if (see Figure 4):
-u ≺ w 1 ≺ w 2 ≺ v, -f u ≻ u and f v ≻ v, -gu ≻ w 1 , gv ≺ w 2 , and f w 2 ≺ w 1 .
Finding a simpler algebraic counterpart of the property of not having a double crossing for an ordering seems to be an interesting problem.
f g u v w 1 w 2 • • • •
An extension to group actions on ordered spaces
Let Γ be a group acting by order-preserving bijections on a totally ordered space (Ω, ≤). A crossing for the action of Γ on Ω is a 5-uple (f, g, u, v, w), where f, g belong to Γ and u, v, w are in Ω, such that:
-u ≺ w ≺ v, -g n u ≺ v and f n v ≻ u for every n ∈ N, -there exist M, N in N so that f N v ≺ w ≺ g M u.
Example 1.8. The real line carries a natural total order, and thus our definition applies to groups acting on it by orientation preserving homeomorphisms. The notion of crossing for this case is exactly the same as that of elements in transversal position in [10, Definition 3.24]. Example 1.9. If Γ is endowed with an ordering , one may take (Ω, ≤) = (Γ, ) as a totally ordered set. The action of Γ by left translations on it preserves the order: a crossing for this action corresponds to a crossing for (Γ, ), in the terminology of §1.1. Note that this example generalizes the preceding one for countable groups, since every countable ordered group may be canonically (up to semiconjugacy) realized as a group of orientation preserving homeomorphisms of the real line [10, §2.1].
For another relevant example recall that, given ordered group (Γ, ), a subset S is -convex if for every f 1 ≺ f 2 in S, every f ∈ Γ satisfying f 1 ≺ f ≺ f 2 belongs to S. When S is a subgroup, this is equivalent to that for all positivef ∈ S, every f ∈ Γ such that 1 ≺ f ≺f belongs to S. Example 1.10. Let (Γ, ) be an ordered group, and let Γ 0 be a -convex subgroup. The space of left cosets Ω = Γ/Γ 0 carries a natural total order ≤, namely f Γ 0 < gΓ 0 if f h 1 ≺ gh 2 for some h 1 , h 2 in Γ 0 (the reader will easily check that this definition is independent of the choice of h 1 and h 2 in Γ 0 ). The action of Γ by left translations on Ω preserves this order. (Note that taking Γ 0 as being the trivial subgroup, this example reduces to the preceding one.) Whenever this action has no crossings, we will say that Γ is a -Conradian extension of Γ 0 . Remark 1.11. Let (Γ, ) be an ordered group, and let Γ 0 be a -convex subgroup. Given any ordering * on Γ 0 , the extension of * by is the ordering * on Γ for which 1 ≺ * f if and only if either f ∈ Γ 0 and 1 ≺ * f , or f / ∈ Γ 0 and 1 ≺ f . The reader can easily check that Γ 0 is still a * -convex subgroup of Γ. Moreover, Γ is a -Conradian extension of Γ 0 if and only if it is a * -Conradian extension of it.
For a general order-preserving action of a group Γ on a totally ordered space (Ω, ≤), the action of an element f ∈ Γ is said to be cofinal if for all x < y in Ω there exists n ∈ Z such that f n (x) > y. Note that if the action of f is not cofinal, then there exist x < y in Ω such that f n (x) < y for every integer n. Proposition 1.12. Let Γ be a group acting by order-preserving bijections on a totally ordered space (Ω, ≤). If the action of Γ on Ω has no crossings, then the set of elements whose action is not cofinal forms a normal subgroup of Γ.
Proof. Let us denote the set of elements whose action is not cofinal by Γ 0 . This set is normal. Indeed, given g ∈ Γ 0 , let x < y in Ω be such that g n (x) < y for all n. For each h ∈ Γ we have g n h −1 (h(x)) < y, and hence (hgh −1 ) n (h(x)) < h(y) (for all n ∈ Z). Since h(x) < h(y), this shows that hgh −1 belongs to Γ 0 .
It follows immediately from the definition that Γ 0 is stable under inversion, that is, g −1 belongs to Γ 0 for all g ∈ Γ 0 . The fact that Γ 0 is stable by multiplication is more subtle. For the proof, given x ∈ Ω and g ∈ Γ 0 , we will denote by I g (x) the convex closure of the set {g n (x): n ∈ Z}, that is, the set formed by the y ∈ Ω for which there exists m, n in Z so that g m (x) ≤ y ≤ g n (x). Note that I g (x) = I g (x ′ ) for all x ′ ∈ I g (x); moreover, I g −1 (x) = I g (x) for all g ∈ Γ 0 and all x ∈ Ω; finally, if g(x) = x, then I g (x) = {x}. We claim that if I g (x) and I f (y) are non-disjoint for some x, y in Ω and f, g in Γ 0 , then one of them contains the other. Indeed, assume that there exist non-disjoint sets I f (y) and I g (x), none of which contains the other. Without loss of generality, we may assume that I g (x) contains points to the left of I f (y) (if this is not the case, just interchange the roles of f and g). Changing f and/or g by their inverses if necessary, we may assume that g(x) > x and f (y) < y, and hence g(x ′ ) > x ′ for all x ′ ∈ I g (x), and f (y ′ ) < y ′ for all y ′ ∈ I y (f ). Take u ∈ I g (x) \ I f (y), w ∈ I g (x) ∩ I f (y), and v ∈ I f (y) \ I g (x). Then one easily checks that (f, g, u, v, w) is a crossing, which is a contradiction.
Let now g, h be elements in Γ 0 , and let x 1 < y 1 and x 2 < y 2 be points in Ω such that g n (x 1 ) < y 1 and h n (x 2 ) < y 2 for all n ∈ Z. Put x = min{x 1 , x 2 } and y = max{y 1 , y 2 }. Then g n (x) < y and h n (x) < y for all n ∈ Z; in particular, y does not belong to neither I g (x) nor I h (x). Since x belongs to both sets, we have either I g (x) ⊂ I h (x) or I h (x) ⊂ I g (x). Both cases being analogous, let us consider only the first one. Then for all x ′ ∈ I g (x) we have I h (x ′ ) ⊂ I g (x ′ ) = I g (x). In particular, h ±1 (x ′ ) belongs to I g (x) for all x ′ ∈ I g (x). Since the same holds for g ±1 (x ′ ), this easily implies that (gh) n (x) ∈ I g (x) for all n ∈ Z. As a consequence, (gh) n (x) < y for all n ∈ Z, thus showing that gh belongs to Γ 0 .
Recall that for an ordered group (Γ, ), a convex jump is a pair (G, H) of distinct -convex subgroups such that H is contained in G, and there is no -convex subgroup between them. The previously developed ideas lead naturally to the following result, which may be viewed as an extension of Conrad's theorem on the structure of convex subgroups for Conradian orderings [3, Theorem 4.1]. However, our proof follows ideas which are rather different from those of Conrad, and is much inspired from [9, Exercise 2.2.46]. Theorem 1.13. Let (Γ, ) be an ordered group, and let (G, H) be a convex jump in Γ. Suppose that G is a Conradian extension of H. Then H is normal in G, and the ordering induced by on the quotient G/H is Archimedean (and hence order isomorphic to a subgroup of (R, +), due to Hölder's theorem [1,5,9]).
Proof. Let us consider the action of G on the space of cosets G/H. Each element in H fixes the coset H, and hence its action is not cofinal. By Proposition 1.12, if we show that the action of each element in G \ H is cofinal, then this will give the normality of H in G. Now given f ∈ G \ H, let G f the smallest convex subgroup of G containing H and f . We claim that G f coincides with the set S f = {g ∈ G : f m ≺ g ≺ f n for some m, n in Z}. Indeed, S f is clearly a convex subset of G containing H and contained in G f . Thus, for showing that G f = S f , we need to show that S f is a subgroup. For this, first note that, in the notation of the proof of Proposition 1.12, the conditions g ∈ S f and I g (H) ⊂ I f (H) are equivalent. Therefore, for each g ∈ S f we have I g −1 (H) = I g (H) ⊂ I f (H), and thus g −1 ∈ S f . Moreover, ifḡ is another element in S f , thenḡgH ∈ḡ(I f (H)) = I f (H), and thus Iḡ g (H) ⊂ I f (H). This means thatḡg belongs to S f , thus concluding the proof that S f and G f coincide.
Each f ∈ G \ H leads to a convex subgroup G f = S f strictly containing H. Since (G, H) is a convex jump, we necessarily have S f = G. Given g 1 ≺ g 2 in G, choose m 1 , n 2 in Z for which f m 1 ≺ g 1 and g 2 ≺ f n 2 . Then we have f n 2 −m 1 g 1 ≻ f n 2 −m 1 f m 1 = f n 2 ≻ g 2 , and hence f n 2 −m 1 (g 1 H) ≥ g 2 H. This easily implies that the action of f is cofinal.
We have then show that H is normal in G. The left invariant total order on the space of cosets G/H is therefore a group ordering. Moreover, given f, g in G, with f / ∈ H, the previous argument shows that there exists n ∈ Z such that f n ≻ g, and thus f n H gH. This is nothing but the Archimedean property for the induced ordering on G/H. Corollary 1.14. Under the hypothesis of Theorem 1.13, up to multiplication by a positive real number, there exists a unique nontrivial group homomorphism τ : G → R such that ker(τ ) = H and τ (g) > 0 for every positive element g ∈ G \ H.
2 On the approximation of a group ordering by its conjugates
Describing the Conradian soul via crossings
The Conradian soul C (Γ) of an ordered group (Γ, ) corresponds to the (unique) subgroup which is -convex, -Conradian, and which is maximal among subgroups verifying these two properties simultaneously. This notion was introduced in [10], where a dynamical counterpart in the case of countable groups was given. To give an analogous characterization in the general case, we consider the set S + formed by the elements h ≻ 1 such that h w for every crossing (f, g, u, v, w) satisfying 1 u. Analogously, we let S − be the set formed by the elements h ≺ 1 such that w h for every crossing (f, g, u, v, w) satisfying v 1. Finally, we let
S = {1} ∪ S + ∪ S − .
A priori, it is not clear that the set S has a nice structure (for instance, it is not at all evident that it is actually a subgroup). However, this is largely shown by the theorem below.
Theorem 2.1. The Conradian soul of (Γ, ) coincides with the set S above.
Before passing to the proof, we give four general lemmas on crossings for group orderings (note that the first three lemmas still apply to crossings for actions on totally ordered spaces). The first one allows us replacing the 'comparison'element w by its 'images' under positive iterates of either f or g. If (f, g, u, v, w) is a crossing, then (f, g, u, v, g n w) and (f, g, u, v, f n w) are also crossings for every n ∈ N.
Lemma 2.2.
Proof. We will only consider the first 5-uple (the case of the second one is analogous). Recalling that gw ≻ w, for every n ∈ N we have u ≺ w ≺ g n w; moreover, v ≻ g M +n u = g n g M u ≻ g n w. Hence, u ≺ g n w ≺ v. On the other hand, f N v ≺ w ≺ g n w, while from g M u ≻ w we get g M +n u ≻ g n w.
Our second lemma allows replacing the 'limiting' elements u and v by more appropriate ones. Lemma 2.3. Let (f, g, u, v, w) be a crossing. If f u ≻ u (resp. f u ≺ u) then (f, g, f n u, v, w) (resp. (f, g, f −n u, v, w)) is also a crossing for every n > 0. Analogously, if gv ≺ v (resp. gv ≻ v), then (f, g, u, g n v, w) (resp. (f, g, u, g −n v, w)) is also crossing for every n > 0.
Proof. Let us only consider the first 5-uple (the case of the second one is analogous). Suppose that f u ≻ u (the case f u ≺ u may be treated similarly). Then f n u ≻ u, which gives g M f n u ≻ g M u ≻ w.
To show that f n u ≺ w, assume by contradiction that f n u w. Then f n u ≻ f N v, which gives u ≻ f N −n v, which is absurd.
The third lemma relies on the dynamical insight of the crossing condition. Lemma 2.4. If (f, g, u, v, w) is a crossing, then (hf h −1 , hgh −1 , hu, hv, hw) is also a crossing for every h ∈ Γ.
Proof. The three conditions to be checked are nothing but the three conditions in the definition of crossing multiplied by h by the left.
A direct application of the lemma above shows that, if (f, g, u, v, w) is a crossing, then the 5uples (f, f n gf −n , f n u, f n v, f n w) and (g n f g −n , g, g n u, g n v, g n w) are also crossings for every n ∈ N. This combined with Lemma 2.3 may be used to show the following.
Lemma 2.5. If (f, g, u, v, w) is a crossing and 1 h 1 ≺ h 2 are elements in Γ such that h 1 ∈ S and h 2 / ∈ S, then there exists a crossing (f ,g,ũ,ṽ,w) such that h 1 ≺ũ ≺ṽ ≺ h 2 .
Proof.
Since 1 ≺ h 2 / ∈ S, there must be a crossing (f, g, u, v, w) such that 1 u ≺ w ≺ h 2 . Let N ∈ N be such that f N v ≺ w. Denote by (f,ḡ,ū,v,w) the crossing (f, f N gf −N , f N u, f N v, f N w). Note thatv = f N v ≺ w ≺ h 2 . We claim that h 1 w = f N w.
Indeed, if f N u ≻ u then f n u ≻ 1, and by the definition of S we must have h 1
w. If f N u ≺ u, then we must have f u ≺ u, so by Lemma 2.3 we know that (f,ḡ, u,v,w) is also a crossing, which allows still concluding that h 1 w. Now for the crossing (f,ḡ,ū,v,w) there exists M ∈ N such thatw ≺ḡ Mū . Let us consider the crossing (ḡ M fḡ −M ,ḡ,ḡ Mū ,ḡ Mv ,ḡ Mw ). Ifḡ Mv ≺v thenḡ Mv ≺ h 2 , and we are done. If not, then we must haveḡv ≻v. By Lemma 2.3, (ḡ M fḡ −M ,ḡ,ḡ Mū ,ḡ Mv ,w) is still a crossing, and sincē v ≺ h 2 , this concludes the proof.
Proof of Theorem 2.1. The proof is divided into several steps.
Claim 0. The set S is convex.
This follows directly from the definition of S. Claim 1. If h belongs to S, then h −1 also belongs to S.
Assume that h ∈ S is positive and h −1 does not belong to S. Then there exists a crossing (f, g, u, v, w) so that h −1 ≺ w ≺ v 1.
We first note that, if h −1 u, then after conjugating by h as in Lemma 2.4, we get a contradiction because (hgh −1 , hf h −1 , hu, hv, hw) is a crossing with 1 hu and hw ≺ hv h. To reduce the case h −1 ≻ u to this one, we first use Lemma 2.4 and we consider the crossing
(g M f g −M , g, g M u, g M v, g M w). Since h −1 ≺ w ≺ g M u ≺ g M w ≺ g M v, if g M v ≺ v then we are done.
If not, Lemma 2.3 shows that (g M f g −M , g, g M u, g M v, w) is also a crossing, which still allows concluding.
For the case where h ∈ S is negative (i.e., its inverse is positive) we proceed similarly but we conjugate by f N instead of g M . Alternatively, since 1 ∈ S and 1 ≺ h −1 , if we suppose that h −1 / ∈ S then Lemma 2.5 provides us with a crossing (f, g, u, v, w) such that 1 ≺ u ≺ w ≺ v ≺ h −1 , which gives a contradiction after conjugating by h. Claim 2. If h andh belong to S, then hh also belongs to S.
First we show that for every positive elements in S, their product still belongs to S. (Note that, by Claim 1, the same will be true for products of negative elements in S.) Indeed, suppose that h,h are positive elements, with h ∈ S but hh / ∈ S. Then, by Lemma 2.5 we may produce a crossing (f, g, u, v, w) such that h ≺ u ≺ v ≺ hh. After conjugating by h −1 we obtain the crossing
(h −1 f h, h −1 gh, h −1 u, h −1 v, h −1 w) satisfying 1 ≺ h −1 u ≺ h −1 w ≺h, which shows thath / ∈ S. Now, if h ≺ 1 ≺h then h ≺ hh.
Hence, if hh is negative then the convexity of S gives hh ∈ S. If hh is positive, thenh −1 h −1 is negative, and sinceh −1 ≺h −1 h −1 , the convexity gives again that h −1 h −1 , and hence hh, belongs to S. The remaining caseh ≺ 1 ≺ h may be treated similarly.
Claim 3. The subgroup S is Conradian.
In order to apply Theorem 1.4, we need to show that there are no crossings in S. Suppose by contradiction that (f, g, u, v, w) is a crossing such that f, g, u, v, w all belong to S. If 1 w then, by Lemma 2.4, we have that (g n f g −n , g, g n u, g n v, g n w) is a crossing. Taking n = M so that g M u ≻ w, this gives a contradiction to the definition of S because 1
w ≺ g M u ≺ g M w ≺ g M v ∈ S.
The case w 1 may be treated in an analogous way by conjugating by powers of f instead of g. Indeed, if C is a subgroup strictly containing S, then there is a positive h in C \ S. By Lemma 2.5, there exists a crossing (f, g, u, v, w) such that 1 ≺ u ≺ w ≺ v ≺ h. If C is convex, then u, v, w belong to C. To conclude that C is not Conradian, it suffices to show that f and g belong to C.
Since 1 ≺ u, we have either 1 ≺ g ≺ gu ≺ v or 1 ≺ g −1 ≺ g −1 u ≺ v.
In both cases, the convexity of C implies that g belongs to C. On the other hand, if f is positive then from f N ≺ f N v ≺ w we get f ∈ C, whereas in the case of a negative f the inequality 1 ≺ u gives 1 ≺ f −1 ≺ f −1 u ≺ v, which still shows that f ∈ C.
Approximation of group orderings: the role of the Conradian soul
For a (left) orderable group Γ, we denote by LO(Γ) the set of all orderings on Γ. This space carries the topology having as a subbasis the family of sets U f = { : f ≻ 1}, where f = 1. Endowed with this topology, LO(Γ) is called the space of (left) orderings of the group Γ. Remark 2.6. As shown in [9], a simple application of Tychonov's theorem shows that LO(Γ) is always compact. Moreover, the 'n = 2' property from Remark 1.5 implies that the subset of Conradian orderings is closed therein (and hence compact). A more dynamical argument for showing this consists in noticing that the condition that (f, g, u, v, w) is a reinforced crossing for prescribed M, N is clearly open in LO(Γ) (c.f., Remark 1.6).
The positive cone of an ordering in LO(Γ) is the set P of its positive elements. Because of the left invariance, P completely determines . The conjugate of by h ∈ Γ is the ordering h having positive cone hP h −1 . In other words, g ≻ h 1 holds if and only if hgh −1 ≻ 1. We will say that may be approximated by its conjugates if it is an accumulation point of its set of conjugates.
Theorem 2.7. If the Conradian soul of an ordered group (Γ, ) is trivial and is not Conradian, then may be approximated by its conjugates.
Proof. Let f 1 ≺ f 2 ≺ . . . ≺ f k be finitely many positive elements in Γ. We need to show that there exists a conjugate of which is different from but for which all the f i 's are still positive. Since 1 ∈ C (Γ) and f 1 / ∈ C (Γ), Theorem 2.1 and Lemma 2.5 imply that there is a crossing (f, g, u, v, w)
such that 1 ≺ u ≺ v ≺ f 1 . Let M, N in N be such that f N v ≺ w ≺ g M u. We claim that 1 ≺ v −1 f i and 1 ≺ w −1 f i for 1 ≤ i ≤ k, but g M f N ≺ v −1 1 and g M f N ≻ w −1 1. Indeed, since 1 ≺ v ≺ f i , we have v ≺ f i ≺ f i v, thus 1 ≺ v −1 f i v. By definition, this means that f i ≻ v −1 1. The inequality f i ≻ w −1 1 is proved similarly. Now note that g M f N v ≺ g M w ≺ v, and so g M f N ≺ v −1 1. Finally, from g M f N w ≻ g M u ≻ w we get g M f N ≻ w −1 1.
Now the preceding relations imply that the f i 's are still positive for both v −1 and w −1 , but at least one of these orderings is different from . This concludes the proof.
Based on the work of Linnell [6], it is shown in [10,Proposition 4.1] that no Conradian ordering is an isolated point of the space of orderings of a group having infinitely many orderings. Together with Theorem 2.7, this shows the next proposition by means of the convex extension procedure (c.f., Remark 1.11).
Proposition 2.8. Let Γ be an orderable group. If is an isolated point of LO(Γ), then its Conradian soul is non-trivial and has only finitely many orderings.
As a consequence of a nice theorem of Tararin, the number of orderings on an orderable group having only finitely many orderings is a power of 2; moreover, all of these orderings are necessarily Conradian [5,9]. By the preceding theorem, if is an isolated point of an space of orderings LO(Γ), then its Conradian soul admits 2 n different orderings for some n ≥ 1, all of them Conradian. Let { 1 , 2 , . . . , 2 n } be these orderings, where 1 is the restriction of to its Conradian soul. Since C (Γ) is -convex, each j induces an ordering j on Γ, namely the convex extension of j by . (Note that 1 coincides with .) All the orderings j share the same Conradian soul [10,Lemma 3.37]. Assume throughout that is not Conradian. Theorem 2.9. With the notation above, at least one of the orderings j is an accumulation point of the set of conjugates of . Corollary 2.10. At least one of the orderings j is approximated by its conjugates.
Proof. Asumming Theorem 2.9, we have k ∈ acc(orb( 1 )) for some k ∈ {1, . . . , 2 n }. Theorem 2.9 applied to this k instead of shows the existence of k ′ ∈ {1, . . . , 2 n } so that k ′ ∈ acc(orb( k )), and hence k ′ ∈ acc(orb( 1 )). If k ′ equals either 1 or k then we are done; if not, we continue arguing in this way... In at most 2 n steps we will find an index j such that j ∈ acc(orb( j )). Theorem 2.9 will follow from the next proposition.
Proposition 2.11. Given an arbitrary finite family G of -positive elements in Γ, there exists h ∈ Γ and 1 ≺h /
∈ C (Γ) such that 1 ≺ h −1 f h / ∈ C (Γ) for all f ∈ G \ C (Γ), but 1 ≻ h −1h h / ∈ C (Γ).
Proof of Theorem 2.9 from Proposition 2.11. Let us consider the directed set formed by the finite sets G of -positive elements. For each such a G, let h G andh G be the elements in Γ provided by Proposition 2.11. After passing to subnets of (h G ) and (h G ) if necessary, we may assume that the restrictions of h −1 G to C (Γ) all coincide with a single j . Now the properties of h G andh G imply:
-f ≻ j 1 and f (≻ j ) h −1 G 1 for all f ∈ G \ C (Γ), -h G ≻ 1, buth G (≺ j ) h −1 G ≺ 1.
This clearly shows the Theorem.
For the proof of Proposition 2.11 we will use three general lemmas.
Lemma 2.12. For every 1 ≺ c / ∈ C (Γ) there is a crossing (f, g, u, v, w) such that u, v, w do not belong to C (Γ) and 1 ≺ u ≺ w ≺ v ≺ c.
Proof. By Theorem 2.1 and Lemma 2.5, for every 1 s ∈ C (Γ) there exists a crossing (f, g, u, v, w) such that s ≺ u ≺ w ≺ v ≺ c. Clearly, v does not belong to C (Γ). The element w is also ouside C (Γ), since in the other case the element a = w 2 would satisfy w ≺ a ∈ C (Γ), which is absurd. Taking M > 0 so that g M u ≻ w, this gives g M u / ∈ C (Γ), g M w / ∈ C (Γ), and g M v / ∈ C (Γ). Consider the crossing (g M f g −M , g, g M u, g M v, g M w). If g M v ≺ v, then we are done. If not, then gv ≻ v, and Lemma 2.3 ensures that (g M f g −M , g, g M u, v, g M w) is also a crossing, which still allows concluding.
Lemma 2.13. Given 1 ≺ c / ∈ C (Γ) there exists 1 ≺ a / ∈ C (Γ) (with a ≺ c) such that, for all 1 b a and allc c, one has 1
≺ b −1c b / ∈ C (Γ).
Proof. Let us consider the crossing (f, g, u, v, w) such that 1 ≺ u ≺ w ≺ v ≺ c and such that u, v, w do not belong to C (Γ). We affirm that the Lemma holds for a = u (actually, it holds for a = w, but the proof is slightly more complicated).
Indeed, if 1 b u, then from b u ≺ v ≺c we get 1 b −1 u ≺ b −1 v ≺ b −1c , and thus the crossing (b −1 f b, b −1 gb, b −1 u, b −1 v, b −1 w) shows that b −1c / ∈ C (Γ). Since 1 b, we conclude that 1 ≺ b −1c b −1c b, and the convexity of S implies that b −1c b / ∈ C (Γ).
Lemma 2.14. For every g ∈ Γ the set g C (Γ) is convex. Moreover, for every crossing (f, g, u, v, w) one has uC (Γ) < wC (Γ) < vC (Γ), in the sense that uh 1 ≺ wh 2 ≺ vh 3 for all h 1 , h 2 , h 3 in C (Γ) (c.f., Example 1.10).
Proof. The verification of the convexity of gC (Γ) is straightforward. Now suppose that uh 1 ≻ wh 2 for some h 1 , h 2 in C (Γ). Then since u ≺ w, the convexity of both left classes uC (Γ) and wC (Γ) gives the equality between them. In particular, there exists h ∈ C (Γ) such that uh = w. Note that such an h must be positive, so that 1 ≺ h = u −1 w. But since (u −1 f u, u −1 gu, 1, u −1 v, u −1 w) is a crossing, this contradicts the definition of C (Γ). Showing that wC (Γ) ≺ vC (Γ) is similar.
Proof of Proposition 2.11. Let us label the elements of G = {f 1 , . . . , f r } so that f 1 ≺ . . . ≺ f r , and let k be such that f k−1 ∈ C (Γ) but f k / ∈ C (Γ). Recall that, by Lemma 2.13, there exists 1 ≺ a / ∈ C (Γ) such that, for every 1 b a, one has 1 ≺ b −1 f k+j b / ∈ C (Γ) for all j ≥ 0. We fix a crossing (f, g, u, v, w) such that 1 ≺ u ≺ v ≺ a and u / ∈ C (Γ). Note that the conjugacy by w −1 gives the crossing (w −1 f w, w −1 gw, w −1 u, w −1 v, 1). Case 1. One has w −1 v a.
In this case, the proposition holds for h = w −1 v andh = w −1 g M +1 f N w. To show this, first note than neither w −1 gw nor w −1 f w belong to C (Γ). Indeed, this follows from the convexity of C (Γ) and the inequalities w −1 g −M w ≺ w −1 u / ∈ C (Γ) and w −1 f −N w ≻ w −1 v / ∈ C (Γ). We also have 1 ≺ w −1 g M f N w, and hence 1 ≺ w −1 gw ≺ w −1 g M +1 f N w, which shows thath / ∈ C (Γ). On the other hand, the inequality w −1 g M +1 f N w(w −1 v) ≺ w −1 v reads as h −1h h ≺ 1. Finally, Lemma 2.2 applied to the crossing (w −1 f w, w −1 gw, w −1 u, w −1 v, 1) shows that (w −1 f w, w −1 gw, w −1 u, w −1 v, w −1 g M +n f N w) is a crossing for every n > 0. For n ≥ M we have w −1 g M +1 f N w(w −1 v) ≺ w −1 g M +n f N w. Since w −1 g M +n f N w ≺ w −1 v, Lemma 2.14 easily implies that w −1 g M +1 f N w(w −1 v)C (Γ) ≺ w −1 vC (Γ), that is, h −1h h / ∈ C (Γ).
Case 2. One has a ≺ w −1 v, but w −1 g m w a for all m > 0.
We claim that, in this case, the proposition holds for h = a andh = w −1 g M +1 f N w. This may be checked in the very same way as in Case 1 by noticing that, if a ≺ w −1 v but w −1 g m w a for all m > 0, then (w −1 f w, w −1 gw, w −1 u, a, 1) is a crossing. Case 3. One has a ≺ w −1 v and w −1 g m w ≻ a for some m > 0. (Note that the first condition follows from the second one. ) We claim that, in this case, the proposition holds for h = a andh = w / ∈ C (Γ). Indeed, we have g m w ≻ ha (and w ≺ ha), and since g m w ≺ v ≺ a, we have wa ≺ a, which means that h −1h h ≺ 1. Finally, from Lemmas 2.2 and 2.14 we get waC (Γ) g m wC (Γ) ≺ vC (Γ) aC (Γ). This implies that a −1 waC (Γ) ≺ C (Γ), which means that h −1h h / ∈ C (Γ).
3 Finitely many or uncountably many group orderings
The goal of this final short section is to use the previously developed ideas to show the following result.
Theorem 3.1. If the space of orderings of an orderable group is infinite, then it is uncountable.
Proof. Let us fix an ordering on an orderable group Γ. We need to analize two different cases. Case 1. The Conradian soul of C (Γ) is non-trivial and has infinitely many orderings.
From [1], we know that the subgroup of B 3 generated by the elements σ 2 1 , σ 2 2 is isomorphic to F 2 , the free group on two generators. Thus we may consider F 2 to be the subgroup of B 3 generated by σ 2 1 and σ 2 2 , and define a positive cone P in F 2 by P = P DD ∩ F 2 . Note that any element of F 2 must always be represented by a braid word having even total exponent, and that the ordering C of F 2 asssociated to the positive cone P is simply the restriction of the P DD ordering to the subgroup σ 2 1 , σ 2 2 .
Proposition A. 4. The ordering C is not an accumulation point of its conjugates in LO(F 2 ). Specifically, no conjugates of C distinct from C lie inside the open set U σ −2 2 ⊂ LO(F 2 ).
Proof. Let β ∈ F 2 ⊂ B 3 be given, and consider the positive cone βP β −1 . To prove the claim, we must show that σ −2 2 ∈ βP β −1 implies βP β −1 = P . First, observe that conjugation of P by any even power of σ 2 does not change P : this follows from the fact that σ −2 2 is the least positive element in the associated ordering C of F 2 . Indeed, for any element g ∈ P , we have σ −2 Remark A. 6. From the work of [5,6], it is known that the ordering C is not an isolated point in LO(F 2 ), but no method of constructing a sequence converging to C is given therein. Given an ordering in LO(F 2 ), the known methods for constructing a sequence converging to involve either approximation using the conjugates of , or approximation by modifying the ordering on the convex jumps in the Conradian soul of . The results of this Appendix show that neither of these methods is sufficient for constructing a sequence of orderings converging to C .
Theorem 1. 4 .
4The ordering is Conradian if and only if (Γ, ) admits no crossing.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Figure 2 :
2The 'n = 2' condition
Claim 4 .
4The subgroup S is maximal among -convex, -Conradian subgroups.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
. . .
. . .
. . .
. . .
. .
. . .
. . .
. . .
. . .
. . .
. .
. . .
. . .
. .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
. . .
. . .
. .
. . .
. . .
. . .
. . .
. . .
. . .
. .
. . .
. . .
. . .
. .
Figure 4: A (reinforced) double crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
. . .
. . .
. . .
. . .
. . .
. . .
. .
. . .
. . .
. . .
. . .
. . .
. .
. . .
. . .
. . .
. . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
, the generators of F 2 , so that S = F 2 .
This case was settled in[10](see Proposition 4.1 therein) using ideas going back to Zenkov[12]and Tararin [5].Case 2. The Conradian soul of C (Γ) has only finitely many orderings.If is Conradian, then Γ = C (Γ) has finitely many orderings. If not, then Theorems 2.7 and 2.9 imply that there exists an ordering * on Γ which is an accumulation point of its conjugates. The closure in LO(Γ) of the set of conjugates of * is then a compact set without isolated points. By a well-known fact in General Topology, such a set must be uncountable. Therefore, Γ admits uncountably many orderings.Appendix: An exotic ordering of the free group on two elements, by Adam ClayAbstractWe construct an ordering of F 2 which is not an accumulation point of its conjugates in LO(F 2 ) and whose Conradian soul is isomorphic to Z. This ordering is realized as the restriction of the Dubrovina-Dubrovin ordering of B 3 to an appropriate free subgroup of B 3 .We begin by defining the Dehornoy ordering of the braid groups (also known as the 'standard' ordering), whose positive cone we shall denote P D[2,3]. Recall that for each integer n ≥ 2, the Artin braid group B n is the group generated by σ 1 , σ 2 , . . . , σ n−1 , subject to the relationsDefinition A.1. Let w be a word in the generators σ i , · · · , σ n−1 (so no σ j occurs for j < i). Then w is said to be: i-positive if the generator σ i occurs in w with only positive exponents, i-negative if σ i occurs with only negative exponents, and i-neutral if σ i does not occur in w.We then define the positive cone of the Dehornoy ordering as Definition A.2. The positive cone P D ⊂ B n of the Dehornoy ordering is the setAn extremely important property of this ordering is that the conjugate βσ k β −1 is always ipositive for some i, for every generator σ k in B n and any braid β ∈ B n . This property is referred to as the subword property[3].There is also a second ordering of interest, discovered by the authors of[4], whose positive cone we shall denote by P DD . Denote by P i ⊂ B n the set of all i-positive braids. Note that the set of all i-negative braids is simply P −1 i .Definition A.3. The positive cone P DD ⊂ B n is the set P DD = P 1 ∪ P −1 2 ∪ · · · ∪ P (−1) n n−1 .
Orderable groups. R Botto-Mura, A Rhemtulla, Lecture Notes in Pure and Applied Mathematics. 27Marcel DekkerR. Botto-Mura and A. Rhemtulla. Orderable groups. Lecture Notes in Pure and Applied Mathematics, Vol. 27. Marcel Dekker, New York-Basel (1977).
Isolated points in the space of left orderings of a group. A Clay, PreprintA. Clay. Isolated points in the space of left orderings of a group. Preprint (2008).
Right-ordered groups. P Conrad, Mich. Math. Journal. 6P. Conrad. Right-ordered groups. Mich. Math. Journal 6 (1959), 267-275.
On braid groups. T Dubrovina, N Dubrovin, Sbornik Mathematics. 192T. Dubrovina and N. Dubrovin. On braid groups. Sbornik Mathematics 192 (2001), 693-703.
Right ordered groups. Siberian School of Algebra and Logic. V Kopitov, N Medvedev, Plenum Publ. CorpNew YorkV. Kopitov and N. Medvedev. Right ordered groups. Siberian School of Algebra and Logic, Plenum Publ. Corp., New York (1996).
The topology on the space of left orderings of a group. P , PreprintP. Linnell. The topology on the space of left orderings of a group. Preprint (2006).
Free lattice-ordered groups represented as o-2 transitive ℓ-permutation groups. S Mccleary, Trans. Amer Math. Soc. 290S. McCleary. Free lattice-ordered groups represented as o-2 transitive ℓ-permutation groups. Trans. Amer Math. Soc. 290 (1985), 81-100.
Amenable groups that act on the line. D Morris-Witte, Algebr. Geom. Topol. 6D. Morris-Witte. Amenable groups that act on the line. Algebr. Geom. Topol. 6 (2006), 2509- 2518.
Groups of circle diffeomorphisms. Forthcoming book. Spanish version published in Ensaios Matemáticos. A Navas, Bull. Braz. Math. Soc. A. Navas. Groups of circle diffeomorphisms. Forthcoming book. Spanish version published in Ensaios Matemáticos, Bull. Braz. Math. Soc. (2007).
On the dynamics of left-orderable groups. A Navas, PreprintA. Navas. On the dynamics of left-orderable groups. Preprint (2007).
Topology on the spaces of orderings of groups. A Sikora, Bull. London Math. Soc. 36A. Sikora. Topology on the spaces of orderings of groups. Bull. London Math. Soc. 36 (2004), 519-526.
On groups with an infinite set of right orders. A Zenkov, English translation: Siberian Math. 38JournalA. Zenkov. On groups with an infinite set of right orders. Sibirsk. Mat. Zh. 38 (1997), 90-92. English translation: Siberian Math. Journal 38 (1997), 76-77.
. address: [email protected]. 3363Andrés Navas Dep. de Matemáticas, Fac. de Ciencia, Univ. de SantiagoAndrés Navas Dep. de Matemáticas, Fac. de Ciencia, Univ. de Santiago, Alameda 3363, Est. Central, Santiago, Chile Email address: [email protected]
. Las Palmeras. 3425Cristóbal Rivas Dep. de Matemáticas, Fac. de Ciencias, Univ. de ChileEmail address: [email protected]óbal Rivas Dep. de Matemáticas, Fac. de Ciencias, Univ. de Chile, Las Palmeras 3425,Ñuñoa, Santiago, Chile Email address: [email protected]
The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. J Crisp, L Paris, Invent. Math. 1451J. Crisp and L. Paris. The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group. Invent. Math., 145(1):19-36, 2001.
Braid groups and left distributive operations. P Dehornoy, Trans. Amer. Math. Soc. 3451P. Dehornoy. Braid groups and left distributive operations. Trans. Amer. Math. Soc., 345(1): 115-150, 1994.
Why are braids orderable. P Dehornoy, I Dynnikov, D Rolfsen, B Wiest, Panoramas et Synthèses. Société Mathématique de France14ParisP. Dehornoy, I. Dynnikov, D. Rolfsen, and B. Wiest. Why are braids orderable?. Volume 14 of Panoramas et Synthèses. Société Mathématique de France, Paris, 2002.
T Dubrovina, N Dubrovin, On braid groups. Sbornik Mathematics. 192T. Dubrovina and N. Dubrovin. On braid groups. Sbornik Mathematics, 192(5):693-703, 2001.
Free lattice-ordered groups represented as o-2 transitive ℓ-permutation groups. S Mccleary, Trans. Amer Math. Soc. 290S. McCleary. Free lattice-ordered groups represented as o-2 transitive ℓ-permutation groups. Trans. Amer Math. Soc., 290:81-100, 1985.
On the dynamics of left-orderable groups. A Navas, V6T 1Z2 Email adress: [email protected] URL. Vancouver, BC CanadaAdam Clay Department of Mathematics, University of British ColumbiaPreprintA. Navas. On the dynamics of left-orderable groups. Preprint, 2007. Adam Clay Department of Mathematics, University of British Columbia, Vancouver, BC Canada V6T 1Z2 Email adress: [email protected] URL: http://www.math.ubc.ca/∼aclay/
| [] |
[
"Chiral Enhancement in the MSSM -An Overview",
"Chiral Enhancement in the MSSM -An Overview"
] | [
"A Crivellin \nAlbert Einstein Center for Fundamental Physics\nInstitute for Theoretical Physics\nUniversity of Bern\nCH-3012BernSwitzerland\n"
] | [
"Albert Einstein Center for Fundamental Physics\nInstitute for Theoretical Physics\nUniversity of Bern\nCH-3012BernSwitzerland"
] | [] | In this article I review the origin and the effects of chirally enhanced loop-corrections in the MSSM based on Refs.[1][2][3]. Chiral enhancement is related to fermion-Higgs couplings (or self-energies when the Higgs field is replaced by its vev). I describe the resummation of these chirally-enhanced corrections to all orders in perturbation theory and the calculation of the effective fermion-Higgs and gaugino(higgsino)-fermion vertices. As an application a model with radiative flavorviolation is discussed which can solve the SUSY-CP and the SUSY-flavor problem while it is still capable of explaining the observed deviation from the SM in the Bs −Bs mixing phase. | 10.1393/ncc/i2012-11114-0 | [
"https://arxiv.org/pdf/1106.1187v1.pdf"
] | 118,468,654 | 1106.1187 | 1d224ebfc94694e2339a5acb09859fa043743960 |
Chiral Enhancement in the MSSM -An Overview
6 Jun 2011
A Crivellin
Albert Einstein Center for Fundamental Physics
Institute for Theoretical Physics
University of Bern
CH-3012BernSwitzerland
Chiral Enhancement in the MSSM -An Overview
6 Jun 2011
In this article I review the origin and the effects of chirally enhanced loop-corrections in the MSSM based on Refs.[1][2][3]. Chiral enhancement is related to fermion-Higgs couplings (or self-energies when the Higgs field is replaced by its vev). I describe the resummation of these chirally-enhanced corrections to all orders in perturbation theory and the calculation of the effective fermion-Higgs and gaugino(higgsino)-fermion vertices. As an application a model with radiative flavorviolation is discussed which can solve the SUSY-CP and the SUSY-flavor problem while it is still capable of explaining the observed deviation from the SM in the Bs −Bs mixing phase.
-Introduction
In this article I summarize recent progress in the field of chirally enhanced self-energies in the MSSM done in collaboration with Jennifer Girrbach, Lars Hofer, Ulrich Nierste, Janusz Rosiek and Dominik Scherer. The discussion will skip the technical aspects and subtleties and instead focus on the essential features. The interested reader is referred to Refs. [1][2][3] for a detailed discussion.
In the standard model chirality violation is suppressed by small Yukawa couplings (except for the top quark). However, the MSSM does not necessarily possess the same suppression effects since the bottom Yukawa coupling can be big at large of tan β = v u /v d and also the trilinear A-terms do not necessarily respect the hierarchy of the Yukawa couplings. Thus, chirality-flipping self-energies can be enhanced either by a factor of tan β [4] or by a ratio A q ij /(vY q ij ). This enhancement can compensate for the loop suppression leading to corrections which are of the same order as the corresponding physical quantities, i.e. of order one. These large corrections must be taken into account to all orders in perturbation theory (Sec. 3) leading to effective fermion-gaugino and fermion-Higgs couplings (see Sec. 4).
It is even possible that light-fermion masses and the CKM elements are entirely due to radiative corrections involving the trilinear A-terms [5]. Such a model of radiative flavor-violation can solve the SUSY-CP [6] and the SUSY-flavor problem while still leading to interesting effects in flavor-observables (see Sec. 5).
2. -Self-energies One can decompose any fermion self-energy into chirality-flipping and chirality-conserving parts in the following way:
(1) Σ f ji (p) = Σ f LR ji (p 2 ) + p /Σ f RR ji (p 2 ) P R + Σ f RL ji (p 2 ) + p /Σ f LL ji (p 2 ) P L .
Here i and j are flavor indices running from 1 to 3. Since the SUSY particles are known to be much heavier than the SM fermions it is possible to expand in the external momentum. For our purpose it is even sufficient to work in the limit ✁ p = 0 in which only the chirality-flipping parts Σ f LR ji (0) = Σ f RL⋆ ij (0) remain. It is well known that in the MSSM these self-energies can be enhanced either by a factor tan β [4] or by a factor [2] which compensates for the loop-suppression and leads to order one corrections. The chirality-changing part of the fermion self-energy (at ✁ p = 0, involving sfermions and gauginos(higgsions)) can be written as
A f ij /(Y f ij M SUSY )(2) Σ fλ LR ji = −1 16π 2 6 s=1 N I=1 mλ I Γλ I L⋆ fjfs Γλ I R fifs B 0 m 2 λI , m 2 fs .
Hereλ stands for the SUSY fermions (g,χ 0 ,χ ± ) and N denotes their corresponding number (2 for charginos, 4 for neutralinos and 8 for gluinos). The coupling coefficients Γλ I L(R) fifs and the loop functions B 0 are defined in the appendix of Ref. [1].
The couplings Γλ I L(R) fifs in Eq. (2) involve the corresponding sfermion mixing matrices W f which diagonalize the sfermion mass matrices:
W f ⋆ s ′ s (M 2 f ) s ′ t ′ W f t ′ t = m 2 fs δ st .
In the case of neutralino-quark-squark and chargino-quark-squark vertices they also depend on Yukawa couplings and CKM elements. An interesting feature of the self-energies in Eq. (2) is that they are finite and that they do not vanish in the limit of infinitely heavy SUSY masses. We refer to this approximation in which only such non-decoupling terms for the self-energies are kept as "the decoupling limit". Note, however, that we do not integrate out the SUSY particles but rather keep them as dynamical degrees of freedom.
Let us take a closer look at the quark self-energy with squarks and gluinos as virtual particles( 1 ). To leading order in v/M SUSY , the self-energy is proportional to one chirality flipping element ∆ q LR jk of the squark mass matrix:
Σ qg LR f i = 2α s 3π mg 3 j,k,j ′ ,f ′ =1 6 s,t=1 Λ q LL s f j ∆ q LR jk Λ q RR t ki C 0 m 2 g , m 2 q L m , m 2 q R n .(3)
( 1 ) The gluino contribution is the dominant one in the flavor-conserving case. In the presence of non-minimal sources of flavor-violation it is also usually the dominant one.
Here the off-diagonal elements of the sfermion mass matrices are given by
(4) ∆ u LR ij = −v u A u ij − v d A ′u ij − v d µ Y ui δ ij , ∆ d,ℓ LR ij = −v d A d,ℓ ij − v u A ′d,ℓ ij − v u µ Y di,ℓi δ ij ,
and the matrices Λ f LL,RR
s ij = W f ⋆ i+3
,s W f js take into account the flavor changes due to bilinear terms. Note that in the decoupling limit W f st depends only on the bilinear terms. For equal SUSY masses we can give a simple approximate formula for the self-energy in Eq. (3):
Σ dg LR f i = −1 100 v d A d f i /M SUSY + Y di tan βδ f i (5) Σ ug LR f i = −1 100 v u A u f i /M SUSY + Y ui cot βδ f i (6)
Thus, generic A d -terms which are of the order M SUSY lead to self-energies which are approximatly v d /100. The part of Eq. (6) containing Y b is of the order of 2 GeV for tan β ≈ 50. In the case of up-quarks, only the part of the self-energy proportional to A u can be important: it is of the order of 1.5 GeV for A u ≈ M SUSY .
According to Eq. (4) and Eq. (6) the quark self-energy with a gluino as virtual particle can be divided into a part linear in a Yukawa coupling and a part linear in an A-term. Such a decomposition is possible for all self-energies (in the decoupling limit) because either a Yukawa coupling or a trilinear A-term is needed in order flip the chirality. Thus, we can also decompose the chargino self-energies (we do not consider the neutralino self-energy here, because it is usually subleading) in an analogous way. In the flavor changing case we also have to distinguish whether the flavor-change is due to a CKM element or not which is important when we consider later the CKM renormalization. Thus we decompose the down-quark self-energy as follows:
(7) Σ d LR ii = Σ d LR ii✟ ✟ Y d i + v u Y di ǫ d i Σ d LR f i = Σ d LR f i✘ ✘ ✘ CKM + m d3 V (0)⋆ 3f ǫ F C δ i3 for f = i Here ǫ d i is the part of the flavor-conserving down-quark self-energy proportional to Y di divided by v u Y di , Σ d LR f i✘ ✘ ✘
CKM is the sum of all self-energies where the flavor-change is not due to CKM elements and ǫ d F C arises from the part of the chargino self-energy where the flavor change comes from a CKM element:
(8) ǫ d F C = −1 16π 2 µ Y d3 m d3 3 m,n=1 Y u3⋆ Λ q LL m 33 ∆ u LR⋆ 33 Λ u RR n 33 C 0 |µ| 2 , m 2 q L m , m 2 u R n .
For the discussion of the effective Higgs vertices we also need a decomposition of Σ f LR ji into its holomorphic and non-holomorphic parts. Here non-holomorphic means that the loop-induced coupling is to the other Higgs doublet than the on involved in the Yukawaterm of the superpotential, i.e. for down-quarks the self-energy involves v u and for up-quarks the self-energy contains v d . In the decoupling limit all enhanced holomorphic self-energies are proportional to A-terms and we denote the the sum as Σ f LR ji A , while the non-holomorphic part is denoted as Σ ′f LR ji :
(9) Σ f LR ji = Σ f LR ji A + Σ ′f LR ji .
-Renormalization
Chirally-enhanced self-energies modify the relation between the bare Yukawa couplings Y qi ≡ Y qi and the corresponding physical fermion masses m fi . For quarks we have the relation:
(10) m qi = v q Y qi + Σ q LR ii , (q = u, d).
Eq. (10) implicitly determines the bare Yukawa couplings Y qi for a given set of SUSY parameters. In the up-quark sector the enhanced terms in the self-energy Σ u LR ii are independent of Y ui . Therefore Eq. (10) is easily solved for Y ui and one finds
(11) Y ui = m ui − Σ u LR ii /v u .
In the down-quark sector we have terms proportional to one power of Y di at most (in the decoupling limit) and by solving Eq. (10) we recover the well-known resummation formula [7] with an extra correction due to the A-terms [8]( 2 ):
(12) Y di = m di − Σ d LR ii ✚ Yi v d 1 + tan βǫ d i The flavor-changing self-energies Σ q LR f i induce wave-function rotations ψ f L,R i → U q L,R ij ψ q L,R j
in flavor-space which have to be applied to all external fermion fields. At the two-loop level U q L f i is given by
(13) U q L = 1 − 1 2 |σ q 12 | 2 σ q 12 + mq 1 mq 2 σ q⋆ 21 σ q 13 + mq 1 mq 3 σ q⋆ 31 −σ q⋆ 12 − mq 1 m f 2 σ q 21 1 − 1 2 |σ q 12 | 2 σ q 23 + m f 2 mq 3 σ q⋆ 32 −σ q⋆ 13 − mq 1 mq 3 σ q 31 + σ q⋆ 12 σ q⋆ 23 −σ q⋆ 23 − mq 2 mq 3 σ q 32 1 ,
where we have neglected terms which are quadratic or of higher order in small quark mass ratios and we have defined the abbreviation σ q f i = Σ q f i /m q max(f,i) . The corresponding expressions for U f R is obtained from the one for U q L by replacing σ q ji → σ q⋆ ij . The rotations in Eq. (13) also renormalize the CKM matrix. The bare CKM matrix V (0) , which arises because of the misalignment between the bare Yukawa couplings, is now determined through the physical one by
(14) V (0) = U u L V U d L † .
This equation can be solved analytically by exploiting the CKM hierarchy. First one calculates the effects of Σ d LR f i✘ ✘ ✘ CKM which lead to an additive change in the CKM elements. Then the self-energies containing CKM elements lead to a scaling of new CKM elements V 13 ,Ṽ 23 ,Ṽ 31 ,Ṽ 32 elements by a factor 1/ 1 − ǫ d F C (similar to Eq. (12)).
4. -Effective Vertices 4 .
1. Higgs Vertices. -The effective Higgs vertices are most easily obtained in an effective field theory approach [9] which is an excellent approximation to the full theory. In addition to the flavor-diagonal holomorphic couplings of quarks to the Higgs fields flavor-changing couplings to both Higgs doublets are induced via loops. The resulting effective Yukawa-Lagrangian is that of a general 2HDM of type III and is given (in the super-CKM basis) by:
(15) L ef f Y =Q a f L (Y di δ f i + E d f i )ǫ ab H b d − E ′d f i H a⋆ u d i R −Q a f L (Y ui δ f i + E u f i )ǫ ab H b u + E ′u f i H a⋆ d d i R .
Here a and b denote SU (2) L -indices and ǫ ab is the two-dimensional antisymmetric tensor with ǫ 12 = 1. The loop-induced couplings E (′)q are given by
(16) E d ij = Σ d LR ij A v d , E ′d ij = Σ ′d LR ij v u , E u ij = Σ u LR ij A v u , E ′u ij = Σ ′u LR ij v d .
Diagonalizing the effective quark mass matrices (after electroweak symmetry breaking) and decomposing the Higgs fields into their physical components leads to the following effective neutral Higgs couplings:
Γ H 0 k LR eff u f ui = x k u mu i vu δ f i − E ′u f i cot β + x k⋆ d E ′u f i , Γ H 0 k LR eff d f di = x k d m d i v d δ f i − E ′d f i tan β + x k⋆ u E ′d f i , with E ′q f i = U q L * jf E ′q jk U q R ki ≈ E ′q f i − ∆E ′q f i ,(17)∆E ′q = 0 σ q 12 E ′q 22 (σ q 13 − σ q 12 σ q 23 ) E ′q 33 + σ q 12 E ′q 23 E ′q 22 σ q 21 0 σ q 23 E ′q 33 E ′q 33 (σ q 31 − σ q 32 σ q 21 ) + E ′q 32 σ q 21 E ′q 33 σ q 32 0 .(18)
The new term ∆E ′q is especially interesting: it contains a non-holomorphic flavorconserving part which multiplies a flavor-changing holomorphic term. In this way the holomorphic A-terms can lead to flavor-changing neutral Higgs couplings. The origin of this term can be understood in the following way: Even though the couplings E d ij are holomorphic, they lead to an additional rotation (if E d ij is flavor non-diagonal) which is needed to diagonalize the effective quark mass matrix. These rotations then lead to off-diagonal neutral Higgs couplings even if E ′d ij is flavor conserving. This effect will allow us to explain the B s mixing in our model with radiative flavor-violation.
4
.
2. Gaugino(Higgsino)-Fermion-Vertices. -Effective gaugino(higgsino)-fermion-vertices which include the chirally-enhanced corrections are obtained by inserting the bare values for the Yukawa couplings and the CKM elements into the corresponding Feynman-rules and applying the wave-function rotations in Eq. (13) to all external fermion fields. Since the genuine vertex corrections are not enhanced, these vertices then include all chirally enhanced effects.
-Radiative Flavor-Violation
The smallness of the off-diagonal CKM elements and the Yukawa couplings of the first two generations suggests the idea that these quantities might be due to radiative corrections, i.e. they are zero at tree-level. Indeed, as we have see previously, the selfenergies involving the trilinear A-terms lead to order one effects in the renormalization of the CKM elements and the light fermion masses and it is possible that they are generated by the self-energy radiative corrections [5]. From Eq. (6) we see that this is the case if the A-terms are of the same order as the other SUSY parameters.
However, the third generation fermion masses are too heavy to be loop generated (without unnaturally large values for the A-terms which would violate vacuum stability [10]) and the successful bottom -tau (top -bottom) Yukawa coupling unification in SU(5) (SO(10)) GUTs suggests to keep the third generation fermion masses. Thus we assume the following structure for the Yukawa couplings of the MSSM superpotential:
(19) Y f ij = Y f3 δ i3 δ j3 , V (0) ij = δ ij .
This means that (in the language of [11]) the global [U (3)] 5 flavor symmetry of the gauge sector is broken down to [U (2)] 5 ×[U (1)] 2 by the Yukawa couplings of the third generation.
Here the five U (2) factors correspond to rotations of the left-handed doublets and the right-handed singlets of the first two generation fermions in flavor space, respectively. Let us first consider the quark sector. Here we demand that the light quark masses and the off-diagonal CKM elements are generated by gluino self-energies. Regarding only the first two generations, no direction in flavor space is singled out by the Yukawa term in the superpotential and the Cabbibo angle is generated by a misalignment between A u and A d ( 3 ). Regarding the third generation, the situation is different because their nonzero Yukawa couplings fix the quark-field rotations involving the third generation. Thus V ub,cb,ts,td are generated by a misalignment between the A u , A d and the third generation Yukawa couplings. We will consider the two limiting cases in which the CKM elements arise only from a mismatch between (A d ) A u and (Y d ) Y d which we call CKM generation in the down (up) sector for obvious reasons. This means we require
Σ d LR 23 = m b V cb ≈ −m b V * ts , Σ d LR 13 = m b V ub , (20) or Σ u LR 23 = −m t V cb ≈ m t V * ts , Σ u LR 13 = m t V ⋆ td .(21)
If the CKM matrix is generated in the down-sector the most stringent constraint stems from an enhancement of b → sγ due to the off-diagonal element ∆ d LR
23
in the squark mass matrix. The resulting bounds on the squark and gluino mass are shown in the left plot of Fig. 1. In addition flavor-changing neutral Higgs coupling are induced according to Eq. (17) which gives an additional contribution to B s → µ + µ − . Also B s mixing can be affected but because it is protected by a Peccei-Quinn symmetry a double Higgs penguins contributes only if also Σ d RL 23 is non-zero (see right plot in Fig. 3). In the case of CKM matrix generation in the up-sector, the most stringent constraints stem from ǫ K (see right plot of Fig. 1) which receives additional contributions via a chargino box diagram involving the double mass insertion δ u LR 23 δ u LR
13
. At the same time the rare Kaon decays K + → π + νν and K L → π 0 νν receive sizable corrections (see Fig. 2) which is very interesting for NA62. Radiative mass generation is also possible in the lepton sector. Here the anomalous magnetic moment of the muon probes the soft muon Yukawa coupling because it gets an additive contribution which depends only on the SUSY scale. If one demands that SM contribution plus the supersymmetric one is within the 2σ region of the experimental measurement the smuon mass must lie between 600 GeV and 2200 GeV for M 1 < 1TeV if its Yukawa coupling is loop-generated (see left plot of Fig. 3). If a smuon is found to be lighter, the observed muon mass cannot entirely stem from the soft SUSY-breaking sector and consequently the muon must have a nonzero Yukawa coupling y µ in the superpotential. -Predicted branching ratio for the rare Kaon decay KL → π 0 νν (left) and K + → π + νν (right) assuming that the CKM matrix is generated in the up-sector for mq = mg.
-Conclusions
In the MSSM self-energies can be chirally enhanced by a factor tan β or by a factor A q ij /(vY q ij ) which can compensate for the loop-suppression. This leads to order one corrections which must be taken into account to all orders in perturbation theory. This goal can be achieved by using effective vertices which include these corrections. The trilinear A-terms can even entirely generate the light fermion masses and the off-diagonal CKM elements via radiative corrections. Such a model of radiative flavor violation can both solve the SUSY CP problem and is consistent with FCNC constraints for SUSY masses of the order of 1 TeV. In addition in the case of CKM generation in the downsector B s → µ + µ − and B s −B s mixing receive additional contributions via Higgs penguins which and can even generate a sizable phase in B s mixing. In the case of CKM generation in the up sector the branching ratio of the rare Kaon decays K + → π + νν and K L → π 0 νν can be enhanced compared to the SM prediction. * * * I thank the organizers, especially Gino Isidori, for the invitation to the "La Thuille conference". This work is supported by the Swiss National Foundation. I am grateful to
( 2 )
2This equation can be directly transferred to the lepton sector by replacing fermion index d for ℓ, except for the vev.
Fig. 1 .
1-Left: Allowed regions in the mg − mq plane. Constraint from b → sγ assuming that the CKM matrix is generated in the down sector. We demand that the gluino contributions should not exceed the SM one. Yellow(lightest): µ tan β = 30 TeV, red: µ tan β = 0 TeV and blue: µ tan β = −30 TeV. Right: Allowed regions in the mg − mq plane. Constraints from Kaon mixing for different values of M2 assuming that the CKM matrix is generated in the up sector. Yellow(lightest): M2 = 1000 GeV, green: M2 = 750 GeV, red: M2 = 500 GeV and blue: M2 = 250 GeV.
( 3 )
3This also implies that the quark-squark gluino vertex is flavor-diagonal for transitions between the first two generations in the super-CKM basis. m q
Fig. 3 .
3-Left: Allowed region in the M1-mμ plane assuming that the muon Yukawa coupling is generated radiatively by v d A ℓ 22 and/or vuA ℓ′ 22 . Here mμ is the lighter smuon mass. Yellow (lightest): aµ ± 2σ, red: aµ ± 1σ, blue (darkest): aµ. Right: correlations between Bs → µ + µ − and Bs−Bs mixing for ǫ b = 0.0075, mH = 400GeV and tan β = 12 in the complex VR = Σ d RL 23 /m d 3 plane. Red: Allowed region from Bs −Bs mixing (95% confidence level). The contour-lines show Br[Bs → µ + µ − ] × 10 9 . Lars Hofer and Lorenzo Mercolli for proofreading the manuscript. The Albert Einstein Center for Fundamental Physics is supported by the "Innovations-und Kooperationsprojekt C-13 of the Schweizerische Universitätskonferenz SUK/CRUS".
. A Crivellin, arXiv:1012.4840Phys. Rev. D. 8356001hep-phA. Crivellin, Phys. Rev. D 83 (2011) 056001 [arXiv:1012.4840 [hep-ph]].
. A Crivellin, L Hofer, J Rosiek, arXiv:1103.4272hep-phA. Crivellin, L. Hofer and J. Rosiek, arXiv:1103.4272 [hep-ph].
. A Crivellin, U Nierste, arXiv:0810.1613Phys. Rev. D. 7935018hep-phA. Crivellin and U. Nierste, Phys. Rev. D 79 (2009) 035018 [arXiv:0810.1613 [hep-ph]].
. A Crivellin, U Nierste, arXiv:0908.4404Phys. Rev. D. 8195007hep-phA. Crivellin and U. Nierste, Phys. Rev. D 81 (2010) 095007 [arXiv:0908.4404 [hep-ph]].
. A Crivellin, J Girrbach, arXiv:1002.0227Phys. Rev. D. 8176001hepphA. Crivellin and J. Girrbach, Phys. Rev. D 81 (2010) 076001 [arXiv:1002.0227 [hep- ph]].
. L Hofer, U Nierste, D Scherer, arXiv:0907.5408JHEP. 091081hep-L. Hofer, U. Nierste and D. Scherer, JHEP 0910 (2009) 081 [arXiv:0907.5408 [hep-
. J Girrbach, S Mertens, U Nierste, S Wiesenfeldt, arXiv:0910.2663JHEP. 100526hep-phJ. Girrbach, S. Mertens, U. Nierste and S. Wiesenfeldt, JHEP 1005 (2010) 026 [arXiv:0910.2663 [hep-ph]].
. A Crivellin, J Girrbach, U Nierste, arXiv:1010.4485Phys. Rev. D. 8355009hep-phA. Crivellin, J. Girrbach and U. Nierste, Phys. Rev. D 83 (2011) 055009 [arXiv:1010.4485 [hep-ph]].
. A Crivellin, L Hofer, U Nierste, D Scherer, arXiv:1105.2818hep-phA. Crivellin, L. Hofer, U. Nierste and D. Scherer, [arXiv:1105.2818 [hep-ph]].
. L J Hall, R Rattazzi, U Sarid, arXiv:hep-ph/9306309Phys. Rev. D. 507048L. J. Hall, R. Rattazzi and U. Sarid, Phys. Rev. D 50 (1994) 7048 [arXiv:hep-ph/9306309].
. W Buchmuller, D Wyler, Phys. Lett. B. 121321W. Buchmuller and D. Wyler, Phys. Lett. B 121 (1983) 321.
. F Borzumati, G R Farrar, N Polonsky, S D Thomas, arXiv:hep-ph/9902443Nucl. Phys. B. 55553F. Borzumati, G. R. Farrar, N. Polonsky and S. D. Thomas, Nucl. Phys. B 555 (1999) 53 [arXiv:hep-ph/9902443].
. M S Carena, D Garcia, U Nierste, C E M Wagner, arXiv:hep-ph/9912516Nucl. Phys. B. 57788M. S. Carena, D. Garcia, U. Nierste and C. E. M. Wagner, Nucl. Phys. B 577 (2000) 88 [arXiv:hep-ph/9912516].
. J Guasch, P Hafliger, M Spira, arXiv:hep-ph/0305101Phys. Rev. D. 68115001J. Guasch, P. Hafliger and M. Spira, Phys. Rev. D 68 (2003) 115001 [arXiv:hep-ph/0305101].
. G Isidori, A Retico, arXiv:hep-ph/0208159JHEP. 020963G. Isidori and A. Retico, JHEP 0209 (2002) 063 [arXiv:hep-ph/0208159].
. J A Casas, A Lleyda, C Munoz, arXiv:hep-ph/9507294Nucl. Phys. B. 4713J. A. Casas, A. Lleyda and C. Munoz, Nucl. Phys. B 471 (1996) 3 [arXiv:hep-ph/9507294].
. J H Park, arXiv:1011.4939Phys. Rev. D. 8355015hep-phJ. h. Park, Phys. Rev. D 83 (2011) 055015 [arXiv:1011.4939 [hep-ph]].
. G Ambrosio, G F Giudice, G Isidori, A Strumia, arXiv:hep-ph/0207036Nucl. Phys. B. 645155G. D'Ambrosio, G. F. Giudice, G. Isidori and A. Strumia, Nucl. Phys. B 645 (2002) 155 [arXiv:hep-ph/0207036].
. R S Chivukula, H Georgi, Phys. Lett. B. 18899R. S. Chivukula and H. Georgi, Phys. Lett. B 188 (1987) 99.
| [] |
[
"VOLUME PRESERVING SUBGROUPS OF A AND K AND SINGULARITIES IN UNIMODULAR GEOMETRY",
"VOLUME PRESERVING SUBGROUPS OF A AND K AND SINGULARITIES IN UNIMODULAR GEOMETRY"
] | [
"W Domitrz ",
"J H Rieger "
] | [] | [] | For a germ of a smooth map f from K n to K p and a subgroup G Ωq of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form Ωq in K q (q = n or p) we study the G Ωqmoduli space of f that parameterizes the G Ωq -orbits inside the G-orbit of f . We find, for example, that this moduli space vanishes for G Ωq = A Ωp and A-stable maps f and for G Ωq = K Ωn and K-simple maps f . On the other hand, there are A-stable maps f with infinite-dimensional A Ωn -moduli space.1991 Mathematics Subject Classification. 32S05, 32S30, 58K40. | null | [
"https://arxiv.org/pdf/0804.2596v2.pdf"
] | 14,728,277 | 0804.2596 | 497aa27b02b5991daec16ef48b4e32273da3e71e |
VOLUME PRESERVING SUBGROUPS OF A AND K AND SINGULARITIES IN UNIMODULAR GEOMETRY
23 Mar 2009
W Domitrz
J H Rieger
VOLUME PRESERVING SUBGROUPS OF A AND K AND SINGULARITIES IN UNIMODULAR GEOMETRY
23 Mar 2009
For a germ of a smooth map f from K n to K p and a subgroup G Ωq of any of the Mather groups G for which the source or target diffeomorphisms preserve some given volume form Ωq in K q (q = n or p) we study the G Ωqmoduli space of f that parameterizes the G Ωq -orbits inside the G-orbit of f . We find, for example, that this moduli space vanishes for G Ωq = A Ωp and A-stable maps f and for G Ωq = K Ωn and K-simple maps f . On the other hand, there are A-stable maps f with infinite-dimensional A Ωn -moduli space.1991 Mathematics Subject Classification. 32S05, 32S30, 58K40.
Introduction
We are going to study singularities arising in unimodular geometry. A singular subvariety of a space with a fixed volume form may be given by some parametrization or by defining equations. This leads to the following (multi-)local classification problems. (1) The classification of germs of smooth maps f : (K n , 0) → (K p , Ω p , 0) (K = C or R) up to A Ωp -equivalence (i.e., for the subgroup of A in which the left coordinate changes preserve a given volume form Ω p in the target), and also of multi-germs of such maps up to A Ωp -equivalence. (2) The classification of varietygerms V = f −1 (0) ⊂ (K n , Ω n , 0) up to K Ωn -equivalence of f : (K n , Ω n , 0) → (K p , 0) (i.e., for the subgroup of K in which the right coordinate changes preserve a given volume form Ω n in the source). More generally, we will consider volume preserving subgroups G Ωq of any of the Mather groups G = A, K, L, R and C preserving a (germ of a) volume form Ω q in the source (for q = n) or target (for q = p). (See the survey [50] for a discussion of the groups G and their tangent spaces LG, or see the beginning of §3 below for a brief reminder.)
These subgroups G Ωq of G fail to be geometric subgroups of A and K in the sense of Damon [11,12], hence the usual determinacy and unfolding theorems do not hold for G Ωq . In this situation moduli and even functional moduli often appear already in codimension zero, and e.g. for R Ωn this is indeed the case: a Morse function has a functional modulus (and hence infinite modality) in the volume preserving case [49]. Hence it might appear surprising that Martinet wrote 30 years ago in his book (see p. 50 of the English translation [37]) on the A Ωp classification problem in unimodular geometry that the groups involved "are big enough that there is still some hope of finding a reasonable classification theorem". It turns out that Martinet was right -the results of this paper imply, for example, that over C the classifications of stable map-germs for A Ωp and for A agree, and hence Mather's [40] nice pairs of dimensions (n, p). Furthermore, the classifications of simple complete intersection singularities agree for K Ωn and for K. Over R a G-orbit (G = A or K) corresponds to one or two orbits in the volume preserving (hence orientation preserving) case, otherwise the results are the same.
We will now summarize our main results. For any of the above Mather groups G, let G f denote the stabilizer of a map-germ f in G and let G e , as usual, denote the extended pseudo group of non-origin preserving diffeomorphisms. The differential of the orbit map of f (sending g ∈ G to g · f ) defines a map γ f :
LG → LG · f with kernel LG f . Let LG q f be the projection of LG f onto the source (for q = n) or the target factor (for q = p). Notice that, for example, the group G = R can be viewed as a subgroup R × 1 of A with Lie algebra LR ⊕ 0 -allowing such trivial factors 1 enables us to define the projections LG q f for all Mather groups G, which will be convenient for the uniformness of the exposition. For a given volume form Ω q in (K q , 0) we have a map div : M q · θ q → C r sending a vector field (vanishing at 0) to its divergence, where r = q for all G Ωq except K Ωp (we use here the following standard notation: C q denotes the local ring of smooth function germs on (K q , 0) with maximal ideal M q , and θ q denotes the C q module of vector fields on (K q , 0)). For K Ωp we consider linear vector fields in (K p , 0) with coefficients in C n , the divergence of such a vector field is an element of C n . We will show that for the (infinitesimal) G Ωq moduli space M(G Ωq , f ) we have the following isomorphism M(G Ωq , f ) :=
LG · f
LG Ωq · f ∼ = C r div(LG q f )
.
For K Ωn the vector space on the right is in turn isomorphic to the nth cohomology group of a certain subcomplex of the de Rham complex associated with any finitely generated ideal I in C n (defined in Section 4), taking I = f 1 , . . . , f p (the ideal generated by the component functions f i of f ). For A Ωp we obtain an analogous isomorphism by taking the vanishing ideal I of the discriminant (for n ≥ p) or the image (for n < p) of f , provided LA p f (also known as Lift(f )) is equal to Derlog of the discriminant or image of f . Furthermore, if LG has the structure of a C r -module (this is the case for all G Ωq except A Ωn ) then dim M(G Ωq , f ) is equal to the number of G Ωq moduli of f (for A Ωn this equality becomes a lower bound). This will be shown in the following way. The notion of G Ωq -equivalence of maps f and g (for a given volume form Ω q ) is easily seen to be equivalent to the following notion of G q f -equivalence of volume forms Ω q and Ω ′ q (for a given map f ): Ω ′ q ∼ G q f Ω q if and only if for some h ∈ G q f we have that h * Ω ′ q = Ω q . It then turns out that a pair Ω q and Ω ′ q (that in the case of R defines the same orientation) can be joined by a path of G q f equivalent volume forms if and only if Ω ′ q − Ω q = d(ξ⌋Ω) for some ξ ∈ LG q f and any volume form Ω in (K q , 0). And the number of G q f moduli of volume forms (and hence of G Ωq moduli of f ) is given by the dimension of the space Λ q /{d(ξ⌋Ω) : ξ ∈ LG q f } (here Λ q denotes the space of q-forms in (K q , 0)), which turns out to be equal to dim C q /div(LG q f ). If, furthermore, M(G Ωq , f ) = 0 then, over C, we have at the formal level (and also in the smooth category, provided the sufficient vanishing condition w.q.h. for M(G Ωq , f ) below holds)
G Ωq · f = G · f
Over R, the orbit G · f consists of one or two G Ωq -orbits, due to orientation as mentioned above. More precisely, if G + denotes the subgroup of G for which the elements of the q-factor of G are orientation-preserving then G Ωq · f = G + · f .
For the most interesting groups G Ωq we have the following sufficient conditions for the vanishing of M(G Ωq , f ), namely certain weak forms of quasihomogeneity. We call f weakly quasihomogeneous for G Ωq if f is q.h. for weights w i ∈ Z and weighted degrees δ j such that the following conditions hold.
• For G Ωq = A Ωp : all δ j ≥ 0 and j δ j > 0.
• For G Ωq = K Ωn : all w i ≥ 0 and i w i > 0.
• For G Ωq = K Ωp : j δ j = 0. Notice that any f with some zero component function (up to the relevant Gequivalence) is w.q.h. for A Ωp and K Ωp (and also for L Ωp and C Ωp ), and any f such that df (0) has positive rank is w.q.h. for K Ωn and K Ωp . These "trivial forms of weak quasihomogeneity" correspond to the fact that diffeomorphisms of a proper submanifold in (K q , 0) can be extended to volume preserving diffeomorphisms of (K q , Ω q , 0). Furthermore, if f is G Ωq -w.q.h. then the statement about equality of G-and G Ωq -orbits over C (and the corresponding one over R) in the previous paragraph holds in the smooth category (where smooth means complex-analytic over C and C ∞ or real-analytic over R, as usual). For a G Ωq -w.q.h. map f the above (generalized) weights and weighted degrees yield a generalized Euler vector field in (K q , 0) (q = n or p) that allows us to integrate the (a priori formally defined) vector fields at the infinitesimal level to give the required smooth diffeomorphisms.
For f not G Ωq -w.q.h. we are interested in upper and lower bounds for the dimension of M(G Ωq , f ) and in the question whether the G-finiteness of f implies the finiteness of M(G Ωq , f ). We have several results in this direction.
(1) For any G Ωq for which there is a version of weak quasihomogeneity we have the following easy upper bound (in the formal category) for G-semiquasihomogeneous (s.q.h.) maps f = f 0 + h, where f 0 q.h. (and hence G Ωq -w.q.h.) and G-finite and h has positive degree (relative to the weights of f 0 ). The normal space N G · f 0 := M n · θ f0 /LG · f 0 (where θ f0 denotes the C n -module of sections of f * 0 T K p ) decomposes into a part of non-positive filtration and a part of positive filtration, denoted by (N G · f 0 ) + . Denoting the number of G-moduli of positive filtration of f by m(G, f ) we have the inequality
dim M(G Ωq , f ) + m(G, f ) ≤ dim(N G · f 0 ) + .
(Note that the same inequality holds for the extended pseudo-groups G e , G Ωq ,e .) For G Ωq = A Ωp all our examples support the following conjecture: for f as above, the upper bound is actually an equality. For A-s.q.h. map-germs f : (K n , 0) → (K p , Ω p , 0) with n ≥ p − 1 and (n, p) in the nice range of dimensions or of corank one (outside the nice range) the validity of this conjecture would have an interesting consequence. Following Damon and Mond [13] we denote by µ ∆ (f ) the discriminant (for n ≥ p) or image (for p = n + 1) Milnor number of f (the discriminants and images ∆(f ) in these dimensions are hypersurfaces in the target, and ∆(f t ) of a stable perturbation f t of f has the homotopy type of a wedge of µ ∆ (f ) spheres). For a q.h. map-germ f 0 we have cod(A e , f 0 ) = µ ∆ (f 0 ) for n ≥ p by the main result in [13] and for p = n + 1 by Mond's conjecture (see Conjecture I in [10], for n = 1, 2 this conjecture has been proved by Mond and others). Now if our conjecture is true we obtain for s.q.h. maps f = f 0 + h the following interesting consequence of these results:
cod(A Ωp,e , f ) = µ ∆ (f ).
For (n, p) = (1, 2) the invariant µ ∆ (f ) is just the classical δ-invariant, hence we recover the formula cod(A Ωp,e , f ) = δ(f ) of Ishikawa and Janeczko [29] in the special case of s.q.h. curves (their formula holds for any A-finite curve-germ).
Notice that for f = f 0 + h we have µ ∆ (f ) = µ ∆ (f 0 ) (because any deformation by terms of positive filtration is topologically trivial). Our conjecture implies that the coefficients of each of the dim(N A e · f 0 ) + terms of h are moduli for A Ωp,e (some of them may be moduli for A e too), hence cod(A Ωp,e , f ) = cod(A e , f 0 ) = µ ∆ (f 0 ), which gives the formula above.
(2) For G Ωq = K Ωn we have more general results (in the analytic category) which, for example, imply the following. For any K-finite map f the moduli space M(K Ωn , f ) is finite dimensional. Furthermore, if f −1 (0) lies in a hypersurface h −1 (0) having (at worst) an isolated singular point at the origin then dim M(K Ωn , f ) ≤ µ(h) (notice that if f = (g 1 , . . . , g p ) defines an ICIS then we can take a generic C-linear combination h = i a i g i having finite Milnor number µ(h)).
(3) For G Ωq = A Ωp the moduli space M(A Ωp , f ) is finite dimensional for maps f whose image (or discriminant) has (at worst) an isolated singularity at the origin. This applies to A-finite maps f : (C n , 0) → (C p , 0) with p ≥ 2n or p = 2 (and any n). For the other pairs of dimensions (n, p) we only have the finiteness results for A-s.q.h. maps (see (1) above).
(4) For G Ωq = A Ωp and K Ωn we have the following criterion for dim M(G Ωq , f ) ≥ 1: suppose f 0 is q.h. and the restriction of γ f0 :
LG → LG · f 0 to the filtration-0 parts of the modules in source and target has 1-dimensional kernel, then the parameter u of a deformation f = f 0 + u · M by some non-zero element M ∈ (N G · f 0 ) + is a modulus for G Ωq . Using this criterion in combination with the existing Aand K-classifications in the literature we conclude the following. Suppose f : (C n , 0) → (C p , 0) is A-simple and n ≥ p or p = 2n or (n, p) = (2, 3), (1, p) (and any corank) or (n, p) = (3, 4) and corank 1 then: f is w.q.h. if and only if dim M(A Ωp , f ) = 0.
Or suppose that f has K-modality at most one, rank(df (0)) = 0 and n ≥ p then: f is q.h. if and only if dim M(K Ωn , f ) = 0.
The contents of the remaining sections of this papers are as follows. §1. Brief summary of earlier related works: by considering the moduli spaces M(G Ωq , f ) parameterizing the G Ωq -orbits inside G · f one can relate the seemingly unrelated earlier works on volume-preserving diffeomorphisms in singularity theory.
§2. H-isotopic volume forms: for a subgroup H of the group of diffeomorphisms Theorem 2.8 gives a criterion for a pair of volume forms to be H-isotopic, and Proposition 2.13 gives a sufficient condition on LH under which all pairs of volume forms are H-isotopic. The results will be applied to the subgroups H = G q f defined above.
§3. The moduli space M(G Ωq , f ): the space parameterizing the G Ωq -orbits in a given G-orbit is isomorphic to C r /div(LG q f ) (Theorem 3.4) and it vanishes for G Ωq -w.q.h. maps f (Proposition 3.8). These results imply, for example, that (over C) the stable orbits for A Ωp and A and the simple orbits for K Ωn and K agree (see Remark 3.10).
§4. A cohomological description of M(G Ωq , f ) and some finiteness results: for finitely generated ideals I in C n we define a subcomplex (Λ * (I), d) of the de Rham complex whose nth cohomology vanishes for w.q.h. ideals I (Theorem 4.4). For I = f * M p (not necessarily w.q.h.) H n (Λ * (I)) is isomorphic to M(K Ωn , f ) and is finite if I contains the vanishing ideal of a variety W with (at worst) an isolated singular point at 0, see Theorem 4.13 (for a hypersurface germ W we have H n (Λ * (I)) ≤ µ(W ), see Theorem 4.14). These finiteness results imply for example: M(K Ωn , f ) is finite if f defines an ICIS, and M(A Ωp , f ) is finite for p ≥ 2n and A-finite f .
§5. The foliation of A-orbits by A Ωp -orbits: in those dimensions (n, p), for which the classification of A-simple orbits is known, an A-simple germ f is w.q.h. if and only if M(A Ωp , f ) = 0. The classifications of the A Ωp -simple orbits in dimensions (n, 2) and (n, 2n), n ≥ 2, are described in Propositions 5.2, 5.3 and 5.4. In §5.3 the foliation of s.q.h. but not w.q.h. A-orbits by A Ωp -orbits is investigated for A-unimodal germs into the plane, and in §5.4 weak quasihomogeneity is defined for multigerms under A Ωp -equivalence.
§6. The foliation of K-orbits by K Ωn -and K Ωp -orbits: a K-unimodal germ f of rank 0 is q.h. if and only if M(K Ωn , f ) = 0, and M(K Ωn , f ) = 0 implies M(K Ωp , f ) = 0 (recall that germs f of positive rank are trivially w.q.h., hence their K-, K Ωn -and K Ωp -orbits coincide). Examples of rank 0 germs f defining an ICIS of codimension greater than one are presented for which dim M(K Ωn , f ) < µ(f )−τ (f ). For hypersurfaces we have dim M(K Ωn , f ) = µ(f ) − τ (f ) (by a result of Varchenko [48]), in all our higher codimensional examples we have dim M(K Ωn , f ) ≤ µ(f ) − τ (f ) (and for s.q.h. germs f it is easy to see that this inequality holds in general).
§7. The groups G Ωq = A Ωp , K Ωn , K Ωp : in the final section we consider the remaining groups G Ωq for which there are G-finite singular maps (as opposed to functions). Examples indicate that already G-stable, singular and not trivially w.q.h. maps f have positive modality for these groups G Ωq (for A Ωn the fold map even has infinite modality).
Brief summary of earlier related works
Having defined the moduli space M(G Ωq , f ) we can now conveniently describe the known results within this framework. Most of these results are on functions (hypersurface singularities), and (as explained above) one can either fix f and classify volume forms in the presence of a hypersurface defined by f (up to G q f = R n f , A n f or K n f -equivalence) or fix a volume form and classify functions up to G Ωq = R Ωn , A Ωn or K Ωn -equivalence. Much less is known for maps (see §1.2).
1.1. Results on functions (hypersurface singularities). First, consider R Ωnequivalence for functions f : (K n , Ω n , 0) → K, n ≥ 2. The isochore Morse-Lemma from the late 1970s by Vey [49] and Colin de Verdière and Vey [9] gives a normal form for an A 1 singularity involving a functional modulus. More recently isochore versal deformations were studied in [8] and [22]. The following result by Francoise [19,20] generalizes the isochore Morse-Lemma:
let b 1 = 1, b 2 , . . . , b µ(f ) be a base for N R e · f then M(R Ωn , f ) ∼ = K{(h i • f )b i : h i ∈ C 1 , i = 1, . . . , µ(f )}.
Hence f has precisely µ(f ) functional moduli (the h i are arbitrary smooth functiongerms in one variable).
Second, for A Ωn it is clear that (keeping the above notation) (
h 1 • f )1 ∈ LL e · f , hence M(A Ωn , f ) ∼ = K{(h i • f )b i : h i ∈ C 1 , i = 2, . . . , µ(f )}.
This moduli space vanishes for an A 1 singularity, and non-Morse functions f have µ(f ) − 1 functional moduli. Finally, for K Ωn the situation is much better. The following generalization of the corresponding K n f classification of volume forms has been studied, for example, by Arnol'd [1], Lando [32,33], Kostov and Lando [31] and Varchenko [48]: given a hypersurface f −1 (0) and a non-vanishing function-germ h, classify n-forms of the type f a hdx 1 ∧ . . . ∧ dx n up to diffeomorphisms that preserve f −1 (0). For a = 0 we have the special case of volume forms, and in this case the result of Varchenko gives
M(K Ωn , f ) ∼ = f, ∇f / ∇f ,
which has dimension µ(f ) − τ (f ). Both Francoise and Varchenko made extensive use of results of Brieskorn [5], Sebastiani [47] and Malgrange [35] on the de Rham complex of differential forms on a hypersurface with isolated singularities.
We will see that this dimension formula for M(K Ωn , f ) does, in general, not hold for map-germs f defining an ICIS of codimension greater than one. The obvious counter-examples are weakly quasihomogeneous maps f that are not quasihomogeneous: for such f the dimension of the moduli space is zero, but µ(f ) − τ (f ) > 0. More subtle counter-examples (Example 6.2 below) are the members of Wall's Kunimodal series F W 1,i of space-curves (which are not weakly quasihomogeneous): here the dimensions of the moduli spaces are equal to one and µ − τ is equal to two.
1.2.
Results for maps. Motivated by Arnold's classification of A 2k singularities of curves in a symplectic manifold [3] Ishikawa and Janeczko [29] have (in our notation) classified all A Ωp -simple map-germs f : (C, 0) → (C 2 , Ω p , 0). Notice that the volume-preserving diffeomorphisms of C 2 are also symplectomorphisms. Looking at their classification we observe that M(A Ωp , f ) = 0 if f is the germ of a q.h. curve. Furthermore, it is shown in [29] that cod(A Ωp,e , f ) = δ(f ), hence the A-finiteness of f (which is equivalent to δ(f ) < ∞) implies the finiteness of the moduli space M(A Ωp , f ).
Notice that for p = 1 any volume-preserving diffeomorphism of (K p , 0) is the identity. For functions the groups G Ωq , where q = n, are therefore the only ones of interest, and the results in §1.1 (which could be reproved using our approach) completely settle the classification problem for function-germs in the volume-preserving case. We will therefore concentrate on maps of target dimension p > 1 (but all general results also hold for p = 1, of course).
H-isotopic volume forms
In this section we study H-isotopies joining pairs of volume forms for subgroups H of D q := Diff(K q , 0). In the subsequent sections we will always apply these results to the subgroups H = G q f introduced in the introduction, but it might be worth mentioning that the results of this section have some additional applications, for example to singularities of vector fields (and the proofs remain valid for subgroups H of the group of diffeomorphisms of an oriented, compact, smooth q-dimensional manifold).
Let Λ k denote the space (of germs) of smooth differential k-forms on (K q , 0), and denote the subset of Λ q of (germs of) volume forms by Vol. For a given subgroup H ⊂ D q we consider a C q -module M in the Lie algebra LH of H (and M = LH if LH itself is a C q -module). In the following Ω and Ω i always denote (germs of) volume forms in (K q , 0). Definition 2.1. We say that Ω 0 and Ω 1 are H-diffeomorphic if there is a diffeomorphism Φ ∈ H such that Φ * Ω 1 = Ω 0 Definition 2.2. We say that Ω 0 and Ω 1 are H-isotopic if there is a smooth family of diffeomorphisms Φ t ∈ H for t ∈ [0, 1] such that Φ * 1 Ω 1 = Ω 0 and Φ 0 = Id. Remark 2.3. Two H-isotopic volume forms Ω 0 and Ω 1 are obviously H-diffeomorphic. The converse is not true in general. For example dx 1 ∧ dx 2 and −dx 1 ∧ dx 2 are diffeomorphic but not isotopic, since any diffeomorphism mapping one to the other changes orientation.
Definition 2.4. We say that Ω 0 and Ω 1 are M -equivalent if there is a vector field X ∈ M such that Ω 0 − Ω 1 = d(X⌋Ω) (for any volume form Ω). Theorem 2.6. If Ω 0 and Ω 1 are M -equivalent volume forms, which for K = R define the same orientation, then Ω 0 and Ω 1 are H-isotopic.
Proof. We use Moser's homotopy method [42]. Let Ω t = Ω 0 + t(Ω 1 − Ω 0 ) for t ∈ [0, 1]. It is easy to see that if Ω 0 and Ω 1 define the same orientation then Ω t ∈ Vol for any t ∈ [0, 1]. We are looking for a family of diffeomorphisms Φ t ∈ H, t ∈ [0, 1], such that
(2.1) Φ * t Ω t = Ω 0 and Φ 0 = Id. Differentiating (2.1) we obtain Φ * t (L Yt Ω t + Ω 1 − Ω 0 ) = 0, where Y t • Φ t = d dt Φ t , which implies that (2.2) d(Y t ⌋Ω t ) = Ω 0 − Ω 1 .
But Ω 0 and Ω 1 are M -equivalent, hence there exists a vector field X ∈ M such that Ω 0 − Ω 1 = d(X⌋Ω) for some volume form Ω. We want to find a family of vector fields Y t satisfying the following condition:
(2.3) Y t ⌋Ω t = X⌋Ω.
But Ω t = g t Ω for some non-vanishing smooth function g t . Hence Y t = (1/g t )X is a solution of (2.3) and Y t ∈ M , because X ∈ M and M is a module. The vector field Y t vanishes at the origin, hence its flow exists on some neighborhood of the origin for all t ∈ [0, 1]. Integrating Y t we obtain a smooth family of diffeomorphisms Φ t ∈ H for t ∈ [0, 1] such that Φ 0 = Id and Φ * t Ω t = Ω 0 , which implies that Ω 0 and Ω 1 are H-isotopic.
Next, we will show that for subgroups H of D q with LH a submodule of the C q -module θ q the existence of an H-isotopy between a pair of volume forms is equivalent to the LH-equivalence of this pair, provided that LH is closed with respect to integration in the following sense.
Definition 2.7. We say LH is closed with respect to integration if for any smooth family X t ∈ LH, t ∈ [0, 1], the integral 1 0 X t dt belongs to LH. Theorem 2.8. Let LH be a submodule of θ q , which is closed with respect to integration. Over K = R we also assume that Ω 0 and Ω 1 define the same orientation.
Then Ω 0 and Ω 1 are LH-equivalent if and only if Ω 0 and Ω 1 are H-isotopic.
Proof. The "only if" part follows directly from Theorem 2.6.
For the converse, we require the following lemma Lemma 2.9. Let Φ t be a smooth family of diffeomorphisms and let X t be a family of vector fields such that d
dt Φ t = X t • Φ t . Then d dt Φ −1 t = −(Φ * t X t ) • Φ −1 t . Proof of Lemma 2.9. Differentiating Φ −1 t • Φ t = Id we obtain 0 = d dt (Φ −1 t • Φ t ) = d dt (Φ −1 t ) • Φ t + d(Φ −1 t ) d dt Φ t , which implies that d dt (Φ −1 t ) = −d(Φ −1 t )(X t • Φ t ) • Φ −1 t . But, by definition, Φ * t X t = d(Φ −1 t )(X t • Φ t ).
Returning to the proof of the theorem, we assume that Ω 0 and Ω 1 are H-isotopic.
Then there exists, for all t ∈ [0, 1], a smooth family of diffeomorphisms Φ t ∈ H such that Φ 0 = Id and Φ * 1 Ω 0 = Ω 1 . Let (Φ t ) ′ = d dt Φ t = X t • Φ t ,
then
Ω 1 − Ω 0 = Φ * 1 Ω 0 − Ω 0 = 1 0 (Φ * t Ω 0 ) ′ dt = 1 0 (Φ * t L Xt Ω 0 )dt = 1 0 Φ * t d(X t ⌋Ω 0 )dt = d 1 0 Φ * t (X t ⌋Ω 0 )dt = d 1 0 (Φ * t X t )⌋Φ * t Ω 0 )dt = d 1 0 (Φ * t X t )⌋h t Ω 0 )dt for some smooth family of positive functions h t .
Thus
Ω 1 − Ω 0 = d 1 0 h t Φ * t X t dt⌋Ω 0 .
Lemma 2.9 implies Φ * t X t ∈ LH, and using the fact that LH is a module we also have h t Φ * t X t ∈ LH. And LH is closed with respect to integration, hence 1 0 h t Φ * t X t dt belongs to LH too. Therefore Ω 0 and Ω 1 are LH-equivalent, as desired.
Definition 2.10. The divergence of a vector field X ∈ θ q with respect to a given volume form Ω is, by definition, the smooth function div Ω (X) = d(X⌋Ω)/Ω. When the volume form Ω is understood from the context then we simply write div(X). And we have a map div : θ q → C q defined by X → div(X).
Corollary 2.11. Under the assumption of Theorem 2.8 the number of H-moduli of volume forms is equal to
dim K C q div(LH) .
Proof. It is easy to see that spaces C q /div(LH) and Λ q /{d(X⌋Ω) : X ∈ LH} are isomorphic. By Theorem 2.8 the number of H-moduli of volume forms is equal to the dimension of Vol/ ∼ LH . But it is easy to see that the spaces Λ q /{d(X⌋Ω) :
X ∈ LH} and Vol/ ∼ LH are equal if there exists a X ∈ LH such that d(X⌋Ω) is a volume form. Otherwise Λ q /{d(X⌋Ω) : X ∈ LH} \ Vol/ ∼ LH is a linear subspace of positive codimension in Λ q /{d(X⌋Ω) : X ∈ LH}. This implies that dim K Λ q {d(X⌋Ω) : X ∈ LH} = dim K Vol/ ∼ LH .
Next, we describe two sufficient conditions for the existence of a single Mequivalence class of volume forms in (K q , 0) (recall M is a C q -module in LH). For the first sufficient condition we require the following
Definition 2.12. A linear vector field E w = q i=1 w i x i ∂ ∂x i .
with integer coefficients w i is called a generalized Euler vector field (for coordinates (x 1 , . . . , x q ) ∈ K q and weights w = (w 1 , . . . , w q )).
We first consider generalized Euler vector fields with non-negative weights w i (for positive weights we obtain the usual Euler vector fields). For K Ωp -equivalence we also require linear vector fields with negative coefficients (see Theorem 3.9 below). Proposition 2.13. Let X be the germ of a smooth vector field on (K q , 0) which is locally diffeomorphic to a generalized Euler vector field with non-negative weights and positive total weight. If X generates a C q -module in LH then any two germs of volume forms (which over K = R define the same orientation) are H-isotopic.
Proof. Let E w be (the germ of) the Euler vector field for a coordinate system (x, y) = (x 1 , . . . , x k , y 1 , . . . , y q−k ) with weights w = (w 1 , . . . , w k , 0, · · · , 0), where w 1 , · · · , w k are positive and let Ω 0 be the germ of the volume-form
dx 1 ∧ . . . ∧ dx k ∧ dy 1 ∧ . . . ∧ dy q−k . By Theorem 2.6, it is enough to show that for any smooth q-form ω on (K q , 0) there exists a smooth function-germ g on (K q , 0) such that ω = d(gE w ⌋Ω 0 ).
Let G t (x, y) = (e w1t x 1 , . . . , e w k t x k , y 1 , . . . , y q−k ) for t ≤ 0. It is easy to see that
(G t ) ′ := d dt G t = E w • G t , G 0 = Id, lim t→−∞ G t (x, y) = (0, y)
for any (x, y) ∈ K q . Thus
(2.4) ω = G * 0 ω − lim t→−∞ G * t ω = 0 −∞ (G * t ω) ′ dt.
But ω = f Ω 0 for some smooth function-germ f and
(G * t ω) ′ = G * t L Ew ω = G * t d(E w ⌋ω) = d(G * t (E w ⌋ω)), hence (G * t ω) ′ = d(G * t (E w ⌋f Ω 0 )) = d((f • G t )G * t (E w ⌋Ω 0 )). One then checks by a direct calculation that G * t (E w ⌋Ω 0 ) = e t P k i=1 wi (E w ⌋Ω 0 ). Therefore (G * t ω) ′ = d((f • G t )e t P k i=1 wi (E w ⌋Ω 0 )). Combining this with (2.4) we obtain ω = d( 0 −∞ ((f • G t )e t P k i=1 wi )dt(E w ⌋Ω 0 )) = d(g(E w ⌋Ω 0 )),
where g is a function-germ on (K q , 0) defined as follows:
g(x, y) = 0 −∞ (e t P k i=1 wi (f (G t (x, y)))dt. The function-germ g is smooth, because 0 −∞ (e t P k i=1 wi (f (G t (x, y)))dt = 1 0 (s α f (F s (x, y))ds, where α = ( k i=1 w i ) − 1 and F s (x 1 , . . . , x k , y 1 , . . . , y q−k ) = (s w1 x 1 , . . . , s w k x k , y 1 , . . . , y q−k )
for any (x, y) = (x 1 , . . . , x k , y 1 , . . . , y q−k ) and s ∈ [0, 1]. Multiplying the weights by a sufficiently large constant we may assume that α > 1.
We conclude this section by stating a second sufficient condition for the existence of a single M -orbit of volume forms. Here we assume that LH contains a module M q X, where X is the germ of a non-vanishing vector field and M q is the maximal ideal of C q . Proposition 2.14. If X ∈ θ q , X(0) = 0, and the C q -module M q X is contained in LH then any two germs of volume forms (which over K = R define the same orientation) are H-isotopic.
Proof. X(0) = 0 implies that X is diffeomorphic to ∂/∂x 1 . Any germ of a q-form has in such a coordinate system, for some f ∈ C q , the following form
f (x)dx 1 ∧ dx 2 ∧ · · · ∧ dx q = d( x1 0 f (t, x 2 , · · · , x q )dt ∂ ∂x 1 ⌋dx 1 ∧ dx 2 ∧ · · · ∧ dx q ).
And x1 0 f (t, x 2 , · · · , x q )dt∂/∂x 1 belongs to M q ∂/∂x 1 . Thus any two germs of volume forms (which over R define the same orientation) are H-isotopic, by Theorem 2.6.
3. The moduli space M(G Ωq , f )
In this section we study smooth map-germs f : (K n , 0) → (K p , 0) (for K = C smooth means complex-analytic, for K = R smooth means either C ∞ or realanalytic). We set R := D n and L := D p (one can compose f with elements of D n on the right and with elements of D p on the left, which explains this notation).
Let G be one of the Mather groups A, K, R, L or C (all of which can be considered as subgroups of A or K, e.g. R × 1 ⊂ A) acting on the space of smooth map-germs f : (K n , 0) → (K p , 0). And let x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y p ) be coordinates on K n and K p , respectively. The differential of the orbit map g → g · f (g ∈ G and the action on f depends on the definition of G)
γ f : LG −→ LG · f has kernel LG f (where G f is the stabilizer of f in G). Recall that for G = A the map γ f is given by LA = M n θ n ⊕ M p θ p → M n θ f , (a, b) → tf (a) − ωf (b),
where tf (a) = df (a) and wf (b) = b • f , and for G = K it is given by
LK = M n θ n ⊕ gl p (C n ) → M n θ f , (a, B) → tf (a) − B · f.
The kernel of γ f inherits a C r module structure from LG, where r = p (or r = n) for G a subgroup of A (or K). Projecting onto source or target factors
LG n f ←− LG f −→ LG p f
preserves this C r module structure. Denoting the factors of G f by G n f and G p f their Lie algebras are the above projections. We also denote the factors of G by G n and G p (hence e.g. for G = A we have G n = R). Superscripts always denote projections onto one of the factors.
Consider subgroups G Ωn and G Ωp of G in which the diffeomorphisms (or families of diffeomorphisms for G = C, see below) preserve a given volume form Ω n or Ω p in the source or target, respectively. For r = n or p and a given volume form Ω r on K r let div : M q θ q → C r be the map that sends a vector field (vanishing at 0 in K n or K p ) to its divergence.
For K-equivalence in combination with a volume form in the target there are two ways to define the C Ωp component. But both version yield identical K Ωp -orbits (just as the alternative definitions of K yield the same K-orbits).
(1) In the original definition of K by Mather, C consists of diffeomorphisms H = (φ(x), ϕ(x, y)) ∈ D n+p , with ϕ(x, 0) = 0 for all x ∈ (K n , 0), and the action on f is given by H · f := ϕ(x, f • φ(x)). We can think of H as a n-parameter family of diffeomorphisms {ϕ x }, x ∈ K n , acting on f by sending x to ϕ x • f (x). If Ω p is a volume form on (K p , 0) we require that each ϕ x preserves Ω p (i.e. ϕ *
x Ω p = Ω p for all x ∈ (K n , 0)). In this way we obtain a subgroup C Ωp of C, and K Ωp := R · C Ωp (semi-direct product).
(2) In the linearized version of K we set C := GL p (C n ) and restrict to C Ωp = SL p (C n ), then LC Ωp = sl p (C n ) consists of p × p matrices over C n with zero trace. And, again, K Ωp := R · C Ωp . Then div can be considered as a map B → traceB as follows: the map gl p (C n ) → M n θ f , sending B to B · f (multiplication of f as a column vector of its component functions by a matrix B = (b ij )), can also be written
B · f = X B • f , where X B = p i=1 (b i1 (x)y 1 + . . . + b ip (x)y p )∂/∂y i is a linear vector field in K p with coefficients b ij ∈ C n . Hence divX B = traceB ∈ C n .
For any of the above volume preserving subgroups G Ωq of G we have the following Proposition 3.1. For q = n or p, and div : M q θ q → C r (where r = n for G q f = K p f and r = q in all other cases), we have an isomorphism
M(G Ωq , f ) := LG · f LG Ωq · f ∼ = C r div(LG q f ) .
Proof. Let π :
LG → LG q be the projection onto one of the factors, so that for
u = (a, b) we have v := π(u) is equal to a ∈ M n θ n or b, where either b ∈ M p θ p (for G = A) or b = X B for some B ∈ gl p (C n ) (for G = K).
(Recall that in the latter case div(X B ) = traceB.) Then consider the epimorphism
β : LG −→ C r , u → div(v).
Factoring out the kernel we obtain an isomorphism β :
LG
LG Ωq −→ C r .
We also have a well-defined map γ :
LG
LG Ωq −→ M n · θ f LG Ωq · f sending [(a, b)] to [tf (a)−ωf (b)] (for G a subgroup of A) and [(a, B)] to [tf (a)−B·f ] or, equivalently, [(a, X B )] to [tf (a) − X B • f ] (for G a subgroup of K). We see that imγ = LG · f
LG Ωq · f and thatβ(ker γ) = div(LG q f ). Factoring out the kernel of γ yields an isomorphism γ onto imγ so thatβ •γ −1 is the desired isomorphism.
Remark 3.2. For G = A the vector fields (a, b) ∈ LA f , b ∈ LA p
f and a ∈ LA n f are also said to be f -related, liftable and lowerable, respectively.
Notice that LG q f inherits a C r module structure, where r = n or p, from LG f and LG. In fact, we have
Lemma 3.3. LG f is a C r -submodule of LG (r = p or n for G a subgroup of G = A or K, respectively), which is closed under integration. The same is true for the factors LG q f of LG f . Proof.
The statements about the module structure are obvious. And for 1-parameter families of vector Proof. In all cases, except LA n f , the component LG q f of LG f is a module over the ring C r appearing as the target of the map div : M q θ q → C r . And LG q f is closed under integration, by the above lemma, hence Corollary 2.11 applies. For LA n f we notice that Proposition 3.1 is a statement about vector spaces (a C r module structure is not required).
fields v t = (a t , b t ) (for G = A) or (a t , X Bt ) (for G = K), t ∈ [0, 1], in the kernel of γ f we have 0 = 1 0 γ f (v t )dt = γ f ( 1 0 v t dt), hence 1 0 v t dt ∈ LG f . And it is clear that the q-component of 1 0 v t dt belongs to LG q f .M(G Ωq , f ) := LG · f LG Ωq · f ∼ = C r div(LG q f ) is
Remark 3.5. At this point it is perhaps useful to briefly recall the following. The G-modality of a map-germ f is, roughly speaking, the least m such that a small neighborhood of f can be covered by a finite number of m-parameter families of G-orbits. (More precisely, we consider the j k (G)-orbits in some neighborhood of j k f in a finite-dimensional jet-space J k (n, p) for some k for which all these j k (G)orbits are G-sufficient -recall that the G-determinacy degree of f in general fails to be upper semicontinuous under deformations of f , see [50] for a survey of results on G-determinacy.) Map-germs f of G-modality 0, 1, 2, . . . are said to be G-simple, G-unimodal, G-bimodal and so on. An m-G-modal family depends on no more than m parameters (moduli), for G = R and function-germs it depends on exactly m moduli [21]. For a subgroup G Ωq of a Mather group G and an m-parameter family of map-germs f λ the dimension of M(G Ωq , f λ ) is equal to the number of G Ωq -moduli of f λ , and also to the number of G q f λ -moduli of volume forms in (K q , 0), for each fixed vector λ ∈ K m of G-moduli of f λ .
We are now interested in classes of map-germs f for which the moduli spaces M(G Ωq , f ) vanish. For the groups G Ωq = A Ωp , K Ωn and K Ωp such classes of maps are given by the following weak forms of quasihomogeneity.
Definition 3.6. A map-germ f : (K n , 0) → (K p , 0), which is q.h. for weights w i ∈ Z (1 ≤ i ≤ n) and weighted degrees δ j (1 ≤ j ≤ p), is said to be weakly quasihomogeneous (w.q.h.) for the group G Ωq if the following conditions hold.
• For G Ωq = A Ωp : all δ j ≥ 0 and j δ j > 0.
• For G Ωq = K Ωn : all w i ≥ 0 and i w i > 0.
• For G Ωq = K Ωp : j δ j = 0. Remark 3.7. (i) The condition w.q.h. depends on the group G Ωq , when the group is clear from the context we will simply say that f is w.q.h.
(ii) For any subgroup G Ωq = A Ωn of G we have the following "trivial versions of w.q.h" for f : (1) for q = p and f G-equivalent to some map-germ having a zero component function, and (2) for q = n and df (0) of positive rank. For G Ωq = A Ωp , K Ωn and K Ωp it is easy to see that "trivially w.q.h." is a special case of w.q.h.: for (1) we give the zero component function positive weighted degree (and set all weights w i or all other degrees δ j to zero), and for (2) we have (up to G-equivalence) f = (x 1 , g(x 2 , . . . , x n )), so we take w 1 = 1 and w i = 0, i > 1.
We then have the following Proof. We will show that
LG
q · f ⊂ LG Ωq · f (here LG q · f is one of the factors of LG · f ), so that LG Ωq · f = LG · f . For G Ωq = A Ωp we have to show that LL · f ⊂ LA Ωp · f .
Clearly it is enough to check this inclusion for the elements of LL · f that do not belong to LL Ωp · f . Let y α = l y α l l and |α| ≥ 0. The following elements of LA Ωp ·f yield ωf (y α y i ·∂/∂y i ) ∈ LL · f , i = 1, . . . , p:
ωf (−(1 + α j )y 1 y α · ∂/∂y 1 + (1 + α 1 )y j y α · ∂/∂y j ), j = 2, . . . , p together with
tf (f * (y α ) n i=1 w i x i · ∂/∂x i ) − p j=2 δ j · ωf − 1 + α j 1 + α 1 y α y 1 · ∂/∂y 1 + y α y j · ∂/∂y j = (1 + α 1 ) −1 p j=1
(1 + α j )δ j · ωf (y α y 1 · ∂/∂y 1 ).
Notice that j (1 + α j )δ j = 0, for any exponent vector α, is equivalent to f being w.q.h. for the group A Ωp . For G Ωq = K Ωn we have to show that LR · f ⊂ LK Ωn · f . Exchanging the roles of the source and target vector fields, we see that the following elements of LK Ωn · f yield tf (x α x i · ∂/∂x i ) ∈ LR · f , i = 1, . . . , n:
tf (−(1 + α j )x 1 x α · ∂/∂x 1 + (1 + α 1 )x j x α · ∂/∂x j ), j = 2, . . . , n together with x α n i=1 δ i f i · ∂/∂y i − n j=2 tf w j − 1 + α j 1 + α 1 x 1 x α · ∂/∂x 1 + x j x α · ∂/∂x j = (1 + α 1 ) −1 n j=1 (1 + α j )w j · tf (x 1 x α · ∂/∂x 1 ).
Notice that j (1 + α j )w j = 0, for any exponent vector α, is equivalent to f being w.q.h. for the group K Ωn .
For G Ωq = K Ωp we have to show that LC · f ⊂ LK Ωp · f . Notice that LC Ωp = sl p (C n ) consists of elements B of gl p (C n ) with trace 0, hence we have a C n -module structure. Therefore, if E ij denotes a p × p matrix with entry (i, j) equal to 1 and all other entries 0 then it is enough to show that E ii · f ∈ LK Ωp · f , for i = 1, . . . , p.
(Notice that this implies that LC · f ⊂ LK Ωp · f , both for the linearized version GL p (C n ) of C and for Mather's original C, because of the C n -module structure.) Taking for j = 2, . . . , p
−f 1 · ∂/∂y 1 + f j · ∂/∂y j (corresponding to (E jj − E 11 ) · f with (E jj − E 11 ) ∈ LC Ωp ) and tf n i=1 w i x i · ∂/∂x i − (−δ 2 − . . . − δ p )E 11 + δ 2 E 22 + . . . + δ p E pp · f = p j=1 δ j f 1 · ∂/∂y 1 we see that E ii · f ∈ LK Ωp · f (i = 1, . . . , p) provided that j δ j = 0.
Finally, by the remark in the introduction, there is nothing to prove in the "trivially w.q.h. cases" (arbitrary diffeomorphisms in a proper subspace can be extended to volume preserving diffeomorphisms of the total space (K q , 0)).
The proposition says that, at the infinitesimal level, the tangent spaces of the G-orbit and of the G Ωq -orbit of f coincide. For K = R let G + be the subgroup of G for which the diffeomorphisms of the G q factor of G are orientation preserving. We then have at the level of orbits the following Theorem 3.9. Let f : (K n , 0) → (K p , 0) be w.q.h. for one of the groups G Ωq = A Ωp , K Ωn or K Ωp (or "trivially w.q.h." for any group) then:
(i) any two volume forms Ω, Ω ′ on K q (so that, in the case of K = R, Ω| 0 and
Ω ′ | 0 define the same orientation in T 0 R q ) are G q f -isotopic. (ii) f ′ ∼ G f (for K = C) and f ′ ∼ G + f (for K = R) imply f ′ ∼ GΩ q f (for some given volume form Ω q on K q ).
Proof. Using the weights w i (for q = n) or weighted degrees δ j (for q = p) in the definition of a G Ωq -w.q.h. map f we can define generalized Euler vector fields in C q . For G Ωq = A Ωp and K Ωn the vector fields have non-negative coefficients, hence Proposition 2.13 implies statement (i). For K Ωp we can have negative coefficients and we deduce statement (i) by a slightly modified argument (see below). The equivalence of (i) and (ii) is clear (over C the G-orbits are connected).
For K Ωp -equivalence the weighted degrees δ i of f yield a generalized Euler vector field E δ = p i=1 δ i y i ∂/∂y i in (K p , 0). We first claim that any volume form Ω p is K p f -equivalent to some linear volume form g(x)dy 1 ∧· · · dy p parameterized by g ∈ C n with g(0) = 0. Let Ψ be an origin-preserving diffeomorphism of (K p , 0) such that, for Ω p = h(y)dy 1 ∧ · · · ∧ dy p , we have Ψ * Ω p = dy 1 ∧ · · · ∧ dy p . Its inverse has the form
Ψ −1 (y) = ( p i=1 φ 1i (y)y i , · · · , p i=1 φ pi (y)y i ). We have Ψ −1 • f (x) = Φ x • f (x) for the following family Φ x of diffeomorphisms of (K p , 0) parameterized by x ∈ (K n , 0) Φ x (y) = ( p i=1 φ 1i (f (x))y i , · · · , p i=1 φ pi (f (x))y i ). Hence Ψ • Φ x • f = f (i.e., Ψ • Φ x ∈ K p f ) and Φ * x Ψ * Ω p = g(x)dy 1 ∧ · · · ∧ dy p , where g(x) = det(dΦ x )
. Clearly g(0) = 0, which implies the above claim.
It is therefore sufficient to consider the equivalence of parameterized linear volume forms. Notice that E δ generates a C n -submodule of LK p f and
g(x)dy 1 ∧ · · · ∧ dy p = d g(x) p i=1 δ i E δ ⌋dy 1 ∧ · · · ∧ dy p (recall that p i=1 δ i = 0)
, hence any pair of such volume forms is LK p f -equivalent. Furthermore, by the argument in the proof of Theorem 2.6, such a pair of volume forms (which, for K = R, is required to define the same orientation) is K p f -isotopic.
"Non-trivial applications" of the above result -namely to weakly quasihomogeneous map-germs f that are neither quasihomogeneous nor trivially weakly quasihomogeneous -will be considered later. For quasihomogeneous and trivially quasihomogeneous germs f we have the following immediate applications.
Remark 3.10.
(1) Quasihomogeneous case: all A-stable and all K-simple mapgerms f are quasihomogeneous. Hence the classifications, over C, of stable germs for A and A Ωp and of simple germs for K, K Ωn and K Ωp agree -over R, each A-stable or K-simple orbit corresponds to one or two stable or simple orbits for the volume preserving subgroups.
(2) Trivially weakly quasihomogeneous case: (i) the classifications of map-germs f , with df (0) of positive rank, for the groups K, K Ωn and K Ωp agree. (ii) For mapgerms f : (K n , 0) → (K p , 0) whose image lies in a proper submanifold of (K p , 0) (such f have, up to a target coordinate change, a zero component function) the Aand A Ωp -orbits, the Land L Ωp -orbits, and the Cand C Ωp -orbits agree. Notice, for example, that the A and A Ωp classifications of simple curve-germs agree for p ≥ 7 (Arnol'd [2] has shown that all stably simple curves can be realized in 6-space, hence all A-simple curves in higher dimensions have zero component functions).
A cohomological description of M(G Ωq , f ) and some finiteness results
The results on M(K Ωn , f ) can be reformulated for ideals, and for this reformulation we obtain a further isomorphism in terms of cohomology. This cohomological description yields some finiteness results in the non-w.q.h. case. Let I ⊂ C n be a finitely generated ideal (recall: for C n = O n all ideals are f.g., for C n = E n , the ring of C ∞ function germs, there are non-f.g. ideals like M ∞ n ). We say that I and J are where, for h ∈ I, we set Y h := dh · Y . For I = g 1 , . . . , g p and f := (g 1 , . . . , g p ) :
(K n , 0) → (K p , 0) we have the following: φ * g 1 , . . . , g p := g 1 • φ, . . . , g p • φ = g 1 , . . . , g p if and only if f = B · (f • φ) for some B ∈ GL p (C n ). Hence Derlog(I) = LK n f , and setting D Ωn := {h ∈ D n : h * Ω n = Ω n } (for a given volume form Ω n in (K n , 0)) we have the following isomorphisms for the (infinitesimal) D Ωn -moduli space of I:
M(D Ωn , I) := C n div(Derlog(I)) ∼ = C n div(LK n f ) ∼ = LK · f LK Ωn · f .
This moduli space is also isomorphic to the nth cohomology group of the following complex (Λ * (I), d). Defining for k = 0, . . . , n the vector spaces
Λ k (I) := {α + dβ ∈ Λ k : dI ∧ α ⊂ IΛ k+1 , dI ∧ β ⊂ IΛ k }
we obtain a subcomplex (Λ * (I), d) of the de Rham complex (Λ * , d). Sometimes we shall simply write Λ * (I) = (Λ * (I), d) and similarly for the other complexes defined below (the differential is always the same d).
The nth cohomology group of the complex (Λ * (I), d) is
H n ((Λ * (I), d) = Λ n /dΛ n−1 (I) = Λ n /{dα ∈ Λ n : dI ∧ α ⊂ IΛ n }.
For a given volume form Ω n the map
Derlog(I) ∋ X → X⌋Ω n ∈ {α ∈ Λ n−1 : dI ∧ α ⊂ IΛ n }
is an isomorphism. Notice that the tangent space to Λ n can be identified with C n , and recall that div Ωn (X) = d(X⌋Ω n )/Ω n . Hence we see that Definition 4.2. We say that an ideal I in C n is w.q.h. if it has a set of generators g 1 , . . . , g p such that the corresponding map f = (g 1 , . . . , g p ) is K Ωn -w.q.h. (notice that this is a natural generalization of homogeneous ideals).
Remark 4.3.
If the ideal I is w.q.h. then the variety defined by I is "quasihomogeneous with respect to a smooth submanifold" in the sense of [17].
We can now reformulate Theorem 3.9 as follows Theorem 4.4. Let I be a w.q.h. ideal in C n = O n or E n . For C n = E n we assume that I is finitely generated, and (over R) D + n denotes the group of orientation preserving diffeomorphisms. Then we have the following:
(i) any two volume forms on K n (which, in the case K = R, define the same orientation in T 0 R n ) can be joined (via pullback) by a 1-parameter family of diffeomorphisms φ t such that φ * t I = I (i.e., by a (D n ) I -isotopy). (ii) For a given volume form Ω n , let D Ωn be the subgroup of D n whose elements preserve Ω n . Then φ * I = J for some φ ∈ D n (for K = C) or some φ ∈ D + n (for K = R) implies h * I = J for some h ∈ D Ωn . Remark 4.5. For A-equivalence we have the following cohomological description. Given a map-germ f : (C n , 0) → (C p , 0), let ∆ f be the discriminant (for n ≥ p) or the image (for n < p) of f . If f satisfies the necessary and sufficient condition (namely, GTQ for n ≥ p or NHS for n < p) for the equality Derlog(I(∆ f )) = Lift(f ) of Theorem 2 in [7] then we have the following isomorphism: 0). For the precise definitions of GTQ (generically a trivial unfolding of a q.h. germ) and NHS (no hidden singularities) we refer to [7].
M(A Ωp , f ) ∼ = H p ((Λ * (I(∆ f ), d)), here I(∆ f ) is the vanishing ideal of ∆ f ⊂ (C p ,
Notice that f w.q.h. (for A Ωp ) implies that the ideal I(∆ f ) is weakly quasihomogeneous. But there are w.q.h. maps f that fail to be GTQ. We give two examples illustrating these facts.
Example 4.6. The map f : (C 3 , 0) → (C 2 , 0) given by f (x, y, x) = (x, xy+y 5 +y 7 z) is w.q.h. with weights (4, 1, −2) and weighted degrees (4,5). The discriminant of f is the origin in (C 2 , 0). The critical set is the z-axis, which consists of A-unstable points, hence f fails to be A-finite.
Example 4.7. In [7] f :
(C 3 , 0) → (C 2 , 0), f (u, x, y) = (u, x 4 + y 4 + ux 2 y 2 )
is presented as an example of a non-GTQ map-germ. But f is weakly quasihomogeneous (for the weights (0, 1, 1)). Notice that, again, f fails to be A-finite.
One may not care much about such degenerate examples of infinite A-codimension.
In Section 5 we describe more subtle examples of weakly quasi-homogeneous mapgerms that are A-finite and even A-simple.
Next, we will derive some finiteness results for H n (Λ * (I)) when I is not necessarily w.q.h., and apply these to deduce G Ωq -finiteness from G-finiteness of f : (C n , 0) → (C p , 0) (for certain G and (n, p)). We assume here that K = C and that all germs (at 0) are C-analytic. For I = g 1 , . . . , g s we denote the ideal of maximal minors of the Jacobian of g = (g 1 , . . . , g s ) (viewed as a map-germ) by J(g), and we set ∇g i := J(g i ). Recall that g 1 , . . . , g s , J(g) is the vanishing ideal of the set of K-unstable points of g, so that (by the Nullstellensatz) g is K-finite if and only if M r n ⊂ g 1 , . . . , g s , J(g) , for some r < ∞, or iff g has (at worst) an isolated singular point at 0. Also notice that g 1 , . . . , g s , J(g) ⊂ g 1 , . . . , g s , ∇g 1 , · · · , ∇g s implies that, for K-finite g, the ideal on the RHS of this inclusion has finite colength.
We will relate the complex Λ * (I) to the following subcomplex of the de Rham complex:
(A * 0 (I), d), where A k 0 (I) = {α + dβ ∈ Λ k : α ∈ IΛ k , β ∈ IΛ k−1 }.
If I is the vanishing ideal of a variety V then this complex is called the complex of zero algebraic restrictions to V (see [18], [17], [16]). The cohomology of the quotient complex (Λ * /A * 0 (I(V )), d) has been studied in detail in earlier works (see [43], [4], [5], [47], [26], [27]). Notice that the k-th cohomology H k (Λ * /A * 0 (I)) of this quotient complex and the (k + 1)-th cohomology H k+1 (A * 0 (I)) of the above subcomplex are related by the map
d : {ω ∈ Λ k : dω ∈ A k+1 0 (I)} dΛ k−1 + A k 0 (I) −→ {γ ∈ A k+1 0 (I) : dγ = 0} dA k 0 (I) ,
which is an isomorphism by the exactness of the de Rham complex of germs of differential forms on C n . We are interested in H n (A * 0 (I)). First notice the following fact. Proposition 4.8. If an ideal I in O n has generators g 1 , . . . , g s , where each g i is K-equivalent to a K Ωn -w.q.h. function-germ, then H n (A * 0 (I)) = 0. Remark 4.9. The hypothesis that each g i is K-equivalent to some function-germ that is q.h. for non-negative weights and total positive weight (and hence K Ωnw.q.h.) does not require that the map g = (g 1 , . . . , g s ) is K Ωn -w.q.h. (the source diffeomorphisms in the K-equivalences can be different for each g i ).
Proof. It is enough to show that any n-form in IΛ n is the differential of a (n − 1)form in IΛ n−1 . Let ω = s i=1 g i ω i , where the ω i are n-forms. Any n-form on C n is closed and each g i = k i Φ * h i , where k i is a non-vanishing function-germ, Φ i is a diffeomorphism-germ and h i is w.q.h. with non-negative weights, at least one of which is positive. We then apply the following lemma to each h Proof of Lemma 4.10. If h generates the vanishing ideal of {h = 0} then this is a corollary of the relative Poincare lemma for varieties that are quasi-homogeneous with respect to a smooth submanifold [17]. More generally (for h not necessarily radical) we use the same method as in the proof of Proposition 2.13.
i (Φ −1 i ) * (k i ω i ) separately.
Let E w be (the germ of) the Euler vector field for h and let G t be the flow of E w . Then G * t h = e δt h, where δ is quasi-degree of h. By direct calculation we obtain
(4.1) ω = 0 −∞ (G * t ω) ′ dt = d(hβ), where β = 0 −∞ e δt G * t (E w ⌋ω)dt is a smooth (n − 1)-form.
To conclude the proof of the proposition, we have from Lemma 4.10
g i ω i = Φ * i (h i (Φ −1 i ) * (k i ω i )) = Φ * i (d(h i β i )) = d(g i α i ), where α i = 1 ki Φ * i β i . Hence ω = s i=1 g i ω i = d( s i=1 g i α i )
, as desired. We can now relate the dimensions of nth cohomology groups of the two complexes in question.
Theorem 4.11. For g 1 , · · · , g s ∈ I we have dim H n (Λ * (I)) ≤ dim O n g 1 , · · · , g s , ∇g 1 , · · · , ∇g s + dim H n (A * 0 ( g 1 , · · · , g s )).
Proof. For J := g 1 , · · · , g s ⊂ I, clearly J Λ n−1 ⊂ Λ n−1 (I), which implies that
dim H n (Λ * (I)) = dim Λ n /d(Λ n−1 (I)) ≤ dim Λ n /d(J Λ n−1 ), where dim Λ n /d(J Λ n−1 ) = dim Λ n /A n 0 (J ) + dim A n 0 (J )/d(J Λ n−1 ). Furthermore, from A n 0 (J ) = { s i=1 g i ω i + dg i ∧ σ i : ω i ∈ Λ n , σ i ∈ Λ n−1 , i = 1, · · · , s}
we see that Λ n /A n 0 (J ) is isomorphic to O n / g 1 , · · · , g s , ∇g 1 , · · · , ∇g s . Finally, d(J Λ n−1 ) = d(A n−1 0 (J )) implies that A n 0 (J )/d(J Λ n−1 ) and H n (A * 0 (J )) are equal.
Theorem 4.11 and Proposition 4.8 imply the following corollary Corollary 4.12. If g 1 , · · · , g s ∈ I satisfy the conditions of Proposition 4.8 then
dim H n (Λ * (I)) ≤ dim O n ∇g 1 , · · · , ∇g s .
Proof. Proposition 4.8 implies that dim H n (A * 0 ( g 1 , · · · , g s )) = 0, and g i ∈ ∇g i (because g i is w.q.h.).
We can now deduce the following finiteness results. Proof. Let I(W ) be generated by g 1 , · · · , g s . Clearly g 1 , · · · , g s ∈ I and from Theorem 4.11 we have dim H n (Λ * (I)) ≤ dim O n g 1 , · · · , g s , ∇g 1 , · · · , ∇g s + dim H n (A * 0 (I(W ))).
From the hypothesis on W we then obtain the finiteness of the dimensions on the right: H n (A * 0 (I(W ))) is finite by a result of Bloom and Herrera [4] and the colength of g 1 , · · · , g s , ∇g 1 , · · · , ∇g s in O n is also finite for such W (see our earlier remark).
Theorem 4.14. Let g be the vanishing ideal of a hypersurface having an isolated singularity at 0. If g is contained in I then dim H n (Λ * (I)) ≤ µ(g), where µ(g) is the Milnor number of g.
Proof.
For g ⊂ I we obtain from Theorem 4.11 dim H n (Λ * (I)) ≤ dim O n g, ∇g + dim H n (A * 0 ( g )).
The desired bound then follows from the following formula of Brieskorn [5] and Sebastiani [47]: dim H n (A * 0 ( g )) = µ(g) − τ (g), where τ (g) := dim O n / g, ∇g is the Tjurina number of g.
Remark 4.15. Theorem 4.13 implies that for a finitely generated ideal I = g 1 , . . . , g p corresponding to a K-finite map f = (g 1 , . . . , g p ) the dimension of H n (Λ * (I)) is finite dimensional. For the ideal of an ICIS we have a more precise bound. For a C-linear combination h = p i=1 a i g i we have h ⊂ I, hence dim H n (Λ * (I)) ≤ µ(h) (for µ(h) < ∞ we apply Theorem 4.14, and otherwise the upper bound is trivial). Furthermore, for a generic projection π : C p → C, (y 1 , . . . , y p ) → p i=1 a i y i the Milnor number of h = π • g, where g = (g 1 , . . . , g p ), is finite (recall the usual method for calculating the Milnor number of an ICIS).
The above finiteness results can be generalized to the case of subgroups H of the group of germs of C-analytic diffeomorphisms of C q . Using the isomorphism θ q ∋ X → X⌋Ω ∈ Λ q we can prove in the same way the following result. Theorem 4.16. Let J be an ideal in O q generated by g 1 , · · · , g s . If J θ q is contained in LH then
dim O q div(LH)
≤ dim O q g 1 , · · · , g s , ∇g 1 , · · · , ∇g s + dim H q (A * 0 ( g 1 , · · · , g s )).
In particular we obtain the following
dim O p /div(LA p f ) is finite. (b) If imf ⊂ g −1 (0)
, for some hypersurface germ g −1 (0) with an isolated singularity at 0, then dim O p /div(LA p f ) ≤ µ(g). Remark 4.18. Suppose that f : (C n , 0) → (C p , 0) is an A-finite map-germ with target dimension p ≥ 2n. Then imf is a variety-germ with an isolated singularity at 0, hence A-finiteness implies A Ωp -finiteness (in the sense that the moduli-space M(A Ωp , f ) is finite dimensional, by the above result). This generalizes the corresponding result in [29] for plane curves.
Also, for map-germs f : (C n , 0) → (C 2 , 0), n ≥ 2, for which LA p f = Lif t(f ) is equal to Derlog of the discriminant we have that the A-finiteness of f implies the A Ωp -finiteness (notice, the discriminant is a curve with isolated singularities).
For p < 2n the image (for n < p) or the discriminant (for n ≥ p ≥ 3) of an Afinite singular map-germ f in general has non-isolated singularities (except perhaps for a generalized fold map f ). Hence the above finiteness result cannot be applied.
The foliation of A-orbits by A Ωp -orbits
In this section we study the foliation of A-orbits of map germs f : (K n , 0) → (K p , Ω p , 0) by A Ωp -orbits. Our main objective here is the classification of A Ωpsimple orbits inside the A-simple orbits, and in dimensions (n, 2) and (n, 2n), n ≥ 2, we give explicit lists (see §5.1 and §5.2). We also consider A Ωp -orbits of positive modality that are s.q.h. but not w.q.h (see §5.3) and w.q.h. multigerms (see §5.4).
For the pairs (n, p) for which the A-simple orbits are known -i.e., for n ≥ p, (1, p) any p, p = 2n, (2, 3) (any corank) and for (3, 4) (of corank 1), see the references below -we find that:
(1) an A-simple germ is A Ωp -simple if and only if it does not lie in the closure of the orbit of any non-weakly quasihomogeneous germ, (2) for n < 2p and for p = 2n, with n ≤ 3, an A-simple germ is (3,4) [28], (n, 2) (n ≥ 2) [44,46] and (3, 3) [36]. The survey in [25] describes the simple singularities of projections of complete intersections, this a priori finer classification corresponds to the A-classification for n ≥ p.)
After explaining the techniques for verifying the above claim, we will describe two particular cases in detail. First, the classification of A Ωp -simple orbits in dimensions (n, 2), n > 1, because for p = 2 the volume preserving and the symplectic classifications agree. Combining this classification with the one by Ishikawa and Janeczko [29] for curves (i.e., for (1, 2)) yields all simple map-germs into the symplectic plane. And second, the classification of A Ωp -simple orbits in dimensions (n, 2n), where "non-trivial" weakly quasihomogeneous germs (that are not quasihomogeneous nor "trivially w.q.h.") start appearing.
Notice that the condition w.q.h. (for A Ωp ) in Proposition 3.8 and Theorem 3.9 is a sufficient condition for the absence of A Ωp -moduli, we do not know whether it is necessary. However, for all A-simple germs in the dimension ranges (n, p) in which the A-simple classification is known (see above) the condition w.q.h. is necessary and sufficient for the absence of A Ωp -moduli. This obviously implies the criterion above: an A-simple germ f is A Ωp -simple if and only if f is only adjacent to w.q.h. germs. All known examples of A-simple map-germs f that fail to be w.q.h. are of the form f = f 0 + h, where f 0 is quasihomogeneous, h is a monomial vector of positive filtration (weighted degree) and the restriction of γ f0 : LA → LA · f to the filtration-0 parts (of the filtered modules in source and target) has 1-dimensional kernel. In this situation the coefficient of h is a modulus for A Ωp (see Lemma 5.1 below).
Consider
LA Ωp · f ⊂ LA · f = tf (M n · θ n ) + wf (M p · θ p ). For the subgroup A Ωp = R × L Ωp of A we have to restrict the homomorphism wf : θ p → θ f , wf (b) = b • f to divergence free vector fields b, hence LL Ωp · f is no longer a C p - module. Let Λ d denote the K-vector space of homogeneous divergence free vector fields in K p of degree d. Notice that Λ d is the kernel of the epimorphism div : (θ p ) (d) := M d p · θ p M d+1 p · θ p → H (d−1) := M d−1 p M d p ,
which maps a vector field on K p of degree d to its divergence. Hence
dim Λ d = dim(θ p ) (d) − dim H (d−1) = (p − 1) p+d−1 d + p+d−2 d .
The dim Λ d vector fields
· f = f * ⊕ d≥1 Λ d .
The criterion in the next easy lemma is sufficient for detecting in the existing classifications of A-simple orbits those which are foliated by an r-parameter family, r ≥ 1, of A Ωp -orbits.
Lemma 5.1. Consider a map-germ f u : (K n , 0) → (K p , 0) of the form f u = f + u · M , where f is a quasi-homogeneous germ, u ∈ K and M = X α · ∂/∂y j / ∈ LA · f = LA Ωp · f is a monomial vector of positive weighted degree (with respect to the weights of f ). Then we have the following:
(i) The coefficient u is not a modulus for A-equivalence.
(ii) For a set of weights for which f is weighted homogeneous, let (θ n ) 0 , (θ p ) 0 and (θ f ) 0 denote the filtration-0 parts of the modules of source-, target-vector fields and vector fields along f , respectively. If the kernel of the linear map
γ f : (θ n ) 0 ⊕ (θ p ) 0 → (θ f ) 0 , (a, b) → tf (a) − wf (b), of K-vector spaces is 1-dimensional then u is an A Ωp -modulus of f u .
Proof. Let f be weighted-homogeneous for the weights w 1 , . . . , w n , and associate to the target variables the weights δ 1 , . . . , δ p . Then the weighted degree of ∂/∂y i is −δ i so that f has filtration 0 and M has filtration r > 0.
For A-equivalence we consider the following element of LA · f u :
tf u ( n i=1 w i x i · ∂/∂x i ) − wf u ( p j=1 δ j y j · ∂/∂y j ) = ruM.
From Mather's lemma (Lemma 3.1 in [38]) we conclude that the connected components of K \ {0} of the parameter axis lie in a single A-orbit, hence u is not a modulus for A.
For the second statement we observe that dim ker γ f = 1 implies that this kernel is spanned by the pair of Euler vector fields (E w , E δ ) in source and target (which is unique up to a multiplication by an element of K * ). And M /
∈ LA · f implies that the only generator of M in LA · f u must be of the form tf u (a) − wf u (b) with (a, b) a non-zero multiple of (E w , E δ ). But E δ has non-zero divergence, hence this generator does not belong to LA Ωp · f u . Now Mather's lemma implies that u is a modulus for A Ωp .
5.1.
A Ωp -simple, hence symplectically simple, maps from n-space to the plane. The following classification, in combination with Ishikawa and Janeczko's classification of plane curves [29], provides a complete list of simple map-germs into the plane C 2 , up to source diffeomorphisms and target symplectomorphisms (volume preserving diffeomorphisms of C 2 are symplectomorphisms).
Proposition 5.2. Any A Ωp -simple map-germ f : (C n , 0) → (C 2 , 0), n ≥ 2, is equivalent to one of the following normal forms (here Q = n−2 i=1 z 2 i for n > 2 and Q = 0 for n = 2): (x, y); (x, y 2 + Q); (x, xy + y 3 + Q); (x, y 3 + x k y + Q), k > 1; (x, xy + y 4 + Q).
Proof. Any A-simple germ in dimensions (n, 2), n ≥ 2, which does not appear in the above list, is adjacent to one of the following germs (for n > 2, up to a suspension by Q defined above): (x, xy + y 5 + y 7 ), (x, xy 2 + y 4 + y 5 ) or (x 2 + y 3 , y 2 + x 3 ) (see the adjacency diagrams in [44] and [46]). These three germs fail to be weakly quasihomogeneous and they satisfy the hypotheses of Lemma 5.1, hence they have at least one modulus for A Ωp . In fact, the parameter a in (x, xy + y 5 + ay 7 ), (x, xy 2 + y 4 + ay 5 + . . .) and (x 2 + ay 3 , y 2 + x 3 ) is a modulus for A Ωp .
5.2.
A Ωp -simple maps from n-space to 2n-space. In the same way we obtain the A Ωp -simple germs in dimensions (n, 2n), n ≥ 2 (notice that n = 1 again corresponds to the classification in [29]). Except for the appearance of a series of w.q.h. germs (see the last two normal forms in Proposition 5.4 below, corresponding to type 22 k and 23 in [30]), which are not q.h. nor trivially w.q.h., this classification follows from the classification of A-orbits (and some information about adjacencies between these orbits) in [30], using the same arguments as in dimensions (n, 2). The classifications in dimensions (2,4) and (n, 2n), n ≥ 3, are as follows. Proposition 5.3. Any A Ωp -simple map-germ f : (C 2 , 0) → (C 4 , 0) is equivalent to one of the following normal forms: (x, y, 0, 0); (x, xy, y 2 , y 2k+1 ), k ≥ 1; (x, y 2 , y 3 , x k y), k ≥ 2; (x, y 2 , y 3 +x k y, x l y), l > k ≥ 2; (x, y 2 , x 2 y +y 2k+1 , xy 3 ), k ≥ 2; (x, y 2 , x 2 y, y 5 ); (x, y 2 , x 3 y + y 5 , xy 3 ); (x, xy, xy 2 + y 3k+1 , y 3 ), k ≥ 1; (x, xy, xy 2 + y 3k+2 , y 3 ), k ≥ 1; (x, xy + y 3k+2 , xy 2 , y 3 ), k ≥ 1; (x, xy, y 3 , y 4 ); (x, xy, y 3 , y 5 ).
Proposition 5.4. Any A Ωp -simple map-germ f : (C n , 0) → (C 2n , 0), n ≥ 3, is equivalent to one of the following normal forms (here x denotes x 1 , . . . , x n−1 , and notice that the last two normal forms are only A Ωp -simple for n ≥ 4): (x, y, 0, . . . , 0) (x, x 1 y, . . . , x n−1 y, y 2 , y 2k+1 ), k ≥ 1 (x, x 2 y, . . . , x n−1 y, y 2 , y 3 , x k 1 y), k ≥ 2 (x, x 2 y, . . . , x n−1 y, y 2 , y 3 + x k 1 y, x l 1 y), l > k ≥ 2 (x, x 2 y, . . . , x n−1 y, y 2 , x 2 1 y + y 2k+1 , x 1 y 3 ), k ≥ 2 (x, x 2 y, . . . , x n−1 y, y 2 , x 2 1 y, y 5 ) (x, x 2 y, . . . , x n−1 y, y 2 , x 3 1 y + y 5 , x 1 y 3 ) (x, x 3 y, . . . , x n−1 y, y 2 , x 2 1 y, x 2 2 y, y 3 + x 1 x 2 y) (x, x 3 y, . . . , x n−1 y, y 2 , x 2 1 y, x 2 2 y, y 3 ) (x, x 3 y, . . . , x n−1 y, y 2 ,
x 1 x 2 y, (x 2 1 + x 3 2 )y, y 3 + x 2 2 y) (x, x 3 y, . . . , x n−1 y, y 2 , x 1 x 2 y, (x 2 1 + x 3 2 )y, y 3 + x 3 2 y) (x, x 3 y, . . . , x n−1 y, y 2 , x 1 x 2 y, (x 2 1 + x 3 2 )y, y 3 ) (x, xy, x 1 y 2 + y 3k+1 , y 3 ), k ≥ 1 (x, xy, x 1 y 2 + y 3k+2 , y 3 ), k ≥ 1 (x, x 1 y + y 3k+2 , x 2 y, .
. . , x n−1 y, x 1 y 2 , y 3 ), k ≥ 1 (x, x 1 y, x 2 y + y 3k+2 , x 3 y, . . . , x n−1 y, x 1 y 2 + y 3l+1 , y 3 ), l > k ≥ 1 (x, x 1 y, x 2 y + y 3k+2 , x 3 y, . . . , x n−1 y, x 1 y 2 + y 3l+2 , y 3 ), l > k ≥ 1 (x, x 1 y + y 3l+2 , x 2 y + y 3k+2 , x 3 y, . . . , x n−1 y, x 1 y 2 , y 3 ), l > k ≥ 1 (x, xy, y 3 , y 4 ) (x, xy, y 3 , y 5 ) (x, x 1 y + y 3 , x 2 y, . . . , x n−1 y, x 1 y 2 + y 2k+1 , x 2 y 2 + y 4 ), for k = 2 and n ≥ 4 (x, x 1 y + y 3 , x 2 y, . . . , x n−1 y, x 1 y 2 + y 5 , y 4 ), for n ≥ 4.
Proof. Except for the germs of type 22 k and 23 in dimensions (n, 2n), n ≥ 4 (these are the last two germs in the second list above), all A-simple germs in [30] are either quasihomogeneous or they satisfy the hypotheses of Lemma 5.1 and hence have at least one A Ωp -modulus.
Consider, then, the series 22 k of map germs (C n , 0) → (C 2n , 0), n ≥ 3 given by: . . . , x n−1 , x 1 y + y 3 , x 2 y, . . . , x n−1 y, x 1 y 2 + y 2k+1 , x 2 y 2 + y 4 ), k ≥ 2.
g k = (x 1 ,
The germs 22 k are not semi-quasihomogeneous: if we write g k = f + y 2k+1 · e 2n−1 then the weighted homogeneous initial part f is not A-finite. For n = 3 all the germs 22 k are A-simple, for n ≥ 4 only 22 2 is A-simple (the germs 22 ≥3 do not have an A-modulus, but they lie in the closure of non-simple A-orbits), see [30]. Now consider A Ωp -equivalence. Writing f u = f + u · y 2k+1 · e 2n−1 we see that dim ker γ f = n − 2. For n = 3 part (ii) of Lemma 5.1 therefore implies that the coefficient u is an A Ωp -modulus. For n ≥ 4 the germs f u are weakly quasihomogeneous (take weights w(x 1 ) = w(x 2 ) = w(y) = 0 and w(x i ) = 1, i ≥ 3) and A Ωp -equivalent to g k (for u = 0).
For the germ of type 23 the argument is the same.
5.3.
Semi-quasihomogeneous, but not weakly quasihomogeneous, singularities. Non-w.q.h. maps have a decomposition f = f 0 + h with f 0 q.h. and h of positive degree (relative to the weights of f 0 ). The normal space N A · f 0 := M n · θ f0 /LA · f 0 decomposes into a part of non-positive filtration and a part of positive filtration, denoted by (N A · f 0 ) + . Using the fact that LA Ωp · f 0 = LA · f 0 and Mather's lemma we obtain the following formal pre-normal form for an element of an A Ωp -orbit inside A · f :
f ′ = f 0 + hi∈B(f0)+ a i h i ,
where B(f 0 ) + denotes a base for (N A · f 0 ) + as a K-vector space. Notice that for semi-quasihomogeneous maps f the above sum is finite (because f 0 is A-finite), otherwise it is infinite.
Preliminary empirical examples indicate that in the s.q.h. case (where f 0 is Afinite) the above pre-normal for f ′ is in fact a (formal) normal form for A Ωp . In this case the coefficients a i are independent moduli for A Ωp (some a i might also be moduli for A). If this observation holds in general for s.q.h. maps in dimensions (n, p), n ≥ p − 1, (and Conjecture I in [10] is true) then such maps f satisfy the formula cod(A Ωp,e , f ) = µ ∆ (f ) as pointed out in the introduction (here µ ∆ denotes the discriminant Milnor number (for n ≥ p) or the image Milnor number (for p = n + 1)). Also notice that for n ≥ p we have cod(A Ωp,e , f ) ≤ µ ∆ (f ), independent of the correctness of the above conjectures.
Let us consider some examples in dimensions (n, 2), n ≥ 2.
Example 5.5. The A-simple non-w.q.h. germs in dimensions (n, 2) have the following formal normal forms for A Ωp (the normal forms ( * ) are not s.q.h. and Q denotes a sum of squares in additional variables): (x, xy + y 5 + ay 7 + Q); (x, xy 2 + y 5 + ay 6 + by 9 + Q); (x, x 2 y + y 4 + ay 5 + Q); ( * ) (x, xy 2 + y 4 + k≥2 a k y 2k+1 + Q); ( * ) (x 2 + ay 2l+1 , y 2 + x 2m+1 ), l ≥ m ≥ 1. The first three normal forms f are s.q.h. and their A Ωp,e -codimensions are equal to the A e -codimensions of their initial parts f 0 , and these are given by 3, 5 and 4, respectively. And from [13] we have the formula µ ∆ (f ) = µ(Σ f ) + d(f ) (relating the discriminant Milnor to the Milnor number of the critical set and the double-fold number), which gives for the three normal forms 3 = 0 + 3, 5 = 1 + 4 and 4 = 2 + 2, respectively.
The two series of non-s.q.h. maps f (marked by ( * )) are GTQ in the sense of [7] and the Milnor numbers of their discriminant curves ∆ f (not to be confused with the discriminant Milnor numbers of f ) are 2k+7 and 2(l+m)+3, respectively. These Milnor numbers are upper bounds for the A Ωp -moduli space of f (by Remark 4.18). Formal calculations (at the infinitesimal level using Mather's lemma) actually show that dim M(A Ωp , (x 2 + y 2l+1 , y 2 + x 2m+1 )) = 1, modulo M ∞ n θ f , and that we can take the above (formal) normal form for A Ωp -equivalence with the parameter a as the modulus. For the other non-s.q.h. map we only know that a 2 is a modulus and that we can take a 3 = a 4 = 0 (provided a 2 = 0), for a k , k > 4, the corresponding calculations of LA Ωp · f + M k+1 n θ f seem very tedious. Finally, a brief remark on our computation of µ(∆ f ) for the above two series. We use the formulas 2δ = µ + r − 1 (relating the δ-invariant, the number of branches r and µ of a planar curve-germ) and δ(∆ f ) = c(f )+d(f )+δ(Σ f ) (where c(f ) and d(f ) are the numbers of cusps and double folds, respectively, in a stable perturbation f t of f , hence δ(∆ ft ) = c(f ) + d(f )). For f = (x, xy 2 + y 4 + y 2k+1 ) we obtain δ(∆ f ) = 3 + k + 1 (see Table 1 in [44]), hence µ(∆ f ) = 2k + 7 (notice that the discriminant has r = 2 branches). This contradicts the claim in part (c) of example 1 in [7] that ∆ f has an E 6k+1 singularity.
Example 5.6. The A-unimodal germs in dimensions (n, 2) lie in the closure of the orbits of one of the following A-unimodal s.q.h. germs (see [45], and Q is again a sum of squares in additional variables): (x, y 4 + x 3 y + ax 2 y 2 + x 3 y 2 + Q), a = −3/2 (x, xy + y 6 + y 8 + ay 9 + Q) (x, xy + y 3 + ay 2 z + z 3 + z 5 + Q). For A Ωp -equivalence the corresponding normal forms f are: (x, y 4 + x 3 y + ax 2 y 2 + bx 3 y 2 + Q) (x, xy + y 6 + ay 8 + by 9 + cy 14 + Q) (x, xy + y 3 + ay 2 z + z 3 + bz 5 + Q). All A-unimodal germs therefore have A Ωp -modality at least two. Also, for the above f = f 0 + h we again have cod(A Ωp,e , f ) = cod(A e , f 0 ) = µ ∆ (f ). . . = deg f s j = δ j , j = 1, . . . , p. Also, if the above weights w i j are positive integers then we say that f is q.h. as a multigerm.
Weakly quasihomogeneous multigerms. Before leaving the subject of
Using Mather's [39] characterization of A-stability of multigerms in terms of multitransversality to K-orbits of multigerms, it is not hard to see that all A-stable multigerms are q.h. and hence A Ωp -w.q.h., which implies that the classifications of A-stable and A Ωp -stable orbits (over C) also agree for multigerms. 6. The foliation of K-orbits by K Ωn -and K Ωp -orbits
In this section we consider the volume-preserving versions of the classification of ICIS or, in other words, of K-finite maps f : C n → C p , n ≥ p. Recall that all K-simple f and all f whose differential has non-zero rank are w.q.h. for both K Ωn and K Ωp . Hence we will consider K-unimodal germs f of rank 0 and concentrate on the more interesting group K Ωn (the condition w.q.h. for K Ωp is weaker than that for K Ωn , hence dim M(K Ωn , f ) = 0 implies dim M(K Ωp , f ) = 0). The relevant K-classifications are therefore those in dimensions (n, p) = (3, 2) and (4, 2) (see [51]) and (2, 2) (see [14]) and (3, 3) (see [15]). Recall that the K Ωn -classification of hypersurfaces f −1 (0) has been settled by the result of Varchenko [48], which gives dim M(K Ωn , f ) = µ(f ) − τ (f ).
Looking at the lists in [51,14,15] we see (using our results) that a K-unimodal map-germ f is w.q.h. for K Ωn if and only if it is quasihomogeneous. We can therefore state:
(1) A K-unimodal map-germ f of rank 0 is K Ωn -unimodal if and only if it is q.h. and does not lie in the closure of a non-q.h. K-orbit. [51,14,15]. These results give upper bounds for dim M(K Ωn , f ), and for certain f some of these upper bound will coincide with the following lower bound (which is analogous to Lemma 5.1 in the A Ωp case).
Lemma 6.1. Consider a map-germ f u : (C n , 0) → (C p , 0) of the form f u = f + u · M ,
where f is a quasi-homogeneous germ, u ∈ C and M = X α · ∂/∂y j / ∈ LK · f = LK Ωn · f is a monomial vector of positive weighted degree (with respect to the weights of f ). For a set of weights for which f is weighted homogeneous, let (θ n ) 0 , (gl p (O n )) 0 and (θ f ) 0 denote the filtration-0 parts of the relevant modules. If the kernel of the linear map The germs f in the example (which are not s.q.h.) show that this inequality can be strict. Example 6.2. Consider the K-unimodal space-curves F W 1,i from [51], given by f = (g 1 , g 2 ) = (xy + z 3 , xz + y 2 z 2 + y 5+i ), i > 0.
γ f : (θ n ) 0 ⊕ (gl p (O n )) 0 → (θ f ) 0 , (a, B) → tf (a) − B · f,
Writing f = f 0 + (0, y 5+i ), where f 0 is q.h. for w = (7, 2, 3), δ = (9, 10) and where (0, y 5+i ) has filtration 2i > 0, and applying Lemma 6.1 we have dim M(K Ωn , f ) ≥ 1.
The component functions of f are both q.h. (for weights w 1 = (1, 2, 1) and w 2 = (7 + i, 2, 3 + i), respectively) and O 3 / ∇g 1 , ∇g 2 ∼ = C, hence dim M(K Ωn , f ) ≤ 1 by Corollary 4.12. Therefore dim M(K Ωn , f ) = 1 and the family f a = (xy + z 3 , xz + y 2 z 2 + ay 5+i ) parameterizes the K Ωn -orbits inside K · f .
A weaker upper bound for dim M(K Ωn , f ) follows from Remark 4.15 (which does not require that the component functions are w.q.h.): take a projection π onto the first target coordinate, then π • f = g 1 and dim M(K Ωn , f ) ≤ µ(g 1 ) = 2.
Finally, notice that dim M(K Ωn , f ) = 1 is smaller than the difference of µ(f ) = 16 + i and τ (f ) = 14 + i, where τ (f ) denotes the dimension of T 1 f = N K e · f . Recall that for hypersurfaces h −1 (0) we have dim M(K Ωn , h) = µ(h) − τ (h) (by [48]), in all our examples of higher codimensional ICIS g −1 (0) we have dim M(K Ωn , g) ≤ µ(g) − τ (g). Notice that for a "suspension" G = (z, g) of g (z an extra variable) µ(G) − τ (G) = µ(g) − τ (g), but dG(0) has positive rank hence dim M(K Ωn , G) = 0, the difference between both sides of the inequality above can therefore be arbitrarily large.
But the examples f = F W 1,i show that even in the rank 0 case the Varchenko formula does not hold for ICIS of codimension greater than 1.
Also notice that the series F W 1,i , i > 0, lies in the closure of the K-orbit of the s.q.h. germ g λ = (xy + z 3 , xz + y 2 z 2 + λy 5 + y 6 ), λ = 0, −1/4, where µ(g λ ) = 16 and τ (g λ ) = 15. Omitting the higher filtration y 6 -term we obtain type F W 1,0 in Wall's list [51], which is q.h. and µ(F W 1,0 ) = τ (F W 1,0 ) = 16. Notice that F W 1,1 (with µ(F 1,1 ) = 17 and τ (F 1,1 ) = 15) corresponds to the exceptional parameter λ = 0 in the modular stratum λ∈C\{0,−1/4} K · g λ (which seems to be missing in Wall's list) and does not lie in the closure of the orbit of F W 1,0 . Example 6.3. Consider the K-unimodal equidimensional maps of type h λ,q from [15], given by f = f λ := (xz + xy 2 + y 3 , yz, x 2 + y 3 + λz q ) = f 0 + (0, 0, y 3 + λz q ), q > 2.
The initial part f 0 is q.h. of type w = (1, 1, 2), δ = (3, 3, 2) and f il(0, 0, y 3 ) = 1, f il(0, 0, z q ) = 2(q − 1) > 1. Applying Lemma 6.1 to f ′ = f 0 + (0, 0, ay 3 ) we see that a is a K Ωn -modulus of f ′ and hence of f , hence dim M(K Ωn , f ) ≥ 1. The component functions of f = (g 1 , g 2 , g 3 ) are q.h. for distinct sets of weights, namely for w 1 = (1, 1, 2), any w 2 and w 3 = (3q, 2q, 6). Now O 3 / ∇g 1 , ∇g 2 , ∇g 3 ∼ = C, so that dim M(K Ωn , f ) = 1 (by Corollary 4.12 and the above lower bound -the upper bound also follows from µ(g 1 + g 2 + g 3 ) = 1, by Remark 4.15). And for each f λ = (xz + xy 2 + y 3 , yz, x 2 + y 3 + λz q ), λ ∈ C, the family f λ a = (xz + xy 2 + y 3 , yz, x 2 + ay 3 + λz q ) parameterizes the K Ωn -orbits inside K · f λ . Example 6.4. Finally, consider the K-unimodal equidimensional maps of type G k,l,m from [14], given by f = (g 1 , g 2 ) = (x 2 + y k , xy l + y m ) = f 0 + (0, y m ), where k = 2(m−l) and either k ≤ l, l+1 < m < l+k−1 (case (a)) or l < k < 2l−1, k < m < 2l (case (b)). As above we check that the coefficient of (0, y m ) is a K Ωnmodulus, hence dim M(K Ωn , f ) ≥ 1. And again the g i are q.h. for distinct sets of weights, but now O 2 / ∇g 1 , ∇g 2 ∼ = C{1, y, . . . , y r }, where r = k − 1 in case (a) and r = l in case (b). Hence 1 ≤ dim M(K Ωn , f ) ≤ r.
We can also obtain an upper bound using Remark 4.15: take the generic projection π onto the first target coordinate, then g 1 = π • f and dim M(K Ωn , f ) ≤ µ(g 1 ) = k − 1. This gives the same upper bound in case (a), but in case (b) we have l ≤ k − 1. In this final section we make some remarks on the remaining volume preserving subgroups G Ωq of A or K. First of all we remark that placing volume forms both in the source and the target of a map f leads to moduli even for invertible linear maps f : C n → C n (the modulus being the determinant of f ).
For function-germs the only relevant groups are those with a volume form to be preserved in the source, and what is known for these had been described in Section 1.
For map-germs R-equivalence is too fine already in the absence of a volume form, hence the remaining cases of interest (not considered in the previous sections) are the groups A Ωn , L Ωp and C Ωp for pairs of dimensions (n, p), p > 1, for which singular G-finite (G = A, L or C) map-germs f exist. And we can also discard those map-germs f that are trivially w.q.h. for the relevant group.
The following (in some sense "simplest" singular but non-w.q.h.) examples indicate that for the above three groups we immediately obtain moduli.
Example 7.1. For A Ωn the fold map f = (x, y 2 ) has infinite modality. We have LA n f = K{(x l y 2k , 0), (0, x l y 2k+1 ); l, k ≥ 0, l + k ≥ 1}, where the elements of LA n f are also known as lowerable vector fields (we write these source vector fields as vectors). It follows that dimension of C n /div(LA n f ), which is a lower bound for the number of A Ωn -moduli, is infinite for the fold f . Example 7.2. For L Ωp perhaps the first interesting example of a singular germ that fails to be trivially w.q.h. is the planar cusp f = (x 2 , x 3 ). We claim that in this case C p /div(LL p f ) ∼ = K{1, y 1 }, hence the L Ωp -modality of f is two (f is L-simple and the dimension of the L Ωp -moduli space is two).
Taking coordinates (y 1 , y 2 ) in the target, we see that the kernel of
LL −→ M n θ f , u → u • f
is (as a K-vector space) generated by elements u i rsl := y r 1 y s 2 (y 2l 2 − y 3l 1 )∂/∂y i , where i = 1, 2, r, s ≥ 0 and l ≥ 1. Set G i rsl := div(u i rsl ), then (2l + s + 1)G 1 r+1,s,l − (r + 1)G 2 r,s+1,l = cy 3l+r 1 y s 2 and (s + 1)G 1 r+1,s,l − (3l + r + 1)G 2 r,s+1,l = cy r 1 y 2l+s 2 where c = −6l 2 −2l(r+1)−3l(s+1) = 0. Finally, we have G 1 001 = −3y 2 1 , G 2 001 = 2y 2 , G 1 011 = −3y 2 1 y 2 and G 2 101 = 2y 1 y 2 , and the claim follows. Example 7.3. For C Ωp we first remark that C-finite germs f can only appear for n ≤ p. As an example for a singular germ f , which fails to be trivially w.q.h., we can consider the fold f = (x, y 2 ). A quick calculation yields C n /div(LC p f ) ∼ = K{1, y}. Hence f has two C Ωp -moduli, which can also be checked by comparing the normal spaces for C and C Ωp . Notice that N C · f is spanned by (0, y) and (y, 0), whereas N C Ωp · f is spanned by these two elements together with (x, 0) and (xy, 0).
Remark 2 . 5 .
25Definition 2.4 does not depend on the choice of a volume form Ω. If Ω ′ is another volume form then Ω = f Ω ′ for some non-vanishing function f . Then Ω 1 − Ω 0 = d(X⌋Ω) = d(f X⌋Ω ′ ) and f X ∈ M (M being a module).
equal to the number of G Ωq -moduli of f and also to the number of G q f -moduli of volume forms in (K q , 0). (For A Ωn the above statement holds in the formal category, in the smooth category the number of moduli is at least dim M(A Ωn , f ).)
Proposition 3. 8 .
8Let f be w.q.h. for one of the groups G Ωq = A Ωp , K Ωn or K Ωp (or "trivially w.q.h." for any group). Then M(G Ωq , f ) = 0.
D n -equivalent if and only if there is a diffeomorphism germ φ ∈ D n such that φ * I = J . The stabilizer of I is (D n ) I = {φ : φ * I = I}, and L(D n ) I = Derlog(I) = {Y ∈ θ n : Y I ⊂ I},
H
n ((Λ * (I), d)) ∼ = C n /div(Derlog(I)) ∼ = M(D Ωn , I).Furthermore, Theorem 2.8 implies the following Proposition 4.1. Two volume forms (defining the same orientation) are (D n ) Iisotopic if and only if they define the same cohomology class in H n ((Λ * (I), d)).
Lemma 4 . 10 .
410If h is w.q.h. then for any n-form ω there exists an (n − 1)-form β such that hω = d(hβ).
Theorem 4 . 13 .
413Let W be a variety-germ with an isolated singularity at 0. If the vanishing ideal of W is contained in I then dim H n (Λ * (I)) < ∞.
Corollary 4. 17 .
17Consider the image imf of a complex-analytic map-germ f : (C n , 0) → (C p , 0), and recall that LA p f = Lif t(f ). (a) If imf ⊂ W , for some variety-germ W with an isolated singularity at 0, then
A Ωpsimple if and only if it does not lie in the closure of the orbit of any non-quasihomogeneous germ. (The classifications of A-simple orbits can be found in the following papers: (n, p) = (1, 2) [6], (1, 3) [23], (1, p) (p ≥ 3) [2], (n, 2n) (n ≥ 2) [30], (2, 3) [41],
= d, i = 1, . . . , p and (setting h yi := ∂h/∂y i )−h yj ∂/∂y 1 + h y1 ∂/∂y j , h = l y α l l , α 1 , α j ≥ 1, l α l = d + 1, j = 2, . . . , pare clearly linearly independent and hence form a basis for Λ d . The tangent space to the L Ωp -orbit at f is then given by LL Ωp
A
Ωp -classification we make a final remark. All the results on A Ωp -equivalence can be easily extended to multigerms f = (f 1 , . . . , f s ) : (K n , S) → (K p , Ω p , 0) at an s-tuple S = {q 1 , . . . , q s } ⊂ K n of points in the source. Such an f is A Ωpw.q.h. if each component f i = (f i 1 , . . . , f i p ) is A Ωp -w.q.h. as a monogerm for (possibly different) sets of weights {w i 1 , . . . , w i n } but of the same weighted degrees deg f 1 j = .
( 2 )
2For a K-unimodal map-germ f of rank 0 such that f −1 (0) defines a ICIS of positive dimension and of codimension greater than one we have the following: (i) f is q.h. if and only if M(K Ωn , f ) = 0, and (ii) for f nonw.q.h. the dimension of M(K Ωn , f ) is one or two. (3) For map-germs f of positive rank we recall that the Kand K Ωn -classifications agree. We will now apply our finiteness results for M(K Ωn , f ) ∼ = H n (Λ * (f * M p )) to some examples of non-q.h. (and non-w.q.h.) map-germs f from the classifications in
of K-vector spaces is 1-dimensional then u is an K Ωn -modulus of f u . Hence the dimension of M(K Ωn , f u ) is positive.In our first example we consider positive dimensional complete intersections, defined by K-finite maps f , that are not hypersurfaces. In all our examples we have (for positive dimensional) ICIS that dim M(K Ωn , f ) ≤ µ(f ) − τ (f ), and for a s.q.h. germ f = f 0 + h this inequality holds in general. (For such f = f 0 + h we have dim M(K Ωn , f ) = dim M(K Ωn,e , f ) ≤ τ (f 0 )−τ (f ) and τ (f 0 ) = µ(f 0 ) = µ(f ).)
7 .
7The groups G Ωq = A Ωp , K Ωn , K Ωp : examples of G-stable maps f of positive and infinite G Ωq -modality
We can now deduce from Proposition 3.1 and Corollary 2.11 the following Theorem 3.4. For all volume preserving subgroups G Ωq of G, except for A Ωn , the dimension of
Acknowledgements. The authors wish to express their thanks to Goo Ishikawa, Stanislaw Janeczko, Andrew du Plessis, Maria Ruas and Michail Zhitomirskii for helpful conversations about this work.
. V I , J. Soviet Math. 47V. I. Arnol'd, Remarks on Poisson structures on a plane and on other powers of volume elements, J. Soviet Math. 47 (1989), 2509-2516.
Arnol'd, Simple singularities of curves. V I , Proc. Steklov Inst. Math. 226V. I. Arnol'd, Simple singularities of curves, Proc. Steklov Inst. Math. 226 (1999), 20-28.
Arnol'd, First steps of local symplectic algebra. V I , D. Fuchs birthday volume. ProvidenceAmer. Math. SocV. I. Arnol'd, First steps of local symplectic algebra, Adv. in Soviet Math., D. Fuchs birthday volume, Providence, Amer. Math. Soc., 1999.
De Rham cohomology of an analytic space. T Bloom, M Herrera, Invent. Math. 7T. Bloom and M. Herrera, De Rham cohomology of an analytic space, Invent. Math. 7 (1969), 275-296.
Die Monodromie der isolierten Singularitäten von Hyperflächen. E Brieskorn, Manuscripta Math. 2E. Brieskorn, Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math. 2 (1970), 103-161.
Simple singularities of maps (C, 0) → (C 2 , 0). J W Bruce, T Gaffney, J. London Math. Soc. 26J. W. Bruce and T. Gaffney, Simple singularities of maps (C, 0) → (C 2 , 0), J. London Math. Soc. 26 (1982), 464-474.
Discriminants and liftable vector fields. J W Bruce, A A Du Plessis, L C Wilson, J. Algebraic Geometry. 3J.W. Bruce, A.A. du Plessis and L.C. Wilson, Discriminants and liftable vector fields, J. Algebraic Geometry 3 (1994), 725-753.
Singular Lagrangian manifolds and semi-classical analysis, Duke Math. Y Colin De Verdière, J. 116Y. Colin de Verdière, Singular Lagrangian manifolds and semi-classical analysis, Duke Math. J. 116 (2003), 263-298.
. Y , Colin De Verdière, J Vey, Le Lemme De Morse Isochore, Topology. 18Y. Colin de Verdière and J. Vey, Le lemme de Morse isochore, Topology 18 (1979), 283-293.
Vanishing topology of codimension 1 multi-germs over R and C. T Cooper, D Mond, R Wik Atique, Compositio Math. 131T. Cooper, D. Mond and R. Wik Atique, Vanishing topology of codimension 1 multi-germs over R and C, Compositio Math. 131 (2002), 121-160.
The unfolding and determinacy results for subgroups of A and K. J Damon, Proc. of Symposia in Pure Mathematics 40. of Symposia in Pure Mathematics 40Providence, R.I.American Mathematical SocietyPart 1J. Damon, The unfolding and determinacy results for subgroups of A and K, in Proc. of Symposia in Pure Mathematics 40, Part 1 (American Mathematical Society, Providence, R.I., 1983), 233-254.
The unfolding and determinacy results for subgroups of A and K. J Damon, Memoirs Amer. Math. Soc. 50306J. Damon, The unfolding and determinacy results for subgroups of A and K, Memoirs Amer. Math. Soc. 50, no. 306 (1984).
A-codimension and the vanishing topology of discriminants. J Damon, D Mond, Inventiones Math. 106J. Damon and D. Mond, A-codimension and the vanishing topology of discriminants, Inven- tiones Math. 106 (1991), 217-242.
Contact unimodular germs from the plane to the plane. A Dimca, C G Gibson, Quart. J. Math. Oxford. 34A. Dimca and C.G. Gibson, Contact unimodular germs from the plane to the plane, Quart. J. Math. Oxford 34 (1983), 281-295.
Classification of equidimensional contact unimodular map germs. A Dimca, C G Gibson, Math. Scand. 56A. Dimca and C.G. Gibson, Classification of equidimensional contact unimodular map germs, Math. Scand. 56 (1985), 15-28.
Local symplectic algebra of quasi-homogeneous curves. W Domitrz, to appear in Fundamenta MathematicaeW. Domitrz, Local symplectic algebra of quasi-homogeneous curves, to appear in Fundamenta Mathematicae.
Relative Poincare lemma, contractibility, quasihomogeneity and vector fields tangent to a singular variety. W Domitrz, S Janeczko, M Zhitomirskii, Ill. J. Math. 48W. Domitrz, S. Janeczko, M. Zhitomirskii, Relative Poincare lemma, contractibility, quasi- homogeneity and vector fields tangent to a singular variety, Ill. J. Math. 48 (2004), 803-835.
Symplectic singularities of varieties: the method of algebraic restrictions. W Domitrz, S Janeczko, M Zhitomirskii, J. reine angew. Math. 618W. Domitrz, S. Janeczko and M. Zhitomirskii, Symplectic singularities of varieties: the method of algebraic restrictions, J. reine angew. Math. 618 (2008), 197-235.
Modèle local simultané d'une fonction et d'une forme de volume. J-P Francoise, Astèrisque S. M. F. J-P. Francoise, Modèle local simultané d'une fonction et d'une forme de volume, Astèrisque S. M. F. 59-60 (1978), 119-130.
Réduction simultanée d'un croisement normal et d'un volume. J-P Francoise, Bol. Soc. Bras. Mat. 13J-P. Francoise, Réduction simultanée d'un croisement normal et d'un volume, Bol. Soc. Bras. Mat. 13 (1982), 79-83.
A M Gabrielov, Bifurcations, Dynkin diagrams and the moduli number for isolated singularities. 8A.M. Gabrielov, Bifurcations, Dynkin diagrams and the moduli number for isolated singu- larities, Functional Anal. Appl. 8 (1974), 7-13.
An isochore versal deformation theorem. M D Garay, Topology. 43M.D. Garay, An isochore versal deformation theorem, Topology 43 (2004), 1081-1088.
Simple singularities of space curves. C G Gibson, C A Hobbs, Math. Proc. Camb. Phil. Soc. 113C.G. Gibson and C.A. Hobbs, Simple singularities of space curves, Math. Proc. Camb. Phil. Soc. 113 (1993), 297-310.
Classification des singularités isolées d'intersections complètes simples. M Giusti, C.R. Acad. Sci. Paris. Sér. A-B. 284M. Giusti, Classification des singularités isolées d'intersections complètes simples, C.R. Acad. Sci. Paris. Sér. A-B 284 (1977), 167-169.
Singularities of projections of complete intersections. V V Goryunov, J. Soviet Math. 27V.V. Goryunov, Singularities of projections of complete intersections, J. Soviet Math. 27 (1984), 2785-2811.
Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten. G.-M Greuel, Math. Ann. 214G.-M. Greuel, Der Gauss-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann. 214 (1975), 235-266.
. G.-M Greuel, Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann. 250G.-M. Greuel , Dualität in der lokalen Kohomologie isolierter Singularitäten, Math. Ann. 250 (1980), 157-173.
On the classification and geometry of corank 1 map-germs from three-to four-space. K Houston, N P Kirk, Singularity Theory. Cambridge University Press263K. Houston and N.P. Kirk, On the classification and geometry of corank 1 map-germs from three-to four-space. In Singularity Theory, 325-351, London Math. Soc. Lecture Note Series 263 (Cambridge University Press, 1999).
Symplectic bifurcations of plane curves and isotropic liftings. G Ishikawa, S Janeczko, Quart. J. Math. Oxford Ser. 2G. Ishikawa and S. Janeczko, Symplectic bifurcations of plane curves and isotropic liftings, Quart. J. Math. Oxford Ser. (2) 54 (2003), 73-102.
Real double points of deformations of A-simple map-germs from R n to R 2n. C Klotz, O Pop, J H Rieger, Math. Proc. Camb. Phil. Soc. 142C. Klotz, O. Pop and J.H. Rieger, Real double points of deformations of A-simple map-germs from R n to R 2n , Math. Proc. Camb. Phil. Soc. 142 (2007), 341-363.
Versal deformations of powers of volume forms. V P Kostov, S K Lando, Computational algebraic geometry. 109Birkhäuser BostonV. P. Kostov and S. K. Lando, Versal deformations of powers of volume forms, Computational algebraic geometry (Nice, 1992), 143-162, Progr. Math., 109, Birkhäuser Boston, Boston, MA, 1993.
Deformations of differential forms. S K Lando, Osob. Gladkikh Otobrazh. s Dop. Strukt. 209Trudy Mat. Inst. Steklov.. RussianS. K. Lando, Deformations of differential forms, Trudy Mat. Inst. Steklov. 209 (1995), Osob. Gladkikh Otobrazh. s Dop. Strukt., 167-199 (Russian).
Singularities of the differential forms of the complex degree and their deformations. S K Lando, Singularity theory. Trieste; River Edge, NJWorld Sci. PublishingS. K. Lando, Singularities of the differential forms of the complex degree and their deforma- tions, Singularity theory (Trieste, 1991), 289-305, World Sci. Publishing, River Edge, NJ, 1995.
Normal forms of powers of volume forms. S K Lando, Funktsional. Anal. i Prilozhen. 19RussianS. K. Lando, Normal forms of powers of volume forms, Funktsional. Anal. i Prilozhen 19 (1985), 78-79 (Russian).
Intégrales asymptotiques et monodromie. B Malgrange, Ann. Sci.École Norm. Sup. 20B. Malgrange, Intégrales asymptotiques et monodromie, Ann. Sci.École Norm. Sup. 20 (1974), 147-176.
On the geometry of simple germs of corank 1 from R 3 to R 3. W L Marar, F Tari, Math. Proc. Cambridge Philos. Soc. 119W.L. Marar and F. Tari, On the geometry of simple germs of corank 1 from R 3 to R 3 , Math. Proc. Cambridge Philos. Soc. 119 (1996), 469-481.
Singularities of smooth functions and maps. J Martinet, Lect. Note. 58Cambridge Univ. PressJ. Martinet, Singularities of smooth functions and maps, London Math. Soc., Lect. Note 58, Cambridge Univ. Press 1982.
IV: classification of stable germs by R-algebras. J N Mather, Stability Of C ∞ Mappings, Publ. Math. I.H.E.S. 37J.N. Mather, Stability of C ∞ mappings, IV: classification of stable germs by R-algebras, Publ. Math. I.H.E.S. 37 (1969), 223-248.
. J N Mather, Stability Of C ∞ Mappings, Adv. Math. 4J.N. Mather, Stability of C ∞ mappings, V: transversality, Adv. Math. 4 (1970), 301-336.
VI: the nice dimensions. J N Mather, Stability Of C ∞ Mappings, Proc. of the Liverpool Singularities Symposium I. of the Liverpool Singularities Symposium IBerlinSpringer Verlag192J.N. Mather, Stability of C ∞ mappings, VI: the nice dimensions, In Proc. of the Liverpool Singularities Symposium I, 207-253, Lecture Notes in Math. 192 (Springer Verlag, Berlin, 1970).
On the classification of germs of maps from R 2 to R 3. D Mond, Proc. London Math. Soc. 50D. Mond, On the classification of germs of maps from R 2 to R 3 , Proc. London Math. Soc. 50 (1985), 333-369.
On the volume elements on a manifold. J Moser, Trans. Amer. Math. Soc. 120J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 280-296.
Das Lemma von Poincaré für holomorphe Differentialformen auf komplexen Räumen. H.-J Reiffen, Math. Z. 101H.-J. Reiffen, Das Lemma von Poincaré für holomorphe Differentialformen auf komplexen Räumen, Math. Z. 101 (1967), 269-284.
Families of maps from the plane to the plane. J H Rieger, J. London Math. Soc. 36J.H. Rieger, Families of maps from the plane to the plane, J. London Math. Soc. 36 (1987), 351-369.
A-unimodal map-germs into the plane. J H Rieger, Hokkaido Math. J. 33J.H. Rieger, A-unimodal map-germs into the plane, Hokkaido Math. J. 33 (2004), 47-64.
Classification of A-simple germs from k n to k 2. J H Rieger, M A S Ruas, Compositio Math. 79J.H. Rieger and M.A.S. Ruas, Classification of A-simple germs from k n to k 2 , Compositio Math. 79 (1991), 99-108.
Preuve d'une conjecture de Brieskorn. M Sebastiani, Manuscripta Math. 2M. Sebastiani, Preuve d'une conjecture de Brieskorn, Manuscripta Math. 2 (1970), 301-308.
Local classification of volume forms in the presence of a hypersurface. A N Varchenko, Funktsional. Anal. i Prilozhen. 194RussianA. N. Varchenko, Local classification of volume forms in the presence of a hypersurface, Funktsional. Anal. i Prilozhen 19 (1985), no. 4, 23-31 (Russian).
. J Vey, Sur Le Lemme De, Morse , Inventiones Math. 40J. Vey, Sur le lemme de Morse, Inventiones Math. 40 (1977), 1-10.
Finite determinacy of smooth map-germs. C T C Wall, Bull. London Math. Soc. 13C.T.C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc. 13 (1981), 481-539.
Classification of unimodal isolated singularities of complete intersections. C T C Wall, Proc. Symposia in Pure Math. 40C.T.C. Wall, Classification of unimodal isolated singularities of complete intersections, Proc. Symposia in Pure Math. Vol. 40 (1983), Part 2, 625-640.
Warszawa, Poland E-mail address: [email protected] Institut für Mathematik. Universität Halle, D-06099 Halle (Saale), Germany E-mail address1Faculty of Mathematics and Information Science, Warsaw University of TechnologyFaculty of Mathematics and Information Science, Warsaw University of Technol- ogy, Pl. Politechniki 1, 00-661 Warszawa, Poland E-mail address: [email protected] Institut für Mathematik, Universität Halle, D-06099 Halle (Saale), Germany E-mail address: [email protected]
| [] |
[
"On non-resistive limit of 1D MHD equations with no vacuum at infinity",
"On non-resistive limit of 1D MHD equations with no vacuum at infinity"
] | [
"Zilai Li \nSchool of Mathematics and Information Science\nHenan Polytechnic University\n454000JiaozuoP.R. China\n",
"Huaqiao Wang [email protected] \nCollege of Mathematics and Statistics\nChongqing University\n401331ChongqingP.R. China\n",
"Yulin Ye [email protected] \nSchool of Mathematics and Statistics\nHenan University\n475004KaifengP.R. China\n"
] | [
"School of Mathematics and Information Science\nHenan Polytechnic University\n454000JiaozuoP.R. China",
"College of Mathematics and Statistics\nChongqing University\n401331ChongqingP.R. China",
"School of Mathematics and Statistics\nHenan University\n475004KaifengP.R. China"
] | [] | In this paper, we consider the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for both the compressible resistive MHD equations and non-resistive MHD equations are also established, respectively.On non-resistive limit of 1D MHD equations with no vacuum at infinity classical solution of 3-D MHD equations with small energy but possibly large oscillations. Later, the result was improved by Hong-Hou-Peng-Zhu [8] just provided ((γ−1) 1 9 +ν − 1 4 )E 0 is suitably small. When the resistivity is zero, then the magnetic equation is reduced from the heat-type equation to the hyperbolic-type equation, the problem becomes more challenging, hence the results are few. Kawashima [12] obtained the classical solutions to 3-D MHD equations when the initial data are of small perturbations in H 3 -norm and away from vacuum. Xu-Zhang [22] proved a blow-up criterion of strong solutions for 3-D isentropic MHD equations with vacuum. Fan-Hu [4] established the global strong solutions to the initial boundary value problem of 1-D heat-conducting MHD equations. With more general heat-conductivity, Zhang-Zhao [30] established the global strong solutions and also obtained the non-resistivity limits of the solutions in L 2 -norm. Jiang-Zhang [10] obtained the non-resistive limit of the strong solution and the "magnetic boundary layer" estimates to the initial boundary value problem of 1-D isentropic MHD equations as the resistivity ν → 0. Yu [27] obtained the global existence of strong solutions to the initial boundary value problem of 1-D isentropic MHD equations. For the Cauchy problem with large initial data and vacuum, Li-Wang-Ye [16] established the global well-posedness of strong solutions to the 1D isentropic MHD equations with vacuum at infinity, that isρ = 0.However, for the Cauchy problem with no vacuum at infinity, the global well-posedness of strong solutions and the non-resistive limits when the resistivity coefficient ν → 0 are still unknown. The goal of this paper is trying to answer these problems. Now we give some comments on the analysis of this paper. The non-resistive limit of global strong solutions to 1D MHD equations (1.1)-(1.2) can be obtained by global uniform a priori estimates which are independent of resistivity ν. Thus, to obtain the a priori ν (resistivity coefficient)-independent estimates, some of the main new difficulties will be encountered due to the absence of resistivity and the initial density and initial magnetic which approach non-zero constants at infinity.It turns out that the key issue in this paper is to derive upper bound for the density, magnetic field and the time-dependent higher norm estimates which are independent of resistivity ν. This is achieved by modifying upper bound estimate for the density developed in [24] and [26] in the theory of Cauchy problem with vacuum and initial magnetic approach zero at infinity. However, in comparison with the Cauchy problem with vacuum at infinity in [24] and [16], some new difficulties will be encountered. The first difficulty lies in no integrability for the density itself just from the elementary energy estimate (see Lemma 3.1), which is required when deriving the upper bound of the density. To overcome this difficulty, we use the technique of mathematical frequency decomposition to divide the momentum ξ into two parts: | 10.1515/anona-2021-0209 | [
"https://arxiv.org/pdf/2102.02962v1.pdf"
] | 231,839,776 | 2102.02962 | 994be577e67e26ec0c4b3c812b16635783879f66 |
On non-resistive limit of 1D MHD equations with no vacuum at infinity
5 Feb 2021 February 8, 2021
Zilai Li
School of Mathematics and Information Science
Henan Polytechnic University
454000JiaozuoP.R. China
Huaqiao Wang [email protected]
College of Mathematics and Statistics
Chongqing University
401331ChongqingP.R. China
Yulin Ye [email protected]
School of Mathematics and Statistics
Henan University
475004KaifengP.R. China
On non-resistive limit of 1D MHD equations with no vacuum at infinity
5 Feb 2021 February 8, 20211D compressible MHD equationsCauchy problemglobal strong solu- tionsnon-resistive limit
In this paper, we consider the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for both the compressible resistive MHD equations and non-resistive MHD equations are also established, respectively.On non-resistive limit of 1D MHD equations with no vacuum at infinity classical solution of 3-D MHD equations with small energy but possibly large oscillations. Later, the result was improved by Hong-Hou-Peng-Zhu [8] just provided ((γ−1) 1 9 +ν − 1 4 )E 0 is suitably small. When the resistivity is zero, then the magnetic equation is reduced from the heat-type equation to the hyperbolic-type equation, the problem becomes more challenging, hence the results are few. Kawashima [12] obtained the classical solutions to 3-D MHD equations when the initial data are of small perturbations in H 3 -norm and away from vacuum. Xu-Zhang [22] proved a blow-up criterion of strong solutions for 3-D isentropic MHD equations with vacuum. Fan-Hu [4] established the global strong solutions to the initial boundary value problem of 1-D heat-conducting MHD equations. With more general heat-conductivity, Zhang-Zhao [30] established the global strong solutions and also obtained the non-resistivity limits of the solutions in L 2 -norm. Jiang-Zhang [10] obtained the non-resistive limit of the strong solution and the "magnetic boundary layer" estimates to the initial boundary value problem of 1-D isentropic MHD equations as the resistivity ν → 0. Yu [27] obtained the global existence of strong solutions to the initial boundary value problem of 1-D isentropic MHD equations. For the Cauchy problem with large initial data and vacuum, Li-Wang-Ye [16] established the global well-posedness of strong solutions to the 1D isentropic MHD equations with vacuum at infinity, that isρ = 0.However, for the Cauchy problem with no vacuum at infinity, the global well-posedness of strong solutions and the non-resistive limits when the resistivity coefficient ν → 0 are still unknown. The goal of this paper is trying to answer these problems. Now we give some comments on the analysis of this paper. The non-resistive limit of global strong solutions to 1D MHD equations (1.1)-(1.2) can be obtained by global uniform a priori estimates which are independent of resistivity ν. Thus, to obtain the a priori ν (resistivity coefficient)-independent estimates, some of the main new difficulties will be encountered due to the absence of resistivity and the initial density and initial magnetic which approach non-zero constants at infinity.It turns out that the key issue in this paper is to derive upper bound for the density, magnetic field and the time-dependent higher norm estimates which are independent of resistivity ν. This is achieved by modifying upper bound estimate for the density developed in [24] and [26] in the theory of Cauchy problem with vacuum and initial magnetic approach zero at infinity. However, in comparison with the Cauchy problem with vacuum at infinity in [24] and [16], some new difficulties will be encountered. The first difficulty lies in no integrability for the density itself just from the elementary energy estimate (see Lemma 3.1), which is required when deriving the upper bound of the density. To overcome this difficulty, we use the technique of mathematical frequency decomposition to divide the momentum ξ into two parts:
Introduction and Main Results
Compressible magnetohydrodynamics (MHD) is used to describe the macroscopic behavior of the electrically conducting fluid in a magnetic field. The application of MHD has a very wide of physical objects from liquid metals to cosmic plasmas. The system of the resistive MHD equations has the form: ρ t + (ρu) x = 0, (ρu) t + (ρu 2 + P (ρ) + 1 2 b 2 ) x = (µu x ) x , b t + (ub) x = νb xx , (1.1) in R × [0, ∞). Here, ρ, u, P (ρ) and b denote the density, velocity, pressure and magnetic field, respectively. µ > 0 is the viscosity, the constant ν > 0 is the resistivity coefficient acting as the magnetic diffusion coefficient of the magnetic field. In this paper, we consider the isentropic compressible MHD equations in which the equation of the state has the form P (ρ) = Rρ γ , γ > 1.
For simplicity, we set R = 1. We focus on the initial condition:
(ρ, u, b)| t=0 = (ρ 0 (x), u 0 (x), b 0 (x)) → (ρ, 0,b) as |x| → +∞, (1.2) whereρ andb are both non-zero constants. Note that we can always normalizeρ such thatρ ≥ 1. However, it is well known that the resistivity coefficient ν is inversely proportional to the electrical conductivity, therefore it is more reasonable to ignore the magnetic diffusion which means ν = 0, when the conducting fluid considered is of highly conductivity, for example the ideal conductors. So instead of equations (1.1), when there is no resistivity, the system reduces to the so called compressible, isentropic, viscous and non-resistive MHD equations: (1.3) in R × [0, +∞), with the following initial condition:
ρ t + (ρũ) x = 0, (ρũ) t + (ρũ 2 + P (ρ) + 1 2b 2 ) x = (µũ x ) x , b t + (ũb) x = 0,
(ρ,ũ,b)| t=0 = (ρ 0 ,ũ 0 ,b 0 ) → (ρ, 0,b) as |x| → +∞. (1.4) Because of the tight interaction between the dynamic motion and the magnetic field, the presence of strong nonlinearities, rich phenomena and mathematical challenges, many physicists and mathematics are attracted to study in this field. Before stating our main theorems, we briefly recall some previous known results on compressible MHD equations. Firstly, we begin with the MHD equations with magnetic diffusion. For one-dimensional case, Vol'pert-Hudjaev [21] proved the local existence and uniqueness of strong solutions to the Cauchy problem and Kawashima-Okada [13] obtained the global smooth solutions with small initial data. For large initial data and the density containing vacuum, Ye-Li [26] proved the global existence of strong solutions to the 1D Cauchy problem with vacuum at infinity. When considering the full MHD equations and the heat conductivity depends on the temperature θ, Chen-Wang [3] studied the free boundary value problem and established the existence, uniqueness and Lipschitz dependence of strong solutions. Recently, Fan-Huang-Li [6] obtained the global strong solutions to the initial boundary value problem to the planner MHD equations with temperature-dependent heat conductivity. Later, with the effect of self-gravitation as well as the influence of radiation on the dynamics at high temperature regimes taken into account, Zhang-Xie [28] obtained the global strong solutions to the initial boundary value problem for the nonlinear planner MHD equations. For multi-dimensional MHD equations, Lv-Shi-Xu [18] considered the 2-D isentropic MHD equations and proved the global existence of classical solutions provided that the initial energy is small, where the decay rates of the solutions were also obtained. Vol'pert-Hudjaev [21] and Fan-Yu [5] obtained the local classical solution to the 3-D compressible MHD equations with the initial density is strictly positive or could contain vacuum, respectively. Hu-Wang [9] derived the global weak solutions to the 3-D compressible MHD equations with large initial data. Recently, Li-Xu-Zhang [14] established the global existence of
ξ =ˆρudx =ˆ √ ρ − √ρ √ ρudx + √ρˆ√ ρudx = ξ 1 + ξ 2 .
It is crucial to obtain the upper bound of ξ 1 and ξ 2 . For getting ξ 1 L ∞ , we use the technique of mathematical frequency decomposition to get the estimate of √ ρ − √ρ L 2 by the elementary energy estimates, and then using Hölder's inequality, we can obtain the upper bound of ξ 1 (see (3.23)). For obtaining ξ 2 L ∞ , due to the Sobolev embedding theory, we need some Lp integrability of ξ 2 . However, we can not obtain it just from ξ 2x L 2 directly, because the Poincáre type inequality is no longer valid in the whole space R. To overcome this difficulty, we use the Caffarelli-Kohn-Nirenberg weighted inequality and the G-N inequality to obtain the upper bound of ξ 2 (see (3.24)). It is worth noting that the non-resistive MHD equation (1.3) looks similar to the compressible model for gas and liquid two-phase fluids. Hence, in some previous results, it is technically assumed that ρ,b ≥ 0 and 0 ≤bρ < ∞ in (1.3), which implies the magnetic fieldb is bounded provided the densityρ is bounded. However, this is not physical and realistic in magnetohydrodynamics. Moreover, compared with that for the Cauchy problem with vacuum at infinity in [16], since the magnetic fieldb →b, as |x| → +∞, the method of getting the upper bound of magnetic field b in [16] can not be used here anymore. So another difficulty is how to get the uniform (independent of ν) upper bound of the magnetic field b without the assumption similar as that in two-phase fluids. To overcome this difficulty, we will make full use of the structure of the momentum equation and effective viscous flux (see Lemma 3.4 and Lemma 3.5).
Notations: We denote the material derivative of u and effective viscous flux bẏ
u u t + uu x and F µu x − P (ρ) − P (ρ) + b 2 −b 2 2 ,
and define potential energy by
Φ(ρ) = ρˆρ ρ P (s) − P (ρ) s 2 ds = 1 γ − 1 ρ γ −ρ γ − γρ γ−1 (ρ −ρ) .
Sometimes we write´R f (x)dx as´f (x) for simplicity.
The main results of this paper can be stated as: As a by-product, the first result is the global existence of strong solutions to 1D non-resistive MHD equations (1.3)-(1.4) when the the initial density and initial magnetic approach non-zero constants at infinity.
Theorem 1.1. Suppose that the initial data (ρ 0 ,ũ 0 ,b 0 )(x) satisfies ρ 0 −ρ ∈ H 1 (R),b 0 −b ∈ H 1 (R),ũ 0 ∈ H 2 (R), 1 2ρ 0ũ 2 0 + Φ(ρ 0 ) + (b 0 −b) 2 2 |x| α ∈ L 1 (R) (1.5)
for 1 < α ≤ 2, and the compatibility condition
µũ 0x − P (ρ 0 ) − 1 2b 2 0 x = ρ 0g (x), x ∈ R (1.6)
with someg satisfyingg ∈ L 2 (R). Then for any T > 0, there exists a unique global strong solution (ρ,ũ,b) to the Cauchy problem (1.3) and (1.4) such that
0 ≤ρ ≤ C, (ρ −ρ,b −b) ∈ L ∞ (0, T ; H 1 (R)),ρ t ∈ L ∞ (0, T ; L 2 (R)), b t ∈ L 2 (0, T ; L 2 (R)), 1 2ρũ 2 + Φ(ρ) + (b −b) 2 2 |x| α ∈ L ∞ (0, T ; L 1 (R)),
u ∈ L ∞ (0, T ; H 2 (R)), ρũ t ∈ L ∞ (0, T ; L 2 (R)),ũ t ∈ L 2 (0, T ; H 1 (R)).
The second result is the global existence and non-resistive limits of strong solutions to 1D MHD equations (1.1)-(1.2) when the the initial density and initial magnetic approach non-zero constants at infinity.
Theorem 1.2. Suppose that the initial data (ρ 0 , u 0 , b 0 )(x) satisfies ρ 0 −ρ ∈ H 1 (R), b 0 −b ∈ H 1 (R), u 0 ∈ H 2 (R), 1 2 ρ 0 u 2 0 + Φ(ρ 0 ) + (b 0 −b) 2 2 |x| α ∈ L 1 (R) (1.7)
for 1 < α ≤ 2, and the compatibility condition
µu 0x − P (ρ 0 ) − 1 2 b 2 0 x = √ ρ 0 g(x), x ∈ R (1.8)
with some g satisfying g ∈ L 2 (R). Then for each fixed ν > 0, there exist a positive constant C and a unique global strong solution (ρ, u, b) to the Cauchy problem (1.
1)-(1.2) such that 0 ≤ ρ(x, t) ≤ C, ∀(x, t) ∈ R × [0, T ], (1.9) sup 0≤t≤T 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 (1 + |x| α ) L 1 (t) +ˆT 0 µ u x (1 + |x| α 2 ) 2 L 2 + ν b x (1 + |x| α 2 ) 2 L 2 dt ≤ C, (1.10) and sup 0≤t≤T ( u L 2 + u xx L 2 + u x L 2 + b x L 2 + ρ x L 2 + √ ρu L 2 ) (t) +ˆT 0 u xt 2 L 2 + ν b xx 2 L 2 dt ≤ C.
(1.11)
Moreover, as ν → 0, we have
(ρ, u, b) → (ρ,ũ,b) strongly in L ∞ (0, T ; L 2 ), νb x → 0, u →ũ, u x →ũ x strongly in L 2 (0, T ; L 2 ), (1.12) and sup 0≤t≤T ρ −ρ 2 L 2 + u −ũ 2 L 2 + b −b 2 L 2 +ˆT 0 µ (u −ũ) x 2 L 2 dt ≤ Cν,(1.13)
where C is a positive constant independent of ν.
Remark 1.1. In Theorems 1.1 and 1.2, we do not need the artificial assumption similarly as that in two-phase fluids. Moreover, if ignoring the magnetic field, then MHD system reduces to the compressible Navier-Stokes equations. So, Theorems 1.1 and 1.2 can be seen as an extension of that in [24]. The rest of the paper is organized as follows. In Section 2, we recall some preliminary lemmas which will be used later. Section 3 is devoted to establishing global ν-independent estimates for (1.1) and (1.2), which will be used to justify the non-resistive limit. Sections 4 and 5 are devoted to proving Theorem 1.1 and Theorem 1.2, respectively.
Preliminaries
In this section, we will recall some known facts and elementary inequalities that will be used frequently later.
The following well-known inequality will be used frequently later.
Lemma 2.1 (Gagliardo-Nirenberg inequality [7,19]). For any f ∈ W 1,m (R)∩L r (R), there exists some generic constant C > 0 which may depend on q, r such that
f L q ≤ C f 1−θ L r ∇f θ L m , (2.1) where θ = ( 1 r − 1 q )( 1 r − 1 m + 1) −1 , if m = 1, then q ∈ [r, ∞), if m > 1, then q ∈ [r, ∞].
The following Caffarelli-Kohn-Nirenberg weighted inequality is the key to deal with the Cauchy problem in this paper.
Lemma 2.2 (Caffarelli-Kohn-Nirenberg weighted inequality [1]). (1) ∀h ∈ C ∞ 0 (R), it holds that |x| κ h r ≤ C |x| α |∂ x h| θ p |x| β h 1−θ q (2.2) where 1 ≤ p, q < ∞, 0 < r < ∞, 0 ≤ θ ≤ 1, 1 p + α > 0, 1 q + β > 0, 1 r + κ > 0 and satisfy 1 r + κ = θ 1 p + α − 1 + (1 − θ) 1 q + β ,(2.
3)
and
κ = θσ + (1 − θ)β with 0 ≤ α − σ if θ > 0 and 0 ≤ α − σ ≤ 1 if θ > 0 and 1 p + α − 1 = 1 r + κ.
Proof. The proof of (1) can be found in [1]. Here, we omit the details.
By direct calculation, the potential energy Φ(ρ,ρ) has the following properties:
Lemma 2.3.
Observing the function of the potential energy Φ(ρ,ρ), we will easily find the following properties for positive constants c 1 , c 2 , C 1 , C 2 :
(1) if 0 ≤ ρ ≤ 2ρ, c 1 (ρ −ρ) 2 ≤ Φ(ρ,ρ) ≤ c 2 (ρ −ρ) 2 ; (2) if ρ > 2ρ, ρ γ −ρ γ ≤ C 1 (ρ −ρ) γ ≤ C 2 Φ(ρ,ρ).
3 Global ν-independent estimates for (1.1) and (1.
2)
The main purpose of this section is to derive the global ν-independent a priori estimates of the solutions (ρ, u, b) to the system (1.1) and (1.2), which is used to justify the nonresistive limit. Hence, in this section, we assume (ρ, u, b) is a smooth solution of the system (1.1) and (1.2) on R × [0, T ] with 0 < T < ∞. For the sake of simplicity, we denote by C the generic positive constant, which may depend on γ, µ, T , but is independent of ν.
First of all, we can prove the following elementary energy estimates.
Lemma 3.1. Let (ρ, u, b) be a smooth solution of (1.1) and (1.2). Then for any T > 0, it holds that
sup 0≤t≤T 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 L 1 (R) +ˆT 0 µ u x 2 L 2 + ν b x 2 L 2 dt ≤ C. (3.1)
Proof. By the defination of the potential energy Φ, and using (1.1) 1 , we deduce
Φ t + (uΦ) x + (ρ γ −ρ γ )u x = 0. (3.2)
Adding the equation (1.1) 2 multiplied by u, (1.1) 3 multiplied by b, into (3.2), then integrating the resulting equation over R × [0, T ] with respect to the variables x and t, we haveˆ
1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 dx +ˆT 0 µ u x 2 L 2 + ν b x 2 L 2 dt ≤ˆ 1 2 ρ 0 u 2 + Φ(ρ 0 ) + (b 0 −b) 2 2 dx ≤ ρ 0 L ∞ u 0 2 L 2 + C( ρ 0 −ρ L 2 + b 0 −b 2 L 2 ) ≤ C( ρ 0 −ρ H 1 +ρ)( u 0 2 L 2 + 1) + C b 0 −b 2 L 2 ≤ C.
Then the proof of Lemma 3.1 is completed.
To obtain the upper bound of the density ρ, we need the following weighted energy estimates.
Lemma 3.2. Let (ρ, u, b) be a smooth solution of (1.1) and (1.2). Then for any T > 0 and some index 1 < α ≤ 2, it holds that
sup 0≤t≤T 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 |x| α L 1 (R) +ˆT 0 µ u x |x| α 2 2 L 2 + ν b x |x| α 2 2 L 2 dt ≤ C. (3.3)
Proof. Multiplying the equation (1.1) 2 by u|x| α and integrating the resulting equation over R with respect to x, we have
1 2 d dtˆρ u 2 |x| α + µˆu 2 x |x| α = 1 2ˆρ u 3 α|x| α−2 x − µˆα|x| α−2 xuu x −ˆ P (ρ) + b 2 2 x u|x| α . (3.4)
It follows from the integration by parts that
−ˆ P (ρ) + b 2 2 x u|x| α = −ˆ((P (ρ) − P (ρ)) x + bb x ) u|x| α = −ˆ (P (ρ) − P (ρ)) x + (b −b)(b −b) x +b(b −b) x u|x| α =ˆ P (ρ) − P (ρ) + (b −b) 2 2 +b(b −b) u x |x| α + uα|x| α−2 x . (3.5)
Then, combining (3.4) and (3.5) gives
1 2 d dtˆρ u 2 |x| α + µˆu 2 x |x| α = 1 2ˆρ u 3 α|x| α−2 x − µˆα|x| α−2 xuu x +ˆ P (ρ) − P (ρ) + (b −b) 2 2 +b(b −b) u x |x| α + uα|x| α−2 x . (3.6)
To deal with the last term on the right-hand side of (3.6), first, multiplying (3.2) by |x| α and integrating over R with respect to x, yields
d dtˆΦ (ρ)|x| α −ˆuΦ(ρ)α|x| α−2 x +ˆ(ρ γ −ρ γ )|x| α u x = 0, (3.7)
and then multiplying the equation (1.1) 3 by (b−b)|x| α and integrating over R with respect to x, we have
1 2 d dtˆ( b −b) 2 |x| α + 1 2ˆ (b −b) 2 x u|x| α +ˆ(b −b) 2 u x |x| α +bˆ(b −b)u x |x| α = νˆb xx (b −b)|x| α ,
which together with the integration by parts implies that
1 2 d dtˆ( b −b) 2 |x| α + νˆb 2 x |x| α = −ˆ( b −b) 2 2 (u x |x| α − uα|x| α−2 x) −bˆ(b −b)u x |x| α − νˆb x (b −b)α|x| α−2 x.1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 |x| α dx + µˆu 2 x |x| α + νˆb 2 x |x| α =ˆ 1 2 ρu 2 + Φ(ρ) + (b −b) 2 uα|x| α−2 x +ˆ(ρ γ −ρ γ )uα|x| α−2 x − µˆα|x| α−2 xuu x +bˆ(b −b)uα|x| α−2 x − νˆ(b −b)b x α|x| α−2 x =: I 1 + I 2 + I 3 + I 4 + I 5 .
(3.9)
Next, we estimate the terms I 1 -I 5 as follows:
I 1 ≤ˆ 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 |u||x| α−1 ≤ˆ 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 |x| α α−1 α 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 1 α |u| ≤ C u L ∞ 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 |x| α 1− 1 α L 1 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 1 α L 1 ≤ C(1 + u x L 2 ) 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 |x| α 1− 1 α L 1 ≤ C(1 + u x L 2 ) 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 |x| α L 1 + 1 ,(3.10)
where α > 1 and we have used Hölder's inequality, Lemma 3.1 and the following facts:
ρˆu 2 =ˆ(ρ − ρ + ρ)u 2 ≤ˆ (ρ − ρ)| {0≤ρ≤2ρ} + (ρ − ρ)| {ρ>2ρ} u 2 +ˆρu 2 ≤ C (ρ −ρ)| {0≤ρ≤2ρ} L 2 u 2 L 4 + (ρ −ρ)| {ρ>2ρ} L γ u 2 L 2γ γ−1 + 1 ≤ C Φ(ρ) 1 2 L 1 u 3 4 L 2 u x 1 4 L 2 2 + Φ(ρ) 1 γ L 1 u 1− 1 2γ L 2 u x 1 2γ L 2 2 + 1 ≤ ε u 2 L 2 + C 1 + u x 2 L 2 ,
where we have used the properties of the potential energy Φ in Lemma 2.3. This together with G-N inequality implies that
u 2 L 2 ≤ C(1 + u x 2 L 2 ),(3.11)
and
u L ∞ ≤ C u 1 2 L 2 u x 1 2 L 2 ≤ C(1 + u x L 2 ). (3.12)
Applying the property of Φ(ρ), Hölder's inequality, the C-K-N weighted inequality (2.3), Lemma 3.1, Young's inequality and (3.12), we obtain
I 2 =ˆ (ρ γ −ρ γ )| {0≤ρ≤2ρ} + (ρ γ −ρ γ )| {ρ>2ρ} uα|x| α−2 x ≤ Cˆ |ρ −ρ|| {0≤ρ≤2ρ} + (ρ −ρ) γ | {ρ>2ρ} |u||x| α−1 ≤ Cˆ Φ 1 2 (ρ)| {0≤ρ≤2ρ} + Φ(ρ)| {ρ>2ρ} |u||x| α−1 ≤ Cˆ Φ 1 2 (ρ)| {0≤ρ≤2ρ} |x| α 2 |u||x| α 2 −1 + Cˆ Φ(ρ)| {ρ>2ρ} |x| α α−1 α Φ 1 α (ρ)|u| ≤ C Φ(ρ)|x| α 1 2 L 1 |x| α 2 −1 u L 2 + Φ(ρ)|x| α 1− 1 α L 1 Φ(ρ) 1 α L 1 u L ∞ ≤ +C(1 + u x 2 L 2 )(1 + Φ(ρ)|x| α L 1 ), (3.13)
where we have used the fact:
|x| α 2 −1 u L 2 ≤ C u α 2 L 2 u x 1− α 2 L 2 ≤ C ( u L 2 + u x L 2 ) ≤ C(1 + u x L 2 ),
(3.14)
here, the index 1 < α ≤ 2.
Similarly, using the C-K-N weighted inequality (2.3), we have
I 3 ≤ C |x| α 2 u x L 2 |x| α 2 −1 u L 2 ≤ ε |x| α 2 u x 2 L 2 + C 1 + u x 2 L 2 . (3.15)
For the term I 4 , it follows from the Hölder inequality and C-K-N weighted inequality that
I 4 ≤ C (b −b)|x| α 2 L 2 |x| α 2 −1 u L 2 ≤ C (b −b)|x| α 2 2 L 2 + u x 2 L 2 + 1 . (3.16)
Similarly, using the C-K-N weighted inequality (2.3), one has
I 5 ≤ Cν |x| α 2 b x L 2 (b −b)|x| α 2 −1 L 2 ≤ Cν |x| α 2 b x L 2 b −b α 2 L 2 b x 1− α 2 L 2 ≤ εν |x| α 2 b x 2 L 2 + Cν 1 + b x 2 L 2 ,(3.1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 |x| α + µˆ|x| α u 2 x + νˆ|x| α b 2 x ≤ C(1 + u x 2 L 2 ) 1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 |x| α L 1 + C(1 + u x 2 L 2 + ν b x 2 L 2 ).
This together with Gronwall's inequality, (1.7) and Lemma 3.1 giveŝ
1 2 ρu 2 + Φ(ρ) + (b −b) 2 2 |x| α + µˆT 0ˆ| x| α u 2 x + νˆT 0ˆ| x| α b 2 x ≤ C(T ).
Then, the proof of Lemma 3.2 is completed.
The upper bound of the density ρ can be shown in the similar manner as that in [24]. However, for completeness of the paper, we give the details here.
0 ≤ ρ(x, t) ≤ C, ∀(x, t) ∈ R × [0, T ], and sup 0≤t≤T ρ −ρ L 2 (R) ≤ C.
Proof. Let ξ =´x −∞ ρudy, then the momentum equation (1.1) 2 can be rewritten as
ξ tx + ρu 2 + P (ρ) − P (ρ) + b 2 −b 2 2 x = (µu x ) x .
Integrating the above equality with respect to x over (−∞, x) yields that
ξ t + ρu 2 + P (ρ) − P (ρ) + b 2 −b 2 2 = µu x ,(3.18)
which together with (1.1) 1 gives
ξ t + ρu 2 + P (ρ) − P (ρ) + b 2 −b 2 2 + µ ρ t + uρ x ρ = 0. (3.19)
Next, we define the particle trajectory X(x, t) as follows:
dX(x,t) dt = u(X(x, t), t), X(x, 0) = x, (3.20) which implies dξ dt = ξ t + uξ x = ξ t + ρu 2 .
Then combining the above equality and (3.19), we infer that
d dt (ξ + µ ln ρ) (X(x, t), t) + P (ρ) + b 2 2 (X(x, t), t) − P (ρ) +b 2 = 0, which together with P (ρ) + b 2 2 (X(x, t), t) ≥ 0 yields d dt (ξ + µ ln ρ) (X(x, t), t) ≤ P (ρ) +b 2 2 (X(x, t), t) ≤ C. (3.21)
Thus, integrating it over [0, T ] with respect to t, we have (ξ + µ ln ρ) (X(x, t), t) ≤ (ξ + µ ln ρ)(x, 0) + C.
By direct calculation, we obtain
ln ρ ≤ 1 µ (ξ 0 + µ ln ρ 0 + C − ξ) ≤ 1 µ ξ 0 + µ ln ρ 0 + C −ˆx −∞ ρudy ≤ 1 µ ξ 0 + µ ln ρ 0 + C −ˆx −∞ √ ρu( √ ρ − √ρ )dy − √ρˆx −∞ √ ρudy ≤ 1 µ (ξ 0 + ln ρ 0 + C − ξ 1 − ξ 2 ). (3.22)
Firstly, it follows from Lemma 2.3 and Lemma 3.1 that ξ 1 L ∞ can be estimated as
|ξ 1 | = |ˆx −∞ √ ρu( √ ρ − √ρ )dy| ≤ √ ρu L 2 ( √ ρ − √ρ )| {0≤ρ≤2ρ} L 2 + ( √ ρ − √ρ )| {ρ>2ρ} L 2 ≤ C √ ρu L 2 (ρ −ρ)| {0≤ρ≤2ρ} L 2 + √ ρ −ρ| {ρ>2ρ} L 2 ≤ C √ ρu L 2 (ρ −ρ)| {0≤ρ≤2ρ} L 2 + (ρ −ρ) {ρ>2ρ} 1 2 L 1 ≤ C √ ρu L 2 (ρ −ρ)| {0≤ρ≤2ρ} L 2 + (ρ −ρ) γ | {ρ>2ρ} 1 2 L 1 ≤ C √ ρu L 2 Φ(ρ) 1 2 L 1 + Φ(ρ) 1 2 L 1 ≤ C.
(3.23)
Secondly, using the Gagliardo-Nirenberg inequality, the Caffarelli-Kohn-Nirenberg weighted inequality, Lemma 3.1 and Lemma 3.2, we can bound ξ 2 L ∞ as
ξ 2 L ∞ ≤ C ξ 2 p p+2 Lp ξ 2x 2 p+2 L 2 ≤ C( ξ 2x η L 2 |x| κ ξ 2 1−η L q )p p+2 ξ 2x 2 p+2 L 2 ≤ C( ξ 2x η L 2 |x| α 2 ξ 2x 1−η L 2 )p p+2 ξ 2x 2 p+2 L 2 ≤ C ξ 2x 1−(1−η)p p+2 L 2 |x| α 2 ξ 2x (1−η)p p+2 L 2 = C ξ 2x 1− 1 α L 2 |x| α 2 ξ 2x 1 α L 2 ≤ C( √ ρu 1− 1 α L 2 |x| α 2 √ ρu 1 α L 2 ) ≤ C.
(3.24)
Here the indexes 1 ≤p < ∞, q > 1, η ∈ (0, 1) and satisfy
1 p = ( 1 2 − 1)η + 1 q + κ (1 − η), 1 q + κ = 1 2 + α 2 − 1 > 0, which gives α > 1, η =p (α − 1) − 2 αp > 0 ⇒p > 2 α − 1 . (3.25)
Similarly as that for (3.23) and (3.24), we can obtain |ξ 0 | ≤ C. It follows from Lemma 2.3 and Lemma 3.1 that To deal with the last term on the right-hand side of (3.27), we discuss it in the following two cases: Case 1: if 1 < γ ≤ 2, by Lemma 2.3, Lemma3.1 and the fact that ρ ≤ C, it holds
ρ −ρ L 2 = (ρ −ρ)| 2 {ρ≤2ρ} 1 2 L 1 + (ρ −ρ)| {ρ≥2ρ} L 2 ≤ Φ(ρ) 1 2 L 1 + (ρ −ρ)| {ρ≥2ρ} L 2 ≤ C + (ρ −ρ)| {ρ≥2ρ} L 2 .(ρ −ρ)| {ρ≥2ρ} L 2 (R) ≤ ρ −ρ γ 2 L γ (R) ρ −ρ 1− γ 2 L ∞ (R) ≤ Φ(ρ) 1 2 L 1 (R) ρ −ρ 1− γ 2 L ∞ (R) ≤ C(T ).
(3.28) Case 2: if γ > 2, using the fact that ρ −ρ >ρ ⇒ 1 (ρ−ρ) γ−2 < 1 ρ γ−2 and Lemma 3.1, we have
(ρ −ρ)| {ρ≥2ρ} L 2 (R) ≤ (ρ −ρ) 2 | {ρ≥2ρ} 1 2 L 1 ≤ (ρ −ρ) γ | {ρ≥2ρ} 1 (ρ −ρ) γ−2 | {ρ≥2ρ} 1 2 L 1 ≤ (ρ −ρ) γ | {ρ≥2ρ} 1 ρ γ−2 1 2 L 1 ≤ 1 ρ γ−2 Φ(ρ) 1 2 L 1 ≤ C.sup 0≤t≤T µ u x 2 L 2 + ν b x 2 L 2 + b −b 4 L 4 +ˆT 0 b −b 6 L 6 + µν (b −b)b x 2 L 2 dt +ˆT 0 ν 2 b xx 2 L 2 dt +ˆT 0 √ ρu 2 L 2 dt ≤ C and sup 0≤t≤T ( u L 2 + u L ∞ ) ≤ C.
Proof. The proof of Lemma 3.4 will be divided into four steps.
Step 1. Multiplying the equation (1.1) 2 byu and integrating the resulting equation over R with respect to x yields
µ 2 d dtˆu 2 x +ˆρu 2 = −µˆu x (uu x ) x −ˆP (ρ) x (u t + uu x ) −ˆ b 2 2 x (u t + uu x ) =: J 1 + J 2 + J 3 .
(3.30)
Firstly, by integration by parts, we find
J 1 = −µˆu 3 x − µˆu u 2 x 2 x = −µˆu 3 x + µˆu 2 x 2 · u x = − µ 2ˆu 3 x .
(3.31) Similarly, we have
J 2 = −ˆ(P (ρ) − P (ρ)) x (u t + uu x ) = d dtˆ( P (ρ) − P (ρ)) u x −ˆ[(P (ρ) − P (ρ)) t + (P (ρ) − P (ρ)) x u] u x = d dtˆ( P (ρ) − P (ρ)) u x + γˆρ γ u 2 x ,(3.32)
and
J 3 = −ˆb(b −b) x (u t + uu x ) = −ˆ (b −b) 2 2 x (u t + uu x ) −bˆ(b −b) x (u t + uu x ) = d dtˆ (b −b) 2 2 +b(b −b) u x −ˆ (b −b) 2 2 t + (b −b) 2 2 x u u x −bˆ (b −b) t + u(b −b) x u x = d dtˆ (b −b) 2 2 +b(b −b) u x −ˆ (b −b) +b (νb xx − bu x )u x ,(3.33)
where we have used the following facts:
(P (ρ) − P (ρ)) t + u(P (ρ) − P (ρ)) x + γρ γ u x = 0,
(b −b) 2 2 t + u (b −b) 2 2 x + b(b −b)u x = νb xx (b −b), and (b −b) t + u(b −b) x + bu x = νb xx .
Substituting (3.31)-(3.33) into (3.30), using Hölder's and Cauchy-Schwarz's inequalities, we have
d dtˆµ 2 u 2 x − (P (ρ) − P (ρ)) u x − (b −b) 2 2 u x −b(b −b)u x dx +ˆρu 2 = − µ 2ˆu 3 x − γˆρ γ u 2 x − νˆb xx (b −b)u x +ˆ(b −b) 2 u 2 x + 2bˆ(b −b)u 2 x −bˆνb xx u x +b 2ˆu2 x ≤ C u x 3 L 3 + u x 2 L 2 + ν b xx L 2 b −b L 6 u x L 3 + u x 2 L 3 b −b 2 L 6 + u x 2 L 3 b −b L 3 + ν b xx L 2 u x L 2 ≤ εν 2 b xx 2 L 2 + C u x 3 L 3 + b −b 6 L 6 + ( b −b 1 2 L 2 b −b 1 2 L 6 ) 3 + u x 2 L 2 ≤ εν 2 b xx 2 L 2 + C u x 3 L 3 + b −b 6 L 6 + u x 2 L 2 + 1 .F L 3 ≤ F 5 6 L 2 F x 1 6 L 2 ≤ C µu x − (P (ρ) − P (ρ)) − (b −b) 2 2 +b(b −b) 5 6 L 2 ρu 1 6 L 2 ≤ C u x L 2 + ρ −ρ L 2 + b −b 2 L 4 + b −b L 2 5 6 ρu 1 6 L 2 ≤ C u x L 2 + b −b 1 2 L 2 b −b 3 2 L 6 + 1 5 6 ρu 1 6 L 2 ≤ C u x L 2 + b −b 3 2 L 6 + 1 5 6 √ ρu 1 6 L 2 .
(3.35)
Moreover, we can obtain
u x L 3 = F + P (ρ) − P (ρ) + b 2 −b 2 2 µ L 3 ≤ C F L 3 + ρ −ρ L 3 + (b −b) 2 2 +b(b −b) L 3 ≤ C F L 3 + ρ −ρ 2 3 L 2 ρ −ρ 1 3 L ∞ + b −b 2 L 6 + b −b L 3 ≤ C u x L 2 + b −b 3 2 L 6 + 1 5 6 √ ρu 1 6 L 2 + b −b 2 L 6 + 1 , (3.36) which implies u x 3 L 3 ≤ C u x L 2 + b −b 3 2 L 6 + 1 5 6 √ ρu 1 6 L 2 3 + b −b 6 L 6 + 1 ≤ ε √ ρu 2 L 2 + C u x 4 L 2 + b −b 6 L 6 + 1 .
(3.37)
Then, substituting (3.37) into (3.34) and choosing ε sufficiently small, we have
d dtˆµ 2 u 2 x − (P (ρ) − P (ρ)) u x − (b −b) 2 2 u x −b(b −b)u x +ˆρu 2 ≤ εν 2 b xx 2 L 2 + ε √ ρu 2 L 2 + C u x 4 L 2 + b −b 6 L 6 + 1 .
(3.38)
Step 2. To control the term b −b 6 L 6 on the right-hand side of (3.38), we rewrite the magnetic field equation as
(b −b) t + u(b −b) x + (b −b)u x +bu x = νb xx .
Then multiplying the above equation by (b −b) 3 and integrating it over R with respect to
x, we have 1 4 d dtˆ( b −b) 4 + 3νˆ(b −b) 2 b 2 x = −ˆ 3 4 (b −b) 4 +b(b −b) 3 u x = −ˆ 3 4 (b −b) 4 +b(b −b) 3 F + P (ρ) − P (ρ) + b 2 −b 2 2 µ = −ˆ 3 4 (b −b) 4 +b(b −b) 3 F + P (ρ) − P (ρ) + (b−b) 2 2 +b(b −b) µ = − 3 4µˆ( b −b) 4 (F + P (ρ) − P (ρ)) −b µˆ( b −b) 3 (F + P (ρ) − P (ρ)) − 3 8µˆ( b −b) 6 − 5b 4µˆ( b −b) 5 −b 2 µˆ( b −b) 4 , which gives 1 4 d dtˆ( b −b) 4 + 3 8µˆ( b −b) 6 +b 2 µˆ( b −b) 4 + 3νˆ(b −b) 2 b 2 x = − 3 4µˆ( b −b) 4 (F + P (ρ) − P (ρ)) −b µˆ( b −b) 3 (F + P (ρ) − P (ρ)) − 5b 4µˆ( b −b) 5 =: K 1 + K 2 + K 3 ,(3.
39) Next, we estimate K 1 −K 3 term by term. Using (3.35), Lemmas 3.3 and Young's inequality, we have
K 1 ≤ C b −b 4 L 6 ( F L 3 + P (ρ) − P (ρ) L 3 ) ≤ C b −b 4 L 6 u x L 2 + b −b 3 2 L 6 + 1 5 6 √ ρu 1 6 L 2 + ρ −ρ L 3 ≤ ε b −b 6 L 6 + ε √ ρu 2 L 2 + C(1 + u x 4 L 2 ). (3.40)
Similarly, we can get
K 2 ≤ b −b 3 L 6 ( F L 2 + P (ρ) − P (ρ) L 2 ) ≤ b −b 3 L 6 u x L 2 + ρ −ρ L 2 + b −b 2 L 4 + b −b L 2 ≤ b −b 3 L 6 u x L 2 + ρ −ρ L 2 + b −b 1 2 L 2 b −b 3 2 L 6 + b −b L 2 ≤ ε b −b 6 L 6 + C u x 4 L 2 + 1 ,(3.41)
and
K 3 ≤ C b −b 5 L 5 ≤ C b −b 1 2 L 2 b −b 9 2 L 6 ≤ ε b −b 6 L 6 + C.d dtˆ( b −b) 4 + 3 8µˆ( b −b) 6 +b 2 µˆ( b −b) 4 + 3νˆ(b −b) 2 b 2 x ≤ ε √ ρu 2 L 2 + C 1 + u x 4 L 2 .
(3.43)
Step 3. To control the term ν 2 b xx 2 L 2 on the right hand-side of (3.38), we multiply the equation (1.1) 3 by νb xx and integrate it over R to get
ν 2 d dtˆb 2 x + ν 2ˆb2 xx = νˆb xx ub x + νˆb xx (b −b)u x +bνˆb xx u x = − ν 2ˆb 2 x u x + νˆb xx (b −b)u x +bνˆb xx u x =: H 1 + H 2 + H 3 .
(3.44)
For the term H 1 , by the effective viscous flux, it shows that
H 1 = − ν 2ˆb 2 x F + P (ρ) − P (ρ) + b 2 −b 2 2 µ ≤ − ν 2µˆb 2 x F + ν 2µˆb 2 x P (ρ) +b 2 2 ≤ C (1 + F L ∞ ) ν b x 2 L 2 ≤ C 1 + √ ρu L 2 + u x L 2 + b −b 3 2 L 6 ν b x 2 L 2 ≤ ε √ ρu 2 L 2 + b −b 6 L 6 + C 1 + u x 2 L 2 + ν b x 2 L 2 ν b x 2 L 2 ,(3.45)
where we have used the following inequality:
F L ∞ ≤ C F 1 2 L 2 F x 1 2 L 2 ≤ C u x L 2 + ρ −ρ L 2 + b −b L 4 + b −b L 2 1 2 √ ρu 1 2 L 2 ≤ C 1 + √ ρu L 2 + u x L 2 + b −b 3 2 L 6 .
(3.46)
For the terms H 2 and H 3 , using Hölder's inequality, Cauchy's inequality and (3.37), we have
H 2 ≤ ν b xx L 2 b −b L 6 u x L 3 ≤ εν 2 b xx 2 L 2 + C b −b 6 L 6 + u x 3 L 3 ≤ εν 2 b xx 2 L 2 + ε √ ρu 2 L 2 + C 1 + u x 4 L 2 + b −b 6 L 6 ,(3.47)
and
H 3 ≤ Cν b xx L 2 u x L 2 ≤ εν 2 b xx 2 L 2 + C u x 2 L 2 .d dtˆb 2 x + ν 2ˆb2 xx ≤ 2ε √ ρu 2 L 2 + C 1 + u x 2 L 2 + ν b x 2 L 2 ν b x 2 L 2 + C 1 + u x 4 L 2 + b −b 6 L 6 .
(3.49)
Step 4. Adding the equation (3.43) multiplied by 8µ 3 (2C + 1) and (3.38) into (3.49), we have d dtˆ
µ 2 u 2 x + ν 2 b 2 x + 2µ(2C + 1) 3 (b −b) 4 − ψ(ρ, u, b) +ˆρu 2 + ν 2ˆb2 xx +ˆ (b −b) 6 + 8b 2 (2C + 1) 3 (b −b) 4 + 8µν(2C + 1)(b −b) 2 b 2 x ≤ C + C 1 + u x 2 L 2 + ν b x 2 L 2 u x 2 L 2 + ν b x 2 L 2 ,(3.50)
where ψ(ρ, u, b) = (P (ρ) − P (ρ)) u
x + (b−b) 2 2 u x +b(b −b)u x .
Integrating (3.50) over [0, T ] with respect to t, we obtain
µu 2 x + νb 2 x + (b −b) 4 +ˆT 0ˆ ρu 2 + (b −b) 6 + µν(b −b) 2 b 2 x + ν 2 b 2 xx ≤ C + CˆT 0 1 + u x 2 L 2 + ν b x 2 L 2 u x 2 L 2 + ν b x 2 L 2 +ˆψ(ρ, u, b) −ˆψ(ρ 0 , u 0 , b 0 ).
(3.51) By Lemma 3.1, Lemma 3.3 and Cauchy-Schwarz's inequality, we can obtain
ψ(ρ, u, b) −ˆψ(ρ 0 , u 0 , b 0 ) ≤ C P (ρ) − P (ρ) L 2 + b −b 2 L 4 + b −b L 2 u x L 2 + C ≤ C ρ −ρ L 2 + b −b 2 L 4 + 1 u x L 2 + C ≤ εµ u x 2 L 2 + C(1 + b −b 4 L 4 ),
which together with (3.51) gives (choosing ε > 0 small enough)
µu 2 x + νb 2 x + (b −b) 4 +ˆT 0ˆ ρu 2 + (b −b) 6 + µν(b −b) 2 b 2 x + ν 2 b 2 xx ≤ C(1 + b −b 4 L 4 ) + CˆT 0 1 + u x 2 L 2 + ν b x 2 L 2 u x 2 L 2 + ν b x 2 L 2 .
(3.52) Now, integrating (3.43) over [0, T ] with respect to t, and then adding the resulting inequality multiplied by 4C into (3.52) show that
µu 2 x + νb 2 x + (b −b) 4 +ˆT 0ˆ ρu 2 + (b −b) 6 + µν(b −b) 2 b 2 x + ν 2 b 2 xx ≤ C + CˆT 0 1 + u x 2 L 2 + ν b x 2 L 2 u x 2 L 2 + ν b x 2 L 2 ,
which together with Gronwall's inequality and Lemma 3.1 yieldŝ
µu 2 x + νb 2 x + (b −b) 4 dx +ˆT 0ˆ( b −b) 6 + µν(b −b) 2 b 2 x dxdt +ˆT 0ˆν 2 b 2 xx dxdt +ˆT 0ˆρu 2 dxdt ≤ C(T ).
Thus, it follows from (3.11) and (3.12) that
u 2 L 2 ≤ C(1 + u x 2 L 2 ) ≤ C, and u L ∞ ≤ C u 1 2 L 2 u x 1 2 L 2 ≤ C(1 + u x L 2 ) ≤ C.
Then, we complete the proof of Lemma 3.4.
To get the uniform upper bound of the magnetic field b, we need to re-estimate the b x L 2 independent of ν as follows.
Lemma 3.5. Let (ρ, u, b) be a smooth solution of (1.1)-(1.2). Then for any T > 0, it holds that
sup 0≤t≤T b L ∞ (R) + ρ x 2 L 2 (R) + b x 2 L 2 (R) +ˆT 0 µ u xx 2 L 2 (R) + ν b xx 2 L 2 (R) dt ≤ C.
Proof. Differentiating the equality (1.1) 3 with respect to x, then multiplying the resulting equation by b x and integrating by parts over R, we have
1 2 d dtˆb 2 x + νˆb 2 xx = −2ˆb 2 x u x −ˆbb x u xx −ˆub x b xx = − 3 2ˆb 2 x u x −ˆbb x u xx .
(3.53)
To control the second term on the right-hand side of (3.53), multiplying the momentum equation (1.1) 2 by u xx gives
µˆu 2 xx =ˆρuu xx + γˆρ γ−1 ρ x u xx +ˆbb x u xx . (3.54)
Combining (3.53) and (3.54) yields
1 2 d dtˆb 2 x + νˆb 2 xx + µˆu 2 xx = − 3 2ˆb 2 x u x +ˆρuu xx + γˆρ γ−1 ρ x u xx = − 3 2ˆb 2 x F + P (ρ) − P (ρ) + b 2 −b 2 2 µ +ˆρuu xx + γˆρ γ−1 ρ x u xx ≤ − 3 2ˆb 2 x F − P (ρ) + −b 2 2 µ +ˆρuu xx + γˆρ γ−1 ρ x u xx ≤ C F L ∞ b x 2 L 2 + b x 2 L 2 + √ ρu L 2 u xx L 2 + ρ x L 2 u xx L 2 ≤ ε u xx 2 L 2 + C ( F L ∞ + 1) b x 2 L 2 + C ρ x 2 L 2 + √ ρu 2 L 2 ≤ ε u xx 2 L 2 + C √ ρu 2 L 2 + b −b 6 L 6 + 1 b x 2 L 2 + C ρ x 2 L 2 + √ ρu 2 L 2 ,(3.55)
where we have used (3.46) and Lemma 3.4. Next, differentiating the density equation (1.1) 1 with respect to x, multiplying the resulting equation by ρ x , and integrating it over R implies that
1 2 d dtˆρ 2 x = −2ˆρ 2 x u x −ˆuρ x ρ xx −ˆρρ x u xx = − 3 2ˆρ 2 x u x −ˆρρ x u xx = − 3 2ˆρ 2 x F + P (ρ) − P (ρ) + b 2 −b 2 2 µ −ˆρρ x u xx ≤ − 3 2µˆρ 2 x F + 3 2µˆ P (ρ) +b 2 2 ρ 2 x + C ρ x L 2 u xx L 2 ≤ ε u xx 2 L 2 + C ( F L ∞ + 1) ρ x 2 L 2 ≤ ε u xx 2 L 2 + C √ ρu 2 L 2 + b −b 6 L 6 + 1 ρ x 2 L 2 ,(3.d dtˆ b 2 x + ρ 2 x + νˆb 2 xx + µˆu 2 xx ≤ C √ ρu 2 L 2 + b −b 6 L 6 + 1 1 + ρ x 2 L 2 + b x 2 L 2 ,
which together Gronwall's inequality and Lemma 3.4 giveŝ
ρ 2 x + b 2 x +ˆT 0ˆ νb 2 xx + µu 2 xx ≤ C(T ).
Thus, it follows from Lemma 3.1 and the G-N inequality (2.1) that
b L ∞ ≤ b −b L ∞ +b ≤ C b −b 1 2 L 2 b x 1 2 L 2 +b ≤ C(T ).
Thus, the proof of Lemma 3.5 is finished.
With the help of the Lemma 3.5, we can get the estimates of the first order derivative with respect to t of the density and the magnetic field, respectively. Lemma 3.6. Let (ρ, u, b) be a smooth solution to (1.1)-(1.2). Then for any T > 0, it holds that sup
0≤t≤T ρ t L 2 (R) + b t L 2 (0,T ;L 2 (R)) ≤ C.
Proof. By direct calculation, we have
ρ t L 2 ≤ C ( ρu x L 2 + uρ x L 2 ) ≤ C ( ρ L ∞ u x L 2 + u L ∞ ρ x L 2 ) ∈ L ∞ (0, T ), and b t L 2 ≤ C ( bu x L 2 + ub x L 2 + νb xx L 2 ) ≤ C ( b L ∞ u x L 2 + u L ∞ b x L 2 + νb xx L 2 ) ≤ C (1 + νb xx L 2 ) ∈ L 2 (0, T ),
where we have used Lemma 3.4 and Lemma 3.5.
The following estimates of the second order derivative of the velocity plays an important role in the analysis of the non-resistive limit. Proof. Differentiating the momentum equation (1.1) 2 with respect to t, we have
ρ tu + ρu t + P (ρ) + b 2 2 xt = µu xxt .
Multiplying the above equation byu and integrating the resulting equation over R yields 1 2
d dtˆρu 2 + µˆu 2 xt = 1 2ˆρ tu 2 −ˆ P (ρ) + b 2 2 xtu − µˆu xt u 2 x + uu xx =: L 1 + L 2 + L 3 .
(3.57)
Now, we estimate the terms L 1 − L 3 as
L 1 = − 1 2ˆ( ρu) xu 2 =ˆρuuu x ≤ C √ ρu L 2 u L ∞ u x L 2 ≤ C √ ρu L 2 u xt L 2 + u x 2 L 4 + u L ∞ u xx L 2 (3.58) ≤ C √ ρu L 2 u xt L 2 + u x 3 2 L 2 u xx 1 2 L 2 + u xx L 2 ≤ ε u xt 2 L 2 + C √ ρu 2 L 2 + u xx 2 L 2 + 1
, and similarly, we also have
L 2 =ˆ P (ρ) + b 2 2 tu x ≤ C ρ γ−1 ρ t + bb t L 2 u x L 2 ≤ C ( ρ t L 2 + b t L 2 ) u xt + u 2 x + uu xx L 2 ≤ C (1 + b t L 2 ) u xt L 2 + u x 3 2 L 2 u xx 1 2 L 2 + u L ∞ u xx L 2 ≤ ε u xt 2 L 2 + C u xx 2 L 2 + b t 2 L 2 + 1 ,(3.59)
and
L 3 ≤ C u xt L 2 u x 2 L 4 + u L ∞ u xx L 2 ≤ ε u xt 2 L 2 + C 1 + u xx 2 L 2 , (3.60)
where we have used the Gagliardo-Nirenberg inequality, Lemma 3.4 and Lemma 3.5. Thus, substituting (3.58)-(3.60) into (3.57) and choosing ε > 0 small enough, we have
1 2 d dtˆρu 2 + µˆu 2 xt ≤ C 1 + √ ρu 2 L 2 + u xx 2 L 2 + b t 2 L 2 ,
which combining with Gronwall's inequality, Lemma 3.4 and Lemma 3.5 gives
1 2ˆρu 2 +ˆT 0ˆµ u 2 xt ≤ C(T ).
This together with the momentum equation yields that
µ u xx L 2 ≤ C ρu L 2 + P (ρ) + b 2 2 x L 2 ≤ C ( √ ρu L 2 + ρ x L 2 + b x L 2 ) ≤ C, and √ ρu t L 2 ≤ C √ ρu − √ ρuu x L 2 ≤ C √ ρu L 2 + √ ρ L ∞ u L ∞ u x L 2 ≤ C.
We complete the proof of Lemma 3.7.
Proof of the Theorem 1.1
This section is devoted to the proof of Theorem 1.1. To do this, we also need the following global a priori estimates of the solution (ρ,ũ,b) to the problem (1 .3) and (1.4). Let (ρ,ũ,b) be a smooth solution of (1.3), (1.4) and (1.5), then for some index 1 < α ≤ 2, we have
Proposition 4.1.0 ≤ρ(x, t) ≤ C, ∀(x, t) ∈ R × [0, T ], (4.1) sup 0≤t≤T 1 2ρũ 2 + Φ(ρ) + (b −b) 2 2 (1 + |x| α ) L 1 +ˆT 0 µ ũ x (1 + |x| α 2 ) 2 L 2 dt ≤ C,(4.
2) and
sup 0≤t≤T ũ H 2 + b x L 2 + ρ x L 2 + ρu L 2 +ˆT 0 µ ũ xt 2 L 2 ≤ C. (4.3)
Proof. In fact, repeating the arguments in the proofs of Lemma 3.1-Lemma 3.7 step by step, we can easily obtain (4.1)-(4.3). Here, we omit the details.
The existence and uniqueness of global strong solution in Theorem 1.1 can be obtained from the local existence in time and the global (in time) a priori estimates in Proposition 4.1 by a standard continuity argument c.f. [16] .
Proof of the Theorem 1.2
The existence and uniqueness of global strong solution in Theorem 1.2:
The existence and uniqueness of global strong solution in Theorem 1.2 can be obtained by the local existence in time and the global (in time) a priori estimates in Sections 3 by a standard continuity argument c.f. [26].
The non-resistive limit in Theorem 1.2: To justify the non-resistive limit as ν → 0, we consider the the difference of these two solutions (ρ −ρ, u −ũ, b −b) which satisfy the following:
(ρ −ρ) t + (ρ −ρ)u x +ρ(u −ũ) x + (ρ −ρ) x u +ρ x (u −ũ) = 0, ρ(u −ũ) t + ρu(u −ũ) x − µ(u −ũ) xx = −(ρ −ρ)(ũ t +ũũ x ) − ρ(u −ũ)ũ x − (P (ρ) − P (ρ)) x − b 2 −b 2 2 x , (b −b) t + u x (b −b) +b(u −ũ) x + u(b −b) x + (u −ũ)b x = νb xx . (5.1)
Firstly, multiplying the equation (5.1) 1 by 2(ρ −ρ), integrating the resultant over R, and integrating by parts, it follows from Hölder's inequality, the Gagliardo-Nirenberg inequality, the Cauchy inequality, Lemma 3.4 and Proposition 4.1 that
d dtˆ( ρ −ρ) 2 = −ˆ(ρ −ρ) 2 u x − 2ˆρ(ρ −ρ)(u −ũ) x − 2ˆρ x (ρ −ρ)(u −ũ) ≤ u x L ∞ ρ −ρ 2 L 2 + ρ L ∞ ρ −ρ L 2 (u −ũ) x L 2 + ρ x L 2 ρ −ρ L 2 u −ũ L ∞ ≤ C u x 1 2 L 2 u xx 1 2 L 2 ρ −ρ 2 L 2 + (u −ũ) x L 2 + u −ũ 1 2 L 2 (u −ũ) x 1 2 L 2 ρ −ρ L 2 ≤ ε (u −ũ) x 2 L 2 + C (1 + u xx L 2 ) ρ −ρ 2 L 2 + 1 . (5.
2) Secondly, we multiply (5.1) 2 by 2(u −ũ), integrate the resultant over R, integrate by parts, and use Hölder's inequality, the Gagliardo-Nirenberg inequality, Young's inequality, the fact that u −ũ 2 L 2 ≤ C 1 + (u −ũ) x 2 L 2 (see (3.11)) and Proposition 4.1 to get
d dtˆρ (u −ũ) 2 + 2µˆ(u −ũ) 2 x = −ˆ2(ρ −ρ)(u −ũ)(ũ t +ũũ x ) −ˆ2ρ(u −ũ) 2ũ x −ˆ2 (P (ρ) − P (ρ)) x (u −ũ) −ˆ(b 2 −b 2 ) x (u −ũ) ≤ C u L 2 ρ −ρ L 2 u −ũ L ∞ + ũ x L ∞ √ ρ(u −ũ) 2 L 2 + ( ρ −ρ L 2 + b −b L 2 ) (u −ũ) x L 2 ≤ C u L 2 u −ũ 1 2 L 2 (u −ũ) x 1 2 L 2 ρ −ρ L 2 + ũ x 1 2 L 2 ũ xx 1 2 L 2 √ ρ(u −ũ) 2 L 2 + ( ρ −ρ L 2 + b −b L 2 ) (u −ũ) x L 2 ≤ ε (u −ũ) x 2 L 2 + C 1 + ρ −ρ 2 L 2 + b −b 2 L 2 + √ ρ(u −ũ) 2 L 2 1 + u 2 L 2 + ũ xx L 2 .L 2 b x 1 2 L 2 , we deduce d dtˆ( b −b) 2 = −ˆ(b −b) 2 u x − 2ˆb(b −b)(u −ũ) x − 2ˆ(b −b)(u −ũ)b x + 2νˆ(b −b)b xx ≤ C u x L ∞ b −b 2 L 2 + b L ∞ b −b L 2 (u −ũ) x L 2 + b −b L 2 b x L 2 u −ũ L ∞ + ν b −b L 2 b xx L 2 ≤ ε (u −ũ) x 2 L 2 + C (1 + u xx L 2 ) b −b 2 L 2 + ν 2 b xx 2 L 2 .
(5.4)
Then combining (5.2)-(5.4) and choosing ε > 0 sufficiently small, we have d dtˆ (ρ −ρ) 2 + ρ(u −ũ) 2 + (b −b) 2 + 2µˆ(u −ũ) 2
x ≤ C 1 + u xx L 2 + ũ xx L 2 + u 2
L 2 ρ −ρ 2 L 2 + b −b 2 L 2 + √ ρ(u −ũ) 2 L 2 + 1 + ν 2 b xx 2 L 2 ,
which together with Gronwall's inequality, Lemma 3.5 and Proposition 4.1 yieldŝ
(ρ −ρ) 2 + ρ(u −ũ) 2 + (b −b) 2 + 2µˆ(u −ũ) 2 x ≤ Cν 2ˆT 0 b xx 2 L 2 dt exp CˆT 0 1 + u xx L 2 + ũ xx L 2 + u 2 L 2 dt ≤ Cν,(5.5)
where we have used Lemmas 3.5 and 3.7, Proposition 4.1 and the following fact:
u L 2 ≤ C √ρu L 2 ≤ C ( √ρ − ρ)ũ L 2 + ρũ L 2 ≤ C ρ −ρ L 2 u L ∞ + 1 ≤ ε u L 2 + C( u x L 2 + 1),
which gives u L 2 ≤ C( (ũ t +ũũ x ) x L 2 + 1)
≤ C ũ xt L 2 + ũ x 2 L 4 + ũũ xx L 2 + 1 ≤ C ( ũ xt L 2 + ũ xx L 2 + 1) ∈ L 2 (0, T ).
Moreover, we havêρ (u −ũ) 2 =ˆ(ρ −ρ)(u −ũ) 2 +ˆρ(u −ũ) 2 ≤ C ρ −ρ L 2 u −ũ 2 L 4 + Cν ≤ C ρ −ρ L 2 u −ũ
Remark 1. 2 .
2In Theorem 1.2, we give that the global strong solution of resistive MHD equation (1.1)-(1.2) converges to that of non-resistive MHD equation (1.3)-(1.4) in L 2norm as ν → 0, moreover, the convergence rates are also justified.
Lemma 3. 3 .
3Let (ρ, u, b) be a smooth solution of (1.1) and (1.2). Then for any T > 0, it holds that
substituting (3.23), (3.24) and (3.26) into (3.21), we have ln ρ ≤ C, which gives ρ ≤ C.
combining (3.28), (3.29) with (3.27), we can obtain ρ −ρ L 2 ≤ C. This completes the proof of Lemma 3.3.
Lemma 3 . 4 .
34Let (ρ, u, b) be a smooth solution of (1.1)-(1.2). Then for any T > 0, it holds that
substituting (3.40)-(3.42) into (3.39) and choosing ε > 0 small enough yields 1 4
substituting (3.45), (3.47) and (3.48) into (3.44) and choosing ε > 0 small enough, we have ν 2
Lemma 3. 7 .u
7Let (ρ, u, b) be a smooth solution to (1.1)-(1.2). Then for any T > 0, xx L 2 (R) + √ ρu t L 2 (R) ≤ C.
multiplying the equation (5.1) 3 by 2(b −b), integrating the resultant over R, integrating by parts and using Proposition 4.1, the third equation of (1.3), the fact that b L ∞ ≤ C b 1 2
have proved the results of Theorem 1.2.
Li is supported by the NSFC (No.11671319, No.11931013), Fund of HPU (No.B2016-57). H. Wang is supported by the National Natural Science Foundation of China (Grant No.11901066), the Natural Science Foundation of Chongqing (Grant No.cstc2019jcyj-msxmX0167) and Project No.2019CDXYST0015 and No. 2020CDJQY-A040 supported by the Fundamental Research Funds for the Central Universities. Y. Ye is supported by the National Natural Science Foundation of China (No.11701145) and China Postdoctoral Science Foundation (No.2020M672196).
First order interpolation inequality with weights. L Caffarelli, R Kohn, L Nirenberg, Compositio Mathematica. 53L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights. Compositio Mathematica 53 (1984), 259-275.
On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions. F Catrina, Z Q Wang, Comm. Pure Appl. Math. LIV. F. Catrina, Z.Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp con- stants, existence (and nonexistence), and symmetry of extremal functions. Comm. Pure Appl. Math. LIV (2001), 229-258.
Globla solutions of nonlinear magnetohydrodynamics with large initial data. G Q Chen, D H Wang, J. Differential Equations. 182G.Q. Chen, D.H. Wang, Globla solutions of nonlinear magnetohydrodynamics with large initial data. J. Differential Equations 182 (2002), 344-376.
Global strong solutions to the 1-D compressible magnetohydrodynamic equations with zero resistivity. J S Fan, Y X Hu, J. Math. Phys. 562ppJ.S. Fan, Y.X. Hu, Global strong solutions to the 1-D compressible magnetohydro- dynamic equations with zero resistivity. J. Math. Phys. 56 (2015), no. 2, 023101, 13 pp.
Strong solution to the compressible magnetohydrodynamic equations with vacuum. J S Fan, W H Yu, Nonlinear Anal. Real World Appl. 101J.S. Fan, W.H. Yu, Strong solution to the compressible magnetohydrodynamic equa- tions with vacuum. Nonlinear Anal. Real World Appl. 10 (2009), no. 1, 392-409.
Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. J S Fan, S X Huang, F C Li, Kinet. Relat. Models. 104J.S. Fan, S.X. Huang, F.C. Li, Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinet. Relat. Models 10 (2017), no. 4, 1035-1053.
Ulteriori propriet di alcune classi di funzioni in più vari-abili. E Gagliardo, Ricerche Mat. 8E. Gagliardo, Ulteriori propriet di alcune classi di funzioni in più vari-abili. Ricerche Mat., 8 (1959), 24-51.
Global existence for a class of large solutions to three-dimensional compressible magnetohydrodynamic equations with vacuum. G Y Hong, X F Hou, H Y Peng, C J Zhu, SIAM J. Math. Anal. 494G.Y. Hong, X.F. Hou, H.Y. Peng, C.J. Zhu, Global existence for a class of large solutions to three-dimensional compressible magnetohydrodynamic equations with vacuum. SIAM J. Math. Anal. 49 (2017), no. 4, 2409-2441.
Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch. Ration. X P Hu, D H Wang, Mech. Anal. 1971X.P. Hu, D.H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows. Arch. Ra- tion. Mech. Anal. 197 (2010), no. 1, 203-238.
On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics. S Jiang, J W Zhang, Nonlinearity. 309S. Jiang, J.W. Zhang, On the non-resistive limit and the magnetic boundary-layer for one-dimensional compressible magnetohydrodynamics. Nonlinearity 30 (2017), no. 9, 3587-3612.
Global classical solution of the Cauchy problem to 1D compressible Navier-Stokes equations with large initial data. Q S Jiu, M J Li, Y L Ye, J. Differential Equations. 2572Q.S. Jiu, M.J. Li, Y.L. Ye, Global classical solution of the Cauchy problem to 1D compressible Navier-Stokes equations with large initial data. J. Differential Equations 257 (2014), no. 2, 311-350.
Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. S Kawashima, Kyoto UniversityPhD thesisS. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. PhD thesis, Kyoto University, (1983).
Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. S Kawashima, M Okada, Proc. Japan Acad. Ser. A Math. Sci. 58S. Kawashima, M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics. Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 384-387.
Global classical solutions to 3D compressible magnetohydrodynamic equations with large oscillations and vacuum. H L Li, X Y Xu, J W Zhang, SIAM J. Math. Anal. 453H.L. Li, X.Y. Xu, J.W. Zhang, Global classical solutions to 3D compressible magne- tohydrodynamic equations with large oscillations and vacuum. SIAM J. Math. Anal. 45 (2013), no. 3, 1356-1387.
On local classical solutions to the Cauchy problem of the twodimensional barotropic compressible Navier-Stokes equations with vacuum. J Li, Z L Liang, J. Math. Pures Appl. 1024J. Li, Z.L. Liang, On local classical solutions to the Cauchy problem of the two- dimensional barotropic compressible Navier-Stokes equations with vacuum. J. Math. Pures Appl. 102 (2014), no. 4, 640-671.
Global strong solutions to the Cauchy problem of 1D compressible MHD equations with no resistivity. Z L Li, H Q Wang, Y L Ye, Commun. Math Sci. 183Z.L. Li, H.Q. Wang, Y.L. Ye, Global strong solutions to the Cauchy problem of 1D compressible MHD equations with no resistivity. Commun. Math Sci. 18 (2020), no. 3, 851-873.
Local existence of classical solutions to the two-dimensional viscous compressible flows with vacuum. Z Luo, Commun. Math. Sci. 102Z. Luo, Local existence of classical solutions to the two-dimensional viscous compress- ible flows with vacuum. Commun. Math. Sci. 10 (2012), no. 2, 527-554.
Global existence and large-time asymptotic behavior of strong solutions to the compressible magnetohydrodynamic equations with vacuum. B Q Lv, X D Shi, X Y Xu, Indiana Univ. Math. J. 653B.Q. Lv, X.D. Shi, X.Y. Xu, Global existence and large-time asymptotic behavior of strong solutions to the compressible magnetohydrodynamic equations with vacuum. Indiana Univ. Math. J. 65 (2016), no. 3, 925-975.
On elliptic partial differential equations. L Nirenberg, Ann. Scuola Norm. Sup. Pisa. 13L. Nirenberg, On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.
On the existence of global solutions of twodimensional Navier-Stokes equations of a compressible viscous fluid. V A Vaigant, A V Kazhikhov, Russian) SibirskV.A. Vaigant, A.V. Kazhikhov, On the existence of global solutions of two- dimensional Navier-Stokes equations of a compressible viscous fluid. (Russian) Sibirsk.
. Mat, Zh, Siberian Math. J. 366translation inMat. Zh., 36 (1995), no. 6, 1283-1316, ii; translation in Siberian Math. J. 36 (1995), no. 6, 1108-1141.
On the Cauchy problem for composite systems of nonlinear differential equations. A I Volpert, S I Hudjaev, Math. USSR-Sb. 16A.I. Volpert, S.I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations. Math. USSR-Sb. 16 (1972), 517-544.
A blow-up criterion for 3D non-resistive compressible magnetohydrodynamic equations with initial vacuum. X Y Xu, J W Zhang, Nonlinear Anal. Real World Appl. 12X.Y. Xu, J.W. Zhang, A blow-up criterion for 3D non-resistive compressible magne- tohydrodynamic equations with initial vacuum. Nonlinear Anal. Real World Appl. 12 (2011), 3442-3451.
Existence and uniqueness of global weak solution to a two-phase flow model with vacuum. L Yao, C J Zhu, Math. Ann. 3494L. Yao, C.J. Zhu, Existence and uniqueness of global weak solution to a two-phase flow model with vacuum. Math. Ann. 349 (2011), no. 4, 903-928.
Global classical solution to 1D compressible Navier-Stokes equations with no vacuum at infinity. Y L Ye, Math. Methods Appl. Sci. 394Y.L. Ye, Global classical solution to 1D compressible Navier-Stokes equations with no vacuum at infinity. Math. Methods Appl. Sci. 39 (2016), no. 4, 776-795.
Local well-posedness to the Cauchy problem of the 3-D compressible Navier-Stokes equations with density-dependent viscosity. Y L Ye, Q S Jiu, C S Dou, Acta Math. Sci. Ser. B (Engl. Ed.). 343Y.L. Ye, Q.S. Jiu, C.S. Dou, Local well-posedness to the Cauchy problem of the 3-D compressible Navier-Stokes equations with density-dependent viscosity. Acta Math. Sci. Ser. B (Engl. Ed.) 34 (2014), no. 3, 851-871.
Global strong solution to the Cauchy problem of 1D compressible MHD equations with large initial data and vacuum. Y L Ye, Z L Li, Z. Angew. Math. Phys. 701ppY.L. Ye, Z.L. Li, Global strong solution to the Cauchy problem of 1D compressible MHD equations with large initial data and vacuum. Z. Angew. Math. Phys. 70 (2019), no. 1, Art. 38, 20 pp.
Global classical large solutions with vacuum to 1D compressible MHD with zero resistivity. H B Yu, Acta Appl. Math. 128H.B. Yu, Global classical large solutions with vacuum to 1D compressible MHD with zero resistivity. Acta Appl. Math. 128 (2013), 193-209.
Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics. J W Zhang, F Xie, J. Differential Equations. 2457J.W. Zhang, F. Xie, Global solution for a one-dimensional model problem in thermally radiative magnetohydrodynamics. J. Differential Equations 245 (2008), no. 7, 1853- 1882.
Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics. J W Zhang, J N Zhao, Commun. Math. Sci. 84J.W. Zhang, J.N. Zhao, Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics. Commun. Math. Sci. 8 (2010), no. 4, 835-850.
On the global solvability and the non-resistive limit of the one-dimensional compressible heat-conductive MHD equations. J W Zhang, X K Zhao, J. Math. Phys. 58317J.W. Zhang, X.K. Zhao, On the global solvability and the non-resistive limit of the one-dimensional compressible heat-conductive MHD equations. J. Math. Phys. 58 (2017), no. 3, 031504, 17 pp
| [] |
[
"SOLAR CONSTRAINTS ON ASYMMETRIC DARK MATTER",
"SOLAR CONSTRAINTS ON ASYMMETRIC DARK MATTER"
] | [
"Ilídio Lopes \nDraft version\n\n",
"Joseph Silk \nDraft version\n\n"
] | [
"Draft version\n",
"Draft version\n"
] | [
"The Astrophysical Journal"
] | The dark matter content of the Universe is likely to be a mixture of matter and antimatter, perhaps comparable to the measured asymmetric mixture of baryons and antibaryons. During the early stages of the Universe, the dark matter particles are produced in a process similar to baryogenesis, and dark matter freeze-out depends on the dark matter asymmetry and the annihilation cross section (s-wave and p-wave annihilation channels) of particles and antiparticles. In these η−parametrised asymmetric dark matter models (ηADM), the dark matter particles have an annihilation cross section close to the weak interaction cross section, and a value of dark matter asymmetry η close to the baryon asymmetry η B . Furthermore, we assume that dark matter scattering of baryons, namely, the spin-independent scattering cross section, is of the same order as the range of values suggested by several theoretical particle physics models used to explain the current unexplained events reported in the DAMA/LIBRA, CoGeNT and CRESST experiments. Here, we constrain η−parametrised asymmetric dark matter by investigating the impact of such a type of dark matter on the evolution of the Sun, namely, the flux of solar neutrinos and helioseismology. We find that dark matter particles with a mass smaller than 15 GeV, a spin-independent scattering cross section on baryons of the order of a picobarn, and an η−asymmetry with a value in the interval 10 −12 − 10 −10 , would induce a change in solar neutrino fluxes in disagreement with current neutrino flux measurements. This result is also confirmed by helioseismology data. A natural consequence of this model is suppressed annihilation, thereby reducing the tension between indirect and direct dark matter detection experiments, but the model also allows a greatly enhanced annihilation cross section. All the cosmological η−asymmetric dark matter scenarios that we discuss have a relic dark matter density Ωh 2 and baryon asymmetry η B in agreement with the current WMAP measured values, Ω DM h 2 = 0.1109 ± 0.0056 and η B = 0.88 × 10 −10 . | 10.1088/0004-637x/757/2/130 | [
"https://arxiv.org/pdf/1209.3631v1.pdf"
] | 118,549,116 | 1209.3631 | d99078407accbce8bdde4bfc13e3b1fa3c071c40 |
SOLAR CONSTRAINTS ON ASYMMETRIC DARK MATTER
17 Sep 2012
Ilídio Lopes
Draft version
Joseph Silk
Draft version
SOLAR CONSTRAINTS ON ASYMMETRIC DARK MATTER
The Astrophysical Journal
75713017 Sep 201210.1088/0004-637X/757/2/130Preprint typeset using L A T E X style emulateapj v. 03/07/07Subject headings: dark matter-elementary particles-stars:evolution-stars:interiors-Sun:interior
The dark matter content of the Universe is likely to be a mixture of matter and antimatter, perhaps comparable to the measured asymmetric mixture of baryons and antibaryons. During the early stages of the Universe, the dark matter particles are produced in a process similar to baryogenesis, and dark matter freeze-out depends on the dark matter asymmetry and the annihilation cross section (s-wave and p-wave annihilation channels) of particles and antiparticles. In these η−parametrised asymmetric dark matter models (ηADM), the dark matter particles have an annihilation cross section close to the weak interaction cross section, and a value of dark matter asymmetry η close to the baryon asymmetry η B . Furthermore, we assume that dark matter scattering of baryons, namely, the spin-independent scattering cross section, is of the same order as the range of values suggested by several theoretical particle physics models used to explain the current unexplained events reported in the DAMA/LIBRA, CoGeNT and CRESST experiments. Here, we constrain η−parametrised asymmetric dark matter by investigating the impact of such a type of dark matter on the evolution of the Sun, namely, the flux of solar neutrinos and helioseismology. We find that dark matter particles with a mass smaller than 15 GeV, a spin-independent scattering cross section on baryons of the order of a picobarn, and an η−asymmetry with a value in the interval 10 −12 − 10 −10 , would induce a change in solar neutrino fluxes in disagreement with current neutrino flux measurements. This result is also confirmed by helioseismology data. A natural consequence of this model is suppressed annihilation, thereby reducing the tension between indirect and direct dark matter detection experiments, but the model also allows a greatly enhanced annihilation cross section. All the cosmological η−asymmetric dark matter scenarios that we discuss have a relic dark matter density Ωh 2 and baryon asymmetry η B in agreement with the current WMAP measured values, Ω DM h 2 = 0.1109 ± 0.0056 and η B = 0.88 × 10 −10 .
INTRODUCTION
During the last three decades, overwhelming evidence has been found for the existence of dark matter in the Universe. This achievement is the result of a careful analysis of the available cosmological observational data, combined with a variety of well structured theoretical physical models coming from quite different and complementary research fields in particle physics, cosmology and astrophysics. Such studies have led the way to identification of the basic gravitational effects of dark matter, and their contribution to the formation of structure in the universe. In spite of this, the fundamental nature of the dark matter particles remains a mystery.
Current observational studies suggest that the matter present in the Universe is composed predominantly of dark matter particles and baryons (Munshi et al. 2011). Recent measurements of the cosmic microwave background by the WMAP team (Larson et al. 2011) give precise measurements of the dark matter density and the baryonic density: Ω DM h 2 = 0.1109 ± 0.0056 and Ω B h 2 = 0.02258 +0.00057 −0.00056 . These observational studies also measure the imbalance between baryons and antibaryons, i.e., the baryon asymmetry, which is found to be equal to (0.88 ± 0.021) × 10 −10 .
Among the most popular candidates for dark matter are a group of particles that occur naturally in supersymmetric extensions of the standard model of particle physics, usually called Weakly Interacting Massive Particles (WIMPs), the neutralino, the lightest supersymmetric stable particle being the typical example.
Significant constraints in the properties of WIMP candidates and similar particles have been made using stars (Bertone et al. 2005), such as the Sun (e.g., Lopes & Silk 2010a,b;Taoso et al. 2010;Cumberbatch et al. 2010), sun-like stars (Casanellas & Lopes 2011b,a) and neutron stars (e.g., Kouvaris 2011;Kouvaris & Tinyakov 2011Bertone & Fairbairn 2008). Although, WIMP candidates can be either dirac or majorana particles, usually, WIMPs are considered to be of the later type, as such these particles do not have an asymmetry such as the baryon asymmetry. Recently, several authors (e.g. Gudnason et al. 2006;Foadi et al. 2009;Hooper et al. 2005) following previous work (e.g. Nussinov 1985;Kaplan 1992) have proposed a new type of matter, known as asymmetric dark matter, which has an asymmetry identical to baryons (Kaplan et al. 2009). Like WIMPs, such particles are non-relativistic massive particles that interact with baryons on the weak scale, thereby having a sizeable scattering cross-section with baryons (Kaplan et al. 2009), but unlike WIMPs, such a type of dark matter is produced in the primordial Universe by a mechanism similar to baryogenesis. This type of dark matter, much the same as baryons, is considered to have an asymmetry that we choose to represent by the parameter η. Hence, this dark matter is composed of an unqual amount of matter and antimatter (e.g. Farrar & Zaharijas 2006). Furthermore, these particles have a mass of the order of a few GeV (e.g. Kang et al. 2011;Cohen et al. 2010). We will refer to this type of dark matter as η−parametrised asymmetric dark matter (ηADM) or simply as η−asymmetric dark matter.
Several experiments committed to direct dark matter searches show evidence of positive particle detection, although these results are still very controversial and not universally accepted: the DAMA/LIBRA (Bellini et al. 2012;Bernabei et al. 2008) and CoGeNT (Aalseth et al. 2011) experiments find evidence of a particle candidate with a mass of the order of a few GeV (likely between 5 and 12 GeV), and a spin-independent scattering cross section of baryons of the order of 10 −40 cm 2 . The CRESS (Brown et al. 2012) experiment also points to unexplained events consistent with direct detection of a light mass particle. Unfortunately, other direct detection experiments, such as CDMS (Ahmed et al. 2011) and XENON10/100 (Aprile et al. 2011;Angle et al. 2011) find no indication of the existence of such particles. Nevertheless, several theoretical explanations based on the existence of asymmetric dark matter have been suggested to explain and accommodate all these positive and negative detections (e.g. Chang et al. 2010;Farina et al. 2011;Hooper & Kelso 2011;Del Nobile et al. 2011).
In this paper, we investigate the origin of such light η−asymmetric dark matter particles, and discuss how this type of matter influences the evolution of stars. We look for their impact on the present structure of the Sun. The Sun, by means of two groups of observables, helioseismology and solar neutrino fluxes, provides one of the most powerful tests of stellar evolution (Turck-Chieze & Couvidat 2011) and of alternative theories of modern physics and cosmology (Casanellas et al. 2012;Turck-Chieze et al. 2012;. Although both groups of observables are equally relevant for such types of studies, we have chosen preferentially to study the impact on solar neutrino fluxes, as neutrinos are the most sensitive probe of changes in the structure of the solar core. In particular, we will compare the neutrino fluxes of a "Sun" evolving in a halo of asymmetric dark matter with the current measurements of solar neutrino fluxes, namely, the 8 B and 7 Be neutrino fluxes. Moreover, we will complete the study with a succinct helioseismology diagnostic.
In the reminder of this paper, we consider that the dark matter asymmetry is identical to the baryon asymmetry, and likewise leads to an unbalanced amount of particles and antiparticles. This η−asymmetry occurs well before the epoch of thermal decoupling of the dark matter. We do not discuss here any of the possible mechanisms for the generation of such asymmetry, but rather treat this quantity as a free parameter. This asymmetric dark matter framework is used to study the impact of such matter in the evolution of the Sun for a wide range of particle masses, annihilation cross sections and dark matter asymmetries.
In Sec. 2 we present the basic properties of η−parametrised asymmetric dark matter and explain how the relic dark matter density of the present day Universe is computed. In Sec.(s) 3, 4 and 5 we discuss the changes caused in the Sun by the presence of η−parametrised asymmetric dark matter in its core, as well as the impact on the flux of solar neutrinos and helioseismology. In the last section, we summarize our results and draw some conclusions.
ASYMMETRIC DARK MATTER IN THE EARLY
UNIVERSE
In the early stages of the Universe, the cosmic fluid rapidly reached thermal equilibrium, as a consequence of standard and dark matter particles having a very high rate of interactions. In this work, we allow dark matter to be constituted by a mixture of particles and antiparticles, χ andχ with a mass m χ , and g χ being the number of internal degrees of freedom (Drees et al. 2006). The Universe is considered to have a temperature T and an effective number of relativistic degrees of freedom g ⋆ (Iminniyaz et al. 2011). Unlike symmetric dark matter, χ andχ particles are not necessarily identical. Following the usual convention, the asymmetric nature of the dark matter is defined by the parameter η which is equal to the difference of particle and antiparticle populations. Without loss of generality, we conventionally define η ≥ 0, i.e., particles are more abundant than anti-particles. Here η stays constant throughout the evolution of the Universe. The asymmetry η is similar to the asymmetry for baryons η B , both of which originate in the early phases of the Universe. If not stated otherwise, in the numerical examples, the population of χ andχ particles is constituted by elementary particles with mass m χ ∼ 10 GeV, g ⋆ = 90 and g χ = 2. This is equivalent to a particle population formed by Dirac fermions (Dent et al. 2010;Drees et al. 2006).
As the Universe expands, the interactions between all particles become sparse, the temperature of the plasma drops, the Universe gets cooler, up to the moment that the χχ annihilation rate drops below the Hubble expansion rate, leading to dark matter becoming decoupled from the rest of the cosmic fluid. The dark matter density, Ω DM , has frozen out and has been constant ever since. The relic densities of χ andχ particles are determined by solving the coupled system of Boltzmann equations that follows the time evolution of the number density of particles and antiparticles in the expanding universe (Iminniyaz et al. 2011).
The present-day dark matter density, Ω DM , depends on the relic densities of particles and antiparticles, Ω χ and Ωχ, which in turn depend on the mass of the particle, m χ , the χχ annihilation cross section, σ and the dark matter asymmetry η.
Primarily, the value of Ω DM relies on the properties of the χχ annihilation cross section. In most cases, the thermal averaging of the χχ annihilation cross section times the relative velocity of colliding particles v can be expanded as σv = a + bv 2 + O(v 4 ). If the annihilation initial state is unsuppressed, the first term of the previous equation dominates (a = 0). This process is known as the s-wave annihilation channel. Alternatively, if the initial state is suppressed, a = 0 and b = 0, this is the p-wave annihilation channel. This expansion is valid for all known examples of s-wave channel, p-wave channel or Fig. 1.-Asymmetric dark matter particle models: The figure shows iso-curves of the relic dark matter density Ω DM h 2 as a function of the asymmetry parameter η and the annihilation cross section σv : (a) s-wave annihilation channel (a = 0,b = 0) and (b) p-wave annihilation channel (a = 0,b = 0). The results are for dark matter particles with mχ = 10 GeV, gχ = 2 and g⋆ = 90 (see text). In the figure, indicated with blue lines are the η−asymmetric dark matter particle cosmological models compatible with the present day dark matter density. In the figure, indicated with blue lines are the asymmetric dark matter particle cosmological models compatible with the present day dark matter density. This cosmological model is computed for two types of annihilation channels, (a, η) or (b, η) (see text). We consider the observational window to be such that 0.10 ≤ Ω DM h 2 ≤ 0.12 (or −1.0 ≤ log (Ω DM h 2 ) ≤ −0.92). This corresponds to the region between the blue lines. The observed value of Ω DM h 2 is shown as a green line. The current measurements of the cosmological density and baryonic asymmetry are Ω DM h 2 = 0.1109 ± 0.0056 and η B = (0.88 ± 0.021)× 10 −10 , or log (Ω DM h 2 ) ∼ −0.9551 and log (η B ) ∼ −10.06 (Larson et al. 2011;Hütsi et al. 2011). both channels, up to an accuracy of a few percent. The expression σv can be simplified further, if the reacting particles are non-relativistic, σv = a + 6bx −1 + O(x −2 ) where x defines the ratio of m χ over the temperature of the Universe T (Iminniyaz et al. 2011).
In the case of symmetric dark matter particles (χ = χ), the relic abundance Ω χ is mainly determined by the annihilation rate σv . Furthermore, Ω χ is equal to the dark matter value Ω DM . Alternatively, in the case of asymmetric dark matter (χ = χ), χ andχ particle contributions have to be added such that Ω DM = Ω χ + Ωχ, as there are more particles than antiparticles (or the reverse). The antiparticles are annihilated away more efficiently, with large numbers of particles left behind without a partner to annihilate them. Consequently, the relic dark matter abundance is determined not only by the σv as in the symmetric dark matter case, but also by the asymmetry parameter η.
. With the objective of computing the evolution of the Sun in different cosmological scenarios, we start by determining the relic densities of particles and antiparticles, Ω χ and Ωχ for specific η−asymmetric dark matter models. The computation is done by following closely the numerical procedure of Iminniyaz et al. (2011). Figure 1 shows the dark matter density Ω DM for light asymmetric particles with m χ ∼ 10 GeV computed for several values of η and σv . Two types of χχ annihilation rates are considered, pure s-wave and a pure p-wave annihilation channels. Annihilation channels with non-vanishing a and b terms are qualitatively similar to the previous ones.
In general, such results show that there are two limiting asymmetric dark matter scenarios, one for very high and other for very low values of η: (i) for the high values of η, the dark matter is "strongly" asymmetric as the relic density is dominated by particles over antiparticles, Ωχ ≪ Ω χ ≈ Ω DM ; this scenario is identical to the baryogenesis process related to ordinary baryons. (ii) for the low values of η, the dark matter is "weakly" asymmetric or symmetric (η ≈ 0), Ωχ ≈ Ω χ ≈ Ω DM ; the final relic abundances of particles and antiparticles are comparable. This generally corresponds to the standard thermal WIMP scenario.
If we consider the present measurements of Ω DM (Hütsi et al. 2011), we find that the number of asymmetric dark matter scenarios that are compatible with observations is relatively reduced, i.e., only η − σv dark matter models with a relic density 0.1053 ≤ Ω DM h 2 ≤ 0.1165. Nevertheless, for reasons of convenience 8 , in this work we choose the observational interval to be such that 0.10 ≤ Ω DM h 2 ≤ 0.12 (region between blue lines in Figure 1).
If we restrict our analysis to the asymmetric models of particles with a mass of 10 GeV compatible with dark matter density observations, it is possible to determine a critical value of σv c for which there is a change of dark matter regime (see figure 1). The critical value σv c is approximately 1.6 × 10 −24 cm 3 s −1 and 1.2 × 10 −23 cm 3 s −1 for s-wave and p-wave annihilation channels. There is also a maximum value of asymmetry η c , above which the asymmetric dark matter models have a Ω DM larger than the current observational value. η c is equal to 3.6 × 10 −11 for the s-wave annihilation channel and 2 × 10 −11 for the p-wave annihilation channel. If a model has σv < σv c , Ω DM becomes uniquely dependent on η (independent of σv ). This corresponds to cosmological models for which the χχ annihilation rate is so efficient that the relic abundance depends only on the initial η asymmetry. Conversely, if a model has σv > σv c , the dark matter density becomes purely determined by the value of σv (independent of η). It follows that the dark matter density content is determined by the low value of the annihila-tion rate.
It is worth noticing that the dependence of Ω DM with the particle mass is quite small for asymmetric dark matter models of light particles (m χ ≤ 20 GeV). In cosmological dark matter models for particles with a mass between 5 GeV and 20 GeV, σv c for both channels is identical to the case of a particle with a mass of 10 GeV. For the same range of masses, η c varies between 8 − 2 × 10 −11 for a s-wave (a = 0) channel and 7 − 2 × 10 −11 for p-wave channel (b = 0).
3. CAPTURE OF η−PARAMETRISED ASYMMETRIC DARK MATTER BY THE SUN
The accumulation of particles (χ) and antiparticles (χ) inside the star depends on the star's gravitational field and the interaction of the dark matter particles with baryons.
In our description of the impact of the η−asymmetric dark matter on stellar evolution, we follow closely earlier discussions of the impact of "classical" asymmetric dark matter 9 with the evolution of the Sun and other stars Cumberbatch et al. 2010;Taoso et al. 2010;Casanellas & Lopes 2009;Lopes et al. 2011).
In the following we discuss the capture and accumulation of η−parametrised asymmetric dark matter by the Sun. Although, such particles interact with baryons in a similar way as WIMPs, the fact that we have two distinct populations of particles, leads to a quite different impact on the evolution of the star. Like WIMPs, the amount of η−asymmetric dark matter captured by the star depends explicitly on the mass of the dark matter particle m χ , the cross-section for scattering with baryons, namely, the spin-dependent scattering crosssection σ SD , and the spin-independent scattering crosssection σ SI (Lopes et al. 2011). However, unlike WIMPs, η−asymmetric dark matter, depends also on the two channels (p-wave and s-wave) for annihilation σv and the η−asymmetry parameter. The total number of particles N χ and antiparticles Nχ that accumulates inside the Sun at a certain epoch is computed by solving the system of coupled equations:
dN i dt = C i − C a N χ Nχ,(1)
with i being χ orχ. The constant C i gives the rate of capture of particles (antiparticles) from the dark matter halo, and C a gives the annihilation rate of particles and antiparticles in the Sun. This approach is entirely different from other studies of accumulation of dark matter inside stars, in part because previous work only studied the capture of the population N χ of dark matter particles. In such cases, the system of coupled equations (1) is substituted by one single equation. In the following, we will consider dark matter particles with a mass larger than 5 GeV , for which the evaporation of particles is negligible (Gould 1990). Furthermore, we neglect the rate of capture of dark matter particles by scattering off other dark matter particles that have already been captured within the Sun (Zentner 2009). The capture rate for particles and antiparticles by the Sun's gravitational field and their scatterings off baryons is computed numerically from the integral expression of Gould (1987), implemented as indicated in Gondolo et al. (2004). The description of how this capture process is implemented in our code is discussed in Lopes et al. (2011). The scattering of particles and antiparticles on baryon nuclei are identical. It follows that, similarly to the symmetric dark matter case, σ SD is only relevant for hydrogen nuclei; and σ SI is important for the scattering of dark matter particles on heavy nuclei. If the value of σ SI is larger or equal to σ SD , the capture of dark matter particles is dominated by the collisions with heavy nuclei, rather than by collisions with hydrogen (Lopes et al. 2011).
If not stated otherwise, we assume that the density of dark matter in the halo is 0.38 GeVcm −3 (Catena & Ullio 2010), the amount of particles and antiparticles in the halo is in the same proportion as in the local Universe, the stellar velocity of the Sun is 220 kms −1 and the Maxwellian velocity dispersion of dark matter particles is 270 kms −1 (e.g., Bertone et al. 2005). Recent measurements of local dark matter density yet to be published propose a value of 0.3 GeVcm −3 (Bovy & Tremaine 2012) and 0.85 GeVcm −3 (Garbari et al. 2012). The annihilation rate C a is computed from
C a = N −1 χ N −1 χ σv (r) n χ (r)nχ(r) 4π 2 r 2 dr,(2)
where σv (r) is the χχ annihilation rate as defined in the previous section, and n χ (r) and nχ(r) are the number density of particles and antiparticles. In this paper, in contrast to previous work, we assume that the dark matter particles and antiparticles present inside the Sun annihilate by the same physical process as the one that occurs in the early Universe, therefore, i.e., σv = a + 6bx −1 + O(x −2 ) where x is now the ratio of mass of the dark matter particle over the local temperature of the Sun's plasma. It follows that inside the Sun the s-wave (a = 0) annihilation channel is identical to the one present in the early Universe, but the p-wave (b = 0) annihilation channel is quite different, as the temperature inside the Sun is larger than the temperature in the early Universe.
As the star evolves, after some time, t is larger than τ A = C a C χ , the rate of accumulation of particles inside the SunṄ χ is proportional to the difference between the capture rate of the particles and antiparticles, i.e., N χ = D χ with D χ = C χ − Cχ. It follows that the total number of particles and antiparticles is given by N χ ≈ Nχ + D χ t and Nχ ≈ Cχ/(C a D χ t) (Griest & Seckel 1987). The value of τ A depends of the values of dark matter parameters. For a typical dark matter halo of density 0.38 GeVcm −3 constituted by particles with a mass of 10GeV and annihilation rate 10 −24 cm 3 s −1 , τ A of the order of 1 million year. Particles and antiparticles, once captured by the Sun during its evolution, end up in thermal equilibrium with baryons. The particle (and antiparticle) population follows a Maxwellian velocity distribution. The number density of particles (antiparticles), n i (r) ≈ N i e −mχφ(r)/Tc where T c in the central temperature of the star and φ(r) is the gravitational potential (Casanellas & Lopes 2009;Giraud-Heraud et al. 1990).
IMPACT OF η−PARAMETRISED ASYMMETRIC DARK
MATTER ON THE SUN'S EVOLUTION
The code used to compute the evolution of asymmetric dark matter is a modified version of the stellar evolution code CESAM (Morel 1997). The basic reference model without dark matter is calibrated to produce a solar standard model (Turck-Chieze & Lopes 1993) identical to others in the literature (Guzik & Mussack 2010;Serenelli et al. 2009;Bahcall et al. 2005b;Turck-Chieze et al. 2010). The microscopic physics (updated equation of state, opacities, nuclear reactions rates, and an accurate treatment of microscopic diffusion of heavy elements) are in full agreement with the standard picture. The solar mixture of Asplund et al. (2005) is used in the computation of models. The modified evolution models are calibrated to the present solar radius, luminosity, mass, and age (e.g. Turck-Chieze & Couvidat 2011). The models are required to have a fixed value of the photospheric ratio (Z/X) ⊙ , where X and Z are the mass fraction of hydrogen and the mass fraction of elements is heavier than helium. The value of (Z/X) ⊙ is determined according to the solar mixture Asplund et al. (2005).
The star evolves from the beginning of the pre-main sequence until its present age. Each solar model has more than 2000 layers, and it takes more than 80 time steps to arrive to the present age. For each set of dark matter parameters, a solar-like calibrated model is obtained by automatically adjusting the helium abundance and the convection mixing length parameter until the total luminosity and the solar radius are within 10 −5 of the present solar values. Typically a calibrated solar model is obtained after a sequence of 10 intermediate solar models, although the models with a large concentration of dark matter need more than 20 intermediate models. The increase of the number of iterations is related with the rapid variation of the structure in the center of the Sun. The values of the calibrating parameters, mixing-length parameter and the initial content of Helium changes slightly relatively to the standard solar model, depending upon the accumulation of dark matter in the core of the star. The maximum variation obtained for these parameters due to the accumulation of dark matter in the centre of the Sun is an increase of the mixing-length parameter of 5% and the initial content of Helium is decreased by 0.5%.
The η−asymmetric dark matter particles impact the evolution of the Sun, by a physical process identical to WIMPs. Likewise they provide an effective mechanism for the transport of energy, for which the efficiency depends locally on the average mean free path of the dark matter particles between successive collisions with baryons. Two distinct regimes of the transport of en- ergy are usually considered: local and non-local transport corresponding to small and large average mean free paths. Both energy transport mechanisms are implemented in our code (Lopes et al. 2011). This physical process leads to the reduction of the temperature gradient Lopes & Silk 2010b). In the more extreme case, the high frequency of collisions between baryons and dark matter particles forms an isothermal core .
This new type of asymmetric dark matter can have a much larger effect on the evolution of the Sun than the usual symmetric dark matter (Lopes & Silk 2010b), because the star can accumulate a much larger amount of particles (and antiparticles) in its core than in the latter case. This is due to the fact that asymmetric dark matter depends, among other parameters, on two major parameters that determine the concentration of dark matter inside the star: dark matter asymmetry, which determines the unbalanced amount of particles and antiparticles; and the annihilation cross-section, which establishes the annihilation efficiency of particles and antiparticles. This is consistent with the results shown in Figure 1. Figure 2 shows the variation for the 8 B and 7 Be neutrino fluxes relative to the solar standard models. The large reduction in 8 B and 7 Be neutrino fluxes is a direct consequence of the reduction of the temperature inside the solar core.
This diminution of temperature is also visible in the sound speed profile near the centre of the Sun. Figure 3 shows the profile of sound speed computed for a few η−asymmetric dark matter scenarios. It also shows the observed sound speed profile obtain from the helioseismology data of the BISON and GONG observational networks ). This sound speed is consistent with a previous sound speed inversion computed from high accuracy data obtained by the GOLF and MDI instruments of the SOHO mission (Turck-Chieze et al. 1997). In the solar core, the impact of η−parametrised dark matter on the sound speed is much larger than can be accommodated by the current solar standard model (Turck-Chieze & Couvidat 2011). Moreover, although the sound speed is not inverted in the very central region of the Sun ), the reduction in the sun's speed profile caused by the presence of dark matter is much larger than previously tentative sound speed inversions of the Sun's core (Turck-Chieze & Couvidat 2011). As shown in figure 3, these dark matter models produce a decrease in the square of the sound speed of the order of 3%-5%, well above 1% of the uncertainty of the current solar model. This reduction in temperature is accompanied by significant changes in the Sun's core structure, which leads to visible effects on other type of seismic diagnostics, such as the small acoustic mode separation (Lopes & Turck-Chieze 1994;Turck-Chieze & Lopes 1993). Cumberbatch et al. (2010) have found that dark matter particles that accumulate inside the Sun's core and produces a temperature variation of this order of magnitude, also have a small acoustic mode separation quite different from helioseismology data.
The relationship between the solar neutrino fluxes and the sound speed with the temperature is easily established, if we notice that the 8 B and 7 Be neutrino fluxes depend on the central temperature T c as T 24 c and T 10 c (Bahcall & Ulmer 1996;Bahcall 2002), and the square of sound speed can be expressed as C 2 s ≈ T c /µ c where µ c molecular weight in the centre (Turck-Chieze & Lopes 1993). This different sensitivity to temperature explains why the diagnostics provided by Φ( 8 B), Φ( 7 Be) and C 2 s are quite distinct. We have also computed the solar neutrino fluxes of other pp-nuclear reactions and the results follow a similar behaviour. Other seismic diagnostic are possible, such as comparing the observed acoustic oscillations to the theoretical acoustic oscillations, or their respective small separations. Nevertheless, the sensitivity of seismic quantities to changes in the Sun's core struc-ture is always less visible than in solar neutrino fluxes.
DISCUSSION
The current standard solar model (Turck-Chieze & Lopes 1993;Turck-Chieze et al. 2010;Serenelli et al. 2011) is in agreement with the Φ( 8 B) and Φ( 7 Be) neutrino fluxes measured by SNO and Borexino detector (Aharmim et al. 2010;Bellini et al. 2010Bellini et al. , 2011Bellini et al. , 2012Arpesella et al. 2008). The current 8 B neutrino flux observational determination in the case of no-neutrino oscillations (or electron neutrinos; including the theoretical uncertainty of neutrino flux solar neutrinos in the solar standard model) is Φ( 8 B) = 5.05 +0.19 −0.20 × 10 6 cm −2 s −1 for the SNO experiment (Aharmim et al. 2010) and Φ( 8 B) = 5.88 ± 0.65 × 10 6 cm −2 s −1 for the Borexino experiment (Bellini et al. 2010). The Borexino experiment measures 7 Be solar neutrino flux Φ( 7 Be) = 4.87 ± 0.24 × 10 9 cm −2 s −1 , under the assumption of the MSW-LMA scenario of solar neutrino oscillations (Bellini et al. 2011;Arpesella et al. 2008).
The Φ( 8 B) neutrino measurements made by Borexino and SNO experiments suggest that the Sun's core is slightly hotter than expected, since the Φ( 8 B) measured value is higher than the value predicted by the current solar standard model. This discrepancy is validated by the Borexino measurement of Φ( 7 Be) which is sensitive to a region slightly off the Sun's centre. The first measurement of the pep neutrinos was done recently by the Borexino experiment, Φ(pep) = 1.6 ± 0.3 × 10 8 cm −2 s −1 (Bellini et al. 2011). The experimental value also suggests that the Sun is hotter than the solar standard model predicts. This diagnostic is quite reliable once we realise that Φ(pep) is strongly dependent on the luminosity of the star. All these independent solar neutrino experiment results suggest that the solar standard model is cooler than the actual Sun.
The internal structure of the Sun is well known by means of solar neutrinos and helioseismology data. Therefore, the theoretical uncertainty of the standard solar model is consistently taken into account by comparing the predictions of these two probes with observations. Several authors have shown that an important source of uncertainty on the calculation of the solar neutrino fluxes (electronic flavour) comes from the unclear measurements of heavy element abundances in the Sun's surface (Bahcall et al. 2005b,a;Asplund et al. 2009). One other source of uncertainty is related with the determination of several pp-reaction rates and electron screening (Mussack & Dappen 2011;Serenelli et al. 2011;Weiss et al. 2001). Similarly, other standard and no-standard physical processes that contribute for the evolution of the star can also be an important source of uncertainty. Among others we can refer to the following indirect processes: convective overshoot, low-z accretion, mass loss, solar rotation and meridional circulation (Guzik & Mussack 2010;Turck-Chieze et al. 2010;Turck-Chieze & Couvidat 2011). These suggestions have been made mostly to resolve the discrepancy between the theoretical sound speed as obtained with the new solar abundances (Asplund et al. , 2005 and the sound speed obtained from the acoustic oscillations. As previously mentioned such physical processes have a much smaller effect on solar neutrino fluxes than the accumu-lation of dark matter in the Sun's core.
In the study of the impact of dark matter in the Sun, we will choose to consider an interval of theoretical uncertainty that takes into account such processes. Therefore, considering both the theoretical and experimental uncertainties, we choose to rule out models that predict a 8 B neutrino flux which deviates more than 30% from our solar standard model. Similarly, a 7 Be neutrino flux of solar models which deviates more than 15% from our solar standard model can also be ruled out. This is consistent with the fact that Φ( 8 B) is two times more sensitive to the central temperature than Φ( 7 Be) . The choice of this threshold is in agreement with other authors (e.g., Taoso et al. 2010). Furthermore, such intervals of uncertainties on 8 B and 7 Be also include the solar structure variations (e.g. temperature) in the Sun's interior, and in particular in the Sun's core that can be attributed to some of the physical processes previously mentioned. Figure 2 shows the neutrino flux variations in more than 30 evolution models of the Sun computed for different values of η and σv . This result is identical for both (s-wave and p-wave) annihilation channels. In the η vs. σv plane, the iso-contours corresponding to Φ ν ( 8 B) and Φ ν ( 7 Be) show neutrino flux variations relatively to the solar standard model. We choose to present the case of dark matter particles that have m χ = 10 GeV, spin-dependent scattering cross-section σ SD = 10 −40 cm 2 , and spin-independent scattering cross-section σ SI = 10 −36 cm 2 . The neutrino flux variation shown follows the decrease of the local halo density of particles ρ χ , which follows the relic density Ω χ specific to each set of values (η, σv ). The observed difference in sensitivity between Φ ν ( 8 B) and Φ ν ( 7 Be) neutrino fluxes, is caused by the fact that 8 B neutrinos are produced in a more central region than 7 Be neutrinos.
If we adopt the 30% and 15% fixed thresholds for the 8 B and 7 Be neutrino fluxes, we conclude that we can rule out η−asymmetric dark matter particles with a mass of 10GeV, that have an η ≥ 10 −12 and an annihilation rate (both annihilation channels) larger than 10 −23 cm 3 s −1 . An analysis of other solar neutrino fluxes such as pp neutrinos re-enforces the conclusions reached in this study.
CONCLUSIONS
In this paper, we proposed a new strategy to study the impact of dark matter in the evolution of Sun and stars. We started by computing the basic cosmological primitive model that is responsible by the formation of the dark matter asymmetric particles, assuming that dark matter like baryons has an asymmetry, analogous to the baryon asymmetry. As a consequence the population of dark matter particles and antiparticles in the current Universe is fixed by the measured dark matter density Ω DM h 2 . The amount of particles and antiparticles depends on the dark matter asymmetry parameter η, as well as on the dark matter annihilation (p-and s-wave) channels. Considering that the Sun is formed in a dark matter halo that has the same amount of particles and anti-particles, we computed the evolution of the Sun in such conditions.
Using the solar standard model as a reference model for the internal structure of the Sun, we have studied the impact of asymmetric dark matter on the production of solar neutrino fluxes. Following a procedure identical to the analysis proposed in Lopes & Silk (2010a,b), we describe the impact of asymmetric dark matter on the Sun's core.
Several evolution models of the Sun were computed for a range of light dark matter particles. Particles with masses smaller than 5 GeV are not considered since evaporation becomes important in this mass range and a large number of dark matter particles escape the gravitational field of the star, significantly reducing the impact on the Sun's core (Gould 1990). Similarly, particles with a mass above 20 GeV produce a very small dark matter core and their effect in the Sun's structure is almost negligible (Lopes et al. 2011). Accordingly, dark matter particles with masses between 5 and 20 GeV, produce a temperature decrease in the Sun's core that visibly affects the neutrino fluxes, although these depend on the values of specific parameters, such as the dark matter asymmetry and annihilation cross section. The impact of more massive dark particles in the evolution of a star like the Sun should be significantly smaller than in the case discussed in this work. The reason is the fact that the concentration of dark matter in the centre of the star, is usually characterized by a dark matter core radius (Casanellas & Lopes 2009) which is inversely proportional to the square root of the mass of the dark matter particle. The choice of other parameters of the scattering cross section will not change fundamentally such results.
We note incidentally that high annihilation rates for light dark matter particles are ruled out if accompanied by lepton production from observations of Cosmic Microwave Background temperature fluctuations and of searches for gamma rays from nearby dwarfs. But these limits are irrelevant once asymmetric dark matter is important, and most importantly for us, assume specific annihilation channels. Our solar constraints come from accumulation of dark matter particles by the Sun, and complement other annihilation limits. The annihilations are of course important for fixing the relic abundance. However a key point of our paper is that we consider asymmetric dark matter for which the dependence on specific annihilation channels is very weak. Furthermore, the relic abundance is fixed by specifying the annihilation rate via freeze-out physics.
Indeed, freeze-out may be far more complicated than given by the simple connection that specifies the "thermal" freeze-out cross-section in terms of the observed relic abundance. Many models violate this condition. Our models test independently of freeze-out the very important parameter space of annihilation rate today (in the low energy universe) whereas relic abundances probe the annihilation rate in a model-dependent way at freezeout (at high energies). Moreover the Sun probes asymmetric dark matter at the present solar radius, and complements the use of dwarfs in the halo and of Cosmic Microwave Background in the early universe for testing Majorana-type dark matter.
In this work we have computed the 8 B and 7 Be neutrino flux variations for several η−parametrised asymmetric dark matter particles, corresponding to different values of η and σv . The values of η and σv were chosen in such a way that Ω DM for each of the asymmetric dark matter scenarios considered is consistent with current observational measurements (cf. Figure 1). We have presented the results of light dark matter particles (typically 10 GeV), with the same characteristics as the candidates "observed" by DAMA/LIBRA and CoGeNT experiments. We find that such type of particles produces large variations on the flux of solar neutrinos. The 8 B neutrino flux variation takes values between 35% and 65% for s-wave and p-wave annihilation channels, clearly above the uncertainty in the current solar models estimated to be of the order of 15% (Turck-Chieze & Couvidat 2011). This conclusion holds even for our more conservative threshold of 30% on 8 B neutrino flux or a threshold of 15% on 7 Be neutrino flux, as discussed in the previous section. Likewise, dark matter particles with m χ = 15 GeV show 8 B neutrino flux variations of the order of 16% up to 40%, which can be rejected if we consider the 15% threshold of uncertainty in the physics of the solar standard model. Therefore, it is reasonable to consider that light dark matter particles that have a scattering cross section of the order of a pico barn, as suggested by several theoretical models to explain the DAMA/LIBRA and CoGeNT experiments, produce neutrino fluxes that are in disagreement with the current neutrino flux observations. On this basis, assuming that stellar evolution is affected by dark matter as discussed here, it is reasonable to assume that such types of particles cannot exist in the current Universe. Forthcoming solar neutrino flux experiments could restrain even further the parameters of the η−asymmetric dark matter candidates proposed by the recent dark matter models.
Fig. 2 .
2-Asymmetric dark matter model impact on the solar neutrino fluxes: The asymmetric dark matter particles have a mass of 10 GeV and a scattering cross section of baryons such that σ SD = 10 −40 cm 2 and σ SI = 10 −36 cm 2 . The asymmetric dark matter cosmological models have a value of σv and η for which the cosmological models have Ω DM h 2 value consistent with the observational window, i.e., 0.10 ≤ Ω DM h 2 ≤ 0.12 or −1.0 ≤ log (Ω DM h 2 ) ≤ −0.92(08). This corresponds to asymmetric dark matter models for which Ω DM h 2 is in between the blue linesFigure 1(see text).
Comparison of the sound speed radial profile between the solar standard model and different solar models evolved within η−asymmetric dark matter halos: The red dotted curve corresponds to the difference between inverted profiles and our solar standard modelTurck-Chieze et al. 1997). The dark matter particles have a mass of 10 GeV and a scattering cross section o baryons such that σ SD = 10 −40 cm 2 and σ SI = 10 −36 cm 2 . The annihilation dark matter particles in the early Universe and inside the Sun have an annihilation σv such σv = a + bv 2 (see main text for details). The η−asymmetric dark matter particles have the following properties: (i) Left figure, s-wave annihilation channel a = 0, b = 0, (a) ρχ = 0.21 GeVcm −3 and a = 1.72 10 −24 cm 3 s −1 (red curve); (b) ρχ = 0.34 GeVcm −3 and a = 2.25 10 −24 cm 3 s −1 (blue curve) (c) ρχ = 0.38 GeVcm −3 and a = 4.61 10 −24 cm 3 s −1 (green curve) (ii) Right figure, p-wave annihilation channel a = 0, b = 0, (a) ρχ = 0.19 GeVcm −3 and b = 1.11 10 −23 cm 3 s −1 (red curve); (b) ρχ = 0.22 GeVcm −3 and b = 1.11 10 −23 cm 3 s −1 (blue curve) (c) ρχ = 0.38 GeVcm −3 and b = 4.61 10 −22 cm 3 s −1 (green curve) The dark matter halo has a total dark matter density, ρ DM = 0.38 GeVcm −3 , such that ρ DM = ρχ + ρχ The density of particles and antiparticles, ρχ and ρχ are proportional to Ωχ and Ωχ.
The recent measurements of Ω DM h 2 are such that 0.1053 ≤ Ω DM h 2 ≤ 0.1165 or −0.9776 ≤ log (Ω DM h 2 ) ≤ −0.9337, nevertheless for clarity of argument we choose a slightly larger interval of Ω DM h 2 which includes the observational window.
In the literature, we find another description of asymmetric dark matter,that is distinct of the type of dark matter discussed in this work. Although both types of asymmetric dark matter accumulate by identical processes inside the star, their effects in the evolution of stars is rather different. We decided to call η−asymmetric dark matter to highlight the difference
This work was supported by grants from "Fundação para a Ciência e Tecnologia" and "Fundação Calouste Gulbenkian" and the NSF (grant OIA-1124453). We would like to thank the referee for the useful comments and insightful suggestions that improved the quality of the paper.
. Aalseth, Phys. Rev. Lett. 106131301Aalseth et al., 2011, Phys. Rev. Lett., 106, 131301
. Aharmim, Phys. Rev. C. 8155504Aharmim et al., 2010, Phys. Rev. C, 81, 55504
. Ahmed , Phys. Rev. Lett. 106131302Ahmed et al. 2011, Phys. Rev. Lett., 106, 131302
. Angle , Phys. Rev. Lett. 10751301Angle et al. 2011, Phys. Rev. Lett., 107, 51301
. Aprile, Phys. Rev. Lett. 107131302Aprile et al. 2011, Phys. Rev. Lett., 107, 131302
. C Arpesella, G Bellini, J Benziger, S Bonetti, B Caccianiga, F Calaprice, F Dalnoki-Veress, D D'angelo, A Etenko, H De Kerret, Physics Letters B. 658101Arpesella, C., Bellini, G., Benziger, J., Bonetti, S., Caccianiga, B., Calaprice, F., Dalnoki-Veress, F., D'Angelo, D., Etenko, A., & de Kerret, H. 2008, Physics Letters B, 658, 101
M Asplund, N Grevesse, A J Sauval, Magnetic Fields Across the Hertzsprung-Russell Diagram. 33625Asplund, M., Grevesse, N., & Sauval, A. J. 2005, Magnetic Fields Across the Hertzsprung-Russell Diagram, 336, 25
. M Asplund, N Grevesse, A J Sauval, P Scott, Annual Review of Astronomy and Astrophysics. 47481Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, Annual Review of Astronomy and Astrophysics, 47, 481
. J Bahcall, S Basu, M Pinsonneault, A M Serenelli, ApJ. 6181049Bahcall, J., Basu, S., Pinsonneault, M., & Serenelli, A. M. 2005a, ApJ, 618, 1049
. J N Bahcall, Phys. Rev. C. 6525801Bahcall, J. N. 2002, Phys. Rev. C, 65, 25801
. J N Bahcall, A M Serenelli, S Basu, ApJ. 62185Bahcall, J. N., Serenelli, A. M., & Basu, S. 2005b, ApJ, 621, L85
. J N Bahcall, A Ulmer, Physical Review D (Particles. 534202Bahcall, J. N. & Ulmer, A. 1996, Physical Review D (Particles, 53, 4202
. S Basu, W J Chaplin, Y Elsworth, R New, A M Serenelli, ApJ. 6991403Basu, S., Chaplin, W. J., Elsworth, Y., New, R., & Serenelli, A. M. 2009, ApJ, 699, 1403
. Bellini, Phys. Rev. Lett. 10851302Bellini,et al. 2012, Phys. Rev. Lett., 108, 51302
. Bellini, Phys. Rev. Lett. 107141302Bellini et al. 2011, Phys. Rev. Lett., 107, 141302
. Bellini, Phys. Rev. D. 8233006Bellini, et al. 2010, Phys. Rev. D, 82, 33006
. R Bernabei, P Belli, F Cappella, R Cerulli, C Dai, A Angelo, H He, A Incicchitti, H Kuang, J Ma, The European Physical Journal C-Particles and Fields. 56333Bernabei, R., Belli, P., Cappella, F., Cerulli, R., Dai, C., d'Angelo, A., He, H., Incicchitti, A., Kuang, H., & Ma, J. 2008, The European Physical Journal C-Particles and Fields, 56, 333
. G Bertone, M Fairbairn, Phys. Rev. D. 7743515Bertone, G. & Fairbairn, M. 2008, Phys. Rev. D, 77, 43515
. G Bertone, D Hooper, J Silk, Physics Reports. 405279Bertone, G., Hooper, D., & Silk, J. 2005, Physics Reports, 405, 279
. Jo Bovy, Scott Tremaine, arXiv:1205.4033BAstrophysical Journal. in pressBovy, Jo and Tremaine, Scott 2012, Astrophysical Journal, in press, eprint arXiv:1205.4033B
. A Brown, S Henry, H Kraus, C Mccabe, Phys. Rev. D. 8521301Brown, A., Henry, S., Kraus, H., & McCabe, C. 2012, Phys. Rev. D, 85, 21301
. J Casanellas, I Lopes, The Astrophysical Journal Letters. 73351ApJCasanellas, J. & Lopes, I. 2009, ApJ, 705, 135 -. 2011a, The Astrophysical Journal Letters, 733, L51
. Monthly Notices of the Royal Astronomical Society. 410535-. 2011b, Monthly Notices of the Royal Astronomical Society, 410, 535
. J Casanellas, P Pani, I Lopes, V Cardoso, ApJ. 74515Casanellas, J., Pani, P., Lopes, I., & Cardoso, V. 2012, ApJ, 745, 15
. S Chang, J Liu, A Pierce, N Weiner, I Yavin, Journal of Cosmology and Astroparticle Physics. 0818Chang, S., Liu, J., Pierce, A., Weiner, N., & Yavin, I. 2010, Journal of Cosmology and Astroparticle Physics, 08, 018
. R Catena, P Ullio, Journal of Cosmology and Astroparticle Physics. 084Catena, R. & Ullio, P. 2010, Journal of Cosmology and Astroparticle Physics, 08, 004
. T Cohen, D J Phalen, A Pierce, K M Zurek, Phys. Rev. D. 8256001Cohen, T., Phalen, D. J., Pierce, A., & Zurek, K. M. 2010, Phys. Rev. D, 82, 56001
. D T Cumberbatch, J A Guzik, J Silk, L S Watson, S M West, Phys. Rev. D. 82103503Cumberbatch, D. T., Guzik, J. A., Silk, J., Watson, L. S., & West, S. M. 2010, Phys. Rev. D, 82, 103503
. E Del Nobile, C Kouvaris, F Sannino, Phys. Rev. D. 8427301Del Nobile, E., Kouvaris, C., & Sannino, F. 2011, Phys. Rev. D, 84, 27301
. J B Dent, S Dutta, R J Scherrer, Physics Letters B. 687275Dent, J. B., Dutta, S., & Scherrer, R. J. 2010, Physics Letters B, 687, 275
. M Drees, H Iminniyaz, M Kakizaki, Phys. Rev. D. 73123502Drees, M., Iminniyaz, H., & Kakizaki, M. 2006, Phys. Rev. D, 73, 123502
. M Farina, D Pappadopulo, A Strumia, T Volansky, Journal of Cosmology and Astroparticle Physics. 1110Farina, M., Pappadopulo, D., Strumia, A., & Volansky, T. 2011, Journal of Cosmology and Astroparticle Physics, 11, 010
. G R Farrar, G Zaharijas, Phys. Rev. Lett. 9641302Farrar, G. R. & Zaharijas, G. 2006, Phys. Rev. Lett., 96, 41302
. R Foadi, M T Frandsen, F Sannino, Phys. Rev. D. 8037702Foadi, R., Frandsen, M. T., & Sannino, F. 2009, Phys. Rev. D, 80, 37702
. S Garbari, C Liu, J I Read, G Lake, 10.1111/j.1365-2966.2012.21608.xarXiv:1206.0015Monthly Notices of the Royal Astronomical Society. accepted for publicationGarbari, S. and Liu, C., and Read, J. I. and Lake, G. Monthly Notices of the Royal Astronomical Society, accepted for publication, doi: 10.1111/j.1365-2966.2012.21608.x [arXiv:1206.0015]
. Y Giraud-Heraud, J Kaplan, F Martin, C Tao, S Turck-Chieze, Solar Physics. 12821Giraud-Heraud, Y., Kaplan, J., Martin, F., Tao, C., & Turck- Chieze, S. 1990, Solar Physics, 128, 21
. P Gondolo, J Edsjö, P Ullio, L Bergstrom, M Schelke, E A Baltz, Journal of Cosmology and Astroparticle Physics. 8Gondolo, P., Edsjö, J., Ullio, P., Bergstrom, L., Schelke, M., & Baltz, E. A. 2004, Journal of Cosmology and Astroparticle Physics, 07, 008
. A Gould, Astrophysical Journal. 321302Astrophysical JournalGould, A. 1987, Astrophysical Journal, 321, 571 -. 1990, Astrophysical Journal, 356, 302
. K Griest, D Seckel, Nuclear Physics B. 283681Griest, K. & Seckel, D. 1987, Nuclear Physics B, 283, 681
. S B Gudnason, C Kouvaris, F Sannino, Phys. Rev. D. 73115003Gudnason, S. B., Kouvaris, C., & Sannino, F. 2006, Phys. Rev. D, 73, 115003
. J A Guzik, K Mussack, ApJ. 7131108Guzik, J. A. & Mussack, K. 2010, ApJ, 713, 1108
. D Hooper, C Kelso, Phys. Rev. D. 8483001Hooper, D. & Kelso, C. 2011, Phys. Rev. D, 84, 83001
. D Hooper, J March-Russell, S M West, Physics Letters B. 605228Hooper, D., March-Russell, J., & West, S. M. 2005, Physics Letters B, 605, 228
. G Hütsi, J Chluba, A Hektor, M Raidal, Astronomy and Astrophysics. 53526Hütsi, G., Chluba, J., Hektor, A., & Raidal, M. 2011, Astronomy and Astrophysics, 535, 26
. H Iminniyaz, M Drees, X Chen, Journal of Cosmology and Astroparticle Physics. 073Iminniyaz, H., Drees, M., & Chen, X. 2011, Journal of Cosmology and Astroparticle Physics, 07, 003
. Z Kang, J Li, T Li, T Liu, J Yang, arXiv:2011arXiv1102.5644KKang, Z., Li, J., Li, T., Liu, T., & Yang, J. 2011, arXiv:2011arXiv1102.5644K.
. D Kaplan, Phys. Rev. Lett. 68741Kaplan, D. 2002, Phys. Rev. Lett., 68, 741
. D E Kaplan, M A Luty, K M Zurek, Phys. Rev. D. 79115016Kaplan, D. E., Luty, M. A., & Zurek, K. M. 2009, Phys. Rev. D, 79, 115016
. C Kouvaris, arXiv:1111.4364Kouvaris, C. 2011, eprint arXiv:1111.4364
. C Kouvaris, P Tinyakov, Phys. Rev. Lett. 8291301Phys. Rev. DKouvaris, C. & Tinyakov, P. 2010, Phys. Rev. D, 82, 63531 -. 2011, Phys. Rev. Lett., 107, 91301
. D Larson, J Dunkley, G Hinshaw, E Komatsu, M R Nolta, C L Bennett, B Gold, M Halpern, R S Hill, N Jarosik, A Kogut, M Limon, S S Meyer, N Odegard, L Page, K M Smith, D N Spergel, G S Tucker, J L Weiland, E Wollack, E L Wright, The Astrophysical Journal Supplement Series. 19216Larson, D., Dunkley, J., Hinshaw, G., Komatsu, E., Nolta, M. R., Bennett, C. L., Gold, B., Halpern, M., Hill, R. S., Jarosik, N., Kogut, A., Limon, M., Meyer, S. S., Odegard, N., Page, L., Smith, K. M., Spergel, D. N., Tucker, G. S., Weiland, J. L., Wollack, E., & Wright, E. L. 2011, The Astrophysical Journal Supplement Series, 192, 16
. I Lopes, J Casanellas, D Eugénio, Phys. Rev. D. 8363521Lopes, I., Casanellas, J., & Eugénio, D. 2011, Phys. Rev. D, 83, 63521
. I Lopes, J Silk, The Astrophysical Journal Letters. 33095ScienceLopes, I. & Silk, J. 2010a, Science, 330, 462 -. 2010b, The Astrophysical Journal Letters, 722, L95
. I Lopes, S Turck-Chieze, Astronomy and Astrophysics. 290845Lopes, I. & Turck-Chieze, S. 1994, Astronomy and Astrophysics, 290, 845
. I P Lopes, J Silk, Phys. Rev. Lett. 88151303Lopes, I. P. & Silk, J. 2002, Phys. Rev. Lett., 88, 151303
. I P Lopes, J Silk, S H Hansen, Monthly Notices of the Royal Astronomical Society. 331361Lopes, I. P., Silk, J., & Hansen, S. H. 2002, Monthly Notices of the Royal Astronomical Society, 331, 361
. S Nussinov, Physics Letters B. 16555Nussinov, S. 1985, Physics Letters B, 165, 55
A & A Supplement series. P Morel, 124597Morel, P. 1997, A & A Supplement series, 124, 597
. D Munshi, A Heavens, A Cooray, P Valageas, Monthly Notices of the Royal Astronomical Society. 4143173Munshi, D., Heavens, A., Cooray, A., & Valageas, P. 2011, Monthly Notices of the Royal Astronomical Society, 414, 3173
. K Mussack, W Dappen, ApJ. 72996Mussack, K. & Dappen, W. 2011, ApJ, 729, 96
. P Scott, M Fairbairn, J Edsjö, Monthly Notices of the Royal Astronomical Society. 39482Scott, P., Fairbairn, M., & Edsjö, J. 2009, Monthly Notices of the Royal Astronomical Society, 394, 82
. A M Serenelli, S Basu, J W Ferguson, M Asplund, The Astrophysical Journal Letters. 705123Serenelli, A. M., Basu, S., Ferguson, J. W., & Asplund, M. 2009, The Astrophysical Journal Letters, 705, L123
. A M Serenelli, W C Haxton, C Pena-Garay, ApJ. 74324Serenelli, A. M., Haxton, W. C., & Pena-Garay, C. 2011, ApJ, 743, 24
. M Taoso, F Iocco, G Meynet, G Bertone, P Eggenberger, Phys. Rev. D. 8283509Taoso, M., Iocco, F., Meynet, G., Bertone, G., & Eggenberger, P. 2010, Phys. Rev. D, 82, 83509
S Turck-Chieze, S Basu, A S Brun, J Christensen-Dalsgaard, A Eff-Darwich, I Lopes, F Perez Hernandez, G Berthomieu, J Provost, R K Ulrich, F Baudin, P Boumier, J Charra, A H Gabriel, R A Garcia, G Grec, C Renaud, J M Robillot, T Roca-Cortes, Solar Physics. 175247Turck-Chieze, S., Basu, S., Brun, A. S., Christensen-Dalsgaard, J., Eff-Darwich, A., Lopes, I., Perez Hernandez, F., Berthomieu, G., Provost, J., Ulrich, R. K., Baudin, F., Boumier, P., Charra, J., Gabriel, A. H., Garcia, R. A., Grec, G., Renaud, C., Robillot, J. M., & Roca-Cortes, T. 1997, Solar Physics, 175, 247
. S Turck-Chieze, S Couvidat, Rep. Prog. Phys. 746901Turck-Chieze, S. & Couvidat, S. 2011, Rep. Prog. Phys., 74, 6901
. S Turck-Chieze, R A Garcia, I Lopes, J Ballot, S Couvidat, S Mathur, D Salabert, J Silk, The Astrophysical Journal Letters. 74612Turck-Chieze, S., Garcia, R. A., Lopes, I., Ballot, J., Couvidat, S., Mathur, S., Salabert, D., & Silk, J. 2012, The Astrophysical Journal Letters, 746, L12
. S Turck-Chieze, I Lopes, Astrophysical Journal. 408347Turck-Chieze, S. & Lopes, I. 1993, Astrophysical Journal, 408, 347
. S Turck-Chieze, A Palacios, J P Marques, P A P Nghiem, ApJ. 7151539Turck-Chieze, S., Palacios, A., Marques, J. P., & Nghiem, P. A. P. 2010, ApJ, 715, 1539
. A Weiss, M Flaskamp, V N Tsytovich, Astronomy and Astrophysics. 3711123Weiss, A., Flaskamp, M., & Tsytovich, V. N. 2001, Astronomy and Astrophysics, 371, 1123
. A R Zentner, Phys. Rev. D. 8063501Zentner, A. R. 2009, Phys. Rev. D, 80, 63501
| [] |
[
"Structure of the Bogoliubov-Valatin canonical basis set",
"Structure of the Bogoliubov-Valatin canonical basis set"
] | [
"Aurel Bulgac \nDepartment of Physics\nUniversity of Washington\n98195-1560Seattle, WashingtonUSA\n",
"Matthew Kafker \nDepartment of Physics\nUniversity of Washington\n98195-1560Seattle, WashingtonUSA\n"
] | [
"Department of Physics\nUniversity of Washington\n98195-1560Seattle, WashingtonUSA",
"Department of Physics\nUniversity of Washington\n98195-1560Seattle, WashingtonUSA"
] | [] | We discuss the mathematical properties of the Bogoliubov-Valatin basis set of quasiparticle wave functions for a fermion system, with particular emphasis on the properties of the canonical basis set. The properties of the canonical basis set, apart from their definition, are largely unknown. In particular, what is the physically required size of the canonical wave function set in order to correctly describe superfluid systems. While the cardinality of the set of quasiparticle wave functions for an isolated system in vacuum is c = |R 3 |, the basis set for a finite system in a finite volume is countable, with cardinality ℵ0 = |Z| = |N|. However, the size of the canonical basis set for an isolated system in a finite volume or for a periodic system is typically much smaller than the size of the entire basis set, and it is determined by the level of the spatial resolution. We show how one can get insight into the character of the canonical wave functions and we justify the minimum number of canonical wave functions needed for a given system. | null | [
"https://arxiv.org/pdf/2203.04843v1.pdf"
] | 247,319,043 | 2203.04843 | 4a2fdc8df47bfd2c838d85bedac0fa1e006e5c5f |
Structure of the Bogoliubov-Valatin canonical basis set
9 Mar 2022
Aurel Bulgac
Department of Physics
University of Washington
98195-1560Seattle, WashingtonUSA
Matthew Kafker
Department of Physics
University of Washington
98195-1560Seattle, WashingtonUSA
Structure of the Bogoliubov-Valatin canonical basis set
9 Mar 2022(Dated: March 10, 2022)
We discuss the mathematical properties of the Bogoliubov-Valatin basis set of quasiparticle wave functions for a fermion system, with particular emphasis on the properties of the canonical basis set. The properties of the canonical basis set, apart from their definition, are largely unknown. In particular, what is the physically required size of the canonical wave function set in order to correctly describe superfluid systems. While the cardinality of the set of quasiparticle wave functions for an isolated system in vacuum is c = |R 3 |, the basis set for a finite system in a finite volume is countable, with cardinality ℵ0 = |Z| = |N|. However, the size of the canonical basis set for an isolated system in a finite volume or for a periodic system is typically much smaller than the size of the entire basis set, and it is determined by the level of the spatial resolution. We show how one can get insight into the character of the canonical wave functions and we justify the minimum number of canonical wave functions needed for a given system.
The mathematical framework for describing superfluid fermionic systems was formulated in terms of quasiparticles by Bogoliubov [1] and Valatin [2]. Zumino [3] and Bloch and Messiah [4] have shown that one can introduce a particular set of quasiparticles, with similar properties to the set used by Bardeen et al. [5] (BCS), the canonical set of states. The ground state of a superfluid or superconducting system has a particularly simple form expressed in terms of this set of single-particle wave functions, well suited for calculations. The microscopic calculations however suffered for a long time from a very annoying divergence of the order parameter, which was handled typically in an ad hoc manner, using physical arguments, devoid however of accurate estimates of what might have been missing. The presence of this divergence prevented Oliveira et al. [6] to extend the Density Functional Theory (DFT) [7] from normal to superconductor systems, in a manner similar to the Kohn-Sham DFT formulation [8], with local pairing fields. The nature of the divergence was clarified in 1980 [9], which later lead to an extension of the DFT framework in the Kohn-Sham spirit to both finite and infinite superfluid systems [10][11][12][13]. In particular, for decades in nuclear physics the pairing correlations were treated with arbitrary cutoffs on the number and the character of the single-particle orbitals in modeling pairing phenomena. The typical argument used in nuclear calculations of superfluid nuclei was that the energy of the ground state converged quite rapidly as a function of the chosen cutoff. Anderson [14], in discussing the treatment of electronic systems, characterized this kind of situation as the "Quantum Chemists' Fallacy No. 1 and 2," of which even Wigner was partially guilty, as "you may get pretty good energetics out of a qualitatively wrong state." The perfect example is the case of a superconductor, in which in spite of the fact that the contribution from the condensation energy * [email protected] † [email protected] is negligible, the wave function with pairing correlations leads to qualitative changes, which otherwise in a typical chemical physicist approach would have been completely overlooked. We show here how using the canonical basis set one can get an insight into how many single-particle states and why one should introduce them in an accurate treatment of pairing correlations.
The creation α † k and annihilation α k quasiparticle operators are represented with a unitary transformation from the field operators as follows [15]
α † k = dξ u k (ξ)ψ † (ξ) + v k (ξ)ψ(ξ) ,(1)α k = dξ v * k (ξ)ψ † (ξ) + u * k (ξ)ψ(ξ) ,(2)
and with the reverse relations
ψ † (ξ) = k u * k (ξ)α † k + v k (ξ)α k ,(3)ψ(ξ) = k v * k (ξ)α † k + u k (ξ)α k .(4)
Here ψ † (ξ) and ψ(ξ) are the field operators for the creation and annihilation of a particle with coordinate ξ = (r, σ, τ ) (spatial, spin, and isospin coordinates), and the integral implies also a summation over discrete variables when appropriate. In a finite volume, with periodic boundary conditions for example, the index k is always discrete. For a finite isolated system in vacuum [9,16] the sum over k stands for a summation over the discrete indices and an integral over the continuous ones respectively.
The Hermitian number density and the skew-arXiv:2203.04843v1 [nucl-th] 9 Mar 2022 symmetric anomalous density matrices are defined as
n(ξ, ζ) = Φ|ψ † (ζ)ψ(ξ)|Φ = k v * k (ξ)v k (ζ),(5)n(ξ, ζ) = Φ|ψ(ξ)ψ † (ζ)|Φ = k u k (ξ)u * k (ζ), (6) κ(ξ, ζ) = Φ|ψ(ζ)ψ(ξ)|Φ = k v * k (ξ)u k (ζ),(7)n(ξ, ζ) + n(ξ, ζ) = δ(ξ − ζ),(8)
where the quasiparticle vacuum is defined as
α k |Φ = 0, |Φ = N k α k |0 , Φ|α k α † l |Φ = δ kl ,(9)
and where N is a normalization factor (determined up to an arbitrary phase), α k |0 = 0 for all k, and |0 is the vacuum state. For any k, if the norm dξ|v k (ξ)| 2 = 0 or α k |0 = 0 the corresponding factor α k should be skipped in the definition of |Φ . The new density matrix n(ξ, ζ) is used in the discussion of the canonical basis set below. The anti-commutation relations for the field operators ψ † (ξ), ψ(ξ) and for the quasiparticle operators α † k , α k imply that [15] dξ
[u * k (ξ)u l (ξ) + v * k (ξ)v l (ξ)] = δ kl ,(10)dξ [u k (ξ)v l (ξ) + v k (ξ)u l (ξ)] = 0,(11)k [u k (ξ)u * k (ζ) + v * k (ξ)v k (ζ) = δ(ξ − ζ),(12)k [u k (ξ)v * k (ζ) + v * k (ξ)u k (ζ)] = 0.(13)
Eq. (12) means that the quasiparticle wave functions u k (ξ), v k (ξ) form (in general) an over complete set, as for an arbitrary function g(ξ) one has the decomposition
g(ξ) = k u * k (ξ) dζ u k (ζ)g(ζ) + k v k (ξ) dζ v * k (ζ)g(ζ).(14)
Additionally one can show that [15] dζ [n(ξ, ζ)n(ζ, η) + κ(ξ, ζ)κ † (ζ, η)] = n(ξ, η), (15) dζ n(ξ, ζ)κ(ζ, η) = dy κ(ξ, ζ)n * (ζ, η).
For a finite system the quasiparticle components v k (ξ) always have a finite norm [9] dξ |v k (ξ)| 2 < ∞,
unlike the quasiparticle components u k (ξ), which can be either normalizable or not in an infinite volume and the index k can be either discrete or continuous respectively.
One can consider an arbitrary unitary transformation UU † = I (where I is the identity operator) of the quasiparticle wave functions
v l = k U kl v k , v k = l U * klṽl ,(18)u l = k U kl u k , u k = l U * klũl ,(19)
which leaves the normal and anomalous density matrices unchanged. This type of transformation for quasiparticle wave functions was suggested in Refs. [17,18] in order to simultaneously diagonalize the overlap matrices v k |v l and u k |u l . We construct the canonical basis as follows
dζ n(ξ, ζ)φ k (ζ) = n k φ k (ξ), 0 ≤ n k ≤ 1,(20)dξ φ k (ξ)φ * l (ξ) = δ kl ,(21)where k n k = N(22)
is the total particle number. Only these canonical occupation probabilities are invariant with respect to arbitrary unitary transformations U mentioned above. One can then show that
n(ξ, ζ) = k n k φ * k (ξ)φ k (ζ),(23)κ(ξ, ζ) = k n k (1 − n k )φ * k (ξ)φ k (ζ),(24)
where
φ * k (ξ) = 1 n k (1 − n k ) dζ κ(ξ, ζ)φ k (ζ),(25)φ k |φ * k = 0, φ * k |φ * k = 1,(26)dζ n(ξ, ζ)φ * k (ζ) = n k φ * k (ξ)(27)
where only 0 < n k < 1 are allowed in Eqs. (24,25) and Eqs. (26,27) follow from Eqs. (15,16,20). In the Hartree-Fock (HF) approximation the situation is much simpler, the anomalous density vanishes and the occupation probabilities are defined in the representation which simultaneously diagonalizes the number density matrix and the mean field, and in that particular representation the occupation probabilities have a straightforward physical interpretation. In the presence of pairing correlations one can introduce a generalized density matrix [15], which commutes with the generalized mean field. However, in that representation the normal number density has the form given by Eq. (5), where however v k |v l = n k δ kl . One can define the occupation probabilities either as n k = v k |v k in the representation in which the generalized mean field is diagonal, or instead use the canonical occupation probabilities n k from Eq. (20) and define the single particle energies as e k = φ k |H|φ k , where H is the normal mean field single-particle Hamiltonian within the Hartree-Fock-Bogoliubov (HFB) and Superfluid Local Density Approximation (SLDA) frameworks, and in which case φ k |H|φ l = 0 if k = l. The simple relationship between the HF occupation probabilities and the single-particle energies becomes thus more difficult to interpret physically and justify within HFB and SLDA frameworks.
Since the total particle number is not well defined within HFB and SLDA, as the gauge symmetry is broken, one has to restore this symmetry. In the canonical representation the gauge symmetry is significantly easier to restore [17,18]. The many-body wave function acquires the well-known BCS form
|Φ = Π k (u k + v k a † k a † k )|0 [5]
, where a k |0 = 0 and u 2 k + v 2 k = 1, and various other gauge symmetry restored observables can be easily extracted [18]. The quasiparticle representation in which the generalized number density matrix and the generalized mean field commute is particularly suited for numerically determining the corresponding static many-body wave function |Φ , only if one uses diagonalization methods, which however can be eschewed [19], as both normal and anomalous densities can be determined without the knowledge of the quasiparticle wave functions (qpwfs) and of the corresponding quasiparticle energies.
The canonical states φ k (ξ) form a complete set
k φ k (ξ)φ * k (ζ) = δ(ξ − ζ).(28)
Since n(ξ, ζ) basically vanishes when either spatial coordinate is well outside the system, any function f (ξ) with support outside the system is automatically an eigenstate of n(ξ, ζ) with n k ≈ 0. The number of qpwfs of a finite nucleus in vacuum, either [u k (ξ), v k (ξ)] or φ k (ξ), form a set with cardinality c, the cardinality of R 3 . If the nucleus is simulated in a finite box then the number of qpwfs is countable and the set has the cardinality ℵ 0 , the cardinality of the integers. Since for a stable nucleus the number density decays exponentially at large distances, the description of a bound nucleus in a sufficiently large simulation box should be sufficient, and a smaller set of quasiparticle wavefunctions with cardinality ℵ 0 is sufficient. An eigenfunction φ k (ξ) with n k > 0 has its support largely inside the system, where the support of n(ξ, ζ) is, and it can oscillate with the maximum momentum p max = 2m|U | that a typical nuclear mean field can support for a bound state, where U is the depth of the mean field. One can then conjecture that the total number of states with n k > 0 is of the order of the total number of bound quantum states a nuclear mean field can sustain where r 0 = 1.2 fm and A is the total number of nucleons, and where we did not account for spin and isospin degrees of freedom. Since both normal number and anomalous densities are constructed from canonical qpwfs, with strictly non-vanishing occupation probabilities 0 < n k ≤ 1, it then follows that only a finite set of such functions is likely needed to represent them. We will show below that this undervalue the size of the canonical basis set with n k > 0. Using Eqs. (8) and (20) it follows that the density matrix n(ξ, ζ) has the same eigenfunctions φ k (ξ)
N max ≈ 4πp 3 max 3 × 4πr 3 0 A 3 × 1 (2π ) 3 ≈ 0.5A,(29)dζ n(ξ, ζ)φ k (ζ) = (1 − n k )φ k (ξ).(30)
This equation is useful to construct the canonical states localized mostly outside the system, unlike Eq. (20). One can introduce the time-reversal canonical orbitals [20], not necessarily identical to those define in Eq. (25),
φ k (ξ) = iσ y φ * k (ξ),(31)
where σ y is the Pauli matrix. We will illustrate the properties of the canonical wave functions with some generic numerical results obtained for a 1-dimensional example, which, however, retains all the qualitative features of a 3-dimensional system. For the sake of simplicity we have chosen a 1-dimensional system with potential and pairing fields where we will use the notation for the spatial coordinate −∞ < x < ∞, V 0 = −50 MeV, ∆ 0 = 3 MeV, R = r 0 A 1/3 = 14.9 fm, a = 0.5 fm, and µ = −5 MeV. (We avoid using a Woods-Saxon potential well in order not to generate singularities of the derivatives of the wave functions at the origin and avoid unphysical long momentum tails of the wave functions.) We solved the non-self consistent SLDA or HFB equations for the qpwfs [12,21], using the Discrete Variable Representation method [22]
V (x) = V 0 1 + cosh(x/a) cosh(R/a) ,(32)∆(x) = ∆ 0 1 + cosh(x/a) cosh(R/a) ,(33)H − µ ∆ ∆ −H + µ u k v k = E k u k v k(34)
in a box of size L = 80 fm and with four different lattice constants dx = 1, 0.5, 0.25, 0.125 fm and where
H = − 2 2m d 2 dx 2 + V (x)(35)
and m is the nucleon mass, in the absence of spin-orbit interaction. The Eq. (34) are for the components u k (x) with spin-up and v k (x) with spin-down. The equations for the components u k (x) with spin-down and v k (x) with spin-up are obtained from these equations by changing the sign of the pairing field ∆(x) only [12,21]. The SLDA equations for cold fermionic gases and nuclei have the same structure in this case. It is straightforward to extend this type of analysis to more complicated geometries, for example the pasta phase in neutron star crusts, or the superconductor-normal metalsuperconductor (SNS) junctions in condensed matter physics. The case discussed here is equivalent to a NSN junction. This analysis equally applies to infinite periodic systems. This 1-dimensional model is equivalent to solving the SLDA equations for a spherical system, but only for swave orbitals, therefore for even orbitals φ k (x) = φ k (−x) and x ≥ 0 in the present formulation. normal number density n(x) here is only for the fermions with spin-down, which in the case of even fermion particle number is identical to the normal number density of the spin-up particles. As shown in Ref. [9] the anomalous density κ(x) has longer exponential tails than the number density n(x). This longer tail of the pairing field becomes particularly important as one approaches the nucleon drip-line. This behavior should be also apparent in the profiles of V (x) and ∆(x), an aspect which we however neglected here and which does not change the qualitative behavior of these densities, see Fig. 1. Fig. 1 also shows that with increasing spatial resolu- tion (dx → 0) the normal density is more accurately reproduced at larger distances. We have also checked that Eqs. (23) and (24) correctly reproduce the normal and anomalous densities as well using the canonical wave functions. In Ref. [9] it was argued that the pairing field ∆(ξ) has longer tails than the mean field potential, as the asymptotic behavior of the normal density is controlled by the behavior |v k (ξ)| 2 closest to the Fermi level, thus with with smaller eigenvalues E k > 0, while that of the anomalous density is controlled by only one such qpwf. It appears however that the different weights with which the canonical basis wave functions enter in Eqs. (23) and (24) can lead to the same expected asymptotic behavior of these densities as well.
The canonical occupation probabilities n k shown in Fig. 2 have a conspicuous behavior not discussed previously in literature. For smaller lattice constant dx the maximum momentum cutoff p cutoff = π dx is larger and then the spectrum of n k extends to higher energies. The profile of n k has two obvious "knees," one close to the Fermi level, the infrared (IR) knee, and a second one at a high energy, the ultraviolet (UV) knee. The canonical wave functions φ k (x) have the expected spatial behavior as long as their support is commensurate with the support of the number density matrix n(x, y) as discussed above, see Eq. (20), the text below, and Figs. 3 and 4 in the case of dx = 0.125 fm. However, as soon as the support of the canonical wave functions φ k (x) is essentially outside the support of the density matrix n(x, y), see Fig. 5, for which the index k is on the right of the UVknee in Fig. 2, the corresponding n k decay significantly faster with k. Both the profiles and the numerical values of n k for these canonical states can be obtained with greater accuracy from Eq. (30). These canonical occupation occupation probabilities do not identically vanish simply due to obvious quantum localization effects, but they are increasingly smaller with increasing resolution and decreasing lattice constant dx. In the limit dx → 0 the UV-knee → ∞ and at the same time the number of canonical states localized outside the system also tends to infinity. These non-localized canonical states however are irrelevant in describing the physical properties of the system.
Around the IR-knee in Fig. 2 the canonical occupation probabilities have the expected BCS behavior [5]. It is clear however that in between the IR-knee and UV-knee there is a region where the canonical occupation probabilities have a power law behavior. Such a behavior, due to the short-range character of the nuclear forces, has been predicted in 1980 by Sartor and Mahaux [23] and recently put clearly in evidence experimentally by O. Hen et al. [24]. Shina Tan [25] has proven analytically the emergence of this behavior for fermions interacting with a zero-range interaction in 3D. The nuclear pairing is typically simulated in theory with a δ-potential, which naturally leads to a local pairing field ∆(ξ) [9], as the case we discuss here, and thus this power law behavior of the canonical occupation probabilities is expected. Tan [25] has shown that asymptotically n k ∝ 1/k 4 . This power law behavior of the number density is directly related to the divergence of the anomalous density matrix Eq. (7). In case of a 3-dimensional system it was shown in Ref. [9] that the anomalous density matrix κ(ξ, ζ) ∝ 1 |r−r | when |r − r | → 0, where r and r are the spatial components of ξ and ζ respectively.
Within SLDA or any treatment of pairing with a local pairing field ∆(ξ) the theory requires regularization and renormalization [10,26]. In particular, without such a procedure the kinetic energy of a 3-dimensional superfluid system diverges as
d 3 k (2π ) 3 n k 2 k 2
2m → ∞, where k stands here for the wave vector. We have checked that ( k ) 2 n k ≈ const. in our example in the region beyond the IR-knee and up to the UV-knee, see Fig. 2, confirming the theoretical prediction of Refs. [23,25]. In the case of pure finite-range nucleon interactions, with no zero-range components, there is an upper momentum cutoff controlled by the interaction range. The expectation value of the average kinetic energy k of each canonical state shown in Fig. 6, for all states localized inside the system, thus up to the UV-knee in Fig. 2, increases as expected theoretically for standing waves in a cavity. Beyond the UV-knee, for the canonical states localized mostly outside the system, see Fig.5, k drops in value, see Fig. 6, and their contribution to the total kinetic energy is commensurate with what one expects from numerical discretization errors (dx = 0) of the continuum.
A closer analysis of Fig. 4 clearly show that some canonical wave functions φ k (x) oscillate much faster than the density n(x, x) and that our estimate of the maximum expected number of relevant canonical wave functions, see Eq.(29), is an understatement when the maximum momentum cutoff π dx > 2m|U |. The assumption made, when deriving the estimate for N max , that the canonical wave functions φ k (x) cannot oscillate with a wave vector greater than ≈ 2m|U | 2 , was not accurate. Coupling of the qpwfs components v k (ξ) to the continuum states, facilitated by ∆(ξ), leads to spatial oscillations with any wave vector. In the limit dx → 0 the cardinality of the set of canonical wave functions φ k (ξ) is either ℵ 0 for a finite system in a finite volume or c for an isolated finite system in vacuum. Therefore, one should use for the best estimate of the number N max the cutoff momentum p max = π dx and from the condition of accommodating a standing wave in our "square well" potential with 2R ≈ 14.9 fm in our numerical example one obtains the approximate position of the UVknee at k max = 2R dx + O a R ≈ 240 for dx = 0.125 fm (as k counts the number of half-wave lengths inside the potential well), in perfect agreement with our numerical identification of the UV-knee in Fig. 2. When coupling a bound state through the pairing field ∆ with the continuum, the strength of the bound state is spread over a large energy range with very long tails, with a Lorentzian shape of the spectral distribution [9]. Moreover, in time-dependent phenomena, even in the absence of a true pairing condensate (when the long range order is lost) and at high excitation energies (with corresponding temperatures well above the pairing phase transition T c ) the remnant pairing field leads to many single-particle transitions and the quantum Boltzmann one-body entropy increases considerably [27]. With this in mind one can now provide a better estimate of the size of the canonical basis set for a 3-dimensional system in a finite simulation box with sides of length L x = N x dx, L y = N y dy, L z = N z dz, (dx = dy = dz), ignoring spin and isospin degrees of freedom,
N max = 4π 3 π dx 3 4π 3 r 3 0 A 1 (2π ) 3 = 2.2A r 0 dx 3 .(36)
At the same time the total number of single-particle quantum states in such a box is
N spwfs = L x L y L z (2π ) 3 2π dx 3 = N x N y N z ,(37)
which is typically significantly larger. For example for a typical simulation box for a heavy nucleus with volume 30 3 fm 3 and dx = 1 fm the total number of qpwfs is N spwfs = 27, 000 N max ≈ 2.2A < 1, 000. This would correspond to a momentum cutoff p cutoff ≈ 600MeV/c, in agreement with current values used for the nuclear forces within the chiral effective theory. When accounting for both u k (ξ) and v k (ξ) components a factor of 2 should be included. Additionally one has to account for the spin and isospin degrees of freedom. This estimate is likely very optimistic, and in practice one would have to consider larger values for the parameter r 0 in Eq. (36), in order to correctly describe the surface diffuseness of a nucleus and its deformation in a fixed basis set for static problems.
FIG. 1 .
1The normal number (black) and anomalous (red) densities for x ≥ 0, for four lattice constants in decreasing order dx = 1, 0.5, 0.25, 0.125. n(x) κ(x) stand for n(x, x) and κ(x, x) respectively.
occupation probabilities for four lattice constants dx in a log-log scale. In the inset we plot n k in the linear scale close to the Fermi surface.
FIG. 4 .
4wave function φ12(x) and occupation probability n12 = 0.99 along with profiles of the number density n(x) and of the anomalous density κ(x) in the case dx = 0.125 fm. Since V (x) = V (−x) and ∆(x) = ∆(−x), all these functions have well defined spatial parities and we represent these quantities only for x ≥ 0. The same asFig. 3for φ44(x) and n44 = 1.9E − 5.
FIG. 5 .
5The same asFig. 3for φ300(x) and n300 = 2.6E −13.
φ k in the case of four different lattice constants dx. k is a monotonic function of k up to the UVknee, see Fig. 2.
AcknowledgementsThe funding from the Office of Science, Grant No. DE-FG02-97ER41014 and also the partial support provided in part by NNSA cooperative Agreement DE-NA0003841 to AB is greatly appreciated.
On a new method in the theory of superconductivity. N N Bogoliubov, Il Nuovo Cimento. 7853N. N. Bogoliubov, "On a new method in the theory of superconductivity," Il Nuovo Cimento 7, 853 (1958).
Comments on the theory of superconductivity. J G Valatin, Il Nuovo Cimento. 7843J. G. Valatin, "Comments on the theory of superconduc- tivity," Il Nuovo Cimento 7, 843 (1958).
Normal Forms of Complex Matrices. B Zumino, 10.1063/1.1724294J. Math. Phys. 31055B. Zumino, "Normal Forms of Complex Matrices," J. Math. Phys. 3, 1055 (1962).
The canonical form of an antisymmetric tensor and its application to the theory of superconductivity. C Bloch, A Messiah, 10.1016/0029-5582(62)90377-2Nucl. Phys. 3995C. Bloch and A. Messiah, "The canonical form of an an- tisymmetric tensor and its application to the theory of superconductivity," Nucl. Phys. 39, 95 (1962).
Theory of superconductivity. J Bardeen, L N Cooper, J R Schrieffer, 10.1103/PhysRev.108.1175Phys. Rev. 1081175J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Theory of superconductivity," Phys. Rev. 108, 1175 (1957).
Densityfunctional theory for superconductors. L N Oliveira, E K U Gross, W Kohn, 10.1103/PhysRevLett.60.2430Phys. Rev. Lett. 602430L. N. Oliveira, E. K. U. Gross, and W. Kohn, "Density- functional theory for superconductors," Phys. Rev. Lett. 60, 2430 (1988).
Inhomogeneous Electron Gas. P Hohenberg, W Kohn, 10.1103/PhysRev.136.B864Phys. Rev. 136864P. Hohenberg and W. Kohn, "Inhomogeneous Electron Gas," Phys. Rev. 136, B864 (1964).
Self-Consistent Equations Including Exchange and Correlation Effects. W Kohn, L J Sham, 10.1103/PhysRev.140.A1133Phys. Rev. 1401133W. Kohn and L. J. Sham, "Self-Consistent Equations In- cluding Exchange and Correlation Effects," Phys. Rev. 140, A1133 (1965).
Hartree-Fock-Bogoliubov approximation for finite systems (1980). A Bulgac, arXiv:nucl-th/9907088A. Bulgac, "Hartree-Fock-Bogoliubov approximation for finite systems (1980)," (), arXiv:nucl-th/9907088.
Renormalization of the Hartree-Fock-Bogoliubov Equations in the Case of a Zero Range Pairing Interaction. A Bulgac, Y Yu, 10.1103/PhysRevLett.88.042504Phys. Rev. Lett. 8842504A. Bulgac and Y. Yu, "Renormalization of the Hartree- Fock-Bogoliubov Equations in the Case of a Zero Range Pairing Interaction," Phys. Rev. Lett. 88, 042504 (2002).
Energy Density Functional Approach to Superfluid Nuclei. Y Yu, A Bulgac, 10.1103/PhysRevLett.90.222501Phys. Rev. Lett. 90222501Y. Yu and A. Bulgac, "Energy Density Functional Ap- proach to Superfluid Nuclei," Phys. Rev. Lett. 90, 222501 (2003).
Local-density-functional theory for superfluid fermionic systems: The unitary Fermi gas. A Bulgac, 10.1103/PhysRevA.76.040502Phys. Rev. A. 7640502A. Bulgac, "Local-density-functional theory for super- fluid fermionic systems: The unitary Fermi gas," Phys. Rev. A 76, 040502 (2007).
Time-Dependent Density Functional Theory for Fermionic Superfluids: from Cold Atomic gases, to Nuclei and Neutron Star Crust. A Bulgac, 10.1002/pssb.201800592Physica Status Solidi B. 2561800592A. Bulgac, "Time-Dependent Density Functional Theory for Fermionic Superfluids: from Cold Atomic gases, to Nuclei and Neutron Star Crust," Physica Status Solidi B 256, 1800592 (2019).
Basic Notions of Condensed Matter Physics. P A Anderson, Benjamin/Cummins Publishing Company Inc. LondonP. A. Anderson, Basic Notions of Condensed Matter Physics (Benjamin/Cummins Publishing Company Inc. London, 1984).
The Nuclear Many-Body Problem. P Ring, P Schuck, Springer-VerlagBerlin Heidelberg New York1st ed.P. Ring and P. Schuck, The Nuclear Many-Body Problem, 1st ed. (Springer-Verlag, Berlin Heidelberg New York, 2004).
Pairing in nuclei in the coordinate representation. S T Belyaev, A V Smirnov, S V Tolokonnikov, S A Fayans, Sov. J. Nucl. Phys. 4545783S. T. Belyaev, A. V. Smirnov, S. V. Tolokonnikov, and S. A. Fayans, "Pairing in nuclei in the coordinate repre- sentation," Sov. J. Nucl. Phys. 45, 783 (45).
Projection of good quantum numbers for reaction fragments. A Bulgac, 10.1103/PhysRevC.100.034612Phys. Rev. C. 10034612A. Bulgac, "Projection of good quantum numbers for re- action fragments," Phys. Rev. C 100, 034612 (2019).
Restoring broken symmetries for nuclei and reaction fragments. A Bulgac, 10.1103/PhysRevC.104.054601Phys. Rev. C. 10454601A. Bulgac, "Restoring broken symmetries for nuclei and reaction fragments," Phys. Rev. C 104, 054601 (2021).
Coordinate-space solver for superfluid many-fermion systems with the shifted conjugate-orthogonal conjugategradient method. S Jin, A Bulgac, K Roche, G Wlazłowski, 10.1103/PhysRevC.95.044302Phys. Rev. C. 9544302S. Jin, A. Bulgac, K. Roche, and G. Wlazłowski, "Coordinate-space solver for superfluid many-fermion systems with the shifted conjugate-orthogonal conjugate- gradient method," Phys. Rev. C 95, 044302 (2017).
. A Bohr, B R Mottelson, Nuclear Structure. IBenjamin IncA. Bohr and B. R. Mottelson, Nuclear Structure, Vol. I (Benjamin Inc., New York, 1969).
Framework for Polarized Superfluid Fermion Systems (2020). A Bulgac, arXiv:2005.06763A. Bulgac, "Framework for Polarized Superfluid Fermion Systems (2020)," (), arXiv:2005.06763.
Use of the discrete variable representation basis in nuclear physics. A Bulgac, M M Forbes, 10.1103/PhysRevC.87.051301Phys. Rev. C. 8751301A. Bulgac and M. M. Forbes, "Use of the discrete variable representation basis in nuclear physics," Phys. Rev. C 87, 051301(R) (2013).
Self-energy, momentum distribution, and effective masses of a dilute Fermi gas. R Sartor, C Mahaux, 10.1103/PhysRevC.21.1546Phys. Rev. C. 211546R. Sartor and C. Mahaux, "Self-energy, momentum dis- tribution, and effective masses of a dilute Fermi gas," Phys. Rev. C 21, 1546 (1980).
Momentum sharing in imbalanced Fermi systems. O Hen, 10.1126/science.1256785Science. 346614O. Hen et al., "Momentum sharing in imbalanced Fermi systems," Science 346, 614 (2014).
Large momentum part of a strongly correlated Fermi gas. S Tan, 10.1016/j.aop.2008.03.005Ann. Phys. 3232971S. Tan, "Large momentum part of a strongly correlated Fermi gas," Ann. Phys. 323, 2971 (2008).
Local density approximation for systems with pairing correlations. A Bulgac, 10.1103/PhysRevC.65.051305Phys. Rev. C. 6551305A. Bulgac, "Local density approximation for systems with pairing correlations," Phys. Rev. C 65, 051305 (2002).
Pure quantum extension of the semiclassical Boltzmann-Uehling-Uhlenbeck equation. A Bulgac, 10.1103/PhysRevC.105.L021601Phys. Rev. C. 10521601A. Bulgac, "Pure quantum extension of the semiclassical Boltzmann-Uehling-Uhlenbeck equation," Phys. Rev. C 105, L021601 (2022).
| [] |
[
"Low voltage and time constant organic synapse-transistor",
"Low voltage and time constant organic synapse-transistor"
] | [
"Simon Desbief \nInstitute for Electronics Microelectronics and Nanotechnology\nAlma Mater Studiorum-Università degli Studi di Bologna\nDipartimento di Chimica \"G. Ciamician\"\nCNRS and University of Lille\nAv. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy\n",
"Adrica Kyndiah ",
"David Guerin \nInstitute for Electronics Microelectronics and Nanotechnology\nAlma Mater Studiorum-Università degli Studi di Bologna\nDipartimento di Chimica \"G. Ciamician\"\nCNRS and University of Lille\nAv. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy\n",
"Denis Gentili ",
"Mauro Murgia ",
"Stéphane Lenfant \nInstitute for Electronics Microelectronics and Nanotechnology\nAlma Mater Studiorum-Università degli Studi di Bologna\nDipartimento di Chimica \"G. Ciamician\"\nCNRS and University of Lille\nAv. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy\n",
"Fabien Alibart \nInstitute for Electronics Microelectronics and Nanotechnology\nAlma Mater Studiorum-Università degli Studi di Bologna\nDipartimento di Chimica \"G. Ciamician\"\nCNRS and University of Lille\nAv. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy\n",
"Tobias Cramer \nDipartimento di Fisica e Astronomia\nUniversità di Bologna\nViale Berti Pichat 6/2BolognaItaly\n",
"Fabio Biscarini \nLife Science Dept\nUniv. Modena and Reggio Emilia\nVia Campi 18341125ModenaItaly\n",
"Dominique Vuillaume \nInstitute for Electronics Microelectronics and Nanotechnology\nAlma Mater Studiorum-Università degli Studi di Bologna\nDipartimento di Chimica \"G. Ciamician\"\nCNRS and University of Lille\nAv. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy\n",
") "
] | [
"Institute for Electronics Microelectronics and Nanotechnology\nAlma Mater Studiorum-Università degli Studi di Bologna\nDipartimento di Chimica \"G. Ciamician\"\nCNRS and University of Lille\nAv. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy",
"Institute for Electronics Microelectronics and Nanotechnology\nAlma Mater Studiorum-Università degli Studi di Bologna\nDipartimento di Chimica \"G. Ciamician\"\nCNRS and University of Lille\nAv. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy",
"Institute for Electronics Microelectronics and Nanotechnology\nAlma Mater Studiorum-Università degli Studi di Bologna\nDipartimento di Chimica \"G. Ciamician\"\nCNRS and University of Lille\nAv. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy",
"Institute for Electronics Microelectronics and Nanotechnology\nAlma Mater Studiorum-Università degli Studi di Bologna\nDipartimento di Chimica \"G. Ciamician\"\nCNRS and University of Lille\nAv. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy",
"Dipartimento di Fisica e Astronomia\nUniversità di Bologna\nViale Berti Pichat 6/2BolognaItaly",
"Life Science Dept\nUniv. Modena and Reggio Emilia\nVia Campi 18341125ModenaItaly",
"Institute for Electronics Microelectronics and Nanotechnology\nAlma Mater Studiorum-Università degli Studi di Bologna\nDipartimento di Chimica \"G. Ciamician\"\nCNRS and University of Lille\nAv. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy"
] | [] | We report on an artificial synapse, an organic synapse-transistor (synapstor) working at 1 volt and with a typical response time in the range 100-200 ms. This device (also called NOMFET, Nanoparticle Organic Memory Field Effect Transistor) combines a memory and a transistor effect in a single device. We demonstrate that short-term plasticity (STP), a typical synaptic behavior, is observed when stimulating the device with input spikes of 1 volt. Both significant facilitating and depressing behaviors of this artificial synapse are observed with a relative amplitude of about 50% and a dynamic response < 200 ms. From a series of in-situ experiments, i.e. measuring the current-voltage characteristic curves in-situ and in real time, during the growth of the pentacene over a network of gold nanoparticles, we elucidate these results by analyzing the relationship between the organic film morphology and the transport properties. This synapstor works at a low energy of about 2 nJ/spike. We discuss the implications of these results for the development of neuro-inspired computing architectures and interfacing with biological neurons. | 10.1016/j.orgel.2015.02.021 | [
"https://arxiv.org/pdf/1505.04282v1.pdf"
] | 15,513,606 | 1505.04282 | 61ea88d4e1eadacb14bc9a5e9342f2fc8b16fc81 |
Low voltage and time constant organic synapse-transistor
Simon Desbief
Institute for Electronics Microelectronics and Nanotechnology
Alma Mater Studiorum-Università degli Studi di Bologna
Dipartimento di Chimica "G. Ciamician"
CNRS and University of Lille
Av. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy
Adrica Kyndiah
David Guerin
Institute for Electronics Microelectronics and Nanotechnology
Alma Mater Studiorum-Università degli Studi di Bologna
Dipartimento di Chimica "G. Ciamician"
CNRS and University of Lille
Av. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy
Denis Gentili
Mauro Murgia
Stéphane Lenfant
Institute for Electronics Microelectronics and Nanotechnology
Alma Mater Studiorum-Università degli Studi di Bologna
Dipartimento di Chimica "G. Ciamician"
CNRS and University of Lille
Av. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy
Fabien Alibart
Institute for Electronics Microelectronics and Nanotechnology
Alma Mater Studiorum-Università degli Studi di Bologna
Dipartimento di Chimica "G. Ciamician"
CNRS and University of Lille
Av. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy
Tobias Cramer
Dipartimento di Fisica e Astronomia
Università di Bologna
Viale Berti Pichat 6/2BolognaItaly
Fabio Biscarini
Life Science Dept
Univ. Modena and Reggio Emilia
Via Campi 18341125ModenaItaly
Dominique Vuillaume
Institute for Electronics Microelectronics and Nanotechnology
Alma Mater Studiorum-Università degli Studi di Bologna
Dipartimento di Chimica "G. Ciamician"
CNRS and University of Lille
Av. Poincaré, Via P. Gobetti 101, Via Selmi 2, 40127F-59652cedex, 40129Villeneuve d'Ascq, Bologna, BolognaItaly. 3)France., Italy
)
Low voltage and time constant organic synapse-transistor
2) Consiglio Nazionale delle Ricerche, Istituto per lo Studio dei Materiali Nanostrutturati (CNR-ISMN)organic synapsenanoparticlememorymemristortransistor
We report on an artificial synapse, an organic synapse-transistor (synapstor) working at 1 volt and with a typical response time in the range 100-200 ms. This device (also called NOMFET, Nanoparticle Organic Memory Field Effect Transistor) combines a memory and a transistor effect in a single device. We demonstrate that short-term plasticity (STP), a typical synaptic behavior, is observed when stimulating the device with input spikes of 1 volt. Both significant facilitating and depressing behaviors of this artificial synapse are observed with a relative amplitude of about 50% and a dynamic response < 200 ms. From a series of in-situ experiments, i.e. measuring the current-voltage characteristic curves in-situ and in real time, during the growth of the pentacene over a network of gold nanoparticles, we elucidate these results by analyzing the relationship between the organic film morphology and the transport properties. This synapstor works at a low energy of about 2 nJ/spike. We discuss the implications of these results for the development of neuro-inspired computing architectures and interfacing with biological neurons.
Introduction.
There is a growing interest in neuro-inspired devices based on emerging technologies beyond the existing silicon CMOS implementation of artificial neural networks (e.g. see recent reviews in a special issue 1 and in Refs. 2,3 ). Organic bioelectronics is emerging as a viable technological approach aiming at interfacing organic devices with cells and neurons. [4][5][6] We have recently demonstrated how we can use charge trapping/detrapping in an array of gold nanoparticles (NPs) at the SiO2/pentacene interface to design a synapse-transistor (synapstor) mimicking the dynamic plasticity of a biological synapse. 7,8 This device called NOMFET (Nanoparticle Organic Memory Field Effect Transistor) combines a memory and a transistor effect in a single device.
NOMFETs have been initially designed for neuro-inspired computing architectures (artificial neural networks). This device (which is memristor-like) 8 mimics short-term plasticity (STP) 7 and temporal correlation plasticity (STDP, spike-timing dependent plasticity) 8 of biological spiking synapses, two "functions" at the basis of learning processes. A compact model was developed 9 and we demonstrated an associative memory, which can be trained to exhibit a Pavlovian response. 10 However, these devices were limited to work with spikes in the range of few tens of volts and time scale of 1-100s. Both fields (neuro-inspired computing and bioelectronics) require devices working at lower voltages (to save energy consumption during computing and because action potentials in synapse and neurons have amplitudes of around 100 mV) and higher speed (e.g. synapse and neurons work at around kHz).
Here, we report on a synapstor working at 1 volt and with a typical response time constant in the range 100-200 ms. To gain a better insight of the NOMFET behavior, we performed a series of in-situ experiments where the response of the device is studied during the growth of the organic semiconductor thin film. We analyzed the relationship between the organic film morphology and the transport properties by measuring the current-voltage characteristic curves, in-situ and in real time, during the growth of the pentacene over the NP network. Second, we performed temperature-dependent measurements of the charge carrier mobility to assess the effect of NPs on charge transport properties. Finally, based on the conclusions drawn from these experiments, we report on an optimized NOMFET fabrication process getting a larger hole mobility of the NOMFET (∼ 0.1 cm 2 /Vs, instead of 10 -3 -10 -2 cm 2 /Vs in our previous work) 7 and a faster response time constant (hundreds of ms instead of few seconds).
Materials and Methods
Si/SiO2 wafer substrate and source/drain electrodes.
The NOMFETs were processed using a bottom-gate electrode configuration. We used highlydoped (resistivity ∼10 -3 Ω.cm) n-type silicon covered with a thermally grown 200 nm thick silicon dioxide (grown at 1100°C during 135 min in a dry oxygen flow (2L/min) and followed by a post-oxidation annealing at 900°C during 30 min under a nitrogen flow (2L/min) in order to reduce the presence of defects into the oxide). Electrodes (titanium/gold (20/200 nm)) were deposited on the substrate by vacuum evaporation and patterned by e-beam lithography for linear source and drain gold electrodes (length L=1 to 50 µm and width W=1000 µm) and by optical lithography for interdigitated source and drain gold electrodes (length L= 20 and 40 µm, width W=11200 and 22400 µm). Before use, these wafers were cleaned following the protocol : i) sonication in acetone for 15 min and isopropanol (15 min), dried in N2. ii) Piranha solution (H2SO4 /H2O2, 2/1 v/v) for 15 minutes (Caution: preparation of the piranha solution is highly exothermic and reacts violently with organics). iii) Rinse with DI water, copiously. iv) Etch with HF 2% for 5s, rinse with DI water, copiously, dry in N2 and bake at 120° (hot plate), or clean by ultraviolet irradiation in an ozone atmosphere (ozonolysis) for 30 minutes. The cleaned substrates are used immediately (see surface functionalization below).
Au NPs synthesis.
Colloidal solutions of citrate-capped Au NP (10 ± 3 nm in diameter) were synthesis as follows. 11 Charge stabilized Au nanoparticles were synthesized by the reduction of chloroauric acid in water. To obtain a 100 mL aqueous solution of Au nanoparticles, a solution with 1 mL of HAuCl44H2O (1% w/v) in 79 mL of H2O was first prepared. A 20 mL reducing solution with 4 mL of trisodium citrate (1% w/v) and 80 µL of tannic acid (1% w/v) in 16 mL of H2O was then added rapidly to the Au solution (all solutions at 60 °C). The mixed solution was boiled for 10 min before being cooled down to room temperature. A continuous stirring was applied throughout the process. The resulting reddish solution contained typically 10nm Au. The NP size (10 ± 3 nm in diameter) and density is calculated from SEM images of NPs arrays on the surface (as the one in Fig. S1 in SI).
SiO2 surface functionalization and NP deposition.
Immediately after the cleaning (see above), the SiO2 gate dielectric was functionalized by selfassembled monolayer (SAM) to anchor gold nanoparticles (NPs) on the surface. 12 The SiO2 surface was functionalized by immersion in a solution of (3-aminopropyl)-trimethoxysilane (APTMS) molecules diluted in methanol at a concentration 1µL/mL for 1h. 13 The reaction took place in a glove-box with a controlled atmosphere (nitrogen, with less than 1 ppm of oxygen and water vapor). Excess, non-reacted, molecules were removed by rinse in methanol under sonication. This sample was subsequently dried under nitrogen stream. Static water contact angle was 19°, a common value for hydrophilic NH2-terminated surfaces. 13 This sample was then immediately dipped in a colloidal solution of citrate-stabilized Au-NP for 24h. This procedure yields an array of NPs with a density of about 4-5x10 10 NP/cm 2 (see details in the Supporting Information, Fig. S1). The sample was then cleaned with deionized water and isopropanol, and dried under nitrogen stream. The NPs deposited on SiO2 do not form a continuous film or large clusters, rather they are adsorbed as individual entities. They do not coalesce (mainly due to coulombic repulsion between the negatively charged citrate layer), and they exhibit a characteristic length scales, i.e. a density, that we can adjust with the concentration of the Au NPs in the solution and the time of deposition. 7 Indeed, we have previously demonstrated that an optimized density of NPs for the NOMFETs is around 5x10 10 NP/cm 2 . 7
OTS functionalization.
Octadecyltricholorosilane (OTS) molecules were used as follow. 14,15 The silanization reaction was carried out in a glovebox under nitrogen atmosphere but non-anhydrous solvents were used to favor hydrolysis of -SiCl3 functions. The freshly cleaned substrate was immersed for 2h in a 10 -3 M solution of OTS in a mixture of n-hexane and dichloromethane (70:30 v/v). The device was rinsed thoroughly by sonication in dichloromethane (2 times) then blown with dry nitrogen.
Pentacene evaporation.
For in-situ experiments. Pentacene was evaporated at a rate of 0.5 ML/min (∼0.125 Å/s, 1 ML of pentacene ≈ 15 Å). The substrate was kept at room temperature. About 20 ML (30 nm) were evaporated. For device measurements (ex-situ), a 35 nm thick pentacene film was evaporated at a rate of 0.1 Å/s. The substrate was kept at 60°C. 4 devices were fabricated in parallel during the same pentacene evaporation run: i)A reference pentacene transistor without NPs (and without APTMS SAM on SiO2) -referred to as P5. ii) A second pentacene transistor with an OTS functionalized SiO2 gate dielectric -referred to as OTS-P5. iii) A "standard" NOMFET, with APTMS and NPs -referred to as APTMS-NP-P5. iv) The "optimized" NOMFET with the OTS treatment and referred to as APTMS-NP-OTS-P5.
AFM measurements.
The surface morphology of the organic layer was determined by imaging with a Bruker ICON atomic force microscope (AFM) in tapping mode. Silicon cantilevers from Bruker (NCHV model) were used to acquire AFM images of 5 x 5 µm² at 1Hz with a resolution of 512 x 512 pixels.
Contact angle measurements.
The wettability of the surface at different stages of the fabrication was assessed by measuring the water contact angle. We measured the static water contact angle with a remote-computer controlled goniometer system (DIGIDROP by GBX, France). We deposited a drop (10-30 μL) of deionized water (18 MΩ.cm-1) on the surface, and the projected image was acquired and stored by the computer. Contact angles were extracted by contrast contour image analysis software.
These angles were determined a few seconds after application of the drop. These measurements were carried out in a clean room (ISO 6) where the relative humidity (50%) and the temperature (22 °C) are controlled. The precision with these measurements are ±2°.
Electrical measurements.
For in-situ measurements, just before pentacene evaporation, samples were transferred in the home-built vacuum evaporation chamber, 16 the bottom gate and the bottom source/drain electrodes were electrically connected, and drain current-gate voltage (ID-VG) curves were recorded in-situ from VG= 30V to -30V both in linear (VD=-1V) and saturation regime (VD=-30V) as pentacene was sublimed at a rate of 0.5 ML/min (substrate at room-temperature, 1 ML of pentacene ≈1.5 nm). When pentacene thickness reached about 20 MLs, deposition was stopped and temperature dependent measurements were carried out by recording ID-VG scans every 30 s while the temperature was varied from -160°C to RT. Hole mobility in linear and saturation regimes was extracted as usual from standard transistor equations. A low-current source-measure unit (SMU) Keithley 6430 is used to apply the gate-source voltage (VGS) and to measure the gate current (IG), and a dual SMU Keithley 2612 is used to apply the drain-source voltages (VDS) and to measure the respective drain-source currents (IDS).
The NOMFET electrical ex-situ characteristics were measured with an Agilent 4155C semiconductor parameter analyzer, the input pulses were delivered by a pulse generator (Tabor 5061 or Keithley 2636A). The electrodes of the NOMFET were contacted with a shielded micromanipulator probe station (Suss Microtec) in the dark. . This finding indicates that efficient transport paths are formed when the NPs start to get covered by the pentacene layer. Therefore, we infer that a more disordered pentacene film is obtained with NPs, as also previously revealed by AFM images with smaller pentacene grain size for the NOMFET compare to pentacene transistor without NPs (see Fig. 2 in Ref. 7 and more images in the next section).
In-situ experiments.
Complementary to these measurements we performed temperature dependent characterization of the transistor once the final layer thickness of 20 ML is achieved. Fig. 2 depicts the mobility as a function of inverse temperature. The Arrhenius plot shows strongly temperature activated behavior which can be described by a single effective barrier of EA = 123 meV. We assign this barrier to a trap state in the HOMO-LUMO gap, near the HOMO. 20-22 Extrapolation to infinite temperature yields a trap-free mobility of μ0 = 0.26 cm 2 /Vs. Without the NPs, we measured an activation energy EA = 65 meV (see Fig. S3 in the supporting information). This latter value is in agreement with previous reports for pentacene OFET (in the range 20-80 meV). 20, 23-26 Our findings point to two possible reasons for the weak performance of the NOMFET grown on citrate capped NPs. First, the pentacene morphology is disordered due to the presence of NPs, and only transport paths spanning at a distance from the dielectric interface can evolve. The electronic coupling and charge mobility are reduced along such a path. The extrapolated trap free mobility is an order of magnitude lower than typical values found in unperturbed pentacene transistors. 27 Second, deep trap states further reduce mobility and lead to activated transport behavior with higher energy barrier. In the NOMFETs, once a sufficiently thick pentacene film is deposited (above about 10 ML, Fig. 1), each nanoparticle is surrounded by pentacene, thus each NP acts as a "shallow trap" for charge carriers moving across the pentacene thin film, explaining the increased activation energy EA with the NPs.
Since low charge carrier mobility in NOMFET is the main cause of the slow dynamics of its synaptic behavior, 7, 8 we modify the fabrication protocol adding a silanization treatment after the NPs deposition. The findings from the in-situ experiment prompted this strategy. Reduced mobility was already observed in the APTMS modified dielectric without the presence of NPs. 28 Therefore, both the surface chemistry that exposes the highly polarizable groups and the morphological changes of the semiconductor as induced by the presence of NPs (given that a sufficiently thick pentacene layer is deposited), contribute to the decrease of charge transfer velocity. Surface functionalization by molecules such as octadecyltrichlorosilane (OTS) is known to improve mobility in OFET as they bind to polar groups at the surface and form an oriented monolayer which exposes non-polar alkyl chains, thereby decreasing the surface energy.
29-31 We treated the NPs/APTMS substrates in a solution of OTS (see details in Materials and
Methods) according to an established protocol. 14,15,[32][33][34] Fig.1 (red dots), clearly confirms the improved performance. Percolation sets in earlier at ΘC1,NP/OTS = 5.2 ML, but remains still beyond the sub-monolayer regime. Here the percolation process follows a simpler pattern that is described by a single power-law in the initial critical regime. The final increase in mobility saturates slowly and only at Θ = 18 ML a stable value of μ = 3 x 10 -2 cm 2 / Vs is obtained which exceeds by more than an order of magnitude the non-OTS passivated device. Temperature dependent measurements demonstrate also for the OTS passivated NOMFET a thermally activated transport, but the activation barrier is clearly reduced to EA = 71 meV, close to the one for pentacene transistor (see Fig. S2, Supporting Information).
Extrapolation to infinite temperature results in μ0 = 0.25 cm 2 /Vs, as for the non-OTS passivated device. The comparison of the two types of NOMFETs provides a clear interpretation of the improvements in electrical performance (see section below). Most important, we achieved a smoothening of the trap states that limited transport in the non-OTS passivated device. The exposure of the polar citric acid groups at the NP surface leads to strong electrostatic interactions with carriers and consequently trapping. OTS reacts directly with these groups 36,37 making the NP surface hydrophobic by exposing the alkyl chain. As a consequence the mobility rises by more than one order of magnitude when operated at room temperature (see below). However, both kind of surfaces demonstrate critical limitations of transport when compared to transistors grown on smooth dielectric surfaces: the NPs perturb the morphology in the first monolayers and give rise to a hindered transport path with reduced electronic coupling. This finding is independent of the NP functionalization and in both investigated cases we find maximum "trapfree" mobilities μ0 = 0.25 cm 2 /Vs by extrapolation.
Synaptic performances of the NOMFETs.
In parallel with these in-situ experiments, we fabricated NOMFET with and without the OTS treatments to assess improvements of their synaptic behavior (devices referred to as "APTMS-NP-P5" and "APTMS-NP-OTS-P5", respectively). The NP deposition and OTS protocols were the same as for the in-situ measurement, then a 30 nm thick pentacene was directly evaporated to complete the device (see Materials and Methods). Fig. 3 shows typical ID-VG measurements in saturation and the ID-VD characteristics. For comparison, we also fabricated transistors without the NPs during the same run of pentacene deposition (pentacene directly on SiO2, referred to as "P5" and pentacene on OTS functionalized SiO2, referred to as "OTS-P5"). Fig. 4 shows the extracted mobilities for the four types of device. For the OTS-treated NOMFET ("APTMS-NP-OTS-P5"), we obtained a mobility of ∼ 0.1 cm 2 /Vs whatever the channel length in the 1-50 μm range, on a part with the mobility of pentacene OFET (without OTS, Fig. 4 device "P5"). Albeit smaller than the one for OTS-treated pentacene OFET (0.4-0.5 cm 2 /Vs, Fig. 4, "OTS-P5"), this value is strongly improved compared to standard NOMFETs (without OTS treatment), which is around 10 -3 cm 2 /Vs (Fig. 4, "APTMS-NP-P5") consistent with our previous results. 7, 8 AFM images of the pentacene (Fig. 5) clearly shows that the OTS-treatment increases the grain size of the pentacene film of the NOMFETs, as already reported for pentacene OFET. 29,38 To measure the synaptic properties, the NOMFET is used as a pseudo two-terminal device (inset Figure 6): the source (S) and gate (G) electrodes are connected together and used as the input terminal of the device, and the drain (D) is used as the output terminal (virtually grounded through the amperometer). A sequence of spikes with amplitude Vspike and at different frequencies is applied at the G/S input and we measure the output drain current. Figure 6-a shows a typical result for the optimized NOMFET (APTMS-NP-OTS-P5) developed in this work. As in previous reports, 7,8 we clearly observe the STP (short term plasticity) behavior, mimicking a biological synapse. 39 In brief, the output current (or equivalently the NOMFET conductance) shows a depressing behavior (decrease of the current with the number of spikes) for the spike at the highest frequencies and a facilitating behavior (increase of the current) for the lowest ones, similar to the synaptic weight (the signal transmittance through the synapse) of a biological synapse. The major improvements, compared to our previous results ( Fig. 6-b), are twofold: i) STP behavior is obtained with spikes of amplitude Vspike = -1V instead of -10 to -20V (for a channel length L=1 μm) in Ref. 7 ; ii) the characteristic response time constant is lowered to hundreds of ms, while we measured few seconds (at same channel length L = 1 μm). 7 This time constant is obtained by fitting (red squares Fig. 6) an analytical and iterative model that takes into account the effect of charge (during the spike) and discharge (between two successive spikes) of the NPs on the drain current as detailed elsewhere. 7 Roughly, the RC charge/discharge time constant of the NOMFET is governed by the channel resistance R of the NOMFET, which scales as L/μ and the capacitance of the NPs network, N·CNP, with N the number of NPs, CNP the self-capacitance of a NP CNP = 2πεD (with D the NP diameter and ε the dielectric constant). Here, at constant L, N and D, the decrease of NOMFET response time constant by a factor 10 is consistent with the increase of mobility by around the same factor (best mobility ≈10 -2 cm 2 /Vs in our previously fabricated NOMFET). 7, 8 A more precise comparison is difficult because of device-to-device dispersion, and the density of NP can only be roughly controlled around 5x10 10 NP/cm 2 (Fig. S1) as it may vary from device to device by about a factor 2.
In term of energy, the energy per spike Espike=Vspike Ispike δ, with Vspike the applied spike amplitude, Ispike the current measured during the spike (Fig. 6) and δ the spike duration, we obtain Espike ∼ 2 nJ for the NOMFET with OTS (Vspike=1V, δ=10ms and taking an average Ispike of 200 nA, Fig. 6-a) and Espike ∼ 10 nJ for the NOMFET without OTS (Vspike=10V, δ=100ms and taking an average Ispike of 10 nA, Fig. 6-b). Thus, we have gained a factor 5 with the OTS treatment. Note that between spikes, the energy consumption is zero, all the device terminals being grounded.
Conclusion and perspectives.
In conclusion, we have developed a hybrid metal NP/organic synapstor working at 1 volt and with a typical response time constant in the range 100-200 ms. We have established a relationship between the device performances and i) the presence of Au NPs at the pentacene/ SiO2 interface and ii) the chemical surface treatment before the pentacene deposition. Further reduction of the spike amplitude can be obtained by using high-k dielectric gate or electrolyte gating. 40 Further reduction of the time response will be also possible using recently reported organic semiconductors 41-43 with a higher mobility (> 1 cm 2 /Vs) then pentacene, or solutionprocessed (i.e. sol-gel) inorganic oxide semiconductors (up to 7 cm 2 /Vs). 44 Fig. S1 shows a SEM image of the NPs on surface before the evaporation of pentacene. From image analysis, an average size of the NPs is 10 ± 3 nm, and a density of about 5x10 10 NP/cm 2 .
SUPPORTING INFORMATION
Figure 1
1shows the evolution of the hole mobility (in saturation regime) as a function of the thickness of the deposited pentacene film. While mobility in linear and saturation regimes are slightly different (μsat > μlin by a factor of about 1.5-2), their evolution with pentacene thickness is the same. The in-situ measurements allow us to evidence the formation of the channel layer and charge transport (holes) along the dielectric interface functionalized with NPs, as the control in thickness is a fraction of monolayer. First we investigate pentacene deposition and channel formation on the NP functionalized substrates.Fig. 1shows how the mobility of this transistor evolves as a function of the nominal pentacene layer thickness Θ (expressed in monolayers ML by taking into account 1ML of pentacene ≈ 1.5nm) during the in-situ experiment. Below a critical thickness of ΘC1,NP = 5.7 ML (square symbol, NOMFET without OTS treatment) no transistor current can be measured and the mobility cannot be extracted. At ΘC1,NP percolation of conducting islands sets in and a first continuous pathway connects source and drain electrode.We measure current modulation in the transfer characteristics that allows us to extract the mobility µ as a function of Θ≥ ΘC1,NP. Upon addition of pentacene the transistor current and the mobility both increase rapidly over orders of magnitude. The increase does not follow a single power law but contains an inflection point at ΘC2,NP = 8.4 ML beyond which a second phase of steep increase in mobility is observed.In Fig 1, bothregions are fitted by a power law ∼(Θ-ΘC) γ , with the critical coverage ΘC indicating the creation of a percolation path between the electrodes, and the critical exponent γ related to the lattice geometry and dimensionality of the underlying conducting network.17 From the fit we find an increase in exponent from 1.1 in the first phase to 1.25 in the second phase of mobility increase, demonstrating a change in growth mode at Θ = 8.4 ML (i.e. about 12 nm) caused by the presence of NPs (diameter of about 10 nm). Finally, saturation of the curve is slowly attained at Θ ≈ 20 ML of deposited pentacene to yield a saturated charge mobility of µ = 3x10 -3 cm 2 /Vs. This mobility value is on agreement with the one we previously measured on NOMFET.7 The observed evolution of the mobility with Θ clearly results from the more disordered organisation of the semiconducting film grown on the NPs. In pentacene transistors grown on a smooth dielectric interface, percolation is observed before the completion of the first monolayer (ΘC,2D ~ 0.7 ML) following the theory of 2D percolation18, 19 (seeFig. S2, Supplementary Information for in situ μ vs Θ measurements without NPs). Here, the shift of the percolation threshold above 5 MLs demonstrates a transition of pentacene growth to a 3D growth mode induced by the NPs adsorbed on the surface. Further complexity in the formation of additional transport paths leads to the formation of the inflection point. Only when the nominal thickness exceeds 8 ML a second critical regime emerges where an additional steep increase of mobility is observed. This second threshold happens at a thickness which exceeds the diameter of the citrate capped NPs (about 10 nm)
Chlorosilane head groups of the OTS molecules are prone to react with both the terminal amine group of APTMS and the -OH and COOgroups of citrate capping the NPs. 35-37 Moreover, OTS can polymerized laterally, though siloxane bonds, forming a capping layer over the surface and NPs. Evidence of the correct grafting of OTS is the observation of an increase of the water contact angle from ∼30-40° for APTMS/NPs surface to ∼90-100° after silanization. This latter value is consistent with 100-110° reported for OTS monolayer on a smooth surface (SiO2), 34 (we observe contact angles of 106-109° for APTMS/OTS bilayer on flat SiO2). Now we compare the above results with in-situ experiments performed after passivation of the NP surface with OTS. The increase of mobility as a function of Θ, as shown in
Fig. 1 .
1Mobility in saturation µsat of pentacene NOMFETs as a function of nominal layer thickness Θ measured in-situ during the deposition of the semiconducting film. Both data sets were obtained on substrates functionalized with APTMS and 10 nm gold nanoparticles. The red curve was measured using a substrate that had been made hydrophobic by a final OTS functionalization.
Fig. 2 .
2Temperature dependence of mobility in saturation µsat for pentacene NOMFETs with 20 ML thickness grown on APTMS/NP (red) or APTMS/NP/OTS surface (blue). Both curves show a temperature activated behavior and follow the Arrhenius behavior. From the slope we determine activation energies as indicated in the plot.
Fig. 3 .
3(a) : ID-VG (at VD = -20 V) for standard NOMFET (APTMS-NP-P5, L=5 µm) without OTS (■) and optimized NOMFET (APTMS-NP-OTS-P5, L=5 µm) with the OTS treatment (•).
Fig. 4 .Fig. 5
45Mobility in saturation for pentacene (P5 ■), pentacene on OTS (OTS-P5 •) OFETs, for NOMFETs with (APTS-NP-OTS-P5 ▼) and without OTS (APTS-NP-P5 ▲) Tapping mode-AFM images of the pentacene films for 2 OFETs (top) : pentacene on SiO2 (left) and pentacene on OTS-treated SiO2 (right); and 2 NOMFETs (bottom) : APTMS-NP-P5 (left) and APTMS-NP-OTS-P5 (right) structures. Image analysis (PSD : power spectral density) gives the following average grain size : 1.25 µm, 1.67 µm, 0.55 µm and 0.83 µm, respectively.
Fig. 6 .
6(a) Typical STP response of an optimized NOMFET (L= 1 µm, NPs of 10 nm) subjected to a sequence of spikes at various frequencies (5/50/10/5/20 Hz, see top part of the figure, pulse width = 10 ms) with a pulse amplitude of -1V. The red dots are fits with an analytical model (see Ref. 7 ) from which we extract the characteristic response time constant of the NOMFET (here : 187 ms). (b) STP for a standard NOMFET (L= 1 µm, NPs of 10 nm) without the OTS treatment
Fig. S1 .
S1SEM image (1 µm x 1µm) of the NPs network in the source-drain channel before the pentacene deposition.
Fig
. S2 shows the evolution of the hole mobility in saturation regime as a function of the thickness of the deposited pentacene film. We compare the behavior with and without the NPs.Without the NP, the onset of mobility starts at about 0.8 monolayer (ML) of pentacene in agreement with previous work, 1 corresponding to the onset of the percolation path in 1 st pentacene monolayer. The steep increase up to 2ML indicates the formation of the spatially confined channel where field effect induced carriers are generated. Then the slow increase of mobility reaching a plateau of 2-3x10 -3 cm 2 /Vs at about 15 ML is the fingerprint of a 2D-3D growth transition. 1, 2 31 Fig. S2. Evolution of the mobility in saturation versus the pentacene coverage for OFET (withoutNPs) and NOMFET with NPs (in this latter case, same data as inFig. 1).
Figure
S3 shows the Arrhenius plot of mobility in saturation measured, in-situ, at the end of the pentacene growth, i.e. for a film thickness of about 20 ML (30 nm). The presence of NPs leads to an increase of the activation energy to EA = 125 meV in contrast to EA = 65 meV in pure pentacene films. This latter value is in agreement with previous reports for pentacene OFET (in the range 20-80 meV). 3-7 Fig. S3. Arrhenius plot of the temperature dependence of the mobility in saturation for devices with and without NPs. With NPs, data are the same as in Fig. 2 for the APTMS/NP/P5 device.
Acknowledgments.
Synaptic electronics. A Demming, J K Gimzewski, D Vuillaume, Nanotechnology. 20133824Demming, A.; Gimzewski, J. K.; Vuillaume, D., Synaptic electronics. Nanotechnology 2013, 24 (38).
Memristive devices for computing. J J Yang, D B Strukov, D R Stewart, Nature Nanotechnology. 81Yang, J. J.; Strukov, D. B.; Stewart, D. R., Memristive devices for computing. Nature Nanotechnology 2012, 8 (1), 13-24.
The missing memristor found. D B Strukov, G S Snider, D R Stewart, R S Williams, Nature. 4537191Strukov, D. B.; Snider, G. S.; Stewart, D. R.; Williams, R. S., The missing memristor found. Nature 2008, 453 (7191), 80-83.
The Rise of Organic Bioelectronics. J Rivnay, R M Owens, G G Malliaras, Chem. Mater. 26679Rivnay, J.; Owens, R. M.; Malliaras, G. G., The Rise of Organic Bioelectronics. Chem. Mater 2014, 26, 679.
Organic ultra-thin film transistors with a liquid gate for extracellular stimulation and recording of electric activity of stem cell-derived neuronal networks. T Cramer, B Chelli, M Murgia, M Barbalinardo, E Bystrenova, D M De Leeuw, F Biscarini, Phys Chem Chem Phys. 3897Cramer, T.; Chelli, B.; Murgia, M.; Barbalinardo, M.; Bystrenova, E.; De Leeuw, D. M.; Biscarini, F., Organic ultra-thin film transistors with a liquid gate for extracellular stimulation and recording of electric activity of stem cell-derived neuronal networks. Phys Chem Chem Phys 2013, 15, 3897.
Electronic control of Ca2+ signalling in neuronal cells using an organic electronic ion pump. J Isaksson, P Kjall, D Nilsson, N D Robinson, M Berggren, A Richter-Dahlfors, Nat Mater. 69Isaksson, J.; Kjall, P.; Nilsson, D.; Robinson, N. D.; Berggren, M.; Richter-Dahlfors, A., Electronic control of Ca2+ signalling in neuronal cells using an organic electronic ion pump. Nat Mater 2007, 6 (9), 673-9.
An Organic Nanoparticle Transistor Behaving as a Biological Spiking Synapse. F Alibart, S Pleutin, D Guerin, C Novembre, S Lenfant, K Lmimouni, C Gamrat, D Vuillaume, Advanced Functional Materials. 202Alibart, F.; Pleutin, S.; Guerin, D.; Novembre, C.; Lenfant, S.; Lmimouni, K.; Gamrat, C.; Vuillaume, D., An Organic Nanoparticle Transistor Behaving as a Biological Spiking Synapse. Advanced Functional Materials 2010, 20 (2), 330-337.
. F Alibart, S Pleutin, O Bichler, C Gamrat, T Serrano-Gotarredona, LinaresAlibart, F.; Pleutin, S.; Bichler, O.; Gamrat, C.; Serrano-Gotarredona, T.; Linares-
A Memristive Nanoparticle/Organic Hybrid Synapstor for Neuroinspired Computing. B Barranco, D Vuillaume, Advanced Functional Materials. 20123Barranco, B.; Vuillaume, D., A Memristive Nanoparticle/Organic Hybrid Synapstor for Neuroinspired Computing. Advanced Functional Materials 2012, 22 (3), 609-616.
Functional Model of a Nanoparticle Organic Memory Transistor for Use as a Spiking Synapse. O Bichler, W S Zhao, F Alibart, S Pleutin, D Vuillaume, C Gamrat, IEEE Transactions on Electron Devices. 5711Bichler, O.; Zhao, W. S.; Alibart, F.; Pleutin, S.; Vuillaume, D.; Gamrat, C., Functional Model of a Nanoparticle Organic Memory Transistor for Use as a Spiking Synapse. IEEE Transactions on Electron Devices 2010, 57 (11), 3115-3122.
. O Bichler, W Zhao, F Alibart, S Pleutin, S Lenfant, D Vuillaume, C Gamrat, Bichler, O.; Zhao, W.; Alibart, F.; Pleutin, S.; Lenfant, S.; Vuillaume, D.; Gamrat, C.,
Pavlov's Dog Associative Learning Demonstrated on Synaptic-Like Organic Transistors. Neural Computation. 25Pavlov's Dog Associative Learning Demonstrated on Synaptic-Like Organic Transistors. Neural Computation 2013, 25, 549-566.
Spectroscopy of Molecular Junction Networks Obtained by Place Exchange in 2D. L Bernard, Y Kamdzhilov, M Calame, S J Vandermolen, J Liao, C Schönenberger, Bernard, L.; Kamdzhilov, Y.; Calame, M.; vanderMolen, S. J.; Liao, J.; Schönenberger, C., Spectroscopy of Molecular Junction Networks Obtained by Place Exchange in 2D
. Nanoparticle Arrays. J. Phys. Chem. C. 50Nanoparticle Arrays. J. Phys. Chem. C 2007, 111 (50), 18445-18450.
Gold nanoparticles: Assembly, supramolecular chemistry, quantum-size related properties, and applications toward biology, catalysis, and nanotechnology. M.-C Daniel, D Astruc, Daniel, M.-C.; Astruc, D., Gold nanoparticles: Assembly, supramolecular chemistry, quantum-size related properties, and applications toward biology, catalysis, and nanotechnology.
. Chem. Rev. 1041Chem. Rev. 2004, 104 (1), 293-346.
An improved method for the assembly of amino-terminated monolayers on SiO2 and the vapor deposition of gold layers. D F Siqueira Petri, G Wenz, P Schunk, T Schimmel, Langmuir. 1513Siqueira Petri, D. F.; Wenz, G.; Schunk, P.; Schimmel, T., An improved method for the assembly of amino-terminated monolayers on SiO2 and the vapor deposition of gold layers. Langmuir 1999, 15 (13), 4520-4523.
Evidence of a transition temperature for optimum deposition of grafted monolayer coatings. J B Brzoska, N Shahidzadeh, F Rondelez, Nature. 360Brzoska, J. B.; Shahidzadeh, N.; Rondelez, F., Evidence of a transition temperature for optimum deposition of grafted monolayer coatings. Nature 1992, 360, 719-721.
On the formation and structure of self-assembling monolayers. R Maoz, J Sagiv, J. Colloid and Interface Sciences. 1002Maoz, R.; Sagiv, J., On the formation and structure of self-assembling monolayers. J. Colloid and Interface Sciences 1984, 100 (2), 465-496.
A high-vacuum deposition system for in situ and real-time electrical characterization of organic thin-film transistors. S D Quiroga, A Shehu, C Albonetti, M Murgia, P Stoliar, F Borgatti, F Biscarini, Review of Scientific Instruments. 822Quiroga, S. D.; Shehu, A.; Albonetti, C.; Murgia, M.; Stoliar, P.; Borgatti, F.; Biscarini, F., A high-vacuum deposition system for in situ and real-time electrical characterization of organic thin-film transistors. Review of Scientific Instruments 2011, 82 (2), -.
Electronic properties of doped semiconductots. B I Shklovskii, A L Efros, SpringerBerlinShklovskii, B. I.; Efros, A. L., Electronic properties of doped semiconductots. Springer: Berlin, 1984.
Spatially correlated charge transport in organic thin film transistors. F Dinelli, M Murgia, P Levy, M Cavallini, F Biscarini, A Shehu, S D Quiroga, P Angelo, C Albonetti, F Borgatti, M Murgia, 116802. 19Phys. Rev. Lett. 1192Dinelli, F.; Murgia, M.; Levy, P.; Cavallini, M.; Biscarini, F., Spatially correlated charge transport in organic thin film transistors. Phys. Rev. Lett. 2004, 92 (11), 116802. 19. Shehu, A.; Quiroga, S. D.; D'Angelo, P.; Albonetti, C.; Borgatti, F.; Murgia, M.;
Transport properties of organic field effect transistors modified by quantum dots. A Scorzoni, P Stoliar, F Biscarini, Y Chen, A M Goldman, 246602. 20Physical Review Letters. 10424Applied Physics LettersScorzoni, A.; Stoliar, P.; Biscarini, F., Layered Distribution of Charge Carriers in Organic Thin Film Transistors. Physical Review Letters 2010, 104 (24), 246602. 20. Nishioka, M.; Chen, Y.; Goldman, A. M., Transport properties of organic field effect transistors modified by quantum dots. Applied Physics Letters 2008, 92 (15), 153308-3.
Trapping centers engineering by including of nanoparticles into organic semiconductors. M Weis, J Lin, T Manaka, M Iwamoto, Journal of Applied Physics. 10411Weis, M.; Lin, J.; Manaka, T.; Iwamoto, M., Trapping centers engineering by including of nanoparticles into organic semiconductors. Journal of Applied Physics 2008, 104 (11), 114502-7.
Trapping effect of metal nanoparticle mono-and multilayer in the organic field-effect transistor. K Lee, M Weis, J Lin, D Taguchi, E Majková, T Manaka, M Iwamoto, Journal of Applied Physics. 109664512Lee, K.; Weis, M.; Lin, J.; Taguchi, D.; Majková, E.; Manaka, T.; Iwamoto, M., Trapping effect of metal nanoparticle mono-and multilayer in the organic field-effect transistor. Journal of Applied Physics 2011, 109 (6), 064512.
Pentacene thin film transistors on inorganic dielectrics: Morphology, structural properties, and electronic transport. D Knipp, R A Street, A Völkel, J Ho, Journal of Applied Physics. 931Knipp, D.; Street, R. A.; Völkel, A.; Ho, J., Pentacene thin film transistors on inorganic dielectrics: Morphology, structural properties, and electronic transport. Journal of Applied Physics 2003, 93 (1), 347-355.
High mobility top-gated pentacene thin-film transistors. C Newman, R Chesterfield, M Panzer, C Frisbie, Journal of Applied Physics. 9884506Newman, C.; Chesterfield, R.; Panzer, M.; Frisbie, C., High mobility top-gated pentacene thin-film transistors. Journal of Applied Physics 2005, 98, 084506.
Gated four-probe measurements on pentacene thin-film transistors: Contact resistance as a function of gate voltage and temperature. T Minari, T Nemoto, S Isoda, P V Pesavento, R J Chesterfield, C R Newman, C D Frisbie, 034506. 26J. Appl. Phys. 9937312J. Appl. Phys.Minari, T.; Nemoto, T.; Isoda, S., Temperature and electric-field dependence of the mobility of a single-grain pentacene field-effect transistor. J. Appl. Phys. 2006, 99 (3), 034506. 26. Pesavento, P. V.; Chesterfield, R. J.; Newman, C. R.; Frisbie, C. D., Gated four-probe measurements on pentacene thin-film transistors: Contact resistance as a function of gate voltage and temperature. J. Appl. Phys. 2004, 96 (12), 7312.
Organic small molecule field-effect transistors with Cytop™ gate dielectric: Eliminating gate bias stress effects. W L Kalb, T Mathis, S Haas, A F Stassen, B Batlogg, Applied Physics Letters. 909Kalb, W. L.; Mathis, T.; Haas, S.; Stassen, A. F.; Batlogg, B., Organic small molecule field-effect transistors with Cytop™ gate dielectric: Eliminating gate bias stress effects. Applied Physics Letters 2007, 90 (9), -.
Control of carrier density by self-assembled monolayers in organic field-effect transistors. S Kobayashi, T Nishikawa, T Takenobu, S Mori, T Shimoda, T Mitani, H Shimotani, N Yoshimoto, S Ogawa, Y Iwasa, Nature Materials. 3Kobayashi, S.; Nishikawa, T.; Takenobu, T.; Mori, S.; Shimoda, T.; Mitani, T.; Shimotani, H.; Yoshimoto, N.; Ogawa, S.; Iwasa, Y., Control of carrier density by self-assembled monolayers in organic field-effect transistors. Nature Materials 2004, 3, 317-322.
Stacked pentacene layer organic thin-film transistors with improved characteristics. Y.-Y Lin, D J Gundlach, S F Nelson, T N Jackson, IEEE Electron Dev. Lett. 1812Lin, Y.-Y.; Gundlach, D. J.; Nelson, S. F.; Jackson, T. N., Stacked pentacene layer organic thin-film transistors with improved characteristics. IEEE Electron Dev. Lett. 1997, 18 (12), 606-608.
Pentacene-based organic thin film transistors. Y Lin, D J Gundlach, S F Nelson, T N Jackson, IEEE Trans. Electron Devices. 448Lin, Y.; Gundlach, D. J.; Nelson, S. F.; Jackson, T. N., Pentacene-based organic thin film transistors. IEEE Trans. Electron Devices 1997, 44 (8), 1325-1331.
Low-voltage organic transistors with an amorphous molecular gate dielectric. M Halik, H Klauk, U Zschieschang, G Schmid, C Dehm, M Schütz, S Maisch, F Effenberger, M Brunnbauer, F Stellacci, Nature. 431Halik, M.; Klauk, H.; Zschieschang, U.; Schmid, G.; Dehm, C.; Schütz, M.; Maisch, S.; Effenberger, F.; Brunnbauer, M.; Stellacci, F., Low-voltage organic transistors with an amorphous molecular gate dielectric. Nature 2004, 431, 963-966.
Oleophobic monolayers: I. Films adsorbed from solution in non-polar liquids. W C Bigelow, D L Pickett, W A Zisman, J. Colloid Sci. 1Bigelow, W. C.; Pickett, D. L.; Zisman, W. A., Oleophobic monolayers: I. Films adsorbed from solution in non-polar liquids. J. Colloid Sci. 1946, 1, 513-538.
Silanization of solid substrates : a step toward reproducibility. J B Brzoska, I Ben Azouz, F Rondelez, Langmuir. 1011Brzoska, J. B.; Ben Azouz, I.; Rondelez, F., Silanization of solid substrates : a step toward reproducibility. Langmuir 1994, 10 (11), 4367-4373.
An introduction to ultrathin organic films : from Langmuir-Blodgett to Selfassembly. A Ulman, C Wiltzsch, J Wagler, G Roewer, E Kroke, Dalton Transactions. 28Academic pressSol-gel analogous aminolysisammonolysis of chlorosilanes to chlorine-free SiUlman, A., An introduction to ultrathin organic films : from Langmuir-Blodgett to Self- assembly. Academic press: Boston, 1991. 35. Wiltzsch, C.; Wagler, J.; Roewer, G.; Kroke, E., Sol-gel analogous aminolysis- ammonolysis of chlorosilanes to chlorine-free Si/(C)/N-materials. Dalton Transactions 2009, (28), 5474-5477.
Self-replicating amphiphilic monolayers. R Maoz, S Matlis, E Dimasi, B M Ocko, J Sagiv, Nature. 384Maoz, R.; Matlis, S.; DiMasi, E.; Ocko, B. M.; Sagiv, J., Self-replicating amphiphilic monolayers. Nature 1996, 384, 150-153.
Hydrogen-bonded multilayers of self-assembling silanes: Structure elucidation by combined Fourier transform infra-red spectroscopy and X-ray scattering techniques. R Maoz, J Sagiv, D Degenhardt, H Möhwald, P Quint, Supramolecular Science. 21Maoz, R.; Sagiv, J.; Degenhardt, D.; Möhwald, H.; Quint, P., Hydrogen-bonded multilayers of self-assembling silanes: Structure elucidation by combined Fourier transform infra-red spectroscopy and X-ray scattering techniques. Supramolecular Science 1995, 2 (1), 9-24.
Contribution à l'étude des transistors à effet de champ organiques : Effet de la fonctionnalisation du diélectrique de grille par des monocouches auto-assemblées. M Ternisien, University of LillePhDTernisien, M. Contribution à l'étude des transistors à effet de champ organiques : Effet de la fonctionnalisation du diélectrique de grille par des monocouches auto-assemblées. PhD, University of Lille, 2008.
A quantitative description of short-term plasticity at excitatory synapses in layer 2/3 of rat primary visual cortex. J A Varela, K Sen, J Gibson, J Fost, L F Abbott, S B Nelson, J. Neurosci. 1720Varela, J. A.; Sen, K.; Gibson, J.; Fost, J.; Abbott, L. F.; Nelson, S. B., A quantitative description of short-term plasticity at excitatory synapses in layer 2/3 of rat primary visual cortex. J. Neurosci. 1997, 17 (20), 7926-7940.
. S Desbief, in preparationDesbief, S., in preparation.
. L Zhang, A Fonari, Y Liu, A.-L M Hoyt, H Lee, D Granger, S Parkin, Russell, Zhang, L.; Fonari, A.; Liu, Y.; Hoyt, A.-L. M.; Lee, H.; Granger, D.; Parkin, S.; Russell,
Bistetracene: An Air-Stable, High-Mobility Organic Semiconductor with Extended Conjugation. T P Anthony, J E Brédas, J.-L Coropceanu, V Briseno, A L , J Am Chem Soc. 136T. P.; Anthony, J. E.; Brédas, J.-L.; Coropceanu, V.; Briseno, A. L., Bistetracene: An Air-Stable, High-Mobility Organic Semiconductor with Extended Conjugation. J Am Chem Soc 2014, 136, 9248-9251.
Record High Electron Mobility of 6. B Sun, W Hong, Z Yan, H Aziz, Y Li, 32Sun, B.; Hong, W.; Yan, Z.; Aziz, H.; Li, Y., Record High Electron Mobility of 6.3 cm2V
−1s−1 Achieved for Polymer Semiconductors Using a New Building Block. 26−1s−1 Achieved for Polymer Semiconductors Using a New Building Block. Adv. Mater. 2014, 26 (17), 2636-2642.
. C Luo, A K K Kyaw, L A Perez, S Patel, M Wang, B Grimm, G C Bazan, Luo, C.; Kyaw, A. K. K.; Perez, L. A.; Patel, S.; Wang, M.; Grimm, B.; Bazan, G. C.;
Impact of dopant species on the interfacial trap density and mobility in amorphous In-X-Zn-O solutionprocessed thin-film transistors. E J Kramer, A J Heeger, M Benwadih, J A Chroboczek, G Ghibaudo, R Coppard, D Vuillaume, Journal of Applied Physics. 1421214501Nano Lett. REFERENCES 1Kramer, E. J.; Heeger, A. J., General Strategy for Self-Assembly of Highly Oriented Nanocrystalline Semiconducting Polymers with High Mobility. Nano Lett 2014, 14, 2764-2771. 44. Benwadih, M.; Chroboczek, J. A.; Ghibaudo, G.; Coppard, R.; Vuillaume, D., Impact of dopant species on the interfacial trap density and mobility in amorphous In-X-Zn-O solution- processed thin-film transistors. Journal of Applied Physics 2014, 115 (21), 214501. REFERENCES 1.
Spatially correlated charge transport in organic thin film transistors. F Dinelli, M Murgia, P Levy, M Cavallini, F Biscarini, Phys. Rev. Lett. 11116802Dinelli, F.; Murgia, M.; Levy, P.; Cavallini, M.; Biscarini, F., Spatially correlated charge transport in organic thin film transistors. Phys. Rev. Lett. 2004, 92 (11), 116802.
Layered Distribution of Charge Carriers in Organic Thin Film Transistors. A Shehu, S D Quiroga, P Angelo, C Albonetti, F Borgatti, M Murgia, A Scorzoni, P Stoliar, F Biscarini, Physical Review Letters. 10424246602Shehu, A.; Quiroga, S. D.; D'Angelo, P.; Albonetti, C.; Borgatti, F.; Murgia, M.; Scorzoni, A.; Stoliar, P.; Biscarini, F., Layered Distribution of Charge Carriers in Organic Thin Film Transistors. Physical Review Letters 2010, 104 (24), 246602.
Pentacene thin film transistors on inorganic dielectrics: Morphology, structural properties, and electronic transport. D Knipp, R A Street, A Völkel, J Ho, J. Appl. Phys. 931Knipp, D.; Street, R. A.; Völkel, A.; Ho, J., Pentacene thin film transistors on inorganic dielectrics: Morphology, structural properties, and electronic transport. J. Appl. Phys. 2003, 93 (1), 347-355.
Gated four-probe measurements on pentacene thin-film transistors: Contact resistance as a function of gate voltage and temperature. P V Pesavento, R J Chesterfield, C R Newman, C D Frisbie, J. Appl. Phys. 127312Pesavento, P. V.; Chesterfield, R. J.; Newman, C. R.; Frisbie, C. D., Gated four-probe measurements on pentacene thin-film transistors: Contact resistance as a function of gate voltage and temperature. J. Appl. Phys. 2004, 96 (12), 7312.
High mobility top-gated pentacene thin-film transistors. C Newman, R Chesterfield, M Panzer, C Frisbie, Journal of Applied Physics. 9884506Newman, C.; Chesterfield, R.; Panzer, M.; Frisbie, C., High mobility top-gated pentacene thin-film transistors. Journal of Applied Physics 2005, 98, 084506.
Temperature and electric-field dependence of the mobility of a single-grain pentacene field-effect transistor. T Minari, T Nemoto, S Isoda, J. Appl. Phys. 99334506Minari, T.; Nemoto, T.; Isoda, S., Temperature and electric-field dependence of the mobility of a single-grain pentacene field-effect transistor. J. Appl. Phys. 2006, 99 (3), 034506.
Transport properties of organic field effect transistors modified by quantum dots. M Nishioka, Y Chen, A M Goldman, Applied Physics Letters. 9215Nishioka, M.; Chen, Y.; Goldman, A. M., Transport properties of organic field effect transistors modified by quantum dots. Applied Physics Letters 2008, 92 (15), 153308-3.
| [] |
[
"Sample Selection for Fair and Robust Training",
"Sample Selection for Fair and Robust Training"
] | [
"Yuji Roh [email protected] \nKAIST\nUniversity of Wisconsin-Madison\nKAIST\n\n",
"Kangwook Lee [email protected] \nKAIST\nUniversity of Wisconsin-Madison\nKAIST\n\n",
"Steven Euijong Whang [email protected] \nKAIST\nUniversity of Wisconsin-Madison\nKAIST\n\n",
"Changho Suh \nKAIST\nUniversity of Wisconsin-Madison\nKAIST\n\n"
] | [
"KAIST\nUniversity of Wisconsin-Madison\nKAIST\n",
"KAIST\nUniversity of Wisconsin-Madison\nKAIST\n",
"KAIST\nUniversity of Wisconsin-Madison\nKAIST\n",
"KAIST\nUniversity of Wisconsin-Madison\nKAIST\n"
] | [] | Fairness and robustness are critical elements of Trustworthy AI that need to be addressed together. Fairness is about learning an unbiased model while robustness is about learning from corrupted data, and it is known that addressing only one of them may have an adverse affect on the other. In this work, we propose a sample selection-based algorithm for fair and robust training. To this end, we formulate a combinatorial optimization problem for the unbiased selection of samples in the presence of data corruption. Observing that solving this optimization problem is strongly NP-hard, we propose a greedy algorithm that is efficient and effective in practice. Experiments show that our algorithm obtains fairness and robustness that are better than or comparable to the state-of-the-art technique, both on synthetic and benchmark real datasets. Moreover, unlike other fair and robust training baselines, our algorithm can be used by only modifying the sampling step in batch selection without changing the training algorithm or leveraging additional clean data. | null | [
"https://arxiv.org/pdf/2110.14222v1.pdf"
] | 239,998,264 | 2110.14222 | 2505543a8458e6bcc0f3c3e715b419593c1b59f7 |
Sample Selection for Fair and Robust Training
Yuji Roh [email protected]
KAIST
University of Wisconsin-Madison
KAIST
Kangwook Lee [email protected]
KAIST
University of Wisconsin-Madison
KAIST
Steven Euijong Whang [email protected]
KAIST
University of Wisconsin-Madison
KAIST
Changho Suh
KAIST
University of Wisconsin-Madison
KAIST
Sample Selection for Fair and Robust Training
Fairness and robustness are critical elements of Trustworthy AI that need to be addressed together. Fairness is about learning an unbiased model while robustness is about learning from corrupted data, and it is known that addressing only one of them may have an adverse affect on the other. In this work, we propose a sample selection-based algorithm for fair and robust training. To this end, we formulate a combinatorial optimization problem for the unbiased selection of samples in the presence of data corruption. Observing that solving this optimization problem is strongly NP-hard, we propose a greedy algorithm that is efficient and effective in practice. Experiments show that our algorithm obtains fairness and robustness that are better than or comparable to the state-of-the-art technique, both on synthetic and benchmark real datasets. Moreover, unlike other fair and robust training baselines, our algorithm can be used by only modifying the sampling step in batch selection without changing the training algorithm or leveraging additional clean data.
Introduction
Trustworthy AI is becoming essential for modern machine learning. While a traditional focus of machine learning is to train the most accurate model possible, companies that actually deploy AI including Microsoft [2021], Google [2020], and IBM [2020] are now pledging to make their AI systems fair, robust, interpretable, and transparent as well. Among the key objectives, we focus on fairness and robustness because they are closely related and need to be addressed together. Fairness is about learning an unbiased model while robustness is about learning a resilient model even on noisy data, and both issues root from the same training data. A recent work [Roh et al., 2020] shows that improving fairness while ignoring noisy data may lead to a worse accuracy-fairness tradeoff. Likewise, focusing on robustness only may have an adverse affect on fairness as we explain below.
There are several possible approaches for supporting fairness and robustness together. One is an in-processing approach where the model architecture itself is modified to optimize the two objectives. In particular, the state-of-the-art approach called FR-Train [Roh et al., 2020] uses adversarial learning to train a classifier and two discriminators for fairness and robustness. However, such an approach requires the application developer to use a specific model architecture and is thus restrictive. Another approach is to preprocess the data [Kamiran and Calders, 2011] and remove any bias and noise before model training, but this requires modifying the data itself. In addition, we are not aware of a general data cleaning method that is designed to explicitly improve model fairness and robustness together.
Instead, we propose an adaptive sample selection approach for the purpose of improving fairness and robustness. This approach can easily be deployed as a batch selection method, which does not require Figure 1: The ProPublica COMPAS dataset [Angwin et al., 2016] has bias where the set (y = 1, z = 1) has the smallest size in Figure 1a. However, when training a logistic regression model on this dataset, the same set has the highest loss at every epoch as shown in Figure 1b. Blindly applying clean selection at any point will thus discard (y = 1, z = 1) samples the most, which results in even worse bias and possibly worse fairness. modifying the model or data. Our techniques build on top of two recent lines of sample selection research: clean selection for robust training [Rousseeuw, 1984, Song et al., 2019, Shen and Sanghavi, 2019 and batch selection for fairness [Roh et al., 2021]. Clean selection is a standard approach for discarding samples that have high-loss values and are thus considered noisy. More recently, batch selection for fairness has been proposed where the idea is to adaptively adjust sampling ratios among sensitive groups (e.g., black or white populations) so that the trained model is not discriminative.
We note that state-of-the-art clean selection techniques like Iterative Trimmed Loss Minimization (ITLM) [Shen and Sanghavi, 2019] are not designed to address fairness and may actually worsen the data bias when used alone. For example, Figure 1 shows how the ProPublica COMPAS dataset (used by U.S. courts to predict criminal recidivism rates [Angwin et al., 2016]) can be divided into four sets where the label y is either 0 or 1, and the sensitive attribute z is either 0 or 1. The sizes of the sets are biased where (y = 1, z = 1) is the smallest. However, the model loss on (y = 1, z = 1) is the highest among the sets as shown in Figure 1b. Hence, blindly applying clean selection to remove high-loss samples will discard (y = 1, z = 1) samples the most, resulting in a worse bias that may negatively affect fairness as we demonstrate in Sec. 2.
We thus formulate a combinatorial optimization problem for fair and robust sample selection that achieves unbiased sampling in the presence of data corruption. To avoid discrimination on a certain group, we adaptively adjust the maximum number of samples per (y, z) set using the recently proposed FairBatch system [Roh et al., 2021], which solves a bilevel optimization problem of unfairness mitigation and standard empirical risk minimization through batch selection. We show that our optimization problem for sample selection can be viewed as a multidimensional knapsack problem and is thus strongly NP-hard [Garey and Johnson, 1979]. We then propose an efficient greedy algorithm that is effective in practice. Our method supports prominent group fairness measures: equalized odds [Hardt et al., 2016] and demographic parity [Feldman et al., 2015]. For data corruption, we currently assume noisy labels [Song et al., 2020], which can be produced by random flipping or adversarial attacks.
Experiments on synthetic and benchmark real datasets (COMPAS [Angwin et al., 2016] and Adult-Census [Kohavi, 1996]) show that our method obtains accuracy and fairness results that are better than baselines using fairness algorithms [Zafar et al., 2017a,b, Roh et al., 2021] and on par with and sometimes better than FR-Train [Roh et al., 2020], which leverages additional clean data.
Notation Let θ be the model weights, x ∈ X be the input feature to the classifier, y ∈ Y be the true class, andŷ ∈ Y be the predicted class whereŷ is a function of (x, θ). Let z ∈ Z be a sensitive attribute, e.g., race or gender. Let d = (x, y) be a training sample that contains a feature and label. Let S be a selected subset from the entire dataset D, where the cardinality of D is n. In the model training, we use a loss function θ (·) where a smaller value indicates a more accurate prediction.
Unfairness in Clean Selection
We demonstrate how clean selection that only focuses on robust training may lead to unfair models. Throughout this paper, we use ITLM [Shen and Sanghavi, 2019] Figure 2: Performances of logistic regression with and without ITLM (ITLM and LR, respectively) on the ProPublica COMPAS dataset while varying the noise rate using label flipping [Paudice et al., 2018]. Although ITLM improves accuracy (higher is better), the fairness worsens according to the equalized odds and demographic parity disparity measures (lower is better).
method for its simplicity, although other methods can be used as well. We first introduce ITLM and demonstrate how ITLM can actually worsen fairness.
ITLM Cleaning samples based on their loss values is a traditional robust training technique [Rousseeuw, 1984] that is widely used due to its efficiency. Recently, ITLM was proposed as a scalable approach for sample selection where lower loss samples are considered cleaner. The following optimization problem is solved with theoretical guarantees of convergence:
min S:|S|= τ n si∈S θ (s i )
where S is the selected set, s i is a sample in S, and τ is the ratio of clean samples in the entire data.
Potential Unfairness
We show how applying ITLM may lead to unfairness through a simple experiment. We use the benchmark ProPublica COMPAS dataset [Angwin et al., 2016] where the label y indicates recidivism, and the sensitive attribute z is set to gender. We inject noise into the training set using label flipping techniques [Paudice et al., 2018]. For simplicity, we assume that ITLM knows the clean ratio τ of the data and selects τ n samples. We train a logistic regression model on the dataset, and ITLM iteratively selects clean samples during the training.
We show the accuracy and fairness performances of logistic regression with and without ITLM while varying the noise rate (i.e., portion of labels flipped) in Figure 2. As expected, ITLM increasingly improves the model accuracy for larger noise rates as shown in Figure 2a. At the same time, however, the fairness worsens according to the two measures we use in this paper (see definitions in Sec. 5)equalized odds [Hardt et al., 2016] disparity and demographic parity [Feldman et al., 2015] disparity -as shown in Figures 2b and 2c, respectively. The results confirm our claims in Sec. 1.
Framework
We provide an optimization framework for training models on top of clean selection with fairness constraints. The entire optimization involves three joint optimizations, which we explain in a piecewise fashion: (1) how to perform clean selection with fairness constraints, (2) how to train a model on top of the selected samples, and (3) how to generate the fairness constraints. In Sec. 4 we present a concrete algorithm for solving all the optimizations.
Clean Selection with Fairness Constraints
The first optimization is to select clean samples such that the loss is minimized while fairness constraints are satisfied. For now, we assume that the fairness constraints are given where the selected samples of a certain set (y = y, z = z) cannot exceed a portion λ (y,z) of the selected samples of the class y = y. Notice that z λ (y,z) = 1, ∀y ∈ Y. The fairness constraints are specified as upper bounds because using equality constraints may lead to infeasible solutions. We explain how the λ values are determined in Sec. 3.3. We thus solve the following optimization for sample selection:
min p n i=1 θ (d i ) p i (1) s.t. n i=1 p i ≤ τ n , j∈I (y,z) p j ≤ λ (y,z) |S y |, ∀(y, z) ∈ Y × Z,(2)p i ∈ {0, 1}, i = 1, ..., n
where p i indicates whether the data sample d i is selected or not, I (y,z) is an index set of the (y, z) class, and S y is the selected samples for y = y. Note that |S y | = j∈Iy p j . In Eq. 2, the first constraint comes from ITLM while the second constraint is for the fairness.
An important detail is that we use inequalities in all of our constraints, unlike ITLM. The reason is that we may not be able to find a feasible solution of selected samples if we only use equality constraints. For example, if λ (1,0) = λ (1,1) = 0.5, but we have zero (y = 1, z = 0) samples, then the fairness equality constraint for (y = 1, z = 0) can never be satisfied.
Model Training
The next optimization is to train a model on top of the selected samples from Sec. 3.1. Recall that the sample selection optimization only uses inequality constraints, which means that for some set (y = y, z = z), its size |S (y,z) | may be smaller than λ (y,z) |S y |. However, we would like to train the model as if there were λ (y,z) |S y | samples for the intended fairness performance. We thus reweight each set (y = y, z = z) by λ (y,z) |Sy| |S (y,z) | and perform weighted empirical risk minimization (ERM) for model training. We then solve the bilevel optimization of model training and sample selection in Sec. 3.1 as follows:
min θ si∈S λ (y,z)∈(Y,Z) I S (y,z) (s i ) λ (y,z) |S y | |S (y,z) | θ (s i )(3)
S λ = arg min S={di|pi=1}:Eq. 2
Eq. 1
where λ is the set of λ (y,z) for all z ∈ Z, y ∈ Y.
Fairness Constraint Generation
To determine the λ y,z values, we adaptively adjust sampling ratios of sensitive groups using the techniques in FairBatch [Roh et al., 2021]. For each batch selection, we compute the fairness of the intermediate model and decide which sensitive groups should be sampled more (or less) in the next batch. The sampling rates become the λ (y,z) values used in the fairness constraints of Eq. 2.
FairBatch supports all the group fairness measures used in this paper. As an illustration, we state FairBatch's optimization when using equalized odds disparity (see Sec. A.1 for the optimization using demographic parity disparity):
min λ max{|L (y,z ) − L (y,z) |}, z = z , z, z ∈ Z, y ∈ Y(5)
where L (y,z) = 1/|S λ(y,z) | S λ(y,z) θ (s i ), and S λ(y,z) is a subset of S λ from Eq. 4 for the (y = y, z = z) set. FairBatch adjusts λ in each epoch via a signed gradient-based algorithm to the direction of reducing unfairness. For example, if the specific sensitive group z is less accurate on the y = y samples in the epoch t (i.e., z is discriminated in terms of equalized odds), FairBatch selects more data from the (y = y, z = z) set to increase the model's accuracy on that set:
λ (t+1) (y,z) = λ (t) (y,z) − α · sign(L (y,z ) − L (y,z) ), z = z(6)
where α is the step size of λ updates. To integrate FairBatch with our framework, we adaptively adjust λ as above for an intermediate model trained on the selected set S λ from Eq. 4.
Algorithm 1: Greedy-Based Clean and Fair Sample Selection
Input: loss θ (d), fair ratio lambda λ, clean ratio τ
profit = max( θ (d)) − θ (d) sortIdx = Sort(profit) S ← [ ]
for idx in sortIdx do if d idx does not violate any constraint in Eq. 8 w.r.t. λ and τ then Append d idx to S Output :S Algorithm 2: Overall Fair and Robust Model Training Input: train data (x train , y train ), clean ratio τ , loss
function θ (·) d ← (x train , y train ) θ ← initial model parameters λ = {λ y,z |y ∈ Y, z ∈ Z} ← random sampling ratios for each epoch do S = Algorithm1( θ (d), λ, τ y ) Draw minibatches from S w.r.t. λ y,z |S y |/|S (y,z) | for each minibatch do
Update model parameters θ according to the minibatch Update ∀λ y,z ∈ λ using the update rule as in Eq. 6 Output :model parameters θ
Algorithm
We present our algorithm for solving the optimization problem in Sec. 3. We first explain how the clean selection with fairness constraints problem in Sec. 3.1 can be converted to a multi-dimensional knapsack problem, which has known solutions. We then describe the full algorithm that includes the three functionalities: clean and fair selection, model training, and fairness constraint generation.
The clean and fair selection problem can be converted to an equivalent multi-dimensional knapsack problem by (1) maximizing the sum of (max (2) re-arranging the fairness constraints so that the right-hand side expressions become constants (instead of containing the variable S y ) as follows:
( θ (d)) − θ (d i ))'s instead of minimizing the sum of θ (d i )'s where d is the set of d i 's andmax n i=1 (max( θ (d)) − θ (d i )) p i (7) s.t. n i=1 p i ≤ τ n, n i=1 w i p i ≤ τ n, where w i = 1 if d i / ∈ D y 1 − λ (y,z) if d i ∈ D y and d i / ∈ D z 2 − λ (y,z) if d i ∈ D (y,z) , ∀(y, z) ∈ Y × Z , (8) p i ∈ {0, 1}, i = 1, ..., n
where D (y,z) is a subset for the (y, z) class. More details on the conversion are in Sec. A.2.
The multi-dimensional knapsack problem is known to be strongly NP-hard [Garey and Johnson, 1979]. Although exact algorithms have been proposed, they have exponential time complexities [Kellerer et al., 2004] and are thus impractical. There is a polynomial-time approximation scheme (PTAS) [Caprara et al., 2000], but the actual computation time increases exponentially for more accurate solutions [Kellerer et al., 2004]. Since our method runs within a single model training, we cannot tolerate such runtimes and thus opt for a greedy algorithm that is efficient and known to return reasonable solutions in practice [Akçay et al., 2007]. Algorithm 1 takes a greedy approach by sorting all the samples in descending order by their (max( θ (d)) − θ (d i )) values and then sequentially selecting the samples that do not violate any of the constraints. The computational complexity is thus O(n log n) where n is the total number of samples.
We now present our overall model training process for weighted ERM in Algorithm 2. For each epoch, we first select a clean and fair sample set S using Algorithm 1. We then draw minibatches from S according to the (y = y, z = z)-wise weights described in Sec. 3.2 and update the model parameters θ via stochastic gradient descent. A minibatch selection with uniform sampling can be considered as an unbiased estimator of the ERM, so sampling each (y = y, z = z) set in proportion to its weight results in an unbiased estimator of the weighted ERM. Finally, at the end of the epoch, we update the λ values using the update rules as in Eq. 6. We discuss about convergence in Sec. A.3.
Experiments
We evaluate our proposed algorithm. We use logistic regression for all experiments. We evaluate our models on separate clean test sets and repeat all experiments with 5 different random seeds. We use PyTorch, and all experiments are run on Intel Xeon Silver 4210R CPUs and NVIDIA Quadro RTX 8000 GPUs. More detailed settings are in Sec. B.1.
Fairness Measures
We focus on two representative group fairness measures: (1) equalized odds (EO) [Hardt et al., 2016], whose goal is to obtain the same accuracy between sensitive groups conditioned on the true labels and (2) demographic parity (DP) [Feldman et al., 2015], which is satisfied when the sensitive groups have the same positive prediction ratio. We evaluate the fairness disparities over sensitive groups as follows: EO disparity = max z∈Z,y∈Y | Pr(ŷ = 1|z = z, y = y) − Pr(ŷ = 1|y = y)| and DP disparity = max z∈Z | Pr(ŷ = 1|z = z) − Pr(ŷ = 1)|. Note that the classifier is perfectly fair when the fairness disparity is zero.
Datasets We use a total of three datasets: one synthetic dataset and two real benchmark datasets. We generate a synthetic dataset using a similar method to Zafar et al. [2017a]. The synthetic dataset has 3,200 samples and consists of two non-sensitive features (x 1 , x 2 ), one sensitive feature z, and one label class y. Each sample (x 1 , x 2 , y) is drawn from the following Gaussian distributions: (x 1 , x 2 )|y = 1 ∼ N ([1; 1], [5, 1; 1, 5]) and (x 1 , x 2 )|y = 0 ∼ N ([−1; −1], [10, 1; 1, 3]). We add the sensitive feature z to have a biased distribution:
Pr(z = 1) = 7 Pr((x 1 , x 2 )|y = 1)/[7 Pr((x 1 , x 2 )|y = 1) + Pr((x 1 , x 2 )|y = 0)] where (x 1 , x 2 ) = (x 1 cos(π/5) − x 2 sin(π/5),
x 1 sin(π/5) + x 2 cos(π/5)). We visualize the synthetic dataset in Sec. B.2. We utilize two real datasets, ProPublica COMPAS [Angwin et al., 2016] and AdultCensus [Kohavi, 1996], and use the same pre-processing in IBM Fairness 360 [Bellamy et al., 2019]. COMPAS and AdultCensus consist of 5,278 and 43,131 samples, respectively. The labels in COMPAS indicate recidivism, and the labels in AdultCensus indicate each customer's annual income level. For both datasets, we use gender as the sensitive attribute. Our experiments do not use any direct personal identifier, such as name or date of birth.
Noise Injection We use two methods for noise injection: (1) label flipping [Paudice et al., 2018], which minimizes the model accuracy (Label Flipping) and (2) targeted label flipping [Roh et al., 2020], which flips the labels of a specific group (Group-Targeted Label Flipping). When targeting a group in (2), we choose the one that, when attacked, results in the lowest model accuracy on all groups. The two methods represent different scenarios for label flipping attacks. In Secs. 5.1, 5.2, and 5.4, we flip 10% of labels in the training data, and in Sec. 5.3, we flip 10% to 20% of labels (i.e., varying the noise rate) in the training data.
Baselines We compare our proposed algorithm with four types of baselines: (1) vanilla training using logistic regression (LR); (2) robust only training using ITLM [Shen and Sanghavi, 2019];
(3) fair only training using FairBatch [Roh et al., 2021]; and (4) fair and robust training where we evaluate two baselines and a state-of-the-art method called FR-Train [Roh et al., 2020]. The two baselines are as follows where each runs in two steps:
• ITLM→FB: Runs ITLM and then FairBatch on the resulting clean samples. • ITLM→Penalty: Runs ITLM and then an in-processing fairness algorithm [Zafar et al., 2017a,b] on the clean samples. The fairness algorithm adds a penalty term to the loss function for reducing the covariance between the sensitive attribute and the predicted labels.
FR-Train [Roh et al., 2020] performs adversarial training between a classifier and fair and robust discriminators. For the robust training, FR-Train relies on a separate clean validation set. Since FR-Train benefits from additional data, its performances can be viewed as an upper bound for our algorithm, which does not utilize such data.
In Sec. B.3, we also compare with other two-step baselines that improve the fairness both during and after clean sample selection.
Hyperparameters We choose the step size for updating λ (i.e., α in Eq. 6) within the candidate set {0.0001, 0.0005, 0.001} using cross-validation. We assume the clean ratio τ is known for any dataset. If the clean ratio is not known in advance, it can be inferred using cross-validation Tao, 2015, Yu et al., 2018]. For all baselines, we start from a candidate set of hyperparameters and [Roh et al., 2021]; and (4) fair and robust training: ITLM→FB, ITLM→Penalty [Zafar et al., 2017a,b], and FR-Train [Roh et al., 2020]. We flip 10% of labels in the training data. Experiments are repeated 5 times. We highlight the best and second-best performances among the fair and robust algorithms. use cross-validation to choose the hyperparameters that result in the best fairness while having an accuracy that best aligns with other results. More hyperparameters are described in Sec. B.1.
Accuracy and Fairness
We first compare the accuracy and fairness results of our algorithm with other methods on the synthetic dataset in Table 1. We inject noise into the synthetic dataset either using label flipping or targeted label flipping. For both cases, our algorithm achieves the highest accuracy and fairness results compared to the baselines. Compared to LR, our algorithm has higher accuracy and lower EO and DP disparities, which indicates better fairness. Compared to ITLM and FB, our algorithm shows much better fairness and accuracy, respectively. FB's accuracy is noticeably low as a result of a worse accuracy-fairness tradeoff in the presence of noise. We now compare with the fair and robust training methods. Both ITLM→FB and ITLM→Penalty usually show worse accuracy and fairness compared to our algorithm, which shows that noise and bias cannot easily be mitigated in separate steps.
Compared to FR-Train, our algorithm surprisingly has similar accuracy and better fairness, which suggests that sample selection techniques like ours can outperform in-processing techniques like FR-Train that also rely on clean validation sets. We also perform an accuracy-fairness trade-off comparison between our algorithm and FR-Train in Sec. B.4, where the trends are consistent with the results in Table 1.
We now make the same comparisons using real datasets. We show the COMPAS dataset results here in Table 2 and the AdultCensus dataset results in Sec. B.5 as they are similar. The results for LR, ITLM, and FB are similar to those for the synthetic dataset. Compared to the fair and robust algorithms ITLM->FB, ITLM->Penalty, and FR-Train, our algorithm usually has the best or second-best accuracy and fairness values as highlighted in Table 2. Unlike in the synthetic dataset, FR-Train obtains very high accuracy values with competitive fairness. We suspect that FR-Train is benefiting from its clean validation set, which motivates us to perform a more detailed comparison with our algorithm in the next section.
Detailed Comparison with FR-Train
We perform a more detailed comparison between our algorithm and FR-Train. In Sec. 5.1, we observed that FR-Train takes advantage of its clean validation set to obtain very high accuracy values on the COMPAS dataset. We would like to see how that result changes when using validation sets that are less clean. In Table 3, we compare with two scenarios: (1) FR-Train has no clean data, so the validation set is just as noisy as the training set with 10% of its labels flipped and (2) the validation set starts as noisy, but is then cleaned using ITLM, which is the best we can do if there is no clean data. As a result, our algorithm is comparable to the realistic scenario (2) for FR-Train. We also compare runtimes (i.e., wall clock times in seconds) of our algorithm and FR-Train on the synthetic and COMPAS datasets w.r.t. equalized odds. As a result, Table 4 shows that our algorithm is faster on the synthetic dataset, but a bit slower on the COM-PAS dataset. Our algorithm's runtime bottleneck is the greedy algorithm (Algorithm 1) for batch selection. FR-Train trains slowly because it trains three models (one classifier and two discriminators) together. Hence, there is no clear winner.
Varying the Noise Rate
We compare the algorithm performances when varying the noise rate (i.e., portion of labels flipped) in the data. Figure 3 shows the accuracy and fairness (w.r.t. demographic parity disparity) of LR, ITLM, FR-Train, and our algorithm when varying the noise rate in the synthetic data. Even if the noise rate increases, the relative performances among the four methods do not change significantly. Our algorithm still outperforms LR and ITLM in terms of accuracy and fairness and is comparable to FR-Train having worse accuracy, but better fairness. The results w.r.t. equalized odds disparity are similar and can be found in Sec. B.6.
Ablation Study
In Table 5, we conduct an ablation study to investigate the effect of each component in our algorithm. We consider three ablation scenarios: (1) remove the fairness constraints in Eq. 2, which reduce discrimination in sample selection, (2) remove the weights in Eq. 3, which ensure fair model training, and (3) remove both functionalities (i.e., same as ITLM). As a result, each ablation scenario leads to either worse accuracy and fairness or slightly-better accuracy, but much worse fairness. We thus conclude that both functionalities are necessary.
Related Work
Fair & Robust Training We cover the literature for fair training, robust training, and fair and robust training, in that order.
For model fairness, many definitions have been proposed to address legal and social issues [Narayanan, 2018]. Among the definitions, we focus on group fairness: equalized odds [Hardt et al., 2016] and demographic parity [Feldman et al., 2015]. The algorithms for obtaining group fairness can be categorized as follows: (1) pre-processing techniques that minimize bias in data [Kamiran and Calders, 2011, Zemel et al., 2013, Feldman et al., 2015, du Pin Calmon et al., 2017, Choi et al., 2020, Jiang and Nachum, 2020, (2) in-processing techniques that revise the training process to prevent the model from learning the bias [Kamishima et al., 2012, Zafar et al., 2017a,b, Agarwal et al., 2018, Cotter et al., 2019, and (3) post-processing techniques that modify the outputs of the trained model to ensure fairness [Kamiran et al., 2012, Hardt et al., 2016, Pleiss et al., 2017, Chzhen et al., 2019. However, these fairness-only approaches do not address robust training as we do. Beyond group fairness, there are other important fairness measures including individual fairness [Dwork et al., 2012] and causality-based fairness [Kilbertus et al., 2017, Kusner et al., 2017, Zhang and Bareinboim, 2018, Nabi and Shpitser, 2018, Khademi et al., 2019. Extending our algorithm to support these measures is an interesting future work.
Robust training focuses on training accurate models against noisy or adversarial data. While data can be problematic for many reasons [Xiao et al., 2015], most of the robust training literature assumes label noise [Song et al., 2020] and mitigates it by using (1) model architectures that are resilient against the noise [Chen and Gupta, 2015, Jindal et al., 2016, Bekker and Goldberger, 2016, Han et al., 2018a, (2) loss regularization techniques [Goodfellow et al., 2015, Pereyra et al., 2017, Tanno et al., 2019, Hendrycks et al., 2019, Menon et al., 2020, and (3) loss correction techniques [Patrini et al., 2017, Chang et al., 2017, Ma et al., 2018, Arazo et al., 2019. However, these works typically do not focus on improving fairness.
A recent trend is to improve both fairness and robustness holistically [Lee et al., 2021]. The closest work to our algorithm is FR-Train [Roh et al., 2020], which performs adversarial training among a classifier, a fairness discriminator, and a robustness discriminator. In addition, FR-Train relies on a clean validation set for robust training. In comparison, our algorithm is a sample selection method that does not require modifying the internals of model training and does not rely on a validation set either.
There are other techniques that solve different problems involving fairness and robustness. A recent study observes that feature selection for robustness may lead to unfairness [Khani and Liang, 2021], and another study shows the limits of fair learning under data corruption [Konstantinov and Lampert, 2021]. In addition, there are techniques for (1) fair training against noisy or missing sensitive group information [Hashimoto et al., 2018, Lamy et al., 2019, Lahoti et al., 2020, Awasthi et al., 2020, Celis et al., 2021, (2) fair training with distributional robustness [Mandal et al., 2020, Rezaei et al., 2021, (3) robust fair training under sample selection bias [Du and Wu, 2021], and (4) fairness-reducing attacks [Chang et al., 2020, Solans et al., 2020. In comparison, we treat fairness and robustness as equals and improve them together.
Sample Selection While sample selection is traditionally used for robust training, it is recently being used for fair training as well. The techniques are quite different where robust training focuses on selecting or fixing samples based on their loss values [Han et al., 2018b, Jiang et al., 2018, Chen et al., 2019, Shen and Sanghavi, 2019, while fair training is more about balancing the sample sizes among sensitive groups to avoid discrimination [Roh et al., 2021]. In comparison, we combine the two approaches by selecting samples while keeping a balance among sensitive groups.
Conclusion
We proposed to our knowledge the first sample selection-based algorithm for fair and robust training.
A key formulation is the combinatorial optimization problem of unbiased sampling in the presence of data corruption. We showed that this problem is strongly NP-hard and proposed an efficient and effective algorithm. In the experiments, our algorithm shows better performances in accuracy and fairness compared to other baselines. In comparison to the state-of-the-art FR-Train, our algorithm is competitive and easier to deploy without having to modify the model training or rely on a clean validation set.
Societal Impact & Limitation Although we anticipate our research to have a positive societal impact by considering fairness in model training, it may also have some negative impacts. First, our fairness-aware training may result in worse accuracy. Although we believe that fairness will be indispensable in future machine learning systems, we should not unnecessarily sacrifice accuracy. Second, choosing the right fairness measure is challenging, and a poor choice may lead to undesirable results. In applications like criminal justice and finance where fairness issues have been thoroughly discussed, choosing the right measure may be straightforward. On the other hand, there are new applications like marketing where fairness has only recently become an issue. Here one needs to carefully consider the social context to understand what it means to be fair.
We also discuss limitations. Fairness is a broad concept, and our contributions are currently limited to prominent group fairness measures. There are other important notions of fairness like individual fairness, and we believe that extending our work to address them is an interesting future work. In addition, our robust training is currently limited to addressing label noise. However, ITLM is also known to handle adversarial perturbation on the features, so we believe our algorithm can be extended to other types of attacks as well. Another avenue of research is to use other robust machine learning algorithms like Ren et al. [2018] in addition to ITLM.
A Appendix -Framework and Algorithm
A.1 Lambda Optimization for Demographic Parity
Continuing from Sec. 3.3, we state FairBatch's optimization for demographic parity disparity:
min λ max{| |S λ(y,z ) | |S λ(z ) | L (y,z ) − |S λ(y,z) | |S λ(z) | L (y,z) |}, z = z , z, z ∈ Z, y ∈ Z(9)
where L (y,z) = 1/|S λ(y,z) | S λ(y,z) θ (s i ), and S λ(y,z) is a subset of S λ from Eq. 4 for the (y = y, z = z) set. More details are described in Roh et al. [2021].
A.2 Fairness Constraints in the Multidimensional Knapsack Problem
Continuing from Sec. 4, we describe how we rearrange the fairness constraints so that the right-hand side expressions of Eq. 2 become constants instead of containing the variable S y . We first express Eq. 2 as a summation using the indicator function 1 D (·) and then move the right-hand side expression to the left-hand side:
j∈I (y,z) p j ≤ λ (y,z) |S y | ⇐⇒ n i=1 1 D (y,z) (d i ) p i ≤ λ (y,z) |S y | = λ (y,z) n i=1 1 Dy (d i ) p i ⇐⇒ n i=1 1 D (y,z) (d i ) p i − λ (y,z) n i=1 1 Dy (d i ) p i ≤ 0
where p i indicates whether the data sample d i is selected or not, I (y,z) is an index set of the (y, z) class, D (y,z) is a subset for the (y, z) class, 1 D (y,z) (d i ) is an indicator function that returns 1 if d i ∈ D (y,z) and 0 otherwise, and S y is the selected samples for y = y.
By considering each case formulated by the indicator functions, we can rewrite the inequality with example weights v i : y,z) . Finally, we add 1 for the above weights v i to make the new weights w i that are always positive:
n i=1 v i p i ≤ 0, where v i = 0 if d i / ∈ D y −λ (y,z) if d i ∈ D y and d i / ∈ D z 1 − λ (y,z) if d i ∈ D (n i=1 w i p i ≤ τ n, where w i = 1 if d i / ∈ D y 1 − λ (y,z) if d i ∈ D y and d i / ∈ D z 2 − λ (y,z) if d i ∈ D (y,z) .
A.3 Convergence of the Algorithm
Continuing from Sec. 4, we discuss the convergence of our algorithm. Currently, our algorithm does not have theoretical guarantees for convergence. However, both ITLM and FairBatch do have convergence guarantees under some assumptions. Hence, we suspect that our algorithm will converge under reasonable circumstances as well. Indeed in our experiments, we did not run into convergence issues so far. In more general applications, averaging the model predictions over the last few epochs can be a reasonable choice. synthetic dataset, we split the data into 2000, 1000, and 200 samples for the training, test, and validation sets, respectively. For the real datasets, we use 20% of the entire data as a test set and split the remaining data into 10:1 for the training and validation sets. Note that the validation set is only used in FR-Train. We choose the learning rate from the candidate set {0.0001, 0.0005} using cross-validation. For ITLM-related algorithms (i.e., ITLM, ITLM→FB, ITLM→Penalty, and Ours), we utilize warm-starting in training, where we train the first 100 epochs without fair or robust training.
B.2 Synthetic Data
Continuing from Sec. 5, we visualize the synthetic dataset. Figure 4 shows the synthetic dataset we utilized in Sec. 5. As we explained in the experimental setting, the synthetic dataset has 3,200 samples and consists of two non-sensitive features (x 1 , x 2 ), one binary sensitive feature z, and one binary label class y.
B.3 Other Fair and Robust Baselines
Continuing from Sec. 5, we compare two more fair and robust baselines that are variants of ITLM→FB and ITLM→Penalty. First, (ITLM+Penalty)→FB is similar to ITLM->FB except that we improve ITLM using fairness penalty terms. In particular, we add the following covariance term to the optimization: min S:|S|= τ N si∈S [l θ (s i ) + µ |Cov(z i ,ŷ i )|] where z i andŷ i are the sensitive group and the predicted label of the sample s i , respectively. After selecting samples via the optimization in ITLM+Penalty, we run FairBatch on the selected data. The second method (ITLM+Penalty)->Penalty is identical except that it runs Penalty as its second step instead of FairBatch. Table 6 shows the performances of the algorithms when using label flipping and group-targeted label flipping. For both types of label flipping, the new fair and robust baselines (i.e., (ITLM+Penalty)→FB and (ITLM+Penalty)→Penalty) perform better than LR, but worse than our algorithm in terms of accuracy and fairness. These results are similar to those of ITLM->FB and ITLM->Penalty.
B.4 A Trade-off Curve Comparison of Our Algorithm and FR-Train
Continuing from Sec. 5.1, we draw accuracy-fairness disparity trade-off curves of our algorithm and FR-Train. Figure 5 shows results using the synthetic dataset w.r.t. equalized odds (EO) disparity where the experimental settings are identical to Table 1. The trends are consistent with the other results in Table 1, where our algorithm usually shows better fairness (i.e., lower disparity) than FR-Train when the accuracy is similar. Another observation is that FR-Train's trade-off curve is noisy due to its adversarial training. Table 6: Performances on the synthetic test set w.r.t. equalized odds disparity (EO Disp.) and demographic parity disparity (DP Disp.). We compare our algorithm with LR and the two-step fair and robust baselines: ITLM→FB [Roh et al., 2021], ITLM→Penalty [Zafar et al., 2017a,b], (ITLM+Penalty)→FB [Roh et al., 2021], and (ITLM+Penalty)→Penalty [Zafar et al., 2017a,b]. We flip 10% of labels in the training data. Experiments are repeated 5 times.
B.5 Accuracy and Fairness -AdultCensus
Continuing from Sec. 5.1, we compare our algorithm with other methods on the AdultCensus dataset. The overall results are similar to those for the COMPAS dataset (Table 2), where our algorithm shows the best or second-best accuracy and fairness performances among the fair and robust training algorithms. Compared to FR-Train, our algorithm has similar accuracy and better fairness.
B.6 Varying the Noise Rate -Equalized Odds
Continuing from Sec. 5.3, we observe the accuracy and fairness w.r.t. equalized odds of the algorithms when varying the noise rate (i.e., flipping different amounts of labels) in the training data. Figure 6 shows the performances of logistic regression (LR), ITLM, FR-Train, and our algorithm for different noise rates. Similar to the demographic parity disparity results (Figure 3), our algorithm outperforms LR and ITLM while having worse accuracy, but better fairness compared to FR-Train. [Roh et al., 2021]; and (4) fair and robust training: ITLM→FB, ITLM→Penalty [Zafar et al., 2017a,b], and FR-Train [Roh et al., 2020]. We flip 10% of labels in the training data. Experiments are repeated 5 times. We highlight the best and second-best performances among the fair and robust algorithms. (b) Equalized odds disparity. Figure 6: Performances of LR, ITLM, FR-Train, and our algorithm (Ours) on the synthetic data while varying the noise rate using label flipping [Paudice et al., 2018].
Figure 3 :
3Performances of LR, ITLM, FR-Train, and our algorithm (Ours) on the synthetic data while varying the noise rate using label flipping[Paudice et al., 2018].
Figure 4 :
4Continuing from Sec. 5, we provide more details of the experimental settings. The batch sizes of the synthetic, COMPAS, and AdultCensus datasets are 100, 200, and 2000, respectively. For the The synthetic dataset.
Figure 5 :
5Accuracy-EO disp. trade-off curves of our algorithm and FR-Train on the synthetic dataset.
as a representative clean selection10%
15%
20%
25%
Noise Rate
0.4
0.5
Accuracy
ITLM
LR
(a) Accuracy.
10%
15%
20%
25%
Noise Rate
0.1
0.2
0.3
EO Disp.
ITLM
LR
(b) Equalized odds disparity.
10%
15%
20%
25%
Noise Rate
0.1
0.2
0.3
DP Disp.
ITLM
LR
(c) Demographic parity disparity.
Table 1 :
1Performances on the synthetic test set w.r.t. equalized odds disparity (EO Disp.) and
Table 2 :
2Performances on the COMPAS test set w.r.t. equalized odds disparity (EO Disp.) and demographic parity disparity (DP Disp.). Other experimental settings are identical to Table 1. ITLM→FB .520±.001 .064±.013 .523±.003 .039±.009 .538±.008 .315±.074 .520±.013 .130±.105 ITLM→Penalty .523±.003 .074±.027 .525±.001 .059±.008 .517±.014 .436±.028 .514±.007 .094±.022 Ours .521±.000 .044±.000 .544±.000 .025±.000 .545±.009 .084±.012 .521±.000 .024±.000 FR-Train .620±.009 .074±.015 .607±.016 .050±.013 .611±.029 .081±.020 .597±.039 .045±.014Label Flipping
Group-Targeted Label Flipping
Method
Acc.
EO Disp.
Acc.
DP Disp.
Acc.
EO Disp.
Acc.
DP Disp.
LR
.503±.002 .123±.021 .503±.002 .093±.020 .513±.000 .668±.000 .513±.000 .648±.000
ITLM
.536±.000 .145±.003 .536±.000 .094±.003 .539±.002 .573±.019 .539±.002 .547±.015
FB
.509±.001 .083±.013 .503±.002 .046±.010 .525±.004 .093±.003 .507±.004 .059±.010
Table 3 :
3Detailed comparison with FR-Train on the COMPAS test set using 10% label flipping. Train with noisy val. set cleaned with ITLM .531±.033 .087±.018 .530±.024 .059±.018 FR-Train with noisy val. setWe
Table 4 :
4Runtime comparison (in seconds)
of our algorithm and FR-Train on the syn-
thetic and COMPAS test sets w.r.t. equal-
ized odds disparity. Other experimental set-
tings are identical to Table 1.
Method
Synthetic COMPAS
FR-Train
90.986
230.304
Ours
44.746
241.291
Table 5 :
5Ablation study on the synthetic and COMPAS test sets using 10% label flipping. fairness const. .718±.002 .070±.004 .723±.001 .036±.002 .520±.002 .110±.005 .520±.002 .073±.002 W/o ERM weights .727±.001 .233±.014 .727±.000 .251±.009 .531±.004 .152±.018 .529±.007 .058±.020We consider
LR.665±.003 .557±.015 .665±.003 .400±.010 .600±.002 .405±.008 .600±.002 .300±.006 ITLM→FB .718±.003 .199±.020 .725±.002 .089±.032 .707±.001 .108±.030 .704±.003 .067±.027 ITLM→Penalty .651±.051 .172±.046 .674±.012 .068±.014 .706±.001 .080±.004 .688±.004 .044±.004 (ITLM+Penalty)→FB .668±.015 .183±.036 .714±.002 .045±.021 .702±.004 .125±.035 .688±.008 .049±.024 (ITLM+Penalty)→Penalty .685±.030 .213±.030 .694±.017 .069±.012 .700±.013 .182±.062 .695±.003 .058±.002 Ours .727±.005 .064±.005 .720±.001 .006±.001 .726±.001 .040±.002 .720±.001 .039±.007Label Flipping
Group-Targeted Label Flipping
Method
Acc.
EO Disp.
Acc.
DP Disp.
Acc.
EO Disp.
Acc.
DP Disp.
0.710
0.715
0.720
0.725
0.730
Accuracy
0.05
0.10
0.15
EO Disp.
FR-Train
Ours
Table 7 :
7Performances on the AdultCensus test set w.r.t. equalized odds disparity (EO Disp.) and demographic parity disparity (DP Disp.). We compare our algorithm with four types of baselines: (1) vanilla training: LR; (2) robust training: ITLM[Shen and Sanghavi, 2019]; (3) fair training: FB
LR .746±.003 .070±.002 .746±.003 .073±.001 .748±.006 .095±.002 .748±.006 .127±.006 ITLM .785±.018 .087±.031 .785±.018 .092±.016 .785±.011 .144±.023 .785±.011 .105±.006 FB .748±.002 .022±.002 .758±.004 .046±.007 .739±.014 .086±.037 .693±.002 .015±.005 ITLM→FB .772±.024 .047±.008 .776±.023 .073±.010 .773±.013 .047±.006 .769±.014 .053±.005 ITLM→Penalty .776±.023 .082±.015 .774±.024 .054±.018 .755±.026 .161±.018 .757±.003 .013±.003 Ours .771±.015 .029±.005 .782±.015 .049±.012 .761±.006 .047±.018 .760±.007 .034±.016 FR-Train .779±.007 .061±.009 .782±.007 .089±.005 .782±.008 .075±.018 .773±.012 .049±.010Label Flipping
Group-Targeted Label Flipping
Method
Acc.
EO Disp.
Acc.
DP Disp.
Acc.
EO Disp.
Acc.
DP Disp.
10%
15%
20%
Noise Rate
0.4
0.6
Accuracy
LR
ITLM
FR-Train
(w. Clean Val.)
Ours
(a) Accuracy.
10%
15%
20%
Noise Rate
0.00
0.25
0.50
EO Disp.
A reductions approach to fair classification. A Agarwal, A Beygelzimer, M Dudík, J Langford, H M Wallach, ICML. A. Agarwal, A. Beygelzimer, M. Dudík, J. Langford, and H. M. Wallach. A reductions approach to fair classification. In ICML, pages 60-69, 2018.
Greedy algorithm for the general multidimensional knapsack problem. Y Akçay, H Li, S H Xu, Annals of Operations Research. 1501Y. Akçay, H. Li, and S. H. Xu. Greedy algorithm for the general multidimensional knapsack problem. Annals of Operations Research, 150(1):17-29, 2007.
Machine bias: There's software used across the country to predict future criminals. And its biased against blacks. J Angwin, J Larson, S Mattu, L Kirchner, ProPublicaJ. Angwin, J. Larson, S. Mattu, and L. Kirchner. Machine bias: There's software used across the country to predict future criminals. And its biased against blacks. ProPublica, 2016.
Unsupervised label noise modeling and loss correction. E Arazo, D Ortego, P Albert, N O'connor, K Mcguinness, ICML. E. Arazo, D. Ortego, P. Albert, N. O'Connor, and K. McGuinness. Unsupervised label noise modeling and loss correction. In ICML, pages 312-321, 2019.
Equalized odds postprocessing under imperfect group information. P Awasthi, M Kleindessner, J Morgenstern, AISTATS. P. Awasthi, M. Kleindessner, and J. Morgenstern. Equalized odds postprocessing under imperfect group information. In AISTATS, pages 1770-1780, 2020.
Training deep neural-networks based on unreliable labels. A J Bekker, J Goldberger, IEEE ICASSP. A. J. Bekker and J. Goldberger. Training deep neural-networks based on unreliable labels. In IEEE ICASSP, pages 2682-2686, 2016.
AI fairness 360: An extensible toolkit for detecting and mitigating algorithmic bias. R K E Bellamy, K Dey, M Hind, S C Hoffman, S Houde, K Kannan, P Lohia, J Martino, S Mehta, A Mojsilovic, S Nagar, K N Ramamurthy, J T Richards, D Saha, P Sattigeri, M Singh, K R Varshney, Y Zhang, IBM J. Res. Dev. 634/5):4:1-4:15R. K. E. Bellamy, K. Dey, M. Hind, S. C. Hoffman, S. Houde, K. Kannan, P. Lohia, J. Martino, S. Mehta, A. Mojsilovic, S. Nagar, K. N. Ramamurthy, J. T. Richards, D. Saha, P. Sattigeri, M. Singh, K. R. Varshney, and Y. Zhang. AI fairness 360: An extensible toolkit for detecting and mitigating algorithmic bias. IBM J. Res. Dev., 63(4/5):4:1-4:15, 2019.
Approximation algorithms for knapsack problems with cardinality constraints. A Caprara, H Kellerer, U Pferschy, D Pisinger, EJOR. 1232A. Caprara, H. Kellerer, U. Pferschy, and D. Pisinger. Approximation algorithms for knapsack problems with cardinality constraints. EJOR, 123(2):333-345, 2000.
Fair classification with noisy protected attributes: A framework with provable guarantees. L E Celis, L Huang, V Keswani, N K Vishnoi, ICML. 2021L. E. Celis, L. Huang, V. Keswani, and N. K. Vishnoi. Fair classification with noisy protected attributes: A framework with provable guarantees. In ICML, 2021.
On adversarial bias and the robustness of fair machine learning. H Chang, T D Nguyen, S K Murakonda, E Kazemi, R Shokri, ArXivH. Chang, T. D. Nguyen, S. K. Murakonda, E. Kazemi, and R. Shokri. On adversarial bias and the robustness of fair machine learning. ArXiv, 2020.
Active bias: Training more accurate neural networks by emphasizing high variance samples. H.-S Chang, E Learned-Miller, A Mccallum, NeurIPS. H.-S. Chang, E. Learned-Miller, and A. McCallum. Active bias: Training more accurate neural networks by emphasizing high variance samples. In NeurIPS, 2017.
Understanding and utilizing deep neural networks trained with noisy labels. P Chen, B B Liao, G Chen, S Zhang, ICML. P. Chen, B. B. Liao, G. Chen, and S. Zhang. Understanding and utilizing deep neural networks trained with noisy labels. In ICML, pages 1062-1070, 2019.
Webly supervised learning of convolutional networks. X Chen, A Gupta, ICCV. X. Chen and A. Gupta. Webly supervised learning of convolutional networks. In ICCV, pages 1431-1439, 2015.
Fair generative modeling via weak supervision. K Choi, A Grover, T Singh, R Shu, S Ermon, ICML. K. Choi, A. Grover, T. Singh, R. Shu, and S. Ermon. Fair generative modeling via weak supervision. In ICML, 2020.
Leveraging labeled and unlabeled data for consistent fair binary classification. E Chzhen, C Denis, M Hebiri, L Oneto, M Pontil, NeurIPS. E. Chzhen, C. Denis, M. Hebiri, L. Oneto, and M. Pontil. Leveraging labeled and unlabeled data for consistent fair binary classification. In NeurIPS, pages 12760-12770. 2019.
Two-player games for efficient non-convex constrained optimization. A Cotter, H Jiang, K Sridharan, ALT. A. Cotter, H. Jiang, and K. Sridharan. Two-player games for efficient non-convex constrained optimization. In ALT, pages 300-332, 2019.
Robust fairness-aware learning under sample selection bias. W Du, X Wu, arXiv:2105.11570arXiv preprintW. Du and X. Wu. Robust fairness-aware learning under sample selection bias. arXiv preprint arXiv:2105.11570, 2021.
Optimized pre-processing for discrimination prevention. F Du Pin Calmon, D Wei, B Vinzamuri, K N Ramamurthy, K R Varshney, NeurIPS. F. du Pin Calmon, D. Wei, B. Vinzamuri, K. N. Ramamurthy, and K. R. Varshney. Optimized pre-processing for discrimination prevention. In NeurIPS, pages 3995-4004, 2017.
Fairness through awareness. C Dwork, M Hardt, T Pitassi, O Reingold, R S Zemel, ITCS. C. Dwork, M. Hardt, T. Pitassi, O. Reingold, and R. S. Zemel. Fairness through awareness. In ITCS, pages 214-226, 2012.
Certifying and removing disparate impact. M Feldman, S A Friedler, J Moeller, C Scheidegger, S Venkatasubramanian, SIGKDD. M. Feldman, S. A. Friedler, J. Moeller, C. Scheidegger, and S. Venkatasubramanian. Certifying and removing disparate impact. In SIGKDD, pages 259-268, 2015.
Computers and Intractability: A Guide to the Theory of NP-completeness. M R Garey, D S Johnson, Mathematical Sciences Series. W. H. FreemanM. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-completeness. Mathematical Sciences Series. W. H. Freeman, 1979.
Explaining and harnessing adversarial examples. I Goodfellow, J Shlens, C Szegedy, ICLR, 2015. Google. Responsible ai practices. I. Goodfellow, J. Shlens, and C. Szegedy. Explaining and harnessing adversarial examples. In ICLR, 2015. Google. Responsible ai practices. https://ai.google/responsibilities/ responsible-ai-practices, 2020.
Masking: A new perspective of noisy supervision. B Han, J Yao, N Gang, M Zhou, I Tsang, Y Zhang, M Sugiyama, NeurIPS. B. Han, J. Yao, N. Gang, M. Zhou, I. Tsang, Y. Zhang, and M. Sugiyama. Masking: A new perspective of noisy supervision. In NeurIPS, pages 5839-5849, 2018a.
Co-teaching: Robust training of deep neural networks with extremely noisy labels. B Han, Q Yao, X Yu, G Niu, M Xu, W Hu, I Tsang, M Sugiyama, NeurIPS. B. Han, Q. Yao, X. Yu, G. Niu, M. Xu, W. Hu, I. Tsang, and M. Sugiyama. Co-teaching: Robust training of deep neural networks with extremely noisy labels. In NeurIPS, pages 8535-8545, 2018b.
Equality of opportunity in supervised learning. M Hardt, E Price, N Srebro, NeurIPS. M. Hardt, E. Price, and N. Srebro. Equality of opportunity in supervised learning. In NeurIPS, pages 3315-3323, 2016.
Fairness without demographics in repeated loss minimization. T Hashimoto, M Srivastava, H Namkoong, P Liang, ICML. T. Hashimoto, M. Srivastava, H. Namkoong, and P. Liang. Fairness without demographics in repeated loss minimization. In ICML, pages 1929-1938, 2018.
Using pre-training can improve model robustness and uncertainty. D Hendrycks, K Lee, M Mazeika, ICML. D. Hendrycks, K. Lee, and M. Mazeika. Using pre-training can improve model robustness and uncertainty. In ICML, pages 2712-2721, 2019.
. Ibm Ai, IBM. Trusting ai. https://www.research.ibm.com/artificial-intelligence/trusted-ai/, 2020.
Identifying and correcting label bias in machine learning. H Jiang, O Nachum, AISTATS. H. Jiang and O. Nachum. Identifying and correcting label bias in machine learning. In AISTATS, pages 702-712, 2020.
Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels. L Jiang, Z Zhou, T Leung, L.-J Li, L Fei-Fei, ICML. L. Jiang, Z. Zhou, T. Leung, L.-J. Li, and L. Fei-Fei. Mentornet: Learning data-driven curriculum for very deep neural networks on corrupted labels. In ICML, pages 2304-2313, 2018.
Learning deep networks from noisy labels with dropout regularization. I Jindal, M Nokleby, X Chen, IEEE ICDM. I. Jindal, M. Nokleby, and X. Chen. Learning deep networks from noisy labels with dropout regularization. In IEEE ICDM, pages 967-972, 2016.
Data preprocessing techniques for classification without discrimination. F Kamiran, T Calders, Knowl. Inf. Syst. 331F. Kamiran and T. Calders. Data preprocessing techniques for classification without discrimination. Knowl. Inf. Syst., 33(1):1-33, 2011.
Decision theory for discrimination-aware classification. F Kamiran, A Karim, X Zhang, IEEE ICDM. F. Kamiran, A. Karim, and X. Zhang. Decision theory for discrimination-aware classification. In IEEE ICDM, pages 924-929, 2012.
Fairness-aware classifier with prejudice remover regularizer. T Kamishima, S Akaho, H Asoh, J Sakuma, ECML PKDD. T. Kamishima, S. Akaho, H. Asoh, and J. Sakuma. Fairness-aware classifier with prejudice remover regularizer. In ECML PKDD, pages 35-50, 2012.
Knapsack Problems. Springer Nature Book Archives Millennium. H Kellerer, U Pferschy, D Pisinger, H. Kellerer, U. Pferschy, and D. Pisinger. Knapsack Problems. Springer Nature Book Archives Millennium. 2004.
Fairness in algorithmic decision making: An excursion through the lens of causality. A Khademi, S Lee, D Foley, V Honavar, WWW. A. Khademi, S. Lee, D. Foley, and V. Honavar. Fairness in algorithmic decision making: An excursion through the lens of causality. In WWW, pages 2907-2914, 2019.
Removing spurious features can hurt accuracy and affect groups disproportionately. F Khani, P Liang, ACM FAccT. F. Khani and P. Liang. Removing spurious features can hurt accuracy and affect groups disproportionately. In ACM FAccT, pages 196-205, 2021.
Avoiding discrimination through causal reasoning. N Kilbertus, M Rojas-Carulla, G Parascandolo, M Hardt, D Janzing, B Schölkopf, NeurIPS. N. Kilbertus, M. Rojas-Carulla, G. Parascandolo, M. Hardt, D. Janzing, and B. Schölkopf. Avoiding discrimina- tion through causal reasoning. In NeurIPS, pages 656-666, 2017.
Scaling up the accuracy of naive-bayes classifiers: A decision-tree hybrid. R Kohavi, SIGKDD. R. Kohavi. Scaling up the accuracy of naive-bayes classifiers: A decision-tree hybrid. In SIGKDD, pages 202-207, 1996.
Fairness-aware learning from corrupted data. N Konstantinov, C H Lampert, ArXivN. Konstantinov and C. H. Lampert. Fairness-aware learning from corrupted data. ArXiv, 2021.
Counterfactual fairness. M J Kusner, J Loftus, C Russell, R Silva, NeurIPS. M. J. Kusner, J. Loftus, C. Russell, and R. Silva. Counterfactual fairness. In NeurIPS, pages 4066-4076. 2017.
Fairness without demographics through adversarially reweighted learning. P Lahoti, A Beutel, J Chen, K Lee, F Prost, N Thain, X Wang, E Chi, NeurIPS. P. Lahoti, A. Beutel, J. Chen, K. Lee, F. Prost, N. Thain, X. Wang, and E. Chi. Fairness without demographics through adversarially reweighted learning. In NeurIPS, pages 728-740, 2020.
Noise-tolerant fair classification. A Lamy, Z Zhong, A K Menon, N Verma, NeurIPS. A. Lamy, Z. Zhong, A. K. Menon, and N. Verma. Noise-tolerant fair classification. In NeurIPS, 2019.
Machine learning robustness, fairness, and their convergence. J.-G Lee, Y Roh, H Song, S E Whang, SIGKDD. 2021J.-G. Lee, Y. Roh, H. Song, and S. E. Whang. Machine learning robustness, fairness, and their convergence. In SIGKDD, 2021.
Classification with noisy labels by importance reweighting. T Liu, D Tao, IEEE TPAMI. 383T. Liu and D. Tao. Classification with noisy labels by importance reweighting. IEEE TPAMI, 38(3):447-461, 2015.
Dimensionality-driven learning with noisy labels. X Ma, Y Wang, M E Houle, S Zhou, S Erfani, S Xia, S Wijewickrema, J Bailey, ICML. X. Ma, Y. Wang, M. E. Houle, S. Zhou, S. Erfani, S. Xia, S. Wijewickrema, and J. Bailey. Dimensionality-driven learning with noisy labels. In ICML, pages 3355-3364, 2018.
Ensuring fairness beyond the training data. D Mandal, S Deng, S Jana, J Wing, D J Hsu, NeurIPS. D. Mandal, S. Deng, S. Jana, J. Wing, and D. J. Hsu. Ensuring fairness beyond the training data. In NeurIPS, pages 18445-18456, 2020.
Can gradient clipping mitigate label noise? In ICLR. A K Menon, A S Rawat, S J Reddi, S Kumar, A. K. Menon, A. S. Rawat, S. J. Reddi, and S. Kumar. Can gradient clipping mitigate label noise? In ICLR, 2020.
Responsible ai principles from microsoft. Microsoft, Microsoft. Responsible ai principles from microsoft. https://www.microsoft.com/en-us/ai/ responsible-ai, 2021.
Fair inference on outcomes. R Nabi, I Shpitser, AAAI. R. Nabi and I. Shpitser. Fair inference on outcomes. In AAAI, pages 1931-1940, 2018.
Translation tutorial: 21 fairness definitions and their politics. A Narayanan, ACM FAccT. 1170A. Narayanan. Translation tutorial: 21 fairness definitions and their politics. In ACM FAccT, volume 1170, 2018.
Making deep neural networks robust to label noise: A loss correction approach. G Patrini, A Rozza, A Menon, R Nock, L Qu, CVPR. G. Patrini, A. Rozza, A. Krishna Menon, R. Nock, and L. Qu. Making deep neural networks robust to label noise: A loss correction approach. In CVPR, pages 1944-1952, 2017.
Label sanitization against label flipping poisoning attacks. A Paudice, L Muñoz-González, E C Lupu, ECML PKDD. A. Paudice, L. Muñoz-González, and E. C. Lupu. Label sanitization against label flipping poisoning attacks. In ECML PKDD, pages 5-15, 2018.
Regularizing neural networks by penalizing confident output distributions. G Pereyra, G Tucker, J Chorowski, Ł Kaiser, G Hinton, G. Pereyra, G. Tucker, J. Chorowski, Ł. Kaiser, and G. Hinton. Regularizing neural networks by penalizing confident output distributions. 2017.
On fairness and calibration. G Pleiss, M Raghavan, F Wu, J M Kleinberg, K Q Weinberger, NeurIPS. G. Pleiss, M. Raghavan, F. Wu, J. M. Kleinberg, and K. Q. Weinberger. On fairness and calibration. In NeurIPS, pages 5684-5693, 2017.
Learning to reweight examples for robust deep learning. M Ren, W Zeng, B Yang, R Urtasun, ICML. M. Ren, W. Zeng, B. Yang, and R. Urtasun. Learning to reweight examples for robust deep learning. In ICML, pages 4331-4340, 2018.
Robust fairness under covariate shift. A Rezaei, A Liu, O Memarrast, B D Ziebart, AAAIA. Rezaei, A. Liu, O. Memarrast, and B. D. Ziebart. Robust fairness under covariate shift. AAAI, pages 9419-9427, 2021.
FR-Train: A mutual information-based approach to fair and robust training. Y Roh, K Lee, S E Whang, C Suh, ICML. Y. Roh, K. Lee, S. E. Whang, and C. Suh. FR-Train: A mutual information-based approach to fair and robust training. In ICML, pages 8147-8157, 2020.
FairBatch: Batch selection for model fairness. Y Roh, K Lee, S E Whang, C Suh, ICLR. 2021Y. Roh, K. Lee, S. E. Whang, and C. Suh. FairBatch: Batch selection for model fairness. In ICLR, 2021.
Least median of squares regression. P J Rousseeuw, Journal of the American Statistical Association. 79388P. J. Rousseeuw. Least median of squares regression. Journal of the American Statistical Association, 79(388): 871-880, 1984.
Learning with bad training data via iterative trimmed loss minimization. Y Shen, S Sanghavi, ICML. Y. Shen and S. Sanghavi. Learning with bad training data via iterative trimmed loss minimization. In ICML, pages 5739-5748, 2019.
Poisoning attacks on algorithmic fairness. D Solans, B Biggio, C Castillo, ECML PKDD. D. Solans, B. Biggio, and C. Castillo. Poisoning attacks on algorithmic fairness. In ECML PKDD, pages 162-177, 2020.
Selfie: Refurbishing unclean samples for robust deep learning. H Song, M Kim, J.-G Lee, ICML. H. Song, M. Kim, and J.-G. Lee. Selfie: Refurbishing unclean samples for robust deep learning. In ICML, pages 5907-5915, 2019.
Learning from noisy labels with deep neural networks: A survey. H Song, M Kim, D Park, J.-G Lee, ArXivH. Song, M. Kim, D. Park, and J.-G. Lee. Learning from noisy labels with deep neural networks: A survey. ArXiv, 2020.
Learning from noisy labels by regularized estimation of annotator confusion. R Tanno, A Saeedi, S Sankaranarayanan, D C Alexander, N Silberman, CVPR. R. Tanno, A. Saeedi, S. Sankaranarayanan, D. C. Alexander, and N. Silberman. Learning from noisy labels by regularized estimation of annotator confusion. In CVPR, pages 11244-11253, 2019.
Robust optimization for fairness with noisy protected groups. S Wang, W Guo, H Narasimhan, A Cotter, M Gupta, M Jordan, NeurIPS. S. Wang, W. Guo, H. Narasimhan, A. Cotter, M. Gupta, and M. Jordan. Robust optimization for fairness with noisy protected groups. In NeurIPS, pages 5190-5203, 2020.
Learning from massive noisy labeled data for image classification. T Xiao, T Xia, Y Yang, C Huang, X Wang, CVPR. T. Xiao, T. Xia, Y. Yang, C. Huang, and X. Wang. Learning from massive noisy labeled data for image classification. In CVPR, pages 2691-2699, 2015.
An efficient and provable approach for mixture proportion estimation using linear independence assumption. X Yu, T Liu, M Gong, K Batmanghelich, D Tao, CVPR. X. Yu, T. Liu, M. Gong, K. Batmanghelich, and D. Tao. An efficient and provable approach for mixture proportion estimation using linear independence assumption. In CVPR, pages 4480-4489, 2018.
Fairness constraints: Mechanisms for fair classification. M B Zafar, I Valera, M Gomez-Rodriguez, K P Gummadi, AISTATS. M. B. Zafar, I. Valera, M. Gomez-Rodriguez, and K. P. Gummadi. Fairness constraints: Mechanisms for fair classification. In AISTATS, pages 962-970, 2017a.
Fairness beyond disparate treatment & disparate impact: Learning classification without disparate mistreatment. M B Zafar, I Valera, M Gomez-Rodriguez, K P Gummadi, WWW. M. B. Zafar, I. Valera, M. Gomez-Rodriguez, and K. P. Gummadi. Fairness beyond disparate treatment & disparate impact: Learning classification without disparate mistreatment. In WWW, pages 1171-1180, 2017b.
Learning fair representations. R S Zemel, Y Wu, K Swersky, T Pitassi, C Dwork, ICML. R. S. Zemel, Y. Wu, K. Swersky, T. Pitassi, and C. Dwork. Learning fair representations. In ICML, pages 325-333, 2013.
Mitigating unwanted biases with adversarial learning. B H Zhang, B Lemoine, M Mitchell, AIES. B. H. Zhang, B. Lemoine, and M. Mitchell. Mitigating unwanted biases with adversarial learning. In AIES, pages 335-340, 2018.
Fairness in decision-making -the causal explanation formula. J Zhang, E Bareinboim, AAAI. J. Zhang and E. Bareinboim. Fairness in decision-making -the causal explanation formula. In AAAI, pages 2037-2045, 2018.
| [] |
[
"Fundamental properties of red-clump stars from long-baseline H-band interferometry",
"Fundamental properties of red-clump stars from long-baseline H-band interferometry"
] | [
"A Gallenne \nEuropean Southern Observatory\nAlonso de Córdova 310719001Casilla, SantiagoChile\n",
"G Pietrzyński \nNicolaus Copernicus Astronomical Centre\nPolish Academy of Sciences\nBartycka 1800-716WarszawaPoland\n\nDepartamento de Astronomía\nUniversidad de Concepción\n160-CCasilla, ConcepciónChile\n",
"D Graczyk \nDepartamento de Astronomía\nUniversidad de Concepción\n160-CCasilla, ConcepciónChile\n\nMillenium Institute of Astrophysics\nAv. Vicuña Mackenna 4860SantiagoChile\n\nPAN\nCentrum Astronomiczne im. Mikołaja Kopernika\nRabiańska 887-100ToruńPoland\n",
"N Nardetto \nLaboratoire Lagrange\nUMR7293\nUniversité de Nice Sophia-Antipolis\nCNRS\nObservatoire de la Côte d'Azur\nNiceFrance\n",
"A Mérand \nEuropean Southern Observatory\nKarl-Schwarzschild-Str. 285748GarchingGermany\n",
"P Kervella \nLESIA (UMR 8109\nObservatoire de Paris\nPSL\nCNRS\nUPMC\nUniv. Paris-Diderot\n5 place Jules Janssen92195MeudonFrance\n",
"W Gieren \nDepartamento de Astronomía\nUniversidad de Concepción\n160-CCasilla, ConcepciónChile\n\nMillenium Institute of Astrophysics\nAv. Vicuña Mackenna 4860SantiagoChile\n",
"S Villanova \nDepartamento de Astronomía\nUniversidad de Concepción\n160-CCasilla, ConcepciónChile\n",
"R E Mennickent \nDepartamento de Astronomía\nUniversidad de Concepción\n160-CCasilla, ConcepciónChile\n",
"B Pilecki \nNicolaus Copernicus Astronomical Centre\nPolish Academy of Sciences\nBartycka 1800-716WarszawaPoland\n"
] | [
"European Southern Observatory\nAlonso de Córdova 310719001Casilla, SantiagoChile",
"Nicolaus Copernicus Astronomical Centre\nPolish Academy of Sciences\nBartycka 1800-716WarszawaPoland",
"Departamento de Astronomía\nUniversidad de Concepción\n160-CCasilla, ConcepciónChile",
"Departamento de Astronomía\nUniversidad de Concepción\n160-CCasilla, ConcepciónChile",
"Millenium Institute of Astrophysics\nAv. Vicuña Mackenna 4860SantiagoChile",
"PAN\nCentrum Astronomiczne im. Mikołaja Kopernika\nRabiańska 887-100ToruńPoland",
"Laboratoire Lagrange\nUMR7293\nUniversité de Nice Sophia-Antipolis\nCNRS\nObservatoire de la Côte d'Azur\nNiceFrance",
"European Southern Observatory\nKarl-Schwarzschild-Str. 285748GarchingGermany",
"LESIA (UMR 8109\nObservatoire de Paris\nPSL\nCNRS\nUPMC\nUniv. Paris-Diderot\n5 place Jules Janssen92195MeudonFrance",
"Departamento de Astronomía\nUniversidad de Concepción\n160-CCasilla, ConcepciónChile",
"Millenium Institute of Astrophysics\nAv. Vicuña Mackenna 4860SantiagoChile",
"Departamento de Astronomía\nUniversidad de Concepción\n160-CCasilla, ConcepciónChile",
"Departamento de Astronomía\nUniversidad de Concepción\n160-CCasilla, ConcepciónChile",
"Nicolaus Copernicus Astronomical Centre\nPolish Academy of Sciences\nBartycka 1800-716WarszawaPoland"
] | [] | Observations of 48 red-clump stars were obtained in the H band with the PIONIER instrument installed at the Very Large Telescope Interferometer. Limb-darkened angular diameters were measured by fitting radial intensity profile I(r) to square visibility measurements. Half the angular diameters determined have formal errors better than 1.2 %, while the overall accuracy is better than 2.7 %. Average stellar atmospheric parameters (effective temperatures, metallicities and surface gravities) were determined from new spectroscopic observations and literature data and combined with precise Gaia parallaxes to derive a set of fundamental stellar properties. These intrinsic parameters were then fitted to existing isochrone models to infer masses and ages of the stars. The added value from interferometry imposes a better and independent constraint on the R − T eff plane. Our derived values are consistent with previous works, although there is a strong scatter in age between various models. This shows that atmospheric parameters, mainly metallicities and surface gravities, still suffer from a non-accurate determination, limiting constraints on input physics and parameters of stellar evolution models. | 10.1051/0004-6361/201833341 | [
"https://arxiv.org/pdf/1806.09572v1.pdf"
] | 119,331,707 | 1806.09572 | 0e3d72458b886d73ec1c9b2ac25e1cd84c58d135 |
Fundamental properties of red-clump stars from long-baseline H-band interferometry
June 26, 2018 June 26, 2018
A Gallenne
European Southern Observatory
Alonso de Córdova 310719001Casilla, SantiagoChile
G Pietrzyński
Nicolaus Copernicus Astronomical Centre
Polish Academy of Sciences
Bartycka 1800-716WarszawaPoland
Departamento de Astronomía
Universidad de Concepción
160-CCasilla, ConcepciónChile
D Graczyk
Departamento de Astronomía
Universidad de Concepción
160-CCasilla, ConcepciónChile
Millenium Institute of Astrophysics
Av. Vicuña Mackenna 4860SantiagoChile
PAN
Centrum Astronomiczne im. Mikołaja Kopernika
Rabiańska 887-100ToruńPoland
N Nardetto
Laboratoire Lagrange
UMR7293
Université de Nice Sophia-Antipolis
CNRS
Observatoire de la Côte d'Azur
NiceFrance
A Mérand
European Southern Observatory
Karl-Schwarzschild-Str. 285748GarchingGermany
P Kervella
LESIA (UMR 8109
Observatoire de Paris
PSL
CNRS
UPMC
Univ. Paris-Diderot
5 place Jules Janssen92195MeudonFrance
W Gieren
Departamento de Astronomía
Universidad de Concepción
160-CCasilla, ConcepciónChile
Millenium Institute of Astrophysics
Av. Vicuña Mackenna 4860SantiagoChile
S Villanova
Departamento de Astronomía
Universidad de Concepción
160-CCasilla, ConcepciónChile
R E Mennickent
Departamento de Astronomía
Universidad de Concepción
160-CCasilla, ConcepciónChile
B Pilecki
Nicolaus Copernicus Astronomical Centre
Polish Academy of Sciences
Bartycka 1800-716WarszawaPoland
Fundamental properties of red-clump stars from long-baseline H-band interferometry
June 26, 2018 June 26, 2018Astronomy & Astrophysics manuscript no. 33341 c ESO 2018Techniques: high angular resolutioninterferometric ; Stars: fundamental parameterslate-type
Observations of 48 red-clump stars were obtained in the H band with the PIONIER instrument installed at the Very Large Telescope Interferometer. Limb-darkened angular diameters were measured by fitting radial intensity profile I(r) to square visibility measurements. Half the angular diameters determined have formal errors better than 1.2 %, while the overall accuracy is better than 2.7 %. Average stellar atmospheric parameters (effective temperatures, metallicities and surface gravities) were determined from new spectroscopic observations and literature data and combined with precise Gaia parallaxes to derive a set of fundamental stellar properties. These intrinsic parameters were then fitted to existing isochrone models to infer masses and ages of the stars. The added value from interferometry imposes a better and independent constraint on the R − T eff plane. Our derived values are consistent with previous works, although there is a strong scatter in age between various models. This shows that atmospheric parameters, mainly metallicities and surface gravities, still suffer from a non-accurate determination, limiting constraints on input physics and parameters of stellar evolution models.
Introduction
Direct stellar angular diameter measurements are a valuable observable with which to determine a star's fundamental properties, particularly the linear radius and absolute luminosity through the combination of Gaia parallaxes. These properties, together with the mass, are particularly necessary to constrain stellar structure and evolution models. Usually, spectroscopy is used to determine the effective temperatures, surface gravities, and metallicities of a star which, combined with a known distance, provide the stellar radius and luminosity. This is then fitted to evolutionary tracks to yield the stellar mass. However, the spectroscopic atmospheric parameters strongly depend on the atmosphere models used, and do not provide estimates accurate enough to well constrain stellar evolution models. Various models exist in the literature, using different input physics and parameters (e.g. the helium content, mixing length parameter, amongst others), which can only be constrain with more precise measurements (see e.g. Gallenne et al. 2016;Valle et al. 2017). Additional accurate parameters such as angular diameters provide independent constraints on the linear radii and luminosities.
Based on observations made with ESO telescopes at the La Silla-Paranal observatory under programme IDs 092.D-0297, 094.D-0074 and 4100.L-0105
The high angular resolution obtained from long-baseline interferometry (LBI) enables us to spatially resolve the photospheric disks of the apparent biggest stars (typically a diameter > 0.5 mas, i.e. ∼ 10 R at 100 pc). LBI can provide very accurate angular size measurements, as already demonstrated (see e.g. Nordgren et al. 1999;Mozurkewich et al. 2003;Kervella et al. 2004;Baines et al. 2010;Boyajian et al. 2012;Gallenne et al. 2012), and provide a valuable constraint on the input physics of theoretical stellar models.
In this paper, we present the measurements of the angular diameter of 48 F and G-type red-clump giant stars observed with the Very Large Telescope Interferometer (VLTI). The purpose is to accurately determine the absolute properties of such kind of stars through additional observable constraints. Details on the observations and data reduction are presented in Sect. 2, including additional spectroscopic observations. Sect. 3 is dedicated to the determination of limb-darkened angular diameters using atmospheric models. Stellar properties and derived masses and ages are presented in Sect. 4, and we summarize in Sect. 5.
Observations and data reduction
Selected targets
All our red-clump stars were selected from Laney et al. (2012), for which uniform and accurate near-infrared magnitudes have been obtained (∼ 0.005 mag). The initial goal of these interferometric observations was to measure their angular diameter to calibrate the surface brightness-colour (SBC) relation for late-type stars. These diameters were then combined with highquality and homogeneous V-and K-band photometry (Mermilliod et al. 1997;Laney et al. 2012) following the relation
S V = V 0 + 5 log θ LD = a (V − K) 0 + b.
This new accurate calibration of a and b, specific for these stars, is published in Pietrzynski et al. (2018), and enables the determination of angular diameters at a 0.8 % accuracy level (r.m.s. of the SBC relation of 0.018 mag). We then used this relation for late-type eclipsing systems to measure the most accurate distance of the Large Magellanic Cloud to 1 % (Pietrzynski et al. 2018). This provides the best reference point ever obtained for the cosmic distance scale. Furthermore, accurate parallaxes for these red-clump stars can be found in the Gaia DR2 (Gaia Collaboration et al. 2018, and combining them with angular diameter measurements provides unbiased informations on their intrinsic fundamental parameters, such as the linear radius and luminosity. Our selection criteria in the dataset of Laney et al. (2012) were to choose:
targets with declination < 20 • in order to be observable from the VLTI, expected angular diameters > 0.8 mas to be sufficiently resolved by the longest available VLTI baseline, not flagged as binary in the Simbad database Also taking also into account the observability of the targets for visitor mode observations, we finally ended up with a total of 48 targets.
VLTI/PIONIER interferometric observations
We used the Very Large Telescope Interferometer (VLTI ; Haguenauer et al. 2010) with the four-telescope combiner PI-ONIER (Precision Integrated Optics Near-infrared Imaging Ex-peRiment, Le Bouquin et al. 2011) to measure squared visibili-ties (V 2 ) and closure phases (CP) of our red-clump stars. PIO-NIER combines the light coming from four telescopes in the H band, either in a broad band mode or with a low spectral resolution, where the light is dispersed across a few spectral channel. Before Dec. 2014, the fringe dispersion was possible across three or seven channels, then PIONIER was upgraded with a new detector and a new GRISM dispersion mode with six spectral channels.
Our observations were carried out from 2013 to 2015 using the 1.8 m Auxiliary Telescopes with the configurations A1-G1-J3-K0 and A1-G1-I1-K0, providing six projected baselines ranging from 45 to 140 m. PIONIER was set up in dispersed mode for all targets, that is, the fringes are dispersed into three, seven, or six spectral channels.
We monitored the interferometric transfer function with the standard procedure which consists of interleaving the science target by reference stars. The calibrators were selected using the SearchCal software 1 (Bonneau et al. 2006(Bonneau et al. , 2011 provided by the Jean-Marie Mariotti Center 2 (JMMC), and are listed in Table 1, together with the journal of the observations. The data have been reduced with the pndrs package described in Le Bouquin et al. (2011). The main procedure is to compute squared visibilities and triple products for each baseline and spectral channel, and to correct for photon and readout noises.
Wavelength calibration
There are several sources of systematic in interferometry, but most of them are known and reduced to less than 0.1 %. For the interferometric observable, the visibility, which is a function of the spatial frequency B/λ, the main systematic is the accuracy on the wavelength calibration. The VLTI baseline lengths during the observations are known at an accuracy of better then 0.01 % thanks to a metrology laser, but the accuracy of the spectral calibration is instrument-dependent. In the case of PIONIER, this calibration is linked to scanning piezos used for the optical path delay modulation.
We performed during two hours several spectral calibrations on a cloudy night, with PIONIER in order to check the stability and repeatability (of the wavelength of each spectral channel). Over the two hours duration of the test, without moving the optics, the repeatability was found to be precise to ∼ 0.02 %. The comparison with the spectral calibration taken at the beginning of night (6h before) is precise to ∼ 0.06 %, while the comparison with the end of the night calibration, after moving the optics, gives an immediate repeatability precise to ∼ 0.02 %. However, there is an additional systematic error because the accuracy is limited by the calibration of the scanning piezo, which is usually assumed to be accurate to about 1 %. To better quantify this, we performed specific observations with GRAVITY, the second generation interferometric instrument of the VLTI (Eisenhauer et al. 2011), which has a dedicated internal reference laser source allowing a wavelength accuracy better than 0.02 %. GRAVITY can therefore be used to cross-calibrate PIONIER through observations of a same target.
Our calibration observations consisted of observing the very well known binary star TZ For, for which the orbital parameters were derived with exquisite accuracy by Gallenne et al. (2016). we used the usual interferometric observing method by interleaving the TZ For observations by calibrator stars in order to monitor the transfer function of each instrument. The observing sequence with the instruments for each half night was P-G-P-G (we will call a sequence P-G a dataset). Data were reduced with the corresponding instrument pipeline, and the relative astrometric position of the secondary for each observation determined using the CANDID tool (see Sect. 2.4). In Fig. 1 we plot the relative difference between our observed projected separations for each instrument and the ones calculated from the orbital solutions of Gallenne et al. (2016). We see a very good agreement in results between the two instruments and with the calculated position. For the first half night, we measured a relative difference of 0.22 % for the first data set (first P-G), and 0.28 % for the second one. For the first half night we found 0.29 %. For the second half night, we determined a relative difference of 0.31 % for the first data set, and 0.22 % for the second one. For the whole half night we also found 0.29 %. Combining all the data, we measured a relative difference of 0.35 %, which we adopt as the systematic uncertainty for the PIONIER wavelength calibration. We note that this is in very good agreement with the 0.4 % we previously determined in Kervella et al. (2017) from a different method.
In Fig. 1 we also see a slight systematic negative offset. Although this has no impact on our previous wavelength calibration analysis, this particularly shows that the orbital solutions need to be slightly revised. As this is not the goal of the paper, we determined new orbital solutions including the new PIONIER and GRAVITY measurements in Appendix A.
Checking for binarity
Although our selected targets are not identified as binary stars, detecting unknown orbiting high-contrast companions are still possible when using high-angular resolution techniques. We first had a visual analysis of the data to see any variations in the V 2 and CP measurements and cross-checked this with possible variations in the signal of the calibrators we used. A binary calibrator would lead to a bias estimate of the observables of the science target. We flagged all suspicious calibrators and reran the calibration process.
We then used the CANDID tool 3 (Gallenne et al. 2015) on all stars of our sample to detect possible components which might bias the angular diameter determinations or/and the photometry. This is particularly important for the calibration of the SBC relation as any binary would result as an outlier. In Table 1 we reported the stars for which a companion might have been detected with more than a 3σ level. We show an example is shown in Fig. 2 for HD45415, for which a companion is detected with a flux ratio f ∼ 1.4 % in H. However, the possible companions are not strongly constrained with our observations, as we only have one or two brackets per epoch and a small (u, v) coverage. Undetected astrometric faint companions which can bias the visible or IR photometry are still possible.
Spectroscopic observations
We collected high-resolution echelle spectra from HARPS spectrograph located in La Silla Observatory (Mayor et al. 2003) and CHIRON spectrograph located in Cerro Tololo Observatory (Tokovinin et al. 2013). HARPS was used in EGGS mode offering a spectral resolution of R ∼ 80000 and CHIRON was also used with a resolution of R ∼ 80000. Both instruments cover the spectral range 3900 − 6900 Å. Calibrated spectra were obtained using the dedicated provided pipelines.
From the reduced spectra we performed two analyses to determined the atmospheric stellar parameters. First, we determined the effective temperature T eff , the surface gravity log g, the metallicity [Fe/H] and the microturbulent velocity v t as described in Villanova et al. (2010), in other words, using the local thermodynamic equilibrium programme MOOG (Sneden 1973) and the equivalent widths (EQW) of the Fe i and Fe ii spectral lines. As a first step, atmospheric models were calculated using ATLAS9 models (Kurucz 1970) and initial estimates from the literature. Then, T eff , log g, and v t were adjusted and new atmospheric models were calculated in an interactive way, in order to remove trends in excitation potential and EQWs versus abundance for T eff , and v t , respectively, and to satisfy the ionization equilibrium for log g. The [Fe/H] value of the model was changed at each iteration, according to the output of the abundance analysis. A second determination of the effective temperature was also derived using the formalism of Kovtyukh & Gorlova (2000) based on spectral lines depth ratios and a calibration for giant stars (Kovtyukh et al. 2006). A third estimate of the temperature was determined using T eff − (V − K) calibrations (Houdashelt et al. 2000;Ramírez & Meléndez 2005;Worthey & Lee 2011). Finally, we retrieved additional measurement of temperatures, gravities, metallicities and velocities from the literature, when available. We used averages and standard deviations as our final values and uncertainties.
Limb-darkened angular diameters
We determine the limb-darkened angular diameters, θ LD , for each star by fitting the calibrated squared visibilities. Assuming a circular symmetry, we followed the formalism of Mérand et al. (2015) which consist in extracting the radial intensity profile I(r) of the spherical SATLAS models (Neilson & Lester 2013), which was converted to a visibility profile using a Hankel transform:
V λ (x) = 1 0 I λ (r)J 0 (rx)rdr 1 0 I λ (r)rdr ,(1)
where λ is the wavelength, x = πθ LD B/λ, B is the interferometric baseline projected on to the sky, J 0 the Bessel function of the first kind, and r = 1 − µ 2 , with µ = cos(θ), θ being the angle between the line of sight and a surface element of the star. The SATLAS grid models span effective temperatures from 3000 to 8000 K in steps of 100 K, effective gravities from -1 to 3 in steps of 0.25, and masses from 0.5 to 20 M . For all stars, we chose the model with the closest temperature and gravity, and for a stellar mass of 1 M (typical for such stars). Effective temperatures and gravities were determined as explained in Sect. 2.5.
In Fig. 3 we plotted the visibility curve for two stars, and all the measured angular diameters are listed in Table 2. Errors were determined using the bootstrapping technique (with replacement) on all baselines. The listed diameters correspond to the median of the distribution and the maximum value between the 16th and 84th percentile as uncertainty. Changing the models with T eff ± 200 K, log g ± 0.5 and M ± 2.5 M change the diameters by at most 0.3 %, which we added as error for each value to be conservative. Finally, we also added 0.35 % due to the systematic uncertainty from the wavelength calibration of PIONIER. The final overall angular diameter accuracy is better than 2.7 %, with a median value of 1.2 %.
Stellar properties
Stellar radii and luminosities
From the measured angular diameters and Gaia parallaxes (or from Hipparcos if not in the Gaia DR2, see Table 1), we can derive the stellar radii and the luminosities through the following equations
R[R ] = 107.523 θ LD [mas] π[mas] ,(2)L L = R R 2 T eff T eff, 4(3)
The values are listed in Table 2. For the conversions, we adopted the nominal solar and astronomical constants from IAU 2015 Resolution B3 (Prša et al. 2016) and CODATA values (Mohr et al. 2016). Gaia parallaxes were corrected from the zero point offset of ∼ 0.03 mas, and we quadratically added to the uncertainties a (conservative) systematic error of ±0.1 mas (Gaia Collaboration et al. 2018).
Ages and masses
We used the PARSEC (Bressan et al. 2012) and BaSTI (Pietrinferni et al. 2004) isochrone models to estimate the stellar masses and ages. These models are well suited as they include the horizontal and asymptotic giant branch evolutionary phases, and contain a wide range of initial masses and metallicities. In addition, it enable us to test the uncertainty of age and mass estimate induced by different stellar models.
PARSEC models are computed for a scaled-solar composition with Z = 0.0152, follow a helium initial content relation Y i = 0.2485 + 1.78 Z i , and include moderate convective core overshooting. The BaSTI models are computed for a scaled-solar composition with Z = 0.0198, a model composition following ∆Y/∆Z ∼ 1.4 with Y = 0.245 at Z = 0, and also include convective core overshooting. Both models assume the Reimers massloss rate η = 0.2.
For our fitting procedure, we computed several isochrones from the PARSEC database tool 4 , with ages ranging from t = 0.1 to 13 Gyr by step of 0.01 Myr, and metallicities from Z = 0.003 to 0.06 (i.e. −0.7 < [Fe/H] < +0.6, using [Fe/H] ∼ log (Z/Z )), by step of 0.001. The BaSTI isochrones are pre-computed in their database 5 , we downloaded models for t = 0.1 − 10 Myr by step of 0.01 Myr and Z = 0.002, 0.004, 0.008, 0.01, 0.0198, 0.03 and 0.04 (i.e. −1.0 < Fe/H < 0.3). These models are also for a scaled-solar composition (with Z = 0.0198) and also include overshooting. HD176704 has Fe/H = 0.36 dex, above the range of the BaSTI isochrones, but we rounded it down to 0.3. We chose grids fine enough in age to avoid interpolation (which might cause problems); the closest age is therefore always chosen in our fitting procedure.
The PARSEC output tables provide the luminosities, effective temperatures, effective gravities, and masses. We computed the linear radius from the table values following the equation
log R R = 1 2 log L L − 2 log T eff + 2 log T .(4)
The surface gravities for BaSTI were determined using Newton's law of universal gravitation
log g = log M M + 4 log T eff − 4 log T − log L L + log g ,(5)
with the solar constants from Prša et al. (2016). Then, from these grids, we performed our isochrone fits by adopting fixed values of metallicity, and searched for the best age fit in luminosity, effective temperature, radii and effective gravity following a χ 2 statistic, that is, minimizing Our fitting procedure was the following. For the PARSEC isochrones, we first chose the closest grid in Z for a given metallicity (given in Table 2). We note that the grid was not interpolated as our downloaded tracks are also fine enough in metallicity. Then, we searched for the global χ 2 minimum in age and mass by fitting all isochrones for that given metallicity. A second fit is then performed around that global minimum values. For the BaSTI models, which are unfortunately not fine enough in metallicity, we first interpolated all isochrones to the given metallicity. We then also searched for the global χ 2 minimum in age and mass by fitting all isochrones for that given metallicity. A second fit was also performed around that global minimum values.
χ 2 = ∆L σ L 2 + ∆T σ T 2 + ∆ log g σ log g 2 + ∆R σ R 2 (6)
To assess the uncertainties for the PARSEC and BaSTI models, we repeated the process with Z ± σ. The final age and mass corresponding to each isochrone model are listed in Table 3. We also listed the average and standard deviation between both models, together with the corresponding evolutionary status of the star. Figure 4 shows an example of a fitted isochrones for the star HD26464.
In most cases, both models give similar age and mass values, within the uncertainties. We found masses in the range 0.97 < M/M < 2.39 and ages 0.72 < t < 11.05 Gyr, which is consistent with what we expect from such stars. We can see that the masses are better determined than the ages. Comparison of the parameters between the two isochrone sets reveals no obvious systematic trends, but in some cases there are significant differences in age.
The average masses determined from isochrones fitting can be compared to the ones calculated from the surface gravity, that is, from Newton's law of universal gravitation with g the surface gravity and R the stellar radius. This is presented in Fig. 5. No specific trend or offset is detected and they are in rather good agreement with each other, within 1-2σ.
M g M = g g R R 2 ,(7)
A&A proofs: manuscript no. 33341 (Liu et al. 2007;Jofré et al. 2015;Alves et al. 2015;Jones et al. 2011;Mishenina et al. 2006;Mikolaitis et al. 2017;Feuillet et al. 2016;Allende Prieto & Lambert 1999;Proust & Foy 1988;Mishenina et al. 2006). E(B − V) were determined as explained in Suchomska et al. (2015).
Our derived values are also consistent with previous works of Luck (2015) who used three different isochrone models (Bertelli et al. 1994;Demarque et al. 2004;Dotter et al. 2008) to estimate the mass and age of some stars in our sample. The comparison with our derived mean values are listed in Table 4. We can see that in some cases, models give very different results in age, while the mass values are less scattered. We mention here that accurate angular diameter measurements help in better constraining isochrones via multi-observables fitting, although it still depends on accurate metallicity determinations. 1.71 ± 0.10 1.72 ± 0.05 1.75 ± 0.20 1.63 ± 0.06 1.73 ± 0.02 1.68 ± 0.04 HD3750
6.31 ± 1.19 1.16 ± 0.07 7.33 ± 1.65 1.11 ± 0.10 6.82 ± 0.51 1.14 ± 0.03 HD4211 4.64 ± 1.01 1.26 ± 0.09 6.50 ± 1.22 1.12 ± 0.06 5.57 ± 0.93 1.19 ± 0.07 HD5722
2.24 ± 0.05 1.57 ± 0.01 2.17 ± 0.12 1.50 ± 0.04 2.20 ± 0.04 1.53 ± 0.04 HD8651
3.55 ± 0.33 1.32 ± 0.05 4.08 ± 0.31 1.26 ± 0.03 3.82 ± 0.27 1.29 ± 0.03 HD9362
2.61 ± 0.75 1.45 ± 0.16 2.33 ± 0.51 1.47 ± 0.14 2.47 ± 0.14 1.46 ± 0.01 HD10142 7.08 ± 0.05 1.04 ± 0.10 3.50 ± 0.10 1.32 ± 0.01 5.29 ± 1.79 1.18 ± 0.14 HD11977
1.78 ± 0.89 1.70 ± 0.31 0.87 ± 0.05 2.08 ± 0.04 1.32 ± 0.46 1.89 ± 0.19 HD12438
2.61 ± 0.71 1.36 ± 0.14 3.33 ± 1.90 1.28 ± 0.22 2.97 ± 0.36 1.32 ± 0.04 HD13468
1.04 ± 0.06 2.04 ± 0.06 0.93 ± 0.05 2.03 ± 0.04 0.99 ± 0.05 2.04 ± 0.01 HD15220 12.60 ± 0.05 0.92 ± 0.01 9.50 ± 0.10 1.10 ± 0.01 11.05 ± 1.41 ± 0.05 1.96 ± 0.01 1.25 ± 0.05 1.91 ± 0.01 1.33 ± 0.08 1.94 ± 0.03 HD74622 12.11 ± 0.67 0.97 ± 0.01 9.50 ± 0.01 1.00 ± 0.01 10.81 ± 1.31 0.99 ± 0.02 HD75916
2.82 ± .26 1.56 ± 0.07 2.92 ± 0.42 1.47 ± 0.09 2.87 ± 0.05 1.52 ± 0.05 HD176704 2.61 ± 0.28 1.62 ± 0.06 3.17 ± 0.42 1.48 ± 0.07 2.89 ± 0.28 1.55 ± 0.07 HD177873 2.93 ± 0.97 1.48 ± 0.18 2.67 ± 0.62 1.51 ± 0.14 2.80 ± 0.13 1.50 ± 0.01 HD188887 3.29 ± 0.36 1.45 ± 0.03 5.83 ± 1.65 1.23 ± 0.14 4.56 ± 1.27 1.34 ± 0.11 HD191584 5.62 ± 0.05 1.25 ± 0.01 6.50 ± 0.05 1.22 ± 0.01 6.06 ± 0.44 1.23 ± 0.01 HD204381 1.00 ± 0.25 2.15 ± 0.17 0.77 ± 0.17 2.26 ± 0.19 0.88 ± 0.12 2.20 ± 0.06 HD219784 3.98 ± 0.37 1.31 ± 0.05 5.83 ± 0.62 1.15 ± 0.03 4.91 ± 0.93 1.23 ± 0.08 HD220572 1.64 ± 0.09 1.81 ± 0.04 1.50 ± 0.05 1.83 ± 0.01 1.57 ± 0.07 1.82 ± 0.03
Age-metallicity relation
We plotted our derived ages and metallicities in the agemetallicity diagram, and compared them with other studies for the solar neighbourhood. We first compared our derived ages with the work of Takeda et al. (2016) for giant stars. We notice a very good agreement with their sample, as seen in Fig. 6. In a wider context, our values for giants stars are also similar to the work for F-, G-, and K-type stars dwarfs (Casagrande et al. 2011;Ibukiyama & Arimoto 2002). Our work supports the previous conclusions about the metallicity in our neighbourhood, that is, a little metallicity evolution in the past 10 Gyr and a large scatter at all ages. Although the scatter of the relation seems to increase with age, the trend tends to be almost flat.
Conclusion
We report accurate angular diameters measurements of nearby giant stars. Our sample includes a total of 48 stars for which the diameter is measured to better than 2.7 %. These observations were initially carried out to improve the calibration of the surface brightness-colour relation of late-type stars and to measure absolute stellar dimension of late-type eclipsing binaries to less than 1 %. We then used such systems to measure the most accurate Large Magellanic Cloud distance at a level of 1 % (Pietrzynski et al. 2018). Combining our angular diameters measurements with Hipparcos and Gaia DR2 parallaxes and spectroscopic effective temperatures, we determined linear radii and absolute luminosities with an average accuracy of 3 % and 6 %, respectively. We also fitted PARSEC and BaSTI model isochrones to derive the age and mass of these giant stars. The added value of interferometry is that the constraint on the mass and age imposed by the R − T eff plane is much tighter than using L−T eff only. We found an overall good agreement between our estimated masses and literature values, while age estimates are rather scattered. Although we have accurate knowledge of the stellar angular diameters, our analysis still requires accurate determinations of the other stellar parameters such as the metallicity in order to be able to constrain different input physics and parameters from stellar evolution models. The stars of our sample will soon be observed by the TESS (Ricker et al. 2014) and PLATO (Rauer et al. 2014) satellites, that will provide detailed asteroseismic frequency spectra. Together with our high-precision interferometric angular diameters and Gaia distances, this will enable a more accurate determination of their physical parameters (see e.g. Kervella et al. 2003;Thévenin et al. 2005;Cunha et al. 2007;Huber et al. 2012). This will also provide a stringent test of the asteroseismic scaling relations (see e.g. Huber et al. 2011;Gaulme et al. 2016, and reference therein).
Acknowledgements. The authors would like to thank all the people involved in the VLTI project. A. G. acknowledges support from FONDECYT grant 3130361. The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-15-CE31-0012-01 (project Unlock-Cepheids). P.K., A.G., and W.G. acknowledge support of the French-Chilean exchange programme ECOS-Sud/CONICYT (C13U01). W.G., R.E.M. and G.P. gratefully acknowledge financial support for this work from the BASAL Centro de Astrofisica y Tecnologias Afines (CATA) PFB-06/2007. R.E.M. acknowledges grant VRID 218.016.004-1.0.W.G. also acknowledges financial support from the Millenium Institute of Astrophysics (MAS) of the Iniciativa Cientifica Milenio del Ministerio de Economia, Fomento y Turismo de Chile, project IC120009. We acknowledge financial support from the Programme National de Physique Stellaire (PNPS) of CNRS/INSU, France. The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 695099). This work made use of the SIMBAD and VIZIER astrophysical database from CDS, Strasbourg, France and the bibliographic information from the NASA Astrophysics Data System. This research has made use of the Jean-Marie Mariotti Center SearchCal and ASPRO services, co-developed by FIZEAU and LAOG/IPAG, and of CDS Astronomical Databases SIMBAD and VIZIER. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www. cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.
Appendix A: Revised orbit of TZ Fornacis
Using the new interferometric measurements of this paper, we revised the orbital solutions of Gallenne et al. (2016) performing the exact same analysis, that is, simultaneously fitting the radial velocities of both components and the astrometry. The revised parameters are listed in Table A.1, which are in very good agreement (within 1σ) with our previously determined values. Here, we have taken into account the systematic uncertainty from the wavelength calibration determined in this paper, reducing the uncertainty on the semi-major axis and the distance. The relative difference between the observed and new calculated projected separations are displayed in Fig. A.1.
Fig. 1 .
1Relative difference of the observed and calculated projected separations.
The observations were executed over two half nights on 31 October and 1 November 2017, alternating between both GRAVITY (hereafter G) and PIONIER (hereafter P). For each instrument
Fig. 2 .
2χ 2 map of the local minima (left) and detection level map (right) of hd45415. The yellow lines represent the convergence from the starting points to the final fitted position. The maps were reinterpolated in a regular grid for clarity.
Fig. 3 .
3Example of calibrated squared visibilities and fitted limb-darkened angular diameter model. Our measurements are represented with black dots, and the best fitted model in red.
Fig. 4 .
4Fitted PARSEC and BaSTI isochrones for HD26464.
Fig. 5 .
5Masses M iso derived from isochrone fitting compared to calculated ones M g . The dotted line denotes a 1:1 relation.
Fig. 6 .
6Age-metallicity relation for our red clump stars compared to other Galactic works.
Fig. A. 1 .
1Relative difference of the observed and newly calculated projected separations.
Table 2 .
2Measured and derived intrinsic stellar parameters.Notes. Data from the literature were also used to the average estimates of T eff , log g and[Fe/H] HD
θ LD
log T eff
log g
[Fe/H]
E(B − V)
log R/R
log L/L
(mas)
(K)
(dex)
360
0.906 ± 0.015 3.678 ± 0.002 2.62 ± 0.11 −0.12 ± 0.05
0.009
1.036 ± 0.009 1.736 ± 0.019
3750
1.003 ± 0.020 3.660 ± 0.006 2.28 ± 0.32 −0.04 ± 0.07
0.002
1.022 ± 0.010 1.636 ± 0.019
4211
1.100 ± 0.011 3.656 ± 0.007 2.29 ± 0.23 −0.07 ± 0.06
0.004
1.089 ± 0.019 1.757 ± 0.037
5722
0.995 ± 0.019 3.689 ± 0.002 2.60 ± 0.06 −0.17 ± 0.03
0.010
1.038 ± 0.011 1.785 ± 0.021
8651
1.228 ± 0.013 3.674 ± 0.003 2.50 ± 0.16 −0.23 ± 0.03
0.002
1.013 ± 0.007 1.673 ± 0.013
9362
2.301 ± 0.021 3.680 ± 0.002 2.61 ± 0.10 −0.28 ± 0.10
0.000
1.033 ± 0.005 1.737 ± 0.011
10142 0.964 ± 0.006 3.674 ± 0.003 2.44 ± 0.04 −0.15 ± 0.01
0.007
1.024 ± 0.006 1.699 ± 0.013
11977 1.528 ± 0.013 3.693 ± 0.004 2.71 ± 0.27 −0.24 ± 0.05
0.002
1.048 ± 0.006 1.819 ± 0.013
12438 1.091 ± 0.016 3.696 ± 0.006 2.43 ± 0.32 −0.66 ± 0.09
0.004
1.025 ± 0.013 1.787 ± 0.026
13468 0.886 ± 0.010 3.688 ± 0.002 2.65 ± 0.04 −0.13 ± 0.05
0.009
1.048 ± 0.008 1.800 ± 0.015
15220 1.185 ± 0.016 3.651 ± 0.010 2.19 ± 0.04
0.26 ± 0.05
0.007
1.030 ± 0.016 1.615 ± 0.033
15248 0.949 ± 0.019 3.669 ± 0.003 2.45 ± 0.04
0.06 ± 0.05
0.010
1.041 ± 0.010 1.710 ± 0.020
15779 1.185 ± 0.014 3.684 ± 0.002 2.67 ± 0.04
0.00 ± 0.06
0.006
1.024 ± 0.007 1.737 ± 0.015
16815 2.248 ± 0.014 3.672 ± 0.006 2.65 ± 0.10 −0.34 ± 0.02
0.000
1.048 ± 0.005 1.738 ± 0.009
17652 1.835 ± 0.014 3.680 ± 0.001 2.67 ± 0.10 −0.34 ± 0.10
0.001
1.019 ± 0.007 1.710 ± 0.014
17824 1.391 ± 0.015 3.700 ± 0.002 2.95 ± 0.03
0.08 ± 0.06
0.002
0.947 ± 0.009 1.647 ± 0.018
18784 1.036 ± 0.014 3.673 ± 0.003 2.35 ± 0.08 −0.12 ± 0.07
0.014
1.026 ± 0.008 1.695 ± 0.016
23319 2.033 ± 0.014 3.662 ± 0.003 2.56 ± 0.05
0.03 ± 0.06
0.001
1.056 ± 0.006 1.712 ± 0.013
23526 0.915 ± 0.021 3.687 ± 0.004 2.68 ± 0.13 −0.15 ± 0.04
0.017
1.046 ± 0.012 1.792 ± 0.024
23940 1.093 ± 0.021 3.682 ± 0.004 2.43 ± 0.18 −0.42 ± 0.08
0.002
0.986 ± 0.014 1.652 ± 0.027
26464 1.089 ± 0.012 3.682 ± 0.006 2.85 ± 0.10
0.11 ± 0.14
0.008
1.067 ± 0.008 1.813 ± 0.015
30814 1.310 ± 0.010 3.689 ± 0.006 2.82 ± 0.24
0.04 ± 0.07
0.006
1.015 ± 0.007 1.739 ± 0.014
35369 2.012 ± 0.016 3.692 ± 0.004 2.76 ± 0.21 −0.18 ± 0.02
0.000
1.062 ± 0.008 1.845 ± 0.017
36874 1.118 ± 0.011 3.664 ± 0.004 2.47 ± 0.07 −0.04 ± 0.04
0.002
1.029 ± 0.006 1.669 ± 0.013
39523 1.939 ± 0.016 3.669 ± 0.008 2.56 ± 0.22
0.15 ± 0.20
0.001
1.050 ± 0.007 1.728 ± 0.014
39640 1.251 ± 0.017 3.689 ± 0.003 2.70 ± 0.08 −0.11 ± 0.03
0.006
1.031 ± 0.008 1.769 ± 0.016
39910 1.090 ± 0.008 3.659 ± 0.008 2.39 ± 0.21
0.18 ± 0.09
0.015
1.051 ± 0.006 1.693 ± 0.013
40020 1.012 ± 0.023 3.669 ± 0.008 2.43 ± 0.24
0.09 ± 0.08
0.013
1.050 ± 0.011 1.728 ± 0.023
43899 1.264 ± 0.017 3.658 ± 0.007 2.04 ± 0.24 −0.12 ± 0.08
0.010
1.067 ± 0.007 1.718 ± 0.015
45415 1.080 ± 0.061 3.679 ± 0.003 2.75 ± 0.08 −0.02 ± 0.05
0.015
1.022 ± 0.026 1.715 ± 0.051
46116 1.145 ± 0.031 3.685 ± 0.003 2.48 ± 0.15 −0.38 ± 0.05
0.009
1.008 ± 0.013 1.712 ± 0.027
53629 1.065 ± 0.024 3.647 ± 0.009 2.14 ± 0.15
0.13 ± 0.05
0.017
1.073 ± 0.011 1.687 ± 0.022
54131 1.061 ± 0.021 3.679 ± 0.005 2.72 ± 0.10 −0.10 ± 0.09
0.012
1.006 ± 0.010 1.684 ± 0.020
56160 1.411 ± 0.012 3.646 ± 0.008 2.19 ± 0.10
0.16 ± 0.09
0.010
1.092 ± 0.006 1.720 ± 0.012
60060 0.948 ± 0.010 3.683 ± 0.004 2.58 ± 0.14 −0.11 ± 0.03
0.018
1.038 ± 0.007 1.761 ± 0.014
60341 1.190 ± 0.022 3.665 ± 0.006 2.42 ± 0.26
0.06 ± 0.07
0.010
1.062 ± 0.010 1.737 ± 0.019
62412 0.950 ± 0.014 3.692 ± 0.003 2.76 ± 0.09
0.03 ± 0.03
0.013
1.014 ± 0.008 1.751 ± 0.017
62713 1.446 ± 0.012 3.666 ± 0.004 2.42 ± 0.33
0.09 ± 0.05
0.005
1.015 ± 0.006 1.645 ± 0.013
68312 1.020 ± 0.023 3.704 ± 0.002 2.75 ± 0.05 −0.10 ± 0.01
0.011
0.974 ± 0.012 1.718 ± 0.023
74622 1.020 ± 0.015 3.647 ± 0.005 2.26 ± 0.20 −0.03 ± 0.03
0.013
1.026 ± 0.008 1.593 ± 0.015
75916 1.013 ± 0.021 3.671 ± 0.003 2.47 ± 0.11
0.15 ± 0.05
0.008
1.064 ± 0.010 1.764 ± 0.021
176704 1.317 ± 0.012 3.655 ± 0.004 2.56 ± 0.10
0.36 ± 0.10
0.007
1.049 ± 0.011 1.671 ± 0.021
177873 1.958 ± 0.029 3.667 ± 0.003 2.59 ± 0.10
0.01 ± 0.10
0.002
1.051 ± 0.010 1.722 ± 0.020
188887 1.595 ± 0.011 3.650 ± 0.003 2.45 ± 0.10
0.11 ± 0.10
0.005
1.093 ± 0.005 1.739 ± 0.010
191584 1.024 ± 0.022 3.649 ± 0.008 2.35 ± 0.15
0.22 ± 0.00
0.009
1.047 ± 0.011 1.645 ± 0.021
204381 1.524 ± 0.017 3.703 ± 0.001 2.96 ± 0.10 −0.01 ± 0.10
0.001
0.934 ± 0.008 1.635 ± 0.016
219784 2.117 ± 0.025 3.661 ± 0.003 2.29 ± 0.17 −0.10 ± 0.04
0.001
1.065 ± 0.009 1.725 ± 0.019
220572 1.092 ± 0.013 3.674 ± 0.002 2.64 ± 0.09
0.07 ± 0.01
0.003
1.028 ± 0.007 1.705 ± 0.014
Table 3 .
3Estimated stellar mass, age and evolutionary status of our giant stars.PARSEC
BaSTI
Average
Star
Age
Mass
Age
Mass
Age
Mass
(Gyr)
(M/M )
(Gyr)
(M/M )
(Gyr)
(M/M )
HD360
Table A . 1 .Table 1 .
A11Best-fit orbital elements and parameters. Log of the interferometric observations, together with some stellar information. A1-G1-J3-K0 uniform disk diameter of the calibrators: HD3145=0.852 ± 0.012 mas;HD224821=0.929 ± Article number, page 12 of 12P orb (days) 75.66691 ± 0.00019
T p (HJD)
2452599.29040
e
0.0000 ± 0.0001
K 1 (km s −1 )
38.91 ± 0.01
K 2 (km s −1 )
40.88 ± 0.01
γ 1 (km s −1 )
17.99 ± 0.03
γ 2 (km s −1 )
18.35 ± 0.11
ω ( • )
270.01 ± 0.04
Ω ( • )
65.95 ± 0.04
a (mas)
2.990 ± 0.011
a (AU)
0.5565 ± 0.0001
i ( • )
85.71 ± 0.04
M 1 (M )
2.057 ± 0.001
M 2 (M )
1.958 ± 0.001
d (pc)
186.1 ± 0.7
π (mas)
5.37 ± 0.02
Star
K
H
π
E(B −
V)
Date
Baselines
Sp.
Calibrator b
Bin.
(mag) (mag)
(mas)
channels
HD
HD 360
3.653 3.757
8.97 ±
0.13
0.009
2014-10-05 A1-G1-J3-K0
3
HD 1588, 6482
-
HD 3750
3.485 3.612
10.26 ±
0.11
0.002
2013-12-31 A1-G1-J3-K0
7
HD 3145, HD 224821, HD 902
-
HD 4211
3.295 3.426
9.63 ±
0.40 a
0.004
2014-10-05 0.013 mas;HD902=0.967 ±
0.013 mas; HD11050=0.710 ±
0.009 mas;HD10216=0.834 ±
0.011 mas;HD11643=0.949 ±
0.013 mas; HD20176=0.897 ±
0.012 mas;HD22826=0.742 ±
0.010 mas; HD13668=0.784 ±
0.010 mas;HD18423=1.092 ±
0.014 mas; HD34587=0.884 ±
0.012 mas;HD47001=0.867 ±
0.011 mas; HD15958=0.864 ±
0.012 mas; HD52574=0.675 ±
0.009 mas;HD44956=0.655 ±
0.008 mas; HD3975=0.957 ±
0.013 mas;HD3909=0.861 ±
0.012 mas;HD6482=0.836 ±
0.012 mas; HD14129=1.006 ±
0.014 mas;HD19121=0.890 ±
0.012 mas;HD20791=0.883 ±
0.012 mas; HD15996=0.870 ±
0.011 mas;HD13692=0.923 ±
0.012 mas;HD18290=0.779 ±
0.011 mas; HD15471=0.911 ±
0.013 mas; HD28625=0.754 ±
0.009 mas;HD31887=0.742 ±
0.010 mas;HD32613=0.864 ±
0.012 mas; HD34137=0.798 ±
0.011 mas;HD81720=0.922 ±
0.013 mas; HD56110=0.817 ±
0.011 mas;HD57820=0.959 ±
0.013 mas; HD71465=1.043 ±
0.011 mas;HD70136=0.923 ±
0.013 mas; HD51546=0.865 ±
0.012 mas;HD57911=0.877 ±
0.012 mas; HD49001=1.055 ±
0.014 mas;HD40605=0.978 ±
0.013 mas; HD70097=0.916 ±
0.013 mas;HD37877=0.951 ±
0.013 mas; HD178272=1.005 ±
0.014 mas;HD181110=0.945 ±
0.013 mas;HD184349=1.067 ±
0.015 mas; HD181019=1.030 ±
0.014 mas;HD207229=1.001 ±
0.013 mas;HD210563=0.968 ±
0.013 mas; HD206146=1.126 ±
0.015 mas;HD204609=1.143 ±
0.016 mas; HD176752=1.198 ±
0.017 mas;HD171960=1.121 ±
0.016 mas;HD174774=1.103 ±
0.015 mas; HD189563=0.915 ±
0.013 mas;HD190057=1.037 ±
0.014 mas;HD195659=0.929 ±
0.012 mas; HD220330=0.953 ±
0.013 mas;HD220790=1.007 ±
0.013 mas;HD215905=0.995 ±
0.013 mas; HD1434=0.941 ±
0.013 mas;HD1588=0.892 ±
0.012 mas; HD221370=0.906 ±
0.013 mas;HD214465=1.193 ±
0.016 mas; HD9742=0.942 ±
0.013 mas;HD8901=0.968 ±
0.013 mas; HD19755=0.840 ±
0.012 mas;HD12851=0.920 ±
0.013 mas;HD18696=0.950 ±
0.012 mas; HD10164=1.061 ±
0.014 mas;HD6903=0.628 ±
0.044 mas; HD14690=0.475 ±
0.033 mas;HD13819=0.541 ±
0.038 mas; HD37377=0.890 ±
0.011 mas;HD38885=1.000 ±
0.013 mas; HD27179=0.919 ±
0.013 mas;HD28947=0.722 ±
0.008 mas; HD51801=0.826 ±
0.011 mas;HD39810=0.690 ±
0.008 mas; HD18185=1.141 ±
0.015 mas;HD81502=1.230 ±
0.016 mas; HD18071=1.024 ±
0.014 mas;HD20520=1.267 ±
0.017 mas; HD51682=1.034 ±
0.014 mas;HD68512=1.227 ±
0.017 mas; HD9293=1.232 ±
0.017 mas;HD8963=1.076 ±
0.015 mas; HD13666=1.007 ±
0.014 mas;HD15875=0.915 ±
0.012 mas; HD32707=1.169 ±
0.016 mas;HD36134=1.164 ±
0.016 mas; HD21149=1.056 ±
0.014 mas;HD26934=0.984 ±
0.013 mas; HD14509=0.866 ±
0.012 mas;HD14832=0.732 ±
0.009 mas; HD52603=0.889 ±
0.012 mas;HD62897=0.834 ±
0.011 mas; HD54257=0.839 ±
0.012 mas;HD28322=0.819 ±
0.011 mas; HD42168=0.806 ±
0.011 mas;HD38054=1.266 ±
0.017 mas; HD42026=1.098 ±
0.014 mas;HD37462=1.263 ±
0.017 mas; HD18959=1.140 ±
0.016 mas;HD14728=1.171 ±
0.016 mas; HD53840=0.830 ±
0.010 mas;HD56537=0.567 ±
0.040 mas; HD44769=0.583 ±
0.041 mas;HD55185=0.533 ±
0.037 mas; HD24267=1.064 ±
0.014 mas;
HD71231=0.896 ±
0.012 mas;HD70409=0.842 ±
0.012 mas;
Available at http://www.jmmc.fr/searchcal. 2 http://www.jmmc.fr Article number, page 2 of 12 Gallenne et al.: Fundamental properties of red-clump stars
Available at https://github.com/amerand/CANDID Article number, page 3 of 12 A&A proofs: manuscript no. 33341
http://stev.oapd.inaf.it/cgi-bin/cmd 5 http://basti.oa-teramo.inaf.it/index.html Article number, page 4 of 12 Gallenne et al.: Fundamental properties of red-clump stars
A&A proofs: manuscript no. 33341Table 4. Comparison with previous works with different isochrone models fromLuck (2015Notes. B, D, Y, and G stands for the Bertelli isochrones(Bertelli et al. 1994), the Dartmouth isochrones(Dotter et al. 2008), the Y 2 isochrones(Demarque et al. 2004) and this work, respectively. H-band
. Allende Prieto, C Lambert, D L , A&A. 352555Allende Prieto, C. & Lambert, D. L. 1999, A&A, 352, 555
. S Alves, L Benamati, N C Santos, MNRAS. 4482749Alves, S., Benamati, L., Santos, N. C., et al. 2015, MNRAS, 448, 2749
. E K Baines, M P Döllinger, F Cusano, ApJ. 7101365Baines, E. K., Döllinger, M. P., Cusano, F., et al. 2010, ApJ, 710, 1365
. G Bertelli, A Bressan, C Chiosi, F Fagotto, E Nasi, A&AS. 106275Bertelli, G., Bressan, A., Chiosi, C., Fagotto, F., & Nasi, E. 1994, A&AS, 106, 275
. D Bonneau, J.-M Clausse, X Delfosse, A&A. 456789Bonneau, D., Clausse, J.-M., Delfosse, X., et al. 2006, A&A, 456, 789
. D Bonneau, X Delfosse, D Mourard, A&A. 53553Bonneau, D., Delfosse, X., Mourard, D., et al. 2011, A&A, 535, A53
. T S Boyajian, H A Mcalister, G Van Belle, ApJ. 746101Boyajian, T. S., McAlister, H. A., van Belle, G., et al. 2012, ApJ, 746, 101
. A Bressan, P Marigo, L Girardi, MNRAS. 427127Bressan, A., Marigo, P., Girardi, L., et al. 2012, MNRAS, 427, 127
. L Casagrande, R Schönrich, M Asplund, A&A. 530138Casagrande, L., Schönrich, R., Asplund, M., et al. 2011, A&A, 530, A138
. M S Cunha, C Aerts, J Christensen-Dalsgaard, A&A Rev. 14217Cunha, M. S., Aerts, C., Christensen-Dalsgaard, J., et al. 2007, A&A Rev., 14, 217
. P Demarque, J.-H Woo, Y.-C Kim, S K Yi, ApJS. 155667Demarque, P., Woo, J.-H., Kim, Y.-C., & Yi, S. K. 2004, ApJS, 155, 667
. A Dotter, B Chaboyer, D Jevremović, ApJS. 17889Dotter, A., Chaboyer, B., Jevremović, D., et al. 2008, ApJS, 178, 89
. F Eisenhauer, G Perrin, W Brandner, The Messenger. 14316Eisenhauer, F., Perrin, G., Brandner, W., et al. 2011, The Messenger, 143, 16
. D K Feuillet, J Bovy, J Holtzman, ApJ. 81740Feuillet, D. K., Bovy, J., Holtzman, J., et al. 2016, ApJ, 817, 40
. A G A Brown, Gaia CollaborationA Vallenari, Gaia CollaborationA&A. in press [e-prints:1804.09365Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, in press [e-prints:1804.09365]
. T Prusti, Gaia CollaborationJ H J De Bruijne, Gaia CollaborationA&A. 5951Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
. A Gallenne, P Kervella, A Mérand, A&A. 54187Gallenne, A., Kervella, P., Mérand, A., et al. 2012, A&A, 541, A87
. A Gallenne, A Mérand, P Kervella, A&A. 57968Gallenne, A., Mérand, A., Kervella, P., et al. 2015, A&A, 579, A68
. A Gallenne, G Pietrzyński, D Graczyk, A&A. 58635Gallenne, A., Pietrzyński, G., Graczyk, D., et al. 2016, A&A, 586, A35
. P Gaulme, J Mckeever, J Jackiewicz, ApJ. 832121Gaulme, P., McKeever, J., Jackiewicz, J., et al. 2016, ApJ, 832, 121
P Haguenauer, J Alonso, P Bourget, SPIE Conference Series. 7734Haguenauer, P., Alonso, J., Bourget, P., et al. 2010, in SPIE Conference Series, Vol. 7734
. M L Houdashelt, R A Bell, A V Sweigart, AJ. 1191448Houdashelt, M. L., Bell, R. A., & Sweigart, A. V. 2000, AJ, 119, 1448
. D Huber, T R Bedding, D Stello, ApJ. 743143Huber, D., Bedding, T. R., Stello, D., et al. 2011, ApJ, 743, 143
. D Huber, M J Ireland, T R Bedding, ApJ. 76032Huber, D., Ireland, M. J., Bedding, T. R., et al. 2012, ApJ, 760, 32
. A Ibukiyama, N Arimoto, A&A. 394927Ibukiyama, A. & Arimoto, N. 2002, A&A, 394, 927
. E Jofré, R Petrucci, C Saffe, A&A. 57450Jofré, E., Petrucci, R., Saffe, C., et al. 2015, A&A, 574, A50
. M I Jones, J S Jenkins, P Rojo, C H F Melo, A&A. 53671Jones, M. I., Jenkins, J. S., Rojo, P., & Melo, C. H. F. 2011, A&A, 536, A71
. P Kervella, L Bigot, A Gallenne, F Thévenin, A&A. 597137Kervella, P., Bigot, L., Gallenne, A., & Thévenin, F. 2017, A&A, 597, A137
. P Kervella, N Nardetto, D Bersier, D Mourard, V Coudé Du Foresto, A&A. 416941Kervella, P., Nardetto, N., Bersier, D., Mourard, D., & Coudé du Foresto, V. 2004, A&A, 416, 941
. P Kervella, F Thévenin, D Ségransan, A&A. 4041087Kervella, P., Thévenin, F., Ségransan, D., et al. 2003, A&A, 404, 1087
. V V Kovtyukh, N I Gorlova, A&A. 358587Kovtyukh, V. V. & Gorlova, N. I. 2000, A&A, 358, 587
. V V Kovtyukh, C Soubiran, O Bienaymé, T V Mishenina, S I Belik, MNRAS. 371879Kovtyukh, V. V., Soubiran, C., Bienaymé, O., Mishenina, T. V., & Belik, S. I. 2006, MNRAS, 371, 879
R L Kurucz, SAO special report. Cambridge309Kurucz, R. L. 1970, SAO special report (Cambridge: Smithsonian Astrophysical Observatory), 309
. C D Laney, M D Joner, G Pietrzyński, MNRAS. 4191637Laney, C. D., Joner, M. D., & Pietrzyński, G. 2012, MNRAS, 419, 1637
. , J.-B Le Bouquin, J.-P Berger, B Lazareff, A&A. 53567Le Bouquin, J.-B., Berger, J.-P., Lazareff, B., et al. 2011, A&A, 535, A67
. Y J Liu, G Zhao, J R Shi, G Pietrzyński, W Gieren, MNRAS. 382553Liu, Y. J., Zhao, G., Shi, J. R., Pietrzyński, G., & Gieren, W. 2007, MNRAS, 382, 553
. R E Luck, AJ. 15088Luck, R. E. 2015, AJ, 150, 88
. M Mayor, F Pepe, D Queloz, The Messenger. 11420Mayor, M., Pepe, F., Queloz, D., et al. 2003, The Messenger, 114, 20
. A Mérand, P Kervella, J Breitfelder, A&A. 58480Mérand, A., Kervella, P., Breitfelder, J., et al. 2015, A&A, 584, A80
. J.-C Mermilliod, M Mermilliod, B Hauck, A&AS. 124349Mermilliod, J.-C., Mermilliod, M., & Hauck, B. 1997, A&AS, 124, 349
. Š Mikolaitis, P De Laverny, A Recio-Blanco, A&A. 60022Mikolaitis, Š., de Laverny, P., Recio-Blanco, A., et al. 2017, A&A, 600, A22
. T V Mishenina, O Bienaymé, T I Gorbaneva, A&A. 4561109Mishenina, T. V., Bienaymé, O., Gorbaneva, T. I., et al. 2006, A&A, 456, 1109
. P J Mohr, D B Newell, B N Taylor, Reviews of Modern Physics. 8835009Mohr, P. J., Newell, D. B., & Taylor, B. N. 2016, Reviews of Modern Physics, 88, 035009
. D Mozurkewich, J T Armstrong, R B Hindsley, AJ. 1262502Mozurkewich, D., Armstrong, J. T., Hindsley, R. B., et al. 2003, AJ, 126, 2502
. H R Neilson, J B Lester, A&A. 55498Neilson, H. R. & Lester, J. B. 2013, A&A, 554, A98
. T E Nordgren, M E Germain, J A Benson, AJ. 1183032Nordgren, T. E., Germain, M. E., Benson, J. A., et al. 1999, AJ, 118, 3032
. A Pietrinferni, S Cassisi, M Salaris, F Castelli, ApJ. 612168Pietrinferni, A., Cassisi, S., Salaris, M., & Castelli, F. 2004, ApJ, 612, 168
. G Pietrzynski, D Graczyk, A Gallenne, Nature. 14561Ap&SSPietrzynski, G., Graczyk, D., Gallenne, A., et al. 2018, Nature (submitted) Proust, D. & Foy, R. 1988, Ap&SS, 145, 61
. A Prša, P Harmanec, G Torres, AJ. 15241Prša, A., Harmanec, P., Torres, G., et al. 2016, AJ, 152, 41
. I Ramírez, J Meléndez, ApJ. 626446Ramírez, I. & Meléndez, J. 2005, ApJ, 626, 446
. H Rauer, C Catala, C Aerts, Experimental Astronomy. 38249Rauer, H., Catala, C., Aerts, C., et al. 2014, Experimental Astronomy, 38, 249
G R Ricker, J N Winn, R Vanderspek, Proc. SPIE. SPIEOptical, Infrared, and Millimeter Wave9143914320Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2014, in Proc. SPIE, Vol. 9143, Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave, 914320
. C Sneden, ApJ. 184839Sneden, C. 1973, ApJ, 184, 839
. K Suchomska, D Graczyk, R Smolec, MNRAS. 451651Suchomska, K., Graczyk, D., Smolec, R., et al. 2015, MNRAS, 451, 651
. Y Takeda, A Tajitsu, B Sato, MNRAS. 4574454Takeda, Y., Tajitsu, A., Sato, B., et al. 2016, MNRAS, 457, 4454
. F Thévenin, P Kervella, B Pichon, A&A. 436253Thévenin, F., Kervella, P., Pichon, B., et al. 2005, A&A, 436, 253
. A Tokovinin, D A Fischer, M Bonati, PASP. 1251336Tokovinin, A., Fischer, D. A., Bonati, M., et al. 2013, PASP, 125, 1336
. G Valle, M Dell'omodarme, P G Prada Moroni, S Innocenti, F Van Leeuwen, A&A. 600653A&AValle, G., Dell'Omodarme, M., Prada Moroni, P. G., & Degl'Innocenti, S. 2017, A&A, 600, A41 van Leeuwen, F. 2007, A&A, 474, 653
. S Villanova, D Geisler, G Piotto, ApJ. 72218Villanova, S., Geisler, D., & Piotto, G. 2010, ApJ, 722, L18
. G Worthey, H Lee, ApJS. 1931Worthey, G. & Lee, H.-c. 2011, ApJS, 193, 1
| [
"https://github.com/amerand/CANDID"
] |
[
"Integrated community occupancy models: A framework to assess occurrence and biodiversity dynamics using multiple data sources",
"Integrated community occupancy models: A framework to assess occurrence and biodiversity dynamics using multiple data sources"
] | [
"Jeffrey W Doser \nDepartment of Forestry\nMichigan State University\nEast LansingMIUSA\n\nEcology, Evolution, and Behavior Program\nMichigan State University\nEast LansingMIUSA\n",
"Wendy Leuenberger \nEcology, Evolution, and Behavior Program\nMichigan State University\nEast LansingMIUSA\n\nDepartment of Integrative Biology\nMichigan State University\nEast LansingMIUSA\n",
"T Scott Sillett \nMigratory Bird Center\nSmithsonian Conservation Biology Institute\nNational Zoological Park\nWashingtonDCUSA\n",
"Michael T Hallworth \nVermont Center for Ecostudies\nNorwichVTUSA\n",
"Elise F Zipkin \nEcology, Evolution, and Behavior Program\nMichigan State University\nEast LansingMIUSA\n\nDepartment of Integrative Biology\nMichigan State University\nEast LansingMIUSA\n"
] | [
"Department of Forestry\nMichigan State University\nEast LansingMIUSA",
"Ecology, Evolution, and Behavior Program\nMichigan State University\nEast LansingMIUSA",
"Ecology, Evolution, and Behavior Program\nMichigan State University\nEast LansingMIUSA",
"Department of Integrative Biology\nMichigan State University\nEast LansingMIUSA",
"Migratory Bird Center\nSmithsonian Conservation Biology Institute\nNational Zoological Park\nWashingtonDCUSA",
"Vermont Center for Ecostudies\nNorwichVTUSA",
"Ecology, Evolution, and Behavior Program\nMichigan State University\nEast LansingMIUSA",
"Department of Integrative Biology\nMichigan State University\nEast LansingMIUSA"
] | [] | 1. The occurrence and distributions of wildlife populations and communities are shifting as a result of global changes. To evaluate whether these shifts are negatively impacting biodiversity processes, it is critical to monitor the status, trends, and effects of environmental variables on entire communities. However, modeling the dynamics of multiple species simultaneously can require large amounts of diverse data, and few modeling approaches exist to simultaneously provide species and community level inferences.2. We present an "integrated community occupancy model" (ICOM) that unites principles of data integration and hierarchical community modeling in a single framework to provide inferences on species-specific and community occurrence dynamics using multiple 1 arXiv:2109.01894v1 [q-bio.PE] 4 Sep 2021 data sources. The ICOM combines replicated and nonreplicated detection-nondetection data sources using a hierarchical framework that explicitly accounts for different detection and sampling processes across data sources. We use simulations to compare the ICOM to previously developed hierarchical community occupancy models and single species integrated distribution models. We then apply our model to assess the occurrence and biodiversity dynamics of foliage-gleaning birds in the White Mountain National Forest in the northeastern USA from 2010-2018 using three independent data sources.3. Simulations reveal that integrating multiple data sources in the ICOM increased precision and accuracy of species and community level inferences compared to single data source models, although benefits of integration were dependent on data source quality (e.g., amount of replication). Compared to single species models, the ICOM yielded more precise species-level estimates. Within our case study, the ICOM had the highest out-ofsample predictive performance compared to single species models and models that used only a subset of the three data sources.4. The ICOM provides more precise estimates of occurrence dynamics compared to multispecies models using single data sources or integrated single-species models. We further found that the ICOM had improved predictive performance across a broad region of interest with an empirical case study of forest birds. The ICOM offers an attractive approach to estimate species and biodiversity dynamics, which is additionally valuable to inform management objectives of both individual species and their broader communities. | 10.1111/2041-210x.13811 | [
"https://arxiv.org/pdf/2109.01894v1.pdf"
] | 237,421,153 | 2109.01894 | de7ff41be56e5054c91c3352d9cdb0d89c66c343 |
Integrated community occupancy models: A framework to assess occurrence and biodiversity dynamics using multiple data sources
Jeffrey W Doser
Department of Forestry
Michigan State University
East LansingMIUSA
Ecology, Evolution, and Behavior Program
Michigan State University
East LansingMIUSA
Wendy Leuenberger
Ecology, Evolution, and Behavior Program
Michigan State University
East LansingMIUSA
Department of Integrative Biology
Michigan State University
East LansingMIUSA
T Scott Sillett
Migratory Bird Center
Smithsonian Conservation Biology Institute
National Zoological Park
WashingtonDCUSA
Michael T Hallworth
Vermont Center for Ecostudies
NorwichVTUSA
Elise F Zipkin
Ecology, Evolution, and Behavior Program
Michigan State University
East LansingMIUSA
Department of Integrative Biology
Michigan State University
East LansingMIUSA
Integrated community occupancy models: A framework to assess occurrence and biodiversity dynamics using multiple data sources
Corresponding Author: Jeffrey W. Doser, email: [email protected]; ORCID ID: 0000-0002-8950-9895 Running Title: Integrated community occupancy modelsavianBayesiandata fusiondata integrationhierarchical modelingimperfect detectionjoint likelihood 2
1. The occurrence and distributions of wildlife populations and communities are shifting as a result of global changes. To evaluate whether these shifts are negatively impacting biodiversity processes, it is critical to monitor the status, trends, and effects of environmental variables on entire communities. However, modeling the dynamics of multiple species simultaneously can require large amounts of diverse data, and few modeling approaches exist to simultaneously provide species and community level inferences.2. We present an "integrated community occupancy model" (ICOM) that unites principles of data integration and hierarchical community modeling in a single framework to provide inferences on species-specific and community occurrence dynamics using multiple 1 arXiv:2109.01894v1 [q-bio.PE] 4 Sep 2021 data sources. The ICOM combines replicated and nonreplicated detection-nondetection data sources using a hierarchical framework that explicitly accounts for different detection and sampling processes across data sources. We use simulations to compare the ICOM to previously developed hierarchical community occupancy models and single species integrated distribution models. We then apply our model to assess the occurrence and biodiversity dynamics of foliage-gleaning birds in the White Mountain National Forest in the northeastern USA from 2010-2018 using three independent data sources.3. Simulations reveal that integrating multiple data sources in the ICOM increased precision and accuracy of species and community level inferences compared to single data source models, although benefits of integration were dependent on data source quality (e.g., amount of replication). Compared to single species models, the ICOM yielded more precise species-level estimates. Within our case study, the ICOM had the highest out-ofsample predictive performance compared to single species models and models that used only a subset of the three data sources.4. The ICOM provides more precise estimates of occurrence dynamics compared to multispecies models using single data sources or integrated single-species models. We further found that the ICOM had improved predictive performance across a broad region of interest with an empirical case study of forest birds. The ICOM offers an attractive approach to estimate species and biodiversity dynamics, which is additionally valuable to inform management objectives of both individual species and their broader communities.
Introduction
Populations and communities of a wide range of organisms, including birds (Rosenberg et al., 2019), bats (Rodhouse et al., 2019), and insects (Wagner et al., 2021), have experienced severe declines in their distributions and abundances across large geographical regions as a result of habitat loss, climate change, and other anthropogenic stressors. As a result, there is growing interest in developing enhanced monitoring techniques to estimate species' trends and distributions. While species-level assessments can be valuable, it is critical to quantify the status and dynamics of entire communities to understand global change effects on biodiversity. However, multi-species approaches require large numbers of observations for each species or make assumptions about equal detectability of species during sampling, precluding robust assessments of rare species and overall community dynamics (Sor et al., 2017).
Despite these limitations, many sampling approaches (e.g., autonomous recording units for birds and bats, camera traps for mammals) naturally provide data on the occurrence patterns of multiple species in a community. However, occurrence data for each species are confounded by imperfect detection during sampling. An observer may record a species as absent if it is indeed absent, or the observer may fail to detect the species during sampling, and thus we refer to such data as detection-nondetection data. Replicated sampling can provide additional information on species' detection rates, allowing for the separation of the occurrence process from the detection process to enable ecologically relevant inference. Most commonly, this information is obtained by collecting multi-species detection-nondetection data on several occasions over short time periods when closure-no permanent extinction or colonization-can be reasonably assumed (MacKenzie and . By accounting for imperfect detection of species during sampling, these data (hereafter "replicated data") allow for estimation of species distributions and occurrence patterns within an occupancy modeling framework (MacKenzie et al., 2002). This additional information required to separate detection from occurrence is often more costly and time consuming to collect, which motivates the desire to use nonreplicated detection-nondetection data to model species distribution patterns. Accordingly, there is widespread interest in determining statistically robust ways to incorporate nonreplicated data sources into species and community level analyses to inform effective biodiversity conservation.
Two recent statistical modeling developments, data integration and hierarchical community occupancy models, have led to improved inference on individual species and community dynamics, respectively, to help inform biodiversity assessments. Data integration is a modelbased approach to combining multiple data types (Zipkin and Saunders, 2018;Miller et al., 2019). This approach yields many key benefits compared to single data source models, such as higher accuracy and precision of parameter estimates (Dorazio, 2014), inference across broader spatio-temporal extents , and the opportunity to accommodate sampling biases and imperfect detection for the various data sources (Miller et al., 2019). Data integration is particularly useful for combining large-scale nonreplicated data with replicated data, as the replicated data allow for the explicit modeling of imperfect detection while still using the information contained in the nonreplicated data. The combination of these data types can thus greatly expand the spatial scope of analysis.
Hierarchical community occupancy models provide combined inference on the occurrence of multiple species using a single replicated detection-nondetection data source (Dorazio and Royle, 2005;Gelfand et al., 2005). Species-specific parameters are viewed as random effects arising from a common, community-level distribution with a mean and variance parameter representing the average effect across species in the community and the variation of speciesspecific effects within the community, respectively (Dorazio and Royle, 2005). In addition to providing more precise estimates of species-specific effects (Zipkin et al., 2009), these models can estimate community-level parameters, as well as biodiversity metrics (Guillera-Arroita et al., 2019), all with associated uncertainty (Zipkin et al., 2010). Community occupancy models have led to improved insights into understanding species occurrence patterns for insects (Mata et al., 2017), birds (Zipkin et al., 2009), and mammals (Gallo et al., 2017).
While substantial development of both data integration and hierarchical community models has occurred over the last decade (reviewed in Zipkin and Saunders 2018;Guillera-Arroita et al. 2019;Miller et al. 2019), there has been comparatively less research focused on modeling multiple species using more than one data source. Multi-species integrated population models have been used to assess competition (Péron and Koons, 2012), synchrony (Lahoz-Monfort et al., 2017), and predator-prey dynamics (Barraquand and Gimenez, 2019;Clark, 2021), but these approaches are not designed for assessment of larger communities. Clark et al. (2017) introduced a broad framework (GJAM) for using multiple data sets to estimate the distribution and abundance of multiple species, but their approach does not model the observation process hierarchically, making it difficult to account for differences in sampling protocols and species detections among data sources. Thus, a new modeling approach that combines the benefits of data integration and hierarchical community modeling in a single framework has the potential to yield detailed inferences on individual species occurrence patterns while simultaneously providing inference on community dynamics.
Here we develop an "integrated community occupancy model" to simultaneously estimate occurrence patterns of multiple species within a community as well as community metrics by incorporating multiple available data sources into a single analysis. Our modeling framework combines one or more "replicated" detection-nondetection data sources with one or more "nonreplicated" data sources. The integrated model consists of a single, ecological process model and multiple observation models that explicitly account for different detection processes in each data source. The detection models are linked to the process model using a joint-likelihood framework (Miller et al., 2019). Our ecological process model is a dynamic community occupancy model in which we explicitly model the latent occurrence process as a function of space and/or time varying covariates and a temporal autologistic parameter (Royle and Dorazio, 2008). We validate our model using simulations and assess its ability to estimate species-specific and community level dynamics using different amounts of data sources. We then compare inferences from our integrated community occupancy model to those generated by single species integrated models to evaluate the marginal benefits of the approach. We apply our model to an empirical case study of a community of twelve foliage-gleaning insectivorous birds in the White Mountain National Forest in the northeastern USA to assess patterns and trends in species occurrence and community metrics (i.e., richness, composition) from 2010-2018. Our integrated community occupancy model provides a rigorous approach to elucidate both species specific and community level dynamics that can provide crucial insight on the specific mechanisms driving biodiversity shifts and population declines.
Materials and Methods
Integrated community occupancy model
We develop an "integrated community occupancy model" (ICOM) that leverages one or more replicated detection-nondetection data sources with one or more nonreplicated detection-nondetection data sources to provide inferences on species-specific and community occurrence dynamics. We present the model using one replicated and one nonreplicated data source, but our framework can be extended to incorporate additional data sources if available in an analogous manner.
The ICOM consists of a single ecological process model for individual species, which is shared across the various data sets. Here, we demonstrate the model with a dynamic ecological process that describes species occurrences as a function of spatio-temporally varying covariates and an autologistic parameter that accounts for temporal correlation (Royle and Dorazio, 2008). This process model is then linked to individual likelihoods for each data set via a hierarchical framework that assumes independence among the detection processes and which is conditional on the true latent occurrence state (i.e., whether a species is truly present or absent at sampling locations; Miller et al. 2019). To link the individual species models, we assume species-level occurrence and detection parameters are random effects coming from common community level distributions (Dorazio and Royle, 2005;Gelfand et al., 2005), enabling information sharing across the community to increase precision of species effects and estimation of biodiversity attributes.
Ecological Process Model
Our goal is to model the occurrence dynamics of multiple species at sites j = 1, . . . J within a specified region of interest A. Let z i,j,t denote the true presence (1) or absence (0) of species i at site j during year t, where i = 1, . . . , I and t = 1, . . . , T . We assume z i,j,t arises from a Bernoulli process following
z i,j,t ∼ Bernoulli(ψ i,j,t ),(1)
where ψ i,j,t is the probability species i occurs at site j in year t. For the first year, we model
ψ i,j,t according to logit(ψ i,j,1 ) = β0 i,1 + β i · x j,1 ,(2)
where β0 i,1 is the species-specific occurrence probability (on the logit scale) in the first
year (at average covariate values) and β i is a vector of species-specific regression coefficients that describe the effect of standardized covariates (i.e., mean 0 and standard deviation 1) x j,1 on the occurrence probability of species i. In subsequent years, the occurrence probability for species i in year t at site j depends on whether or not the species was present at the site j in the previous year t − 1 in addition to covariates (which can vary spatially and/or temporally). We accommodate the temporal dependence by incorporating a species-specific autologistic parameter φ i into the occurrence model, such that for t > 1
logit(ψ i,j,t ) = β0 i,t + β i · x j,t + φ i · z i,j,t−1 ,(3)
where β0 i,t + φ i is the species-specific intercept in year t when species i occurred at site j in the previous year t − 1 and β0 i,t is the intercept in year t when species i did not occur at site j in the previous year t − 1. We use the autologistic parameterization of the dynamic community occupancy model as it allows us to assess covariate effects directly on species occurrence probabilities (i.e., the covariate effects remain the same regardless of the value of z i,j,t−1 ) and we can derive species-specific trends post-hoc from the occurrence probabilities (ψ i,j,t ). However, the ecological process model can be readily modified to incorporate relevant biological processes of interest (provided sufficient data are available; Royle and Dorazio 2008). For example, the ecological process model can be modified to explicitly include a trend effect (covariate on year), assess covariate effects on colonization and persistence (Dorazio et al., 2010), or the autologisitc component can be removed if that is not relevant to the target community.
Observation Model: replicated detection-nondetection data
For the replicated data type, we assume K > 1 "sampling replicates" within each year t are available at a subset of sites r = 1, . . . , R. The sampling replicates can be observations from multiple independent surveys, multiple independent observers, spatial subsamples, or sampling intervals from a removal design, which enable separate estimation of occurrence and detection probability (MacKenzie et al., 2002). We assume the R sites are a subset of the total J sites (i.e., R ≤ J), which may cover the entire region of interest A or, more commonly, only a portion of it. Let y i,r,k,t denote the detection (1) or nondetection (0) of species i during replicate k at site r during year t. We model y i,r,k,t as
y i,r,k,t ∼ Bernoulli(p i,r,k,t · z i,j[r],t ),(4)
where p i,r,k,t is the probability of detecting species i during visit k at site r in year t and
z i,j[r]
,t is the true occurrence status of species i in year t at site j corresponding to the rth replicated data site.
Species detection probabilities can vary by site and/or sampling covariates following
logit(p i,r,k,t ) = α0 i,t + α i · w r,k,t(5)
where α0 i,t is the species-specific detection probability (on the logit scale) in year t at average covariate values and α i is a vector of parameters that describe the effect of standardized covariates w r,k,t on the detection probability of species i.
Observation Model: nonreplicated detection-nondetection data
Let v i,m,t be the detection (1) We model the detection-nondetection data v i,m,t according to
v i,m,t ∼ Bernoulli(π i,m,t · z i,j[m],t ),(6)
where π i,m,t is the probability of detecting species i in site m in year t for the nonreplicated data set and z i,j[m],t is the true occurrence status of species i in year t at the site j corresponding to the mth nonreplicated data site. Detection probability π i,m,t can vary by species, site, and time following
logit(π i,m,t ) = γ0 i,t + γ i · s m,t ,(7)
where γ0 i,t is a species and year specific intercept and γ i is a vector of parameters that describe the effect of standardized covariates s m,t on the detection probability of species i.
Nonreplicated data alone are unable to separate occurrence probabilities from detection probabilities as the model structure is generally unidentifiable (Dorazio, 2014
Linking species models across the community
Following the structure of the hierarchical community occupancy model (Dorazio and Royle, 2005;Gelfand et al., 2005), species-specific parameters in both the ecological process model and observation models are treated as random effects arising from community level normal distributions with associated community level mean and variance parameters. For example, β0 i,t , the intercept on occurrence probabilities for species i in year t, is modeled as
β0 i,t ∼ Normal(µ β0t , σ 2 β0t ),(8)
where µ β0t is the hyper-mean for occurrence probability (on the logit scale) of all species in the community in year t (at average covariate values) and σ 2 β0t is the hyper-variance for occurrence probability across all species in the community in year t. Models for all other species-specific effects in the ecological and observation models are defined analogously. By treating species-specific effects as random, we improve estimates for both rare and abundant species (Zipkin et al., 2009) while simultaneously estimating community level effects.
A further benefit of the hierarchical community modeling approach is the ability to easily calculate biodiversity metrics (e.g., alpha, beta diversity) from the latent occurrence state (z i,j,t ) that account for imperfect detection of species. Under a Bayesian framework, we can calculate any biodiversity metric as a derived parameter at each iteration of the MCMC to obtain a full posterior distribution from which we can obtain estimates with fully propagated uncertainty.
For example, we can estimate species richness at each site j and year t at each iteration of the MCMC by summing the latent occurrence state (z i,j,t ) for all species. As a metric of beta diversity, we can calculate the Jaccard index (Magurran, 2013), which describes the similarity between two sites in terms of the number of species that occur at both sites. More specifically, we calculate the Jaccard index between site j and j in year t as
JACCARD j,j ,t = I i=1 z i,j,t · z i,j ,t I i=1 z i,j,t + I i=1 z i,j ,t − I i=1 z i,j,t · z i,j ,t ,(9)
which takes value 0 if the two sites have no species in common and value 1 if the same species occur at the two sites.
Data integration via joint likelihood
We use a joint likelihood framework to integrate the replicated and nonreplicated detectionnondetection data sources into a single model (the ICOM; Miller et al. 2019). To do this, we assume the likelihoods for the individual data sets are independent, conditional on the shared latent ecological process. This assumption can be interpreted as the detection of a species in one data set (conditional on the species being present) is independent of the detection of the species in any other data set (Schaub and Abadi, 2011;Kéry and Royle, 2020). Thus, our full joint likelihood, conditional on the true, shared ecological process, is the product of the individual conditional likelihoods for each data set:
L ICOM (α0, α, γ0, γ | z, β0, β, y, v) = L REP (α0, α | z, β0, β, y) · L NREP (γ0, γ | z, β0, β, v).(10)
Simulation study 1: Assessing benefits of integration
We performed a simulation study to assess whether integration of multiple data sources in an ICOM framework could provide improved accuracy and precision for species and community level parameters compared to individual analyses under a range of realistic parameter values (Supplemental Information S1.1). We simulated data from one replicated data source with K = 3 replicates and two nonreplicated data sources, with the replicated data source having medium community-level detection probability (mean hyper-parameter = 0.5), one nonreplicated data source having low community-level detection probability (mean = 0.22), and the other having high community-level detection probability (mean = 0.78), which allowed us to compare the benefits of integration across varying qualities of nonreplicated data. We generated 100 replicates of each data source under the ICOM framework using a range of community-level ecological parameter values and subsequently drew simulated species data for I = 25 species for T = 6 years. We generated species' occurrence probabilities according to Equations 2 and 3 with a single spatially-varying covariate. We generated detection processes for each data source as a function of a species and year specific intercept, and a species-specific effect of a spatio-temporally varying covariate unique to each data set. We simulated all covariates as normally distributed random variables with mean zero and standard deviation one. We generated each data source at 50 distinct sites that were randomly distributed across the range of covariates, resulting in a total of J = 150 sites. We compared model performance by fitting models individually for each of the seven unique combinations of the three data sources and subsequently computing the average bias (i.e., true simulated value minus estimated value) of the species-specific occurrence parameters across all 25 species and 100 simulations, as well as the average bias of the community level parameters.
Simulation study 2: Assessing benefits of community modeling
To evaluate the benefits of the hierarchical community model approach used in the ICOM, we assessed how species-level estimates from the ICOM compared to estimates from single species integrated distribution models (IDMs). The IDM took the same form as the ICOM except species-specific parameters were no longer random effects from a community level distribution, rather species-specific parameters were estimated individually in a model for each species. We simulated data from one replicated data source with K = 3 replicates and one nonreplicated data source, both with medium community-level detection probabilities (mean = 0.5). We simulated a community of I = 25 species, where each of the two data sources consisted of 50 unique locations sampled over T = 6 years, where the locations were randomly distributed across the range of a spatially varying covariate influencing occurrence. We assumed detection for both data sets was a function of a species and year specific intercept, and a species-specific spatio-temporally varying covariate unique to each data point. We generated species-specific occurrence and detection intercepts and covariate effects from uniform distributions (Supplemental Information S1.1), which allowed us to compare the ICOM to individual IDMs under the scenario when species level effects may not follow a normal distribution. We simulated 100 data sets from the community under realistic parameter values (Supplemental Information S1.1). We assessed model performance across the 100 simulated data sets by comparing the accuracy and precision of estimates from the ICOM to the IDMs.
Case study: foliage-gleaning birds in the White Mountains
We applied the ICOM to characterize temporal trends and spatial variability in individual species occurrence, species richness, and species composition of a community of twelve foliagegleaning birds from 2010-2018 across the White Mountain National Forest using two replicated data sets and one nonreplicated data set (Figure 1). Our two replicated data sets come from
) = β0 i,1 + β1 i · ELEV j + β2 i · ELEV 2 j + β3 i · FOR j ,(11)
where β0 i,1 is the species-specific intercept in year 1, and β1 i , β2 i , and β3 i are species-specific effects of elevation (ELEV, linear and quadratic) and local forest cover within a 250m radius (FOR), respectively. Occurrence in subsequent years is modeled analogously with a year-specific intercept and an autologistic parameter following Equation 3. We extracted elevation data at a 30 × 30 m resolution from the National Elevation Dataset (Gesch et al., 2002) and associated each point count site with the elevation at the center of the point count. We used the National Land Cover Database (Homer et al., 2015) to determine the amount of local forest cover in 2016 within a 250m radius of each point count location. To compute species-specific temporal trend estimates, we performed a post-hoc linear regression using the average occurrence probability of each species during each year as a response variable and year as a covariate. Under a Bayesian framework, we obtain full uncertainty propagation by calculating the trend for each posterior sample of the average occurrence probabilities (Supplemental Information S1.3). All speciesspecific occurrence intercepts and regression coefficients were modeled hierarchically following Equation 8.
We incorporated multiple covariates in the conditional likelihoods of each data type to account for variation in detection rates following the species' detection models described in the "Integrated community occupancy model" section. For the HBEF and NEON data, we included species and year-specific intercepts, a species-specific linear effect of the time of the survey, and species-specific linear and quadratic effects of the day of the survey. For the BBS data, we modeled detection as a function of a species and year specific intercept, species-specific linear and quadratic effects of day of survey, and a random observer effect to account for variation in detection among observers. All detection covariates were modeled hierarchically following Equation 8. Species and year specific intercepts were also modeled hierarchically, but were drawn from a single distribution for all species and years within each data set (Supplemental Information S3).
Goodness of fit and model validation
We assessed model fit for the case study using a Bayesian p-value approach with a Chi-square fit statistic (Supplemental Information S4). We used two-fold cross validation with the log predictive density (Vehtari et al., 2017) as a predictive performance metric to assess the outof-sample predictive performance of the full ICOM compared to six models using subsets of the three data sets for the case study. Assessing out-of-sample predictive performance with occupancy models presents additional complexities since the ecological state of interest is not directly observed (Zipkin et al., 2012). For a given data set, we compared the occurrence predictions at the hold out locations to the occurrence values generated from models that were fit using the data at the hold out locations. To account for model uncertainty, we compared the occurrence predictions individually to latent occurrence values generated from the subset of the seven models that used the data set in the model fitting process (see Supplemental Information S5 for details). We summarize predictive performance for each species and the entire community individually at each data set location, as well as across the entire study region (i.e., White Mountains). We used a similar approach to compare the performance of the ICOM to individual species IDMs (Supplemental Information S5).
Model implementation
We estimated the parameters in all model versions (simulations and case study) with a Bayesian framework using Markov Chain Monte Carlo (MCMC). We fit the models in NIMBLE (de Valpine et al., 2017(de Valpine et al., , 2021 within the R statistical environment (R Core Team, 2020) using vague priors for all hyper-parameters (Supplemental Information S1.3). For all simulations, we ran three chains, each with 20,000 iterations with a burn-in period of 10,000 iterations and a thinning rate of four. For the case study, we ran models for three chains of 450,000
iterations with a burn-in period of 200,000 iterations and a thinning rate of 20, resulting in a total of 52,000 samples from the posterior distribution. We assessed model convergence using the Gelman-Rubin R-hat diagnostic (Brooks and Gelman, 1998) and visual assessment of trace plots using the coda package (Plummer et al., 2006).
Results
Simulations
The ICOM using one replicated data set, one nonreplicated data set with low average detection probability across the community, and one nonreplicated data set with high detection probability yielded unbiased estimates and was generally more precise in community and species-level occurrence parameter estimates than models using smaller combinations of the three data sets or data sets individually (Figure 2, Supplemental Figure S1). Patterns were similar across community effects and species-specific effects, with increases in precision more prominent in species-level effects. Despite the general improvement in estimates found when integrating all three data sources, models using only two data sources, particularly the replicated data source and the nonreplicated data source with high detection probability, also yielded estimates with low bias and high precision. Models using only the nonreplicated data source with low detection probability often failed to converge, while models using only the nonreplicated data source with high detection probability mostly converged but were less precise than models using replicated data (and with essentially unidentifiable intercept values; Figure 2A). Figure 2: Sampling distribution of estimated bias in simulated species level occurrence intercepts (A) and covariate effects (B) under models using different combinations of a replicated (REP) data set and nonreplicated data sets with low (NREP L) and high (NREP H) detection probability. Points represent the median bias (posterior mean -true simulated value) in a species-level effect across 100 simulations for a community of 25 species. Lines represent the 95% quantiles of the bias values. The intercept parameter using only NREP L is not shown as it failed to converge.
The ICOM also led to substantial improvements in precision of parameter estimates compared to single species IDMs (
Case Study
The ICOM estimated variable trends in occurrence for the twelve foliage-gleaning bird species across the White Mountain National Forest, with five species having >75% probability of increasing occurrence rates and three species having >75% probability of decreasing (Supplemental Figure S3) from 2010-2018. Community level parameters revealed that average occurrence probability peaked at medium elevations and higher amounts of local forest cover across the community, although species-specific parameters were highly variable (Supplemental Table S3).
Occurrence probabilities peaked at a variety of elevations across the twelve species (Supplemental Figure S4), which resulted in species richness being maximized at medium elevations (600-800m; Figure 3A). Species composition of the community, as measured by the Jaccard index, largely followed similar patterns ( Figure 3B). Estimated trends in species-specific occurrence probabilities were highly dependent on the data sets included in the model ( Figure 4), suggesting important spatial variability across the White Mountains. For example, while
Red-eyed Vireo occurrence showed consistent trends across models from all data source combinations, trends for the Black-throated Blue Warbler and Black-throated Green Warbler were stable in estimates from most data combinations, but occurrence probabilities were estimated to have declined over this time period in a model using only BBS data.
Integration of all data sources in the ICOM yielded better predictive performance for the community of birds across the three data set locations than models using only a subset of the available data (Table 3). This is likely a result of both a larger number of detections and a wider range of the covariate space when using all three data sources.
Discussion
Understanding species distributions and occurrence dynamics of multiple species in a community is an important task for biodiversity conservation (Guillera-Arroita et al., 2019). Monitoring programs collect different types of data that vary in amount, spatial extent, quality, and information content, and incorporating these varied data into a unified analysis can yield improved estimates on quantities of interest (Zipkin and Saunders, 2018). We developed an "integrated community occupancy model" (ICOM) that uses replicated and nonreplicated detectionnondetection data to simultaneously provide inferences on species-specific and community dynamics. Using simulations and empirical bird data, we showed that the ICOM can provide more accurate and precise estimates of occurrence dynamics than analyses using single data sources ( Figure 2) or single species models (Table 2) as well as improved predictive performance across a region of interest (Table 3, Supplemental Table S6).
In our simulation study, the ICOM using one replicated data set, one nonreplicated data set with low detection probability, and one replicated data set with high detection probability provided unbiased and generally more precise parameter estimates than models using a subset of the three data sources (Figure 2), which aligns with previous single species data integration work (Fletcher Jr et al., 2019). Despite this general improvement in the simulation study, integrating the single replicated data source with the high detection nonreplicated data source yielded comparable accuracy and precision to the model using one replicated and two nonreplicated data sources. This suggests that integrating a data source of particularly low quality (e.g., nonreplicated, potentially large detection variability) with higher quality data sources may not yield any practical benefits (Simmonds et al., 2020). We generated each data source at random locations across the range of the covariate with no systematic bias in sampling locations.
In reality, many sources of detection-nondetection data are spatially biased (e.g., collected along road transects or near locations with high human density), which can lead to a narrow range of habitat variables (e.g., forest cover) that drive species and community occurrence dynamics. Such biases can result in different species being observed across data sets or estimated parameters that are either biased, only span a portion of the covariate range of interest, or are not indicative of the larger spatial areas of interest (Conn et al., 2017). By integrating disparate data sources from multiple locations in the ICOM, we can increase the likelihood that the sampled sites vary along important ecological and environmental gradients, which in turn enables more precise and accurate inference on the environmental conditions driving species occurrence.
In the foliage-gleaning bird case study, we used a two-fold cross validation approach to show that integration of all three data sets yielded the best overall predictive performance across the White Mountain National Forest compared to models using smaller subsets of the three data sources (Table 3). Further, the ICOM generally yielded improved predictive performance for individual species and the overall community compared to single species integrated distribution models (Supplemental Table S5). In contrast to the simulation study, parameter precision was not always highest when integrating all three data sets in the case study (Supplemental Tables S4, S5). Additionally, models with smaller subsets of the three data sources outperformed the ICOM with all three data sets individually for the HBEF and NEON data sets, although the improvements in predictive performance were not large (Table 3). This is likely a result of different covariate ranges among the three data sources and only having NEON data for a subset of the time period of interest. For example, precision of species-specific effects of local forest cover was highest for the model using HBEF and NEON data. Sites at HBEF and NEON have low variability in forest cover, with average forest cover of sites being 94.2% and 97.7%, respectively. However, variability in forest cover is higher at BBS sites, which likely explains why precision is lower when incorporating BBS data in the model. For most intercept parameters at the community and species-specific level, models using either NEON data alone or NEON and BBS were often the most precise. For these models, information for separating detection probability from true occurrence comes solely from NEON data, which are only available for four of the nine years of the study period. This smaller time period likely results in less unexplained variability in occurrence probability and detection across years, which in turn leads to more precise intercept estimates. While the smaller covariate and temporal ranges marginally increased precision, the ICOM using all three data sets enables inference across the entire temporal period of interest as well as across a broader range of covariates, which makes estimates on trends and spatial patterns across the White Mountains more informative.
The large variation in temporal trends when using different combinations of the three data sources suggests spatially-varying occurrence trends for the community of twelve foliage-gleaning birds. NEON data were only available from 2015-2018, which may account for differences in estimated trends from the model using only NEON data compared to other models. BBS data were sampled along road transects and thus have less forest cover than both the NEON and HBEF sites, suggesting that occurrence trends could vary as a result of differences in amount of local forest cover and/or proximity to roads (Furnas, 2020). Estimates from the full ICOM are a weighted average across heterogeneity in trends across the region, where the weights are determined by the amount of the different data sources (Fletcher Jr et al., 2019). In our case study, the HBEF data source comprised a majority of observations (77%; Supplemental Table S2) and thus contributed the most to estimates of model parameters, which we deemed acceptable because the HBEF data are a high-quality replicated data source. Alternatively, a profiling approach could be used within the MCMC sampler to change the weights for each data set similar to the maximum likelihood approach of Fletcher Jr et al. 2019. By including information from multiple data sources within a region, the ICOM yields area-wide averaged species-specific trends. If an area-wide averaged trend is not desired, trends from different spatial locations could be estimated hierarchically in a multi-region framework (Doser et al., 2021b) or treated as spatially varying coefficients to explicitly model the spatial heterogeneity (Finley, 2011).
We envision numerous methodological extensions and ecological applications of the ICOM framework. While our case study used three data sources arising from the same method (i.e., point count surveys), the ICOM can incorporate detection-nondetection data from different data collection approaches (e.g., camera traps, autonomous recording units, citizen science checklists) in an analogous manner. Similar integrated models could be developed to estimate alternative ecological processes such as abundance of multiple species within a community by extending single species integrated models that use distance sampling (Farr et al., 2021), acoustic recordings (Doser et al., 2021a), or capture-recapture data (Chandler and Clark, 2014). If regional species pools differ between data source locations, the ICOM could be adapted to a multiregion framework (Sutherland et al., 2016). Given our model's ability to estimate species and community level effects, the ICOM can be applied to help elucidate individual species sensitivities to various global change drivers, determine factors causing shifts in taxonomic or functional diversity, or forecast future species and community shifts under varying climate and land use change scenarios to help prioritize conservation strategies. In particular, the ICOM will assist multi-species conservation planning by providing estimates for rare species that lack adequate data for common analysis approaches, while simultaneously obtaining inference on an entire community that may elucidate specific species traits linked to occurrence trends and be more indicative of large-scale biodiversity change.
The benefits of data integration for multi-species detection-nondetection data sets will depend on characteristics and goals of each specific study. While the ICOM generally leads to improved inference for species and community level effects, integrating multiple data sets leads to increased computation times and potential difficulties in model convergence. When determining what data sources to use in an ICOM, we recommend considering the following factors: (1) the amount of the different data sources within the area of interest and how they are distributed across the range of ecological and environmental gradients; (2) the precision of estimates required for the analysis objectives; (3) amount of time and computing power available to run the models; (4) the spatial resolution of each data source; and (5) the quality and information content) of each data source (e.g., replicated vs nonreplicated, large detection variability vs.
standardized protocol). For example, if high quality replicated data exist across an adequate number of years and spatial locations that are distributed across potential habitat, a community model using only these data may suffice to accomplish research objectives. Simulation studies based on the specific data sets and sample sizes for a given research study can help to determine the ideal combinations of data required for a given study objective.
As changes in environmental and climate conditions continue globally, continued development of monitoring and analysis techniques that can effectively produce accurate and precise estimates of biodiversity metrics are needed to understand global change impacts and develop appropriate mitigation plans. Our integrated community occupancy models provide a new approach to simultaneously analyze multi-species data from numerous available sources. This framework can be used to elucidate both species specific and community level dynamics, improving understanding of the mechanisms driving biodiversity shifts and informing appropriate management and conservation actions to address global change.
or non-detection (0) of species i at site m in year t for the nonreplicated data source, where m = 1, . . . , M . We assume the M sites are a subset of all J sites of interest (i.e., M ≤ J) within the area of interest A. The replicated data may be available at different sites than the nonreplicated data in the same region A, the same sites, or a subset of the same sites in A.
Figure 1 :
1Study location for the case study. Panel (A) shows the White Mountain National Forest (shaded dark grey region) and the location of the Hubbard Brook Experimental Forest (HBEF; light grey region), the BBS routes (dark blue lines), and the NEON data from Bartlett Forest (purple region). Panel (B) shows the distribution of point count locations in HBEF, and Panel (C) shows the distribution of points in the NEON data set. Note different axis spacings across the three plots.
Figure 3 :
3Estimated average site-level species richness (A) and Jaccard index (B) of a community of twelve foliage-gleaning bird species in the Hubbard Brook Experimental Forest (HBEF). Points are posterior means. Jaccard index values are relative to a single site in HBEF with value 1, with 0 indicating no species in common to the reference site, and 1 indicating identical community composition to the reference site.
Figure 4 :
4Average occurrence probabilities of twelve foliage-gleaning bird species in the White Mountain National Forest from 2010 to 2018 from models using different subsets of the three data sources. Points show posterior mean occurrence probabilities averaged across all sites in a given year. Gray shaded region indicates the 95% credible interval for the model with all three data sets.
(i.e., roads) that at least partially fell within the White Mountain National Forest, resulting in a total of 200 nonreplicated point count locations during each survey year. Integration of these three data sets is particularly valuable as each data source has clear advantages and disadvantages(Table 1) and cover disparate areas within the study region, and thus integration may yield parameter estimates more indicative of the entire White Mountains rather than analyses of the data sources independently. See Supplemental Information S1.2 for additional details on the three data sets.the Hubbard Brook Experimental Forest (HBEF) and the National Ecological Observatory
Network (NEON) at Bartlett Experimental Forest (Barnett et al., 2019; National Ecological
Observatory Network (NEON), 2021), while our nonreplicated data set comes from the North
American Breeding Bird Survey (BBS; Pardieck et al. 2020). At HBEF, observers performed
three replicate surveys at each of 373 sites in each survey year to account for imperfect detection,
while observers at NEON used a removal design at 81 sites to separate detection from occurrence.
BBS observers performed point counts at 50 point count locations (called stops) along four routes
Table 1 :
1Characteristics of the three data sources used to model occurrence dynamics of twelve foliage-gleaning birds in the White Mountain National Forest from 2010-2018. Forest Cover corresponds to the amount of forest within a 250m radius of a point count site. Values for elevation and forest cover are mean (minimum, maximum).HBEF
NEON
BBS
Data type
Replicated
Replicated
Nonreplicated
Years
2010-2018
2015-2018
2010-2018
Number of sites
373
81
200
Elevation (m)
607 (240, 932)
432 (268, 766)
352 (134, 917)
Forest Cover (%)
97.7 (71, 100)
94.2 (75, 100)
70.6 (0, 92)
Survey Location Experimental forest Experimental forest
Roadside
We modeled occurrence dynamics for the following twelve foliage-gleaning bird species:
American Redstart (Setophaga ruticilla), Black-and-white Warbler (Mniotilta varia), Blue-
Table 2 ,
2SupplementalFigure S2). Species-level IDMs provided slightly more accurate estimates of species-specific occurrence parameters for some species as compared to the ICOM, which is a result of Bayesian shrinkage (i.e., borrowing strength) driving species-level parameters closer to the community average for those species with extreme parameter values in the ICOM. However, the true species-level parameters were contained within the 95% Bayesian credible interval of the estimated parameter values across 94.9% of all simulations, indicating that this loss in accuracy is negligible. Further, we simulated species-level effects from a uniform distribution rather than a normal distribution, which led to more extreme species values. Losses in accuracy would be much lower for communities of species where the normal assumption is adequate. Thus, in addition to providing community-level parameter effects, the ICOM provides more precise estimates of species-specific effects compared to IDMs with only minor losses in accuracy for extreme species.
Table 2 :
2Precision and accuracy of species-specific parameter estimates when using the
integrated community occupancy model (ICOM) compared to a single species integrated
distribution model (IDM) for a simulated community of 25 species over six years across
100 simulations with one replicated (REP) data set and one nonreplicated (NREP) data
set. Precision improvement is the percentage improvement in precision when using the
ICOM compared to the IDM, where precision is defined as the difference between the
2.5% and 97.5% quantiles of the posterior means. Bias is the average magnitude of the
posterior means minus the true simulated value. Values are averaged across all 25 species
and six years.
Parameter
Precision
ICOM Bias IDM Bias
Parameter
Improvement (%)
γ0 i
41.9
0.235
0.101
NREP detection intercept
γ1 i
33.4
0.070
0.027
NREP detection covariate
φ i
30.0
0.173
0.121
Auto-logistic
β0 i
29.2
0.173
0.108
Occurrence intercept
β1 i
22.5
0.042
0.017
Occurrence covariate
α0 i
18.8
0.104
0.044
REP detection intercept
α1 i
9.63
0.023
0.012
REP detection covariate
Table 3 :
3Two-fold cross validation results comparing predictive performance across models using different combinations of the three data sets. The fitted model is shown in the firstcolumn and log predictive density measures are shown for the entire community with
each data set and across all data sets. Values in parentheses show the average rank of
the model for an individual species across all models, with 1 indicating the model is the
best for all species and 7 indicating the the model is worst for all species. Bold values
indicate the best performing model for each individual data set. Predictive performance
of the model using only NEON data is only assessed at NEON locations because NEON
data are only available for four of the nine study years.
Model
HBEF
BBS
NEON
All
HBEF
-10174 (3.75) -6200 (4.67)
-947 (4.75)
-17321 (4.5)
NEON
-
-
-792 (2.83)
-
BBS
-13876 (3.75)
-5702 (3.5)
-1148 (5.08) -20726 (3.83)
HBEF+NEON
-10097 (3.58) -5878 (3.33)
-852 (3.67)
-16829 (3.33)
HBEF+BBS
-9732 (2.67) -5717 (3.67)
-934 (4.42)
-16383 (2.67)
NEON+BBS
-12560 (4.5)
-5759 (3.25)
-801 (3.83)
-19121 (4.33)
HBEF+NEON+BBS -9767 (2.75) -5691 (2.58) -836 (3.42) -16294 (2.33)
AcknowledgementsWe declare no conflicts of interest. This manuscript is a contribution of the Hubbard Brook Ecosystem Study. Hubbard Brook is part of the Long-Term Ecological Research network, which is supported by the U.S. National Science Foundation. The Hubbard Brook Experimental Forest is operated and maintained by the USDA Forest Service, Northern Research Station, Newtown Square, PA. This work was supported by National Science Foundation grant DBI-1954406.Data AvailabilityAll data and code associated with this manuscript are posted on GitHub (https://github.com/ zipkinlab/Doser_etal_2021_InReview) and will be archived on Zenodo upon acceptance.Authors' ContributionsJWD and EFZ developed the modeling framework with critical insight provided by WL. TSS and MTH assisted in data management and preparation. JWD performed all analyses and led writing of the manuscript. All authors contributed critically to the drafts and gave final approval for publication.
The terrestrial organism and biogeochemistry spatial sampling design for the national ecological observatory network. D T Barnett, P A Duffy, D S Schimel, R E Krauss, K M Irvine, F W Davis, J E Gross, E I Azuaje, A S Thorpe, D Gudex-Cross, Ecosphere. 1022540Barnett, D. T., Duffy, P. A., Schimel, D. S., Krauss, R. E., Irvine, K. M., Davis, F. W., Gross, J. E., Azuaje, E. I., Thorpe, A. S., Gudex-Cross, D., et al. (2019). The terrestrial organism and biogeochemistry spatial sampling design for the national ecological observatory network. Ecosphere, 10(2):e02540.
Integrating multiple data sources to fit matrix population models for interacting species. F Barraquand, O Gimenez, Ecological modelling. 411108713Barraquand, F. and Gimenez, O. (2019). Integrating multiple data sources to fit matrix popu- lation models for interacting species. Ecological modelling, 411:108713.
General methods for monitoring convergence of iterative simulations. S P Brooks, A Gelman, Journal of Computational and Graphical Statistics. 74Brooks, S. P. and Gelman, A. (1998). General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7(4):434-455.
Spatially explicit integrated population models. R B Chandler, J D Clark, Methods in Ecology and Evolution. 512Chandler, R. B. and Clark, J. D. (2014). Spatially explicit integrated population models. Methods in Ecology and Evolution, 5(12):1351-1360.
Generalized joint attribute modeling for biodiversity analysis: Median-zero, multivariate, multifarious data. J S Clark, D Nemergut, B Seyednasrollah, P J Turner, S Zhang, Ecological Monographs. 871Clark, J. S., Nemergut, D., Seyednasrollah, B., Turner, P. J., and Zhang, S. (2017). Generalized joint attribute modeling for biodiversity analysis: Median-zero, multivariate, multifarious data. Ecological Monographs, 87(1):34-56.
Large carnivore recolonization reshapes population and community dynamics in the Rocky Mountains: Implications for harvest management. T J Clark, University of MontanaPhD thesisClark, T. J. (2021). Large carnivore recolonization reshapes population and community dynamics in the Rocky Mountains: Implications for harvest management. PhD thesis, University of Montana.
Confronting preferential sampling when analysing population distributions: diagnosis and model-based triage. P B Conn, J T Thorson, Johnson , D S , Methods in Ecology and Evolution. 811Conn, P. B., Thorson, J. T., and Johnson, D. S. (2017). Confronting preferential sampling when analysing population distributions: diagnosis and model-based triage. Methods in Ecology and Evolution, 8(11):1535-1546.
NIMBLE: MCMC, Particle Filtering, and Programmable Hierarchical Modeling. P De Valpine, C Paciorek, D Turek, N Michaud, C Anderson-Bergman, F Obermeyer, C Wehrhahn Cortes, A Rodrìguez, D Temple Lang, S Paganin, R package version 0.11.1de Valpine, P., Paciorek, C., Turek, D., Michaud, N., Anderson-Bergman, C., Obermeyer, F., Wehrhahn Cortes, C., Rodrìguez, A., Temple Lang, D., and Paganin, S. (2021). NIMBLE: MCMC, Particle Filtering, and Programmable Hierarchical Modeling. R package version 0.11.1.
Programming with models: writing statistical algorithms for general model structures with NIMBLE. P De Valpine, D Turek, C Paciorek, C Anderson-Bergman, D Temple Lang, R Bodik, Journal of Computational and Graphical Statistics. 26de Valpine, P., Turek, D., Paciorek, C., Anderson-Bergman, C., Temple Lang, D., and Bodik, R. (2017). Programming with models: writing statistical algorithms for general model structures with NIMBLE. Journal of Computational and Graphical Statistics, 26:403-413.
Accounting for imperfect detection and survey bias in statistical analysis of presence-only data. R M Dorazio, Global Ecology and Biogeography. 2312Dorazio, R. M. (2014). Accounting for imperfect detection and survey bias in statistical analysis of presence-only data. Global Ecology and Biogeography, 23(12):1472-1484.
Models for inference in dynamic metacommunity systems. R M Dorazio, M Kery, J A Royle, M Plattner, Ecology. 918Dorazio, R. M., Kery, M., Royle, J. A., and Plattner, M. (2010). Models for inference in dynamic metacommunity systems. Ecology, 91(8):2466-2475.
Estimating size and composition of biological communities by modeling the occurrence of species. R M Dorazio, J A Royle, Journal of the American Statistical Association. 100470Dorazio, R. M. and Royle, J. A. (2005). Estimating size and composition of biological commu- nities by modeling the occurrence of species. Journal of the American Statistical Association, 100(470):389-398.
Integrating automated acoustic vocalization data and point count surveys for estimation of bird abundance. J W Doser, A O Finley, A S Weed, E F Zipkin, Methods in Ecology and Evolution. Doser, J. W., Finley, A. O., Weed, A. S., and Zipkin, E. F. (2021a). Integrating automated acoustic vocalization data and point count surveys for estimation of bird abundance. Methods in Ecology and Evolution.
Trends in bird abundance differ among protected forests but not bird guilds. J W Doser, A S Weed, E F Zipkin, K M Miller, A O Finley, Ecological Applications. 2377Doser, J. W., Weed, A. S., Zipkin, E. F., Miller, K. M., and Finley, A. O. (2021b). Trends in bird abundance differ among protected forests but not bird guilds. Ecological Applications, page e2377.
Integrating distance sampling and presence-only data to estimate species abundance. M T Farr, D S Green, K E Holekamp, E F Zipkin, Ecology. 10213204Farr, M. T., Green, D. S., Holekamp, K. E., and Zipkin, E. F. (2021). Integrating distance sampling and presence-only data to estimate species abundance. Ecology, 102(1):e03204.
Comparing spatially-varying coefficients models for analysis of ecological data with non-stationary and anisotropic residual dependence. A O Finley, Methods in Ecology and Evolution. 22Finley, A. O. (2011). Comparing spatially-varying coefficients models for analysis of ecologi- cal data with non-stationary and anisotropic residual dependence. Methods in Ecology and Evolution, 2(2):143-154.
A practical guide for combining data to model species distributions. R J FletcherJr, T J Hefley, E P Robertson, B Zuckerberg, R A Mccleery, R M Dorazio, Ecology. 10062710Fletcher Jr, R. J., Hefley, T. J., Robertson, E. P., Zuckerberg, B., McCleery, R. A., and Dorazio, R. M. (2019). A practical guide for combining data to model species distributions. Ecology, 100(6):e02710.
Rapid and varied responses of songbirds to climate change in california coniferous forests. B J Furnas, Biological Conservation. 108347Furnas, B. J. (2020). Rapid and varied responses of songbirds to climate change in california coniferous forests. Biological Conservation, 241:108347.
Mammal diversity and metacommunity dynamics in urban green spaces: implications for urban wildlife conservation. T Gallo, M Fidino, E W Lehrer, S B Magle, Ecological Applications. 278Gallo, T., Fidino, M., Lehrer, E. W., and Magle, S. B. (2017). Mammal diversity and meta- community dynamics in urban green spaces: implications for urban wildlife conservation. Ecological Applications, 27(8):2330-2341.
Modelling species diversity through species level hierarchical modelling. A E Gelfand, A M Schmidt, S Wu, J A SilanderJr, A Latimer, A G Rebelo, Journal of the Royal Statistical Society: Series C (Applied Statistics). 541Gelfand, A. E., Schmidt, A. M., Wu, S., Silander Jr, J. A., Latimer, A., and Rebelo, A. G. (2005). Modelling species diversity through species level hierarchical modelling. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(1):1-20.
The national elevation dataset. Photogrammetric engineering and remote sensing. D Gesch, M Oimoen, S Greenlee, C Nelson, M Steuck, D Tyler, 68Gesch, D., Oimoen, M., Greenlee, S., Nelson, C., Steuck, M., and Tyler, D. (2002). The national elevation dataset. Photogrammetric engineering and remote sensing, 68(1):5-32.
Inferring species richness using multispecies occupancy modeling: Estimation performance and interpretation. G Guillera-Arroita, M Kéry, J J Lahoz-Monfort, Ecology and evolution. 92Guillera-Arroita, G., Kéry, M., and Lahoz-Monfort, J. J. (2019). Inferring species richness using multispecies occupancy modeling: Estimation performance and interpretation. Ecology and evolution, 9(2):780-792.
Completion of the 2011 national land cover database for the conterminous united states-representing a decade of land cover change information. Photogrammetric Engineering & Remote Sensing. C Homer, J Dewitz, L Yang, S Jin, P Danielson, G Xian, J Coulston, N Herold, J Wickham, K Megown, 81Homer, C., Dewitz, J., Yang, L., Jin, S., Danielson, P., Xian, G., Coulston, J., Herold, N., Wickham, J., and Megown, K. (2015). Completion of the 2011 national land cover database for the conterminous united states-representing a decade of land cover change information. Photogrammetric Engineering & Remote Sensing, 81(5):345-354.
Applied hierarchical modeling in ecology: Analysis of distribution, abundance, and species richness in R and BUGS. M Kéry, J A Royle, Dynamic and advanced models. Academic Press2Kéry, M. and Royle, J. A. (2020). Applied hierarchical modeling in ecology: Analysis of distri- bution, abundance, and species richness in R and BUGS: Volume 2: Dynamic and advanced models. Academic Press.
. J J Lahoz-Monfort, M P Harris, S Wanless, S N Freeman, B J Morgan, Lahoz-Monfort, J. J., Harris, M. P., Wanless, S., Freeman, S. N., and Morgan, B. J. (2017).
Bringing it all together: multi-species integrated population modelling of a breeding community. Journal of Agricultural, Biological and Environmental Statistics. 222Bringing it all together: multi-species integrated population modelling of a breeding commu- nity. Journal of Agricultural, Biological and Environmental Statistics, 22(2):140-160.
Dealing with detection error in site occupancy surveys: what can we do with a single survey. S R Lele, M Moreno, E Bayne, Journal of Plant Ecology. 51Lele, S. R., Moreno, M., and Bayne, E. (2012). Dealing with detection error in site occupancy surveys: what can we do with a single survey? Journal of Plant Ecology, 5(1):22-31.
Estimating site occupancy rates when detection probabilities are less than one. D I Mackenzie, J D Nichols, G B Lachman, S Droege, J Andrew Royle, C A Langtimm, Ecology. 838MacKenzie, D. I., Nichols, J. D., Lachman, G. B., Droege, S., Andrew Royle, J., and Langtimm, C. A. (2002). Estimating site occupancy rates when detection probabilities are less than one. Ecology, 83(8):2248-2255.
Designing occupancy studies: general advice and allocating survey effort. D I Mackenzie, J A Royle, Journal of applied Ecology. 426MacKenzie, D. I. and Royle, J. A. (2005). Designing occupancy studies: general advice and allocating survey effort. Journal of applied Ecology, 42(6):1105-1114.
Measuring biological diversity. A E Magurran, John Wiley & SonsMagurran, A. E. (2013). Measuring biological diversity. John Wiley & Sons.
Conserving herbivorous and predatory insects in urban green spaces. L Mata, C G Threlfall, N S Williams, A K Hahs, M Malipatil, N E Stork, S J Livesley, Scientific reports. 71Mata, L., Threlfall, C. G., Williams, N. S., Hahs, A. K., Malipatil, M., Stork, N. E., and Livesley, S. J. (2017). Conserving herbivorous and predatory insects in urban green spaces. Scientific reports, 7(1):1-12.
The recent past and promising future for data integration methods to estimate species' distributions. D A Miller, K Pacifici, J S Sanderlin, B J Reich, Methods in Ecology and Evolution. 101Miller, D. A., Pacifici, K., Sanderlin, J. S., and Reich, B. J. (2019). The recent past and promising future for data integration methods to estimate species' distributions. Methods in Ecology and Evolution, 10(1):22-37.
Breeding landbird point counts. dp1.10003.001National Ecological Observatory Network (NEONNational Ecological Observatory Network (NEON) (2021). Breeding landbird point counts (dp1.10003.001).
. K Pardieck, D ZiolkowskiJr, M Lutmerding, V Aponte, M.-A Hudson, Pardieck, K., Ziolkowski Jr, D., Lutmerding, M., Aponte, V., and Hudson, M.-A. (2020).
North american breeding bird survey dataset 1966-2019. U.S. Geological Survey data release. 10.5066/P9J6QUF6North american breeding bird survey dataset 1966-2019. U.S. Geological Survey data re- lease, https://doi.org/10.5066/P9J6QUF6.
Integrated modeling of communities: parasitism, competition, and demographic synchrony in sympatric ducks. G Péron, D N Koons, Ecology. 9311Péron, G. and Koons, D. N. (2012). Integrated modeling of communities: parasitism, competi- tion, and demographic synchrony in sympatric ducks. Ecology, 93(11):2456-2464.
Coda: Convergence diagnosis and output analysis for mcmc. M Plummer, N Best, K Cowles, K Vines, News. 61Plummer, M., Best, N., Cowles, K., and Vines, K. (2006). Coda: Convergence diagnosis and output analysis for mcmc. R News, 6(1):7-11.
R Core Team, R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, AustriaR Core Team (2020). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
Evidence of region-wide bat population decline from long-term monitoring and bayesian occupancy models with empirically informed priors. T J Rodhouse, R M Rodriguez, K M Banner, P C Ormsbee, J Barnett, K M Irvine, Ecology and evolution. 919Rodhouse, T. J., Rodriguez, R. M., Banner, K. M., Ormsbee, P. C., Barnett, J., and Irvine, K. M. (2019). Evidence of region-wide bat population decline from long-term monitoring and bayesian occupancy models with empirically informed priors. Ecology and evolution, 9(19):11078-11088.
Decline of the north american avifauna. K V Rosenberg, A M Dokter, P J Blancher, J R Sauer, A C Smith, P A Smith, J C Stanton, A Panjabi, L Helft, M Parr, Science. 3666461Rosenberg, K. V., Dokter, A. M., Blancher, P. J., Sauer, J. R., Smith, A. C., Smith, P. A., Stanton, J. C., Panjabi, A., Helft, L., Parr, M., et al. (2019). Decline of the north american avifauna. Science, 366(6461):120-124.
Hierarchical modeling and inference in ecology: the analysis of data from populations, metapopulations and communities. J A Royle, R M Dorazio, ElsevierRoyle, J. A. and Dorazio, R. M. (2008). Hierarchical modeling and inference in ecology: the analysis of data from populations, metapopulations and communities. Elsevier.
Integrated population models: a novel analysis framework for deeper insights into population dynamics. M Schaub, F Abadi, Journal of Ornithology. 1521Schaub, M. and Abadi, F. (2011). Integrated population models: a novel analysis framework for deeper insights into population dynamics. Journal of Ornithology, 152(1):227-237.
Is more data always better? a simulation study of benefits and limitations of integrated distribution models. E G Simmonds, S G Jarvis, P A Henrys, N J Isaac, R B Hara, Ecography. 4310Simmonds, E. G., Jarvis, S. G., Henrys, P. A., Isaac, N. J., and O'Hara, R. B. (2020). Is more data always better? a simulation study of benefits and limitations of integrated distribution models. Ecography, 43(10):1413-1422.
Effects of species prevalence on the performance of predictive models. R Sor, Y.-S Park, P Boets, P L Goethals, S Lek, Ecological Modelling. 354Sor, R., Park, Y.-S., Boets, P., Goethals, P. L., and Lek, S. (2017). Effects of species prevalence on the performance of predictive models. Ecological Modelling, 354:11-19.
A multiregion community model for inference about geographic variation in species richness. C Sutherland, M Brambilla, P Pedrini, S Tenan, Methods in Ecology and Evolution. 77Sutherland, C., Brambilla, M., Pedrini, P., and Tenan, S. (2016). A multiregion community model for inference about geographic variation in species richness. Methods in Ecology and Evolution, 7(7):783-791.
Practical bayesian model evaluation using leave-one-out cross-validation and waic. A Vehtari, A Gelman, J Gabry, Statistics and computing. 275Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical bayesian model evaluation using leave-one-out cross-validation and waic. Statistics and computing, 27(5):1413-1432.
. D L Wagner, E M Grames, M L Forister, M R Berenbaum, D Stopak, Wagner, D. L., Grames, E. M., Forister, M. L., Berenbaum, M. R., and Stopak, D. (2021).
Insect decline in the anthropocene: Death by a thousand cuts. Proceedings of the National Academy of Sciences. 1182Insect decline in the anthropocene: Death by a thousand cuts. Proceedings of the National Academy of Sciences, 118(2).
Impacts of forest fragmentation on species richness: a hierarchical approach to community modelling. E F Zipkin, A Dewan, Andrew Royle, J , Journal of Applied Ecology. 464Zipkin, E. F., DeWan, A., and Andrew Royle, J. (2009). Impacts of forest fragmentation on species richness: a hierarchical approach to community modelling. Journal of Applied Ecology, 46(4):815-822.
Evaluating the predictive abilities of community occupancy models using auc while accounting for imperfect detection. E F Zipkin, E H C Grant, W F Fagan, Ecological Applications. 227Zipkin, E. F., Grant, E. H. C., and Fagan, W. F. (2012). Evaluating the predictive abilities of community occupancy models using auc while accounting for imperfect detection. Ecological Applications, 22(7):1962-1972.
Multi-species occurrence models to evaluate the effects of conservation and management actions. E F Zipkin, J A Royle, D K Dawson, S Bates, Biological Conservation. 1432Zipkin, E. F., Royle, J. A., Dawson, D. K., and Bates, S. (2010). Multi-species occurrence models to evaluate the effects of conservation and management actions. Biological Conservation, 143(2):479-484.
Synthesizing multiple data types for biological conservation using integrated population models. E F Zipkin, S P Saunders, Biological Conservation. 217Zipkin, E. F. and Saunders, S. P. (2018). Synthesizing multiple data types for biological con- servation using integrated population models. Biological Conservation, 217:240-250.
Addressing data integration challenges to link ecological processes across scales. E F Zipkin, E R Zylstra, A D Wright, S P Saunders, A O Finley, M C Dietze, M S Itter, M W Tingley, Frontiers in Ecology and the Environment. 191Zipkin, E. F., Zylstra, E. R., Wright, A. D., Saunders, S. P., Finley, A. O., Dietze, M. C., Itter, M. S., and Tingley, M. W. (2021). Addressing data integration challenges to link ecological processes across scales. Frontiers in Ecology and the Environment, 19(1):30-38.
| [] |
[
"Faster Creation of Smaller Test Suites (with SNAP)",
"Faster Creation of Smaller Test Suites (with SNAP)"
] | [
"Jianfeng Chen \nComputer Science\nNorth Carolina State University\nUSA\n",
"Tim Menzies \nComputer Science\nNorth Carolina State University\nUSA\n"
] | [
"Computer Science\nNorth Carolina State University\nUSA",
"Computer Science\nNorth Carolina State University\nUSA"
] | [] | State-of-the-art theorem provers, combined with smart sampling heuristics, can generate millions of test cases in just a few hours. But given the heuristic nature of those methods, not all of those tests may be valid. Also, test engineers may find it too burdensome to run all those tests.Within a large space of tests, there can be redundancies (duplicate entries or similar entries that do not contribute much to overall diversity). Our approach, called SNAP uses specialized sub-sampling heuristics to avoid finding those repeated tests. By avoiding those repeated structures SNAP explores a smaller space of options. Hence, it is possible for SNAP to verify all its tests.To evaluate SNAP, this paper applied 27 real-world case studies from a recent ICSE'18 paper. Compared to prior results, SNAP's test case generation was 10 to 3000 times faster (median to max). Also, while prior work showed that their tests were 70% valid, our method generates 100% valid tests. Most importantly, test engineers would find it relatively easiest to apply SNAP's tests since our test suites are 10 to 750 times smaller (median to max) than those generated using prior work. | null | [
"https://arxiv.org/pdf/1905.05358v1.pdf"
] | 153,312,536 | 1905.05358 | 6fa40973264c08fd88156e7f848dc541b6f179eb |
Faster Creation of Smaller Test Suites (with SNAP)
Jianfeng Chen
Computer Science
North Carolina State University
USA
Tim Menzies
Computer Science
North Carolina State University
USA
Faster Creation of Smaller Test Suites (with SNAP)
Index Terms-SAT solverstest suite generationmutation
State-of-the-art theorem provers, combined with smart sampling heuristics, can generate millions of test cases in just a few hours. But given the heuristic nature of those methods, not all of those tests may be valid. Also, test engineers may find it too burdensome to run all those tests.Within a large space of tests, there can be redundancies (duplicate entries or similar entries that do not contribute much to overall diversity). Our approach, called SNAP uses specialized sub-sampling heuristics to avoid finding those repeated tests. By avoiding those repeated structures SNAP explores a smaller space of options. Hence, it is possible for SNAP to verify all its tests.To evaluate SNAP, this paper applied 27 real-world case studies from a recent ICSE'18 paper. Compared to prior results, SNAP's test case generation was 10 to 3000 times faster (median to max). Also, while prior work showed that their tests were 70% valid, our method generates 100% valid tests. Most importantly, test engineers would find it relatively easiest to apply SNAP's tests since our test suites are 10 to 750 times smaller (median to max) than those generated using prior work.
I. INTRODUCTION
This paper replicates and improves QuickSampler [1], a recent ICSE'18 paper which built test suites by applying theorem provers to logical formula extracted from procedural source code. We found that QuickSampler generated test suites with many repeated entries. After applying some redundancy avoidance heuristics (defined below), our new algorithm (called SNAP) runs much faster than QuickSampler and returns much smaller test suites. This is useful since smaller test suites are simpler to execute and maintain.
To generate tests from programs, they must first be converted into a logic formula. Fig. 1 shows how this might be done. Symbolic/dynamic execution techniques [2], [3] extract the possible execution branches of a procedural program. Each branch B a is a conjunction of conditions B i = C x ∧ C y ∧ ... so the whole program can be summarized as the disjunction B i ∨ B j ∨ .... Using deMorgan's rules 1 these clauses can be converted to conjunctive normal form (CNF) where:
• The inputs to the program are the variables in the CNF; • A test is valid if uses input settings that satisfy the CNF. • A test suite is a set of valid tests. • One test suite is more diverse than another if it uses more variable within the CNF disjunctions. Diverse test suites are better since they cover more parts of the code.
Theorem provers like Z3, pycoSAT, MathSAT, or vZ [4]- [7] can use this CNF as follows:
1 P ∨ Q ≡ (¬P ∧ ¬Q) and ¬(P ∧ Q) ≡ ¬P ∨ ¬Q.
• Generation: tests can be generated by "solving" the CNF; i.e. find settings to variables such that all clauses in CNF evaluates to "true". Given multiple disjunctions inside an CNF, one CNF formula can generate multiple tests. • Verification: check if variable settings satisfy the CNF. • Repair: patch tests that fail verification by (a) removing "dubious" variable settings then (b) asking the theorem prover to appropriately complete the missing bits. Later in this paper, we show how to find the "dubious" settings. In terms of runtimes:
• Verification is fastest (since there are no options to search); • Repair is somewhat slower (some more options to search); • And generation is slowest since the theorem prover must search around all the competing CNF constraints. In practice, generation can be very slow indeed. When translated to CNF, the case studies explored in this paper require 5000 to 500,000 variables within 17,000 to 2.6 million clauses (median to max). Even with state-of-the-art theorem provers like Z3, generating a single test from these clauses can take as much as 20 minutes. Worse still, this process must be repeated many times to find enough tests to build a diverse test suite.
To address the runtime issue, many researchers try to minimize the calls to the theorem prover. In that approach, heuristics are used to generate most of the tests. For example, QuickSampler assumes that adding the deltas between valid tests a, b to a valid test c will produce a new valid test d; i.e.
d = c ⊕ (a ⊕ b)(1)
(where ⊕ means "exclusive or"). Using Eq. 1, QuickSampler generates thousands, or even millions of test cases within hours. But there is a catch -heuristically generated tests may be invalid. According to Dutra et al., 30% of the tests built by QuickSampler are often invalid. The research of this paper began when we observed that many of the test cases generated by QuickSampler contained duplicates. For example. in the blasted case47 case study, QuickSampler generated more than 10 7 samples within 2 hours. On these, there were only approximate 2.62 × 10 5 unique solutions (i.e. one in 40).
Based on this observation, we conjectured: If a slow test generator has some redundancy, then to build a faster generator, avoid that redundancy. To test this conjecture, we built SNAP. Like QuickSampler, SNAP uses a combination of Eq. 1 (sometimes) and Z3 (at other times). SNAP also includes some specialized subsampling heuristics that strive to avoid the redundant tests. Fig. 1: From source code (top left) to constraint solver. Note one detail: as x, y and z are integers, we could use (say) 7 bits to representing them. If so, then tools like SMT [8] could find subsets of continuous ranges that satisfy branch conjunctions. That said, this paper is comparing its methods to those of QuickSampler study. Accordingly, we use the same conventions as QuickSampler; i.e. our constraints are just boolean variables.
. . .∧ ∧ ∧(¬11504∨11510)∧ ∧ ∧(¬11510∨11504∨11507∨11502)∧ ∧ ∧. . .
Those heuristics are described later in this paper. For now, all we need say is that SNAP explores and generates a much smaller sample of solutions than QuickSampler. Hence it runs faster, and generates smaller tests suites.
This paper evaluates SNAP via four research questions. RQ1: How reliable is the Eq. 1 heuristic? One reason we advocate SNAP is that, unlike QuickSampler, our method verifies each test (with Z3). But is that necessary? How often does Eq. 1 produce invalid tests? In our experiments, we can confirm Dutra et al.'s estimate that the percent of invalid tests generated by Eq. 1 is usually 30% or less. However, that median result does not quite characterize the variability of that distribution. We found that in a third of case studies, the percent of valid tests generated by Eq. 1 is 25% to 50% (median to max). Hence we say:
Conclusion #1: Eq. 1 should not be used without a verification of the resulting test.
One useful feature of SNAP is that its test suites are so small that it is possible to quickly verify all our final candidate test suites. That is, unlike QuickSampler, all SNAP's tests are valid.
RQ2: How diverse are the SNAP test cases? Since SNAP explores far fewer tests than QuickSampler, its tests suites could be less diverse. Nevertheless:
Conclusion #2: The diversity of SNAP's test suites are not markedly worse than those of QuickSampler.
RQ3: How fast is SNAP? SNAP was motivated by the observation that QuickSampler built many similar test cases; i.e. much of its analysis seemed redundant. If so, we would expect SNAP to generate test cases much faster than Quick-Sampler (since it avoids redundant analysis). This prediction turns out to be true. In our case studies:
Conclusion #3: SNAP was 10 to 3000 times faster than QuickSampler (median to max).
RQ4: How easy is it to apply SNAP's test cases? Finally, we end on a pragmatic note. The smaller a test suite, the easier it is for programmers to run those tests. Therefore it is important to ask which method produces fewer tests: QuickSampler or SNAP? We find that:
Conclusion #4: SNAP's test cases were 10 to 750 times smaller than those of QuickSampler (median to max).
Hence, we argue that it would be easiest for an industrial practitioner to execute and maintain SNAP's test suite.
In summary, the unique contributions of this paper are:
• A novel mutation based sampling algorithm name SNAP; • Experiments on common case studies that compare SNAP to a recent state-of-the-art sampler in method (QuickSampler, from ICSE'18); • Based on that comparison, we show that SNAP generates much smaller solutions that the diverse as the prior stateof-the-art; and does so far faster. Further, 100% of our tests are valid (while other methods may only generate 70% valid tests, or less). • A reproduction package for this paper, and SNAP. 2 The rest of this paper is structured as follows: §II introduces some related works in solving this problem. §III shows the core algorithm of SNAP. §IV addresses the details of ex-periments and the study cases. §V discusses the experiment results. Following that, §VI and §VII have further discussion and conclusions.
II. RELATED WORK
Using the methods of Fig. 1, software often generates CNF with three or more variables per clause. Since 3-SAT problem is NP-complete [18], so generating tests from these clauses is an inherently slow process. This problem has been explored for decades. One way to solve the theorem proving problem is to simplify or decompose the CNF formulas. A recent example in this arena was GreenTire, proposed by Jia et al. [19]. GreenTire supports constraint reuse based on the logical implication relation among constraints. One advantage of this approach is its efficiency guarantees. Similar to the analytical methods in linear programming, they are always applied to specific class of problem. However, even with the improved theorem prover, such methods may be difficult to be adopted in large models. GreenTire was tested in 7 case studies. Each case study was corresponding to a small code script with tens lines of code, e.g. the BinTree in [20]. For the larger models, such as those explored in this paper, the following methods might do better.
Another approach, which we will call sampling, is to combine theorem provers Z3 with stochastic sampling heuristics. For example, given random selections for b, c, Eq. 1 might be used to generate a new test suite, without calling a theorem prover. Theorem proving might then be applied to some (small) subset of the newly generated tests, just to assess how well the heuristics are working. Table I includes some of related works.
The earliest sampling tools were based on binary decision diagrams (BDDs) [21]. Yuan et al. [9], [11] build a BDD from the input constraint model and then weighted the branches of the vertices in the tree such that a stochastic walk from root to the leaf was able to generate samples with desired distribution.
In other work, Iyer proposed a technique named RACE which has been applied in multiple industrial solutions [10]. RACE (a) builds a high-level model to represent the constraints; then (b) implements a branch-and-bound algorithm for sampling diverse solutions. The advantage of RACE is its implementation simplicity. However, RACE, as well as the BDD-based approached introduced above, return highly biased samples, that is, highly non-uniform samples. For testing, this is not recommended since it means small parts of the code get explored at a much higher frequency than others.
Using a SAT solver WalkSat [22], Wei et al. [12] proposed SampleSAT. SampleSAT combines random walk steps with greedy steps from WalkSat. This method works well in small constraint models. However, due to the greedy nature of WalkSat, the performance of SampleSAT is highly skewed as the size of the constraint model increases.
For seeking diverse samples, universal hashing [23] techniques have been proposed. These algorithms were designed for strong guarantees of uniformity. Meel et al. [17] provided an overview of key ingredients of integration of universal hashing and SAT solvers; e.g. with universal hashing, it is possible to guarantee uniform solutions to a constraint model. These hashing algorithms can be applied to the extreme large models (with near 0.5M variables). More recently, several improved hashing-based techniques have been purposed to balance the scalability of the algorithm as well as diversity (i.e. uniform distribution) requirements. For example, Chakraborty et al. proposed an algorithm named UniGen [15], following by the Unigen2 [16]. UniGen provides strong theoretical guarantees on the uniformity of generated solutions and has applied to constraint models with hundreds of thousands of variables. However, UniGen suffered from a large computation resource requirement. Later work explored a parallel version of this approach. Unigen2 achieved near linear speedup of the number of CPU cores.
To the best of our knowledge, the state-of-the-art technique
III. ABOUT SNAP
As stated in the introduction, SNAP uses the Z3 theorem prover combined with Eq. 1. Also, SNAP uses specialized subsampling heuristics to avoid redundant tests. Just to say the obvious, we have no formal proofs that any of the following are useful. Instead, these heuristics are based on hunches we acquired while working with QuickSampler.
Heuristic #1: Instead of computing some deltas between many tests, SNAP restrains mutation to many deltas between a few tests. Specifically, SNAP builds a pool of 10,000 deltas from N = 100 valid tests (note that this process requires calling the theorem prover only N = 100 times). SNAP uses this pool as a set of candidate "mutators" for existing tests (and by "mutator", we mean an operation that converts an existing test into a new one).
Heuristic #2: SNAP builds new tests by apply Eq. 1 to old tests. To minimize redundancy, SNAP uses old tests that are quite distant. More specifically, SNAP uses the centroids found after applying a k = 5-means clustering algorithm to the N = 100 initial samples.
Heuristic #3: We have an intuition that the more frequently we see a delta, the more likely it might represent a valid change to a test. Hence, when SNAP mutates our centroids, it uses deltas that are seen most frequently.
Heuristic #4: We have another intuition that test cases that pass verification and somehow less interesting than those that fail. Hence, when SNAP finds a new failing test, it repairs it (using the process described below) and focuses the rest of the test generation on that harder case.
Heuristic #5: Z3 is much slower for generating new tests than repairing invalid tests than for verifying that a test is valid. As discussed in the introduction, the reason for this is that the search space of options is much larger for generation that for repairing than for verification. Hence, SNAP needs to verify more than it repairs (and also do repairs more than generating new tests).
Algorithm 1 shows how SNAP uses all these heuristics. In this algorithm, each test is a zero or one (false, true) for all the variables in the CNF of our case studies.
SNAP uses the Z3 theorem prover for steps 1a,3biii, and 3biv. As required by Heuristic#5 , SNAP performs verification more often than repair, which in turn is performed far more often than generation:
• The call to Z3 in step 1a can be the slowest (since this a generate call that must navigate all the constraints of our CNF). Hence, we only do this N = 100 times; • The call to Z3 in step 3biii verification call is much faster since all the variables are set. • The call of Z3 in the step 3biv repair call, is a little slower than step 3biii since (as discussed below), our repair operator introduces some open choices into the test. But note that we only need to repair the minority of new tests that fail verification. How small is that minority? Later in this paper, we can use Fig. 3 to show that repairs are only needed on 30% (median) of all tests.
Algorithm 1 requires a repair function for step 3biv, and a termination function for step 4a. Those two functions are discussed below.
A. Implementing "Repair"
When the new test (found in step3ii) is invalid, SNAP uses Z3 to repair that test. As mentioned in the introduction, SNAP's repair function deletes potentially "dubious" parts of a test case, then calls Z3 to fill in the missing details. In this way, when we repair a test, most of the bits are set and Z3 only has to search a small space.
To find the "dubious" section, we reflect on how step 3bii operates. Recall that the new test is c ⊕ δ where δ = a ⊕ b and a, b are valid tests taken from samples. Since a, b were valid tests, then the "dubious" parts of the test are anything that was not seen in both a and b. Hence, we preserve the bits in c ⊕ δ bits (where the corresponding δ bit was 1), while removing all other bits (where δ bit was 0). For example:
• Assuming we are mutating c =(1,0,0,1,1,0,0,0) using δ =(1,0,1,0,1,0,1,0). • If c ⊕ δ =(0,0,1,1,0,0,1,0) is invalid, then SNAP deletes the "dubious" sections as follows. • SNAP preserves any "1" bits that were seen in δ. • SNAP deletes the other bits; i.e. the 2, 4, 6, 8th bits (0, ¡ e 0,1, ¡ e 1,0, ¡ e 0,1, ¡ e 0). • Z3 is then called to fill out the missing bits of (0?1?0?1?). Heuristic #5 (shown above) is based on the assumption that these last step (where Z3 repairs the vector) is usually faster than generating a completely new solution from scratch.
B. Implementing "Termination"
To implement SNAP's termination criteria (step 4a), we need a working measure of diversity. Recall from the introduction that one test suite is more diverse than another if it uses more of the variable settings with disjunctions inside the CNF. Diverse test suites are better since they cover more parts of the code.
To measure diversity, we used the normalized compression distance (NCD) proposed by Feldt et al. [24]. Feldt et al. showed that a test suite with high NCD implies higher code coverage during the testing 3 .
NCD is based on information theory -the Kolmogorov complexity [25] of a binary string x is the length of the shortest program that outputs x. NCD is based on the observation that the degree to which a string can be compressed by real-world compression programs (such as gzip or bzip2). SNAP uses gzip to estimate Kolmogorov complexity. Let C(x) be the length for the compression of x and C(X) be the compression length of binary string set X's concatenation. The NCD of X is defined as
NCD(X) = C(X) − min x∈X {C(x)} max x∈X {C(X\{x})}(2)
SNAP uses NCD as follows. This algorithm terminates when improvement of NCD was less than X = 5% within T = 10 minutes.
C. Engineering Choices
Our implementation used the following control parameters that were set via engineering judgment:
• X = 5%; • T = 10 minutes; • N = 100 samples; • k = 5 clusters. In future work, it could be insightful to vary these values.
Another area that might bear further investigation is the clustering method used in step 3a. For this paper, we tried different clustering methods. Clustering ran so fast that we were not motivated to explore alternate algorithms. Also, we found that the details of the clustering were less important than pruning away most of the items within each cluster (so that we only mutate the centroid).
IV. EXPERIMENTAL SET-UP A. Code
To explore the research questions shown in the introduction, the SNAP system shown in Algorithm 1 was implemented in C++ using Z3 v4.8.4 (the latest release when the experiment was conducted). A k-means cluster was added using the free edition of ALGLIB [26], a numerical analysis and data processing library delivered for free under GPL or Personal/Academic license. QuickSampler does not integrate the samples verification into the workflow. Hence, in the experiment, we adjusted the workflow of QuickSampler so that all samples are verified before termination. Also, the outputs of QuickSampler were the assignments of independent support. The independent support is a subset of variables which completely determines all the assignments to a formula [1]. In practice, engineers need the complete test case input; consequently, for valid samples, we extended the QuickSampler to get full assignments of all variables from independent support's assignment via propagation.
B. Experimental Rig
We compared SNAP to the state-of-the-art QuickSampler, technique purposed by Dutra et al. at ICSE'18. To ensure a repeatable result, we updated the Z3 solver in QuickSampler into the latest version.
To reduce the observation error and test the performance robustness, we repeated all experiment 30 times with 30 different random seeds. To simulate real practice, such random seeds were used in Z3 solver (for initial solution generation), ALGLIB (for the k-means) and other components. Due to the space limitation, we cannot report results for all 30 repeats. Correspondingly we report the medium or the IQR (75-25th variations) results.
All experiments were conducted in the machines with Xeon-E5@2GHz and 4GB memory, running CentOS. These were multi-core machines but for systems reasons, we only used one core per machine. Table II lists the attributes of all the case studies used in this work. We can see that number of variables ranges from hundreds to more than 486K. The large examples have more than 50K clauses, which is very huge. For exposition purposes, we divided the case studies into three groups: the small case studies with vars < 6K; the medium case studies with 6K < vars < 12K and the large case studies with vars > 12K.
C. Case Studies
For the following reasons, our case studies are the same as those used in the QuickSampler paper:
• We wanted to compare our method to QuickSampler over same case studies; • Their case studies were online available; • Their case studies are used in multiple papers [1], [15]- [17] etc. These case studies are representative of scenarios engineers met in software testing or circuit testing in embedded system design. They include bit-blasted versions of SMTLib case studies, ISCAS89 circuits augmented with parity conditions on randomly chosen subsets of outputs and next-state variables, problems arising from automated program synthesis and constraints arising in bounded theorem proving. For more introduction of the case studies, please see [1], [16].
For pragmatic reasons, certain case studies were omitted from our study. For example, we do not report on diagStencil-Clean.sk 41 36 in the experiment, since the purpose of this paper is to sample a set of valid solutions to meet the diversity requirement; while there are only 13 valid solutions from this model. The QuickSampler spent 20 minutes (on average) to search for one solution.
Also, we do report on the case studies marked with a star(*) in Table II. Based on the experiment, we found that even though the QuickSampler generates tens of millions of samples for these examples, all samples were the assignment to the independent support (defined in §IV-A). The omission of these case studies is not a critical issue. Solving or sampling these examples is not difficult; since they are all very small, as compared to other larger case studies.
V. RESULTS
The rest of this paper use the machinery defined above to answer the four research questions posed in the introduction.
A. RQ1: How Reliable is the Eq. 1 Heuristic?
QuickSampler ran so quickly since it assumed that tests generated using Eq. 1 did not need verification. This research question checks that assumption, as follows.
For each case study, we randomly generated 100 valid solutions, S = {s 1 , s 2 , . . . s 100 } using Z3. Next, we selected three {a, b, c} ∈ S and built a new test case using Eq. 1; i.e. new = c ⊕ (a ⊕ b). Fig. 2 reveals the number of identical deltas seen within these 100 2 of deltas. Among all case studies, we rarely found large sets of unique deltas. This means that among the 100 valid solutions given by Z3, many δs were shared within various pairwise solutions. This is important since, if otherwise, the Eq. 1 heuristic would be dubious.
The percentage of these deltas that proved to be valid in step3biii of Algorithm 1 are shown in Fig. 3. Dutra et al.'s estimate were that the percentage of valid tests generated by Eq. 1 was usually 70% or more. As shown by the median values of Fig. 3, this was indeed the case. However, we also see that in lower third of those results, the percent of valid tests generated by Eq. 1 is very low: 25% to 50% (median to max). This result alone would be enough to make us cautious about using QuickSampler since, when the Eq. 1 heuristics fails, it seems to fail very badly. We recommend:
Conclusion #1: Eq. 1 should not be used without a verification of the resulting test.
By way of comparisons, it is useful to add here that SNAP Table II. verifies every test case it generates. This is practical for SNAP, but impractical for QuickSampler since these two systems typically process 10 2 to 10 8 test cases, respectively. In any case, another reason to recommend SNAP over QuickSampler is that the former delivers tests suites where 100% of all tests are valid.
B. RQ2: How Diverse are the SNAP Test Cases?
As stated in our introduction, diverse test suites are better since they cover more parts of the code. A concern with SNAP is that, since it explores fewer tests than QuickSampler its tests suites could be far less diverse. Fig. 4 compares the diversity of the test suites generated by our two systems. These results are expressed as ratios of the observed NCD values. Results less than one indicate that SNAP's test suites are less diverse than QuickSampler.
In that figure, we see that occasionally, SNAP's faster test suite generation means that the resulting test suites are much less diverse (see s1238 3 2 and parity.sk 11 11). That said, while QuickSampler's tests are more diverse, the overall difference is usually very small.
Also, RQ1 showed us that many of the 11 11 is not reported since their achieved NCD were much worse than QuickSampler's (see Fig. 4). Fig. 6 illustrates the corresponding speedups.
QuickSampler are somewhat inflated since invalid tests would not enter the branches they are meant to cover. Hence, overall, we say: Table II. That is, from left to right, these case studies grow from around 300 to around 3,000,000 clauses.
For the smaller case studies, shown on the left, SNAP is sometimes slower than QuickSampler. Moving left to right, from smaller to larger case studies, it can be seen that SNAP often terminates much faster than QuickSampler. Fig. 6 is a summary of Fig. 5 that divides the execution time for both systems. From this figure it can be seen:
Conclusion #3: SNAP was 10 to 3000 times faster than QuickSampler (median to max).
There are some exceptions to this conclusion, where Quick-Sampler was faster than SNAP (see the right-hand-side of Fig. 6). We note that in most of those these cases, those models are small (17,000 clauses or less). For medium to larger models, with 20,000 to 2.5 million clauses, SNAP is nearly always orders of magnitude faster than QuickSampler. D. RQ4: How Easy is it to Apply SNAP's Test Cases?
Finally, we end on a pragmatic note. The smaller a test suite, the easier it is for programmers to run those tests. Therefore it is important to ask which method produces fewer tests: QuickSampler or SNAP? Table III compares the number of tests (different suggested inputs) generated by QuickSampler and SNAP. As shown by the last column in that table, Conclusion #4: SNAP's test cases were 10 to 750 times smaller than those of QuickSampler (median to max).
Hence, we argue that it would be easiest for an industrial practitioner to execute and maintain the test suites generated by SNAP.
VI. DISCUSSION
A. Why does SNAP Work?
This section reflects on the success of SNAP. Specifically, we ask why can SNAP generate comparable diversity to QuickSampler? We conjecture that, for SE problems: Combining solutions from local samples leads to diverse global solutions.
SNAP is an example of such a "local sampling". Consider how it executes: each round of the sampling (step 3 in Algorithm 1) focuses on a local sample (generated by the clustering method). All these local samples then combined into a global test suite. The success of SNAP's local sampling strategy may be a comment on the nature of software systems. Large SE systems are the result of much work by many teams. In such systems, small parts of software combine into some greater whole. For such systems, local sampling (as done in SNAP) would be useful. There is much other evidence of the benefits of local sampling in SE. Chen et al. proposed a local sampling technique called SWAY [27], [28] that finds optimized configuration for agile requirements engineering [29]. Subsequently, it was seen that the same local sampling approach leads to a new highwatermark in optimizing cloud containers deployments [30]. All of that researches used the same strategy (local sampling) and had similar results (orders of magnitude improvement on the prior state-of-the-art).
Based on this, we make two observations. Firstly, local sampling is a strategy that may be very useful in many future SE applications. Secondly, if SE problems have such a structure (of larger problems composed of numerous smaller ones), then it would not be insightful to assess new methods using randomly generated solutions (e.g. to assess SNAP using randomly generated CNF formula). We say this since such randomly generated models may not contain the structures that have been found so useful in local sampling methods like SNAP and elsewhere [27], [28], [30].
B. Threats to Validity
One threat to validity of this work is the baseline bias. Indeed, there are many other sampling techniques, or solvers, that SNAP might be compared to. However, our goal here was to compare SNAP to a recent state-of-the-art result from ICSE'18. In further work, we will compare SNAP to other methods.
A second threat to validity is internal bias that raises from the stochastic nature of sampling techniques. SNAP requires many random operations. To mitigate the threats, we repeated the experiments for 30 times and reported the medium or IQR of those results.
A third threat is the measurement bias. To determine the diversity of a test suite, in the experiment, we use normalized compression distance (NCD). Prior research has argued for the value of that measure [24]. However, there exist many other diversity measurements for the theorem proving problem and changing the diversity measurement might lead to change of the results. That said, in one research report, it is impossible to explore all options. For the convenient of further exploration, we have released the source code of SNAP in the hope that other researchers will assist us by evaluating SNAP on a broader range of measures.
Another threat is hyperparameter bias. The hyperparameter is the set of configurations for the algorithm. For SNAP, we need to use a set of control parameters shown in §III-C. There now exists a range of mature hyperparameter optimizers [31]- [33] which might be useful for finding better settings for SNAP. This is a clear direction for future work.
VII. CONCLUSION
The experiments of this paper suggest that SNAP is a better test suite generation system than QuickSampler. Our system avoids much of the redundant reasoning that:
• Slows up QuickSampler by a factor of 10 to 3000 (compared to SNAP); • And which also results in test suites that are 10 to 750 times larger than they need to be (again, compared to SNAP). Another reason to prefer SNAP is that since its test suites are so small, we can run verification on 100% of all of SNAP's test. That is, unlike QuickSampler, all our tests are known to be valid.
SNAP has its drawbacks. Specifically, the diversity of its test suites are not always as good as QuickSampler. That said, this difference in diversity is so small (see Fig. 4) that overall, given all the other advantages of SNAP, we can still recommend it.
As shown by the Algorithm 1 pseudocode, SNAP is relatively simple to implement, Hence, if nothing else, we can recommend SNAP as a baseline method against which more elaborate methods might be benchmarked.
Algorithm 1 : SNAP 0 )
10Set up a) let N = 100; i.e. initial sample size; b) let k = 5; i.e. number of clusters; c) let suite= ∅; i.e. the output test suite; d) let samples= ∅; i.e. a temporary work space. 1) Initial samples generation: a) Add N solutions (from Z3) to samples b) Put all samples into suite (since they are valid) 2) Delta preparation (heuristic#1) : a) Find delta δ = (a ⊕ b) for all a, b ∈ samples. b) Weight each delta by how often it repeats 3) Sampling a) Find k centroids in samples using k-means (heuristic#2) ; b) For each centroid c, repeat N times: i) Select stochastically two deltas δ i , δ j at probability equal to their weight (heuristic#3) . ii) Get new candidate via c ⊕ (δ i ∨ δ j ) iii) Verify new candidate using Z3; iv) If invalid, then repair using Z3 (see §III-A) and add to sample (heuristic#4) ; v) Add new to suite; 4) Loop or terminate: a) If diversity improving (see §III-B), go to step 2. b) Else terminate, returning suite.
Fig. 2 :
2Number of identical deltas among 100*100 pair of valid solution deltas for all case studies. Same color scheme asTable II.
Fig. 3 :
3RQ1 results: percentage of valid mutations found it step3biii (computed separately for each case study).
Fig. 4 :
4RQ2 results: Normalized compression distance (NCD) observed when QuickSampler and SNAP terminated on the same case studies. Median results over 30 runs (and the small black lines show the 75th-25th variations). Same color scheme as
Fig. 5 :
5RQ3 results: Time to terminated (seconds), The y-axis is in log scale. The SNAP sampling time for s1238 a 3 2 and parity.sk
Conclusion #2 :
#2The diversity of SNAP's test suites are not markedly worse than those of QuickSampler. C. RQ3: How Fast is SNAP? Fig. 5 shows the execution time required for SNAP and QuickSampler. The y-axis of this plot is a log-scale and shows time in seconds. These results are shown in the same order as
Fig. 6 :
6RQ3 results: Sorted speedup (time(QuickSampler) / time(SNAP)). The speedup > 10 0 implies SNAP terminates earlier than QuickSampler.
The code above has the six branches shown below. Each branch can be modeled as a logical constraint C1 ∨ C2 ∨ C3 . . . ∨ C6. A valid test selects x, y, z such that it satisfies these constraints.By convention, the disjunction ∨Ci is transformed into the conjunction normal form (CNF) C 1 ∧C 2 . . .. A valid assignment to the CNF, i.e. the assignment that fulfills all clauses, is corresponding to a test case, covering some branch of code. When translated into the input required for Z3, these conjunctions look like:Line 1 indicates that that this CNF expression has 11511 variables which are shared between 41411 clauses. Remaining lines the 41411 clauses (and a zero at end of line denotes endof-clause). For example, line 2-5 can be read as1 int mid(int x, int y, int z) {
2 if (x < y) {
3
if (y < z) return y;
4
else if (x < z) return z;
5
else return x;
6 } else if (x < z) return x;
7 else if (y < z) return z;
8 else return y; }
path 1: [C1: x < y < z] L2->L3
path 2: [C2: x < z < y] L2->L3->L4
path 3: [C3: z < x < y] L2->L3->L4->L5
path 4: [C4: y < x < z] L2->L6
path 5: [C5: y < z < x] L2->L6->L7
path 6: [C6: z < y < x] L2->L6->L7->L8
1 p cnf 11511 41411
2 ...
3 -11507 11510 0
4 -11510 11504 11507 11502 0
5 ...
TABLE I :
ISNAP and its related work for solving theorem proving constraints via sampling. / : the absence / presence of corresponding item : only partial case studies (the small case studies) were reported for generating test cases using theorem provers is Quick-Sampler [1]. QuickSampler was evaluated on large real-world case studies, some of which have more than 400K variables. At ICSE'18, it was shown that QuickSampler outperforms aforementioned Unigen2 as well as another similar technique named SearchTreeSampler [14]. QuickSampler starts from a set of valid solutions generated by Z3. Next, it computes the differences between the solutions using Eq. 1. New test cases generated in this manner are not guaranteed to be valid. According to Dutra et al.'s experiments, the percent of valid tests found by QuickSampler can be higher than 70%. The percent of valid tests found by SNAP, on the other hand, is 100%. Further, as shown below, SNAP builds those tests with enough diversity much faster than QuickSampler.Reference Year Citation
Sampling methodology
Case study size
(max|variables|)
Verifying
samples
Distribution/
diversity
reported
[9]
1999 105
Binary Decision Diagram
≈1.3K
[10]
2003 50
Interval-propagation-based
200
[11]
2004 54
Binary Decision Diagram
< 1K
[12]
2004 141
Random Walk + WALKSAT
No experiment conducted
[13]
2011 88
Sampling via determinism
6k
[14]
2012 25
MAXSAT + Search Tree
Experiment details not reported
[15]
2014 29
Hashing based
400K
[16]
2015 28
Hashing based (paralleling)
400K
[17]
2016 29
Universal hashing
400K
[1]
2018 5
Z3 + Eq. 1 flipping
400K
SNAP
2019 this
paper
Z3 + Eq. 1 + local sampling
400K
TABLE II :
IICase studies used in this paper. Sorted by number of variables. Medium sized-problems are highlighted with blue rows while the large ones are in orange rows. Three items (marked with *) are not included in some further reports (see text). See text for details.Size
Case studies
Vars
Clauses
blasted case47
118
328
blasted case110
287
1263
s820a 7 4
616
1703
s820a 15 7
685
1987
s1238a 3 2
685
1850
Small
s1196a 3 2
689
1805
s832a 15 7
693
2017
blasted case 1 b12 2*
827
2725
blasted squaring16*
1627
5835
blasted squaring7*
1628
5837
70.sk 3 40
4669
15864
ProcessBean.sk 8 64
4767
14458
56.sk 6 38
4836
17828
35.sk 3 52
4894
10547
80.sk 2 48
4963
17060
7.sk 4 50
6674
24816
doublyLinkedList.sk 8 37
6889
26918
19.sk 3 48
6984
23867
29.sk 3 45
8857
31557
Medium isolateRightmost.sk 7 481
10024
35275
17.sk 3 45
10081
27056
81.sk 5 51
10764
38006
LoginService2.sk 23 36
11510
41411
sort.sk 8 52
12124
49611
parity.sk 11 11
13115
47506
77.sk 3 44
14524
27573
Large
20.sk 1 51
15465
60994
enqueueSeqSK.sk 10 42
16465
58515
karatsuba.sk 7 41
19593
82417
tutorial3.sk 4 31
486193 2598178
QuickSampler tests are invalid. This means that the diversity numbers reported forbl as te d_ ca se 47
bl as te d_ ca se 11 0
s8 20 a_ 7_ 4
s8 20 a_ 15 _7
s1 23 8a _3 _2
s1 19 6a _3 _2
s8 32 a_ 15 _7
70 .s k_ 3_ 40
Pr oc es sB ea n. XX
56 .s k_ 6_ 38
35 .s k_ 3_ 52
80 .s k_ 2_ 48
7. sk _4 _5 0 do ub ly Li nk ed Li st .X X 19 .s k_ 3_ 48
29 .s k_ 3_ 45 is ol at eR ig ht m os t. XX 17 .s k_ 3_ 45
81 .s k_ 5_ 51 Lo gi nS er vi ce 2. XX so rt .s k_ 8_ 52 pa rit y. sk _1 1_ 11
77 .s k_ 3_ 44
20 .s k_ 1_ 51 en qu eu eS eq SK .X X ka ra ts ub a. sk _7 _4 1 tu to ria l3 .s k_ 4_ 31
0
10 0
10 1
10 2
10 3
10 4
Sampling time (s)
SNAP
QuickSampler
TABLE III :
IIIRQ4: results. Number of unique valid cases in the test suite. Case studies are sorted by number of variables. Same color scheme asTable II.S S
S Q
S Q /
Case studies
SNAP
QuickSampler
S S
blasted case47
2899
71
0.02
isolateRightmost
15480
7510
0.49
LoginService2
404
210
0.52
19.sk 3 48
204
200
0.98
70.sk 3 40
3050
4270
1.40
s820a 15 7
29065
70099
2.41
29.sk 3 45
225
660
2.93
s820a 7 4
37463
124457
3.32
s832a 15 7
27540
96764
3.51
s1196a 3 2
225
1890
8.40
enqueueSeqSK
338
2495
7.38
blasted case110
274
2386
8.71
tutorial3.sk 4 31
336
2953
8.79
81.sk 5 51
227
2814
12.40
sort.sk 8 52
812
10184
12.54
karatsuba.sk 7 41
139
4210
30.29
20.sk 1 51
239
10039
42.00
doublyLinkedList
278
12042
43.32
17.sk 3 45
228
12780
56.05
ProcessBean
1193
75392
63.20
7.sk 4 50
258
18090
70.12
56.sk 6 38
1827
149031
81.57
80.sk 2 48
653
54440
83.37
77.sk 3 44
245
33858 138.20
35.sk 3 52
258
193920 751.63
Source code at https://github.com/ai-se/SatSpaceExpo
Just as an aside, we note that we did not adopt the diversity metric (distribution of samples displayed as histogram) from[1],[16] since computing that metric is very time-consuming. For the case studies of this paper, that calculation required days of CPU.
Efficient sampling of sat solutions for testing. R Dutra, K Laeufer, J Bachrach, K Sen, 2018 IEEE/ACM 40th International Conference on Software Engineering (ICSE). IEEER. Dutra, K. Laeufer, J. Bachrach, and K. Sen, "Efficient sampling of sat solutions for testing," in 2018 IEEE/ACM 40th International Conference on Software Engineering (ICSE). IEEE, 2018, pp. 549-559.
A survey of symbolic execution techniques. R Baldoni, E Coppa, D C Delia, C Demetrescu, I Finocchi, ACM Computing Surveys (CSUR). 51350R. Baldoni, E. Coppa, D. C. Delia, C. Demetrescu, and I. Finocchi, "A survey of symbolic execution techniques," ACM Computing Surveys (CSUR), vol. 51, no. 3, p. 50, 2018.
Guiding dynamic symbolic execution toward unverified program executions. M Christakis, P Müller, V Wüstholz, Proceedings of the 38th International Conference on Software Engineering. the 38th International Conference on Software EngineeringACMM. Christakis, P. Müller, and V. Wüstholz, "Guiding dynamic symbolic execution toward unverified program executions," in Proceedings of the 38th International Conference on Software Engineering. ACM, 2016, pp. 144-155.
Z3: An efficient smt solver. L De Moura, N Bjørner, International conference on Tools and Algorithms for the Construction and Analysis of Systems. SpringerL. De Moura and N. Bjørner, "Z3: An efficient smt solver," in Inter- national conference on Tools and Algorithms for the Construction and Analysis of Systems. Springer, 2008, pp. 337-340.
Picosat essentials. A Biere, Journal on Satisfiability, Boolean Modeling and Computation. 4A. Biere, "Picosat essentials," Journal on Satisfiability, Boolean Model- ing and Computation, vol. 4, pp. 75-97, 2008.
The mathsat 4 smt solver. R Bruttomesso, A Cimatti, A Franzén, A Griggio, R Sebastiani, International Conference on Computer Aided Verification. SpringerR. Bruttomesso, A. Cimatti, A. Franzén, A. Griggio, and R. Sebastiani, "The mathsat 4 smt solver," in International Conference on Computer Aided Verification. Springer, 2008, pp. 299-303.
νz-an optimizing smt solver. N Bjørner, A.-D Phan, L Fleckenstein, International Conference on Tools and Algorithms for the Construction and Analysis of Systems. SpringerN. Bjørner, A.-D. Phan, and L. Fleckenstein, "νz-an optimizing smt solver," in International Conference on Tools and Algorithms for the Construction and Analysis of Systems. Springer, 2015, pp. 194-199.
Equisatisfiable sat encodings of arithmetical operations. M Finke, M. Finke, "Equisatisfiable sat encodings of arithmetical operations," Online] http://www. martin-finke. de/documents/Masterarbeit bitblast Finke. pdf, 2015.
Modeling design constraints and biasing in simulation using bdds. J Yuan, K Shultz, C Pixley, H Miller, A Aziz, Proceedings of the 1999 IEEE/ACM international conference on Computer-aided design. the 1999 IEEE/ACM international conference on Computer-aided designIEEE PressJ. Yuan, K. Shultz, C. Pixley, H. Miller, and A. Aziz, "Modeling design constraints and biasing in simulation using bdds," in Proceedings of the 1999 IEEE/ACM international conference on Computer-aided design. IEEE Press, 1999, pp. 584-590.
Race: A word-level atpg-based constraints solver system for smart random simulation. M A Iyer, IEEE299M. A. Iyer, "Race: A word-level atpg-based constraints solver system for smart random simulation," in null. IEEE, 2003, p. 299.
Simplifying boolean constraint solving for random simulation-vector generation. J Yuan, A Aziz, C Pixley, K Albin, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. 233J. Yuan, A. Aziz, C. Pixley, and K. Albin, "Simplifying boolean con- straint solving for random simulation-vector generation," IEEE Trans- actions on Computer-Aided Design of Integrated Circuits and Systems, vol. 23, no. 3, pp. 412-420, 2004.
Towards efficient sampling: Exploiting random walk strategies. W Wei, J Erenrich, B Selman, AAAI. 4W. Wei, J. Erenrich, and B. Selman, "Towards efficient sampling: Exploiting random walk strategies," in AAAI, vol. 4, 2004, pp. 670-676.
Samplesearch: Importance sampling in presence of determinism. V Gogate, R Dechter, Artificial Intelligence. 1752V. Gogate and R. Dechter, "Samplesearch: Importance sampling in presence of determinism," Artificial Intelligence, vol. 175, no. 2, pp. 694-729, 2011.
Uniform solution sampling using a constraint solver as an oracle. S Ermon, C P Gomes, B Selman, arXiv:1210.4861arXiv preprintS. Ermon, C. P. Gomes, and B. Selman, "Uniform solution sampling using a constraint solver as an oracle," arXiv preprint arXiv:1210.4861, 2012.
Balancing scalability and uniformity in sat witness generator. S Chakraborty, K S Meel, M Y Vardi, Proceedings of the 51st Annual Design Automation Conference. the 51st Annual Design Automation ConferenceACMS. Chakraborty, K. S. Meel, and M. Y. Vardi, "Balancing scalability and uniformity in sat witness generator," in Proceedings of the 51st Annual Design Automation Conference. ACM, 2014, pp. 1-6.
On parallel scalable uniform sat witness generation. S Chakraborty, D J Fremont, K S Meel, S A Seshia, M Y Vardi, International Conference on Tools and Algorithms for the Construction and Analysis of Systems. SpringerS. Chakraborty, D. J. Fremont, K. S. Meel, S. A. Seshia, and M. Y. Vardi, "On parallel scalable uniform sat witness generation," in International Conference on Tools and Algorithms for the Construction and Analysis of Systems. Springer, 2015, pp. 304-319.
Constrained sampling and counting: Universal hashing meets sat solving. K S Meel, M Y Vardi, S Chakraborty, D J Fremont, S A Seshia, D Fried, A Ivrii, S Malik, Workshops at the thirtieth AAAI conference on artificial intelligence. K. S. Meel, M. Y. Vardi, S. Chakraborty, D. J. Fremont, S. A. Seshia, D. Fried, A. Ivrii, and S. Malik, "Constrained sampling and counting: Universal hashing meets sat solving," in Workshops at the thirtieth AAAI conference on artificial intelligence, 2016.
A simplified np-complete satisfiability problem. C A Tovey, Discrete applied mathematics. 81C. A. Tovey, "A simplified np-complete satisfiability problem," Discrete applied mathematics, vol. 8, no. 1, pp. 85-89, 1984.
Enhancing reuse of constraint solutions to improve symbolic execution. X Jia, C Ghezzi, S Ying, Proceedings of the 2015 International Symposium on Software Testing and Analysis. the 2015 International Symposium on Software Testing and AnalysisACMX. Jia, C. Ghezzi, and S. Ying, "Enhancing reuse of constraint solutions to improve symbolic execution," in Proceedings of the 2015 Interna- tional Symposium on Software Testing and Analysis. ACM, 2015, pp. 177-187.
Test input generation for java containers using state matching. W Visser, C S Psreanu, R Pelánek, Proceedings of the 2006 international symposium on Software testing and analysis. the 2006 international symposium on Software testing and analysisACMW. Visser, C. S. Psreanu, and R. Pelánek, "Test input generation for java containers using state matching," in Proceedings of the 2006 international symposium on Software testing and analysis. ACM, 2006, pp. 37-48.
Binary decision diagrams. S B Akers, IEEE Transactions. 6S. B. Akers, "Binary decision diagrams," IEEE Transactions on com- puters, no. 6, pp. 509-516, 1978.
Local search strategies for satisfiability testing. B Selman, H A Kautz, B Cohen, Cliques, coloring, and satisfiability. 26B. Selman, H. A. Kautz, B. Cohen et al., "Local search strategies for satisfiability testing." Cliques, coloring, and satisfiability, vol. 26, pp. 521-532, 1993.
The computational complexity of universal hashing. Y Mansour, N Nisan, P Tiwari, Theoretical Computer Science. 1071Y. Mansour, N. Nisan, and P. Tiwari, "The computational complexity of universal hashing," Theoretical Computer Science, vol. 107, no. 1, pp. 121-133, 1993.
Test set diameter: Quantifying the diversity of sets of test cases. R Feldt, S Poulding, D Clark, S Yoo, 2016 IEEE International Conference on Software Testing, Verification and Validation. ICSTR. Feldt, S. Poulding, D. Clark, and S. Yoo, "Test set diameter: Quanti- fying the diversity of sets of test cases," in 2016 IEEE International Conference on Software Testing, Verification and Validation (ICST).
. IEEE. IEEE, 2016, pp. 223-233.
An introduction to Kolmogorov complexity and its applications. M Li, P Vitányi, Springer Science & Business MediaM. Li and P. Vitányi, An introduction to Kolmogorov complexity and its applications. Springer Science & Business Media, 2013.
Alglib. S Bochkanov, V Bystritsky, Available from: www. alglib. net. 59S. Bochkanov and V. Bystritsky, "Alglib," Available from: www. alglib. net, vol. 59, 2013.
sampling" as a baseline optimizer for search-based software engineering. J Chen, V Nair, R Krishna, T Menzies, IEEE Transactions on Software Engineering. J. Chen, V. Nair, R. Krishna, and T. Menzies, "" sampling" as a baseline optimizer for search-based software engineering," IEEE Transactions on Software Engineering, 2018.
Beyond evolutionary algorithms for search-based software engineering. J Chen, V Nair, T Menzies, Information and Software Technology. 95J. Chen, V. Nair, and T. Menzies, "Beyond evolutionary algorithms for search-based software engineering," Information and Software Technol- ogy, vol. 95, pp. 281-294, 2018.
Xomo: Understanding development options for autonomy. T Menzies, J Richardson, COCOMO forum. T. Menzies and J. Richardson, "Xomo: Understanding development options for autonomy," in COCOMO forum, vol. 2005, 2005.
Riot: A stochastic-based method for workflow scheduling in the cloud. J Chen, T Menzies, 2018 IEEE 11th International Conference on Cloud Computing (CLOUD). IEEEJ. Chen and T. Menzies, "Riot: A stochastic-based method for workflow scheduling in the cloud," in 2018 IEEE 11th International Conference on Cloud Computing (CLOUD). IEEE, 2018, pp. 318-325.
Revisiting unsupervised learning for defect prediction. W Fu, T Menzies, Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering. the 2017 11th Joint Meeting on Foundations of Software EngineeringACMW. Fu and T. Menzies, "Revisiting unsupervised learning for defect pre- diction," in Proceedings of the 2017 11th Joint Meeting on Foundations of Software Engineering. ACM, 2017, pp. 72-83.
Hyperparameter optimization for effort estimation. T Xia, R Krishna, J Chen, G Mathew, X Shen, T Menzies, arXiv:1805.00336arXiv preprintT. Xia, R. Krishna, J. Chen, G. Mathew, X. Shen, and T. Menzies, "Hyperparameter optimization for effort estimation," arXiv preprint arXiv:1805.00336, 2018.
Data-driven search-based software engineering. V Nair, A Agrawal, J Chen, W Fu, G Mathew, T Menzies, L Minku, M Wagner, Z Yu, 2018 IEEE/ACM 15th International Conference on Mining Software Repositories (MSR). IEEEV. Nair, A. Agrawal, J. Chen, W. Fu, G. Mathew, T. Menzies, L. Minku, M. Wagner, and Z. Yu, "Data-driven search-based software engineering," in 2018 IEEE/ACM 15th International Conference on Mining Software Repositories (MSR). IEEE, 2018, pp. 341-352.
| [
"https://github.com/ai-se/SatSpaceExpo"
] |
[] | [
"G Q Ding ",
"J L Qu \nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina\n",
"L M Song \nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina\n",
"Y Huang \nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina\n",
"S Zhang \nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina\n",
"Q C Bu \nInstitut für Astronomie und Astrophysik\nKepler Center for Astro and Particle Physics\nEberhard Karls Universität\nSand 172076TübingenGermany\n",
"M Y Ge \nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina\n",
"X B Li \nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina\n",
"L Tao \nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina\n",
"X Ma \nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina\n",
"Y P Chen \nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina\n",
"Y L Tuo \nInstitute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina\n"
] | [
"Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina",
"Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina",
"Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina",
"Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina",
"Institut für Astronomie und Astrophysik\nKepler Center for Astro and Particle Physics\nEberhard Karls Universität\nSand 172076TübingenGermany",
"Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina",
"Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina",
"Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina",
"Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina",
"Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina",
"Institute of High Energy Physics\nKey Laboratory of Particle Astrophysics\nChinese Academy of Sciences\n100049BeijingChina"
] | [] | Using the observations of the high-energy (HE) detector of the Hard X-ray Modulation Telescope (Insight-HXMT) for Scorpius X-1 in 2018, we search for hard X-ray tails in the X-ray spectra in ∼30-200 keV. The hard X-ray tails are found throughout the Z-track on the hardness-intensity diagram and they harden and fade away from the horizontal branch (HB), through the normal branch (NB), to the flaring branch (FB). Comparing the hard X-ray spectra of Insight-HXMT between Cyg X-1 and Sco X-1, it is concluded that the hard X-ray spectrum of Cyg X-1 shows high-energy cutoff, implying a hot corona in it, but the high-energy cutoff does not reveal in the hard Xray spectrum of Sco X-1. Jointly fitting the HE spectrum with the medium-energy and low-energy spectra of Sco X-1 in ∼2-200 keV, it is suggested that the upscattering Comptonization of the neutron star (NS) emission photons by the energetic free-falling electrons onto the NS or by the hybrid electrons in the boundary layer between the NS and the accretion disk could be responsible for the hard X-ray tails of Sco X-1 on the HB and NB, but neither of the two mechanisms can be responsible for the hard X-ray tail on the FB. Some possible origins for the peculiar hard X-ray tail of FB are argued. | null | [
"https://arxiv.org/pdf/2203.05299v1.pdf"
] | 247,362,928 | 2203.05299 | 7af5ed3d2f188c680827c7754269f8c0e0570655 |
10 Mar 2022
G Q Ding
J L Qu
Institute of High Energy Physics
Key Laboratory of Particle Astrophysics
Chinese Academy of Sciences
100049BeijingChina
L M Song
Institute of High Energy Physics
Key Laboratory of Particle Astrophysics
Chinese Academy of Sciences
100049BeijingChina
Y Huang
Institute of High Energy Physics
Key Laboratory of Particle Astrophysics
Chinese Academy of Sciences
100049BeijingChina
S Zhang
Institute of High Energy Physics
Key Laboratory of Particle Astrophysics
Chinese Academy of Sciences
100049BeijingChina
Q C Bu
Institut für Astronomie und Astrophysik
Kepler Center for Astro and Particle Physics
Eberhard Karls Universität
Sand 172076TübingenGermany
M Y Ge
Institute of High Energy Physics
Key Laboratory of Particle Astrophysics
Chinese Academy of Sciences
100049BeijingChina
X B Li
Institute of High Energy Physics
Key Laboratory of Particle Astrophysics
Chinese Academy of Sciences
100049BeijingChina
L Tao
Institute of High Energy Physics
Key Laboratory of Particle Astrophysics
Chinese Academy of Sciences
100049BeijingChina
X Ma
Institute of High Energy Physics
Key Laboratory of Particle Astrophysics
Chinese Academy of Sciences
100049BeijingChina
Y P Chen
Institute of High Energy Physics
Key Laboratory of Particle Astrophysics
Chinese Academy of Sciences
100049BeijingChina
Y L Tuo
Institute of High Energy Physics
Key Laboratory of Particle Astrophysics
Chinese Academy of Sciences
100049BeijingChina
10 Mar 2022arXiv:2203.05299v1 [astro-ph.HE] Insight-HXMT Detections of Hard X-ray Tails in Scorpius X-1Subject headings: Neutron stars (1108)X-ray binary stars (1811)Low-mass X-ray binary stars (939)Non-thermal radiation sources (1119)
Using the observations of the high-energy (HE) detector of the Hard X-ray Modulation Telescope (Insight-HXMT) for Scorpius X-1 in 2018, we search for hard X-ray tails in the X-ray spectra in ∼30-200 keV. The hard X-ray tails are found throughout the Z-track on the hardness-intensity diagram and they harden and fade away from the horizontal branch (HB), through the normal branch (NB), to the flaring branch (FB). Comparing the hard X-ray spectra of Insight-HXMT between Cyg X-1 and Sco X-1, it is concluded that the hard X-ray spectrum of Cyg X-1 shows high-energy cutoff, implying a hot corona in it, but the high-energy cutoff does not reveal in the hard Xray spectrum of Sco X-1. Jointly fitting the HE spectrum with the medium-energy and low-energy spectra of Sco X-1 in ∼2-200 keV, it is suggested that the upscattering Comptonization of the neutron star (NS) emission photons by the energetic free-falling electrons onto the NS or by the hybrid electrons in the boundary layer between the NS and the accretion disk could be responsible for the hard X-ray tails of Sco X-1 on the HB and NB, but neither of the two mechanisms can be responsible for the hard X-ray tail on the FB. Some possible origins for the peculiar hard X-ray tail of FB are argued.
INTRODUCTION
The hard X-ray emission is usually observed in black hole X-ray binaries (BHXBs) (e.g., Dove et al. 1998;Remillard & McClintock 2006;Montanari et al. 2009;Titarchuk & Seifina 2009;Motta et al. 2021), which could be attributed to a hot corona in them (e.g., Zhang et al. 2000;Yao et al. 2005;Liu et al. 2011;Qiao & Liu 2012;Yan et al. 2020). In BHXBs, the disk emission X-ray photons illuminating the hot corona are inversely Comptonized by the high-energy electrons in it, resulting in the hard X-ray emission. But, it is uncommon to detect the hard X-ray emission in neutron star low mass X-ray binaries (NS-LMXBs). The socalled hard X-ray tails have been seldom detected in Z sources. However, it is not difficult to detect hard X-ray emission in Atoll sources (e.g., Piraino et al. 1999;Fiocchi et al. 2006;Tarana et al. 2007Tarana et al. , 2011Raichur et al. 2011). On the other hand, there does not exist any evidence for a hot corona in NS-LMXBs and there is no consensus on the detection of the hard X-ray tails in NS-LMXBs.
In the past decades, the X-ray satellites with broadband energy ranges have accumulated a great number of observational data for NS-LMXBs, making it possible to search for hard X-ray tails in Z sources. Among these satellites, BeppoSAX and RXTE, as well as INTEGRAL, have contributed a lot on this issue. For example, a hard X-ray power-law (PL) tail having photon index (Γ) ∼2 and contributing ∼1.5% of the source luminosity was significantly detected in the horizontal branch (HB) of Cyg X-2 from a BeppoSAX observation (DiSalvo et al. 2002). A similar PL component was also found in the INTEGRAL spectrum of GX 5-1 (Paizis et al. 2005). In addition, in the BeppoSAX spectrum of the third Cyg-like Z source, i.e. GX 340+0, a hard X-ray tail was found to dominate the spectrum above 30 keV (Lavagetto et al. 2004). As for the first Sco-like Z source, i.e. Scorpius X-1 (Sco X-1), RXTE had accumulated 16-year observations, making it possible to study the properties of the hard X-ray tails of this source in detail. Using the data of RXTE for Sco X-1, D' Amico et al. (2001) and Ding et al. (2021) searched for hard X-ray tails and study their evolution on its hardness-intensity diagram (HID). They found that along the Z-track on the HID, the hard X-ray tail becomes hard and fades away. The hard Xray tail was also detected from the INTEGRAL observations for this source (Di Salvo et al. 2006;. Farinelli et al. (2005Farinelli et al. ( , 2007 detected the transient PL-like hard X-ray tail in one BeppoSAX observation for the second Sco-like Z source, i.e. GX 17+2 an explained it with bulk-motion Comptonization (BMC) or hybrid Comptonization. Also using BeppoSAX observations, Di Salvo et al. (2000) searched for the hard X-ray tail of GX 17+2 and investigated its evolution along the Z-track on the HID. They detected the hard X-ray tail with Γ ∼2.7 in the HB. However, it was not found in the normal branch (NB). Coincidentally, with the RXTE observations for GX 17+2, Ding & Huang (2015) performed the similar investigation and found that the presence of the hard X-ray tail is not confined to a specific branch and it hardens and weakens along the Z-track on the HID, which is similar to the hard tail evolution behavior of Sco X-1 (D' Amico et al. 2001;Ding et al. 2021). As to the third Sco-like Z source, i.e. GX 349+2, from the BeppoSAX observation for it, a hard X-ray tail was also detected in the nonflaring state during its flaring branch (FB) (Di . In addition to the six persistent Z sources, there exists a peculiar Z source, i.e. Cir X-1. Analyzing the BeppoSAX spectra of Cir X-1, Iaria et al. (2001Iaria et al. ( , 2002 discovered relatively soft hard X-ray tails with Γ ∼3 at its periastron and near its apoastron, respectively. Moreover, using the data of RXTE, Ding et al. (2003Ding et al. ( , 2006b) investigated the hard X-ray tail evolution of Cir X-1 and found that it becomes hard along the Z-track on the HID and it is obviously modulated by the orbit phase.
It is noted that the two X-ray satellites launched into sky in recent years will enrich the study on the hard X-ray emission of X-ray binaries (XRBs). One is the AstroSat launched in 2015. The large area xenon proportional counter (LAXPC) of AstroSat works in the energy range of 3-100 keV and provides a total of effective area of 8000 cm 2 at 10 keV. Fitting the LAXPC spectrum of 4U 1538-522, a NS high-mass XRB, Varun et al. (2019) found the cyclotron resonant scattering feature (CRSF) at ∼22 keV. Similarly, with the AstroSat observations for GRO J2058+42, a transient accreting X-ray pulsar, Mukerjee et al. (2020) detected three CRSFs in the energy range of ∼10-45 keV. Another is the Hard X-ray Modulation Telescope (Insight-HXMT), the first X-ray astronomical satellite of China, launched on June 15, 2017 (Zhang et al. 2014. Insight-HXMT consists of three instruments. The high energy (HE) instrument of Insight-HXMT consists of 18 NaI(TI)/CsI(Na) scintillation detectors with a total of effective area of 5100 cm 2 , working in 20-250 keV (Li et al. 2018(Li et al. , 2019. With the Insight-HMXT observation for transient accreting pulsar GRO J1008-57, a CRSF at ∼90 keV was detected, being the highest energy CRSF (Ge et al. 2020). Another cheering achievement of the HE of Insight-HXMT is the discovery of quasiperiodic oscillations (QPOs) above ∼200 keV in MAXI J1820+070, a BHXB, being the QPO phenomenon at the highest energy range in XRBs (Ma et al. 2021). The two breaking achievements show the excellent ability of the HE instrument of Insight-HXMT to perform spectral and timing study for XRBs in hard X-ray energy ranges.
In this work, using the observation data of Insight-HMXT for Sco X-1 in 2018, we search for hard X-ray tails, compare the hard X-ray spectra between this source and Cyg X-1, and perform investigation on the possible mechanisms for producing the hard X-ray tails in this Z source. We describe our data analyses in Section 2, present our results in Section 3, discuss our results in Section 4, and, finally, give our conclusions in Section 5.
DATA ANALYSES
Analysis of Insight-HXMT data
In addition to the HE instrument, Insight-HXMT includes another two payloads, i.e. the medium energy instrument (ME) and the low energy instrument (LE), which work in 5-30 keV and 1-15 keV, consist of 1728 Si-PIN units and 96 swept charge devices, and have detection areas of 952 cm 2 and 384 cm 2 , respectively (Li et al. 2018(Li et al. , 2019. Insight-HXMT has accumulated adequate observation data for the Galaxic XRBs and acquired significant achievements (e.g., Chen et al. 2018;Huang et al. 2018;Ge et al. 2020;Ma et al. 2021;You et al. 2021, etc.).
The data analysis of any instrument of Insight-HXMT goes through three stages: calibration → screening → extracting high level products. At the calibration stage, some unusual events, such as spike or trigger events, are removed and the pulse invariant values are calculated from the raw files. Next, during the screening stage, the screen files, i.e. good event files, are generated with the good time interval (GTI) files which are produced to satisfy some criteria. Finally, at the third stage, the source+background light curves and spectra as well as the background light curves and spectra, together with the spectral response files, are generated. At last, one can get the subtracted-background light curves and spectra. The recommended criteria for producing the GTI files are as follow: (1) the offset angle from the pointing direction <0.04 • ; (2) pointing direction above the earth >10 • ; (3) minimal value of the geomagnetic cutoff rigidity >8; (4) excluding the data in SAA; (5) for LE data, pointing direction above bright the earth >30 • . The tasks in each stage are finished with some specific tools. For example, the calibration of HE data is executed with the tool HEPICAL, the screening of ME data is implemented with the tool MESCREEN, and extracting the LE spectrum is finished with the tool LESPECGEN, etc.. It is known that the deadtime correction should be considered for analyzing X-ray astronomical data. Thanks to the Insight-HXMT team, the deadtime correction file for HE data is provided, and that for ME data is produced in the screening stage, while, the deadtime correction for LE data is unneeded. As for the background, it is worth noting that the Insight-HXMT team has developed the so-called blind-detector method to estimate the background (Li et al. 2018(Li et al. , 2019.
Searching for the hard tails
In this work, we use the Insight-HXMT data analysis software HXMTSOFT v2.04 to analyze the data. We use the shell script HPIPELINE to generate the light curves and spectra, as well as the spectral response files, of all the three instrument of each observation at once.. We search for the hard X-ray tails of Sco X-1 from the HE spectra extracted from the observations for this source in 2018. The Insight-HXMT data for Sco X-1 in 2018 include 35 observations and, correspondingly, we get 35 HE spectra. In order to improve the signal-to-noise ratios (S/Ns) of the HE spectra, we group the raw channels with S/Ns<1.5, for each new channel resulted from grouping those raw channels to have S/Ns>1.5. It is noted that the raw channels with S/Ns<1.5 are restricted in the energy intervals above ∼50-60 keV exclusively. In most cases, after grouping the raw channels above ∼50-60 keV, we can only get a few new channels with S/N>1.5, which are distributed around the energy range of ∼50-60 keV, or even cannot get any new channels with S/N>1.5. In these cases, ignoring the channels below 30 keV and discarding the raw channels with S/Ns<1.5 in the energy bands above ∼50-60 keV, the HE spectra can be fitted well in ∼30-60 keV with a thermal emission model such as bremsstrahlung (BREMSS) or COMPTT. Nonetheless, in a small quantity of cases, after grouping the raw channels above ∼50-60 keV, we can get some new channels with S/Ns>1.5, which are distributed in the energy range of ∼50-200 keV, as shown in Figure 1. Similarly, ignoring the channels below 30 keV and those raw channels with S/Ns<1.5 above ∼200 keV, we perform fittings of these HE spectra in ∼30-200 keV. It is found that the obvious high-energy excesses are present in the energy range above ∼50-60 keV when these HE spectra are fitted with an one-component model such as BREMSS or COMPTT, etc., and, then, statistically acceptable fittings cannot be obtained, while they are fitted statistically well with a two-component model consisting of a BREMSS or a COMPTT, plus a PL, as demonstrated by Table 1. Consulting D'Amico et al. (2001) and Ding et al. (2021), we define detection of a hard X-ray tail in the HE spectra satisfying the two needs: (1) the new channels with S/Ns>1.5, resulted from grouping the raw channels, extend over the energy band of ∼50-60 keV considerably; (2) the F-test probability for adding the PL component into the spectral model is less than 6 × 10 −5 .
RESULT
Among the 35 HE spectra of Sco X-1 of Insight-HXMT, the hard X-ray tails are detected in 8 observations. The HE spectra with detection of a hard X-ray tail are fitted by the two-component model consisting of a BREMSS and a PL. The fitting results, as well as the F-test probability for the PL component to be added into the spectral model, are listed in Table 1. Two HE spectra with hard tail detection are shown in Figure 1. In order to constrain the positions of the detected hard Xray tails in the HID, we produce the HID of the 8 observations with hard tail detection using their ME data. We define the hardness as the ratio of count rate in 12-18 keV to that in 8-12 keV, and the intensity as the count rate in 8-30 keV. The produced HID is displayed by Figure 2. Then, we determine the position of each observation in the HID. As listed in Table 1, among the 8 detected hard X-ray PL tails, 4 and 3 hard tails are belonged to the HB and NB, respectively, and only one is found in the FB. The values of Γ of the HB and NB hard tails are positive, indicating the hard tails in the two branches are relatively hard. However, the Γ value of the FB hard tail is negative, being a peculiar hard tail. Such unusual hard tails of Sco X-1 were also found in the high-energy spectra of RXTE (D'Amico et al. 2001;Ding et al. 2021). Interestingly, the extraordinary hard tails with negative Γ of this source have been always found in the FB. Moreover, it seems that among the Z sources , such peculiar hard tails have been found in Sco X-1 only. In order to investigate the hard tail evolution on the HID, the correlation between the two parameters of the PL component is demonstrated by Figure 3. Figure 3 shows that along the Z-track on the HID, from the HB, via the NB, to the FB, the value of Γ decreases and, meanwhile, the PL flux decreases, indicating that the hard tail of Sco X-1 becomes hard and fades away in the sequence HB → NB → FB, which is consistent with that found by D' Amico et al. (2001) and Ding et al. (2021). Such hard tail evolution behavior was also found in another Z source, i.e. GX 17+2 (Ding & Huang 2015).
DISCUSSION
Comparison of hard X-ray spectrum between Cyg X-1 and Sco X-1
We compare the property of the hard X-ray spectrum of Cyg X-1 with that of Sco X-1 and try to find some clues for the origin of the nonthermal emission of the latter from the mechanism for producing the hard X-ray emission for the former.
The hot corona in Cyg X-1
We extract some HE spectra of Insight-HXMT of Cyg X-1. We group the raw channels of these HE spectra for each new bin generated from grouping to have S/Ns>5 and, then, fit these grouped spectra in 30-220 keV. It is found that in most cases, the statistically acceptable fittings cannot be obtained when the HE spectra of Cyg X-1 are fitted with a PL, but they can be fitted statistically well by the model of a PL with high-energy exponential cutoff (CUTOFFPL in XSPEC, CPL). On the other hand, in a few cases, although the statistically acceptable fittings can be obtained when the HE spectra are fitted with a PL or a CPL, the fittings are statistically better fitting with a CPL than fitting with a PL. From our analysis, it is concluded that generally, the hard X-ray spectrum of Cyg X-1 can be fitted statistically well only with a one-component model of CPL, and occasionally, it can be fitted statistically acceptably with a CPL or a PL, in which the fitting is statistically better with a CPL than with a PL. The fitting results with a CPL of six HE spectra of Cyg X-1 are listed in Table 2 and a representative fitting is shown by panel (a) of Figure 4. The values of χ 2 ν (χ 2 ν = χ 2 /do f ) indicate that these fittings are statistically well. The presence of high-energy cutoff indicates that the hard X-ray emission of Cyg X-1 is originated from thermal emission. Therefore, we fit the six spectra with the thermal Comptonization models in XSPEC, e.g. COMPTT (Titarchuk 1994) or COMPST (Sunyaev & Titarchuk 1980), etc. It is found that each of the six HE spectra can be fitted by these models well and the fitting results with the COMPTT model are listed in Table 2. Panel (b) of Figure 4 shows a fitting of compTT. It is noted that the cutoff energy (E cut ) and the electron temperature (kT e ) span a range of ∼150-200 keV and a range of ∼45-85 keV, respectively. Interestingly, E cut is related to kT e in this way E cut ≈ 2.2 − 3.5 kT e , approximating the anticipated relation E cut = 2 − 3 kT e (Petrucci et al. 2001;Yan et al. 2020). The fact that the temperature of the electrons for Comptonization is so high implies that there exists a hot source in Cyg X-1 to supply these high-temperature electrons. Generally, such a hot source is assumed to be a hot corona in XRBs. Moreover, the hot corona in XRBs should be apart from the accretion disk and it might be above the cool disk. Otherwise, the disk will be evaporated by the hot corona and it will disappear. Such a hot corona in BHXBs could be responsible for the frequently observed hard X-ray emission in them (e.g., Zhang et al. 2000;Yao et al. 2005;Liu et al. 2011;Qiao & Liu 2012;Yan et al. 2020). The soft X-ray photons from the disk are inversely Comptonized by the high-energy electrons in the hot corona to produce hard X-ray emission.
In accreting black hole systems, forming a corona above the accretion disk could be due to the large-scale magnetic field in them. In these systems, along the open magnetic filed lines, the accreting matter will be hurled to a position above the disk (Blandford & Znajek 1977;Blandford & Payne 1982). It is possible for the hurled plasma to accumulate and, then, constitute a corona at the location, where the radiation pressure and magnetic pressure balance the gravitation. The corona located above the disk might be heated and become into a hot corona through the magnetic buoyancy mechanism, in which the magnetic field lines float above the disk and heat the corona through magnetic reconnection (Galeev et al. 1979;Liu et al. 2002Liu et al. , 2003.
A possible warm corona in Sco X-1
As analyzed above, the HE spectrum of Cyg X-1 shows high-energy cutoff, which indicates a hot corona. However, the HE spectrum of Sco X-1 does not present high-energy cutoff, which rules out a hot corona in it. Nevertheless, it is worthy of investigating whether or not there exists a warm corona in it. Therefore, we fit the low energy spectrum of Sco X-1. Generally, there are two types of models to fit the low-energy spectrum of NS-LMXBs, i.e. the so-called Eastern model and Western model. Each of the two models includes two continua. The Eastern model consists of a blackbody (BB) to account for the emission from the NS surface or a Comptonized BB (COMPBB, Nishimura et al. 1986) to describe the Comptonization emission around the NS, plus a multicolored disk (MCD) to be responsible for the emission from the disk (Mitsuda et al. 1984(Mitsuda et al. , 1989). Therefore, this model can be written in two forms, i.e. BB+MCD and COMPBB+MCD. In the Western model, the NS emission is described by a BB and, however, the disk emission is interpreted as an unsaturated Comptonized continuum approximated with a CPL or a Sunyaev-Titarchuk Comptonization (COMPST, Sunyaev & Titarchuk 1980) (White et al. 1985(White et al. , 1986. Thus, the Western model can also be expressed in two forms, i.e. BB+CPL and BB+COMPST. Taking into account the interstellar absorption by fixing the hydrogen column density N H at 0.3 × 10 22 atom cm −2 (Church et al. 2012) and the iron line in the spectrum, we use the Eastern and Western models to fit the LE+ME spectra of Sco X-1 in 2-35 keV. Our analysis suggests that the LE+ME spectrum cannot be fitted by any form of the Eastern model, but it can be fitted by any of the two forms of the Western model statistically and physically well. Moreover, if the COMPST is replaced with a COMPLS (Lamb & Sanford 1979) or a COMPTT in the form of BB+COMPST, the spectrum can also be fitted well. Nevertheless, the fitting is statistically better with the BB+CPL form than with any other forms, because among these fittings, the lest value of χ 2 ν is obtained from the fitting with the BB+CPL form. A representative fitting with the BB+CPL form is demonstrated by Figure 5. The fitting results are consistent with those obtained through from fitting the EXOSAT or RXTE spectra with the same model for the same source (White et al. 1985;Church et al. 2012). The BB temperature (kT bb ) is ∼3 keV, indicating that the BB emission is indeed from the surface of the neutron star (NS), because in NS-LMXBs, generally, the temperature of the NS surface emission is above ∼2 keV, while the temperature of the disk emission is below ∼1.5 keV (e.g., Mitsuda et al. 1984Mitsuda et al. , 1989Ding et al. 2006bDing et al. , 2011Church et al. 2012, etc.). In the Western model, White et al. (1986) proposed that the CPL describes the unsaturated Comptonization and the Comptonization process takes place in the inner disk region. However, Church & Bałucińska-Church (1995 explained the CPL component in this way that the soft X-ray photons from the disk are Comptonized by the electrons from a corona on the disk. They called the corona on the disk as an accretion disk corona (ADC) and developed the Western model into the so-called ADC model. It is noted that the E cut of CPL is ∼4.5 keV, so the temperature of the electrons for Comptonization (kT e ) is only several keV. Therefore, if there does exist a corona in Sco X-1, the corona could be a warm corona rather than a hot corona as that in Cyg X-1. Sequentially, it is unlikely for such a warm corona in Sco X-1 to provide so energetic electrons for Comptonization to produce hard X-ray emission.
As demonstrated by Fig. 1 of Lamb et al. (1973) or Fig. 1 of Ghosh & Lamb (1979), in NS-LMXBs, owing to the small-scale magnetic field, the magnetic field lines form an enclosed structure around the NS and, thus, the plasma can only move in the space near the NS, leading to that the accreting matter cannot be expelled far away. Therefore, in NS-LMXBs, if a corona is formed, it must reside near the accretion disk. The corona should be a warm one, as supported by our analysis above. Otherwise, the disk will be evaporated and disappeared.
4.2. The origin of the hard X-ray emission of Sco X-1 4.2.1. The thermal component in the hard X-ray spectrum of Sco X-1
As shown in Figure 1, the HE spectrum of Sco X-1 consists of a PL component, plus a thermal component. The thermal component dominates the emission in ∼30-50 keV and it certainly originates from the thermal emission in the system. There are two thermal emission sources in NS-LMXBs, i.e. the NS and the accretion disk. In order to distinguish between the two sources, we replace the BREMSS with a BB or a MCD in the BREMSS+PL model to fit the HE spectrum of Sco X-1. Our fitting suggests that the spectrum can be fitted statistically well either by the BB+PL model, as shown with panel (a) of Figure 6, or by the MCD+PL model. The PL parameters obtained from fitting with the two models are similar to those obtained from the fitting with the BREMSS+PL model. The BB temperature (kT bb ) obtained through fitting with the BB+PL model and the inner disk temperature (kT in ) gotten by fitting with the MCD+PL model are ∼3 keV and ∼3.5 keV, respectively. The kT bb is significant, but the kT in is meaningless, because, generally, in NS-LMXBs, the BB temperature of the NS surface emission is around or above ∼2 keV, but the inner disk temperature is below ∼1.5 keV (e.g., Mitsuda et al. 1984Mitsuda et al. , 1989Ding et al. 2006aDing et al. , 2011Church et al. 2012, etc.). Furthermore, we replace the BREMSS with the COMPTT in the BREMSS+PL model and use the COMPTT+PL model to fit the HE spectrum and get statistically acceptable fitting too, which is demonstrated by panel (b) of Figure 6. From this fitting, the obtained seed photon temperature (kT 0 ), the plasma temperature (kT ), and the plasma optical depth are ∼2.2 keV, ∼3.8 keV, and 2.5, respectively, which could be resulted from the Comptonization emission that the photons of the BB emission of the NS surface are thermally Comptonized in the boundary layer between the NS and the disk (Barret et al. 2000). Therefore, it is concluded that the thermal component in the hard X-ray spectrum of Sco X-1 comes from the NS surface emission or the Comptonization emission with the NS surface emission photons as the seed ones. In other words, the thermal component in the HE spectrum is related to the NS emission, instead of the disk emission.
The nonthermal component in the hard X-ray spectrum of Sco X-1
As shown in Figure 1 or Figure 6, the PL component in the HE spectrum of Sco X-1 dominates the flux in the energy band of ∼50-200 keV. Such nonthermal component cannot be explained with the hot-corona mechanism, because, as analyzed above, there does not exist a hot corona in Sco X-1. Such high-energy emission cannot be originated from a possible warm corona in this source too, because the warm corona is only with the temperature of several keV. Therefore, other mechanisms responsible for the hard X-ray tail in Sco X-1 should be considered. In XRBs, the accretion process is characterized by the converging inflow onto the compact star and the Comptonization connected with the energetic electrons in the accreting inflow cannot be neglected. Titarchuk et al. (1997) and Laurent & Titarchuk (1999) investigated such Comptonization in BHXBs and they called it BMC, i.e. bulk-motion Comptonization. The BMC model includes three parameters, i.e. the temperature of the thermal source for providing seed photons (kT bb ), the energy spectral index α (Γ = α + 1, Γ: photon index of the PL component in the spectrum), and logA (A: a parameter to describe Comptonization), as well as a normalization (N). According to the extent that the seed photons are Comptonized, they are divided into two portions. Some seed photons are weakly Comptonized and they escape with the way approximating BB emission, which contribute 1/(1+A) of the total luminosity. However, others are strongly Comptonized and the Comptonzation emission of these seed photons takes A/(1+A) of the total luminosity. The BMC model has been applied to BHXBs (e.g., Titarchuk & Seifina 2009;Shrader et al. 2010, etc.). It was also used to explain the X-ray nonthermal emission in NS-LMXB Z sources (e.g., Farinelli et al. 2007;Ding & Huang 2015;Ding et al. 2021). In this work, we assume that the hard X-ray PL tail in Sco X-1 could be resulted from the upscattering of the photons of the BB emission of the NS surface by the free-falling electrons onto the NS in the converging inflow. Meanwhile, taking into account the the iron line and the disk emission in the spectrum, we form a tri-component model of BMC+LINE+CPL to fit the LE+ME+HE spectrum in ∼2-200 keV. When fitting, in order to account for the flux calibration differences among the LE, ME, and HE instruments, the model is multiplied by a constant, which is fixed for the LE spectral normalization, but free for the ME and HE spectral normalizations. At the same time, the interstellar photoelectric absorption is considered, with N H being fixed at 0.3 × 10 22 atom cm −2 (Church et al. 2012). Panel (a) of Figure 7 demonstrates such a fitting. The value of χ 2 ν , as well as the residual distribution, shows that this fitting is statistically acceptable well. It is noted that the seed photon temperature is ∼3 keV, indicating that the seed photons for BMC do come from the NS surface emission rather than the disk emission. The value of logA is -1.48, meaning that the BMC emission only occupies ∼3% of the total luminosity, and, therefore, the observed luminosity mainly comes from the direct BB emission of the NS. Actually, comparing the individual components in panel (a) of Figure 7 with those in Figure 5, one can see that the segment of BMC (dash line) in ∼2-35 keV in panel (a) of Figure 7, representing the observed direct BB emission of the NS surface, almost happens to correspond the BB component (dash line) in Figure 5, and, meanwhile, the segment of BMC in ∼35-200 keV in panel (a) of Figure 7, demonstrating the BMC emission, just is the extension of the BB component of Figure 5 in the high-energy interval. Moreover, the other two components in the two Figures, i.e. the CPL and LINE components, just correspond to each other. In conclusion, the joint fitting of the LE+ME+HE spectrum in ∼2-200 keV with the BMC+LINE+CPL model suggests an alternative mechanism for producing the hard X-ray nonthermal emission in Sco X-1, which is that the hard X-ray PL emission in Sco X-1 could be generated from upscattering of the photons of the BB emission of the NS surface by the energetic free-falling electrons in converging inflow onto the NS.
Although the fitting of the broadband spectrum with the BMC+LINE+CPL model proposes a scenario to produce the hard X-ray tail in Sco X-1, yet there exists a potential contradiction in this interpretation. As pointed out by Zdziarski et al. (2001) and , if the BMC process does be responsible for the hard X-ray tail in XRBs, the high-energy cutoff lower than ∼100-200 keV should be present in the hard X-ray spectrum of theirs (Titarchuk et al. 1997). However, in this work, our analysis suggests that the hard X-ray tail of Sco X-1 shows a PL-like shape without cutoff in the HE spectrum. Therefore, we investigate other possible mechanisms to generate the hard X-ray tail in Sco X-1. It is a natural assumption that through heating mechanisms, two types of electrons are produced in plasma, one of which consists of the thermal electrons with Maxwellian distribution and the other of which comprises of the nonthermal electrons with PL distribution. Correspondingly, two types of Comptonization processes will take place in the plasma. Coppi (1999) investigated the hybrid Comptonization in the thermal/nonthermal plasmas and developed a model (EQPAIR in XSPEC) to describe such Comptonization emission. In the EQPAIR model, the total luminosity of the source is expressed with the dimensionless compactness, defined as
l = L R σ T m e c 3(1)
Where, L is the observed source luminosity, R is the characteristic radius of the hard X-ray emission region approximated as a sphere, σ T is the Thomson cross section, m e is the electron mass, and c is the speed of light. The source luminosity (l) consists of hard and soft luminosities, i.e. l = l h + l s . The hard luminosity is assumed to be resulted from the hybrid Comptonization, therefore, l h = l nth + l th , where l nth and l th are the nonthermal and thermal Comptonization luminosities, respectively. According to Kirkhoff ′ s law, l nth and l th can also be considered as the powers supplied to heat the nonthermal and thermal electrons, respectively. This model includes twenty parameters, plus a normalization. Among the parameters, there is a critical parameter, i.e. l nth /l h , which describes the fraction of the total heating power to heat the nonthermal electrons. When the value of this parameter is 1 or 0, the hybrid Comptonization is reduced to a pure nonthermal Comptonization or a pure thermal Comptonization. Another important parameter is G inj , which describes the distribution of the nonthermal electrons by γ −G inj , where γ is the Lorentz factor of the nonthermal electrons, taking the value from γ min to γ max . In the version 12.11.1 of XSPEC, with which we perform analysis in this work, the ranges of γ min and γ max are set to be 1.2-1000 and 5-10000, respectively. This model has been applied to interpret the hard X-ray emission of BHXBs (e.g., Gierliński & Done 2003;Caballero-García et al. 2009;Kalemci et al. 2016;Zdziarski et al. 2021, etc.). It was also used to explain the hard X-ray tail of Z sources (Farinelli et al. 2005;D'Ai et al. 2007). In this work, we replace BMC with EQPAIR in the BMC+LINE+CPL model to fit the broadband spectrum of Sco X-1 in ∼2-200 keV, in which a hard X-ray tail is present. The fitting is statistically acceptable well and a fitting is shown by Panel (b) of Figure 7. It is noted that the BB temperature (kT bb ) is around 2.5 keV, indicating that the power supplied to heat plasma particles comes from the NS or the boundary layer between the NS and the disk, rather than the disk, which is consistent with the result of Farinelli et al. (2005) or (D'Ai et al. 2007). While, without doubt, in BHXBs, such power is provided by the disk, because of the fact that the seed photon temperature kT bb for the hybrid Comptonization in BHXBs is lower than ∼1 keV (e.g., Zdziarski et al. 2001;Caballero-García et al. 2009;Kalemci et al. 2016;Cangemi et al. 2021). The value of l nth /l h is about 0.4, meaning that after being powered, the plasma becomes a hybrid and the Comptonization emission of the thermal electrons predominates over that of the nonthermal electrons. This fitting of ours suggests that the hybrid Comptonization taking place around the NS or in the boundary layer between the NS and the disk could be another alternative mechanism to result in the hard X-ray tail of sco X-1, as proposed by D'Ai et al. (2007).
The peculiar hard X-ray tail in the FB of Sco X-1
Usually, a hard X-ray tail of Z sources is fitted by a PL with a positive photon index, but, in this work we detect a hard tail with a negative photon index in the FB (see Table 1). This peculiar hard tail of the same source was also detected previously in the same HID branch with the observations of RXTE (D'Amico et al. 2001;Ding et al. 2021). Neither the bulk-motion Comptonization mechanism nor the hybrid Comptonization mechanism can be used to explain such peculiar hard tail, because the photon index Γ in the BMC model or the EQPAIR model is set to be a positive parameter in XSPEC. Actually, in our practice, the broadband spectrum (2-200 keV) with this unusual hard tail cannot be fitted by the BMC+LINE+CPL model or the EQPAIR+LINE+CPL model. The origin of this peculiar hard tail is worth thinking deeply. Firstly, it is found in the flaring branch. Generally, the FB of Z sources is considered to be resulted from the unstable nuclear burning on the NS surface. Therefore, is this unusual hard tail connected with the unstable nuclear burning on the NS surface? Secondly, as the argument of Ding et al. (2021), is it linked to the jet in NS-LMXBs? Thirdly, it seems that among the Z sources, the unusual hard tail is only detected in Sco X-1, so, does it has to do with the source?
Distinguishing between BHXBs and NS-LMXBs by hard X-ray spectra
This work of ours, as well as some works of others, shows that the hard X-ray spectrum of BHXBs differentiates much from that of the high-luminosity NS-LMXBs, i.e. the Z sources. On one hand, the hard X-ray spectrum of BHXBs reveals high-energy cutoff in most cases, but the cutoff phenomenon does not appear in the hard X-ray spectrum of Z sources. On the other hand, the hard X-ray spectrum of BHXBs must be fitted with the one-component mode of CPL in most cases. Occasionally, it can be fitted by a PL or a CPL, in which the the fitting is statistically better with the CPL than with the PL. Nevertheless, the spectral model of the hard X-ray spectrum of Z sources should consists of two components. One could be a thermal component related to the NS emission and the other must be the nonthermal component described with the PL. Interestingly, the spectrum of the low-luminosity NS-LMXBs, i.e. the Atoll sources, sometimes behaves similarly to that of BHXBs (Rodi et al. 2016).
As demonstrated in the Figure 13 of Barret et al. (2000), the hard X-ray luminosities of BHXBs generally exceed those of low-luminosity NS-LMXBs, i.e. Atoll sources. Here, we compare the hard X-ray luminosities between Cyg X-1 and Z-source Sco X-1. Assuming the distances of Cyg X-1 and Sco X-1 to be 1.86 kpc (Reid et al. 2011) and 2.8 kpc (Bradshaw et al. 1997), respectively, we estimate the hard X-ray luminosities of the two sources. The inferred luminosities in 20-200 keV of Cyg X-1 and Sco X-1 span a range of ∼(8.5-10.1)×10 36 ergs s −1 and a range of ∼(6.5-12.9)×10 36 ergs s −1 , respectively. Therefore, it seems that the hard X-ray luminosity of the high-luminosity Z sources could be comparable to that of BHXBs, which might be owing to the NS emission of Z sources. As analyzed above, in Z sources, the thermal emission of the NS contribute a portion of low-energy hard X-ray flux . So, reducing the hard X-ray interval, we make further comparison. The derived luminosities in 30-200 keV of Cyg X-1 and Sco X-1 are ∼(6.9-8.4)×10 36 ergs s −1 and ∼(1.0-2.6)×10 36 ergs s −1 , respectively. In this range, the hard X-ray luminosity is obviously larger of the former than of the latter. Furthermore, the higher the hard Xray energy band, the larger the luminosity difference. These spectral characteristics could provide a way to distinguish the two types of XRBs.
CONCLUSION
In this work, with the observations of Insight-HXMT for Sco X-1, we search for the PL component in its hard X-ray spectrum and investigate the origin of this nonthermal emission. We detect eight hard X-ray tails of this Z source. The detected hard tails are distributed throughout the Z-track on the HID. On the HID, along the Z-track, the hard tail becomes hard and fades away from HB, via NB, to FB. Studying the hard X-ray spectrum of Cyg X-1 and comparing it with that of Sco X-1, it is concluded that the former shows high-energy cutoff, but the latter does not, indicating that there might exist a hot corona in Cyg X-1, but there is without a hot corona in Sco X-1. However, the spectral analysis in ∼2-35 keV implies a warm corona in Sco X-1. Fitting the broadband spectrum of Sco X-1 suggests two alternative mechanisms responsible for the hard X-ray tails of Sco X-1. One is that the hard tail could be resulted from the Comptonization of the photons of the NS surface emission by the energetic free-falling electrons onto the NS. The other possibility is that the hard tail might be generated through the hybrid Comptonization of the NS surface emission photons by the thermal as well as nonthermal electrons in the plasmas around the NS or in the boundary layer between the NS and the disk. But, any of the two mechanisms cannot be responsible for the hard tail in the FB. Several possibilities for the production of the peculiar hard tail of the FB are argued, including the unstable nuclear burning on the NS surface, linking to the jet, and connecting with the source. Note. -Fitting is performed with the BREMSS+PL model in ∼30-200 keV. The parameter errors are derived at the 90% confidence level (∆χ 2 = 2.7). a The observation of this OBSID spans in the HB and the NB. This interval is the exposure time in the HB. -The HID of the eight OBSIDs of Sco X-1 in which hard X-ray tails are detected. The hardness is defined as the ratio of count rate in 12-18 keV to that in 8-12 keV and the intensity is defined as the count rate in 8-30 keV.
Figure 1 .
1Following D'Amico et al. (2001) and Ding et al. (2021), we fit these HE spectra with the two-component model of BREMSS+PL in ∼30-200 keV and, meanwhile, perform F-test for adding the PL component in the spectral model. The fitting results are listed in
b
Flux in the 30-60 keV range, in units of 10 −9 ergs cm −1 s −1 . c Flux in the 30-200 keV range, in units of 10 −9 ergs cm −1 s −1 .
d F-test probability for adding the PL component in the spectral model.
Fig. 1 .
1-The two representive HE spectra of Sco X-1 in which the hard X-ray tail is present. The spectra are fitted by the two-component model consisting of a BREMSS, plus a PL. The individual components are shown, namely, the BREMSS (the dot line) and the PL (the dash line). Residuals are plotted in units of the standard deviation (σ) with error bars of size one. Panels (a) and (b) demonstrate the results obtained from OBSIDs P010132801001 and P010132800804, respectively.
Fig. 2 .
2Fig. 2.-The HID of the eight OBSIDs of Sco X-1 in which hard X-ray tails are detected. The hardness is defined as the ratio of count rate in 12-18 keV to that in 8-12 keV and the intensity is defined as the count rate in 8-30 keV.
Fig. 3 .
3-The correlation between the photon index (Γ) of the hard tail and its flux (in ∼30-200 keV, in units of 10 9 ergs cm −1 s −1 ). The red and blue solid circles indicate the results of HB and NB, respectively. The green solid circle represents the result of FB. The dash line is drawn with the least square method.
Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .
4567-The fitting of HE spectrum of Cyg X-1. (a) The spectrum is fitted by the CPL model, with χ 2 ν =0.93 for 88 degrees of freedom, Γ = 1.53 +0.06 −0.06 , and E cut = 147 +0.24 −0.28 keV; (b) The spectrum is fitted by the COMPTT model, with χ 2 ν =0.90 for 88 degrees of freedom, kT e = 45.4 +6.5 −4.0 keV, and τ = 1.16 +0.12 −0.16 . The parameter errors are derived at the 90% confidence level (-A representative fitting of the LE+ME spectrum of Sco X-1 with the BB+CPL form, plus the line component. The fitting is performed in 2-35 keV. The black and red colours represent the results obtained from the LE and ME instruments of Insight-HXMT, respectively. Individual components are shown, namely, BB (dash line), LINE (dash-dot line, gaussian line profile), and CPL (dot line). The χ 2 ν is 1.04 for 1361 degrees of freedom. The fitting parameters: kT bb = 3.11 +0.02 −0.03 keV for the BB; E Fe = 6.54 +0.10 −0.10 keV and σ Fe = 0.93 +0.18 −0.15 keV for the LINE; Γ = 1.00 +0.03 −0.04 and E cut = 4.55 +0.24 −0.28 keV for the CPL. The parameter errors are inferred at the 90% confidence level -The fitting of HE spectrum of Sco X-1. (a) The spectrum is fitted by the model consisting of a BB, plus a PL. The χ 2 ν is 0.63 for 33 degrees of freedom, with kT BB = 2.95 +0.12 −0.12 keV for the BB and Γ = 1.43 +0.54 −0.53 for the PL. The individual components are shown, namely, a BB (dot line) and a PL (dash line). (b) The spectrum is fitted by the two-component model of COMPTT+PL. The χ 2 ν is 0.58 for 32 degrees of freedom. The individual components are displayed, i.e. a COMPTT (dot line) and a PL (dash line). The seed photon temperature (kT 0 ) and the electron temperature (kT e ) are 2.15 keV (fixed) and 3.81 +2.13 −0.89 keV, respectively, for the COMPTT. The photon index (Γ) is 1.07 +0.67 −0.74 for the PL. The parameter errors are inferred at the 90% confidence level (-The fitting of the LE+ME+HE spectrum of Sco X-1 in ∼2-200 keV, in which a hard X-ray tail is detected. (a) The spectrum is fitted by the BMC+LINE+CPL model. The individual components are shown, namely, a BMC (red dash line), a LINE (green dash-dot line), and a CPL (blue dot line). The χ 2 ν is 1.06 for 1315 degrees of freedom. The fitting parameters: kT bb =3.04 +0.02
Table 1 :
1Observations of hard X-ray tail detections in Sco X-1BREMSS
PL
Table 2 :
2Fitting result of the HE spectrum of Cyg X-1CPL
COMPTT
Xinjiang Astronomical Observatory, Chinese Academy of Sciences, 150, Science 1-Street, Urumqi, Xinjiang 830011, China; [email protected]
This research has made use of data from the Insight-HXMT mission, the first X-ray satellite of China, and the software provided by the Insight-HXMT team.
. D Barret, J F Olive, L Boirin, ApJ. 533329Barret, D., Olive, J. F., Boirin, L., et al. 2000, ApJ, 533, 329
. R D Blandford, D G Payne, MNRAS. 199883Blandford, R. D., & Payne, D. G. 1982, MNRAS, 199, 883
. R D Blandford, R L Znajek, MNRAS. 179433Blandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433
. C F Bradshaw, E B Fomalont, B J Geldzahler, ApJL. 48455Bradshaw, C. F., Fomalont, E. B., & Geldzahler, B. J. 1997, ApJL, 484, L55
. M D Caballero-García, J M Miller, M D Trigo, ApJ. 6921339Caballero-García, M. D., Miller, J. M., Trigo, M. D., et al. 2009, ApJ, 692, 1339
. F Cangemi, T Beuchert, T Siegert, A&A. 65093Cangemi, F., Beuchert, T., Siegert, T., et al. 2021, A&A, 650, A93
. Y P Chen, S Zhang, J L Qu, ApJ. 86430Chen, Y. P., Zhang, S., Qu, J. L., et al. 2018, ApJ, 864, L30
. M J Church, M Bałucińska-Church, A&A. 300441Church, M. J., & Bałucińska-Church, M. 1995, A&A, 300, 441
. M J Church, M Bałucińska-Church, MNRAS. 348955Church, M. J., & Bałucińska-Church, M. 2004, MNRAS, 348, 955
. M J Church, A Gibiec, M Bałucińska-Church, N K Jackson, A&A. 54635Church, M. J., Gibiec, A., Bałucińska-Church, M., & Jackson, N. K. 2012, A&A, 546, A35
P S Coppi, ASP Conf. Ser. 161, High Energy Processes in Accreting Black Holes. J. Poutanen & R. SvenssonSan Francisco, CAASP375Coppi, P. S. 1999, in ASP Conf. Ser. 161, High Energy Processes in Accreting Black Holes, ed. J. Poutanen & R. Svensson (San Francisco, CA: ASP), 375
. A D'aí, P Życki, T Di Salvo, R Iaria, G Lavagetto, N R Robba, ApJ. 667411D'Aí, A.,Życki, P., Di Salvo, T., Iaria, R., Lavagetto, G., & Robba, N. R. 2007, ApJ, 667, 411
. F D'amico, W A Heindl, R E Rothschild, D E Gruber, ApJ. 547147D'Amico, F., Heindl, W. A., Rothschild, R. E., & Gruber, D. E. 2001, ApJ, 547, L147
. G Q Ding, T T Chen, J L Qu, MNRAS. 500772Ding, G. Q., Chen, T. T., & Qu, J. L. 2021, MNRAS, 500, 772
. G Q Ding, C P Huang, JApA. 36335Ding, G. Q., & Huang, C. P. 2015, JApA, 36, 335
. G Q Ding, J L Qu, T P Li, ApJ. 596219Ding, G. Q., Qu, J. L., & Li, T. P. 2003, ApJ, 596, L219
. G Q Ding, J L Qu, T P Li, AJ. 1311693Ding, G. Q., Qu, J. L., & Li, T. P. 2006a, AJ, 131, 1693
. G Q Ding, S N Zhang, T P Li, J L Qu, ApJ. 645576Ding, G. Q., Zhang, S. N., Li, T. P., & Qu, J. L. 2006b, ApJ, 645, 576
. G Q Ding, S N Zhang, N Wang, J L Qu, S P Yan, AJ. 14234Ding, G. Q., Zhang, S. N., Wang, N., Qu, J. L., & Yan, S. P. 2011, AJ, 142, 34
. Di Salvo, T Farinelli, R Burderi, L , A&A. 386535Di Salvo, T., Farinelli, R., Burderi, L., et al. 2002, A&A, 386, 535
. Di Salvo, T Goldoni, P Stella, L , ApJ. 64991Di Salvo, T., Goldoni, P., Stella, L., et al. 2006, ApJ, 649, L91
. Di Salvo, T Robba, N R Iaria, R , ApJ. 55449Di Salvo, T., Robba, N. R., Iaria, R., et al. 2001, ApJ, 554, 49
. Di Salvo, T Stella, L Robba, N R , ApJL. 544119Di Salvo, T., Stella, L., Robba, N. R., et al. 2000, ApJL, 544, L119
. J B Dove, J Wilms, M A Nowak, B A Vaughan, M C Begelman, MNRAS. 298729Dove, J. B., Wilms, J., Nowak, M. A., Vaughan, B. A., & Begelman, M. C. 1998, MNRAS, 298, 729
. R Farinelli, L Titarchuk, F Frontera, ApJ. 6621167Farinelli, R. Titarchuk, L., & Frontera, F. 2007, ApJ, 662, 1167
. R Farinelli, F Frontera, A A Zdziarski, A&A. 43425Farinelli, R., Frontera, F., Zdziarski, A. A., et al. 2005, A&A, 434, 25
. M Fiocchi, A Bazzano, P Ubertini, P Jean, ApJ. 651416Fiocchi, M., Bazzano, A." Ubertini, P., & Jean, P. 2006, ApJ, 651, 416
. A A Galeev, R Rosner, G S Vaiana, ApJ. 229318Galeev, A. A., Rosner, R., & Vaiana, G. S. 1979, ApJ, 229, 318
. M Y Ge, L Ji, S N Zhang, ApJL. 89919Ge, M. Y., Ji, L., Zhang, S. N., et al. 2020, ApJL, 899, L19
. P Ghosh, F K Lamb, ApJ. 234296Ghosh, P., & Lamb, F. K. 1979, ApJ, 234, 296
. M Gierliński, C Done, MNRAS. 3421083Gierliński, M., & Done, C. 2003, MNRAS, 342, 1083
. Y Huang, J L Qu, S N Zhang, ApJ. 866122Huang, Y., Qu, J. L., Zhang, S. N., et al. 2018, ApJ, 866, 122
. F K Lamb, C J Pethick, D Pines, ApJ. 184271Lamb, F. K., Pethick, C. J., & Pines, D. 1973, ApJ, 184, 271
. P Lamb, P W Sanford, MNRAS. 188555Lamb, P., & Sanford, P. W. 1979, MNRAS, 188, 555
. R Iaria, L Burderi, T Di Salvo, A La Barbera, N R Robba, ApJ. 547412Iaria, R., Burderi, L., Di Salvo, T., La Barbera, A., & Robba, N. R. 2001, ApJ, 547, 412
. R Iaria, T Di Salvo, N R Robba, L Burderi, ApJ. 567503Iaria, R., Di Salvo, T., Robba, N. R., & Burderi, L. 2002, ApJ, 567, 503
. E Kalemci, M C Begelman, T J Maccarone, MNRAS. 463615Kalemci, E., Begelman, M. C., Maccarone, T. J., et al. 2016, MNRAS, 463, 615
. P Laurent, L Titarchuk, ApJ. 511289Laurent, P., & Titarchuk, L. 1999, ApJ, 511, 289
. G Lavagetto, R Iaria, T Di Salvo, NuPhS. 132616Lavagetto, G., Iaria, R., di Salvo, T., et al. 2004, NuPhS, 132, 616
. B F Liu, C Done, R E Taam, ApJ. 72610Liu, B. F., Done, C., & Taam, R. E. 2011, ApJ, 726, 10
. B F Liu, S Mineshige, K Ohsuga, ApJ. 587571Liu, B. F., Mineshige, S., & Ohsuga, K. 2003, ApJ, 587, 571
. B F Liu, S Mineshige, K Shibata, ApJL. 572173Liu, B. F., Mineshige, S., & Shibata, K. 2002, ApJL, 572, L173
. X B Li, X F Li, Y Tan, JHEAp. 2764Li, X. B., Li, X. F., Tan, Y., et al. 2020, JHEAp, 27, 64
X B Li, L M Song, X F Li, Proc. SPIE. SPIE106991069969Li, X. B., Song L. M., Li X.F., et al. 2018, Proc. SPIE, 10699, 1069969
. X F Li, C Z Li, Z Chang, JHEAp. 246Li, X. F., Li, C. Z., Chang, Z., et al. 2019, JHEAp, 24, 6
. X Ma, L Tao, S N Zhang, Nature Astronomy. 594Ma, X., Tao, L., Zhang, S. N., et al. 2021, Nature Astronomy, 5, 94
. K Mitsuda, H Inoue, K Koyama, PASJ. 36741Mitsuda, K., Inoue, H., Koyama, K., et al. 1984, PASJ, 36, 741
. K Mitsuda, H Inoue, N Nakamura, Y Tanaka, PASJ. 4197Mitsuda, K., Inoue, H., Nakamura, N., & Tanaka, Y. 1989, PASJ, 41, 97
. E Montanari, L Titarchuk, F Frontera, ApJ. 6921597Montanari, E., Titarchuk, L., & Frontera, F. 2009, ApJ, 692, 1597
. S E Motta, J Rodriguez, E Jourdain, New Astronomy Reviews (NewAR). 93101618Motta, S. E., Rodriguez, J., Jourdain, E., et al. 2021, New Astronomy Reviews (NewAR), 93, 101618
. K Mukerjee, H M Antia, T Katoch, ApJ. 89773Mukerjee, K., Antia, H. M., & Katoch, T. 2020, ApJ, 897, 73
. J Nishimura, K Mitsuda, M Itoh, PASJ. 38819Nishimura, J., Mitsuda, K., & Itoh, M. 1986, PASJ, 38, 819
. A Paizis, K Ebisawa, T Tikkanen, A&A. 443599Paizis, A., Ebisawa, K., Tikkanen, T., et al. 2005, A&A, 443, 599
. P O Petrucci, F Haardt, L Maraschi, ApJ. 556716Petrucci, P. O., Haardt, F., Maraschi, L., et al. 2001, ApJ, 556, 716
. S Piraino, A Santangelo, E C Ford, P Kaaret, A&A. 34977Piraino, S., Santangelo, A., Ford, E. C., & Kaaret, P. 1999, A&A, 349, L77
. E Qiao, B F Liu, ApJ. 744145Qiao, E., & Liu, B. F. 2012, ApJ, 744, 145
. H Raichur, R Misra, G Dewangan, MNRAS. 416637Raichur, H., Misra, R., & Dewangan, G. 2011, MNRAS, 416, 637
. R A Remillard, J E Mcclintock, ARA&A. 4449Remillard, R. A., & McClintock, J. E. 2006, ARA&A, 44, 49
. M J Reid, J E Mcclintock, R Narayan, ApJ. 74283Reid, M. J., McClintock, J. E., Narayan, R., et al. 2011, ApJ, 742, 83
. M G Revnivtsev, S S Tsygankov, E M Churazov, R A Krivonos, MNRAS. 4451205Revnivtsev, M. G., Tsygankov, S. S., Churazov, E. M., & Krivonos, R. A. 2014, MNRAS, 445, 1205
. J Rodi, E Jourdain, J P Roques, ApJ. 817101Rodi, J., Jourdain, E., & Roques, J. P. 2016, ApJ, 817, 101
. C R Shrader, L Titarchuk, N Shaposhnikov, ApJ. 718488Shrader, C. R., Titarchuk, L., & Shaposhnikov, N. 2010, ApJ, 718, 488
. R A Sunyaev, L G Titarchuk, A&A. 86121Sunyaev, R. A., & Titarchuk, L. G. 1980, A&A, 86, 121
. M G Revnivtsev, S S Tsygankov, E M Churazov, R A Krivonos, MNRAS. 4451205Revnivtsev, M. G., Tsygankov, S. S., Churazov, E. M., & Krivonos, R. A. 2014, MNRAS, 445, 1205
. A Tarana, A Bazzano, P Ubertini, A A Zdziarski, ApJ. 654494Tarana, A., Bazzano, A., Ubertini, P., & Zdziarski, A. A. 2007, ApJ, 654, 494
. A Tarana, T Belloni, A Bazzano, M Méndez, P Ubertini, MNRAS. 416873Tarana, A., Belloni, T., Bazzano, A., Méndez, M., & Ubertini, P. 2011, MNRAS, 416, 873
. L Titarchuk, ApJ. 434570Titarchuk, L. 1994, ApJ, 434, 570
. L Titarchuk, A Mastichiadis, N D Kylafis, ApJ. 487837Titarchuk, L., Mastichiadis, A., & Kylafis, N. D. 1997, ApJ, 487, 837
. L Titarchuk, E Seifina, ApJ. 7061463Titarchuk, L., & Seifina, E. 2009, ApJ, 706, 1463
. L Titarchuk, N Shaposhnikov, ApJ. 7241147Titarchuk, L., & Shaposhnikov, N. 2010, ApJ, 724, 1147
. Maitra Varun, C Pragati, P Harsha, R Biswajit, P , MNRAS. 4841Varun, Maitra, C., Pragati, P., Harsha, R., & Biswajit, P. 2019, MNRAS, 484, L1
. N E White, A Peacock, G Hasinger, MNRAS. 218129White, N. E., Peacock, A., Hasinger, G., et al. 1986, MNRAS, 218, 129
. N E White, A Peacock, B G Taylor, ApJ. 296475White, N. E., Peacock, A., & Taylor, B. G. 1985, ApJ, 296, 475
. Z Yan, Xie, Fu-Guo, W Zhang, ApJL. 88918Yan, Z., Xie, Fu-Guo, & Zhang, W. 2020, ApJL, 889, L18
. Y Yao, S N Zhang, X Zhang, Y Feng, C R Robinson, ApJ. 619446Yao, Y., Zhang, S. N., Zhang, X., Feng, Y., & Robinson, C. R. 2005, ApJ, 619, 446
. B You, Y Tou, C Li, Nature Communications. 121025You, B., Tou, Y., Li, C., et al. 2021, Nature Communications, 12, 1025
. A A Zdziarski, J E Grove, J Poutanen, A R Rao, S V Vadawale, ApJL. 55445Zdziarski, A. A., Grove, J. E., Poutanen, J., Rao, A. R., & Vadawale, S. V. 2001, ApJL, 554, L45
. A A Zdziarski, E Jourdain, P Lubiński, ApJL. 9145Zdziarski, A. A., Jourdain, E., Lubiński, P., et al. 2021, ApJL, 914, L5
S Zhang, F J Lu, S N Zhang, T P Li, Proc. SPIE. SPIE9144914421Zhang, S., Lu, F. J., Zhang, S. N., & Li, T. P. 2014, Proc. SPIE, 9144, 914421
. S N Zhang, W Cui, W Chen, Science. 2871239Zhang, S. N., Cui, W., Chen, W., et al. 2000, Science, 287, 1239
. S N Zhang, T P Li, F J Lu, SCPMA. 63249502Zhang, S. N., Li, T. P., Lu, F. J., et al. 2020, SCPMA, 63, 249502
Fitting is performed in 30-220 keV. When fitting with the COMPTT model, the soft photon (Wien) temperature is fixed at the default value, i.e. 0.1 keV. The parameter errors are inferred at the 90% confidence level. The spectrum is fitted with the CPL model and the COMPTT model, respectively. ∆χ 2 = 2.7Note. -The spectrum is fitted with the CPL model and the COMPTT model, respectively. Fitting is performed in 30-220 keV. When fitting with the COMPTT model, the soft photon (Wien) temperature is fixed at the default value, i.e. 0.1 keV. The parameter errors are inferred at the 90% confidence level (∆χ 2 = 2.7).
keV, σ Fe =0.8 keV (fixed), and N=0.32 +0.03 −0.03 for the LINE. E Fe, Γ=0.96 +0.04E Fe =6.53 +0.09 −0.09 keV, σ Fe =0.8 keV (fixed), and N=0.32 +0.03 −0.03 for the LINE; Γ=0.96 +0.04
The spectrum is fitted by the EQPAIR+LINE+CPL model. The individual components are shown, namely, a EQPAIR (red dash line), a LINE (green dash-dot line), and a CPL (blue dot line). The χ 2 ν is 1.08 for 1314 degrees of freedom. The main fitting parameters for the EQPAIR: l h /l s =0.15 ± 0.03, l bb =2195 ± 1263, kT bb =2.52 ± 0.18 keV, and l nth /l h =0.38 ± 0.19. The fitting parameters for the two other components are similar to those obtained in the fitting of (a). −0.55 for the CPL. (b. The parameter errors are inferred at the 90% confidence level (∆χ 2 = 2.7−0.55 for the CPL. (b) The spectrum is fitted by the EQPAIR+LINE+CPL model. The individual components are shown, namely, a EQPAIR (red dash line), a LINE (green dash-dot line), and a CPL (blue dot line). The χ 2 ν is 1.08 for 1314 degrees of freedom. The main fitting parameters for the EQPAIR: l h /l s =0.15 ± 0.03, l bb =2195 ± 1263, kT bb =2.52 ± 0.18 keV, and l nth /l h =0.38 ± 0.19. The fitting parameters for the two other components are similar to those obtained in the fitting of (a). The parameter errors are inferred at the 90% confidence level (∆χ 2 = 2.7).
| [] |
[
"AN EXPLICIT FORMULA FOR COMPUTING BELL NUMBERS IN TERMS OF LAH AND STIRLING NUMBERS",
"AN EXPLICIT FORMULA FOR COMPUTING BELL NUMBERS IN TERMS OF LAH AND STIRLING NUMBERS"
] | [
"Feng Qi "
] | [] | [] | In the paper, the author finds an explicit formula for computing Bell numbers in terms of Lah numbers and Stirling numbers of the second kind.2010 Mathematics Subject Classification. Primary 11B73; Secondary 11B75, 33B10, 33C15. Key words and phrases. explicit formula; Bell number; Lah number; Stirling number of the second kind; Kummer confluent hypergeometric function. This paper was typeset using A M S-L A T E X. | 10.1007/s00009-015-0655-7 | [
"https://arxiv.org/pdf/1401.1625v1.pdf"
] | 56,340,071 | 1401.1625 | 3a2ac4239542def0f6db53cbb73908f6c0be3482 |
AN EXPLICIT FORMULA FOR COMPUTING BELL NUMBERS IN TERMS OF LAH AND STIRLING NUMBERS
8 Jan 2014
Feng Qi
AN EXPLICIT FORMULA FOR COMPUTING BELL NUMBERS IN TERMS OF LAH AND STIRLING NUMBERS
8 Jan 2014
In the paper, the author finds an explicit formula for computing Bell numbers in terms of Lah numbers and Stirling numbers of the second kind.2010 Mathematics Subject Classification. Primary 11B73; Secondary 11B75, 33B10, 33C15. Key words and phrases. explicit formula; Bell number; Lah number; Stirling number of the second kind; Kummer confluent hypergeometric function. This paper was typeset using A M S-L A T E X.
In combinatorics, Bell numbers, usually denoted by B n for n ∈ {0} ∪ N, count the number of ways a set with n elements can be partitioned into disjoint and non-empty subsets. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s. Every Bell number B n may be generated by
e e x −1 = ∞ k=0 B k k! x k(1)
or, equivalently, by
e e −x −1 = ∞ k=0 (−1) k B k x k k! .(2)
In combinatorics, Stirling numbers arise in a variety of combinatorics problems. They are introduced in the eighteen century by James Stirling. There are two kinds of Stirling numbers: Stirling numbers of the first and second kinds. Every Stirling number of the second kind, usually denoted by S(n, k), is the number of ways of partitioning a set of n elements into k nonempty subsets, may be computed by
S(n, k) = 1 k! k i=0 (−1) i k i (k − i) n ,(3)
and may be generated by
(e x − 1) k k! = ∞ n=k S(n, k) x n n! , k ∈ {0} ∪ N.(4)
In combinatorics, Lah numbers, discovered by Ivo Lah in 1955 and usually denoted by L(n, k), count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets and have an explicit formula
L(n, k) = n − 1 k − 1 n! k! .(5)
Lah numbers L(n, k) may also be interpreted as coefficients expressing rising factorials (x) n in terms of falling factorials x n , where
(x) n = x(x + 1)(x + 2) · · · (x + n − 1), n ≥ 1, 1, n = 0 (6) F. QI and x n = x(x − 1)(x − 2) · · · (x − n + 1), n ≥ 1, 1, n = 0.(7)
In [4, Theorem 2] and its formally published paper [7, Theorem 2.2], the following explicit formula for computing the n-th derivative of the exponential function e ±1/t was inductively obtained:
e ±1/t (n) = (−1) n e ±1/t n k=1 (±1) k L(n, k) 1 t n+k .(8)
The formula (8) have been applied in [2,3,5,6].
In combinatorics or number theory, it is common knowledge that Bell numbers B n may be computed in terms of Stirling numbers of the second kind S(n, k) by B n = n k=1 S(n, k).
(9)
In this paper, we will find a new explicit formula for computing Bell numbers B n in terms of Lah numbers L(n, k) and Stirling numbers of the second kind S(n, k). (10)
Proof. In combinatorics, Bell polynomials of the second kind, or say, partial Bell polynomials, denoted by B n,k (x 1 , x 2 , . . . , x n−k+1 ) for n ≥ k ≥ 1, are defined by
B n,k (x 1 , x 2 , . . . , x n−k+1 ) = 1≤i≤n,ℓi∈N n i=1 iℓi=n n i=1 ℓi=k n! n−k+1 i=1 ℓ i ! n−k+1 i=1 x i i! ℓi .(11)
See [1, p. 134, Theorem A]. The famous Faà di Bruno formula may be described in terms of Bell polynomials of the second kind B n,k (x 1 , x 2 , . . . , x n−k+1 ) by which may be rearranged as (10). The proof of Theorem 1 is complete.
d n d t n f • h(t) = n k=1 f (k) (h(t))B n,k h ′ (t), h ′′ (t), . . . , h (n−k+1) (t) .(12)
Theorem 1 .
1For n ∈ N, Bell numbers B n may be computed in terms of Lah numbers L(n, k) and Stirling numbers of the second kind S(n, k)
See [ 1 , p. 139 ,L
1139Theorem C]. Taking f (u) = e 1/u and h(x) = e x in (12) and making use of ((k, ℓ) 1 e (k+ℓ)x B n,k ( n−k+1 e x , e x , . . . , e x ). Further by virtue of B n,k abx 1 , ab 2 x 2 , . . . , ab n−k+1 x n−k+1 = a k b n B n,k (x 1 , x n , . . . , x . . . , 1 = S(n, k) (14) listed in [1, p. 135], where a and b are complex numbers, we obtain d n e e −x d x n = e eComparing this with the n-th derivative of the generating function (
Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition. L Comtet, D. Reidel Publishing Co23Dordrecht; BostonL. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974. 2, 3
A recurrence formula for the first kind Stirling numbers. F Qi, F. Qi, A recurrence formula for the first kind Stirling numbers, available online at http: //arxiv.org/abs/1310.5920. 2
Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions. F Qi, F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, available online at http://arxiv.org/abs/1302. 6731. 2
Properties of three functions relating to the exponential function and the existence of partitions of unity. F Qi, F. Qi, Properties of three functions relating to the exponential function and the existence of partitions of unity, Available online at http://arxiv.org/abs/1202.0766. 2
Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function. F Qi, C Berg, 10.1007/s00009-013-0272-2.2Mediterr. J. Math. 104Available online atF. Qi and C. Berg, Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function, Mediterr. J. Math. 10 (2013), no. 4, 1685-1696; Available online at http://dx.doi.org/10.1007/ s00009-013-0272-2. 2
Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions. F Qi, S.-H Wang, F. Qi and S.-H. Wang, Complete monotonicity, completely monotonic degree, integral rep- resentations, and an inequality related to the exponential, trigamma, and modified Bessel functions, available online at http://arxiv.org/abs/1210.2012. 2
Properties of three functions relating to the exponential function and the existence of partitions of unity. X.-J Zhang, F Qi, W.-H Li, Int. J. Open Probl. Comput. Sci. Math. 53X.-J. Zhang, F. Qi, and W.-H. Li, Properties of three functions relating to the exponential function and the existence of partitions of unity, Int. J. Open Probl. Comput. Sci. Math. 5 (2012), no. 3, 122-127. 2
Inner Mongolia Autonomous Region, 028043, China; Department of Mathematics. stitute of Mathematics. Tianjin City, 300387, China; Jiaozuo City, Henan Province454010College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City ; College of Science, Tianjin Polytechnic University ; Henan Polytechnic UniversityChina E-mail address: [email protected], [email protected], [email protected] URLCollege of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China; In- stitute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China E-mail address: [email protected], [email protected], [email protected] URL: http://qifeng618.wordpress.com
| [] |
[
"Prepared for submission to JHEP Ambitwistor strings and the scattering equations",
"Prepared for submission to JHEP Ambitwistor strings and the scattering equations"
] | [
"Lionel Mason ",
"David Skinner \nDepartment of Applied Mathematics and Theoretical Physics\nWilberforce RoadCB3 0WACambridgeUnited Kingdom\n",
"\nThe Mathematical Institute\nAndrew Wiles Building, Woodstock RoadOX2 6GGOxfordUnited Kingdom\n"
] | [
"Department of Applied Mathematics and Theoretical Physics\nWilberforce RoadCB3 0WACambridgeUnited Kingdom",
"The Mathematical Institute\nAndrew Wiles Building, Woodstock RoadOX2 6GGOxfordUnited Kingdom"
] | [] | We show that string theories admit chiral infinite tension analogues in which only the massless parts of the spectrum survive. Geometrically they describe holomorphic maps to spaces of complex null geodesics, known as ambitwistor spaces. They have the standard critical space-time dimensions of string theory (26 in the bosonic case and 10 for the superstring). Quantization leads to the formulae for treelevel scattering amplitudes of massless particles found recently by Cachazo, He and Yuan. These representations localize the vertex operators to solutions of the same equations found by Gross and Mende to govern the behaviour of strings in the limit of high energy, fixed angle scattering. Here, localization to the scattering equations emerges naturally as a consequence of working on ambitwistor space. The worldsheet theory suggests a way to extend these amplitudes to spinor fields and to loop level. We argue that this family of string theories is a natural extension of the existing twistor string theories. | 10.1007/jhep07(2014)048 | [
"https://arxiv.org/pdf/1311.2564v1.pdf"
] | 53,666,173 | 1311.2564 | 5dd5d81e829cdc23401a978837569653cd1f3ecb |
Prepared for submission to JHEP Ambitwistor strings and the scattering equations
11 Nov 2013
Lionel Mason
David Skinner
Department of Applied Mathematics and Theoretical Physics
Wilberforce RoadCB3 0WACambridgeUnited Kingdom
The Mathematical Institute
Andrew Wiles Building, Woodstock RoadOX2 6GGOxfordUnited Kingdom
Prepared for submission to JHEP Ambitwistor strings and the scattering equations
11 Nov 2013
We show that string theories admit chiral infinite tension analogues in which only the massless parts of the spectrum survive. Geometrically they describe holomorphic maps to spaces of complex null geodesics, known as ambitwistor spaces. They have the standard critical space-time dimensions of string theory (26 in the bosonic case and 10 for the superstring). Quantization leads to the formulae for treelevel scattering amplitudes of massless particles found recently by Cachazo, He and Yuan. These representations localize the vertex operators to solutions of the same equations found by Gross and Mende to govern the behaviour of strings in the limit of high energy, fixed angle scattering. Here, localization to the scattering equations emerges naturally as a consequence of working on ambitwistor space. The worldsheet theory suggests a way to extend these amplitudes to spinor fields and to loop level. We argue that this family of string theories is a natural extension of the existing twistor string theories.
Introduction
Witten's twistor string theory [1] led to a strikingly compact formula [2] for tree-level scattering amplitudes in four-dimensional Yang-Mills theory in terms of an integral over the moduli space of holomorphic curves in twistor space. More recently, analogous expressions have been found for N = 8 supergravity [3][4][5] and for ABJM theory [6]. This year, in the remarkable series of papers [7][8][9][10], Cachazo, He and Yuan have presented analogous formulae based on the ideas in [3], but now extended to describe scattering of massless particles of spins 0, 1 or 2 in arbitrary dimension. A striking property of these new expressions is that they provide one of the most concrete expressions to date of the Kawai, Lewellen and Tye notion of gravitational amplitudes being the square of Yang-Mills amplitudes [11], and are also closely related to the duality between colour and kinematics found by Bern, Carrasco and Johannson [12].
The formulae of Cachazo et al. are based on holomorphic maps of a Riemann sphere into complex momentum space
P (σ) = n j=1 k j σ − σ j : CP 1 → C d ,(1.1)
where the k j are the null momenta of the n particles taking part in the scattering process, and the σ j are n points on the Riemann sphere. These points are not arbitrary, but are determined in terms of the external kinematics by imposing the scattering equations
k i · P (σ i ) = j =i k i · k j σ i − σ j = 0 . (1.2)
These equations were first obtained by Gross and Mende [13,14], where they were shown to govern the string path integral in the limit of high energy scattering at fixed angle (s 1/α ). They also underpin the twistor string formulae of [2], as first observed by Witten in [15]. This is quite remarkable, since the twistor string contains only massless states and is weakly coupled suggestive of a α → 0 limit rather than α → ∞.
Witten's original twistor string (with an alternative formulation by Berkovits [16] and a heterotic formulation in [17]) was discovered to be equivalent to a certain unphysical non-minimal version of conformal supergravity [18] coupled to N = 4 Yang-Mills. More recently, the gravitational amplitudes found in [4] were discovered to arise from a new twistor string theory [19] for N = 8 supergravity. These twistor strings are specific to these theories and it remains unclear how to extend them to other theories, or whether either has any validity for loop amplitudes. In general one would like to be able to construct analogous string theories for more generic field theories and to have some reasonable expectation that they will, at least in favourable circumstances, lead to the correct loop amplitudes.
In this paper we present a new family of string theories that are better placed to fulfill these aims and that underpin the more recent formulae of Cachazo et al.. To motivate these theories, consider the standard first-order worldline action for a massless particle traversing a d dimensional space-time (M, g) 1
S[X, P ] = 1 2π P µ dX µ − e 2 P µ P µ . (1.3)
In this action, the einbein e is a Lagrange multiplier enforcing the constraint P 2 = 0, and is also the worldline gauge field for the gauge transformations δX µ = αP µ δP µ = 0 δe = dα (1.4) conjugate to this constraint. We learn that P must be null and that we should consider fields X and X that differ by translation along a null direction to be equivalent. Consequently, the solutions to the field equations modulo this gauge redundancy are null geodesics in space-time, parametrized by the scaling of P . The quantization of this action leads to the massless Klein-Gordon equation.
The new chiral string theories we study may be viewed as a natural analogue of (1.3), obtained by complexifying the worldline to a Riemann surface Σ and likewise complexifying the target space so that the X µ are holomorphic coordinates on a complexified space-time with holomorphic metric g. In the simplest case, we merely replace dX in (1.3) by∂X = dσ ∂σX to obtain the bosonic action S[X, P ] = 1 2π Σ P µ∂ X µ − e 2 P µ P µ . (1.5) For the kinetic term of (1.5) to be meaningful, we must interpret P µ not as a scalar field, but as a complex (1,0)-form on the worldsheet, so that (suppressing the target space index) P = P σ (σ)dσ in terms of some local holomorphic worldsheet coordinate σ. It then follows that e must now be a (0,1)-form on Σ with values in T Σ -in other words a Beltrami differential. It is perhaps not surprising that we find in section 3 that the spectrum of the string theory based on (1.5) contains only massless particles. Indeed, as we show in appendix A, (1.5) may also be obtained by taking the α → 0 limit of the conventional bosonic string in a chiral way, so the usual string excitations decouple (the tachyon is also absent). However, the geometrical interpretation is quite different from that of the ordinary string. The constraint P 2 = 0 (as a quadratic differential) and corresponding gauge freedom δX µ = α P µ δP µ = 0 δe =∂α (1.6) survive in this model, again provided we interpret α as transforming as a worldsheet holomorphic vector. Thus, if the fields (X, P ) may be thought of as describing a map into complexified cotangent bundle T * M of complexified space-time, imposing this constraint and gauge symmetry mean that the target space of (1.5) is the space of complex null geodesics. Note that, unlike the particle case, P σ is only defined up to a rescaling (P takes values in the canonical bundle of Σ) so there is no preferred scaling of these geodesics.
In four dimensions, this space of complex null geodesics lies in the product of twistor space and its dual and so has become known as (projective) ambitwistor space, denoted P A. It was studied in the 1970s and 1980s as a vehicle for extending the deformed twistor space constructions for Yang-Mills [20,21]. Such constructions were extended to arbitrary dimensions in the context of gravity by LeBrun [22] and in a supersymmetric context in 10 dimensions by Witten [23]. See also [24,25] for more recent work on ambitwistors in the context of scattering amplitudes in N = 4 super Yang-Mills. Although the connection between spaces of complex null geodesics with twistors is less direct in higher dimensions, we will use the term 'ambitwistor space' throughout as they nevertheless provide a family of twistor-like correspondences that encode space-time fields into holomorphic objects on the space of (perhaps spinning) complexified null geodesics in arbitrary dimensions. In particular, as in the usual twistor correspondence, deformations of the space-time metric may be encoded in deformations of the complex structure of ambitwistor space. Similar to the original twistor string, the fact that these ambitwistor string theories are chiral (holomorphic) allows them to describe space-time gravity by coupling to the complex structure of the target space, here P A. We will see that the integrated vertex operators for the ambitwistor string describe deformations of the complex structure of P A preserving this contact structure and naturally incorporate delta function support on the scattering equations (1.2). Indeed these are necessary to impose the resulting constraint P 2 = 0 everywhere on Σ which is crucial to reduce the target space from T * M to P A.
Since the spectrum of this string theory contains only massless states, and since the constraint P 2 = 0 that reduced the target space from T * M to P A is the same constraint as results from imposing the scattering equations (1.2), one might expect this model to underpin the formulae for scattering massless particles of spin s = 0, 1, 2 found in [9,10]. This turns out to be essentially correct for the spin zero case (after coupling to a worldsheet current algebra). To recover the S-matrices of Yang-Mills and gravity, we must instead start from the worldline action
S[X, P, Ψ] = P µ dX µ + g µν Ψ µ dΨ ν − e 2 P µ P µ − χP µ Ψ µ (1.7)
describing a massless particle with spin. Here, Ψ µ is a wordline fermion and χ imposes a constraint associated to the worldline supersymmetry acting as
δX µ = Ψ µ δΨ µ = P µ δP µ = 0 (1.8)
on the matter fields and
δχ = d δe = χ (1.9)
on the gauge fields. The space of solutions to the field equations modulo these gauge transformation is the space of (parametrized) spinning null geodesics. Quantization of Ψ µ gives the Dirac matrices and the quantization of the constraint Ψ µ P µ = 0 is the massless Dirac equation.
In section 4 we consider a chiral analogue of the spinning ambitwistor string with worldsheet action
S[X, P, Ψ] = Σ P µ∂ X µ + e 2 P µ P µ + 2 r=1 Ψ rµ∂ Ψ ν r + χ r P µ Ψ µ r (1.10)
with two spin vectors Ψ µ r each of which also transforms as worldsheet spinor (so that each Ψ = Ψ σ √ dσ in local coordinates). We will call these theories 'type II ambitwistor strings'. Note that here, in stark contrast to the usual RNS string, both sets of Ψ r fields are left-moving. The path integral over these fermions leads to the Pfaffians in the representation of the tree-level gravitational S-matrix found by Cachazo et al.. As we show in section 5, trading one set of these fermions for a general current algebra as in the heterotic string gives (at leading trace) their representation of Yang-Mills amplitudes where one Pfaffian is replaced by a current correlator. Trading both sets of fermions for general current algebras replaces both Pfaffians by current correlators, giving the amplitudes for scalars in the adjoint of G × G found in [10]. Thus the origin of 'gravity as Yang-Mills squared' in [10] is really the same as in the original KLT construction [11].
We conclude in section 6 with a brief look at some of the many possible directions for future work and new perspectives offered by these ideas. These include a brief look at the Ramond-NS and Ramond-Ramond sectors where we anticipate space-time spinors and form fields to reside, and a discussion of how to extend these amplitudes and the scattering equations to higher genus. In section 6.3 we briefly explain how to define Green-Schwarz ambitwistor string actions that make direct contact with Witten's super ambitwistor space [23] for 10 dimensional space-time. It also seems likely that there is a pure spinor formulation. In section 6.4 we argue that the existing twistor string models are perhaps best thought of as different representations of these theories.
These ideas should also lead to new insights into the BCJ colour kinematics relations. Although these have their origins in standard string theory, see e.g. [26], ambitwistor strings give a simpler context without the towers of massive modes of standard string theory. Ambitwistors may also provide a route towards a conventional field theory formulation of these ideas, perhaps using the scattering equations as in e.g. [27], or an ambitwistor action such as in [24].
The space of complex null geodesics
The target space of the string theories we construct will be the space of complex null geodesics in complexified space-time M . We denote the space of scaled complex null geodesics by A and the space of unscaled complex null geodesics by P A, calling them 'ambitwistor space' and 'projective ambitwistor space', respectively. The terminology follows the four dimensional case where P A can be viewed as the projectivized cotangent bundle of both the twistor and dual twistor spaces of M 2 . However, ambitwistor space is a more versatile notion that exists in any dimension and for any (globally hyperbolic) space-time. It has long been known that gauge and gravitational fields may be encoded in terms of holomorphic structures on P A [20][21][22]. We will discuss the gauge theory case later, but here give a brief review of the gravitational case following LeBrun [22] (see also appendix B).
Given any d dimensional space-time (M R , g R ), its complexification (M, g) is a Riemannian manifold of complex dimension d with a holomorphic metric g. A complex null direction at a point x ∈ M is a tangent vector v ∈ T x M obeying g(v, v) = 0, or equivalently a cotangent vector p ∈ T * x M obeying g −1 (p, p) = 0. The bundle T * N M of complex null directions over M thus sits inside the holomorphic cotangent bundle T * M as
T * N M = (x, p) ∈ T * M | g −1 (p, p) = 0 (2.1)
To obtain the space A of scaled complex null geodesics, we must quotient T * N M by the action of
D 0 = p µ ∂ ∂x µ + Γ ρ µν p ρ ∂ ∂p ν . (2.2)
This vector is the horizontal lift of the space-time derivative p µ ∂ µ to the cotangent bundle T * M using the Levi-Civita connection Γ associated to g. Flowing along D 0 generates a null geodesic -the integral curves of D 0 are the horizontal lifts of geodesics with (null) cotangent vector p µ to the cotangent bundle T * M -so to obtain A we should not count as different two points in T * N M that are joined along this flow. Ambitwistor space is a holomorphic symplectic manifold. To see this, note that the cotangent bundle T * M is naturally a holomorphic symplectic manifold with holomorphic symplectic form ω = dp µ ∧ dx µ . The geodesic spray D 0 of (2.2) is the simply the Hamiltonian vector field associated to the function 1 2 g µν (x)p µ p ν ; that is,
D 0 ω + 1 2 d(p µ p µ ) = 0 . (2.3)
Thus, to both impose the constraint p 2 = 0 and quotient by the action of D 0 is simply to take the symplectic quotient of T * M by D 0 , and so A naturally inherits a holomorphic symplectic structure. As L D 0 ω = 0, the symplectic form is invariant along these null geodesics and we will abuse notation by also using ω to denote the holomorphic symplectic form on A. For a d dimensional space-time, A is 2d − 2 (complex) dimensional and the fact that the symplectic structure is non-degenerate means that ω d−1 = 0. The null geodesics obtained this way come with a natural scaling that may be adjusted by rescaling p → rp for any non-zero complex number r. On T * M , this scaling is generated by the Euler vector field Υ = p µ ∂/∂p µ and, since [Υ, D 0 ] = D 0 , the scaling descends to A. If we further quotient A by the action of Υ we obtain the 2d − 3 (complex) dimensional space P A of unscaled complex null geodesics.
To understand the geometric structure inherited by P A, note that the natural symplectic potential θ = Υ ω = p µ dx µ on T * M obeys L D 0 θ + 1 2 d(p µ p µ ) = 0. Thus, while θ is not invariant along the flow of an arbitrary geodesic, it is invariant along (lifts to T * M of) null geodesics and so descends to A. The projectivization A → P A expresses A as the total space of a line bundle that we denote L −1 → P A; sections of L are functions of homogeneity degree one in p. Finally, since L Υ θ = θ, the symplectic potential θ descends to the 2d − 3 dimensional manifold P A to define a 1-form with values in L, θ ∈ Ω 1 (P A, L). Such a line bundle-valued 1-form is known as a contact structure. Because the symplectic structure ω on A obeys ω d−1 = 0, the contact 1-form θ on P A obeys θ ∧ dθ d−2 = 0 and is said to be non-degenerate. Thus, a d dimensional complex space-time (M, g) has a space of complex null geodesics P A that is a 2d − 3 dimensional complex non-degenerate contact manifold.
While a point of P A by definition corresponds to a complex null geodesic in M , a point in M corresponds to a quadric surface Q x ⊂ P A. This may be viewed as the space of complex null rays through x. For example, in four dimensions Q x ∼ = CP 1 ×CP 1 parametrizing the complex null vectors p αα = λ αλα up to scale. For the real Minkowski slice, we setλα = (λ α ) * which gives the familiar celestial sphere S 2 ⊂ CP 1 × CP 1 . More generally, the correspondences between space-time M and the space of complex null geodesics with or without scaling may be expressed in terms of double fibrations as
A M T * N M π 1 π 2 © d d P A M P T * N M π 1 π 2 © d d (2.4)
where, in the projective case the fibres of π 2 are the unscaled complex lightcones Q x and are compact holomorphic quadrics of complex dimension d − 2, while the fibres of π 1 are the complex null geodesics.
LeBrun [22] shows that, conversely, P A together with its contact structure on is sufficient to reconstruct the original space-time M , together with its torsion-free conformal structure. In outline, to reconstruct M from P A one first notes that the non-degenerate contact structure θ defines a complex structure on P A. To see this, we use the fact that because θ is non-degenerate, θ ∧ dθ d−2 is a non-vanishing 2d − 3 form on the 2d − 3 complex dimensional space. We then simply declare an antiholomorphic vector to be a vector V which obeys V (θ ∧ dθ d−2 ) = 0. Now, supposing we can find at least one holomorphic quadric Q 0 ⊂ P A with normal bundle T P d−1 ⊗ O(−1)| Q 0 , Kodaira theory assures us that we can find a d dimensional family of nearby Q x (see e.g. [22] for details). We then interpret this family as providing the points in space-time M . The conformal structure on M together with its null geodesics may be reconstructed from the intersection of these Q x in P A. LeBrun shows [22] that these geodesics arise from a torsion-free connection precisely when P A admits a contact structure θ that vanishes on restriction to the Q x . Furthermore, arbitrary small deformations of the complex structure of P A which preserve the contact structure θ correspond to small deformations of the conformal structure on M .
We will use a linearized version of this correspondence in order to generate amplitudes, focussing on the gravitational case. See appendix B or [28] for a more detailed discussion of the linear Penrose transform for the ambitwistor correspondence in the case of general spin. Since the conformal structure of M is determined by the contact structure of P A, to describe a fluctuation in the space-time metric we need only consider a perturbation δθ of the contact structure. Up to infinitesimal diffeomorphisms, δθ can be taken to be an antiholomorphic 1-form with values in the contact line bundle. If δθ is∂-exact then it does not genuinely describe a deformation of the contact structure, but rather just a diffeomorphism of P A along a Hamiltonian vector field. Thus non-trivial deformations correspond to elements of the Dolbeault cohomology class [δθ]. In short, δθ ∈ Ω 0,1 (L) ,
[δθ] ∈ H 0,1 (P A, L) .
(2.5)
Pulled back to the non-projective space A, it determines a (0, 1)-form valued Hamiltonian vector field X δθ by
X δθ ω + d(δθ) = 0 , X δθ ∈ H 0,1 (A, T P A) (2.6)
and so X δθ determines a deformation of the complex structure of A and hence P A. To see how this deformation determines a deformation of the conformal structure on M , we first pull it back by π 1 to obtain π * 1 (δθ) on P T * N M . It turns out that there is no first cohomology on P T * N M because as a complex manifold it is essentially the cartesian product of M , which has no cohomology by assumption, and a projective quadric of dimension d − 2, which has no first cohomology in dimension d > 3, and none with this weight for any d. Thus we can write
π * 1 (δθ) =∂j (2.7)
for some j ∈ Γ(P T * N M, L). Now, because δθ was originally defined on P A, its pullback to P T * N M must be constant along the fibres of π 1 and so D 0 (π * 1 (δθ)) = 0. But because [D 0 ,∂] = 0 as D 0 is a holomorphic vector field, we learn that∂(D 0 j) = 0, or in other words that D 0 j is holomorphic. Finally, because D 0 j is homogeneous of degree 2 in p µ and holomorphic, it must actually be quadratic so that
h := D 0 j = δg µν (x) p µ p ν (2.8)
for some symmetric, trace-free tensor µν (x) depending only on x. δg µν describes a variation in the space-time metric, while h itself can be viewed as the deformation of the Hamiltonian constraint g µν p µ p ν = 0. To summarize, the ambitwistor Penrose transform relates deformations of the conformal structure on space-time to elements of H 0,1 (P A, L) on projective ambitwistor space. The case of particles with more general spin is treated in appendix B following [28]. One of the most important differences between this ambitwistor version of the Penrose transform and the (perhaps more familiar) Penrose transform between twistor space and space-time is that here, the field on space-time is not required to satisfy any field equations at this stage. Much work in the 70's and 80's focussed on the expression of the field equations in ambitwistor space (in terms of the existence of supersymmetries [20,23] or (essentially equivalently) formal neighbourhoods [21,28,29]). In the following we will see that for our string models, the space-time massless field equations arise automatically from quantum consistency of the symplectic reduction at the level of the worldsheet path integral.
The key example that we will use to discuss scattering amplitudes is the case where our metric fluctations correspond to momentum eigenstates in flat space. To describe these space-time momentum eigenstates in terms of wavefunctions on ambitwistor space we take δg µν (x) = µν e ik·x whereupon h becomes h = e ik·x µν p µ p ν (2.9) while j = D −1 0 h and δθ are then given by
j = e ik·x µν p µ p ν k · p , δθ =δ(k · p) e ik·x µν p µ p ν . (2.10)
As promised, δθ is a (0,1)-form on P A of homogeneity +1 in p, and so defines an element of H 0,1 (P A, L).
The form of the ambitwistor wavefunction δθ is somewhat similar to the form ∼δ ( λ λ i ) e i[µ,λ i ] of a twistor wavefunction for a four-dimensional momentum eigenstate with four-dimensional momentum k = λ iλi . The main differences are that i) the ambitwistor wavefunction is non-chiral and is defined in arbitrary dimensions, and ii) neither the momentum nor the (symmetric, trace-free) polarization vector are constrained in the ambitwistor wavefunction. In particular, at this stage we do note require k 2 = 0 or k µ µν = 0. This is in keeping with the fact that holomorphic objects on ambitwistor space are not manifestly on-shell objects in space-time. As mentioned above, these constraints will arise from quantum consistency of the string theory, but it is worth noting that the formulae of [9,10] involve polarization vectors µν and momenta k -their representation of amplitudes is also not manifestly on-shell. Finally, we remark that in the context of the ambitwistor string path integral, the factor ofδ(k · p) in the ambitwistor wavefunction for a momentum eigenstate ultimately provides the origin of the constraint to solutions of the scattering equations in the formulae of [9,10].
The bosonic ambitwistor string
We now consider a chiral string theory whose target space is projective ambitwistor space. As discussed in the introduction, the worldsheet action is a natural analogue of the worldline action for a massless scalar particle and may be written as
S bos = 1 2π Σ P µ∂ X µ − e 2 P µ P µ . (3.1)
Note that this is different from the first-order action
S = 1 2π Σ P µ dX µ − 1 2 P µ ∧ * P µ (3.2)
that is equivalent to the usual Polyakov string, because in (3.2) P µ is a general 1-form on the worldsheet, i.e. P µ ∈ Ω 1 ∼ = Ω 1,0 ⊕Ω 0,1 , whereas in (A.3) P µ lives only in Ω 1,0 ∼ = K and the kinetic operator is∂ rather than the full exterior derivative. We interpret P 2 in (A.3) to be a quadratic differential and then e ∈ Ω 0,1 (T Σ ) is a Beltrami differential. Both (A.3) and (3.2) are manifestly invariant under worldsheet reparametrizations. In particular, under a diffeomorphism generated by a smooth worldsheet vector field v ∈ T Σ , the fields in (A.3) transform as
δX µ = v∂X µ , δP µ = ∂(vP µ ) , δe = v∂e − e∂v (3.3)
as usual. However, S bos is also separately invariant under the gauge transformations
δX µ = αP µ , δP µ = 0 , δe =∂α (3.4)
for α a further smooth worldsheet vector. As explained in section 2, together with the associated constraint P 2 = 0, these gauge transformations implement the symplectic reduction from T * M to the space of (scaled) null geodesics A. Furthermore, since P takes values in the line bundle K, it is only defined up to a local rescaling. Thus there is really no preferred scaling so the target space is properly interpreted as P A. Said differently, we are identifying the pullback of the contact line bundle L with K, and then the worldsheet action is simply the pullback to Σ of the contact 1-form θ on P A.
The BRST operator
To perform these gauge redundancies in the quantum theory, we introduce the usual holomorphic reparametrization ghost c and antighost b, which are fermionic sections of T Σ and K 2 , respectively. In addition, we introduce a further set of ghosts and antighosts associated to the gauge symmetry (3.4). We call these new ghostsc andb, and they are again fermionic sections of T Σ and K 2 -that is, despite the tildes, they are again holomorphic on the worldsheet. The fact that we have two sets of the usual holomorphic ghosts but no antiholomorphic ghosts is in keeping with the chiral nature of the model. It will have consequences for the form of the vertex operators that we explore below. At genus zero h 1 (Σ, T Σ ) = 0 so we can use the gauge symmetry δe =∂α to set e = 0. In this gauge, the ghost action takes the standard form
S = 1 2π Σ b∂c +b∂c (3.5)
while the BRST operator is
Q = cT +c 2 P 2 (3.6)
where the worldsheet stress tensor T = P µ ∂X µ + c∂b + 2(∂c)b +b∂c. The central charge is
c = 2d − 26 − 26 = 2(d − 26) . (3.7)
Thus Q 2 = 0 when d = 26 as in the standard bosonic string. However, here we recall that X defines a map into the complexification of space-time.
Vertex operators
As in section 2, the simplest vertex operators correspond to variations in the spacetime metric g → g + δg, where for momentum eigenstates δg µν (X) = µν e ik·X with µν symmetric and trace-free. The corresponding fixed vertex operators are
ccV := cc P µ P ν µν e ik·X ,(3.8)
and may be interpreted as cc times the variation in P 2 under this variation of the spacetime metric. Note that the quadratic differential P µ P ν µν e ik·X is balanced by the ghosts c,c ∈ T Σ to form a scalar operator, and that the trace µ µ is absent because we enforce P 2 = 0. This vertex operator is BRST closed iff the momentum and polarization obey
k 2 = 0 , µν k µ = 0 (3.9)
where, as usual, these conditions come from double contractions with the BRST operator. Similarly, it is BRST exact if µν = k (µ ν) for some ν , which is usual linearized diffeomorphism invariance. Consequently, the vertex operator (3.10) represents an onshell linearized graviton. The corresponding integrated vertex operators take the form
Σ V := Σδ (k · P ) V = Σδ (k · P ) P µ P ν µν e ik·X . (3.10)
The fact that we remove the ghost c from the fixed vertex operator is standard, but the presence of theδ(k · P ) here appears to be non-standard and requires further explanation. Firstly, notice that V is indeed a (1,1)-form on the Riemann surface so that (3.10) is at least well-defined. As usual, Σ V may be interpreted as a deformation of the worldsheet action induced by the deformation δg of the space-time metric. To understand this, recall that our worldsheet action is really just the pullback to Σ of the contact 1-form θ on P A, where the pullback to Σ of the contact line bundle L → P A is identified with the worldsheet canonical line bundle K. From the discussion of section 2 we know that a variation of the space-time metric δg determines and is determined by a deformation of this contact 1-form θ → θ + δθ where δθ defines a class [δθ] ∈ H 0,1 (P A, L). Pulled back to the worldsheet, [δθ] thus lies in H 0,1 (Σ, K) and may be integrated to produce a deformation of the action. The vertex operator (3.10) is just this deformation specified to the case of a momentum eigenstate (2.10) on ambitwistor space. From this point of view, the fixed vertex operator is the Hamiltonian associated to the reduction from P T * N M to P A. Again, the vertex operator (3.10) is BRST closed iff the on-shell conditions (3.9) hold. Thus the field equations are not automatically built into the ambitwistor correspondence, but arise in the usual manner through quantum consistency of the string model.
Perhaps the most important difference between the ambitwistor string (A.3) and the usual string is that the XX OPE in (A.3) is trivial. (This is simplest to see in the gauge e = 0). In particular, e ik·X does not here acquire anomalous conformal weight, so we cannot compensate for the conformal weight of a generic polynomial in P or ∂ r X by allowing k 2 = 0. Consequently, there are no massive states in the spectrum, which is consistent with the ambitwistor string being a chiral α → 0 limit of the usual string (see appendix A).
The path integral and the scattering equations
At genus zero, the three zero-modes for each of c andc require that we insert three fixed vertex operators (3.8) and then arbitrarily many integrated ones (3.10). Thus the n-particle amplitude is given by the worldsheet correlation function
M(1, . . . , n) = c 1c1 V 1 c 2c2 V 2 c 3c3 V 3 V 4 · · · V n .
(3.11)
Consider first the XP system. The vertex operators are polynomial neither in P nor in X, so to evaluate this correlation function it is simplest to incorporate the plane waves e ik i ·X into the action. In the gauge e = 0 this becomes
S[X, P ] = 1 2π Σ P µ∂ X µ + i n i=1 k i · X δ 2 (σ − σ i ) (3.12)
and now contains the entire X dependence inside the path integral. Let us consider integrating out X. The constant zero modes decouple from the kinetic P∂X, so integrating these out leads to a momentum conserving δ-function δ 26 ( k i ) as usual. The non-zero modes are Lagrange multipliers enforcing the field equation
∂P µ = 2πi i k iµ δ 2 (σ − σ i ) (3.13)
on the worldsheet (1,0)-form P µ . At genus zero, this has unique solution
P µ (σ) = dσ n i=1 k iµ σ − σ i ,(3.14)
which may now be substituted into the remaining factors of P µ in the vertex operators.
In particular, using the on-shell conditions k 2 i = 0, the factors ofδ(k i · P (σ i )) impose the scattering equations j =i [13,14], which are sufficient to determine the insertion points σ i in terms of the external momenta. However, unlike the saddle-point approximation used in [13,14], here these scattering equations provide the only contributions to the path integral without taking any kinematic limit. This is the same situation as found in the expressions for massless amplitudes found in [9,10] and is also the same as in the twistor string in four dimensions [15]. Just like the c ghosts, the zero modes ofc give a factor of (σ 12 σ 23 σ 13 )/(dσ 1 dσ 2 dσ 3 ). Including this contribution, the measure
k i · k j σ i − σ j = 0 (3.15) of Gross & Mendei δ (k i · P (σ i )) := σ 12 σ 23 σ 13 dσ 1 dσ 2 dσ 3 n i=4δ (k i · P (σ i )) (3.16)
transforms under Möbius transformations as worldsheet vector at each point, and was shown in [8] to be permutation invariant (on the support of the overall momentum conserving δ-function). Thus we the path integral (3.11) gives
M(1, . . . , n) = δ 26 i k i 1 Vol SL(2; C) i δ (k i · P (σ i )) n j=1 µν j P µ (σ j )P ν (σ j ) , (3.17)
where P µ (σ) is constrained to take its value as in (3.14) and where the factor of 1/Vol SL(2; C) = (σ 12 σ 23 σ 31 )/(dσ 1 dσ 2 dσ 3 ) is the usual c ghost path integral. Unfortunately we do not have a satisfactory interpretation of these amplitudes in relation to a standard space-time theory of gravity 3 . In section 4 we turn to a chiral analogue of a type II RNS string model, which does yield the correct gravitational amplitudes. We return to consider this bosonic model in section 5 where we will see that, after including two worldsheet current algebras, it does provide the correct amplitudes in a certain scalar theory.
Ambitwistor superstrings
In this section we construct the worldsheet theory underlying the representations of gravitational scattering amplitudes found in [9,10]. As mentioned in the introduction, our starting-point is a chiral worldsheet analogue of the wordline action for a massless spinning particle. Thus, in addition to the (P, X) system above, we choose a spin structure √ K on Σ and introduce two additional fermionic fields Ψ µ r (r = 1, 2), each with values in √ K ⊗X * T M . Furthermore, as well as gauging P 2 , we will also gauge the Ψ r · P with analogues of worldsheet gravitini χ r ∈ Ω 0,1 ⊗ √ T Σ . These constraints will have the interpretation of reducing the target space of the model to super ambitwistor space, as we discuss in section 4.1.
The action of the matter fields is taken to be
S f = 1 2π Σ P µ∂ X µ − e 2 P 2 + r=1,2 1 2 Ψ rµ∂ Ψ µ r − χ r P µ Ψ µ r , (4.1)
In addition to the transformations
δX µ = αP µ , δΨ µ = 0 , δP µ = 0 , δe =∂α , δχ r = 0 (4.2)
that trivially extend (3.4), this action also has a degenerate N = 2 worldsheet supersymmetry generated by
δX µ = r Ψ µ r , δΨ µ r = r P µ , δP µ = 0 , δe = 0 , δχ r =∂ r , (4.3)
where r ∈ T 1/2 are a pair of anticommuting worldsheet spinors. We will discuss the meaning of this gauge symmetry presently, but first note that there is also a Z 2 × Z 2 symmetry acting as Ψ r → −Ψ r and χ r → −χ r independently on each set of fermion species r. We will gauge this discrete symmetry, meaning we only consider vertex operators that are invariant under Z 2 × Z 2 . In particular, requiring invariance under the action of this Z 2 × Z 2 means we break the O(2) symmetry of (4.1) that rotates the two fermion species into one another down to the Z 2 ⊂ O(2) that simply exchanges them.
The super ambitwistor correspondence
The underlying geometry of this string leads to an extension 4 of the bosonic ambitwistor correspondence that was described in the section 2. The fields (X µ , P µ , Ψ µ r ) define a map from the worldsheet into the bundle T * S M := (T * ⊕ ΠT ⊕ ΠT )M , where the Π reminds us that the two tangent vectors Ψ µ r are each anticommuting. We let (x µ , p µ , ψ µ r ) denote coordinates on this space. T * S M is naturally a holomorphic symplectic supermanifold with holomorphic symplectic potential
θ S = p µ dx µ + 2 r=1 1 2 g µν (x)ψ µ r dψ ν r (4.4)
and associated symplectic form ω S = dθ S . Note that the fermionic differential 1-forms dψ are commuting. Imposing the constraints p 2 = 0 and p µ ψ µ r = 0 gives what we will call the bundle of super null covectors T * SN M , i.e.,
T * SN M := (x µ , p µ , ψ µ r ) ∈ (T * ⊕ ΠT ⊕ ΠT )M | p 2 = 0 = p µ ψ µ r . (4.5)
As before, with the help of the symplectic form ω S , the functions 1 2 p 2 and p µ ψ µ r define Hamiltonian vector fields D 0 and D r given by
D 0 = p µ ∂ ∂x µ + Γ ρ µν p ρ ∂ ∂p ν D r = ψ µ r ∂ ∂x µ + p µ ∂ ∂ψ µ r ,(4.6)
where D 0 is the same bosonic vector field as before while the D r are fermionic. These vectors obey
{D r , D s } = δ rs D 0 , (4.7)
which is a version of the N = 2 supersymmetry algebra along the super null geodesic.
Similarly to the bosonic case of section 2, we define non-projective super ambitwistor space A S to be the quotient of T * SN M by the action generated by these vectors; it is also the symplectic quotient of T * S M by the same action.
A S := T * SN M/ {D 0 , D r } ∼ = T * S M / / {D 0 , D r } . (4.8)
To obtain projective super ambitwistor space P A S we further quotient by the Euler vector field so that P
A S = A S /{Υ} where Υ = 2p µ ∂ ∂p µ + 2 r=1 ψ µ r ∂ ∂ψ µ r (4.9)
is extended to scale the fermionic directions at half the rate it scales the null momentum p. We denote the line bundle A S → P A S by O(−1) so that the ψ µ r take values in O(1) and p µ and the symplectic potential θ S take values in O(2). We thus identify O(2) as the contact line bundle here. Corresponding to (2.4) in the bosonic case, we now have the double fibrations
A S M T * SN M π 1 π 2 © d d P A S M P T * SN M π 1 π 2 © d d (4.10)
Super ambitwistor space has some additional structure that we will use. Firstly we have the two involutions τ r with τ r ψ s = (−1) δrs ψ s , leaving (x µ , p µ ) invariant. These involutions are the Z 2 × Z 2 symmetry used in the worldsheet action above. We also have that g µν ψ µ 1 ψ ν 2 descends to P A S as a section of O(2) generating the R-symmetry, although it will not generally be preserved under deformations.
As before there is a Penrose transform between cohomology of P A S and fields on space-time. For gravity we will just be concerned with [δθ S ] ∈ H 0,1 (P A S , O(2)), again thought of as a perturbation of the contact 1-form θ S . The principal is much the same as before, but there are some new features we briefly point out here. (A more complete treatment of the Penrose transform in this supersymmetric context may be found in appendix B.2.) Again, to Penrose transform δθ S to obtain fields on space-time, we first pull it back to give π * 1 (δθ) on P T * SN M . Here it becomes cohomologically trivial as there are no H 1 s, so we can write π * 1 (δθ) =∂j where j is determined only up to the addition of a polynomial of weight two in the ψ r and one in p µ . Because it was pulled back from P A S , all three of the vector fields D 0 and D r annihilate π * 1 (δθ), so that D 0 j and D r j are global and holomorphic and can therefore be expanded as polynomials of the appropriate degree for their weight in p µ and ψ µ r . However, because D 0 j = D 2 1 j = D 2 2 j, it is not necessary to consider D 0 j itself. We set 11) and the definitions and commutation relations (4.3) show that the J r obey
J r := D r j ∈ O(3) ,(4.D 1 J 1 = D 2 J 2 , D 2 J 1 + D 1 J 2 = 0 . (4.12)
It is easy to see that these relations are solved if there exists a global U ∈ O(2) such that
J 1 = D 2 U and J 2 = −D 1 U . (4.13)
It is more non-trivial to see that there is a choice of the gauge freedom in j so that such a U always exists whenever δθ is invariant under the involutions τ r . Imposing also oddness under the τ r , we must have that
H S = H µν (x)ψ 1µ ψ 2ν (4.14)
for some tensor H µν (x) that depends only on x but is otherwise arbitrary. In particular, we do not require H µν to be either symmetric or trace-free. As in the bosonic case, H µν also obeys no field equations at this stage. The remaining gauge freedom in j induces the change δH µν = ∂ (µ v ν) for some vector field v µ on M , so that H µν is defined modulo diffeomorphisms.
To describe momentum eigenstates we take H µν (x) = µ 1 ν 2 e ik·x as before, where we have now written the polarization tensor in terms of two vectors µ 1,2 as usual. The corresponding H S is now given by
H S = 1 · ψ 1 2 · ψ 2 e ik·x (4.15) whereupon J 1 = 1 · ψ 1 ( 2 · p + k · ψ 2 2 · ψ 2 ) e ik·x (4.16)
and J 2 is obtained by exchanging 1 ↔ 2 and including a minus sign. These give
j = e ik·x k · p 2 r=1 ( r · p + k · ψ r r · ψ r ) (4.17)
and
δθ =δ(k · p) e ik·x 2 r=1 ( r · p + k · ψ r r · ψ r ) . (4.18)
as the deformation of the super contact structure. It is easy to see that D 0 (π * 1 δθ) = D r (π * 1 δθ) = 0. As before, the Penrose transform implies no field equations classically, although we will again see that they arise quantum mechanically in the next section. Note incidentally that we have a potential ξ for δθ given by
ξ = e ik·x 1 · ψ 1 2 · ψ 2δ (k · p) (4.19)
which obeys D 1 D 2 ξ = δθ S . However, although ξ satisfies D 0 ξ = 0, it does not satisfy D r ξ = 0, and so lives on the larger space P T * SN M/{D 0 } rather than on super ambitwistor space.
Quantization
As before, in order to quantize we introduce the bosonic ghosts γ r ∈ √ T Σ , β r ∈ K 3/2 as well as the fermionic ghosts c andc that we had in the bosonic model. In the gauge where e = χ r = 0, the ghost action takes the standard form
S gh = 1 2π Σ b∂c +b∂c + r=1,2 β r∂ γ r .(4.20)
In particular, all sets of ghosts are holomorphic in this chiral model. The BRST operator is extended to become
Q = cT +c 2 P 2 + 2 r=1 γ r P µ Ψ µ r +b 2 γ r γ r ,(4.21)
where T is now the full stress-energy tensor including contributions from both fermions and ghost systems. This operator generates the gauge transformations
δX µ = c∂X µ +cP µ + r γ r Ψ µ r δΨ µ r = c∂Ψ µ r + 1 2 (∂c)Ψ µ r + γ r P µ δP µ = ∂(cP µ ) (4.22)
reproducing the worldsheet supersymmetries (4.3) together with worldsheet diffeomorphism invariance. Thus, in the notation of the previous section, the action of Q reduces the worldsheet path integral from being over the space of maps into P T * SN M down to the space of maps into P A S . When (the complexification of) M has d (complex) dimensions, we have central charge
c = 2d + d 2 + d 2 − 26 + 11 − 26 + 11 = 3(d − 10) (4.23)
so, as in the usual RNS string, the critical dimension is ten, ensuring that Q 2 = 0 at the quantum level.
Vertex Operators
We now construct vertex operators corresponding to gravitational states on space-time.
We will content ourselves with discussing the NS sector (for both sets of fermions Ψ r ); the Ramond sector is discussed very briefly in section 6.1. As before, the fixed vertex operators take the largely standard form
U = ccδ(γ 1 )δ(γ 2 ) 1 · Ψ 1 2 · Ψ 2 ,(4.24)
where the ghost insertions ccδ(γ 1 )δ(γ 2 ) restrict us to considering worldsheet diffeomorphisms and gauge transformations (4.22) that act trivially at the insertion point of U, and where the rest of the vertex operator is the field H S obtained in (4.15). These vertex operators thus fix the residual symmetry in the D r directions, enforcing that ξ = D −1 1 D −1 2 j is constant at its insertion point. (See section 4.1 for the definition of j and ξ.) They are very similar in appearance to the usual graviton vertex operator of the RNS string, except that here all the fields are holomorphic. In particular, the conformal weight of c andc is compensated for by the rest of the vertex operator, which transforms as a quadratic differential.
The usual descent procedure in the supersymmetric directions transforms U into the vertex operator ccV = cc e ik·X 2 r=1 ( r · P + k · Ψ r r · Ψ r ) .
(4.25)
As in the bosonic case, this fixed vertex operator enforces the gauge condition j =constant at its insertion point, fixing the residual symmetry along D 0 . Finally, the integrated vertex operator is
Σ V = Σδ (k · P ) e ik·X 2 r=1 ( r · P + k · Ψ r r · Ψ r ) (4.26)
and represents a deformation of the action corresponding to the deformation θ S → θ S + δθ S of the contact structure on P A S . This is just the supersymmetric version of the contact structure deformation corresponding to a momentum eigenstate on P A S as given in (4.18). The spectrum arising from these vertex operators includes a graviton in the form of g µν = 1(µ 2ν) e ik·X , together with a scalar dilaton φ = µ 1 2µ e ik·X and a 2-form B µν = 1[µ 2ν] e ik·X which we identify as the ten dimensional Neveu-Schwarz B-field. Altogether, these fields constitute the NS-NS sector of ten dimensional supergravity. Although classically the vertex operators can be defined off-shell, in checking BRST closure one meets double contractions whose vanishes enforces the on-shell conditions k 2 = r · k = 0. These conditions are also ensure that the vertex operators themselves are free from normal ordering ambiguities. As before, the XX OPE is trivial so the ambitwistor string spectrum contains are no massive states.
Gravitational scattering amplitudes
At genus zero, h 1 (Σ, T Σ ) = h 1 (Σ, T 1/2 Σ ) = 0, so the gauge fields e and χ r may all be set to zero using the gauge transformations (4.2)-(4.3). There are three zero modes of each of c andc as before, and in addition each of the γ r ghosts has two zero modes as in the RNS string. To fix these zero modes we insert two U operators and one ccV operator so that n-particle tree-level amplitudes are given by the correlation function
M(1, . . . , n) = U 1 U 2 c 3c3 V 3 V 4 · · · V n .(4.27)
Much of the evaluation of the path integral proceeds as before. In particular, the (X, P )system may be treated as before and we again find an overall factor of momentum conservation (now in ten dimensions) and that P µ is frozen to be
P µ (σ) = dσ n i=1 k iµ σ − σ i . (4.28)
Furthermore, the n − 3 factors ofδ(k i · P (σ i )) combine with thec ghost zero modes to produce the permutation invariant factor
iδ (k i · P (σ i )) = σ 12 σ 23 σ 31 d n σ n i=4δ j =i k i · k j σ i − σ j (4.29)
imposing the scattering equations as before.
The main new ingredient is the contribution from the fermions Ψ r . Each set r = 1, 2 are decoupled both in the action and the vertex operators, so it suffices to treat the contribution from, say, Ψ 1 . To evaluate the correlator, first consider the path integral
[dΨ] exp − 1 2π Σ Ψ µ∂ Ψ µ n i=1 i · Ψ(σ i ) k i · Ψ(σ i ) . (4.30)
It is a standard result that (4.30) yields the Pfaffian of the 2n × 2n antisymmetric matrix
M = A −C T C B ,(4.31)
where the n × n matrices A, B and C have entries
A ij = k i · k j dσ i dσ j σ ij B ij = i · j dσ i dσ j σ ij C ij = i · k j dσ i dσ j σ ij (4.32) for i = j, and A ii = B ii = C ii = 0 . (4.33)
These entries result from contracting either the · Ψ or the k · Ψ at site i to the · Ψ or k · Ψ at site j. As usual, we get the Pfaffian of M rather than its determinant because the action in (4.30) is quadratic in Ψ, rather than bilinear in Ψ andΨ. Now, the form of the vertex operators means we must actually consider a product of terms of the form ( i · P (σ i ) + i · Ψ(σ i )k i · Ψ(σ i )). The additional i · P (σ i ) can be incorporated by notionally replacing the vanishing contraction between i · Ψ(σ i ) and
k i · Ψ(σ i ) with i · P (σ i ) = dσ i j =i i · k j σ ij ,(4.34)
where we have used the fact that P (σ) is frozen by the X path integral. These factors are incorporated into the Pfaffian by replacing the matrix C by a matrix C whose off-diagonal entries agree with those of C , but where now
C ii = i · P (σ i ) = −dσ i j =i i · k j σ ij . (4.35)
In fact, worldsheet supersymmetry (4.3) means that the Pfaffian of the n×n matrix
M = A −C T C B
vanishes to second order. Our actual correlation function (4.27) does not have n integrated vertex operators, but rather involves two vertex operators U at sites 1 and 2. These U operators do not contain a factor of k · Ψ. In this case, the path integral over Ψ instead leads to Pfaff(M 12 12 ), the Pfaffian of the matrix M 12 12 obtained by removing the first two rows and columns from M . The U operators also involve a δ-function δ(γ) in the ghosts that are responsible for fixing the residual worldsheet supersymmetry by forcing the supersymmetry variations to vanish at these insertion points. Upon performing the βγ path integral, these δ-functions produce a factor of √ dσ 1 dσ 2 /σ 12 coming from the two elements of H 0 (Σ, T which transforms as a section of K at each of the n marked points. It was shown in [9] that this factor is indeed permutation invariant -of course, from the current perspective this is just a consequence of having the freedom to fix the residual worldsheet supersymmetry in any way we choose. Note in particular that since both U and V each do involve a factor of · Ψ, this operator appears at every site and so no matter where we place the U operators we never remove any rows and columns from B, as was necessary in [9]. Combining all the pieces, including both sets of fermions Ψ r and their associated ghosts, we obtain finally the amplitude
M(1, . . . , n) = δ 10 k i 1 Vol SL(2; C) Pf (M 1 )Pf (M 2 ) i δ (k i · P (σ i )) , (4.37)
where M 1 is built out of the polarization vectors 1i and M 2 out of the 2i and where P (σ) = dσ i k i /(σ − σ i ). The two Pf s together provide a quadratic differential at each marked point, which becomes a (1,0)-form upon multiplication by δ (k i · P (σ i )).
Dividing by Vol SL(2; C) then transforms this to a holomorphic n − 3 form which may be integrated over a middle dimensional cycle in the moduli space M 0,n of marked rational curves. This is exactly the expression originally discovered in [9] and describes all tree-level scattering amplitudes of massless states in the NS-NS sector of pure (type II) supegravity in ten dimensions.
Yang-Mills amplitudes
To construct amplitudes for Yang-Mills fields from ambitwistor strings, we will replace one set of Ψ fields by a more general level k current algebra. This is somewhat analogous to a heterotic string, although we stress again that all our worldsheet fields will be holomorphic (or left-moving). Thus we have the same fields as before but now with just r = 1, together with a current J a (σ) ∈ K Σ ⊗ g with OPE
J a (σ)J b (σ ) = k δ ab (σ − σ ) 2 + f c ab J c σ − σ + · · · , (5.1)
where f c ab are the structure constants for the gauge group G, with a a Lie algebra index. As usual, the current algebra could be realized in many ways, such as a free fermionic model or a WZW model. We will not need to be specific.
The matter action is
S het = S current + 1 2π Σ P µ∂ X µ + 1 2 Ψ µ∂ Ψ µ + e 2 P 2 + χP µ Ψ µ (5.2)
where S current is the action for the current algebra and the other fields have the same meaning as before. This model has only one copy of the worldsheet supersymmetry and the BRST operator becomes For example, this gives the standard result c = 16 in ten dimensions, but also allows c = 31 when d = 4. The possibility of constructing this theory in various dimensions is striking. We note again that both ambitwistor space and the tree-level formulae of [8][9][10] make sense in any number of dimensions. Of course, modular invariance may be expected to impose strong restrictions on the admissible current algebras at higher genus; we will return to consider these constraints in a subsequent paper.
Q het = cT +c 2 P 2 + γP · Ψ +b 2 γ 2 ,(5.
Yang-Mills amplitudes
An (off-shell) Yang-Mills bundle on space-time is equivalent to a holomorphic vector bundle E → P A on ambitwistor space. To describe perturbative gluons, we consider deformations of the complex structure of this bundle, represented by A a ∈ H 1 (P A, End(E)). Essentially by definition, the deformation of the worldsheet current algebra action is
Σ V a = Σ A a J a , (5.5)
which may be interpreted as the integrated vertex operator for a gluon with ambitwistor wavefunction A a . To describe a momentum eigenstate with polarization vector µ , we choose the wavefunctions
A a =δ(k · P ) e ik·X ( · P + · Ψk · Ψ) T a (5.6)
as in (B.18), where T a ∈ g labels the colour of the external state. The integrated vertex operator thus becomes
V 1 =δ(k · P ) [ · P + · Ψ k · Ψ] e ik·X T a J a (5.7)
and transforms as a (1,1)-form on Σ. The fixed vertex operators for gluons are 5.8) and are worldsheet scalars as expected. The form of the Yang-Mills vertex operators are thus very closely related to the Yang-Mills vertex operators in the standard heterotic string, with differences arising as in the bosonic and type II ambitwistor strings because all the fields are chiral. As usual, these vertex operators are BRST invariant classically for any k and , but quantum corrections mean BRST closure fails unless k 2 = 0 and · k = 0. If µ ∝ k µ then (5.7) and (5.8) are BRST exact. Thus nontrivial vertex operators correspond to on-shell gluons.
U 1 = cc δ(γ) · Ψ e ik·X T a J a(
To compute the scattering of these Yang-Mills states we again need two U insertions to fix the two γ zero modes, one ccV to fix the remaining c andc zero-mode and then the rest of the vertex operator insertions must be Vs. Thus we consider
M het (1, . . . , n) = U 1 1 U 1 2 c 3c3 V 1 3 V 1 4 · · · V 1 n . (5.9)
The current algebra is decoupled from the Ψ and XP system in S het , much of the calculation proceeds as in the type II case. In particular, the path integral over the Ψ field and ghosts gives the Pfaffian as before, though now only one copy. In all, the path integral (5.9) may be evaluated as
δ d i k i d n σ Vol SL(2; C) i δ (k i ·P (σ i )) Pf (M )
tr(T 1 T 2 · · · T n ) σ 12 σ 23 · · · σ n1 + · · · , (5.10)
where the term in square brackets arises from the current correlator. Here, the ellipsis represents a sum over both non-cyclic permutations of the marked points and also multitrace contributions. The Pf and the current algebra provide a quadratic differential at each marked point, which combines with the δ-functions imposing the scattering equations and the 1/Vol SL(2; C) factor to produce a holomorphic n − 3 form that may be integrated over a middle dimensional slice of M 0,n . The leading trace terms in (5.10) coincide exactly with the representation of all Yang-Mills tree amplitudes found in [9]. The multi-trace terms are indicative of coupling to gravity, with the gravitational contribution linking the gauge singlets as in the standard heterotic string. Indeed, this model also contains the (fixed) vertex operator (5.11) (and its associated integrated operator) that describes gravitational states (metric + B-field + dilaton) with polarization H µν , together with a 3-form potential C. Again, in order for these to be BRST invariant vertex operators quantum mechanically, we need k 2 = k µ C µνρ = k µ H µν = k ν H µν = 0. However, because of the presence of the 3-form field C, we no longer have the appropriate spectrum for the NS sector of heterotic gravity. Furthermore, as in the bosonic case, the amplitudes obtained by scattering these states do not agree with those of gravity, even if we turn off C.
ccδ(γ) (H µν P µ Ψ ν + C µνρ Ψ µ Ψ ν Ψ ρ ) e ik·X
Scalar fields from an additional current algebra
In [10] the authors constructed amplitudes for massless scalars transforming in the adjoint of some gauge group G × G. We can duplicate these here if we introduce a further levelk set of currentsJã ∈ K Σ ⊗g in place of the remaining Ψ fields. There is thus no remaining worldsheet supersymmetry and the BRST operator is the same as the bosonic case (3.6), but with the stress tensor including those of the current algebras.
Each of the J a andJã currents have the standard OPE (5.1), while J a (σ)Jã(σ ) ∼ 0. The central charge vanishes provided the contributions c andc from the current algebras obey c +c = 2(26 − d) (5.12)
for a d complex dimensional space-time.
In order to construct amplitudes, we introduce the (1,1)-form vertex operator V 0 =δ(k · P )J a T aJãTã e ik·X . Integrating this vertex operator over the worldsheet provides a deformation to the action that now couples the two current algebras. Via the ambitwistor Penrose transform, the contributionδ(k · P )e ik·X T aTã to the integrated vertex operator is an ambitwistor space representative of the scalar field φ aã = e ik·X T aTã on space-time (see appendix B). We also have the fixed vertex operator ccV 0 = ccJ a T aJãTã e ik·X obtained by the Penrose transform as before (see section 2 and appendix B). For these operators to be Q-invariant, we require that k 2 = 0.
Since the two current algebras commute, their path integrals may be performed independently of eachother (an independently of the XP system). Each factor leads to both single trace and multi-trace terms. Picking out only the leading trace contributions from each factor, we find
M scal (1, . . . , n) = c 1c1 V 0 1 c 2c2 V 0 2 c 3c3 V 0 3 V 0 4 · · · V 0 n = δ d i k i (d n σ) 2 Vol SL(2, C) i δ (k i · P (σ i ))
tr(T 1 · · · T n ) σ 12 σ 23 · · · σ n1 × tr(T α(1) · · ·T α(n) ) σ α(1)α(2) · · · σ α(n)α(1) + · · · (5.13) where the ellipsis denotes both non-cyclic permutations of this 'double' leading trace term, together with multi-trace terms. Again, the quadratic differentials from the two holomorphic current algebras combine with the δ-functions imposing the scattering equations and the 1/Vol SL(2; C) to produce a holomorphic n − 3 form that may be integrated over a real slice of M 0,n . The double leading trace part coincides with the scalar field scattering formulae of [10]. The sum over permutations of this double leading trace term is there argued to give the tree-level amplitudes corresponding to the space-time scalar field theory with action
S[φ aã ] = M 1 2 ∂ µ φ aã ∂ µ φ aã + 1 3 f abcfãbc φ aã φ bb φ cc . (5.14)
However, the string theory also generates multi-trace contributions in the correlator. These perhaps arise from coupling the scalar to gravity in this bosonic string, but are not straightforward to interpret.
Conclusions and further directions
We have presented worldsheet models whose n-point correlation functions at genus zero reproduce the new representations of tree-level gravitational, Yang-Mills and scalar amplitudes presented in [9]. These representations are supported on solutions of the scattering equations (1.2) by virtue of the origin of the wave functions as cohomology classes on ambitwistor space. The amplitudes for particles of different spin came from different string theories, with the scalar, Yang-Mills and gravitational amplitudes arising from the bosonic, 'heterotic' and 'type II' ambitwistor strings, respectively. The bosonic and heterotic models are problematic because the gravitational amplitudes they contain do not seem to correspond to Einstein gravity. (Indeed, we are not yet certain whether their amplitudes agree with any known space-time theory of gravity.) However, the type II model does seem to be consistent. As noted in [9,10], one of the most intriguing features of these scattering equations is that they also determine saddle points in the usual string worldsheet path integral which dominate the limit of high energy, fixed angle scattering studied by Gross & Mende [13]. Classical gravitational and Yang-Mills amplitudes emerge from string theory when the energy scales are small compared to the string tension, while the Gross-Mende limit is the opposite case where all kinematic invariants are very large. It is remarkable that the same equations determine both limits. We hope that the present derivation of the amplitude representations of [10] from a worldsheet model not too distant from the usual RNS string helps provide a starting point to understand this fascinating connection.
We conclude this final section by listing a few possible avenues that seem ripe for further investigation.
Ramond sector vertex operators
The type II ambitwistor string appears to be equivalent to a type II supergravity in 10 dimensions. To be sure of this we need to see that, as well as the NS-NS 5 sector studied in this paper, it also correctly reproduces the (massless) Ramond-Ramond and Ramond-NS sectors. The formulation of these ambitwistor strings is sufficiently close to the standard RNS string that we expect standard technology can be brought to bear.
In particular, we anticipate that the model also contains two space-time gravitinos, associated to the vertex operator
Σδ (k · P ) V α 1 δ(γ 2 ) µ α (P µ + Ψ 2µ k · Ψ 2 ) e ik·X (6.1)
and a similar one obtained by exchanging Ψ 1 ↔ Ψ 2 . Here, V α 1 = e φ/2 P µ γ µ αβ Θ β 1 ∈ K Σ , where φ arise in the bosonization of the βγ ghost system, γ µ αβ = γ µ (αβ) are the ten dimensional Van der Waerden symbols, and Θ α 1 is the spin field for the Ψ 1 system (see e.g. [30,31]). There are likewise Ramond-Ramond sector p-form fields created by vertex operators
Σδ (k · P )V α 1 V β 2 γ µ 1 ...µp αβ µ 1 ...µp e ik·X (6.2)
that involve spin fields for both the Ψ r systems. Once more, the presence of theδ(k · P ) term is dictated by the Penrose transform, and is necessary to construct well-defined vertex operators in the case where both sets of worldsheet fermions are holomorphic. It will be fascinating to see whether the amplitudes involving these fields indeed agree with those of supergravity, and what constraints on these vertex operators are imposed by modular invariance.
Loop amplitudes
One advantage of understanding the expressions found in [10] from the perspective of a worldsheet theory is that it provides a natural way to try to extend these amplitudes beyond tree-level: we simply consider the relevant correlation function on a higher genus Riemann surface. One might have said the same also for Witten's original twistor string, and also for the twistor string developed by one of us [19] for N = 8 supergravity. However, the ambitwistor strings are appreciably closer to the standard RNS string, so it is likely that one can make more rapid progress with the current model. We note however that ten dimensional supergravity is UV divergent even at one loop. It will be interesting to see how this arises from the current models.
Green-Schwarz strings in ambitwistor space
Here we have focussed on the bosonic and RNS string, but our general philosophy applies equally well to models with manifest space-time supersymmetry. One simply views ambitwistor superspace as the space of super null geodesics in superspace, in the original spirit of Witten [23]. Alternatively, in four dimensions, one may make space-time supersymmetry manifest using the close relation between ambitwistors and ordinary twistors. We now briefly survey such models, modelling our discussion on that given by Berkovits for standard string theory [32]. The Green-Schwarz models can be motivated by starting from the Brink-Schwarz superparticle [33] for a null geodesic in (10|16)-dimensional superspace with coordinates (x µ , θ α ). Its action is
S = P µ (dX µ − γ µ αβ θ α dθ β ) − 1 2 eP 2 ,(6.3)
where γ µ αβ is one of the Van de Waerden symbols that arise from decomposing the gamma matrices into their chiral parts. As in the RNS case, this can be elevated to an ambitwistor string action
S[X, θ, P ] = Σ P µ (∂X µ − γ µ αβ θ α∂ θ β ) − 1 2 eP 2 (6.4)
for fields (X, θ) : Σ → C 10|16 , P ∈ K ⊗ C 10 and e a Beltrami differential. Exactly as before, this action is manifestly reparametrization invariant and e is a worldsheet gauge field for transformations δX µ = αP µ , δθ α = 0, δP µ = 0, and δe =∂α, with α a worldsheet vector pointing in the holomorphic directions. As usual, (6.4) is invariant under the space-time supersymmetry transformations δX µ = γ µ αβ α θ β , δθ α = α and δP µ = δe = 0, with α a constant anticommuting parameter. There is also a local κ-symmetry that arises because, when P µ is null, the matrix P µ γ µ αβ has an 8-dimensional kernel so that the action is degenerate in the fermionic variables. Specifically, if κ α satisfies P µ γ µ αβ κ α = 0, the action is invariant under δθ = κ. Thus (6.4) really defines a string theory into Witten's version [23] of superambitwistor space for 10 dimensional space-time, in which a super null geodesic is the (1|8)-dimensional supersymmetric extension of the standard light-ray given parametrically by (X 0 +τ P, θ α 0 +κ α ) where the parameters (τ, κ α ) ∈ C 1|8 satisfy P µ γ µ αβ κ β = 0. The conventional Green-Schwarz action is usually quantized in light-cone gauge, breaking manifest covariance, whereas computing amplitudes in the RNS string requires breaking manifest space-time supersymmetry and the introduction of rather awkward spin fields to describe space-time fermions. These considerations led Berkovits to introduce the pure spinor superparticle and string. We expect that our procedure should also be applicable to the pure spinor formulation of the superparticle, leading to a pure spinor variant of the ambitwistor string.
Twistor and ambitwistor strings
In four dimensions, ambitwistor space -the space of complex null geodesics -is closely related to both standard twistor space and its dual. Indeed, the name 'ambitwistor' originates with this relation. In four dimensions, a null momentum p can be written as a simple bispinor p αα = λ αλα , where λ α andλα are each two component spinors. Given a null geodesic with momentum λ αλα through the point x, we can introduce a twistor Z ∈ C 4 and a dual twistor W ∈ C 4 by W a = (λ α , µα) = (λ α , −ix αα λ α ) ∈ T * , Z a = (μ α ,λα) = (ix αα ,λα) ∈ T . (6.5)
It is easily seen that if (Z, W ) arise from a null geodesic in this way, then they satisfy
Z a W a = 0 , where Z a W a = λ αμ α + µαλα . (6.6)
Conversely, if Z · W = 0 then (Z, W ) arises from such a null geodesic. The pair (Z, W ) has two scalings, one for Z and one for W . The product scaling is clearly that of the original null geodesic, but Υ = Z a ∂ ∂Z a − W a ∂ ∂Wa is redundant. Thus we arrive at the description of ambitwistor space as a symplectic reduction
A 0 = {(Z, W ) ∈ T × T * | Z · W = 0} / Υ ,(6.7)
where we start with the holomorphic symplectic form ω = dW a ∧ dZ a and symplectic potential θ = W a dZ a . In four dimensions, ambitwistor superspace can likewise be introduced by starting with super null geodesics in C 4|16 with coordinates (x αα , θ α A ,θ Aα ), where A = 1, . . . , N is an R-symmetry index. A super null geodesic is the (1|2N )-dimensional subspace described parametrically as (x αα 0 +τ λ αλα , θ α 0A +λ α κ A ,θ Aα 0 +λακ A ) where the κ andκ are anticommuting parameters 6 . Given a super null geodesic we can define a supertwistor and dual supertwistor, each in C 4|N , by
W I := (W a , χ A ) = (W a , θ α 0A λ α ) , Z I := (Z a ,χ A ) = (Z a ,θ Aα 0λα ) . (6.8)
Again, if the supertwistor arises in this way we will have Z · W := Z a W a +χ A χ A = 0. We can therefore define superambitwistor space by
A = {(Z I , W I ) ∈ C 4|N × C 4|N | Z · W = 0} / Υ , (6.9)
where Υ is extended to also scale the fermionic directions in the obvious way. Again it is a symplectic quotient, now by Z · W. For N = 3 the projectivisation P A turns out to be a Calabi-Yau supermanifold. This space was introduced by Witten [20], who showed that on-shell N = 3 super Yang-Mills fields correspond to deformations of trivial holomorphic vector bundles on this space. In terms of these coordinates, an ambitwistor superstring can be obtained by gauging the constraint Z · W = 0. We thus have the worldsheet action
S[Z, W, A] = Σ W I (∂ + A)Z I ,(6.10)
where∂ + A defines a∂-operator a line bundle L → Σ such that
Z : Σ → L ⊗ C 4|N , W : Σ →L ⊗ C 4|N (6.11)
where L ⊗L ∼ = K. When N = 4, this is essentially a chiral version of Berkovits' formulation of twistor strings. Similarly, the N = 8 twistor string of [19] can be understood as belonging to this general family of ambitwistor strings (albeit with additional fields that we do not discuss here). However, the symmetrical presentation now allows us to consider line bundles L of negative as well as positive degree and vertex operators that depend non-trivially on W as well as Z. From this point of view, with maximal supersymmetry we have a doubling of the degrees of freedom unless further constraints (perhaps involving a real structure) are imposed.
A The bosonic Ambitwistor string as an α → 0 limit
As a heuristic motivation, we derive (1.5) as a chiral α → 0 limit of the standard bosonic string. We write the Polyakov string action for a map X : Σ → M R from the Riemann surface Σ to a real d-dimensional space-time (M d R , g) as
S = 1 2πα Σ 1 √ 1 − e ē (∂X ·∂X + e ∂X · ∂X +ē ∂ X ·∂X) . (A.1)
Here the dot denotes inner product with respect to the metric g on M , and the metric h on Σ has been referred to a background choice of complex structure∂ = dσ∂σ by h ij ∂ i ∂ j = Ω(∂ σ∂σ + e ∂ σ ∂ σ +ē∂σ∂σ. We now take α → 0 by introducing Lagrange multipliers P andP , and rescalings e = α e ,ẽ = α −2ē to obtain the equivalent action
S = 1 2π Σ 1 √ 1 − α eē (P ·∂X + α P · ∂X − α 2 P ·P + eP 2 + α ēP 2 ) (A.2)
as can be seen by eliminating P andP (here P 2 = P · P ) etc.). Taking α → 0 we obtain the bosonic classical string action
S b = 1 2π Σ P ·∂X + eP 2 . (A.3)
B Ambitwistor space and the Penrose-Ward transform.
Here we give a few more technical details on the Penrose transform between linear fields on space-time and cohomology classes on Ambitwistor space, both in the bosonic case, and in the case where we have just one Ψ (the heterotic case). When we come to the Penrose-Ward transform we will want to work on its projectivisation, P A. Ambitwistor space in general has cohomology in degree 1 and d − 2, but here we only discuss degree 1 as that is the only case needed in this work although conceivably a role for the higher degree cohomology might emerge at some point.
B.1 The bosonic case
The Penrose transform can be described for H 1 s with values in L n for all n as follows.
Theorem 1 The Penrose transform maps cohmology classes on P A to fields on spacetime as follows. For n ≥ −1 we have
H 1 (P A, L n ) = {A µ 0 ...µn = φ (µ 0 ...µn) 0 }/{∇ (µ 0 a µ 1 ...µn) 0 } . (B.1)
Here (. . .) 0 denotes 'the symmetric trace-free part'. When n < −1 H 1 (P A, L n ) = 0.
Proof: Homogeneity degree n functions O P A (n) on ambitwistor space can be represented as homogeneous degree n functions on the projective cotangent bundle P T * M restricted to P 2 = 0 that are anniilated by the geodesic flow D 0 . Thus we have the short exact sequence:
0 → L n P A → L n P T * N M D 0 → L n+1 P T * N M → 0 . (B.2)
The associated long exact sequence in cohomology degenerates quickly because the cohomology of the projective lightcone vanishes except in degrees 0 and d − 2. The latter wont be of much interest to us as we are just interested in the degree zero and one stretch of the long exact sequence. For degree 0, it is nontrivial when n ≥ 0 where it is given by symmetric trace free tensors with n indices. Thus
0 → H 0 (P T * N M, L n ) D 0 → H 0 (P T * N M, L n+1 ) δ → H 1 (P A, L n ) → 0 . (B.3)
The connecting homeomorphism δ at degree zero to one thus gives the isomorphisms and this is equivalent to (B.1) by contraction of the tensors on the right hand side of (B.1) with n + 1 copies of P . Since P is null, we can only determine trace-free symmetric tensors from their contractions with copies of P . 2
In particular, for n = 0, we obtain off-shell Maxwell fields modulo gauge, and for n = 1 we obtain linearized trace-free metrics (the trace-free condition means that we are really just talking about conformal structures) modulo diffeomorphisms.
It is instructive to see how the transform works explicitly in terms of the Dolbeault representatives we will use. We will just work through the n = 0 case as all the others work very similarly. Starting from space-time, we will have a Maxwell field A = A µ dX µ on M . We can then attempt to find a P dependent gauge transformation a(X, P,P ) so that A − dα descends to P A. Thus we must solve
P µ ∂α ∂X µ = P µ A µ . (B.5)
It is always possible to find a solution α to to this equation holomorphically in P locally. However, if it were holomorphic in P globally, it would, by Liouville's theorem, be independent of P and would represent a gauge transformation to the zero Maxwell field. Thus it must depend nonholomorphically on P , but we can nevertheless assume that it will be holomorphic in X as we work on an analytically trivial subset of complex space-time. We then define
a := ∂α ∂P µ dP µ ∈ H 1 ∂ (P A, O) . (B.6)
The fact that a descends to P A follows by acting on (B.5) with ∂/∂P µ and its∂ closure follows from its∂ closure (indeed exactness) on PT * M .
In the converse direction, given such an a, we can pull it back to P T * N M . On the fibres, a must be cohomologically trivial and so can be expressed as a = dα for some α. Since a is pulled back from P A, we have L D 0 α = 0 and this yields L D 0∂ α =∂Dα = 0. Thus D 0 α is holomorphic in P and X globally in P and so by Liouville's theorem in the P variables adapted to homogeneity degree-1, D 0 a = A µ P µ for some A µ .
For the case of a momentum eigenstate, A = e ik·X µ dX µ we see that the above chain of correspondences is fulfilled by
α = · P k · P e ik·X , so a = e ik·x · P∂ 1 k · P . (B.7)
For a complex variable z,∂ 1 z is a distributional (0, 1)-formδ(z) with delta function support at z = 0 so we may write a = e ik·x · Pδ(k · P ) .
(B.8)
We remark that on the support of the delta function D 0 e ik·X = 0 so it is clear that this representative descends to P A. This is defined irrespective of whether k 2 vanishes or not. The Penrose transform for n = 0, 1 has nonlinear extensions. The case n = 0 coresponds directly to a deformation of the complex structure on the trivial line bundle and naturally extends to nonabelian Yang-Mills fields: given a bundle E with connection A on M , we can define a holomorphic bundle E → P A whose fibre at a null geodesic n ∈ P A is the space of covariantly constant sections of E over the corresponding null geodesic. It can be seen that (E , A) can be reconstructed from E as a holomorphic vector bundle and the correspondence is stable under small deformations. Thus any holomorphic vector bundle on P A that is a deformation of the trivial bundle will give rise to a Yang-Mills field on space-time In the case n = 1, given h ∈ H 1 (P A, L), we can construct the corresponding Hamiltonian vector field X h with respect to the symplectic structure ω yielding X h ∈ H 1 (P A, T 1,0 P A). It thus corresponds to the infinitesimal deformation of the complex structure∂ →∂ + X h . By construction it preserves the existence of the holomorphic symplectic structure. It also preserves the existence of the holomorphic contact structrure as
L X h θ = dθ(X h ) + ω(X h , ·) = dh − dh = 0 . (B.9)
since we have from the Euler relation
ω(Υ, ·) = θ(·) , θ(X h ) = ω(Υ, X h ) = Υ(h) = h . (B.10)
Thus, this is a linearized deformation of the complex structure of P A that preserves the holomorphic contact and symplectic structures on A, and we see from the above that this corresponds precisely to variations of the conformal structure of M , see [28] for the 4-dimensional case. We can understand the role of h more directly by observing that the contact structure determines the complex structure. This is because dθ is nondegenerate on T 1,0 A and so determines T 0,1 A as those complex vector fields that annihilate dθ. Under the deformation determined by h, the deformed contact structure is θ h = θ − h to first order as θ h must annihilate the deformed∂-operator∂ h =∂ + X h and as we have seen θ(X h ) = h. Thus h is the deformation of θ.
B.2 The heterotic extension
We will now take A to be the supersymmetric ambitwistor space appropriate to the heterotic case of dimension (18|8) (the type II version is (18|16) dimensional).
We again construct this super-ambitwistor space to be symplectic reduction. We extend the cotangent bundle coordinates (X, P ) with the d fermionic coordinates Ψ µ and the symplectic potential and 2-form by θ = P µ dx µ + g µν Ψ µ dΨ ν /2 , ω = dθ = dP µ ∧ dX µ + g µν dΨ µ dΨ ν /2 . (B.11)
We now perform the symplectic reduction by both P 2 and P · Ψ. Thus we set P 2 = P · Ψ = 0 and quotient by D 0 = P · ∇ and now also D 1 = Ψ · ∇ + P · ∂/∂Ψ. Thus we can define A to be the quotient of the bundle T * SN M of super null vectors as follows For the projectivisation P A, we take the quotient by the Euler vector Υ = 2P · ∂ ∂P + Ψ · ∂ ∂Ψ , P A = A/Υ , (B.14)
so that, before the quotient by D we are taking the equivalence relation (X, P, Ψ) ∼ (X, λ 2 P, λΨ) making the fibres of P T * SN M → M a weighted projective super space. It is easy to see that Υ preserves D and descends to A and so expresses A as the total space of a line bundle O(−1) → P A with P taking values in O(2) and Ψ in O(1).
We can follow the same strategy for the Penrose transform as in the purely bosonic case. We will just discuss the low lying examples that are relevant in detail.
Theorem 2
We have that H 1 (P A, O(n)) vanishes for n < −1. For n ≥ −1 elements correspond to a polynomial in (P, Ψ) of weight n + 1 whose coefficients are arbitrary holomorphic functions of X, modulo D 1 of an arbitrary polynomial in (P, Ψ) of degree n.
The proof follows the strategy given before and can be obtained from the long exact sequence in cohomology that follows from the short exact sequence This is essentially (B.2) but with D 0 replaced by D 1 . As before we pull a ∈ H 1 (P A, O(n)) back to P T * SN M and deduce that on this space a =∂α for some α(X, P, Ψ) of weight n, defined up to the addition of polynomials in (P, Ψ) of weight n whose coefficients are arbitrary functions of X alone. Because D 1 a = 0,∂D 1 α = 0 so that D 1 α is global and holomorphic, and hence a polynomial of degree n + 1 in (P, Ψ) whose coefficients are arbitrary functions of X. The gauge freedom in α gives the stated gauge freedom in D 1 α.
The simplest case is the weight zero case and we will start with a choice of a ∈ H 1 (P A, O). Since the vectors in D acting on a vanish, we have that D 0 α and D 1 α are holomorphic in P and Ψ respectively of weight 2 and 1. We can therefore expand
D 1 α = Ψ µ A µ .
(B.16)
Since D 0 = D 2 1 we will have D 0 α = P µ A µ + Ψ µ Ψ ν F µν . (B.17)
Thus we have an off-shell Maxwell field A defined up to gauge. The α and a associated to a momentum eigenstate A = e ik·X µ dX µ are α = e ik·X · P + · Ψk · Ψ k · P , a = e ik·X ( · P + · Ψk · Ψ)δ(P · k) . (B.18)
The same strategy can be applied to all O(n) albeit with increasing complexity. For O(−1) it is easy to see that one obtains a scalar field. For O(2) we obtain a rank two tensor (without any symmetry or trace assumption) and a 3-form where w µν = w [µν] . The corresponding Dolbeault representative for such a set of fields of the form e ik·X H µν etc., with H and C constant is h = e ik·Xδ (k · P ) (P µ P ν H µν − P µ Ψ ν Ψ ρ (H µν k ρ + 3C µνρ ) − Ψ µ Ψ ν Ψ ρ Ψ σ k µ C νρσ ) (B.20)
As in the weight zero case, the pullback of h to P T * SN M is trivial with h =∂η where η = e ik·x k · P (P µ (P ν + k · ΨΨ ν )H µν − (3P µ + k · ΨΨ µ )Ψ ν Ψ ρ C µνρ ) (B.21) and we have D 0 η = (P µ (P ν + k · ΨΨ ν )H µν − (3P µ + k · ΨΨ µ )Ψ ν Ψ ρ C µνρ ) e ik·x D 1 η = (P µ Ψ ν H µν + Ψ µ Ψ ν Ψ ρ C µνρ ) e ik·x .
(B.22)
We can interpret these as determining linearized deformations of the constraints underlying the symplectic reduction, with the first representing a deformation of P 2 and the second of P · Ψ. The gauge freedom can be seen to arise from diffeomorphisms of P T * SN M generated by Hamiltonian vector fields of functions of the form P ·v+Ψ µ Ψ ν w µν which corresponds to the natural Lie lift of a vector field on M together with an infinitesimal rotation of the Ψ µ .
As before, unlike Witten's super-ambitwistor construction in 10 dimensions [23], our fields A, h, C are completely off-shell. The on-shell conditions will arise from quantum corrections to the BRST invariance that corresponds to the quotient by D. These will correspond to the application of second order operators ∇ · ∇ and ∇ · ∂/∂Ψ to the representatives above. It is straightforward to see that, as a combination, these operators descend to P A and so can be consistently applied to α and β. The first of these simply gives k 2 = 0 so that k is null. The second gives k · = 0 for α and H µν k ν = 0 together with k µ C µνρ = 0 for β.
Again there are non-linear extensions of these transforms as in the bosonic case. The case of H 1 (P A, O) extends naturally to give an encoding of Yang-Mills fields on space-time in terms of holomorphic vector bundles on P A that are deformations of the trivial bundle. Similarly, h ∈ H 1 (P A, O(2)) naturally corresponds to deformations of the contat structure θ h = θ − h that determines the complex structure as in the bosonic case. The Hamiltonian vector fields using the supersymmetric extension of the symplectic structure of members of H 1 (P A, O(2)) give a direct representation of the associated complex structure deformation; these are the deformations of the complex structure of P A that preserve the symplectic potential and symplectic structure. It would be interesting to understand how the on-shell conditions can be imposed in the non-linear regime.
now have only one set of βγ ghosts, and where the holomorphic stress tensor T includes a contribution from the current algebra. This BRST operator implements the symplectic quotient of T * M generated by D = {D 0 , D 1 }.Unlike the usual heterotic string, because all the fields are chiral, it is possible to balance the central charge of the current algebra against that of the rest of the matter and ghosts and obtain cancellation even away from ten dimensions. The total central charge vanishes provided only the central charge c of the current algebra and the (complex) dimension d of the target space are
H 1 (
1P A, L n ) = H 0 (P T * N M, L n+1 )/D(H 0 (P T * N M, L n )) (B.4)
A
= T * SN M/D , where T * SN M = (X, P, Ψ) ∈ T * ⊕ ΠT M | P 2 = 0 = P · Ψ (B.12)where D is the distribution given byD := {D 0 , D 1 } := P · ∇ , Ψ · ∇ + P · ∂ ∂Ψ .(B.13)
H 1 (
1P A, O(2)) = {H µν , C µνρ = C [µνρ] }/{∇ µ v ν + w µν , ∇ [µ w νρ] } . (B.19)
These expressions are given for flat space. For a general metric g the transformations involve the Christoffel symbols as generated by(2.2).
In fact, in four dimensions, P A sits as a quadric inside the Cartesian product of twistor space and its dual; see 6.4.
Their three particle amplitudes are suggestive of a (Weyl) 3 vertex, while the overall weights in the momenta seem to extend these vertices to n-point amplitudes using a standard 1/k 2 propagator.
This RNS-type extension is somewhat different to the notion of superambitwistor space used in[20,23] where space-time supersymmetry is manifest. See section 6.3 for a brief discussion of a Green-Schwarz ambitwistor string.
In our purely chiral context, by the NS-NS sector, we mean the Neveu-Schwarz sector for each of the two sets of left moving worldsheet fermions Ψ r .
For N = 4, 8 this can be understood by reduction from 10 dimensional κ-symmetry.
Perturbative Gauge Theory as a String Theory in Twistor Space. E Witten, hep-th/0312171Commun. Math. Phys. 252E. Witten, Perturbative Gauge Theory as a String Theory in Twistor Space, Commun. Math. Phys. 252 (2004) 189-258, [hep-th/0312171].
On the Tree-Level S-Matrix of Yang-Mills Theory. R Roiban, M Spradlin, A Volovich, hep-th/0403190Phys. Rev. 7026009R. Roiban, M. Spradlin, and A. Volovich, On the Tree-Level S-Matrix of Yang-Mills Theory, Phys. Rev. D70 (2004) 026009, [hep-th/0403190].
F Cachazo, Y Geyer, arXiv:1206.6511A 'Twistor String' Inspired Formula for Tree-Level Scattering Amplitudes in N = 8 Supergravity. F. Cachazo and Y. Geyer, A 'Twistor String' Inspired Formula for Tree-Level Scattering Amplitudes in N = 8 Supergravity, arXiv:1206.6511.
F Cachazo, D Skinner, arXiv:1207.0741Gravity from Rational Curves. F. Cachazo and D. Skinner, Gravity from Rational Curves, arXiv:1207.0741.
F Cachazo, L Mason, D Skinner, arXiv:1207.4712Gravity in Twistor Space and its Grassmannian Formulation. F. Cachazo, L. Mason, and D. Skinner, Gravity in Twistor Space and its Grassmannian Formulation, arXiv:1207.4712.
A New Integral Formula for Supersymmetric Scattering Amplitudes in Three Dimensions. Y.-T Huang, S Lee, arXiv:1207.4851Phys. Rev. Lett. 109Y.-T. Huang and S. Lee, A New Integral Formula for Supersymmetric Scattering Amplitudes in Three Dimensions, Phys. Rev. Lett. 109 (2012) 191601, [arXiv:1207.4851].
F Cachazo, S He, Y E Y , arXiv:1306.2962Scattering in Three Dimensions from Rational Maps. F. Cachazo, S. He, and Y. E. Y., Scattering in Three Dimensions from Rational Maps, arXiv:1306.2962.
F Cachazo, S He, Y E Y , arXiv:1306.6575Scattering Equations and KLT Orthogonality. F. Cachazo, S. He, and Y. E. Y., Scattering Equations and KLT Orthogonality, arXiv:1306.6575.
F Cachazo, S He, Y E Y , arXiv:1307.2199Scattering of Massless Particles in Arbitrary Dimension. F. Cachazo, S. He, and Y. E. Y., Scattering of Massless Particles in Arbitrary Dimension, arXiv:1307.2199.
F Cachazo, S He, Y E Y , arXiv:1309.0885Scattering of Massless Particles: Scalars, Gluons and Gravitons. F. Cachazo, S. He, and Y. E. Y., Scattering of Massless Particles: Scalars, Gluons and Gravitons, arXiv:1309.0885.
A Relation between Tree Amplitudes of Closed and Open Strings. H Kawai, D Lewellen, S Tye, Nucl.Phys. 2691H. Kawai, D. Lewellen, and S. Tye, A Relation between Tree Amplitudes of Closed and Open Strings, Nucl.Phys. B269 (1986) 1.
Perturbative Quantum Gravity as a Double Copy of Gauge Theory. Z Bern, J J Carrasco, H Johansson, arXiv:1004.0476Phys. Rev. Lett. 10561602Z. Bern, J. J. Carrasco, and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett. 105 (2010) 061602, [arXiv:1004.0476].
String Theory Beyond the Planck Scale. D Gross, P Mende, Nucl. Phys. 303407D. Gross and P. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B303 (1988) 407.
The High Energy Behavior of String Scattering Amplitudes. D Gross, P Mende, Phys. Lett. 197129D. Gross and P. Mende, The High Energy Behavior of String Scattering Amplitudes, Phys. Lett. B197 (1987) 129.
Parity Invariance for Strings in Twistor Space. E Witten, hep-th/0403199Adv. Theor. Math. Phys. 8E. Witten, Parity Invariance for Strings in Twistor Space, Adv. Theor. Math. Phys. 8 (2004) 779-796, [hep-th/0403199].
An Alternative String Theory in Twistor Space for N = 4 Super-Yang-Mills. N Berkovits, hep-th/0402045Phys. Rev. Lett. 9311601N. Berkovits, An Alternative String Theory in Twistor Space for N = 4 Super-Yang-Mills, Phys. Rev. Lett. 93 (2004) 011601, [hep-th/0402045].
Heterotic Twistor-String Theory. L Mason, D Skinner, arXiv:0708.2276Nucl. Phys. 795L. Mason and D. Skinner, Heterotic Twistor-String Theory, Nucl. Phys. B795 (2008) 105-137, [arXiv:0708.2276].
N Berkovits, E Witten, hep-th/0406051Conformal Supergravity in Twistor-String Theory. 9N. Berkovits and E. Witten, Conformal Supergravity in Twistor-String Theory, JHEP 08 (2004) 009, [hep-th/0406051].
D Skinner, arXiv:1301.0868Twistor Strings for N = 8 Supergravity. D. Skinner, Twistor Strings for N = 8 Supergravity, arXiv:1301.0868.
An Interpretation of Classical Yang-Mills Theory. E Witten, Phys.Lett. 77394E. Witten, An Interpretation of Classical Yang-Mills Theory, Phys.Lett. B77 (1978) 394.
Non Selfdual Gauge Fields. J Isenberg, P Yasskin, P Green, Phys. Lett. 78J. Isenberg, P. Yasskin, and P. Green, Non Selfdual Gauge Fields, Phys. Lett. B78 (1978) 462-464.
Spaces of Complex Null Geodesics in Complex Riemannanian Geometry. C Lebrun, Trans. Amer. Math. Soc. 278209C. LeBrun, Spaces of Complex Null Geodesics in Complex Riemannanian Geometry, Trans. Amer. Math. Soc. 278 (1983) 209.
Twistor-Like Transform in Ten Dimensions. E Witten, Nucl. Phys. 266245E. Witten, Twistor-Like Transform in Ten Dimensions, Nucl. Phys. B266 (1986) 245.
An Ambitwistor Yang-Mills Lagrangian. L Mason, D Skinner, hep-th/0510262Phys. Lett. 636L. Mason and D. Skinner, An Ambitwistor Yang-Mills Lagrangian, Phys. Lett. B636 (2005) 60-67, [hep-th/0510262].
Null Polygonal Wilson Loops in Full N = 4 Superspace. N Beisert, S He, B Schwab, C Vergu, arXiv:1203.1443J. Phys. 45N. Beisert, S. He, B. Schwab, and C. Vergu, Null Polygonal Wilson Loops in Full N = 4 Superspace, J. Phys. A45 (2012) 265402, [arXiv:1203.1443].
The Momentum Kernel of Gauge and Gravity Theories. N Bjerrum-Bohr, P Damgaard, T Sondergaard, P Vanhove, arXiv:1010.3933JHEP. 1101N. Bjerrum-Bohr, P. Damgaard, T. Sondergaard, and P. Vanhove, The Momentum Kernel of Gauge and Gravity Theories, JHEP 1101 (2011) 001, [arXiv:1010.3933].
R Monteiro, D O'connell, arXiv:1311.1151The Kinematic Algebras from the Scattering Equations. R. Monteiro and D. O'Connell, The Kinematic Algebras from the Scattering Equations, arXiv:1311.1151.
Conformal Gravity, the Einstein Equations and Spaces of Complex Null Geodesics. R Baston, L Mason, Class. Quant. Grav. 4R. Baston and L. Mason, Conformal Gravity, the Einstein Equations and Spaces of Complex Null Geodesics, Class. Quant. Grav. 4 (1987) 815-826.
. C Lebrun, Thickenings , Conformal Gravity, Commun. Math. Phys. 139C. LeBrun, Thickenings and Conformal Gravity, Commun. Math. Phys. 139 (1991) 1-43.
Conformal Invariance, Supersymmetry and String Theory. D Friedan, E Martinec, S Shenker, Nucl. Phys. 27193D. Friedan, E. Martinec, and S. Shenker, Conformal Invariance, Supersymmetry and String Theory, Nucl. Phys. B271 (1986) 93.
J Polchinski, Superstring Theory and Beyond. Cambridge University Press2J. Polchinski, String Theory. Vol. 2: Superstring Theory and Beyond. Cambridge University Press, 1998.
O Bedoya, N Berkovits, arXiv:0910.2254GGI Lectures on the Pure Spinor Formalism of the Superstring. O. Bedoya and N. Berkovits, GGI Lectures on the Pure Spinor Formalism of the Superstring, arXiv:0910.2254.
. L Brink, J Schwarz, Quantum Superspace, Phys. Lett. 100L. Brink and J. Schwarz, Quantum Superspace, Phys. Lett. B100 (1981) 310-312.
| [] |
[
"Search for high-energy neutrino emission from Mrk 421 and Mrk 501 with the ANTARES neutrino telescope",
"Search for high-energy neutrino emission from Mrk 421 and Mrk 501 with the ANTARES neutrino telescope"
] | [
"M Organokov \nUMR 7178\nUniversité de Strasbourg\nCNRS\nIPHC\nF-67000StrasbourgFrance\n",
"T Pradier \nUMR 7178\nUniversité de Strasbourg\nCNRS\nIPHC\nF-67000StrasbourgFrance\n"
] | [
"UMR 7178\nUniversité de Strasbourg\nCNRS\nIPHC\nF-67000StrasbourgFrance",
"UMR 7178\nUniversité de Strasbourg\nCNRS\nIPHC\nF-67000StrasbourgFrance"
] | [] | ANTARES is the largest high-energy neutrino telescope in the Northern Hemisphere. This contribution presents the results of a search, based on the ANTARES data collected over 17 months between November 2014 and April 2016, for high energy neutrino emission in coincidence with TeV γ-ray flares from Markarian 421 and Markarian 501, two bright BL Lac extragalactic sources highly variable in flux, detected by the HAWC observatory. The analysis is based on an unbinned likelihood-ratio maximization method. The γ-ray lightcurves (LC) for each source were used to search for temporally correlated neutrinos, that would be produced in pp or p-γ interactions. The impact of different flare selection criteria on the discovery neutrino flux is discussed. Plausible neutrino spectra derived from the observed γ-ray spectra in addition to generic spectra E −2 and E −2.5 are tested. * | 10.5281/zenodo.1300943 | [
"https://arxiv.org/pdf/1809.05777v1.pdf"
] | 119,388,982 | 1809.05777 | d2922cf3b0eeada76edf9896cbb498210738f775 |
Search for high-energy neutrino emission from Mrk 421 and Mrk 501 with the ANTARES neutrino telescope
M Organokov
UMR 7178
Université de Strasbourg
CNRS
IPHC
F-67000StrasbourgFrance
T Pradier
UMR 7178
Université de Strasbourg
CNRS
IPHC
F-67000StrasbourgFrance
Search for high-energy neutrino emission from Mrk 421 and Mrk 501 with the ANTARES neutrino telescope
Dated: September 18, 2018on behalf of the ANTARES Collaboration
ANTARES is the largest high-energy neutrino telescope in the Northern Hemisphere. This contribution presents the results of a search, based on the ANTARES data collected over 17 months between November 2014 and April 2016, for high energy neutrino emission in coincidence with TeV γ-ray flares from Markarian 421 and Markarian 501, two bright BL Lac extragalactic sources highly variable in flux, detected by the HAWC observatory. The analysis is based on an unbinned likelihood-ratio maximization method. The γ-ray lightcurves (LC) for each source were used to search for temporally correlated neutrinos, that would be produced in pp or p-γ interactions. The impact of different flare selection criteria on the discovery neutrino flux is discussed. Plausible neutrino spectra derived from the observed γ-ray spectra in addition to generic spectra E −2 and E −2.5 are tested. *
Introduction
The very high energy (VHE; 0.1-100 TeV) extragalactic sky is dominated by emission from blazars [1], a class of a radio-loud AGN [2]. Two blazars, Mrk 421 and Mrk 501, the brightest and closest BL Lac objects known, at luminosity distances d L = 134 Mpc with redshift z = 0.031 and d L = 143 Mpc with redshift z=0.033 respectively. These blazars are the first and the second extragalactic objects discovered in the TeV energy band. This analysis focuses on the search of spatial/temporal correlation between neutrinos (ν) detected by ANTARES and γ-ray emission from flares detected by HAWC from these blazars in the period Nov. 2014 -Apr. 2016, and reported in [3]. As the nearest blazars to Earth, both are excellent sources to test the blazar-neutrino connection scenario, especially during flares where time-dependent neutrino searches may have a higher detection probability.
ANTARES and HAWC
The ANTARES (Astronomy with a Neutrino Telescope and Abyss environmental RESearch) neutrino telescope [4] is a Cherenkov detector designed to search for high-energy neutrinos from astrophysical sources by detecting the Cherenkov light emission of neutrino-induced charged particles in the very deep waters of the Mediterranean Sea and most sensitive for neutrino energies 100 GeV < E ν < 100 TeV. The ANTARES is located in the Mediterranean sea, 40 km off the coast of Toulon, France (42 • 48 N 6 • 10 ), at a depth of 2,475 meters. The HAWC (High-Altitude Water Cherenkov Observatory) gammaray observatory [3] is a Cherenkov detector designed to search for high-energy γ-rays (100 GeV < E γ < 100 TeV) from astrophysical sources by detecting the Cherenkov light emission from charged particles in γ-ray induced air showers. HAWC is located at an elevation of 4,100 m above sea level on the flanks of the Sierra Negra volcano in the state of Puebla, Mexico (18 • 99 N, 97 • 18 W).
The ANTARES data set
The ANTARES data set covers the same period of observation as HAWC: from November 26 th , 2014 until April 20 th , 2016 (MJD: 56988-57497) leading to an effective detector livetime of 503.7 days. The search relies on track-like event signatures, so only CC interactions of muon neutrinos are considered. The muon track reconstruction returns two quality parameters, namely the track-fit quality parameter, Λ, and the estimated angular uncertainty on the fitted muon track direction, β. Cuts on these parameters are used to improve the signal-to-noise ratio. To avoid biasing the analysis, it has been performed according to a blinding policy with the data have been blinded by time-scrambling. The final sensitivities are derived from the blinded dataset.
The HAWC light curves of blazars
HAWC has made clear detections of Mrk 421 and Mrk 501. In this analysis HAWC-300 data of first long-term TeV light curve studies with single-transit intervals are used [3]. The HAWC flare states for Mrk 421 and Mrk 501 with one day binning applied are shown in Fig. 1 and Fig. 2 respectively. These flare states are the Bayesian blocks derived from the LCs [3]. Several flare selection conditions considered: all flare states are taken as they are (long case);only those flare states are taken which pass the defined thresholds (short case): average flux, average flux +1 σ, average flux +2 σ.
Time-dependent search method
A search for neutrino candidades in coincidence with γ-rays from astrophysical sources is performed using an unbinned likelihood-ratio maximization method [5,6,7]. The goal is to determine the relative contribution of background and signal components for a given direction in the sky and at a given time.
ln(L) = N i=1 ln[N S S i + N B B i ] − [N S + N B ](1)
To perform the analysis, the ANTARES data sample is parametrized as two-component mixture of signal and background. The signal is expected to be small so that the full data direction can be used as an estimation of the background. S i and B i are defined as the probability density functions (PDF) respectively for signal and background for an event i, at time t i , energy E i , declination δ i . As a result,
S i = P s (α i ) · P s (E i ) · P s (t i ) and B i = P b (sin(δ i )) · P b (E i ) · P b (t i ).
The parameter α i represents the angular distance between the direction of the event i and direction to the source. Additionally, N S and N B are unknown signal events and known background rate (a priori when building the L) respectively. Since the signal is expected to be small, the total number of events N in the considered data sample can be treated as background.
The energy PDF for the signal events is produced according to the studied energy spectra: E −2.0 , E −2.5 , E −1.0 exp (−E/1 PeV) for both sources and extra E −2.25 for Mrk 501 (as from the fit of the spectral shape of this source performed for same data in [8]). Worth noting that Mrk 421 is well described by the E −2.0 spectrum.
The signal time PDF shape is extracted directly from the γ-ray light curve assuming a proportionality between the γ-ray and the ν fluxes (see Figures 1 and 2).
The test statistics (TS) is evaluated by generating pseudo-experiments simulating background and signal around the considered source:
T S = 2(ln(L max s+b ) − ln(L b ))(2)
Contrarily to other analyses [5,6], it is assumed that neutrinos are emitted all along the LC and not only in the selected peaks. If the only high peaks selected with neutrinos injected solely there, it gives the flux outside of those peaks being artificially treated as zero which subsequently can raise the loss of neutrinos. Therefore, the derived N S required for discovery for the selected peaks is then rescaled as if like neutrinos injected on all flares. Conversion from N S to F , the equivalent source flux, is done through the acceptance of the detector.
Cuts on the cosine of the zenith angle of the reconstructed events cos(θ) > −0.1 and on the direction of the reconstructed events β < 1.0 • are used to improve the signal-to-noise ratio. The Λ cut is optimized for each source on the basis of maximizing Model Discovery Potential (MDP) for 5σ level for each ν spectrum (see Fig. 3). MDP is a probability to make a discovery assuming that the model is correct [9].
E 2 ν Φ ν [GeV cm −2 s −1 ] MRK501 E −2.25 MRK421 E −1.0 · e −E/1PeV MRK501 E −1.0 · e −E/1PeV MRK421 E −2.0 MRK501 E −2.0 MRK421 E −2.5 MRK501 E −2.5
PRELIMINARY
Mrk 421
Mrk 501 10 0 The lowest flux required for discovery was obtained with the case of all flare states selected (see Fig. 5). In contrast, the lowest fluence was obtained with average flux +2 σ threshold (see Fig. 5). Usage of this threshold instead of all flare states makes possible to set a better upper limits in the absence of discovery.
10 1 F 90%CL [GeV cm −2 ] MRK501 E −2.25 MRK421 E −1.0 · e −E/1PeV MRK501 E −1.0 · e −E/1PeV MRK421 E −2.0 MRK501 E −2.0 MRK421 E −2.5 MRK501 E −2.5
Conclusion
The HAWC detector operates nearly continuously and it is currently the most sensitive wide FOV γ-ray telescope in the very promising HE band from 100 GeV to 100 TeV. Therefore, it opens prospects to study the most energetic astrophysical phenomena in the Universe as well as to understand the mechanisms that power them and endeavor to break the mystery of their origin. Taking into account the flare timing information given by γ-ray observations should improve the efficiency of the search for a ν counterpart with ANTARES. The next generation KM3NeT neutrino telescope [10] will provide more than an order of magnitude improvement in sensitivity [11]; therefore, such sources are promising candidates as HE ν emitters for an improved future time-dependent search.
Figure 1 :Figure 2 :Figure 3 :
123Flare states for Mrk 421 vs threshold. The blue dotted line represent the average flux ∼ 0.8 CU (Crab Units); the green dotted lines represent the peak selection thresholds: average flux, average flux + 1 σ, average flux + 2 σ. The bottommost plot shows the long case, the three upper plots show short case for average flux, average flux + 1 σ, average flux + 2 σ respectively. The left axes represent the units of the fluxes, the right-right axis represent the fluxes in corresponding CU. The right-left axes represent the units of fluences shown as shaded grey areas. Flare states for Mrk 501 vs threshold. The blue dotted line represent the average flux ∼ 0.3 CU (Crab Units); the green dotted lines represent the peak selection thresholds: average flux, average flux + 1 σ, average flux + 2 σ. The bottommost plot shows the long case, the three upper plots show short case for average flux, average flux + 1 σ, average flux + 2 σ respectively. The left axes represent the units of the fluxes, the right-right axis represent the fluxes in corresponding CU. The right-left axes represent the units of fluences shown as shaded grey areas. Examples of discovery power (left) and MDP (right) at 5σ level vs Λ cut for Mrk 501 for E −1.0 exp (−E/1 PeV) (pink), E −2.0 (green), E −2.25 (grey), E −2.5 (blue) spectra in case of all flare states selected. The light green color circles represent the MDP 5σ max values.4 Results.The sensitivities at 90% C.L. on neutrino energy fluxes and fluences obtained with 90% C.L. sensitivity fluxes for optimum Λ cut values for each spectrum and gathered inFig. 4.
Figure 4 :
4Neutrino energy flux sensitivities (left) and neutrino fluence sensitivities (right) at 90% C.L. obtained in the analysis with the all neutrino energy spectra and for cases of all flare states selected and flaress with average flux +2 σ threshold selected respectively.
Figure 5 :
5Examples of discovery fluxes comparison at 5σ level (left) and neutrino fluence sensitivities at 90% C.L. (right) vs Λ cut for Mrk 421 for E −2.0 spectrum with different peaks selection thresholds. Light green color circles represent the values with Λ cut that maximizes MDP 5σ .
Target of Opportunity observations of blazars with. M Cerruti, H.E.S.S. Collaboration10.1063/1.4968975arXiv:1610.05523AIP Conf.Proc. 1792. H.E.S.SM. Cerruti et al. [H.E.S.S. Collaboration], "Target of Opportunity observations of blazars with H.E.S.S", AIP Conf.Proc. 1792 (2017) no.1, 050029 DOI:10.1063/1.4968975 arXiv:1610.05523
The AGN phenomenon: open issues. V Beckmann, C R Shrader, 10.22323/1.176.0069PoS. 2012V. Beckmann and C.R. Shrader. "The AGN phenomenon: open issues", PoS INTEGRAL2012 (2012), 069 DOI:10.22323/1.176.0069 arXiv:1302.1397
Daily monitoring of TeV gamma-ray emission from Mrk 421, Mrk 501, and the Crab Nebula with HAWC. A Abeysekara, HAWC Collaboration10.3847/1538-4357/aa729eAstrophys.J. 8412A. Abeysekara et al. [HAWC Collaboration], "Daily monitoring of TeV gamma-ray emission from Mrk 421, Mrk 501, and the Crab Nebula with HAWC", Astrophys.J., 841 (2017) no.2, 100 DOI:10.3847/1538-4357/aa729e arXiv:1703.06968
ANTARES: the first undersea neutrino telescope. M Ageron, ANTARES Collaboration10.1016/j.nima.2011.06.103Nucl.Instrum.Meth. 656M. Ageron et al. [ANTARES Collaboration], "ANTARES: the first undersea neutrino telescope", Nucl.Instrum.Meth. A656 (2011) 11-38, DOI:10.1016/j.nima.2011.06.103 arXiv:1104.1607
Search for muon-neutrino emission from GeV and TeV gamma-ray flaring blazars using five years of data of the ANTARES telescope. S Adrian-Martinez, ANTARES Collaborationhttp:/iopscience.iop.org/article/10.1088/1475-7516/2015/02/020/metaJCAP. 151212S. Adrian-Martinez. et al. [ANTARES Collaboration], "Search for muon-neutrino emission from GeV and TeV gamma-ray flaring blazars using five years of data of the ANTARES telescope", JCAP 1512 (2015) no.12, 014 DOI:10.1088/1475-7516/2015/1 arXiv:1506.07354
Time-dependent search for neutrino emission from x-ray binaries with the ANTARES telescope. A Albert, ANTARES Collaboration10.1088/1475-7516/2017/04/019JCAP. 170404A. Albert et al. [ANTARES Collaboration], "Time-dependent search for neutrino emission from x-ray binaries with the ANTARES telescope", JCAP 1704 (2017) no.04, 019 DOI:10.1088/1475- 7516/2017/04/019 arXiv:1609.07372
Time-dependent search for neutrino emission from Mrk 421 and Mrk 501 observed by the HAWC gamma-ray observatory. T Pradier, ANTARES CollaborationM Organokov, ANTARES CollaborationA Sanchez Losa, ANTARES Collaboration10.22323/1.301.0946PoS. 2017T. Pradier, M. Organokov, A. Sanchez Losa. [ANTARES Collaboration], "Time-dependent search for neutrino emission from Mrk 421 and Mrk 501 observed by the HAWC gamma-ray observatory", PoS ICRC2017 (2017), 946 DOI:10.22323/1.301.0946
Spectral analysis of Markarian 421 and Markarian 501 with HAWC. S Coutiño De León, HAWC Collaboration10.22323/1.301.0606PoS. 2017S. Coutiño de León et al. [HAWC Collaboration], "Spectral analysis of Markarian 421 and Markarian 501 with HAWC," PoS ICRC 2017, 606 (2018) DOI:10.22323/1.301.0606 arXiv:1708.04637.
Search for muon neutrinos from gammaray bursts with the ANTARES neutrino telescope using 2008 to 2011 data. S Adrian-Martinez, ANTARES Collaboration10.1051/0004-6361/201322169arXiv:1307.0304Astron.Astrophys. 5599S. Adrian-Martinez. et al. [ANTARES Collaboration], "Search for muon neutrinos from gamma- ray bursts with the ANTARES neutrino telescope using 2008 to 2011 data", Astron.Astrophys. 559 (2013) A9 DOI:10.1051/0004-6361/201322169 arXiv:1307.0304
Letter of intent for KM3NeT 2.0. S Adrian-Martinez, KM3NeT Collaboration10.1088/0954-3899/43/8/084001arXiv:1601.07459J.Phys. 438S. Adrian-Martinez et al., [KM3NeT Collaboration], "Letter of intent for KM3NeT 2.0", J.Phys. G43 (2016) no.8, 084001 DOI:10.1088/0954-3899/43/8/084001 arXiv:1601.07459
Future neutrino telescopes in water and ice. U Katz, 10.5281/zenodo.1287686U. Katz. (2018, June). "Future neutrino telescopes in water and ice", Zenodo. DOI:10.5281/zenodo.1287685
| [] |
[
"Symmetric Norm Inequalities And Positive Semi-Definite Block-Matrices",
"Symmetric Norm Inequalities And Positive Semi-Definite Block-Matrices"
] | [
"Antoine Mhanna [email protected] \nDept of Mathematics\nLebanese University\nHadath, BeirutLebanon\n"
] | [
"Dept of Mathematics\nLebanese University\nHadath, BeirutLebanon"
] | [] | For positive semi-definite block-matrix M, we say that M is P.S.D. and we writeThe focus is on studying the consequences of a decomposition lemma due to C. Bourrin and the main result is extending the class of P.S.D. matrices M written by blocks of same size that satisfies the inequality:M ≤ A + B for all symmetric norms. | null | [
"https://arxiv.org/pdf/1508.03754v3.pdf"
] | 119,664,188 | 1508.03754 | 01e4984fc357c0622c60aeffe4d993674025d459 |
Symmetric Norm Inequalities And Positive Semi-Definite Block-Matrices
14 Sep 2015
Antoine Mhanna [email protected]
Dept of Mathematics
Lebanese University
Hadath, BeirutLebanon
Symmetric Norm Inequalities And Positive Semi-Definite Block-Matrices
14 Sep 2015arXiv:1508.03754v3 [math.FA]Matrix AnalysisHermitian matricessymmetric norms
For positive semi-definite block-matrix M, we say that M is P.S.D. and we writeThe focus is on studying the consequences of a decomposition lemma due to C. Bourrin and the main result is extending the class of P.S.D. matrices M written by blocks of same size that satisfies the inequality:M ≤ A + B for all symmetric norms.
Introduction
Let A be an n×n matrix and F an m×m matrix, (m > n) written by blocks such that A is a diagonal block and all entries other than those of A are zeros, then the two matrices have the same singular values and for all unitarily invariant norms A = F = A⊕0 , we say then that the symmetric norm on M m induces a symmetric norm on M n , so for square matrices we may assume that our norms are defined on all spaces M n , n ≥ 1.
The spectral norm is denoted by . s , the Frobenius norm by . (2) , and the Ky Fan k−norms by . k . Let M + n denote the set of positive and semi-definite part of the space of n × n complex matrices and M be any positive semi-definite block-matrices; that is,
M = A X X * B ∈ M + n+m , with A ∈ M + n , B ∈ M + m .
2 Decomposition of block-matrices
A X X * B = U A 0 0 0 U * + V 0 0 0 B V * for some unitaries U, V ∈ M n+m .
Proof. Factorize the positive matrix as a square of positive matrices:
A X X * B = C Y Y * D . C Y Y * D
we verify that the right hand side can be written as T * T + S * S so :
C Y Y * D . C Y Y * D = C 0 Y * 0 T * . C Y 0 0 T + 0 Y 0 D S * . 0 0 Y * D S . Since T T * = CC + Y Y * 0 0 0 = A 0 0 0 , SS * = 0 0 0 Y * Y + DD = 0 0 0 B and
AA * is unitarily congruent to A * A for any square matrix A, the lemma follows.
Remark 1. As a consequence of this lemma we have:
M ≤ A + B
for all symmetric norms.
Equations involving unitary matrices are called unitary orbits representations.
Recall that if A ∈ M n , R(A) = A + A * 2 and I(A) = A − A * 2i .
Corollary 2.1. For every matrix in M + 2n written in blocks of the same size, we have the decomposition:
A X X * B = U A+B 2 − R(X) 0 0 0 U * + V 0 0 0 A+B 2 + R(X) V * for some unitaries U, V ∈ M 2n . Proof. Let J = 1 √ 2 I −I I I
where I is the identity of M n , J is a unitary matrix, and we have:
J A X X * B J * = A+B 2 − R(X) A−B 2 + X * −X 2 A−B 2 − X−X * 2 A+B 2 + R(X) N
Now we factorize N as a square of positive matrices:
A X X * B = J * L M M * F . L M M * F J
and let:
δ = J * L M M * F = 1 √ 2 L + M * M + F M * − L F − M ψ = L M M * F J = 1 √ 2 L + M M − L F + M * F − M *
A direct computation shows that:
δ.ψ = 1 2 (L+M * )(L+M )+(M +F )(F +M * ) (L+M * )(M −L)+(M +F )(F −M * ) (M * −L)(L+M )+(F −M )(F +M * ) (M * −L)(M −L)+(F −M )(F −M * ) = Γ * Γ + Φ * Φ (1) where: Γ = 1 √ 2 L + M M − L 0 0 , and Φ = 1 √ 2 0 0 F + M * F − M * toA X X * B = U A+B 2 + I(X) 0 0 0 U * + V 0 0 0 A+B 2 − I(X) V * for some unitaries U, V ∈ M n+m .
Proof. The proof is similar to Corollary 2.1, we have:
J 1 A X X * B J * 1 = A iX −iX * B where J 1 = I 0 0 −iI , and K = JJ 1 A X X * B J * 1 J * = A+B 2 + I(X) * where T T * = A+B 2 + I(X) 0 0 0 and SS * = 0 0 0 A+B 2 − I(X)
finally the congruence property completes the proof.
The existence of unitaries U and V in the decomposition process need not to be unique as one can take the special case; that is, M any diagonal matrix with diagonal entries equals a nonnegative number k, explicitly M = kI = U k 2
I U * + V k 2 I V *
for any U and V unitaries.
A X X * B ≤ 1 2 U A + B + |X − X * | 0 0 0 U * + V 0 0 0 A + B + |X − X * | V * for some unitaries U, V ∈ M n+m .
Proof. This a consequence of the fact that I(X) ≤ |I(X)|.
Symmetric Norms and Inequalities
In [1] they found that if X is hermitian then
M ≤ A + B(2)A X X * B > 0 ⇐⇒ A ≥ XB −1 X * Proof. Write A X X * B = I XB −1 0 I A − XB −1 X * 0 0 B I 0 XB −1 I
where I is the identity matrix, and that complete the proof since for any matrix A, Theorem 3.3. Given A X X * B a matrix in M + 2n written in blocks of same size:
A ≥ 0 ⇐⇒ X * AX ≥ 0, ∀X.1. If A X X * 0 is positive semi-definite, I(X) > 0 or I(X) < 0, then there exist a matrix Y such that M = A Y Y * 0 is positive semi-definite and: A Y Y * 0 > A (4)
for all symmetric norms.
If
0 X X * B is positive semi-definite, I(X) > 0 or I(X) < 0 then there exist a matrix Y such that M = 0 Y Y * B is positive semi-definite and: 0 Y Y * B > B (5)
The same result holds if we replaced
I(X) by R(X) because A iX −iX * B is unitarily congruent to A X X * B .
Proof. Without loss of generality we can consider I(X) > 0 cause A X X * B and
A −X −X * B
are unitarily congruent, we will show the first statement as the second one has a similar proof, from Corollary 2.2 we have:
A X X * 0 ≥ U A 2 0 0 0 U * + U I(X) 0 0 0 U * + V 0 0 0 A 2 V * Since A X X * 0 is congruent to L = A lX lX * 0 for any l ∈ C, L is P.S.D. A is a fixed matrix, we have U A 2 0 0 0 U * + V 0 0 0 A 2 V * k = β A k for some β ≤ 1 finally we set Y = lX where l ∈ R is large enough to have M k > A k , ∀k thus M > A for all symmetric norms.
Notice that there exist a permutation matrix P such that P A X X * 0 P −1 = 0 X * X A and since I(X) > 0 if and only if I(X * ) < 0, the two assertions of Theorem 3.3
are equivalent up to a permutation similarity. Finally we get:
Corollary 3.1. If M = A X X * 0 , A aTheorem 3.4. If X = 0 and B = 0, A ≥ 0, the matrix M = A X X * 0 cannot be positive semi-definite.
Proof. Suppose the converse, so M = A X X * 0 is positive semi-definite, without loss of generality the only case we need to discuss is when R(X) has positive and negative eigenvalues, by Corollary 2.1 we can write:
M = U A 2 − R(X) 0 0 0 U * + V 0 0 0 A 2 + R(X) V * for some unitaries U, V ∈ M 2n . Now if R(X) has −α the smallest negative eigenvalue R(X) + (α + ǫ)I > 0 consequently the matrix H = U A 2 − R(X) 0 0 0 U * + V 0 0 0 A 2 + (α + ǫ)I + R(X) + (α + ǫ)I V * (6) = A + 2(α + ǫ)I X + (α + ǫ)I (X + (α + ǫ)I) * 0 (7)
is positive semi-definite with R(Y ) > 0, where Y = X + (α + ǫ)I, by Corollary 3.1 this is a contradiction.
A natural question would be how many are the nontrivial P.S.D.matrices written by blocks ? The following lemma will show us how to construct some of them.
R if f (x) ≥ g(x)
for all x such that x s = 1 then f (x) ≥ g(x) for any x ∈ C n . So let us
set K = max x s =1
g(x), and L = min
x s =1
f (x) since g(x) and f (x) are continuous functions and {x; x s = 1} is compact, there exist a vector w respectively v such that K = g(w)
,
respectively L = f (v). Now choose t ≥ 1 such that tf (v) > g(w) t , to obtain x * (tA)x ≥ v * (tA)v > w * X(tB) −1 X * w ≥ x * X(tB) −1 X * x
for all x such that x s = 1, thus x * (tA)x > x * X(tB) −1 X * x for any x ∈ C n which completes the proof.
Theorem 3.5. Let A = diag(λ 1 , · · · , λ n ), B = diag(ν 1 , · · · , ν n ) and M = A X X * B a
given positive semi-definite matrix. If X * commute with A and X * X equals a diagonal matrix, then
M ≤ A + B
for all symmetric norms. The same inequality holds if X commute with B and XX * is diagonal.
Proof. It suffices to prove the inequality for the Ky Fan k−norms k = 1, · · · , n, let P = 0 I n I n 0 where I n is the identity matrix of order n, since B X * X A = P A X X * B P −1 and A X X * B have same singular values, we will discuss only the first case; that is, when X * commute with A and X * X is diagonal, as the second case will follows. Let Equivalently the eigenvalues are all the solutions of the n equations:
D := X * X = d 1 0 ··· 0 0 d 2 ··· 0 . . . . . . . . . . . .1) (λ 1 − µ)(ν 1 − µ) − d 1 = 0 2) (λ 2 − µ)(ν 2 − µ) − d 2 = 0 3) (λ 3 − µ)(ν 3 − µ) − d 3 = 0 . . . . . . i) (λ i − µ)(ν i − µ) − d i = 0 . . . . . . n) (λ n − µ)(ν n − µ) − d n = 0
Each equation is of 2 nd degree, if we denote by a i and b i the two solutions of the i th equation we deduce that:
a 1 + b 1 = λ 1 + ν 1 a 2 + b 2 = λ 2 + ν 2 . . . a n + b n = λ n + ν n But A + B = λ 1 + ν 1 0 · · · 0 0 λ 2 + ν 2 · · · 0 . . . . . . . . . . . . 0 0 · · · λ n + ν n
and each diagonal entry of A + B is equal the sum of two nonegative eigenvalues of M , thus we have necessarily: M k ≤ A + B k for all k = 1, · · · , n which completes the proof.
Example 3.1.
Let
M x = x 0 i 2 0 0 99 100 0 − i 2 − i 2 0 99 100 0 0 i 2 0 1 2 If 3 10 ≤ x ≤ 1 2
, M x is positive definite and we have: And the (8) inequality follows from Theorem 3.5.
M x ≤ A + B(8
Let us study the commutation condition in Theorem 3.5. First notice that any square matrix X = (x ij ) ∈ M n will commute with A = diag(a 1 , · · · , a n ) if and only if :
Y ′ =
x 1,1 a 1 x 1,2 a 2 · · · x 1,n a n x 2,1 a 1 x 2,2 a 2 · · · x 2,n a n . . . . . . . . . . . .
x n,1 a 1 x n,2 a 2 · · · x n,n a n
= x 1,1 a 1 x 1,2 a 1 · · · x 1,n a 1 x 2,1 a 2 x 2,2 a 2 · · · x 2,n a 2 . . . . . . . . . . . .
x n,1 a n x n,2 a n · · · x n,n a n
= Y An (i, j) entry of Y ′ is equal to that of Y if and only if x i,j a j = x i,j a i , i.e. either a i = a j or x i,j = 0.
Corollary 3.2. Let A = diag(λ 1 , · · · , λ n ), B = diag(ν 1 , · · · , ν n ) and M = A X X * B a given positive semi-definite matrix. If X * commute with A, or X commute with B, then M ≤ A + B for all symmetric norms.
Proof. As in Theorem 3.5, we will assume without loss of generality that X * commute with A, as the other case is similar. If X * is diagonal the result follows from Theorem 3.5, suppose there is an off diagonal entry x i,j of X * different from 0, from the commutation condition we have a i = a j and the same goes for all such entries, of course if AX = XA then P AXP −1 = P XAP −1 = P AP −1 P XP −1 = P XP −1 P AP −1 = P XAP −1
Take P to be the permutation matrix that will order the same diagonal entries of A in a one diagonal block and keeps the matrix B the same, since M is Hermitian so is P M P −1 because we can consider the permutation matrix as a product of transposition matrices P 1 , · · · , P n wich are orthogonal; in other words P M P −1 = P 1 P 2 · · · P n M P T n · · · P T 2 P T 1 .
Consequently P T = P −1 for any permutation matrix and M = P M P T for all symmetric norms. If H = P M P T , D := P X and X i is some i × i extracted submatrix of X * , we will have the block written matrix
H = P AP T P X X * P T B = aI i O j ··· Os O i bI j ··· Os . . . . . . . . . . . . O i O j ··· rIs X * i O i ··· O i O j X * j ··· O j . . . . . . . . . . . . Os Os ··· X * s X i O j ··· Os O i X j ··· Os . . . . . . . . . . . . O i O j ··· Xs ν 1 0 ··· 0 0 ν 2 ··· 0 . . . . . . . . . . . . 0 0 ··· νn
where we denoted the diagonal matrix of order i whose diagonal entries are equal to a by aI i and the zero block of order i by O i . Let us calculate the roots of the characteristic polynomial of H; that is, the roots of
det (a−λ)I i O j ··· Os O i (b−λ)I j ··· Os . . . . . . . . . . . . O i O j ··· (r−λ)Is ν 1 −λ 0 ··· 0 0 ν 2 −λ ··· 0 . . . . . . . . . . . . 0 0 ··· νn−λ − D * D = 0
we translate this to a system of blocks, while each eigenvalue of H, which is the same as its singular value, will verify one of the following equations:
1) det (a − λ)I i ) ν 1 −λ ··· 0 . . . . . . . . . 0 ··· ν i −λ − X * i X i = 0 2) det (b − λ)I j ) ν i+1 −λ ··· 0 . . . . . . . . . 0 ··· ν i+j −λ − X * j X j = 0 . . . . . . c) det (r − λ)I s ) ν n−s −λ ··· 0 . . . . . . . . . 0 ··· νn−λ − X * s X s = 0 (T )
where c is the number of diagonal blocks we have. Let us have a closer look to any of the equations above, without loss of generality we will take the first one, the same will hold for the others, notice that all eigenvalues λ are nonnegative and we have
M 1 = aI i X * i X i ν 1 ··· 0 . . . . . . . . . 0 ··· ν i = C 1 X * i X * i K 1
is positive semi-definite because it's eigenvalues are a subset of those of M. The key idea is that for this matrix C 1 + K 1 = C 1 + K 1 for all symmetric norms. where
C 1 = aI i and K = ν 1 ··· 0 . . . . . . . . . 0 ··· ν i
. Now back to the system (T ) we associate like we did to M 1 each equation whose number is i to a positive semi-definite matrix M i to obtain by Remark 1
M 1 k ≤ aI i + ν 1 ··· 0 . . . . . . . . . 0 ··· ν i k = aI i k + ν 1 ··· 0 . . . . . . . . . 0 ··· ν i k M 2 k ≤ bI j + ν i+1 ··· 0 . . . . . . . . . 0 ··· ν i+j k = bI j k + ν i+1 ··· 0 . . . . . . . . . 0 ··· ν i+j k . . . . . .A X X * B = Q A X X * B Q * = D o T T * G o ≤ D o + G o(13)
for all symmetric norms. Proof. We consider first that the normal matrix X * has all of its eigenvalues distinct, by Theorem ?? and the normality condition, there exist a unitary matrix U such that U * AU and U * X * U are both diagonal. A direct computation shows that:
U * 0 0 U * A X X * B U 0 0 U = U * AU U * XU U * X * U U * BU = G.
Now U * XU also commute with U * BU, since U * XU is diagonal and all of its diagonal entries are distinct by Remark ?? U * BU must be also diagonal, applying Theorem 3.5
to the matrix G yields to:
A X X * B = G ≤ U * AU + U * BU = A + B ,
for all symmetric norms. The inequality holds for any X normal by a continuity argument.
Lemma 3.2. Let N = a 1 0 ··· 0 0 a 2 ··· 0 . . . . . . . . . . . . 0 0 ··· an D D * b 1 0 ··· 0 0 b 2 ··· 0 . . . . . . . . . . . . 0 0 ··· bn
where a 1 , · · · , a n respectively b 1 , · · · , b n are nonnegative respectively negative real num- x 1 + y 1 = a 1 + b 1 ≥ 0
bers, A = a 1 0 ··· 0 0 a 2 ··· 0 . . . . . . . . . . . . 0 0 ··· an , B = b 1 0 ··· 0 0 b 2 ··· 0 . . . . . . . . . . . .
x 2 + y 2 = a 2 + b 2 ≥ 0 . . . . . .
x n + y n = a n + b n ≥ 0
x 1 y 1 = a 1 b 1 − d 1 < 0 x 2 y 2 = a 2 b 2 − d 2 < 0 . . . . . .
x n y n = a n b n − d n < 0
This implies that each equation of (S) has one negative and one positive solution, their sum is positive, thus the positive root is bigger or equal than the negative one. Since 2. 16 + y 2 + 1 = N 2 (2) > A + B 2 (2) = 4(3 + y) + y 2 + 1
Theorem 3. 2 .
2Let M = A B C D be any square matrix written by blocks of same size, if AC = CA then det(M ) = det(AD − CB) Proof. Suppose first that A is invertible, let us write M as we find that: Z = A, V = C, E = A −1 B, F = D − CA −1 B taking the determinant on each side of (3) we get: det(M ) = det(A(D − CA −1 B)) = det(AD − CB) the result follows by a continuity argument since the Determinant function is a continuous function. Given the matrix M = A X X * 0 a matrix in M + 2n written by blocks of same size, we know that it M is not P.S.D., to see this notice that all the 2 × 2 extracted principle submatrices of M are P.S.D if and only if X = 0 and A is positive semi-definite. Even if a proof of this exists and would take two lines, it is quite nice to see a different constructive proof, a direct consequence of Lemma 2.1.
positive semi-definite matrix, and we have one of the following conditions: 1. R(X) > 0 2. R(X) < 0 3. I(X) > 0 4. I(X) < 0 Then M can't be positive semi-definite. Proof. By Remark 1 any positive semi-definite matrix M written in blocks must satisfy M ≤ A + B for all symmetric norms which is not the case of the matrix M constructed in Theorem 3.3.
Lemma 3. 1 .
1Let A and B be any n × n positive definite matrices, then there exist an integer t ≥ 1 such that the matrix F t = tA X X * tB is positive definite.Proof. Recall from Theorem 3.1 that F 1 is positive definite if and only if A > XB −1 X * , which is equivalent to x * Ax > x * XB −1 X * x for all x ∈ C n . Set f (x) := x * Ax and g(x) := x * XB −1 X * x and let us suppose, to the contrary, that there exist a vector z such that f (z) ≤ g(z) since f (x) and g(x) are homogeneous functions of degre d = 2 over
, as X * commute with A, from Theorem 3.2 we conclude that the eigenvalues of A X X * B are the roots ofdet((A − µI n )(B − µI n ) − D) = 0
k, but the order of the entries of B are arbitrary chosen, thus from Theorem 3.5 M k ≤ A + B k for all k = 1, · · · , n and that completes the proof.
Corollary 3. 3 .
3Let M = A X X * B be a positive semi-definite matrix written by blocks. There exist a unitary V and a unitary U such that A X X * B ≤ U AU * + V BV * := A + B for all symmetric norms. Proof. Let U and V be two unitary matrix such that U AU * = D o and V BV * = G o where D o and G o are two diagonal matrices having the same ordering o, of eigenvalues with respect to their indexes i.e., if λ n ≤ · · · ≤ λ 1 are the diagonal entries of D o , and ν n ≤ · · · ≤ ν 1 are those of G o , then if λ i is in the (j, j) position then ν i will be also. Consequently U AU * + V BV * = D o + G o = D o + G o = A + B , for all the Ky-Fan k−norms and thus for all symmetric norms. To complete the proof notice that if T = U XV * and Q is the unitary matrix U 0 0 V , by Remark 1
and D is any diagonal matrix, then norN neither −N is positive semi-definite. Set (d 1 , · · · , d n ) as the diagonal entries of D * D, if a i + b i ≥ 0 and a i b i − d i < 0 for all i ≤ n, then N > A + B . for all symmetric normsProof. The diagonal of N has negative and positive numbers, thus nor N neither −N is positive semi-definite, now any two diagonal matrices will commute, in particular D * and A, by applying Theorem 3.2 we get that the eigenvalues of N are the roots ofdet((A − µI n )(B − µI n ) − D * D) i − µ)(b i − µ) − d n = 0 . . . . . .n) (a n − µ)(b n − µ) − d denote by x i and y i the two solutions of the i th equation then:
,..
summing over indexes we see that N k > A + B k for k = 1, · · · , n which yields to N > A + B for all symmetric norms It seems easy to construct examples of non P.S.D matrices N written in blocks such that N s > A + B s , let us have a look of such inequality for P.S.D. matrices. Since the eigenvalues of C are all positive with λ 1 ≈ 3.008, λ 2 ≈ 1.7, λ 3 ≈ 0.9, λ 4 ≈ 0.089, C is positive definite and we verify that 3.008 ≈ C s > A + B s The eigenvalues of N y are the numbers: λ 1 = 4, λ 2 = 1, λ 3 = y, λ 4 = 0, thus if y ≥ 0, N y is positive semi-definite and for all y such that 0 ≤ y < 1 we have 1. 4 = N y s > A + B s = 3
Lemma 2.1. For every matrix M in M + n+m written in blocks, we have the decomposition:
finish notice that for any square matrix A, A * A is unitarily congruent to AA * and, ΓΓ * , ΦΦ * have the required form.The previous corollary implies that A+B
2 ≥ R(X) and A+B
2
≥ −R(X).
Corollary 2.2. For every matrix in M +
2n written in blocks of the same size, we have the
decomposition:
if X is normal, X * commute with A and X commute with B, then we have M ≤ A + B for all symmetric norms.Theorem 3.6. Let M =
A X
X * B
≥ 0,
On a decomposition lemma for positive semidefinite block-matrices. J C Bourin, E Y Lee, M Lin, Linear Algebra and its Applications. 437J. C. Bourin, E. Y. Lee, and M. Lin, On a decomposition lemma for positive semi- definite block-matrices, Linear Algebra and its Applications 437, pp.1906-1912, (2012).
Norm and anti-norm inequalities for positive semi-definite matrices. J C Bourin, F Hiai, Internat.J.Math. 63J. C. Bourin, F. Hiai, Norm and anti-norm inequalities for positive semi-definite matrices , Internat.J.Math.63, pp.1121-1138, (2011).
| [] |
[
"Toward Training at ImageNet Scale with Differential Privacy",
"Toward Training at ImageNet Scale with Differential Privacy"
] | [
"Alexey Kurakin \nGoogle Research\n\n",
"Shuang Song \nGoogle Research\n\n",
"Steve Chien \nGoogle Research\n\n",
"Roxana Geambasu \nGoogle Research\n\n\nColumbia University\n\n",
"Andreas Terzis \nGoogle Research\n\n",
"Abhradeep Thakurta \nGoogle Research\n\n"
] | [
"Google Research\n",
"Google Research\n",
"Google Research\n",
"Google Research\n",
"Columbia University\n",
"Google Research\n",
"Google Research\n"
] | [] | Differential privacy (DP) is the de facto standard for training machine learning (ML) models, including neural networks, while ensuring the privacy of individual examples in the training set. Despite a rich literature on how to train ML models with differential privacy, it remains extremely challenging to train real-life, large neural networks with both reasonable accuracy and privacy.We set out to investigate how to do this, using ImageNet image classification as a poster example of an ML task that is very challenging to resolve accurately with DP right now. This paper shares initial lessons from our effort, in the hope that it will inspire and inform other researchers to explore DP training at scale. We show approaches that help make DP training faster, as well as model types and settings of the training process that tend to work better in the DP setting. Combined, the methods we discuss let us train a Resnet-18 with DP to 47.9% accuracy and privacy parameters ε = 10, δ = 10 −6 . This is a significant improvement over "naive" DP training of ImageNet models, but a far cry from the 75% accuracy that can be obtained by the same network without privacy. The model we use was pretrained on the Places365 data set as a starting point. We share our code at https://github. com/google-research/dp-imagenet, calling for others to build upon this new baseline to further improve DP at scale. | null | [
"https://arxiv.org/pdf/2201.12328v2.pdf"
] | 246,411,162 | 2201.12328 | d210e55bd1afab9eba52a604565d09933dab5ad3 |
Toward Training at ImageNet Scale with Differential Privacy
February 8, 2022
Alexey Kurakin
Google Research
Shuang Song
Google Research
Steve Chien
Google Research
Roxana Geambasu
Google Research
Columbia University
Andreas Terzis
Google Research
Abhradeep Thakurta
Google Research
Toward Training at ImageNet Scale with Differential Privacy
February 8, 2022
Differential privacy (DP) is the de facto standard for training machine learning (ML) models, including neural networks, while ensuring the privacy of individual examples in the training set. Despite a rich literature on how to train ML models with differential privacy, it remains extremely challenging to train real-life, large neural networks with both reasonable accuracy and privacy.We set out to investigate how to do this, using ImageNet image classification as a poster example of an ML task that is very challenging to resolve accurately with DP right now. This paper shares initial lessons from our effort, in the hope that it will inspire and inform other researchers to explore DP training at scale. We show approaches that help make DP training faster, as well as model types and settings of the training process that tend to work better in the DP setting. Combined, the methods we discuss let us train a Resnet-18 with DP to 47.9% accuracy and privacy parameters ε = 10, δ = 10 −6 . This is a significant improvement over "naive" DP training of ImageNet models, but a far cry from the 75% accuracy that can be obtained by the same network without privacy. The model we use was pretrained on the Places365 data set as a starting point. We share our code at https://github. com/google-research/dp-imagenet, calling for others to build upon this new baseline to further improve DP at scale.
Introduction
Machine learning (ML) models are becoming increasingly valuable for improved performance across a variety of consumer products, from recommendations to automatic image classification and labeling. However, despite aggregating large amounts of data, it is possible for models to encode -and thus, in theory, "reveal" -characteristics of individual entries from the training set. For example, experiments in controlled settings have shown that language models trained using email datasets may sometimes encode sensitive information included in the training data [9] and may have the potential to reveal the presence of a particular user's data in the training set [36]. As such, it is important to prevent the encoding of such characteristics from individual training entries.
The standard rigorous solution to this problem is differential privacy (DP) [15,14]. DP randomizes a computation over a dataset (such as training of an ML model) to bound the "leakage" of individual entries in the dataset through the output of the computation (the model). The randomness ensures that the model is almost as likely to be output independent of the presence or absence of any individual entry in the training set. This "almost as likely" is quantified by a privacy parameter, ε > 0. If ε is small, the model cannot encode -and thus cannot "reveal" -much information about any individual entry.
Despite significant amounts of research in DP and machine learning, both on the theoretical front [11,29,8,37,7,31,45,35,42,19,6,39] and on the empirical front [25,2,43] over the past decade, training ML models with DP remains challenging in practice, which limits its adoption. First, DP often impacts utility, such as model accuracy. The impact on accuracy comes from the randomness introduced into the computation to eliminate memorization of individual entries in some models, which can help accuracy. Sometimes, the utility loss resulting from DP training is acceptable (or even could represents an improvement), but more often it is dramatically negative, making the resulting model useless. For example, when you simply switch from SGD to DP-SGD [37, 8,2] to train a Resnet-18 or Resnet-50 on ImageNet, you get near-zero accuracy for any reasonable value of ε. Second, existing implementations of DP-SGD are inefficient [40]. Indeed, DP-SGD involves some expensive operations, like computing per-example gradients instead of per-minibatch gradients as non-private SGD. This invalidates certain optimizations that are present in typical ML frameworks. In production, this overhead can be prohibitive, especially if the model needs to be retrained often. These two challenges have been preventing, on one hand, industry from adopting DP more widely and on the other hand, the research community from making progress on readying DP for wider adoption. For example, most DP research papers evaluate DP algorithms on very small datasets, such as MNIST, CIFAR-10, or UCI. This paper shares initial results from our ongoing effort to train a large image classification model on ImageNet using differential privacy while maintaining high accuracy and minimizing computational cost. We split the paper in two parts: (1) the main body, which communicates the main lessons we have learned to date and (2) a hefty appendix that includes significantly more data from our experimentation. Our goal is to inspire and inform other researchers who want to explore DP training at scale, including settings that work (our lessons) and those that may not (also covered in the appendix).
A first lesson we communicate (Section 2.2) is that a substantial amount of exploration -of architectures, techniques, and hyperparameters -is needed to discover a training setting that performs well with DP, and this exploration is impaired by the significant overheads of DP SGD implementations in most ML frameworks. We recommend JAX as a good ML framework to perform such exploration in, because it is surprisingly effective at automatically optimizing otherwise very expensive DP operations. A second lesson comes from our initial exploration of a few training settings for ImageNet, which combined let us DP-train a Resnet-18 to 47.9% accuracy and privacy parameter ε = 10 (Section 3), when pre-trained on Places365 data set. This marks a good improvement compared to "naive" DP training, which achieves few percent accuracy for the same privacy parameter, but it remains far from the 75% accuracy that can be obtained by the same network without privacy.
We share our code at https://github.com/google-research/dp-imagenet. By sharing an early snapshot of the lessons, results, and code from our ongoing project, we hope to energize others to work on improving DP for ambitious tasks such as Ima-geNet, as a proxy for challenging production-scale tasks. Our 47.9%-accuracy/ε = 10 model should serve as a baseline to further improve upon to make DP training closer to practical at scale.
Background
Privacy Notions and DP-SGD
ML Privacy Attacks. ML models are statistical aggregates of their underlying training data. Despite this, they have been shown to contain -and therefore, in theory, expose to attacks -information about the specific examples used to train them. For instance trained ML models, and even their predictions, have been shown to enable membership inference attacks [5,17,36] (e.g., an learns that a particular user was in the training set for a disease detection model), and reconstruction attacks [9,13,16,10] (e.g., an attacker reconstructs social security numbers from a language model).
Differential Privacy (DP). DP randomizes a computation over a dataset (such as training of an ML model) to bound the "leakage" of individual entries in the dataset through the output of the computation (the model). Intuitively, enforcing DP on the training procedure for an ML model ensures that the model is almost as likely to be output independent of the presence or absence of any individual entry in the training set; hence, the model cannot encode, and thus leak, much information about any individual entry. DP is known to address the preceding ML privacy attacks [36, 16,9,26]. At a high level, membership and reconstruction attacks work by finding data points that make the observed model more likely: if those points were in the training set, the likelihood of the observed output increases. DP prevents these attacks, as no specific data point can drastically increase the likelihood of the model outputted by the training procedure.
We state the formal definition of DP here.
Definition 2.1 (Differential privacy [15,14]). A randomized algorithm A is (ε, δ)differentially private if, for any pair of datasets D and D differing in exactly one data point (called neighboring datasets) i.e., one data point is added or removed, and for all events S in the output range of A, we have
Pr[A(D) ∈ S] ≤ e ε · Pr[A(D ) ∈ S] + δ,
where the probability is over the randomness of A.
For meaningful privacy guarantees, ε is assumed to be a small constant, and δ 1/n where n is the size of D. DP-SGD. DP-SGD is currently the most widely-used differentially private machine learning algorithm in practice. Along with its practical success, it also provides optimal privacy/utility trade-offs analytically [8,7]. The full algorithm is described in Algorithm 1.
Algorithm 1 Differentially private stochastic gradient descent (DP-SGD)
Require: Data set D = {d 1 , · · · , d n } with d i ∈ D, loss function: :
R p × D → R,
clipping norm: C, number of iterations: T , noise multiplier: σ 1: Randomly initialize θ 0 . 2: for t = 0, . . . , T − 1 do
g t ← d∈Bt clip (∇ (θ t ; d)), where clip(v) = v · min 1, C v 2 .
5:
θ t+1 ← one step of first order optimization with gradient g t + N 0, (σC) 2 6: end for
7: return 1 T T t=1 θ t or θ T .
The privacy guarantee of DP-SGD comes from that of Gaussian mechanism with privacy amplification by subsampling and privacy composition. In Step 4, clipping upper-bounds the 2 norm of each gradient in the mini-batch to be C, and thus the 2 norm of g t changes by at most C when we add or remove one example from the minibatch B t . Suppose σ t is the noise multiplier needed to achieve ε t -differential privacy in step t when B t is arbitrarily chosen. If, instead, B t is formed by sampling k examples u.a.r. and i.i.d. from D, then privacy amplification by sampling [28,8,2,44] allows one to scale down the noise to σ t · (k/n) while achieving the same privacy guarantee. Similar subsampling schemes and analyses can be found in [18,20,31,51]. We can then accumulate the privacy loss in each step using composition, and obtain the final privacy cost of releasing all intermediate models
{θ t } T t=1 as ε = O √ T ε t .
In this paper we use DP-SGD privacy analysis to compute (ε, δ)-DP guarantees for our experiments. We report ε at δ = 10 −6 ≈ 1 DATASET SIZE in most of the results. However using smaller δ would only result in minor increase of ε, see Appendix I.
Automatically Fast DP Training with JAX
The previous section describes how DP-SGD works at a high level. To understand why it is slow, let us compare SGD with DP-SGD in more detail. One step of SGD works as follows: (0) Draw a batch of examples from the dataset. (1) (Forward Pass) Compute the loss function on the batch. (2) (Backward Pass) Then compute an average of the gradients with respect to the model parameters, and apply the average gradients to the model parameters using a chosen learning rate and optimizer. DP-SGD differs as follows. In each step, gradients are computed individually for each example instead of for the whole batch. Then, these per-example gradients are clipped to be within an 2 -norm, C, to control the sensitivity of the gradient average computation. After clipping, gradients of all examples are added together and Gaussian noise is added to the gradient vector, per the Gaussian mechanism described above. Finally, this clipped and noised gradient vector is applied to model parameters using the chosen learning rate and optimizer. This per-example gradient clipping slows down DP-SGD compared to SGD for the following reason. By default, ML frameworks -such as Tensorflow and PyTorchdo not directly compute per-example gradients and only provide aggregate, per-batch gradients. A naïve way to obtain per-example gradients is to loop over the examples in the batch, or to configure a batch size equal to one example. Unfortunately, this approach loses all the benefits of parallel batch computations and results in N times slowdown compared to regular SGD, where N is the batch size.
However, the lack of parallelization is not an inherent issue of per-example clipping. If one manually writes the mathematical expressions for per-example gradient clipping, then it can be done in a parallel way at a cost of approximately one Forward and two Backward Passes. The Forward Pass proceeds as usual. At a first Backward Pass, norms of per-example gradients are accumulated. At a second Backward Pass, normalized gradients are computed. Thus, beyond this "1.5x" overhead, the overhead of a DP-SGD instantiation within a specific ML framework is probably caused by a suboptimal implementation of these operations inside the ML framework, rather than some inherent inefficiency of DP-SGD. This idea is inspired by [21].
Some DP-SGD libraries [48] take the approach of manually implementing necessary per-example operations in an efficient way. This is valuable, however it can be time-consuming, error prone, and, importantly, difficult to evolve as new techniques for more effective DP-SGD training arise that may require changes within the underlying mathematical formulas. Thus we prefer to avoid such manual approach.
[40] first observed that using JAX, a high-performance computational library based on XLA one can do efficient auto-vectorization and just-in-time compilation of the mathematical expressions needed for evaluating the clipped mini-batch gradient. In this work too, we rely on JAX's autovectorization capabilities to compute the clipped minibatch gradients necessary for the execution of DP-SGD. While [40] empirically demonstrated the observation on CIFAR-10, we do the same on ImageNet. We confirm that JAX can do all the parallelization and optimization necessary for per-example gradient computations automatically. Table 1 shows the performance of DP and non-private training on MNIST and CI-FAR10 (corroborating the observation by [40]). For both datasets we are using small convnets which are commonly used in DP literature [2,34] These performance improvements can be quite important for research explorations with large networks, such as Resnet-18/Resnet-50 on ImageNet, whose runtimes without privacy are already orders of magnitude higher. Table 2 shows the runtime per training epoch in seconds for ImageNet on JAX when run on eight GPUs in parallel, for Resnet-18 and Resnet-50. Compared to non-private training, our JAX implementation is within 2x overhead, which we deem is fairly close to that rough theoretical best of "1.5x," and also within reasonable realm for DP exploration.
Effective Training Settings for DP
Armed with this relatively faster JAX engine for DP-SGD training, we set out to answer the following question: Of the multitude of potential training settings -including model architectures, hyperparameter values, batch sizes, known DP methods, with their own settings -which combination can lead to both good accuracy and privacy on ImageNet? We are still at the beginnings of answering this question, and many more explorations remain to be done, but in this section we summarize a few observations we have made so far. They are:
Obs. 1: Choice of model matters, in particular smaller models tend to work better.
Obs. 2:
More epochs is better than lower noise.
Obs. 3: Hyperparameter tuning makes a big difference.
Obs. 4: Extremely large batch size improves the privacy-utility tradeoff.
Obs. 5: Transfer learning from public data significantly boosts accuracy. Similar observations can be found scattered in prior DP literature, but to our knowledge they have not been all combined and evaluated together. Our contribution thus lies in evaluating these methods on the ambitious ImageNet classification task, where we find they yield a reasonable new baseline for DP-training at scale.
No
Obs. 1: Choice of model matters
A large variety of models have been developed for non-private training on ImageNet and the question of which model design (if any) is more suitable for DP training is still wide open. In our exploration, we wanted to start with not too small but also not too big models, so we chose the Resnet-v2 [23] model family. Within that, we chose to focus on Resnet-50 and Resnet-18: the former is generally considered as a sufficiently powerful model for this dataset; the latter is a smaller model for comparison. We applied known rules to adapt these models for DP training. In particular, any crossbatch procedure during training -such as through batch normalization -breaks privacy analysis of DP-SGD and must be replaced with a per-batch procedure. In Resnet-50 and Resnet-18, we thus replaced batch normalization with group normalization [46].
In evaluating Resnet-50 and Resnet-18, we observe that model size impacts accuracy differently than one might expect in non-private training. In non-private training, one rule of thumb is that complex learning tasks, such as image classification on Ima-geNet, tend to do better with larger, more expressive networks. In DP training, larger is not always better. To obtain a desired level of privacy, ε, DP-SGD calls for adding a fixed amount of noise, σ, to every parameter independently of the model size. Thus, applying DP-SGD to large models causes a larger shift in the parameters of large models compared to smaller ones, which can lead to a negative effect on accuracy. Table 3 compares accuracies of Resnet-18 and Resnet-50 without DP, while Table 4 compares them with DP. Without DP (Table 3), we show accuracies achieved at different epochs, with tuned learning rate (LR) and a fixed LR = 0.4. The larger Resnet-50 tends to do better across the board. With DP (Table 4), we show accuracies achieved at 10 epochs for various ε values with batch size 1024. Resnet-18 tends to outperform Resnet-50 when ε is low (and hence the privacy guarantees are better). On the other hand, when ε is very large, Resnet-50 starts to outperform Resnet-18 again, consistent with the non-private behavior. Further research is needed into whether simply using a smaller network, as we do here, is the right way to reduce dimensionality and the impact of DP noise on NN training. Another option might be to adapt σ to the statistics of individual parameters, as done in AdaClip for non-private training may not work for DP training.
We also investigated the effect of model structure and size in CIFAR-10. Appendix H describes the methodology and results. There too, in the low ε regime, smaller and simpler networks tend to have better accuracy than networks that do best in non-private learning.
There is some limited prior work studying relationship between model size and accuracy with DP training. We found that such results for image models (Figure 1 in [32], Table 20 in [43]) agree with ours -larger models tend to perform worse. However results for language models are quite opposite. As reported in [30] during DP fine-tuning of language models, larger model architecture tends to be beneficial. We hypothesize that this is related to the following facts. First of all, the gradient spectrum of language models differ significantly from that of vision models [3]. Second, (noisy) gradient descent tends to respect low-dimensional subspace induced by the gradients [38]. Thus finetuning of language models might be happening in lower dimensional subspace when larger model size is beneficial. Nevertheless, future research in this direction is needed to fully understand these differences.
Obs. 2: More epochs is better than lower noise
Privacy loss bound ε of a DP-SGD training depends on noise multiplier σ and number of training steps. Higher noise multiplier σ leads to lower ε, but typically worse accuracy. Longer training leads to higher ε as described in Section 2. At the same time, at least in non-private setting, training for longer typically helps to achieve higher accuracy. Thus if one to set a fixed ε and wants to maximize accuracy, a natural question arises: Is it better to train longer with higher noise multiplier or train for fewer epochs with lower noise? Figure 1 shows the accuracy achieved by Resnet-18 when training for increasing number of epochs while keeping the privacy loss ε constant by correspondingly decreasing the noise multiplier σ. In these experiments, training longer and with higher noise multiplier gives better accuracy than training for fewer epochs but with less noise. Nevertheless, accuracy reaches an upper bound somewhere between 40 and 70 epochs.
Obs. 3: Hyperparameter tuning makes a big difference
Training with DP-SGD requires setting extra hyper-parameters, such as the noise multiplier σ and the clipping norm C. Moreover, we observed that additional tuning of learning rate is typically necessary. The noise multiplier σ should be chosen to satisfy desired privacy budget. The clipping norm and learning rate should be tuned jointly for the best result. Appendix D gives results from our exploration of various values for learning rate and clipping norm. Here, we formulate a few observations we made experimentally on how to tune these parameters. We observed that there is typically some threshold valueC, such that the best private accuracy is obtained when the clipping norm is smaller thanC. Below theC threshold, a wide range of values of the clipping norm can be used as long as the learning rate is adjusted accordingly. Specifically, when the clipping norm is decreased k times, the learning rate should be increased k times to maintain similar accuracy. However, a larger learning rate (with smaller clipping norm) may lead to less stable training, thus we recommend keeping the clipping norm close toC.
Overall, we recommend the following procedure to tune the clipping norm and learning rate:
1. Find a good set of hyperparameters in a non-private case (for example from a public dataset). Let α pub be the good non-private learning rate.
2. Sweep over various values of the clipping norm C with fixed learning rate α pub and zero noise σ = 0. Find the smallestC for which DP model accuracy remains close to the non-private model accuracy.
3. Set the clipping norm toC and set the noise multiplier σ based on desired privacy budget. Run a learning rate sweep and find a good learning rateα. TypicallyC andα would be a reasonably good combination of hyperparameters for private training.
4. Further grid search in the vicinity ofC andα may bring additional improvement of the accuracy. 57.2% Privacy loss bound ε 9.8 · 10 8 6.1 · 10 7 3.5 · 10 6 6.7 · 10 4 Table 5: Resnet-18 accuracy and privacy (ε) on ImageNet with increasingly large batches. All experiments used noise multiplier σ = 0.001 · √ 8 · BatchSize 1024 and learning rate 16.
It is important to acknowledge that the preceding procedure for hyper-parameter tuning does not preserve privacy. We leave it for future work to develop such a private procedure, perhaps based on recent results [33].
Obs. 4: Large batch size improves the privacy-utility tradeoff
In theory, it is known that large minibatch size improves utility in DP-SGD, with fullbatch training giving the best outcome [41,6]. (Here, full-batch training refers to using the entire dataset as the batch for each iteration.) Recently, [4] corroborated this observation empirically on language models (e.g., BERT). Since, language models tend to have a different gradient profile (see Figure 5 in [3]) than vision models, it is nonobvious at the outset that a similar phenomenon would hold in the problem setup we consider. Via empirical evaluation, we demonstrate that the observations of [4] extend to vision models too.
In practice, for large datasets like ImageNet, full batch does not fit into GPU/TPU memory, and an attempt to distribute it will result in a prohibitively large number of accelerators. For example, Resnet-50 training on one accelerator will typically fit a batch size of 64 or 128. Thus, full-batch training on ImageNet would require more than 9000 accelerators in distributed data-parallel regime. Fortunately, large batch and full batch training can be simulated by accumulating gradients over several steps before applying them, a process called virtual steps and already implemented in some DP libraries, including Opacus. Table 5 shows that joint scaling of batch size, number of training epochs and noise multiplier can lead to decrease of ε while maintaining a similar accuracy level. A batch size of 16*1024 means to make one gradient step, we accumulate in memory 16 virtual steps each with 1024 examples. As we increase the batch size, we must keep the number of training steps the same to maintain the same accuracy. This results in larger number training epochs as we increase batch size. Table 5 is demonstrational: the ε values shown there are entirely unacceptable for privacy. In Figure 2, we plot the ε value at different batch size, when the noise / batch ratio and the number of steps are fixed. However, the same effect of large-batch size being advantageous can be witnessed for more reasonable values of ε, albeit with much lower accuracy outcomes. For example, with a batch size of 16 * 1024 and 10 epochs, the Resnet-18 will obtain an accuracy of 5.8% for ε = 2.2 · 10 5 . But with a much larger batch size of 1024 * 1024 (full-batch training) and a corresponding larger number of epochs of 640 (to preserve the total number of steps), the Resnet-18 will obtain a roughly equivalent 6.2% accuracy but for a (perhaps) more reasonable ε = 72. A recent work [12] studies full-batch DP-SGD algorithm as an instantiation of Langevin Diffusion, and show a tighter privacy/utility trade-offs as compared to the standard optimization viewpoint [8]. While the impact of this approach is yet to be realized in practice (as the improvement only holds for smooth and strongly convex losses), it is an important research direction to explore. For our paper, however, we remain in the realm of standard optimization view of DP-SGD, and instead combine large-batch training with transfer learning from public data, which offers a significant boost in accuracy.
Obs. 5: Transfer learning from public data significantly boosts accuracy
Pre-training on "public" data followed by DP fine-tuning on private data has previously been shown to improve accuracy on other benchmarks [43,49,30]. We confirm the same effect on ImageNet. A big question with transfer learning is what public data to use for a given task. We were surprised to see that reasonable choices are quite effective. We pre-trained our models on Places365 [50], another image classification dataset, before fine-tuning them with DP-SGD on ImageNet. Places365 contains 1.8M images of various scenes with 365 labels describing the scene. This dataset has only images of landscapes and buildings, not of animals as ImageNet, so it is quite different. Thus we consider the pair (Places365, ImageNet) as a reasonable proxy for real world setups of public and private datasets.
In our experiments, we first non-privately train Resnet-18 on Places365 to 54.96% accuracy (see Appendix F). Then we strip the last linear layer of the model and replace it with a randomly initialized one with the 1000 output classes of the ImageNet classification. Finally, we try different schemes of fine-tuning this model on ImageNet with DP-SGD.
In a first set of experiments, we compared finetuning with training from scratch and explored whether keeping some layers frozen helps to increase accuracy, see table 6. Unsurprisingly, finetuning is always better than training from scratch. These results Table 7: Accuracy of Resnet18 model which was pre-trained on Places365 and finetuned with DP-SGD on ImageNet. Each accuracy number was obtained by running learning rate sweep. Noise multiplier for each experiment was chosen in a way that in the end of the training ε = 10. Bold numbers highlight the best accuracy in each row, i.e. the best accuracy when number of training epochs and frozen blocks is fixed.
epochs 40 epochs 70 epochs
also suggest that freezing more layers tend to be better. It's possible that freezing less layers may work well when batch size and number of epochs are increased, however we didn't explore any further increase of number of epochs with finetuning in this paper.
In the second set of experiments we restrict ourself to freezing most of the network layers, and in this setup studied how batch size and number of epochs affect the final private accuracy, see table 7. These experiments reinforces our observation that longer training is generally better when ε is fixed. At the same we see that increase of batch size only helps to a certain point, after which it is actually hurting accuracy. As was mentioned earlier, increase of batch size generally should be done together with increasing number of epochs. Thus we hypothesize that number of training epochs in table 7 was not large enough to see benefits of batch sizes 256 * 1024 and 1024 * 1024.
Combining everything together
Combining all experiments and observations from previous sections, we managed to train a Resnet-18 on ImageNet to 47.9% accuracy with privacy budget ε = 10 by using the following hyperparameters:
• Start with a model pre-trained on Places365.
• Finetune this model on ImageNet with DP-SGD for 70 epochs with batch size 64 * 1024.
• Use cosine decay learning rate schedule with a warmup for 1 epochs. Maximum learning rate is 7.68.
• Use Nesterov momentum optimizer with standard decay 0.9.
• Use weight decay loss term with a coefficient 10 −4 .
• Set clipping norm C = 1 and choose noise multiplier σ to satisfy privacy budget.
Conclusion
In this work we make a first attempt to train a model on ImageNet dataset with differential privacy. To achieve this we study how various techniques and training parameters can affect accuracy of DP-SGD training. Combination of all of this findings enables us to train Resnet-18 on ImageNet to 47.9% accuracy with private budget ε = 10. While it may look relatively low compared to typical Resnet accuracy obtained in non-private training, one should keep in mind that "naive" Resnet training with DP-SGD on Im-ageNet will typically result in either only few percent accuracy or very high epsilon (which means lack of privacy).
[ [39] Shuang Song, Om Thakkar, and Abhradeep Thakurta. Characterizing private clipped gradient descent on convex generalized linear problems. arXiv preprint arXiv:2006.06783, 2020.
[40] Pranav Subramani, Nicholas Vadivelu, and Gautam Kamath. Enabling fast differentially private sgd via just-in-time compilation and vectorization. arXiv preprint arXiv:2010.09063, 2020.
A Resnet training without DP
We trained Resnet-18 and Resnet-50 models without differential privacy for various number of epochs, as shown table 3. We used the following training setup in these experiments:
• Nesterov momentum optimizer with momentum 0.9.
• Learning rate warmup for the first 5 epochs to maximum learning rate, followed by cosine decay to zero. In most of the non-private experiments we used the same maximum learning rate 0.4 which was originally tuned for 90 epochs Resnet-50 training. In some experiments we did additional tuning of the learning rate, which may result in a few percent accuracy boost.
• Weight decay loss term with coefficient 10 −4 . We tried different values of weight decay coefficient and found that 10 −4 is the best one.
• Total batch size was 1024 and training was done on 8 v100 GPUs or 8-core TPU.
B Resnet training with DP-SGD
For DP-SGD training we generally used similar setup as for non-private training (see appendix A): Nesterov momentum optimizer, learning rate warmup followed by cosine decay, weight decay 10 4 and training on 8 GPUs or 8 TPU cores. The main differences is that we used DP-SGD version of the optimizer [2], which requires to set two extra parameters: clipping norm C and noise multiplier σ. Another difference is that we have to re-tune learning rate compared to non-private training. Additionally when training for 10 epochs we typically used only one epoch to warmup learning rate.
It should be noted that in our implementation Gaussian noise is added independently per GPU or per TPU core. This means that total noise added to the gradients has standard deviation of σ = √ 8σ 1 where σ 1 is a standard deviation of noise added to single replica. That's why many tables in this report are showing σ/ √ 8 which is equal to standard deviation of per-replica noise.
We note that the ε of DP-SGD is computed through Rényi differential privacy (RDP) analysis, and the conversion between RDP to DP can be affected by the choice of RDP orders. In this paper, we use a function [1] in TF-Privacy to compute the DP ε. Due to the choice of RDP orders there, ε might be overestimated, especially in the low privacy / large ε regime.
C Experiments with fixed epsilon and different number of epochs
We did a series of experiments where we set ε to a desired value and then vary number of training epochs and other parameters. To run these experiments we need to compute noise multiplier σ based on desired ε, batch size, number of training epochs, etc... However privacy accountants from DP-SGD libraries typically provide a way to compute ε given noise multiplier, but not the other way around. To overcome this difficulty we used a routine from Tensorflow Privacy which computes ε based on noise multiplier and did a binary search with branching by geometric mean to find σ corresponding to the desired ε.
Plots with results of the experiments with fixed ε and different number of epochs are provided in the Figure 1 in the main body of the paper. Table 8 contains detailed numerical results of these experiments.
Num epochs ε = 4.57 ε = 71.5 ε = 9.7 · 10 6 10/
D Hyperparameter tuning for DP-SGD
While longer training can generally help increase utility, we found that training for 10 epochs is generally provide a reasonable idea of what we can expect in terms of accuracy. Thus to be able to do large hyperparameter sweeps within limited compute budget we restricted most of the hyperparameter sweeps to 10 epochs. For 10 epoch training of Resnet-18 we did a joint sweep of learning rate and clipping norm for various value of noise multiplier σ. For each value of σ we did one fine grained sweep of learning rate from {1, 2, 4, 8, 16, 40} and another coarse sweep where learning rate was increasing as a power of 10.
Results for non-private training with gradient clipping are provided in figure 3. As could be seen from the heatmap, model reaches highest accuracy for all C larger than 10. We can conclude that when C > 10 clipping is no longer happening and it becomes effectively equivalent to training without clipping.
Results with non zero noise multiplier σ are provided in figure 4. In all of these experiments the best accuracy is obtained when C < 10. This could be explained by the fact that standard deviation of Gaussian noise added to the gradients is computed as a product of C and σ. Thus if C is much large than the norm of the gradients then model updates will be dominated by the noise which will result in low utility. Another observation, that the subset of hyperparameters which correspond to the best utility lies on the curve learning rate · C = const. This could be explained by the fact that if C is decreased k times then learning rate has to be increased also k times in order to keep updates roughly the same. Nevertheless, we observed that extremely high learning rates together with extremely low C typically lead to lower accuracy and less stable training.
E Details of large batch training sweep
We explored various combinations of batch size and number of training epochs to study how large batch training can help improving utility with DP-SGD, see table 9. In these experiments we used the following set of hyperparameters:
• Total Gaussian noise added on every training step has standard deviation σ = 0.001 · √ 8 · BatchSize 1024 . • Clipping norm C was set to 1 in all experiments.
• Number of learning rate warmup epochs was set to 1 10 of total number of training epochs.
• Learning rate was set to 16.0 in all experiments. While typically, it's recommended to scale learning rate when increasing the batch size [22,47,24], it didn't seem to work well in this experiment. Specifically we have tried different learning rates from the set There is an input 7x7 convolution before the first block group and fully connected logits layer after the last block group. Each block group contains 2 residual blocks. Each residual blocks contains two convolutions with normalization layers and residual connection. Note that some of the residual blocks had projection and/or pooling operation on their residual connection.
G DP-SGD finetuning experiments
We performed finetuning experiments on Resnet18 while optionally freezing some of the layers. Due to the structure of Resnet18 (see figure 5) we decided that it's most convenient to freeze layers at a block group level. So we did experiments for the following configurations of frozen/trainable layers:
• None frozen block groups. Entire network is trainable.
• 1 frozen block groups. Input convolution and block group 1 are frozen.
• 2 frozen block groups. Input convolution, block groups 1 and 2 are frozen.
• 3 frozen block groups. Input convolution, block groups 1, 2 and 3 are frozen.
• 4 frozen block groups. Everything except the last fully connected layer is frozen. Only last fully connected layer (logits layer) is trainable.
In the first set of experiments we did initial learning rate sweep using batch size 1024 to identify feasible range of learning rates for finetuning, and then run a series of DP-SGD experiments with frozen layers and batch size 1024*1024, see table 11. As could be seen from the table, when number of frozen block groups is less than 4 accuracy stays pretty low. Thus we focused further experiments on cases of 4 or 3 frozen block groups.
For 3 and 4 frozen block groups we did non private finetuning to estimate upper bound of accuracy which is possible to achieve, see
H Experiments on CIFAR-10
We aim to examine the effect of model size on the privacy-utility tradeoff. We consider a class of neural networks which includes the one used for differentially private CIFAR-10 classification in a few previous work [34,27]. This class of networks, which we denote as simpleVGG, can be abstracted in the following way. A simpleVGG consists of multiple "blocks" and a linear layer, connected by max-pooling layers; each "block" consists of multiple convolution layers with the same number of channels. In the experiments, we will consider three sets of simpleVGG, in which we vary the number of convolution channels, number of layers per block, and the number of blocks, respectively. We fix the fully connected layer to size 128, and use tanh as the activation function. We use simpleVGG-32(2)-64(2)-128(2)-128 to denote a network with three blocks, each having two convolution layers with 32, 64 and 128 channels, and a fully connected layer of size 128 in the end. This is the neural network that has been used in previous work [34,27].
We train each neural network on CIFAR-10 for 100 epochs with batch size 500. We consider three different privacy levels for each setting: noise σ = 0.5, 1.5 and σ = 3.5, corresponding to ε = 47.41, 3.45, and 1.20 at δ = 10 −5 . For each setting, we keep the clip norm to be 1.0 and use a momentum optimizer with learning rate tuned from {0.001, 0.002, 0.005, . . . , 0.1, 0.2, 0.5}.
In Figure 6, we plot the final test accuracy of 7 different simpleVGGs under different privacy levels, each with the learning rate that achieves the best final test accuracy. We compare networks with different number of convolution channels, different number of convolution layers per block, and different number of blocks in the three subplots. Clearly, in the non-private setting, accuracy increases with the complexity of the network. As σ grows, the gap becomes smaller and the simpler network gets the best accuracy when we vary the number of channels and number of convolution layers per block. In most of the text of the paper we provide (ε, δ)-DP guarantees for fixed δ = 10 −6 , i.e. δ ≈ 1 DATASET SIZE . In practice, these privacy guarantees come from DP-SGD privacy accountant and could be recomputed for different values of δ. Figure 7 shows ε computed at different values of δ for our best training run, which achieves ≈ 48% final top-1 accuracy. 14 Epsilon Figure 7: Relationship between ε and δ for our best training run, which achieves ≈ 48% top-1 accuracy.
Figure 1 :
1Resnet-18 accuracy while training for different number of epochs but keeping constant ε. We tuned learning rate independently and reported the best accuracy for each combination of ε and number of epochs. Specific accuracy numbers for these plots are available in Appendix C.
Figure 2 :
2ε value as batch size increases, with noise / batch size ratio and number of steps kept fixed.
Figure 3 :
3Sweep of clipping norm and learning rate when σ = 0. Left plot correspond to coarse sweep of hyperparameters, right plot correspond to more fine-grained sweep of learning rate. Values in the table is model accuracy.
Figure 4 :
4Sweep of clipping norm and learning rate with various values of noise. Left column correspond to coarse sweep of both learning rate and clipping norm, right column correspond to more fine grained sweeps of learning rate. Values in the
Figure 5 :
5Structure of Resnet-v2-18. Resnet18 contains 4 block groups.
Figure 6 :
6Test accuracy for simpleVGGs on CIFAR-10. Averaged over 3 repeated runs. I Values of ε at different δ.
. Our results measure the time per training epoch in seconds, averaged over entire training (15 epochs forDataset
DP/No DP
JAX (ours)
Resnet-18 Resnet-50
ImageNet
DP
555.05
546.69
ImageNet
No DP
275.5
365.96
Table 2: Average time per training epoch in seconds for ImageNet on JAX. Exper-
iments run on eight V100 GPUs. It could be seen that DP-SGD training time for
Resnet-18 and Resnet-50 is similar despite the fact that Resnet-18 is a smaller model.
We didn't investigate this performance difference in details, but suspect that it might
be related to the fact that Resnet-50 using bottleneck residual blocks, while Resnet-18
is using non-bottleneck blocks.
MNIST, 90 epochs for CIFAR10). To minimize influence of data loader on the perfor-
mance we cached datasets in memory. For non-private training, JAX and PyTorch do
not show a consistent advantage over one another in these experiments, although they
both show an advantage over Tensorflow. For DP training, JAX is consistently faster
than both Opacus and TF-Privacy: 24 − 44% improvement per epoch over Opacus and
76 − 86% over TF-Privacy.
Table 3 :
3Comparison of Resnet-18 and Resnet-50 top-1 accuracy without DP, depend-
ing on number of training epochs.
[35]. Overall, though, our results point to the importance of adapting model architecture to DP training, since what tends to work Resnet-18 3.7% 6.9% 11.3% 45.7% 55.4% 56.0% 56.3% 56.4% Resnet-50 2.4% 5.0% 7.7% 44.3% 58.8% 57.8% 58.2% 58.6% Table 4: Comparison of the best Resnet-18 and Resnet-50 top-1 accuracies obtained at 10 epochs and batch size 1024, for various values of the privacy loss bound ε and fixed δ = 10 −6 . The different ε values are obtained by applying to the gradient vector noise from Gaussian distributions with different standard deviations, σ ∈ {0.56, 0.42, 0.28, 2.8 × 10 −2 , 2.8 × 10 −3 , 2.8 × 10 −4 , 2.8 × 10 −5 , 2.8 × 10 −6 }, respectively. Clipping norm is fixed C = 1. Each accuracy number for Resnet-18 is obtained by sweeping over learning rates in {1, 2, 4, 8, 16, 40}. Each accuracy number for Resnet-50 is obtained by sweeping over learning rates in {0.4, 0.8, 1.6, 4, 8, 16, 40}.DP
privacy loss bound ε
4.6
13.2
71
≈ 10 7
10 9
10 11
10 13
10 15
Table 6 :
6Comparison of finetuning vs training from scratch with differential privacy.First row is training from scratch, other rows are finetuning with different number of
frozen layers. Privacy budget was set to ε = 10 in all experiments. We used batch
size 4 * 1024 and each accuracy number was obtained by sweeping learning rate in
{0.016, 0.048, 0.16, 0.48, 1.6, 4.8, 16.0}. Bold number highlights the best accuracy in
each column.
Frozen
Batch
block
size →
4*1024 16*1024 64*1024 256*1024 1024*1024
groups Number of
epochs ↓
10
32.5%
39.6%
33.0%
18.6%
3.3%
3
40
38.9%
44.0%
44.9%
36.4%
17.0%
70
40.7%
45.0%
47.9%
41.7%
18.4%
10
33.5%
36.1%
37.0%
33.6%
23.1%
4
40
36.3%
37.2%
37.8%
37.0%
33.1%
70
36.9%
37.7%
38.0%
38.1%
34.7%
Nicolas Papernot, Abhradeep Thakurta, Shuang Song, Steve Chien, andÚlfar Erlingsson. Tempered sigmoid activations for deep learning with differential privacy. arXiv preprint arXiv:2007.14191, 2020.[35] Venkatadheeraj Pichapati, Ananda Theertha Suresh, Felix X Yu, Sashank J Reddi, and Sanjiv Kumar. Adaclip: Adaptive clipping for private sgd.29] Daniel Kifer, Adam Smith, and Abhradeep Thakurta. Private convex empirical
risk minimization and high-dimensional regression. In Conference on Learning
Theory, pages 25-1, 2012.
[30] Xuechen Li, Florian Tramer, Percy Liang, and Tatsunori Hashimoto. Large lan-
guage models can be strong differentially private learners. In International Con-
ference on Learning Representations, 2022.
[31] H Brendan McMahan, Daniel Ramage, Kunal Talwar, and Li Zhang.
Learning differentially private recurrent language models. arXiv preprint
arXiv:1710.06963, 2017.
[32] Nicolas Papernot, Steve Chien, Shuang Song, Abhradeep Thakurta, and Ulfar
Erlingsson. Making the shoe fit: Architectures, initializations, and tuning for
learning with privacy, 2020.
[33] Nicolas Papernot and Thomas Steinke. Hyperparameter tuning with renyi differ-
ential privacy, 2021.
[34] arXiv preprint
arXiv:1908.07643, 2019.
[36] Reza Shokri, Marco Stronati, Congzheng Song, and Vitaly Shmatikov. Member-
ship inference attacks against machine learning models. In 2017 IEEE Symposium
on Security and Privacy (SP), pages 3-18, 2017.
[37] Shuang Song, Kamalika Chaudhuri, and Anand D Sarwate. Stochastic gradient
descent with differentially private updates. In 2013 IEEE Global Conference on
Signal and Information Processing, pages 245-248. IEEE, 2013.
[38] Shuang Song, Thomas Steinke, Om Thakkar, and Abhradeep Thakurta. Evading
the curse of dimensionality in unconstrained private glms. In Arindam Banerjee
and Kenji Fukumizu, editors, Proceedings of The 24th International Conference
on Artificial Intelligence and Statistics, volume 130 of Proceedings of Machine
Learning Research, pages 2638-2646. PMLR, 13-15 Apr 2021.
Table 8 :
8Resnet-18 accuracy while training for different number of epochs but keeping constant ε. For each of the experiments we did a learning rate sweep in{0.1, 0.2, 0.5, 1, 2, 4, 8, 16, 40} and reported the best accuracy. Row 10/1 correspond
to training for 10 epochs with 1 epoch of learning rate warmup. Experiments in all
other row used 5 epochs of learning rate warmup.
table is model accuracy.training model for 10 epochs with DP-SGD. We observed that learning rate 16 was the best or very close to the best for all considered batch sizes. · 10 8 1.5 · 10 7 2.2 · 10 5 1.1 · 10 3Batch
size →
1024
4*1024 16*1024 64*1024 256*1024 1024*1024
Num
epochs ↓
10
56%
35.5%
5.8%
1.6%
0.44%
0.14%
9.8 23
5.6
20
56.4%
53.1%
18.8%
2 · 10 9
3.0 · 10 7 4.4 · 10 5
40
61.7%
57.5%
39.5%
10%
1.3%
0.44%
3.9 · 10 9 6.1 · 10 7 8.9 · 10 5 4.2 · 10 3
48
11.9
80
54.3%
1.8 · 10 6
160
57.9%
37.5%
8%
0.95%
3.5 · 10 6 1.7 · 10 4
120
28
640
57.2%
36.2%
6.2%
6.7 · 10 4
326
72.3
Table 9 :
9ImageNet training using extremely large batches. Rows correspond to different number of training epochs, columns -to batch size. Each cell contains two numbers, top one is the accuracy and bottom one is ε. If cell is empty then corresponding experiment was not run.F Pre-training on Places365 datasetWe tried to train Resnet-18 on Places365 for different number of epochs and with different learning rates, all other hyperparameters were the same as in case of ImageNet training (see appendix A). Our results are summarized in table 10. Since accuracy increase going from 40 to 80 epochs was very small we didn't try longer training and simply picked the training run with the best accuracy as a starting point for all finetuning experiments. Best accuracy 51.41 53.16 54.26 54.96Table 10: Accuracy of Resnet-18 trained on Places365 dataset. Each accuracy number was obtained by sweeping learning rate in {0.02, 0.04, 0.08, 0.2, 0.4, 0.8, 1.6}.Num epochs
10
20
40
80
table 12 .
12Frozen block groups None1
2
3
4
Best accuracy
1.8% 2.1% 2.7% 3.3% 23.2%
Table 11 :
11Private finetuning experiments with frozen layers. Training was done for 10 epochs with privacy budget ε ≈ 6. In this experiments we used Nesterov momentum optimizer with cosine learning rate decay and learning rate warmup for 1 epoch. We used batch size 1024 * 1024. Each accuracy number was obtained by sweeping learning rate in {4.096, 12.288, 40.96, 122.88, 409.6, 1228.8, 4096.0} Best non private accuracy 66.7% 41.7%Frozen block groups
3
4
Table 12 :
12Best results of non-private finetuning of Resnet18 model. All experiment were run for 80 epochs with batch size 1024 and by sweeping learning rate in {0.004, 0.012, 0.04, 0.12, 0.4, 1.2, 4.0}.
Tensorflow Privacy library. Privacy analysis in Tensorflow Privacy library. https: //github.com/tensorflow/privacy/blob/ a749ce4e3041003383be524f5efc3961ec6c1568/tensorflow_ privacy/privacy/analysis/compute_dp_sgd_privacy_lib. py#L49. Accessed: 2022-02-07.
Deep learning with differential privacy. Martín Abadi, Andy Chu, Ian J Goodfellow, H Brendan Mcmahan, Ilya Mironov, Kunal Talwar, Li Zhang, Proc. of the 2016 ACM SIGSAC Conf. on Computer and Communications Security (CCS'16). of the 2016 ACM SIGSAC Conf. on Computer and Communications Security (CCS'16)Martín Abadi, Andy Chu, Ian J. Goodfellow, H. Brendan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang. Deep learning with differential privacy. In Proc. of the 2016 ACM SIGSAC Conf. on Computer and Communications Se- curity (CCS'16), pages 308-318, 2016.
Efficient full-matrix adaptive regularization. Naman Agarwal, Brian Bullins, Xinyi Chen, Elad Hazan, Karan Singh, Cyril Zhang, Yi Zhang, Proceedings of the 36th International Conference on Machine Learning. the 36th International Conference on Machine LearningNaman Agarwal, Brian Bullins, Xinyi Chen, Elad Hazan, Karan Singh, Cyril Zhang, and Yi Zhang. Efficient full-matrix adaptive regularization. In Proceed- ings of the 36th International Conference on Machine Learning, pages 102-110, 2019.
Large-scale differentially private BERT. Rohan Anil, Badih Ghazi, Vineet Gupta, Ravi Kumar, Pasin Manurangsi, Rohan Anil, Badih Ghazi, Vineet Gupta, Ravi Kumar, and Pasin Manurangsi. Large-scale differentially private BERT, 2021.
Membership privacy in microRNA-based studies. Michael Backes, Pascal Berrang, Mathias Humbert, Praveen Manoharan, ACM SIGSAC Conference on Computer and Communications Security. Michael Backes, Pascal Berrang, Mathias Humbert, and Praveen Manoharan. Membership privacy in microRNA-based studies. In ACM SIGSAC Conference on Computer and Communications Security, 2016.
Stability of stochastic gradient descent on nonsmooth convex losses. Raef Bassily, Vitaly Feldman, Cristóbal Guzmán, Kunal Talwar, NeurIPS. Raef Bassily, Vitaly Feldman, Cristóbal Guzmán, and Kunal Talwar. Stability of stochastic gradient descent on nonsmooth convex losses. In NeurIPS, 2020.
Private stochastic convex optimization with optimal rates. Raef Bassily, Vitaly Feldman, Kunal Talwar, Abhradeep Guha Thakurta, Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems. Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d'Alché-Buc, Emily B. Fox, and Roman GarnettNeurIPS; Vancouver, BC, CanadaRaef Bassily, Vitaly Feldman, Kunal Talwar, and Abhradeep Guha Thakurta. Pri- vate stochastic convex optimization with optimal rates. In Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d'Alché-Buc, Emily B. Fox, and Roman Garnett, editors, Advances in Neural Information Processing Systems 32: Annual Conference on Neural Information Processing Systems 2019, NeurIPS 2019, 8-14 December 2019, Vancouver, BC, Canada, pages 11279-11288, 2019.
Private empirical risk minimization: Efficient algorithms and tight error bounds. Raef Bassily, Adam Smith, Abhradeep Thakurta, Proc. of the 2014 IEEE 55th Annual Symp. on Foundations of Computer Science (FOCS). of the 2014 IEEE 55th Annual Symp. on Foundations of Computer Science (FOCS)Raef Bassily, Adam Smith, and Abhradeep Thakurta. Private empirical risk min- imization: Efficient algorithms and tight error bounds. In Proc. of the 2014 IEEE 55th Annual Symp. on Foundations of Computer Science (FOCS), pages 464-473, 2014.
The secret sharer: Evaluating and testing unintended memorization in neural networks. Nicholas Carlini, Chang Liu, Úlfar Erlingsson, Jernej Kos, Dawn Song, Proceedings of the 28th USENIX Conference on Security Symposium. the 28th USENIX Conference on Security SymposiumNicholas Carlini, Chang Liu,Úlfar Erlingsson, Jernej Kos, and Dawn Song. The secret sharer: Evaluating and testing unintended memorization in neural net- works. In Proceedings of the 28th USENIX Conference on Security Symposium, 2019.
Dawn Song,Úlfar Erlingsson, Alina Oprea, and Colin Raffel. Nicholas Carlini, Florian Tramèr, Eric Wallace, Matthew Jagielski, Ariel Herbert-Voss, Katherine Lee, Adam Roberts, Tom Brown, USENIX Security. Extracting training data from large language modelsNicholas Carlini, Florian Tramèr, Eric Wallace, Matthew Jagielski, Ariel Herbert- Voss, Katherine Lee, Adam Roberts, Tom Brown, Dawn Song,Úlfar Erlingsson, Alina Oprea, and Colin Raffel. Extracting training data from large language mod- els. In USENIX Security, 2021.
Differentially private empirical risk minimization. Kamalika Chaudhuri, Claire Monteleoni, Anand D Sarwate, Journal of Machine Learning Research. 12Kamalika Chaudhuri, Claire Monteleoni, and Anand D Sarwate. Differentially private empirical risk minimization. Journal of Machine Learning Research, 12(Mar):1069-1109, 2011.
Differential privacy dynamics of langevin diffusion and noisy gradient descent. Rishav Chourasia, Jiayuan Ye, Reza Shokri, Rishav Chourasia, Jiayuan Ye, and Reza Shokri. Differential privacy dynamics of langevin diffusion and noisy gradient descent, 2021.
Revealing information while preserving privacy. Irit Dinur, Kobi Nissim, Proceedings of the Twenty-second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems. the Twenty-second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database SystemsIrit Dinur and Kobi Nissim. Revealing information while preserving privacy. In Proceedings of the Twenty-second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, 2003.
Our data, ourselves: Privacy via distributed noise generation. Cynthia Dwork, Krishnaram Kenthapadi, Frank Mcsherry, Ilya Mironov, Moni Naor, Advances in Cryptology-EUROCRYPT. Cynthia Dwork, Krishnaram Kenthapadi, Frank McSherry, Ilya Mironov, and Moni Naor. Our data, ourselves: Privacy via distributed noise generation. In Advances in Cryptology-EUROCRYPT, pages 486-503, 2006.
Calibrating noise to sensitivity in private data analysis. Cynthia Dwork, Frank Mcsherry, Kobbi Nissim, Adam Smith, Proc. of the Third Conf. on Theory of Cryptography (TCC). of the Third Conf. on Theory of Cryptography (TCC)Cynthia Dwork, Frank McSherry, Kobbi Nissim, and Adam Smith. Calibrating noise to sensitivity in private data analysis. In Proc. of the Third Conf. on Theory of Cryptography (TCC), pages 265-284, 2006.
Exposed! A survey of attacks on private data. Annual Review of Statistics and Its Application. Cynthia Dwork, Adam Smith, Thomas Steinke, Jonathan Ullman, Cynthia Dwork, Adam Smith, Thomas Steinke, and Jonathan Ullman. Exposed! A survey of attacks on private data. Annual Review of Statistics and Its Applica- tion, 2017.
Robust traceability from trace amounts. Cynthia Dwork, Adam Smith, Thomas Steinke, Jonathan Ullman, Salil Vadhan, IEEE 56th Annual Symposium on Foundations of Computer Science. Cynthia Dwork, Adam Smith, Thomas Steinke, Jonathan Ullman, and Salil Vad- han. Robust traceability from trace amounts. In IEEE 56th Annual Symposium on Foundations of Computer Science, 2015.
Amplification by shuffling: From local to central differential privacy via anonymity. Úlfar Erlingsson, Vitaly Feldman, Ilya Mironov, Ananth Raghunathan, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms. the Thirtieth Annual ACM-SIAM Symposium on Discrete AlgorithmsSIAMKunal Talwar, and Abhradeep ThakurtaÚlfar Erlingsson, Vitaly Feldman, Ilya Mironov, Ananth Raghunathan, Kunal Tal- war, and Abhradeep Thakurta. Amplification by shuffling: From local to central differential privacy via anonymity. In Proceedings of the Thirtieth Annual ACM- SIAM Symposium on Discrete Algorithms, pages 2468-2479. SIAM, 2019.
Private stochastic convex optimization: Optimal rates in linear time. Vitaly Feldman, Tomer Koren, Kunal Talwar, Proc. of the Fifty-Second ACM Symp. on Theory of Computing (STOC'20). of the Fifty-Second ACM Symp. on Theory of Computing (STOC'20)2020Vitaly Feldman, Tomer Koren, and Kunal Talwar. Private stochastic convex opti- mization: Optimal rates in linear time. In Proc. of the Fifty-Second ACM Symp. on Theory of Computing (STOC'20), 2020.
Hiding among the clones: A simple and nearly optimal analysis of privacy amplification by shuffling. Vitaly Feldman, Audra Mcmillan, Kunal Talwar, arXiv:2012.12803arXiv preprintVitaly Feldman, Audra McMillan, and Kunal Talwar. Hiding among the clones: A simple and nearly optimal analysis of privacy amplification by shuffling. arXiv preprint arXiv:2012.12803, 2020.
Efficient per-example gradient computations. Ian Goodfellow, arXiv:1510.01799arXiv preprintIan Goodfellow. Efficient per-example gradient computations. arXiv preprint arXiv:1510.01799, 2015.
Priya Goyal, Piotr Dollár, Ross B Girshick, Pieter Noordhuis, Lukasz Wesolowski, Aapo Kyrola, Andrew Tulloch, Yangqing Jia, and Kaiming He. Accurate, large minibatch SGD: training imagenet in 1 hour. Priya Goyal, Piotr Dollár, Ross B. Girshick, Pieter Noordhuis, Lukasz Wesolowski, Aapo Kyrola, Andrew Tulloch, Yangqing Jia, and Kaiming He. Ac- curate, large minibatch SGD: training imagenet in 1 hour, 2017.
Identity mappings in deep residual networks. Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun, Computer Vision -ECCV 2016. Bastian Leibe, Jiri Matas, Nicu Sebe, and Max WellingKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In Bastian Leibe, Jiri Matas, Nicu Sebe, and Max Welling, editors, Computer Vision -ECCV 2016, 2016.
Train longer, generalize better: closing the generalization gap in large batch training of neural networks. Elad Hoffer, Itay Hubara, Daniel Soudry, Elad Hoffer, Itay Hubara, and Daniel Soudry. Train longer, generalize better: closing the generalization gap in large batch training of neural networks, 2018.
Towards practical differentially private convex optimization. Roger Iyengar, P Joseph, Dawn Near, Om Song, Abhradeep Thakkar, Lun Thakurta, Wang, 2019 IEEE Symposium on Security and Privacy (SP). Roger Iyengar, Joseph P Near, Dawn Song, Om Thakkar, Abhradeep Thakurta, and Lun Wang. Towards practical differentially private convex optimization. In 2019 IEEE Symposium on Security and Privacy (SP), 2019.
Evaluating differentially private machine learning in practice. Bargav Jayaraman, David Evans, USENIX Security. Bargav Jayaraman and David Evans. Evaluating differentially private machine learning in practice. In USENIX Security, 2019.
Practical and private (deep) learning without sampling or shuffling. Peter Kairouz, Brendan Mcmahan, Shuang Song, Om Thakkar, Abhradeep Thakurta, Zheng Xu, ICML. 2021Peter Kairouz, Brendan McMahan, Shuang Song, Om Thakkar, Abhradeep Thakurta, and Zheng Xu. Practical and private (deep) learning without sampling or shuffling. In ICML, 2021.
What can we learn privately?. Homin K Shiva Prasad Kasiviswanathan, Kobbi Lee, Sofya Nissim, Adam D Raskhodnikova, Smith, 49th Annual IEEE Symp. on Foundations of Computer Science (FOCS). Shiva Prasad Kasiviswanathan, Homin K. Lee, Kobbi Nissim, Sofya Raskhod- nikova, and Adam D. Smith. What can we learn privately? In 49th Annual IEEE Symp. on Foundations of Computer Science (FOCS), pages 531-540, 2008.
Private empirical risk minimization beyond the worst case: The effect of the constraint set geometry. Kunal Talwar, Abhradeep Thakurta, Li Zhang, abs/1411.5417CoRRKunal Talwar, Abhradeep Thakurta, and Li Zhang. Private empirical risk mini- mization beyond the worst case: The effect of the constraint set geometry. CoRR, abs/1411.5417, 2014.
Differentially private learning with adaptive clipping. Om Thakkar, Galen Andrew, H Brendan Mcmahan, abs/1905.03871CoRROm Thakkar, Galen Andrew, and H. Brendan McMahan. Differentially private learning with adaptive clipping. CoRR, abs/1905.03871, 2019.
Differentially private learning needs better features (or much more data). Florian Tramèr, Dan Boneh, International Conference on Learning Representations. Florian Tramèr and Dan Boneh. Differentially private learning needs better fea- tures (or much more data). In International Conference on Learning Representa- tions, 2021.
Subsampled renyi differential privacy and analytical moments accountant. Yu-Xiang Wang, Borja Balle, Shiva Prasad Kasiviswanathan, PMLRProceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics. Kamalika Chaudhuri and Masashi Sugiyamathe Twenty-Second International Conference on Artificial Intelligence and Statistics89Yu-Xiang Wang, Borja Balle, and Shiva Prasad Kasiviswanathan. Subsam- pled renyi differential privacy and analytical moments accountant. In Kamalika Chaudhuri and Masashi Sugiyama, editors, Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, volume 89 of Proceedings of Machine Learning Research, pages 1226-1235. PMLR, 16-18 Apr 2019.
Bolt-on differential privacy for scalable stochastic gradient descentbased analytics. Xi Wu, Fengan Li, Arun Kumar, Kamalika Chaudhuri, Somesh Jha, Jeffrey F Naughton, Proceedings of the 2017 ACM International Conference on Management of Data, SIGMOD. Semih Salihoglu, Wenchao Zhou, Rada Chirkova, Jun Yang, and Dan Suciuthe 2017 ACM International Conference on Management of Data, SIGMODXi Wu, Fengan Li, Arun Kumar, Kamalika Chaudhuri, Somesh Jha, and Jeffrey F. Naughton. Bolt-on differential privacy for scalable stochastic gradient descent- based analytics. In Semih Salihoglu, Wenchao Zhou, Rada Chirkova, Jun Yang, and Dan Suciu, editors, Proceedings of the 2017 ACM International Conference on Management of Data, SIGMOD, 2017.
. Yuxin Wu, Kaiming He, Group normalization. Yuxin Wu and Kaiming He. Group normalization, 2018.
Scaling SGD batch size to 32k for imagenet training. Yang You, Igor Gitman, Boris Ginsburg, Yang You, Igor Gitman, and Boris Ginsburg. Scaling SGD batch size to 32k for imagenet training, 2017.
A Yousefpour, I Shilov, A Sablayrolles, D Testuggine, K Prasad, M Malek, J Nguyen, S Ghosh, A Bharadwaj, J Zhao, G Cormode, I Mironov, arXiv:2109.12298Opacus: User-friendly differential privacy library in pytorch. arXiv preprintA. Yousefpour, I. Shilov, A. Sablayrolles, D. Testuggine, K. Prasad, M. Malek, J. Nguyen, S. Ghosh, A. Bharadwaj, J. Zhao, G. Cormode, and I. Mironov. Opacus: User-friendly differential privacy library in pytorch. arXiv preprint arXiv:2109.12298, 2021.
Differentially private fine-tuning of language models. Da Yu, Saurabh Naik, Arturs Backurs, Sivakanth Gopi, A Huseyin, Gautam Inan, Janardhan Kamath, Kulkarni, Yin Tat Lee, Andre Manoel, Lukas Wutschitz, Sergey Yekhanin, and Huishuai ZhangDa Yu, Saurabh Naik, Arturs Backurs, Sivakanth Gopi, Huseyin A. Inan, Gau- tam Kamath, Janardhan Kulkarni, Yin Tat Lee, Andre Manoel, Lukas Wutschitz, Sergey Yekhanin, and Huishuai Zhang. Differentially private fine-tuning of lan- guage models, 2021.
Places: A 10 million image database for scene recognition. Bolei Zhou, Agata Lapedriza, Aditya Khosla, Aude Oliva, Antonio Torralba, IEEE Transactions on Pattern Analysis and Machine Intelligence. Bolei Zhou, Agata Lapedriza, Aditya Khosla, Aude Oliva, and Antonio Torralba. Places: A 10 million image database for scene recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2017.
Poission subsampled rényi differential privacy. Yuqing Zhu, Yu-Xiang Wang, International Conference on Machine Learning. PMLRYuqing Zhu and Yu-Xiang Wang. Poission subsampled rényi differential privacy. In International Conference on Machine Learning, pages 7634-7642. PMLR, 2019.
. Simplevgg-32, 64128SimpleVGG-32(1)-64(1)-128(1)-128
. Simplevgg-32, 64128SimpleVGG-32(2)-64(2)-128(2)-128
. Simplevgg-32, 64128SimpleVGG-32(3)-64(3)-128(3)-128
. SimpleVGG. 322128SimpleVGG-32(2)-128
. Simplevgg-32, 64128SimpleVGG-32(2)-64(2)-128
. Simplevgg-32, 64128SimpleVGG-32(2)-64(2)-128(2)-128
| [
"https://github.com/google-research/dp-imagenet."
] |
[
"Factorization, resummation and sum rules for heavy-to-light form factors",
"Factorization, resummation and sum rules for heavy-to-light form factors"
] | [
"Yu-Ming Wang \nFakultät für Physik\nUniversität Wien\nBoltzmanngasse 51090ViennaAustria\n\nSchool of Physics\nNankai University\n300071TianjinChina\n"
] | [
"Fakultät für Physik\nUniversität Wien\nBoltzmanngasse 51090ViennaAustria",
"School of Physics\nNankai University\n300071TianjinChina"
] | [] | Precision calculations of heavy-to-light form factors are essential to sharpen our understanding towards the strong interaction dynamics of the heavy-quark system and to shed light on a coherent solution of flavor anomalies. We briefly review factorization properties of heavy-to-light form factors in the framework of QCD factorization in the heavy quark limit and discuss the recent progress on the QCD calculation of B → π form factors from the light-cone sum rules with the B-meson distribution amplitudes. Demonstration of QCD factorization for the vacuum-to-B-meson correlation function used in the sum-rule construction and resummation of large logarithms in the short-distance functions entering the factorization theorem are presented in detail. Phenomenological implications of the newly derived sum rules for B → π form factors are further addressed with a particular attention to the extraction of the CKM matrix element |V ub |. | 10.1051/epjconf/201612900027 | [
"https://www.epj-conferences.org/articles/epjconf/pdf/2016/24/epjconf_qcd2016_00027.pdf"
] | 59,406,119 | 1609.01216 | a5fb621ce3907fc247431a84d33db6c941c4dcfb |
Factorization, resummation and sum rules for heavy-to-light form factors
Yu-Ming Wang
Fakultät für Physik
Universität Wien
Boltzmanngasse 51090ViennaAustria
School of Physics
Nankai University
300071TianjinChina
Factorization, resummation and sum rules for heavy-to-light form factors
Precision calculations of heavy-to-light form factors are essential to sharpen our understanding towards the strong interaction dynamics of the heavy-quark system and to shed light on a coherent solution of flavor anomalies. We briefly review factorization properties of heavy-to-light form factors in the framework of QCD factorization in the heavy quark limit and discuss the recent progress on the QCD calculation of B → π form factors from the light-cone sum rules with the B-meson distribution amplitudes. Demonstration of QCD factorization for the vacuum-to-B-meson correlation function used in the sum-rule construction and resummation of large logarithms in the short-distance functions entering the factorization theorem are presented in detail. Phenomenological implications of the newly derived sum rules for B → π form factors are further addressed with a particular attention to the extraction of the CKM matrix element |V ub |.
Introduction
Heavy-to-light form factors serve as fundamental inputs of describing many exclusive heavy hadron decays which are of great phenomenological interest to the ongoing and forthcoming collider experiments. Extensive efforts have been devoted to develop systematic theoretical frameworks for the precision calculation of heavy-to-light form factors in QCD (see, for instance [1][2][3][4][5][6][7]). In addition to the nonperturbative hadronic distribution amplitudes (DA), the symmetry-conserving "soft" form factors have to be introduced in QCD factorization for B → π form factors due to the emergence of rapidity divergences in the corresponding convolution integrals. An alternative approach to compute B → π form factors is QCD light-cone sum rules (LCSR) constructed from the vacuum-to-B-meson correlation functions with the aid of the dispersion relations and parton-hadron duality approximation [8][9][10][11]. In the following, we will first establish QCD factorization for the vacuum-to-B-meson correlation function at one loop employing the method of regions and perform the resummation of large logarithms in the hard and jet functions with the renormalization-group approach in section 2, where the resummation improved LCSR for B → π form factors and a new determination of the CKM matrix element |V ub | are also presented.
The LCSR for the B → π form factors at O(α s )
The construction of LCSR for the B → π form factors can be achieved with the following correlation function [12]:
Π μ (n · p,n · p) = d 4 x e ip·x 0|T d (x) n γ 5 u(x),ū(0) γ μ b(0) |B(p + q) = Π(n · p,n · p) n μ + Π(n · p,n · p)n μ ,(1)
where p + q = m B v and the two light-cone vectors satisfy the relations n · v =n · v = 1. To facilitate the perturbative calculation of short-distance functions in the factorization formula of (1), we need to establish the power counting scheme for the external momentum p μ
n · p m 2 B + m 2 π − q 2 m B = 2E π ,n · p ∼ O(Λ QCD ) .(2)
Applying the light-cone operator-produce-expansion (OPE) at space-like p 2 yields the tree-level factorization formula
Π(n · p,n · p) =f B (μ) m B ∞ 0 dω φ − B (ω ) ω −n · p − i 0 + O(α s ) , Π(n · p,n · p) = O(α s ) .(3)
in the heavy quark limit, where the B-meson light-cone DA in the coordinate space is defined as
0|d β (τn) [τn, 0] b α (0)|B(p + q) = − if B (μ) m B 4 1+ v 2 2φ + B (τ) + φ − B (τ) −φ + B (τ) n γ 5 αβ .(4)
Applying the standard definitions of B → π form factors and the pion decay constant
π(p)|ūγ μ b|B(p B ) = f + Bπ (q 2 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ p B + p − m 2 B − m 2 π q 2 q ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ μ + f 0 Bπ (q 2 ) m 2 B − m 2 π q 2 q μ , π(p)|d n γ 5 u|0 = −i n · p f π ,(5)
it is straightforward to derive the hadronic dispersion relation for the correlator (1)
Π μ (n · p,n · p) = f π n · p m B 2 (m 2 π − p 2 ) n μ n · p m B f + Bπ (q 2 ) + f 0 Bπ (q 2 ) + n μ m B n · p − m B n · p m B f + Bπ (q 2 ) − f 0 Bπ (q 2 ) + +∞ ω s dω 1 ω −n · p − i0 ρ h (ω , n · p) n μ +ρ h (ω , n · p)n μ ,(6)
where the threshold parameter ω s in the pion channel scales as Λ 2 /m b . Matching the hadronic and OPE representations of the vacuum-to-B-meson correlation function leads to
f + Bπ (q 2 ) =f B (μ) m B f π n · p exp m 2 π n · p ω M ω s 0 dω e −ω /ω M φ − B (ω ) + O(α s ) , f 0 Bπ (q 2 ) = n · p m B f + Bπ (q 2 ) + O(α s ) .(7)
Now we are in a position to demonstrate QCD factorization for Π μ (n · p,n · p) at one loop using the diagrammatical approach. Technically, this amounts to justifying the equivalence between the soft Figure 1. Next-to-leading-order QCD correction to the correlation function Π μ (n · p,n · p). contribution to the one-loop QCD diagrams displayed in Figure 1 and the infrared subtraction shown in Figure 2 in the heavy quark limit and extracting the hard and hard-collinear contributions to the correlation function at leading power in Λ/m b . We will apply the standard strategies to establish the diagrammatical factorization for QCD Green functions at one-loop accuracy [13,14]:
d b q p B u (a) (b) (c) (d)d b v (a) (b) (c)
• Identify the leading regions of the QCD amplitudes with the power counting scheme (2);
• Evaluate the loop integrals with the method of regions and extract the "bare" perturbative kernels;
• Apply the ultraviolet renormalization and perform the infrared subtraction;
• Substitute the momentum-space light-cone projector of the B-meson.
Taking the weak vertex diagram displayed in Figure 1(a) as an example, the corresponding QCD amplitude can be written as [12] Π (1)
μ, weak = g 2 s C F 2 (n · p − ω) d D l (2π) D 1 [(p − k + l) 2 + i0][(m b v + l) 2 − m 2 b + i0][l 2 + i0] d(k) n γ 5 n γ ρ ( p− k+ l) γ μ (m b v+ l + m b ) γ ρ b(v) .(8)
Employing the power counting scheme (2) one can readily identify that the leading power contributions to Π (1) μ, weak come from the hard, hard-collinear and soft regions. Expanding the QCD amplitude Π (1) μ, weak in terms of the powers of Λ/m b in the soft region and keeping the leading-order terms yield On the other hand, the one-loop correction to the B-meson DA can be computed with the Wilson-line Feynman rules
Π (1),s μ, weak = g 2 s C F 2 (n · p − ω) d D l (2π) D 1 [n · (p − k + l) + i0][v · l + i0][l 2 + i0] d(k) n γ 5 n γ μ b(v) .(9)Φ αβ ,(1) bd, a (ω, ω ) = i g 2 s C F d D l (2π) D 1 [n · l + i0][v · l + i0][l 2 + i0] ×[δ(ω − ω −n · l) − δ(ω − ω)] [d(k)] α [b(v)] β ,(10)
from which the infrared subtraction term can be deduced as Φ (1) bd, a ⊗ T (0) = Π (1), s μ, weak , verifying QCD factorization for the vacuum-to-B-meson correlation function at one loop diagrammatically. By proceeding in a similar way, the hard-colliner contribution from the weak vertex diagram can be extracted as follows:
Π (1),hc μ,weak = g 2 s C F 2(n · p − ω) d D l (2π) D 2 m b n · (p + l) [n · (p + l)n · (p − k + l) + l 2 ⊥ + i0][m b n · l + i0][l 2 + i0] d(k) n γ 5 n γ μ b(p b ) .(11)
which can be further computed with the loop integrals collected in the Appendix A of [12]. Finally, expanding the QCD amplitude (8) in the hard region gives rise to
Π (1), h μ, weak = α s C F 4 πf B (μ) m B φ − bd (ω) n · p − ω n μ 1 2 + 1 2 ln μ n · p + 1 + 2 ln 2 μ n · p + 2 ln μ m b − ln 2 r − 2 Li 2 −r r + 2 − r r − 1 ln r + π 2 12 + 3 + n μ 1 r − 1 1 + r r ln r ,(12)
with r = n · p/m b andr = 1 − r. Along the same vein, one can evaluate the leading power contributions to the remaining diagrams in Figure 1 and the resulting factorization formulae for the correlation function are given by
Π =f B (μ) m B k=± C (k) (n · p, μ) ∞ 0 dω ω −n · p J (k) μ 2 n · p ω , ω n · p φ (k) B (ω, μ) , Π =f B (μ) m B k=± C (k) (n · p, μ) ∞ 0 dω ω −n · p J (k) μ 2 n · p ω , ω n · p φ (k) B (ω, μ) ,(13)
where the hard and hard-collinear functions at one loop read [12] C (+) =C (+) = 1,
C (−) = α s C F 4 π 1 r r r ln r + 1 , C (−) = 1 − α s C F 4 π 2 ln 2 μ n · p + 5 ln μ m b − ln 2 r − 2 Li 2 −r r + 2 − r r − 1 ln r + π 2 12 + 5 ,(14)
and The hard coefficients presented in (14) are in compatible with the perturbative matching coefficients of the weak currentq γ μ b from QCD to soft-collinear effective theory (SCET) [15], and the hardcollinear functions displayed in (15) coincide with the corresponding jet functions computed with the SCET Feynman rules [11].
J (+) = 1 rJ (+) = α s C F 4 π 1 −n · p ω ln 1 − ω n · p , J (−) = 1 ,J (−) = 1 + α s C F 4 π ln 2 μ 2 n · p(ω −n · p) − 2 lnn · p − ω n · p ln μ 2 n · p(ω −n · p) − ln 2n · p − ω n · p − 1 + 2n · p ω lnn · p − ω n · p − π 2 6 − 1 .(15)
It is evident that there is no common value of μ that can avoid the parametrically large logarithms of order ln(m b /Λ) in the hard functions, the jet functions and the B-meson DA. The summation of these large logarithms in the hard coefficient functions can be accomplished by solving the following renormalization-group equations
d d ln μC (−) (n · p, μ) = −Γ cusp (α s ) ln μ n · p + γ(α s ) C (−) (n · p, μ) , d d ln μf B (μ) =γ(α s )f B (μ) ,(16)
where the three-loop cusp anomalous dimension and the two-loop γ (γ) are needed to achieve the nextto-leading-logarithmic (NLL) resummation. Applying the standard procedure for the construction of QCD sum rules leads to
f π e −m 2 π /(n·p ω M ) n · p m B f + Bπ (q 2 ) , f 0 Bπ (q 2 ) = U 2 (μ h2 , μ)f B (μ h2 ) ω s 0 dω e −ω /ω M U 1 (n · p, μ h1 , μ) C (−) (n · p, μ h1 ) φ − B,eff (ω , μ) + r φ + B,eff (ω , μ) ± n · p − m B m B φ + B,eff (ω , μ) + C (−) (n · p, μ) φ − B (ω , μ) ,(17)
where the explicit expressions of φ ± B,eff (ω , μ) can be found in [12]. Having at our disposal the NLL resummation improved LCSR presented in (17), it is straightforward to plot the q 2 dependence of the B → π form factors f +,0
Bπ (q 2 ) in Figure 3 where the theoretical predictions from the sum rules with the pion DA [4] are also shown for a comparison. The observed discrepancy of the q 2 shape for the vector form factor f + Bπ (q 2 ) predicted from the two distinct sum rules could be attributed to the systematical uncertainties generated by the different parton-hadron duality ansatz and to the yet unaccounted higher order/power corrections in the perturbative calculations of the corresponding correlation functions. Expressing the differential branching fraction of B → πμν μ in terms of the form factor f + Bπ (q 2 ) dΓ dq 2
(B → πμν μ ) = G 2 F |V ub | 2 24π 3 | p π | 3 | f + Bπ (q 2 )| 2 .(18)
and employing the experimental measurements for the integrated decay rate [16,17] lead to |V ub | = 3.05 +0.54 −0.38 | th. ± 0.09| exp. × 10 −3 ,
which is consistent with the exclusive determination of |V ub | from the leptonic B → τν τ decay [18].
Conclusion
To summarize, we demonstrated QCD factorization for the vacuum-to-B-meson correlation function used in the construction of the B → π form factors at one-loop accuracy explicitly with the method of regions and achieved the resummation of large logarithms in the short-distance functions at NLL by solving the renormalization-group equations in the momentum space. Phenomenological applications of the newly derived sum rules with the B-meson DA were also discussed in brief, focusing on the extraction of exclusive |V ub | from the semi-leptonic B → πμν μ decay.
Figure 2 .
2One-loop effective diagrams for the infrared subtraction.
Figure 3 .
3The momentum-transfer dependence of the B → π form factors f +,0 Bπ (q 2 ) from the NLL resummation improved LCSR(17)with the B-meson DA (pink curves) and from the NLO QCD sum rules with the pion DA (blue curves).
. M Beneke, T Feldmann, hep-ph/0008255Nucl. Phys. B. 5923M. Beneke and T. Feldmann, Nucl. Phys. B 592 (2001) 3 [hep-ph/0008255].
. P Ball, R Zwicky, hep-ph/0406232Phys. Rev. D. 7114015P. Ball and R. Zwicky, Phys. Rev. D 71, 014015 (2005) [hep-ph/0406232].
. G Duplancic, A Khodjamirian, T Mannel, B Melic, N Offen, arXiv:0801.1796JHEP. 080414hep-phG. Duplancic, A. Khodjamirian, T. Mannel, B. Melic and N. Offen, JHEP 0804, 014 (2008) [arXiv:0801.1796 [hep-ph]].
. A Khodjamirian, T Mannel, N Offen, Y.-M Wang, arXiv:1103.2655Phys. Rev. D. 8394031hep-phA. Khodjamirian, T. Mannel, N. Offen and Y.-M. Wang, Phys. Rev. D 83, 094031 (2011) [arXiv:1103.2655 [hep-ph]].
. Y M Wang, Y Li, C D Lü, arXiv:0804.0648Eur. Phys. J. C. 59861hep-phY. M. Wang, Y. Li and C. D. Lü, Eur. Phys. J. C 59, 861 (2009) [arXiv:0804.0648 [hep-ph]].
. H N Li, Y L Shen, Y M Wang, arXiv:1201.5066Phys. Rev. D. 8574004hep-phH. n. Li, Y. L. Shen and Y. M. Wang, Phys. Rev. D 85, 074004 (2012) [arXiv:1201.5066 [hep-ph]].
. C D Lü, Y M Wang, H Zou, A Ali, G Kramer, arXiv:0906.1479Phys. Rev. D. 8034011hep-phC. D. Lü, Y. M. Wang, H. Zou, A. Ali and G. Kramer, Phys. Rev. D 80, 034011 (2009) [arXiv:0906.1479 [hep-ph]].
. A Khodjamirian, T Mannel, N Offen, hep-ph/0504091Phys. Lett. B. 62052A. Khodjamirian, T. Mannel and N. Offen, Phys. Lett. B 620, 52 (2005) [hep-ph/0504091].
. A Khodjamirian, T Mannel, N Offen, hep-ph/0611193Phys. Rev. D. 7554013A. Khodjamirian, T. Mannel and N. Offen, Phys. Rev. D 75, 054013 (2007) [hep-ph/0611193].
. F De Fazio, T Feldmann, T Hurth, hep-ph/0504088Nucl. Phys. B. 7331F. De Fazio, T. Feldmann and T. Hurth, Nucl. Phys. B 733, 1 (2006) [hep-ph/0504088].
. F De Fazio, T Feldmann, T Hurth, arXiv:0711.3999JHEP. 080231hep-phF. De Fazio, T. Feldmann and T. Hurth, JHEP 0802, 031 (2008) [arXiv:0711.3999 [hep-ph]].
. Y M Wang, Y L Shen, arXiv:1506.00667Nucl. Phys. B. 898563hep-phY. M. Wang and Y. L. Shen, Nucl. Phys. B 898, 563 (2015) [arXiv:1506.00667 [hep-ph]].
. Y M Wang, Y L Shen, arXiv:1511.09036JHEP. 1602179hep-phY. M. Wang and Y. L. Shen, JHEP 1602, 179 (2016) [arXiv:1511.09036 [hep-ph]].
. Y M Wang, arXiv:1606.03080hep-phY. M. Wang, arXiv:1606.03080 [hep-ph].
. C W Bauer, S Fleming, D , I W Stewart, hep- ph/0011336Phys. Rev. D. 63114020C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, Phys. Rev. D 63, 114020 (2001) [hep- ph/0011336].
. J P Lees, BaBar CollaborationarXiv:1208.1253Phys. Rev. D. 8692004hepexJ. P. Lees et al. [BaBar Collaboration], Phys. Rev. D 86, 092004 (2012) [arXiv:1208.1253 [hep- ex]].
. A Sibidanov, Belle CollaborationarXiv:1306.2781Phys. Rev. D. 8832005hep-exA. Sibidanov et al. [Belle Collaboration], Phys. Rev. D 88, 032005 (2013) [arXiv:1306.2781 [hep-ex]].
. B Kronenbitter, Belle CollaborationarXiv:1503.05613Phys. Rev. D. 9251102hep-exB. Kronenbitter et al. [Belle Collaboration], Phys. Rev. D 92, 051102 (2015) [arXiv:1503.05613 [hep-ex]].
| [] |
[
"Development of a fast electromagnetic shutter for compressive sensing imaging in scanning transmission electron microscopy",
"Development of a fast electromagnetic shutter for compressive sensing imaging in scanning transmission electron microscopy"
] | [
"A Béché \nEMAT\nUniversity of Antwerp\nGroenenborgerlaan 1712020AntwerpBelgium\n",
"B Goris \nEMAT\nUniversity of Antwerp\nGroenenborgerlaan 1712020AntwerpBelgium\n",
"B Freitag \nFEI Electron Optics\nNL-5600 KAEindhovenThe Netherlands\n",
"J Verbeeck \nEMAT\nUniversity of Antwerp\nGroenenborgerlaan 1712020AntwerpBelgium\n"
] | [
"EMAT\nUniversity of Antwerp\nGroenenborgerlaan 1712020AntwerpBelgium",
"EMAT\nUniversity of Antwerp\nGroenenborgerlaan 1712020AntwerpBelgium",
"FEI Electron Optics\nNL-5600 KAEindhovenThe Netherlands",
"EMAT\nUniversity of Antwerp\nGroenenborgerlaan 1712020AntwerpBelgium"
] | [] | The concept of compressive sensing was recently proposed to significantly reduce the electron dose in scanning transmission electron microscopy (STEM) while still maintaining the main features in the image. Here, an experimental setup based on an electromagnetic shutter placed in the condenser plane of a STEM is proposed. The shutter blanks the beam following a random pattern while the scanning coils are moving the beam in the usual scan pattern. Experimental images at both medium scale and high resolution are acquired and then reconstructed based on a discrete cosine algorithm. The obtained results confirm the predicted usefulness of compressive sensing in experimental STEM even though some remaining artifacts need to be resolved. | 10.1063/1.4943086 | [
"https://arxiv.org/pdf/1509.06656v1.pdf"
] | 119,197,906 | 1509.06656 | 0209ebcacdc6ade64a846064f6f80ae523f2ec32 |
Development of a fast electromagnetic shutter for compressive sensing imaging in scanning transmission electron microscopy
A Béché
EMAT
University of Antwerp
Groenenborgerlaan 1712020AntwerpBelgium
B Goris
EMAT
University of Antwerp
Groenenborgerlaan 1712020AntwerpBelgium
B Freitag
FEI Electron Optics
NL-5600 KAEindhovenThe Netherlands
J Verbeeck
EMAT
University of Antwerp
Groenenborgerlaan 1712020AntwerpBelgium
Development of a fast electromagnetic shutter for compressive sensing imaging in scanning transmission electron microscopy
Compressive sensingScanning transmission electron microscope (STEM)Electromagnetic shut- ter
The concept of compressive sensing was recently proposed to significantly reduce the electron dose in scanning transmission electron microscopy (STEM) while still maintaining the main features in the image. Here, an experimental setup based on an electromagnetic shutter placed in the condenser plane of a STEM is proposed. The shutter blanks the beam following a random pattern while the scanning coils are moving the beam in the usual scan pattern. Experimental images at both medium scale and high resolution are acquired and then reconstructed based on a discrete cosine algorithm. The obtained results confirm the predicted usefulness of compressive sensing in experimental STEM even though some remaining artifacts need to be resolved.
I. INTRODUCTION
One of the most challenging topics in modern transmission electron microscopy (TEM) is to perform experiments on soft or beam sensitive materials as they suffer from irradiation damage that can range from structural modification to the complete destruction of the sample. Such sample modifications are even more problematic when 3D or/and analytical characterizations are involved. In order to overcome this issue, a wide range of different approaches are being tested in the TEM community like reduction of the kinetic energy of the fast electrons 1-3 , improving the detection efficiency of cameras 4-7 , time resolved approaches 8-10 and many more. Relatively recently, the concept of compressive sensing was proposed to significantly reduce the electron dose while maintaining all the important features in a TEM or Scanning TEM (STEM) image [11][12][13][14] . Compressive sensing is based on the assumption that an image contains a significant amount of redundancy and not all pixels in the image are independent. The condition for a proper image reconstruction is that the original image has a sparse representation in a specific basis which can be chosen prior to image reconstruction [15][16][17] . Therefore the image can be approximated from a small subset of pixels randomly taken from the completely sampled image. Depending on the redundancy, such reconstructed image can be very close to the fully sampled image while considerably reducing the required dose. The process is somewhat comparable to image compression algorithms that try to represent images with less information by exploiting the redundancy present in a typical image.
Over the last few years, some studies have indeed proposed to apply compressive sensing to transmission electron microscopy images using different types of reconstruction algorithms based on Bayesan dictionary learning technique 12 , wavelet frame based 13 or total variation a) Corresponding author: [email protected] inpainting 14 . All this work was done on virtually masked images starting from a fully sampled experimental or stored image and applying a digital mask to it, taking out a number of often randomly selected pixels. This indeed shows a great potential for compressive sensing but should be seen as an idealized simulation assuming that experimental random pixel measurements would be possible. Implementing compressive sensing in practice is complicated by the fact that typical scan engines in STEM microscopes are not designed to be driven in a non-regular pattern as would be required. An alternative is to make use of a beam shutter that can switch the electron beam on and off while keeping the conventional regular scanning pattern. In this paper we present an experimental realization of such a beam shutter based on electromagnetic deflection and demonstrate that we obtain experimental compressive sensing in a STEM in both medium and high resolution. By shuttering the beam using a pseudorandom generator, it was possible to acquire images with a limited number of pixels and reconstruct them using the discrete cosine transform. This demonstrates that compressive sensing became an experimentally viable technique in STEM opening up the predicted advantages of the technique for experimental research.
II. EXPERIMENTAL SETUP
The aim in compressive sensing acquisition is to illuminate only parts of the sample. Scanning imaging modes are especially well suited for this operation as images are acquired in a pixel by pixel way. Mainly two strategies can be followed to achieve compressive sensing in such mode: (i) blanking the beam in between two illuminated pixels or (ii) driving the scanning coils in a specific way to jump from one pixel to a non-adjacent next pixel. Stevens et al. 12 achieved successful acquisitions with the second strategy in a SEM. Doing this in a STEM is more complicated due to the higher beam location precision and scanning speed that are typically required. Although less interesting in terms of total ac- quisition speed, the blanking method has the distinctive advantage of a simpler hardware setup and will allow us to test compressive sensing in STEM. In a TEM, the pre-specimen beam blanker is located at the gun level but suffers from relatively slow response time, making it unattractive for the current purpose of blanking the beam during scanning. Our first challenge was then to realize a sufficiently fast beam shutter compatible with a typical microsecond range dwell times in STEM. The most accessible locations in the illumination system of a modern TEM are the condenser aperture holders. In the design of our TEM microscope, an FEI Titan3 equipped with both probe and image Cs correctors and a Gatan Image Filter (GIF), the C1 apertures are located in the high vacuum region of the gun making it improper for fast access and convenient operation. The C2 aperture further down the illuminating system was consequently the best place to place the shutter. The first step was to design a completely new aperture holder with four electrical feedthrough contacts. A picture of the custom built holder is displayed in Figure 1. The four aperture slots are clearly visible together with four electrical contacts. We deflect the electron beam making use of a simple solenoid wrapped around the first condenser aperture. The solenoid produces a quasi-homogeneous magnetic field at the plane of the aperture which deflects the electrons due to the Lorentz force. At the given winding density, it turns out that a current of 250 mA is capable of deflecting a 300 keV focused probe about 350 nm away from the sample area. A selected area (SA) aperture was introduced in the path of the beam too prevent high angle diffraction signal from reaching the detector when the shutter blanks the beam. The solenoid has a series resistance of R s = 0.75Ω a self-inductance of L s = 0.3 H and an estimated capacitance of 12 pF 18,19 . These parameters set the maximum switching speed which can be estimated as τ = (L s C s ) leading maximum estimated switching speeds of 2 ns, much shorter than typical STEM dwell times.
As the beam deflector was successfully implemented, we had to synchronize the STEM scan engine with the beam deflection in order to properly shutter the beam at given pixel positions. An Arduino 20 microcontroller unit linked to a switched current source was used to drive the deflector coil synchronized with the shutter signal of the GIF CCD. The microcontroller was then programmed to open or close the shutter based on a (pseudo) random 21 generator in synchronization with the scan engine. A pixel will be illuminated if the random generator with uniform probability distribution between 1 and 100 draws a number higher than X, with X the average amount of unblanked pixels we want to obtain over the whole scan. Figure 2 displays a schematic of our experimental setup.
In order to correctly reconstruct the acquired sparsely sampled image, the reconstruction algorithm needs to have access to the acquisition mask, namely knowing when the beam was blanked or unblanked. As the storage space on the microcontroller was limited, we used a workaround to obtain the shutter mask by acquiring a zero loss (ZL) EELS map together with the HAADF STEM image. In addition to obtaining the applied shutter mask, this workaround allowed us to check how efficiently the electron was shuttered by studying the intensity of the zero loss peak which should ideally be high for unblanked pixels (the sample is electron transparent) and zero for blanked pixels. In order to obtain a ZL peak of sufficient intensity, the single pixel exposure time, for typical High Resolution STEM illumination settings, was set to 0.5 ms. Even though this value is relatively high compared to typical dwell times used in STEM, it nevertheless allows us to prove the setup works and to verify the efficiency of shuttering. In a later stage the described workaround will disappear and the dwell time will be only limited by the maximum speed at which the shutter can be driven which should be well in the microsecond to nanosecond range. In order to reconstruct the images from the subsampled projections, an interpolation is required filling the missing pixels in the images. This interpolation corresponds to solving the following equations:
x = argmin x Φ x − b 2 with Ψ x 1 < λ (1)
Wherex corresponds to the reconstructed image, b equals the measured pixels and Φ is a subsampling operator that selects the imaged pixels. The operator Ψ represents the sparsifying transform that can be chosen prior to the re-construction and λ is a parameter that can be adjusted according to the sparsity of the image after the sparsifying transform. In this work, a discrete cosine transform is selected which is well suited for images showing a local periodicity such as high resolution STEM projections. The reconstruction is implemented in Matlab using the spgl1 algorithm 22,23 . More elaborate transforms can easily be implemented but we focused here on the experimental realization of the shutter.
III. RESULTS
The effect of compressive sensing on STEM image acquisition and reconstruction was investigated using two samples with rather different properties. On the one hand, medium resolution STEM imaging was investigated on a standard gold cross grating sample. This sample has the advantage of presenting a very high density of gold nanoparticles with quite complex agglomerated shapes. On the other hand, HRSTEM was investigated on a complex perovskite oxide sample consisting of an NbGaO 3 substrate covered with 6 atomic layers of SrTiO 3 and a 10 nm of LaSrMnO 3 24 . The lattice parameter is well above the theoretical resolution limit of our instrument, thus insuring optimal conditions for the image reconstruction. Both samples also have the advantage of being relatively beam hard allowing for the extra acquisition time needed for the ZL spectrum mapping workaround. One could argue that both samples are rather far from the beam sensitive samples one would expect when discussing compressive sensing, they allow us however to study the feasibility of this new imaging technique without beam damage issues complicating the interpretation of the results. Both samples were imaged with three different acquisition schemes: no shuttering, 50% shuttering and 80% shuttering. The total dose is then effectively reduced respectively by a factor of 2 and 5 in the different cases. The total frame size was set to 256x256 pixels with a dwell time of 0.5 ms in standard HRSTEM illumination conditions, being an acceleration voltage of 300 kV, a convergence angle of 20 mrad with a beam current of 50 pA using a 20 µm C2 aperture. The simultaneous ZL EELS map was acquired with a collection angle of 35 mrad using a dispersion of 0. 25 For the 50% shuttered case, the main features of the image are retrieved in the reconstructed image, even though the resolution decreased somewhat. The 80% shuttered image reveals a lack of detail and only the larger image features are reconstructed while the finer structural details are lost. Note that in terms of redundancy, the cross grating is probably a very challenging case for compressive sensing for the same reasons that this is an excellent sample to align an electron microscope providing irregular features without favoring certain directions over others.
For the HRSTEM sample displayed in Figure 4, both 50% and 80% shuttering cases are still revealing acceptable high resolution information. The contrast of the lightest atoms tends to significantly decrease in the 80% shuttering case but all atoms remain visible. The presence of the stripes in the middle of the 50% shuttering case are due to some sample instability during the acquisition and do not reflect any issues with the reconstruction algorithm. Because of the longer acquisition times, the atomic lattice in Figures 4b and 4c is more distorted by sample drift in comparison to Figure 4a.
(b) (c) (d) (a)
IV. DISCUSSION
In order to discriminate shutter imperfections from reconstruction issues, we compare the reconstructed images with the reconstructions obtained from a virtually shuttered image obtained by applying a digital mask on the unshuttered experimental HAADF image as was typically done in papers discussing compressive sensing so far 13,14,25 . The results are shown in Figure 5, together with the reconstructed experimental images. In the case of the cross grating sample, simulated and experimental images look very similar, with the small image features being completely lost in the most sparsely sampled case. One can either incriminate the reconstruction algorithm, which may fail to sufficiently exploit the sparsity in the observed object, or it could be that the object itself simply doesn't have enough redundancy to be accurately represented by sparse sampling. For the HRSTEM case, the difference between the virtually shuttered images and the experimental ones is quite significant pointing towards shutter artifacts. For the virtually shuttered images, in both the 50% and 80% shuttered cases, the reconstructions are approaching the fully sampled image quite closely with the only noticeable difference being a slight contrast reduction at sparser sampling. However, the experimentally shuttered image reconstructions suf-fer from many more artifacts, going from a strong loss of contrast to losing the light atoms all together. These artifacts can have many origins like remaining synchronization and timing issues, sample drift caused by increased exposure times, local sample charging issues affecting probe positioning and possibly others. As sample drift and temporal instabilities of the instrument are related to the total acquisition time, they will disappear when the setup is changed to exploit the full shutter speed as the workaround with the ZL spectral acquisition is removed. Some of the mentioned artifacts could be overcome by using an electrostatic shutter even though this implementation will likely have its own artifacts 26 .
V. CONCLUSION
In this paper, we successfully demonstrate the implementation of compressive sensing in a TEM making use of an electromagnetic beam shutter. Using a small solenoid placed in the condenser system of the microscope, the beam can be independently shuttered for every pixel during a STEM image acquisition. The reconstruction of the images from the experimentally obtained sparsely sampled images shows that compressive sensing works significantly better on e.g. high resolution images with much redundancy as compared to more irregular and less redundant images as demonstrated with a cross grating sample. This is entirely expected but has to be kept in mind when estimating the potential reduction in dose one could obtain from compressive sensing. At the atomic scale, artifacts induced by sample drift significantly alter the result, but can be entirely overcome in the future when lower dwell times are used. The speed of the present setup remains insufficient for realistic use on beam sensitive samples but this limitation is imposed by technological factors which can be overcome in the near future. If these remaining technological challenges are overcome a reduction of dose of at least 5 times can be expected depending on the type and sampling of the object. The implemented solution could offer a cost effective alternative to e.g. a direct electron detection camera or can be combined with it in order to further reduce the dose. Application in 3D tomographic acquisitions seem especially attractive as redundancy between different projections could be exploited. This would be especially important in the case of analytical 3D experiments.
FIG. 1 .
1C2 electric contact aperture holder mounted with a solenoid.
FIG. 2 .
2Schematic of the compressive sensing acquisition setup. The STEM engine drives the STEM coils and synchronized the GIF CCD and the random generator. Depending on the random number generator, the beam is either blanked (dashed trajectory) or not, leading to the absence or presence of signal on the HAADF detector and zero loss peak in the EELS map.
FIG. 3 .
3Experimental acquisition of STEM images at medium scale using (a) no beam shuttering, (b) 50% beam shuttering and (c) 80% beam shuttering conditions. The reconstructed images based on a discrete cosine algorithm are displayed below the experimental images.
FIG. 4 .
4Experimental acquisition of STEM images at medium scale using (a) no beam shuttering, (b) 50% beam shuttering and (c) 80% beam shuttering conditions. The reconstructed images based on a discrete cosine algorithm are displayed below the experimental images.
eV/pixel and a 5 mm GIF entrance aperture. The experimental images on the cross grating sample are regrouped in Figure 3 together with their reconstructions based on the discrete cosine algorithm.
FIG. 5 .
5Comparison between reconstructed images from experiment (top row) and from theory (bottom) row in four cases: (a) cross grating with 50% shuttering time, (b) cross grating with 80% shuttering time, (c) HRSTEM with 50% shuttering time and (d) HRSTEM with 80% shuttering time.
.B, B.G. and J.V. acknowledge funding from the European Research Council under the 7th Framework Program (FP7), ERC Starting Grant No. 278510 VORTEX and No. 335078 COLOURATOM. A.B. and J.V. acknowledge financial support from the European Union under the 7th Framework Program (FP7) under a contract for an Integrated Infrastructure Initiative (Reference No. 312483 ESTEEM2). B.G. acknowledges the Research Foundation Flanders (FWO Vlaanderen) for a postdoctoral research grant. A.B., J.V. acknowledges funding from the GOA project SOLARPAINT and the POC project I13/009 from the University of Antwerp. The Quantem microscope was partially funded by the Hercules Foundation. We thank the Mesa+ laboratory at the University of Twente for the perovskite test sample.
Criteria and Prospects for Realizing Optimum Electron Microscopes. Hh Rose, Microscopy and Microanalysis. 13Hh Rose. Criteria and Prospects for Realizing Optimum Elec- tron Microscopes. Microscopy and Microanalysis, 13(Supplement S02):134-135, August 2007.
Structure and bonding at the atomic scale by scanning transmission electron microscopy. David A Muller, Nature Materials. 84David A. Muller. Structure and bonding at the atomic scale by scanning transmission electron microscopy. Nature Materials, 8(4):263-270, April 2009.
Transmission electron microscopy at 20 kV for imaging and spectroscopy. U Kaiser, J Biskupek, J C Meyer, J Leschner, L Lechner, H Rose, M Stger-Pollach, A N Khlobystov, P Hartel, H Mller, M Haider, S Eyhusen, G Benner, Ultramicroscopy. 1118U. Kaiser, J. Biskupek, J. C. Meyer, J. Leschner, L. Lechner, H. Rose, M. Stger-Pollach, A. N. Khlobystov, P. Hartel, H. Mller, M. Haider, S. Eyhusen, and G. Benner. Transmission electron microscopy at 20 kV for imaging and spectroscopy. Ultrami- croscopy, 111(8):1239-1246, July 2011.
Initial evaluation of a direct detection device detector for single particle cryo-electron microscopy. Anna-Clare Milazzo, Anchi Cheng, Arne Moeller, Dmitry Lyumkis, Erica Jacovetty, James Polukas, Mark H Ellisman, Nguyen-Huu, Bridget Xuong, Clinton S Carragher, Potter, Journal of Structural Biology. 1763Anna-Clare Milazzo, Anchi Cheng, Arne Moeller, Dmitry Lyumkis, Erica Jacovetty, James Polukas, Mark H. Ellisman, Nguyen-Huu Xuong, Bridget Carragher, and Clinton S. Potter. Initial evaluation of a direct detection device detector for single particle cryo-electron microscopy. Journal of Structural Biology, 176(3):404-408, December 2011.
Direct electron detection yields cryo-EM reconstructions at resolutions beyond 3/4 Nyquist frequency. Benjamin E Bammes, Ryan H Rochat, Joanita Jakana, Dong-Hua Chen, Wah Chiu, Journal of Structural Biology. 1773Benjamin E. Bammes, Ryan H. Rochat, Joanita Jakana, Dong- Hua Chen, and Wah Chiu. Direct electron detection yields cryo- EM reconstructions at resolutions beyond 3/4 Nyquist frequency. Journal of Structural Biology, 177(3):589-601, March 2012.
Electron counting and beam-induced motion correction enable near-atomic-resolution single-particle cryo-EM. Xueming Li, Paul Mooney, Shawn Zheng, Christopher R Booth, Michael B Braunfeld, Sander Gubbens, David A Agard, Yifan Cheng, Nature Methods. 106Xueming Li, Paul Mooney, Shawn Zheng, Christopher R. Booth, Michael B. Braunfeld, Sander Gubbens, David A. Agard, and Yi- fan Cheng. Electron counting and beam-induced motion correc- tion enable near-atomic-resolution single-particle cryo-EM. Na- ture Methods, 10(6):584-590, June 2013.
Comparison of optimal performance at 300 keV of three direct electron detectors for use in low dose electron microscopy. G Mcmullan, A R Faruqi, D Clare, R Henderson, Ultramicroscopy. 147G. McMullan, A. R. Faruqi, D. Clare, and R. Henderson. Com- parison of optimal performance at 300 keV of three direct elec- tron detectors for use in low dose electron microscopy. Ultrami- croscopy, 147:156-163, December 2014.
Single-shot dynamic transmission electron microscopy. T Lagrange, M R Armstrong, K Boyden, C G Brown, G H Campbell, J D Colvin, W J Dehope, A M Frank, D J Gibson, F V Hartemann, J S Kim, W E King, B J Pyke, B W Reed, M D Shirk, R M Shuttlesworth, B C Stuart, B R Torralva, N D Browning, Applied Physics Letters. 89444105T. LaGrange, M. R. Armstrong, K. Boyden, C. G. Brown, G. H. Campbell, J. D. Colvin, W. J. DeHope, A. M. Frank, D. J. Gib- son, F. V. Hartemann, J. S. Kim, W. E. King, B. J. Pyke, B. W. Reed, M. D. Shirk, R. M. Shuttlesworth, B. C. Stuart, B. R. Torralva, and N. D. Browning. Single-shot dynamic transmis- sion electron microscopy. Applied Physics Letters, 89(4):044105, July 2006.
Single-electron pulses for ultrafast diffraction. M Aidelsburger, F O Kirchner, F Krausz, P Baum, Proceedings of the National Academy of Sciences. 10746M. Aidelsburger, F. O. Kirchner, F. Krausz, and P. Baum. Single-electron pulses for ultrafast diffraction. Proceedings of the National Academy of Sciences, 107(46):19714-19719, November 2010.
Unlocking the time resolved nature of electron microscopy. Philip E Batson, Proceedings of the National Academy of Sciences. 1088Philip E. Batson. Unlocking the time resolved nature of electron microscopy. Proceedings of the National Academy of Sciences, 108(8):3099-3100, February 2011.
Compressed Sensing and Electron Microscopy. Peter Binev, Wolfgang Dahmen, Ronald Devore, Philipp Lamby, Daniel Savu, Robert Sharpley, Modeling Nanoscale Imaging in Electron Microscopy. Thomas Vogt, Wolfgang Dahmen, and Peter BinevSpringer USPeter Binev, Wolfgang Dahmen, Ronald DeVore, Philipp Lamby, Daniel Savu, and Robert Sharpley. Compressed Sensing and Elec- tron Microscopy. In Thomas Vogt, Wolfgang Dahmen, and Pe- ter Binev, editors, Modeling Nanoscale Imaging in Electron Mi- croscopy, Nanostructure Science and Technology, pages 73-126. Springer US, 2012.
The potential for Bayesian compressive sensing to significantly reduce electron dose in high-resolution STEM images. Andrew Stevens, Hao Yang, Lawrence Carin, Ilke Arslan, Nigel D Browning, Microscopy. 631Andrew Stevens, Hao Yang, Lawrence Carin, Ilke Arslan, and Nigel D. Browning. The potential for Bayesian compressive sens- ing to significantly reduce electron dose in high-resolution STEM images. Microscopy, 63(1):41-51, January 2014.
Wavelet Frame Based Algorithm for 3d Reconstruction in Electron Microscopy. M Li, Z Fan, H Ji, Z Shen, SIAM Journal on Scientific Computing. 361M. Li, Z. Fan, H. Ji, and Z. Shen. Wavelet Frame Based Al- gorithm for 3d Reconstruction in Electron Microscopy. SIAM Journal on Scientific Computing, 36(1):B45-B69, January 2014.
Reduced-dose and high-speed acquisition strategies for multi-dimensional electron microscopy. Zineb Saghi, Martin Benning, Rowan Leary, Manuel Macias-Montero, Ana Borras, Paul A Midgley, Advanced Structural and Chemical Imaging. 11Zineb Saghi, Martin Benning, Rowan Leary, Manuel Macias- Montero, Ana Borras, and Paul A. Midgley. Reduced-dose and high-speed acquisition strategies for multi-dimensional electron microscopy. Advanced Structural and Chemical Imaging, 1(1):1- 10, May 2015.
Stable signal recovery from incomplete and inaccurate measurements. Emmanuel J Cands, Justin K Romberg, Terence Tao, Communications on Pure and Applied Mathematics. 598Emmanuel J. Cands, Justin K. Romberg, and Terence Tao. Sta- ble signal recovery from incomplete and inaccurate measure- ments. Communications on Pure and Applied Mathematics, 59(8):1207-1223, August 2006.
Compressed sensing. D L Donoho, IEEE Transactions on Information Theory. 524D.L. Donoho. Compressed sensing. IEEE Transactions on In- formation Theory, 52(4):1289-1306, April 2006.
An Introduction To Compressive Sampling. E J Candes, M B Wakin, IEEE Signal Processing Magazine. 252E.J. Candes and M.B. Wakin. An Introduction To Compres- sive Sampling. IEEE Signal Processing Magazine, 25(2):21-30, March 2008.
HF resistance and self capac itance of single layer solenoids Part1. R G Medhurst, Wireless Engineering. 24R. G. Medhurst. HF resistance and self capac itance of single layer solenoids Part1. Wireless Engineering, 24:35-43, February 1947.
HF resistance and self capac itance of single layer solenoids Part2. R G Medhurst, Wireless Engineering. 24R. G. Medhurst. HF resistance and self capac itance of single layer solenoids Part2. Wireless Engineering, 24:80-92, March 1947.
. Arduino -Home, Arduino -Home.
Random Number Generators: Good Ones Are Hard to Find. S K Park, K W Miller, Commun. ACM. 3110S. K. Park and K. W. Miller. Random Number Generators: Good Ones Are Hard to Find. Commun. ACM, 31(10):1192-1201, Oc- tober 1988.
Probing the Pareto Frontier for Basis Pursuit Solutions. E Van Den, M Berg, Friedlander, SIAM Journal on Scientific Computing. 312E. van den Berg and M. Friedlander. Probing the Pareto Frontier for Basis Pursuit Solutions. SIAM Journal on Scientific Com- puting, 31(2):890-912, November 2008.
SPGL1: A solver for large-scale sparse reconstruction. SPGL1: A solver for large-scale sparse reconstruction.
Controlled lateral anisotropy in correlated manganite heterostructures by oxygen octahedral coupling. Z Liao, M Huijben, Z Zhong, N Gauquelin, S Macke, R J Green, S Van Aert, J Verbeeck, S Van Tendeloo, K Held, G A Sawatzky, G Koster, G Rinjders, Nature Materials. SubmittedZ. Liao, M. Huijben, Z. Zhong, N. Gauquelin, S. Macke, R.J. Green, S. Van Aert, J. Verbeeck, S. Van Tendeloo, K. Held, G. A. Sawatzky, G. Koster, and G. Rinjders. Controlled lateral anisotropy in correlated manganite heterostructures by oxygen octahedral coupling. Nature Materials, Submitted, 2015.
Sparse imaging for fast electron microscopy. S Hyrum, Jovana Anderson, Brandon Ilic-Helms, Jason Rohrer, Kurt Wheeler, Larson, 8657Hyrum S. Anderson, Jovana Ilic-Helms, Brandon Rohrer, Jason Wheeler, and Kurt Larson. Sparse imaging for fast electron mi- croscopy. volume 8657, pages 86570C-86570C-12, 2013.
Private Discussion. Quentin Ramasse, Quentin Ramasse. Private Discussion. 2015.
| [] |
[
"Nucleon Electromagnetic Form Factors *",
"Nucleon Electromagnetic Form Factors *"
] | [
"Kees De Jager \nJefferson Laboratory\n23606Newport NewsVAUSA\n"
] | [
"Jefferson Laboratory\n23606Newport NewsVAUSA"
] | [] | A review of data on the nucleon electromagnetic form factors in the space-like region is presented. Recent results from experiments using polarized beams and polarized targets or nucleon recoil polarimeters have yielded a significant improvement on the precision of the data obtained with the traditional Rosenbluth separation. Future plans for extended measurements are outlined.* Talk presented at the Bates25 Symposium; Jefferson Lab publication JLAB-PHY-00-01 | 10.1063/1.1291508 | [
"https://arxiv.org/pdf/hep-ex/0003034v1.pdf"
] | 119,107,421 | hep-ex/0003034 | d34d1830b5fcf0803d2b59e6ab3990b0c995bc46 |
Nucleon Electromagnetic Form Factors *
arXiv:hep-ex/0003034 30 Mar 2000
Kees De Jager
Jefferson Laboratory
23606Newport NewsVAUSA
Nucleon Electromagnetic Form Factors *
arXiv:hep-ex/0003034 30 Mar 2000
A review of data on the nucleon electromagnetic form factors in the space-like region is presented. Recent results from experiments using polarized beams and polarized targets or nucleon recoil polarimeters have yielded a significant improvement on the precision of the data obtained with the traditional Rosenbluth separation. Future plans for extended measurements are outlined.* Talk presented at the Bates25 Symposium; Jefferson Lab publication JLAB-PHY-00-01
INTRODUCTION
The nucleon electromagnetic form factors (EMFF) are of fundamental importance for an understanding of their internal structure. These EMFF, which in the Breit frame can be simply related to the spatial distribution of the nucleon charge and magnetization densities, are measured through elastic electron-nucleon scattering.
In where Q is the four-momentum transfer, σ M the Mott cross section for scattering off a point-like particle, m N the mass of the nucleon, θ e the electron scattering angle and E e the electron energy. This equation shows that G E and G M can be determined separately by measuring at fixed Q 2 over a range of (θ e ,E e ) combinations. This procedure is called the Rosenbluth separation. 1 The Sachs form factors can be identified with the Fourier transform of the nucleon charge and magnetization density distributions, such that the slope at Q 2 -> 0 of the EMFF is related to the charge and magnetization radius. There has been considerable debate over the interpretation of the neutron charge radius. 2,3 The charge radius can be split into two components, one of which (the so-called Foldy 4 term) is related to the magnetic moment, not to the rest-frame charge distribution. However, Isgur 5 and Bawin and Coon 6 have shown that this Foldy term is exactly canceled by a contribution from the Dirac form factor, so that the charge radius is indeed determined solely by the rest-frame charge distribution.
Up until the beginning of the previous decade all available proton EMFF data had been collected using the Rosenbluth separation. This experimental procedure requires an accurate knowledge of the electron energy and the total luminosity. In addition, since the contribution to the elastic cross section from the magnetic form factor is weighted with Q 2 , data on G E p suffer from increasing systematic uncertainties at higher These restrictions are clearly presented in the review paper by Bosted et al. 7 The then available world data set was compared to the so-called dipole parametrization, which corresponds to exponentially decreasing radial charge and magnetization densities:
GG G GG GG G Q Q Q GeV c Q GeV
THEORY
A frequently used framework 9 to describe the EMFF is that of Vector Meson Dominance (VMD), in which one assumes that the virtual photon -after having become a quark-antiquark pair -couples to the nucleon as a vector meson. The EMFF can then be expressed in terms of coupling strengths between the virtual photon and the vector meson and between the vector meson and the nucleon, summing over all possible vector mesons. In some cases additional terms are included to account for the effect of unknown or lesser known mesons.
A common restriction of the VMD models is that they do not predict a correct behaviour of the EMFF at high Q 2 -values. The quark-dimensional scaling framework 10 predicts that only valence quark states contribute at sufficiently high Q 2values. Under these conditions the EMFF Q 2 -dependence is determined simply by the number of gluon propagators, causing the Dirac and Pauli form factors to be proportional to Q -4 and Q -6 , respectively, whereas any VMD-model will predict a Q -2 behaviour at large Q 2 -values. Gari and Krümpelmann have constructed a hybrid (EVMD) model which combines the low Q 2 -behaviour of the VMD model with the asymptotic behaviour predicted by pQCD. In their first paper 11 they consider only coupling to the ρ and ω mesons, whereas later 12 the φ meson was also included. VMD models form a subset of models using dispersion relations, which relate form factors to spectral functions. These spectral functions can also be thought of as a superposition of vector meson poles, but include contributions from n-particle production continua. This framework allows then a model-independent fit 13 to all available EMFF data in the space-and the time-like region.
Many attempts have been made to enlarge the domain of applicability of pQCD calculations to moderate Q 2 -values. Kroll et al. 15 have generalized the hard-scattering scheme by assuming nucleons to consist of quarks and diquarks. The diquarks are used to approximate the effects of correlations in the nucleon wave function. This model is equivalent to the hard-scattering formalism of pQCD in the limit Q 2 -> ∞. Chung and Coester 16 have developed a relativistic constituent quark model with effective quark masses and a confinement scale as free parameters.
Lu et al. 17 have recently expanded the cloudy bag model, whereby the nucleon is described as a bag containing three quarks, but including an elementary pion field coupled to them, in such a way that chiral symmetry is restored. Finally, recent developments 18 within the Skewed Parton Distribution formalism indicate a relation between the EMFF behaviour at larger Q 2 -values and the nucleon spin.
NUCLEON FORM FACTORS
Over 20 years ago Akhiezer and Rekalo 19 showed that the accuracy of EMFF measurements could be increased significantly by scattering polarized electrons off a polarized target (or by equivalently measuring the polarization of the recoiling nucleon). In the early nineties a series of measurements 20-25 at the MIT-Bates facility showed the feasibility of that measurement principle.
Neutron Magnetic Form Factor
Significant progress has been made in measurements of G M n at low Q 2 -values by measuring the ratio of quasi-elastic neutron and proton knock-out from a deuterium target. This method is practically insensitive to nuclear binding effects and to fluctuations in the luminosity and detector acceptance. The basic set-up used in all such measurements was very similar: the electron was detected in a magnetic spectrometer with coincident neutron/proton detection in a large scintillator array. The main technical difficulty in such a ratio measurement is the absolute determination of the neutron detection efficiency. For the measurements at Bates 25 and ELSA 26 the efficiency was measured in situ using the D(γ,p)n or p(γ,π + ) reaction with a bremsstrahlung radiator up stream of the experimental target. The hadron detectors used in the experiments at NIKHEF 27 and Mainz 28 were calibrated at the PSI neutron beam using the kinematically complete p(n,p)n reaction. Figure 2 shows the results of those four experiments. The Mainz G M n data are 8-10 % lower than the ELSA ones, despite the quoted uncertainty of appr. 2 %. This discrepancy would require a 16-20% error in the detector efficiency. The contribution from electroproduction in the ELSA set-up, caused by the electron contamination in the bremsstrahlung beam, which could result in a loss of events due to the three-body kinematics in electroproduction, has been extensively investigated. 29 Thus far, the detection inefficiency due to electroproduction has been established at less than 5 %, clearly much smaller than required to explain the discrepancy in the data. Recently, inclusive quasi-elastic scattering of polarized electrons off a polarized 3 He target was measured 30 in Hall A at JLab in a Q 2 -range from 0.1 to 0.6 (GeV/c) 2 . This experiment will provide an independent accurate measurement of G M n in a Q 2range overlapping with that of the ELSA and Mainz data. Measurements of G M n at Q 2 -values up to 5 (GeV/c) 2 are expected in the near future from a JLab experiment that will measure the neutron/proton quasi-elastic cross-section ratio using the CLAS detector. 31
Neutron Electric Form Factor
Since a free neutron target is not available, one has to use neutrons bound in nuclei to study the neutron EMFF. The most precise data on G E n prior to any spin-dependent experiment were obtained from the elastic electron-deuteron scattering experiment by Platchkov et al. 32 The deuteron elastic form factor contains a term of the form G E n G E p .
However, in order to extract G E n from the data, one has to calculate the deuteron wave function, which requires a choice of the nucleon-nucleon potential. Figure 3 shows the G E n values extracted from the Platchkov data with the Paris potential, while the grey band indicates the range of G E n values extracted with the Nijmegen, AV14 and RSC potentials. Clearly, the choice of NN-potentials results in a systematic uncertainty of appr. 50 % in G E n . One should realize that all modern NN-potentials yield consistent results for a large variety of two-and three-nucleon observables. Thus, one might expect that a reevaluation of the Platchkov data using modern high-precision NNpotentials and a consistent treatment of exchange currents will yield a reduced potential dependence. Significant advances have been made in the last decade in the development of electron beams with high polarization and intensity and of reliable polarized targets. This progress has been used in a series of new spin-dependent measurements of G E n , which utilizes the fact that the ratio of the beam-target asymmetry with the target polarization perpendicular and parallel to the momentum transfer is directly proportional to the ratio of the electric and magnetic form factors:
G G A A E n M n e =+ + () ( ) ⊥ // tan / ( ) ττ τ θ 1 2 23
A similar relation can be derived for the reaction 2 Heen (,' ) rr when one measures the polarization of the recoiling neutron directly and after having precessed the neutron spin over 90° with a dipole magnet. Figure 3 shows the results of the pioneering experiments of that technique, performed at Bates, using the reactions 2 Heen (,' ) rr 20 and 3 r r He e e (,' ) [21][22][23] and at Mainz, with the 3 r r He e e n (,' ) reaction. 33 These results have not been corrected for rescattering or nuclear medium effects. Figure 4 shows the most recent results, obtained through the reaction channels 2 r r Heen (,' ) 34 , 2 Heen (,' ) rr 35,36 and 3 r r He e e n (,' ) . 37,38 At low Q 2 -values corrections for nuclear medium and rescattering effects can be sizeable: 65 % for deuterium at 0.15 (GeV/c) 2 and 50 % for 3 He 38 at 0.35 (GeV/c) 2 . These corrections are expected to decrease significantly with increasing Q, although no reliable results are at present available for 3 He above 0.5 (GeV/c) 2 . Thus, there are now data from a variety of reaction channels available in a Q 2 -range up to 0.6 (GeV/c) 2 with an overall accuracy of appr. 20 %, which are in mutual agreement. However, neither the VMD 11 nor the dispersion relation 14 calculations agree with the data. Only the Galster parametrization 40 which uses a modified version of the dipole form factor, is able to describe the data adequately. A more detailed discussion of these recent results is given by Schmieden. 41 Also shown in fig. 4 are the results expected in the near future, from the 3 r r He e e n (,' ) channel at NIKHEF 42 and from the 2 r r Heen (,' ) 43
Proton Electric Form Factor
Arnold et al. 46 have shown that the systematic error in a measurement of G E p , inherent to the Rosenbluth separation, can be significantly reduced by scattering longitudinally polarized electrons off a hydrogen target and measuring the ratio of the transverse to longitudinal polarization of the recoiling proton. fig. 6. The most striking feature of the data is the sharp decline as Q 2 increases. Since it is known that G M p closely follows the dipole parametrization, it follows that G E p falls more rapidly with Q 2 than the dipole form factor G D . A comparison with fig. 1 confirms the expected improvement in accuracy of such a spin-dependent measurement over the Rosenbluth separation. None of the theoretical models shown in fig. 6 is able to adequately describe the new data. An extension 49 of this experiment to a Q 2 -value of 5.6 (GeV/c) 2 has been scheduled for the fall of 2000.
CONCLUSIONS
Recent advances in polarized electron sources, polarized nucleon targets and nucleon recoil polarimeters have made it possible to accurately measure the spindependent elastic electron-nucleon cross section. New data on nucleon electromagnetic form factors with an unprecedented precision have (and will continue to) become available in an ever increasing Q 2 -domain. These data will form tight constraints on models of nucleon structure and will hopefully incite new theoretical efforts. In addition they will significantly improve the accuracy of the extraction of strange form factors from parity-violating experiments. 50
ACKNOWLEDGMENTS
The author expresses his gratitude to Ulf Meissner and Mark Jones for fruitful discussions and for receiving their results prior to publication. This work was supported in part by the U.S. Department of Energy.
Plane Wave Born Approximation (PWBA) the cross section can be expressed in terms of the so-called Sachs form factors G E and G
FIGURE 2 .
2The square of the ratio of G M n to µ n G D as a function of Q 2 , compared to predictions by Gari and Krümpelmann 11 and Höhler9 . The expected precision of JLab experiment E95-001 30 is indicated by the solid squares. Data: diamonds 25 , stars 26 , circle24 , large triangle 27 , triangles.28
FIGURE 3 .
3Older (star21,22 , square23 , cross 20 and diamond33 ) results for G E n as a function of Q2 . The open circles depict the results of Platchkov et al.32 for the Paris potential, the shaded area the systematic uncertainty due to the choice of NN-potential.
and 2 FIGURE 4 .FIGURE 5 .
245Heen (,' ) rr 44 channels at JLab. Finally, in fig. 5 are shown the results expected with the BLAST detector 45 with both the 2 r r Heen (,' ) and the 3 r r He e e n (,' ) reaction channels. Recent (circle 34 , triangle 35,36 , square 37,39 and diamond 38 ) and future (open square 42 , open diamonds 44 and open triangles 43 ) results for G E n as a function of Q 2 , compared to three theoretical calculations (full 11 , dashed 14 and dotted 40 ). Predicted accuracy of G E n data to be obtained with the BLAST detector.
Polarimeter (FPP) in the hadron HRS, consisting of two pairs of straw chambers with a carbon analyzer in between. Instrumental asymmetries are cancelled by taking the difference of the azimuthal distributions of the protons scattered in the analyzer for positive and negative beam helicity. A Fourier analysis of this difference then yields the transverse and normal polarization components at the FPP. The data were analyzed in bins of each of the target coordinates. No dependence on any of these variables was observed.The results for the ratio GG E
Q 2 -values. Data for the neutron resulted mainly from quasi-elastic scattering off the deuteron, because a free neutron target is not available in nature. This additional constraint caused large uncertainties, especially on the data for G E n .012345
Q 2 [GeV/c] 2
0.4
1.0
1.6
µG
E /G
M
Dispersion relations (1976)
EVDM (1985)
EVDM (1992) including φ meson
Constituent Quark (1991)
DESY (1973)
Bonn (1971)
SLAC (1994)
SLAC (1970)
Cambridge (1971)
SLAC (1994)
FIGURE 1. The ratio µ pE
p
M
p
GG /
as a function of Q 2 , determined with the Rosenbluth separation
technique. Data symbols are explained in ref. 48. Theory: full 9 , dotted 11 , dashed 16 and dot-dashed. 12
. M N Rosenbluth, Phys. Rev. 79Rosenbluth, M.N., Phys. Rev. 79, 615-619 (1950).
. Yu A Alexandrov, Neutron News. 5Alexandrov, Yu.A., Neutron News 5, 1-20-22 (1994).
. J Byrne, Neutron News. 5Byrne, J., Neutron News 5, 4-15-17 (1994).
. L L Foldy, Rev. Mod. Phys. 30471Foldy, L.L., Rev. Mod. Phys. 30, 471 (1958).
. N Isgur, Phys. Rev. Lett. 83Isgur, N., Phys. Rev. Lett. 83, 272-275 (1999).
. M Bawin, S A Coon, Phys. Rev. C. 60Bawin, M., and Coon, S.A., Phys. Rev. C 60, 025207-1-3 (1999).
. P Bosted, Phys. Rev. C. 51Bosted, P., et al., Phys. Rev. C 51, 409-411 (1995).
. A F Lung, Phys. Rev. Lett. 70Lung, A.F., et al., Phys. Rev. Lett. 70, 718-721 (1993).
. G Höhler, Nucl. Phys. B. 114Höhler, G., et al., Nucl. Phys. B 114, 505-534 (1976).
. S J Brodsky, G Farrar, Phys. Rev. D. 11Brodsky, S.J., and Farrar, G., Phys. Rev. D 11, 1309-1330 (1975).
. M F Gari, W Krümpelmann, Z. Phys. A. 322689Gari, M.F., and Krümpelmann, W., Z. Phys. A 322, 689 (1985).
. M F Gari, W Krümpelmann, Phys. Lett. B. 274Gari, M.F., and Krümpelmann, W., Phys. Lett. B 274, 159-162 (1992).
. P Mergell, Nucl. Phys. A. 596Mergell, P., et al., Nucl. Phys. A 596, 367-396 (1996).
. U.-G Meissner, Eprint, hep-ph/9907323private communicationMeissner, U.-G, eprint: hep-ph/9907323; private communication (1999).
. P Kroll, M Schürmann, W Schweiger, Z. Phys. A. 338private communicationKroll, P., Schürmann, M., and Schweiger, W., Z. Phys. A 338, 339-348 (1991); private communication (1998).
. P L Chung, F Coester, Phys. Rev. D. 44private communicationChung, P.L., and Coester, F., Phys. Rev. D 44, 229-241 (1991); private communication (1999).
. D H Lu, A W Thomas, A G Williams, Phys. Rev. C. 57Lu, D.H., Thomas, A.W., and Williams, A.G., Phys. Rev. C 57, 2628-2637 (1998).
. X Ji, Phys. Rev. D. 55Ji, X., Phys. Rev. D 55, 7114-7125 (1997);
. Phys. Rev. Lett. 78Phys. Rev. Lett. 78, 610-613 (1997).
. A I Akhiezer, M P Rekalo, Sov. J. Part. Nucl. 3277Akhiezer, A.I., and Rekalo, M.P., Sov. J. Part. Nucl. 3, 277 (1974).
. T Eden, Phys. Rev. C. 50Eden, T., et al., Phys. Rev. C 50, R1749-R1753 (1994).
. C E Woodward, Phys. Rev. Lett. 65Woodward, C.E., et al., Phys. Rev. Lett. 65, 698-700 (1990).
. C E Jones-Woodward, Phys. Rev. C. 44Jones-Woodward, C.E., et al., Phys. Rev. C 44, R571-R574 (1991).
. A K Thompson, Phys. Rev. Lett. 68Thompson, A.K., et al., Phys. Rev. Lett. 68, 2901-2904 (1992).
. H Gao, Phys. Rev. C. 50Gao, H., et al., Phys. Rev. C 50, R546-R549 (1994).
. P Markowitz, Phys. Rev. C. 48Markowitz, P., et al, Phys. Rev. C 48, R5-R9 (1993).
. E E W Bruins, Phys. Rev. Lett. 75Bruins, E.E.W., et al., Phys. Rev. Lett. 75, 21-24 (1995).
. H Anklin, Phys. Lett. B. 336Anklin, H., et al., Phys. Lett. B 336, 313-318 (1994).
. H Anklin, Phys. Lett. B. 428Anklin, H., et al., Phys. Lett. B 428, 248-253 (1998).
. B Schoch, private communicationSchoch, B., private communication (1999).
JLab experiment. H Gao, O Hansen, Gao, H., and Hansen, O., JLab experiment E95-001.
JLab experiment. W Brooks, M F Vineyard, Brooks, W., and Vineyard, M.F., JLab experiment E94-017.
. S Platchkov, Nucl. Phys. A. 510Platchkov, S., et al., Nucl. Phys. A 510, 740-758 (1990).
. M Meyerhoff, Phys. Lett. B. 327Meyerhoff, M., et al., Phys. Lett. B 327, 201-207 (1994).
. I Passchier, Phys. Rev. Lett. 82Passchier, I., et al., Phys. Rev. Lett. 82, 4988-4991 (1999).
. C Herberg, Eur. Phys. Jour. A. 5Herberg, C., et al., Eur. Phys. Jour. A 5, 131-135 (1999).
. M Ostrick, Phys. Rev. Lett. 83Ostrick, M., et al., Phys. Rev. Lett. 83, 276-279 (1999).
. J Becker, Eur. Phys. Jour. A. 6Becker, J., et al., Eur. Phys. Jour. A 6, 329-344 (1999).
. D Rohe, Phys. Rev. Lett. 83Rohe, D., et al., Phys. Rev. Lett. 83, 4257-4260 (1999).
. W Glöckle, private communicationGlöckle, W., private communication (1999).
. S Galster, Nucl. Phys. B. 32221Galster, S., et al., Nucl. Phys. B 32, 221 (1971).
. H Schmieden, these proceedingsSchmieden, H., these proceedings.
. J F J Brand, Van Den, M Ferro-Luzzi, Nikhef, Brand, J.F.J. van den, and Ferro-Luzzi, M., NIKHEF experiment 94-05.
JLab experiment. D Day, Mitchell , J , Day, D., and Mitchell, J., JLab experiment E93-026.
. B D Anderson, S Kowalski, R Madey, JLab experiment E93-038Anderson, B.D., Kowalski, S., and Madey, R., JLab experiment E93-038.
. Bates Large Acceptance Spectrometer Toroid. Bates Large Acceptance Spectrometer Toroid, http://mitbates.mit.edu/blast.
. R Arnold, C Carlson, F Gross, Phys. Rev. C. 23Arnold, R., Carlson, C., and Gross, F., Phys. Rev. C 23, 363-374 (1981).
. B Milbrath, Phys. Rev. Lett. 802221erratumPhys. Rev. Lett.Milbrath, B., et al., Phys. Rev. Lett. 80, 452-455 (1998); erratum, Phys. Rev. Lett. 82, 2221 (1999).
. M K Jones, Phys. Rev. Lett. 841398Jones, M.K., et al., Phys. Rev. Lett. 84, 1398 (2000).
JLab experiment. C Perdrisat, Perdrisat, C., et al., JLab experiment E99-007.
. P Souder, these proceedingsSouder, P., these proceedings.
| [] |
Subsets and Splits